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• T2(L)/T is surjective. By [3], a basis of T A, the symbol /p^ means tp, viewed as a map from R^ to A^r\ and A^ is always considered as an /^'-algebra via tf^. • N of _R-modules is called quasi-split if if ® R/1 : M/IM —> N/IN is injective for any parameter ideal / of HH exists we can do with Oi/c, what was said above about fiL/r> . /t/rto 259
M(z) then M(z"ei) = M(ek2 . . . ek
54
Bonanzinga
Therefore, if m(z) ^ 1, then
If m(z) - 1, then Ip([z"x^i+£^^}) = {(z"ei}'x(ei+£M^"^\ which is one element of the canonical basis of T-2(L}/T. Let now i < M(z). Considering the boundary of z" if i ^ 771(2 then
if i — m(z) then
since ? ^ 1 then z/eM(z)x^i+eM(*^
e
^- Moreover
M(2) =M(eklek2...ekd) = kd = M(z"ei) = M(efc2 . . . e, . . . ekd). Thus, if i ^ m(z)
then
If t = m(z) or m(z) ^ 1, then v([z" x(£™^+s^}} = 0. If t ^ m(«) = 1, it follows that
is an element of the canonical basis of Ti(L}/T. Therefore, a basis of the image of ip is given by the elements
with z < u; i 7^ 1. Now we claim that lp is surjective if and only if condition (2) in the theorem holds, that is, if and only if for every w < v, there exists 1 < i < n and z < u such that w — zei/e\. In fact, the canonical basis of T2(L)/T is given by the cycles [W'X^I+£M!-W^] with w < v. Therefore if condition (2) holds, then [W'X^I+£M^] — [(zei/ei)'x^1+£M(~ze'/e^} is in the image of
Lexsegment ideals in the exterior algebra
3
55
LEXSEGMENT IDEALS WITH LINEAR RESOLUTION
In this section we give sufficient conditions for lexsegment ideals to have a linear resolution. We believe that these conditions are also necessary THEOREM 3.10 Let I be a lexsegment ideal. With the notation of 2.7 the ideal I has a linear resolution in the following cases: (1) / is an initial or final lexsegment ideal; (2) / satisfies 2.1(1); (3) 1 ^ «i 7^ ji, and eim/eM^m-j < u for every m < v.
Proof. Initial and final lexsegments ideals, generated in a fixed degree, are both stable. Thus it follows from [3] that these have a linear resolution. In case (2), / is isomorphic to the final lexsegment ideal generated by (eik+l-(k-i) • • •eid-(k-i)} in n — (A; — 1) variables, hence / has a linear resolution. In order to prove that / has a linear resolution it is enough to show that
is surjective for every s > 2, where L is the ideal generated by all monomials of degree d. By [3], the homology classes of the cycles w'x^kl+ek2+...+£ks+^M(u,))^ with w e Md and 1 < ki < k2 < ... < ks < M(w) form a basis of T s +i(L). It is clear that if w is in Ll(v), then £ J [w'x( kl rek+...+ek3+eM(w))^
jg jn
thg image of
^ If
w
w < v, then the differential d is:
if .7 = 1 if
ks = M(w).
If k\ ^ 1, then w'e^ £ K for every kj, because if w = e^e^ . . . e^d < v < u, then k\ ^ j\ > i\ — 1 and so w'x(£ki+£k2+'"+Sks+£M(w^ is a cycle whose homology class belongs to Ts+i(K). Thus [ w ' a; ^ A; i +£fc 2+-+ e * s + £ M(^))] is in the image of . Hence in order to prove that
56
Bonanzinga
References [1] H. A. Hulett, H. M. Martin, Betti numbers of lexsegment ideals, Preprint 1996. [2] E. De Negri, J. Herzog. Completely lexsegment ideals, Proc. Amer. Math. Soc. 126 (12) (1998), 3467-3473.
[3] A. Aramova, J. Herzog, T. Hibi. Gotzmann theorems for exterior algebras and combinatorics. J. Algebra 191 (1997), 174-211. [4] A. Aramova, L. L. Avramov, J. Herzog. Resolutions of monomial ideals and cohomology over exterior algebras, Trans. AMS. 352 (2) (2000), 579-594.
On the equations defining toric projective varieties EMILIO BRIALES MORALES Departamento de Algebra. Facultad de Matematicas. Universidad de Sevilla (Spain). E-mail: [email protected] ANTONIO CAMPILLO LOPEZ Departamento de Algebra, Geometria y Topologia. Facultad de Ciencias. Universidad de Valladolid (Spain). E-mail: [email protected]
PILAR PISON CASARES Departamento de Algebra. Facultad de Matematicas. Universidad de Sevilla (Spain). E-mail: [email protected]
1
INTRODUCTION
Several recent results ([1], [2], [3], [4]) study the syzygies of toric varieties. In particular, the equations defining an embedded affine toric variety can be described. When the toric variety is projective one, the situation becomes special, since the semigroup defining it has a system of generators which lies on an hyperplane (i.e. there exists a map L with the properties in section 1 below). The purpose of this paper is to give an estimation for the degrees of the equations defining an embedded projective toric variety, and give an effective upper bound for such degrees. Our results show such upper bound can be derived from some general facts stablished in [2]. As an illustration, we compute the bound explicitly in the case of toric projective curves. 2
THE APERY SET
Let 5 be a cancellative finitely generated commutative semigroup with zero element and torsion free. Let A be a finite set of generators for S, j}A = h. Denote G(S) the smallest group containing 5 and let d be its rank. Then G(S) ~ Zd C Qd ~ V(S) :— G(S)
1. L(A) = {1} 2. L(n + m) = L(n) + L(m) 57
58
Morales, Campillo and Casares
3. L(n) = 0 <^ n = 0. Fix, from now on, the data S, A, and L. Notice that the existence of L implies that 5 satisfies the property 5n (-5) = {0} therefore, the cone C(S) is strongly convex. Denote / the number of extremal rays of (7(5). Notice that, since (7(5) generates V(S), one obviously has / > d. Then, there are subsets E c A with jjE = / such that C(E] = C(5), where C(E) is the cone in V(S) generated by E. Following [2], fix a partition A = E U A, where E satisfies the above property. This kind of partition is called convex partition. Let |A = r = h — f .
The Apery set Q of S relative to E is defined to be the set given by Q :={
| q-e$S,VeeE}.
The Apery set Q is finite as proved in Proposition 5.1 of [2]. The terminology Apery comes from the use of the set Q, for the particular case of numerical semigroups, done in [5]. This finiteness property follows from the fact that the semigroup ring Z[5] is a finite integral extension of Z[E], however we will show below an effective proof of it. Set E — {ei, . . . ,e/} and A = {a\, . . . ,ar}. We know that E contains a basis of y(5) as Q- vector space. Since a,j £ (7(5), for any j, we have that
\ijei, with \ij £ Q+. Therefore, for any j, 3qj £ N such that
tijei, with tij € N. If m € Q, m = 2j=i &iai w^tn ft < 9j> f°r finite number of m € Q.
an
y 3- Therefore, there exists only a
REMARK 2.1 In order to find the set Q it is enough: 1. Compute qj, for any j, 1 < j < r.
2. Check whether the elements m — £^=1 /3jUj, with /3j < qj for any j, is in Q. Now, for any t > 0, let H* := {m e 5 i
| L(m) = t}, and denote
g :=gnH*,
and
to := min{t
\ Q* = 0}
PROPOSITION 2.2 With the above notation, if Q* = 0 then Q*' = 0 for all t' > t. In particular, one has Q* = 0 for t > t0.
Proof. It is enough to prove that Qt = 0 =» g*+i = 0.
Suppose that m 6 Qt+l. Then, m = m' 4- a with a 6 A and m' £ #*. Since Q* = 0, we have that m' g Q and therefore there exists e € E such that m' — e 6 5. But then m — e € 5, a contradiction with m £ Q.
Equations defining toric projective varieties
59
REMARK 2.3 In order to find t0 it is enough to use 2.1. Note that 2 in 2.1 could be computed, in practice, by using an integer programming method.
3
SOME HOMOLOGY EXACT SEQUENCES
Fix a field k for coefficients. Denote by E the simplex of parts of A. For any simplicial subcomplex A of E, let us denote by Hi(A), —l
Tm = {FcE n an<
n
=
| m-nF€S},
ur
where np '•— Sn€F ^ ® 0- ^ objective will be to obtain some information of Am by means of Tm. Notice that if A — 0, then A m = Tm. To have a significative discussion from now on, assume that $A = r > 1. DEFINITION 3.1 On the elements of S, define a partial order >Q
m >Q m' <=$• m - m! € 5 \ Q.
If H c 5, we say that m £ H is Q-minimal in the set H if m >Q m', with m' € H, implies that m — m'. PROPOSITION 3.2 The set
D(0):={meS
\ H0(Tm) ? 0}
is finite and it can be determined by an algorithm.
Proof. The finiteness of D(Q) is shown in Proposition 4.1 of [2]. We will give here an effective proof of this fact. __ As above, set E = {ei, . . . , e / } and A = {oi,. .. ,ar}. Notice that if H0(Tm) ^ 0 then, by choosing elements e\ and 62 in different connected component of Tm, one obtains, for j = 1,2: / m-ej£S=$m
f+r
= ^a^ei + ^ a\})ai-f, with a(j} > 1. i=l
«=/+!
Set a^ = (a[j),...,a(^) € N h , for j = 1,2, and let Cj £ N fe , the vector with coordinates equal to zero, excepting the jth one which is equal to 1. Then, a(rt » ej, for j — 1, 2, where » stands for the componentwise partial order. Sea A the matrix whose column vectors are the generators of 5. Notice that m = AaW, for j = 1,2, and a = (a ( 1 ) ,a( 2 >) € N 2h satisfies (A\-A)a = 0, and a » ( e i , e 2 ) -
60
Morales, Campillo and Casares
Then, if one considers the sets
Reie2 := {/3 = (/3 (1) ,/3 (2 >) € N2ft | (.4| - A) ft = 0, with /3 » ( Cl ,e 2 )}, and
Stfeie2 := {m' e 5 | m' = .40W with £ 6 we have that a e J?eie2 and m € ^-Reie 2 - Furthermore, in fact one has, m € M eje2 where M ei e 2 := {m' € £.Reie2 | "i' is Q - minimal in EReiC2}.
In fact, notice that on the semigroup 5 one can write m = m' + m", with m' € M eie2 andm" € S\Q. Ifm" 7^ 0, then one has m" = Ej=i^ e j + Ej=/+i^ a j-/> /?j € N and /3j ^ 0 for some j, 1 < j> < /, which is a contradiction since ei and 63 are in differente connected component of Tm. Thus, to show that -D(O) is finite, it is enough to prove that each set M eie2 is so. For it, notice that the set eie2
:= {/? € Reie3 I /8 is minimal for >},
is finite and set eie2
= {m' € 5 m' = AP(1) with £
We claim that M eiC2
The proof of the proposition follows from the claim. In fact, let n € MeiC2. Set n = A^l\ with /? € ^?eie2- If /? € HReie:!, we claim is obvious. Otherwise, /? = 7+/x, with 7 € HReie2 and /u € N2/l. Then, n = n'+n", with n' = A"f^ and n" = AfJ,^. It is clear by definition that n' £ S/HJ?e,e2, and n" € Q because n is Q-minimal in Mei<,2. This proves the claim. Above effective proof gives rise to the algorithm mentioned in the statement of the proposition. Next remark points out how one can proceed to the computation of D(0).
REMARK 3.3 In order to find the set I>(0) it is enough: 1. Compute the set Q (see 2.1).
2. Compute the sets SH/?eie2, for any 61,62 € E, with e\ ^ 62 using integer programming (see [6]).
3. Check the Q-minimal elements in EHfJ ei62 + Q and obtain MeiC2, for, any ei, e-2 € .E, with e! / e2 .
4. Check the elements m £ |J M eie2 , ei,e 2
such that Ho(Tm) ^ 0 by using linear algebra. Now, let us recover from [2] a construction which shows how the set £>(0) can be combinatorially described.
Equations defining toric protective varieties
61
• For any m e G(S) and I > —I, denote by C^Q™) the vector space which has the set f T
.—
A
|
U T
_. J
i^ "!
£- /^"l
as a basis. • For any chain z in C*t(Qm), denote by Ot(z) the projection on (7j_i(Q m ) of the simplicial boundary of z. By Lemma 2.2 in [2], {C".(Qm), #•} is a chain complex for any m. To understand better the homology of this complex, consider, for any m € 5, the following subset of E:
Km = { L e A m
(LnE^Q) or (L C A and m-niZS
It is easy to check that K m is a simplicial subcomplex of A m , so that one can consider the chain complex C.(Km) and the relative chain complex C,(A m ,K m ). Notice that, by construction, one has an identification C^Q,,,) ~ (7.(A m ,K m ). Notice that if m £ Q, then one has that (7«(QTO) c; A; and KTO = 0. Otherwise, if m 6 5 \ Q, since 3e £ E such that m — e £ S, one obtains L = {e} € K m and Km ^ {0}. Therefore, H-i(Km) — 0 for any m € 5. This allows to deduce, from the exact sequence of complexes,
0 -> C.(Km) -» C.(Am) ->• C7.(Qm) ->• 0, that there is a long exact sequence of homology,
) -* H,(Qm) -»• ... > H_i(K m ) = 0 -)•
Now, in order to understand the homology .H»(K m ), let us consider the simplicial complex given by the following disjoint union of subsets of S:
K m :-K m U{/UJ,I C A, J C E
\ m-m-nj $ S and m-nj-e £ 5,Ve € J}.
Notice that any /U J in the second set of the above union is such that the cardinality of J is at least 2. The complex K m is acyclic, i.e. if;(Km) = 0 for any I > -1 (see Corollary 2.1 in [2]). Thus, the long exact sequence of homology coming from the exact sequence of chain complexes 0 -> C.(K m ) -» C.(Km) ->- C.(Kro,Kro) -> 0
gives rise to an isomorphism p;+i : JJ(+i(K m ,K m ) —> f/j(K m ), for every / > — 1. To study the homology #.(K m ,K m ) let us consider, the chain of simplicial complexes r) -1) K - M( C MTn(°) *^-C •Mm(1' V—C • • - ^_C ±M.( = Km X^-m — m *^-m lvA
lvA
LVA
VI
xv
62
Morales, Campillo and Casares
where Mm , —1 < i < r, is the simplicial subcomplex of Km given by: M$ := K m U {L = / U J e K m
| J C A, J C £, and j}/ < i}.
Now, ff,(K m ,K m ) can be computed (see [2]) by means of the long exact sequences
for -1 < i < j < k < r. In fact, to compute tf.(K m ,K m ) = H.(M$,M(m1}), it will be enough to use the above exact sequences for the concrete values of (i,j,k) given by (—1,0, 1), (— 1, 1,2), ..., (— l,r — l,r), and take into account the following result which is obvious by construction (see proposition 4.3 in [2]).
PROPOSITION 3.4 With the previous notations, for any m £ S, one has: for any I > — 1 and any i, 0 < i < r,
(in this formula, Hi-i(Tm-nt) = 0 if either / - f l < z o r m — n / ^ 5). 4
DEGREES OF THE EQUATIONS
It is known that the degrees of the elements in a minimal generating set of the ideal of 5 are exactly (L(m) | MA m ) 7^ 0} (see [7] and [4]). Moreover, each such degree t appears as many times as the sum of the values /io(A m ) for m € Hl. In order to estimate these degrees, we are going to use the formulae and exact sequences in above section and, in particular, the following four terms one _,
TT (f\
\
.
TT (TS
\f
\
v ZJ
/ A
\
V E7
{f~\
\
Vn
••• —' ^ I v ^ r o / —' ^1 \-"-77ii ^\-mj —' -^Ov^mj —' -^Ov^cmJ —' "J*
PROPOSITION 4.1 Let m € 5 D Hl with t > t0, then H0(Qm) = 0.
Proof. Ifm-aeQ and m e 5 n H*, then m - a € Ql~l. Since t > t0, t - 1 > t0 and so Q^1 = 0 by 2.2. Thus, one has that Co(QTO) = ©m-aeoM 0 } — 0 and TT
ff~\
\ __ A
JiQlV^y^ J
—— U.
Now, on the other hand, by 3.4 one has for / = 0
.M^- 1 ')-
©
H-i(Tm-nt).
Thus, one has
for i > 1, and the following diagram of exact sequences
Equations defining toric projective varieties
63 = 0
- tf!(K m ,K ro ) -»•
( M . M - 1 ) =0
Again by 3.4 one has: For I = 0, i = 1: a€A
For / = 1, i = 1:
For i = 0, i = 0:
Replacing in the first sequence of the above diagram, one obtains
.. -»• ® # 0 (T m _ a ) *-$ H0(Tm) REMARK 4.2 In order to describe the mapping ipm we can do the following: 1. Compute the simplicial complexes Tm and Tm-a with a € A. (Using Integer Programming)
2. Take bases of H0(Tm) and Ho(Tm~a) with a £ A, picking a point {e} in each connected component and considering a generating tree, for example fix {e\ } for one concrete of the components and consider {e^} — {ei}, with e^ over the other components (see [7] for details).
3. Take the natural basis of @a&A H0(Tm^.a) obtained from the bases of H0(Tm-a) computed in 2.
4. Give the linear mapping ipm by means of a matrix using that f {ei}-{e 2 } if m - e i - e 2 £ S (see Proposition 3.2 in [2])
Now, let
ti := min{f
coker((pm) = 0 Vm € H1},
i.e. the minimum t € N such that (pm is surjective for every m 6 H*. (There exists ti by 3.2) REMARK 4.3 In order to find t\ it is enough to check the condition
coker(tpm) = 0 over the set £>(0) computed by 3.3 and using the matrix given in 4.2.
64
Morales, Campillo and Casares
LEMMA 4.4 Let m € H* with t > t0, then ff_i(T m _ a ) = 0, for any a £ A.
Proof. One has tf_i(T m _ a ) ^ 0 if and only if T m _ a = {0}, i.e, if and only if m - a e S and m - a - e £ S, for every e € E. But m - a - e ^ 5, for all e 6 E implies m — a €. Q. Since m — a € H1"1, we have that m — o € Q*"1,* — 1 > £oBut Q*-1 = 0 by 2.2. Therefore, H_i(T m - 0 ) = 0. THEOREM 4.5 Let 5 be a semigroup with the previous conditions. Then, the degrees of the polynomials in a minimal generating set of the ideal of S are less or equal than max(to,ti).
Proof. The degrees we are looking for are {I/(m) | /io(A m ) ^ 0}. Hence, it is enough to prove that if t > max(£ 0 ,ii), then H0(&m) = 0 for any First, since t > to we obtain the exact sequences: .
IT ( c\
\
\
£? /T?
~V
\
\
£/ / A
\
vn
U tf\
\
••• ~f fll(*
I -» jff^M^M^ 1 )) ->• 0. Second, since t > ti, we obtain ffi(M m , Mm ) — 0. Now, the diagram guarantees that Hi(Km,Km) = 0, and the result in the theorem follows from this fact.
5
AN EXAMPLE: THE DEGREE OF THE EQUATIONS OF TORIC PROJECTIVE CURVES
A projective toric curve is the projective scheme Proj(k[S}), where k is the ground field and 5 is the subsemigroup of N2 generated by elements of a set A consisting of elements of type ei - (d,0),e 2 = (0,d),ai = (an,a 12 ),...,a r = (a r i,a r2 ), whereof > 0 is the degree of the curve, Oji+0^2 = dforeachi, and#cd(d, on,...,ai r ) = 1. The choice of the generators gives an embedding of the toric curve into the projective space Pr+1. The homogeneous ideal defining this embedding is nothing but the ideal associated to the semigroup 5 and the generator set A. Notice that the pair S,A satisfies the conditions in section 1, if one chooses the map L given by L(ai,fl2) = (ai + a 2 )/d. Hence the degrees of the homogeneous equations defining the toric projective curve can be estimated from the results in above sections. For it, consider the partition A = ELSA where E = {61,62} and A = {ai, ...,ar}. Then, for each m € 5, the simplicial complex Tm has non trivial reduced homology if either it consists of only the empty set (in that case the —1 reduced homology is isomorphic to k) or it is the complex T consisting of the two points corresponding to ei and 62 but not to the edge joining both points (in that case the 0-th homology is isomorphic to k). Thus, the estimation of the degrees of the equations will be
Equations defining toric projective varieties
65
given in terms of the sets Q and D where Q is the Apery set (which can be viewed as the set of m € S such that Tm = {0}) and D is the set of those m E S such that Tm — T. Note that the set D is empty if and only if the curve is arithmetically Cohen-Macaulay, i.e. if the fc-algebra k[S] is Cohen-Macaulay (see [8]). Now, in the Cohen-Macaulay case, the degree of the equations of the curve are bounded by the integer to defined in section 1. In the non Cohen Macaulay case, one can also define the integer fa to be the largest integer t such that D n H1 is non empty. Notice that one has t\ < fa, so the equations of the projective curve are bounded by the integer max(to,fa). Using the explicit description, given in [2], of the sets Q and D in terms of the numerical semigroup Si generated by the integers d,an, ...,air 4.5, the integers t0 and fa can be easily computed as follows. First, for every b € Z, set l(b) equal to infinity if b £ S and, otherwise, equal to the least number of integers among d,an,...,air (allowing repetitions) with sum equal to b. Then, if B is the set of b £ S such that l(b — d)< l(b), and if B' is the subset of B consisting of those b & B such that b — d € S, one has t0 = I + max{l(b) fa=max{l(b-d)
& € £}, \ b&B'}.
Thus, one concludes the following result: THEOREM 5.1 With assumptions and notations as above, the degrees of the polynomials in a minimal set of homogeneous equations defining a toric projective curve C are bounded by the integer 1 + max{l(b)
| b € B}
if C is arithmetically Cohen Macaulay, and by min{l + max{l(b)
\ b £ B},max{l(b - d) \ b£B'}},
if C is not arithmetically Cohen-Macaulay.
References [1] D.BAYER, B.STURMFELS, Cellular resolutions of monomial modules J. reine angew. Math. 502, (1998), 123-140. [2] A. CAMPILLO, P. GIMENEZ, Syzygies of affine toric varieties. Journal of Algebra 225, 2000, 142-161.
[3] P. PISON-CASARES, A. VIGNERON-TENORIO, First Syzygies of Toric Varieties and Diophantine Equations in Congruence, Preprint of University of Sevilla Seccion Algebra 52 (1999).(Communications in Algebra to appear). [4] BRIALES E.; PISON P.; VIGNERON A. The Regularity of a Toric Variety, Preprint of University of Sevilla Seccion Algebra 53 (1999). [5] R.APERY, Sur les branches superlineaires des courbes algebriques. C.R.Acad.Sci.Paris 222 (1946), 1198-1200.
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[6] P.PISON-CASARES, A. VIGNERON-TENORIO, N-solutions to linear systems over Z. Preprint of University of Sevilla Section Algebra 43 (1998).
[7] E. BRIALES, A. CAMPILLO, C. MARIJUAN, P. PISON, Minimal Systems of Generators for Ideals of Semigroups J. of Pure and Applied Algebra, 124 (1998), 7-30. [8] N.V.TRUNG, L.T.HOA, Affine semigroup and Cohen-Macaulay rings generated by monomials. Trans.Am.Math.Soc. 298 (1986), 145-167.
KRS and determinantal ideals WINFRIED BRUNS, Universitat Osnabruck, FB Mathematik/Informatik, 49069 Osnabriick, Germany, Winf rled.BrunsQmathematik.uni-osnabrueck. de ALDO CONCA, Dipartimento di Matematica e Fisica, Universita di Sassari, Via Vienna 2, 07100 Sassari, Italy, concafissmain.uniss.it
1 INTRODUCTION
Let K be a field and X an m x n matrix of indeterminates. The determinantal ideals in K[X] are the ideals It generated by the i-minors of X, 1 < t < min(m,n), and ideals related to them. The Knuth-Robinson-Schensted correspondence (KRS) is a powerful tool for the computation of Grobner bases of determinantal ideals. For this purpose it has first been used by Sturmfels [16]. Then Herzog and Trung [12] have considerably extended the class of ideals to which KRS can be applied. In a different direction Sturmfels' method has been generalized by Bruns and Conca [4] and Bruns and Kwiecinski [5]. While Herzog and Trung use Grobner bases in order to derive numerical results, the papers [4] and [5] aim at structural information, mainly on powers of determinantal ideals and the corresponding Rees algebras. The crucial point in the application of KRS to Grobner bases is to show the equality in(7) = KRS(7) for the ideals I under consideration. We call these ideals in-KRS. Here in(/) is the initial ideal of I with respect to a so-called diagonal term order on ^[-Y], and KRS(I) is the image of / under the automorphism of the polynomial ring K[X] induced by KRS - in the strict sense KRS is a bijection from the set of standard bitableaux (or standard monomials) S to the set of monomials M. of jK^-X]. Both S and M. are K-bases of A'fX]: for S this is asserted by the straightening law of Doubilet-Rota-Stein. Since in(/) is a monomial ideal, one must assume that I has a basis of standard bitableaux. The first three sections of the paper are an expanded version of the first author's lecture in the conference. Section 2 recapitulates the straightening law, and Section
67
68
Bruns and Conca
3 introduces KRS. Section 4 explains the common ideas underlying the results of [4] and [5]. For this purpose we develop a conceptual framework in which KRS invariants play the central role. Such invariant is a function F : T> —> N defined on the set £> of all bitableaux (or products of minors) that, roughly speaking, is compatible with the straightening law and, moreover, satisfy the condition
F(£) = max{F(A) : A 6 £>, in(A) = KRS(E)}. It is then easy to see that each of the ideals Ik (F) generated by all standard bitableaux E with F(E) > k satisfies the condition in(/fc(F)) = KRS(/fc(F)). Even more is true: Ik(F) is G-KRS, i. e. in addition to being in-KRS, Ik(F) has a Grobner basis of bitableaux. The class of G-KRS ideals is closed under sums, that of in-KRS ideals / is closed under sums and intersections, and therefore one then obtains many G-KRS or at least in-KRS ideals. It has been shown in [4] that the functions 7( introduced by De Concini, Eisenbud and Procesi [9] are KRS invariants. This fact allows one to compute the Grobner bases, or at least the initial ideals of the symbolic powers of the Jj and products J fl • • • Its. The family o^ of KRS invariants found by Greene has been used in [5] for the analysis of the ideal underlying MacPherson's graph construction in the generic case. In Section 5 we show that all ideals that are generated by products of minors and do not "prefer any rows or columns" of the matrix X are in-KRS, at least if char K is 0 or > min(m, n). In characteristic 0 this is exactly the class of ideals that have a standard monomial basis and are stable under the natural action of GL(m, K) x GL(n,K) on AT[X]. In fact all these ideals can be written as sums of intersections of symbolic powers of the ideals It, and the symbolic powers are G-KRS, as stated above. Section 6 characterizes those among all the ideals of Section 5 that are even GKRS. We show that these are essentially the sums of the ideals J(k, d) introduced in [5] and for which Greene's theorem yields the property of being G-KRS. Since each KRS invariant can be derived from a family of G-KRS ideals, this shows that Greene's functions o^ are truly basic KRS invariants, at least if one considers functions F : T> —>• N for which .F(A) only depends on the shape of A. Section 7 complements the results of [4]. We show that the formation of initial ideal and symbolic power commute for the ideals It. This result can be interpreted as a description of the semigroup of monomials in the initial algebra of the symbolic Rees algebra by linear inequalities. In Section 8 we turn to a potential new KRS invariant jg related with the ideal I(X, 6) cogenerated by a minor 8. Except in the case in which Is = It, these do not only depend on shape and therefore constitute an interesting new class of functions. Though [12] gives some information on 7,5, we have not yet been able to show that these are KRS invariants. The application of KRS to determinantal ideals has also been investigated by Abhyankar and Kulkarni [1, 2]. Furthermore, variants of the KRS can be used to study ideals of symmetric matrices of indeterminates (Conca [7]) or ideals generated by Pfaffians of alternating matrices ([12], Bae^ica [3], De Negri [8]). There are now excellent discussions of KRS available in textbooks; see Fulton [10] and Stanley [15].
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2 THE STRAIGHTENING LAW Let K be a field and X an m x n matrix of indeterminates over K. For a given positive integer t < min(m,n), we consider the ideal It — It(X) generated by the i-minors (i. e. the determinants of the t x t submatrices) of X in the polynomial ring R = K[X] generated by all the indeterminates X^. From the viewpoint of algebraic geometry R should be regarded as the coordinate ring of the variety of K-\me&r maps /: Km -> Kn. Then V(It) is just the variety of all / such that rank/ < t, and R/It is its coordinate ring. The study of the determinantal ideals It and the objects related to them has numerous connections with invariant theory, representation theory, and combinatorics. For a detailed account we refer the reader to Bruns and Vetter [6]. Almost all of the approaches one can choose for the investigation of determinantal rings use standard bitableaux and the straightening law. The principle governing this approach is to consider all the minors of X (and not just the 1-minors X^) as generators of the .ft'-algebra R so that products of minors appear as "monomials". The price to be paid, of course, is that one has to choose a proper subset of all these "monomials" as a linearly independent JiT-basis: the standard bitableaux are a natural choice for such a basis, and the straightening law tells us how to express an arbitrary product of minors as a K-linear combination of the basis elements. (In [4], [5] and [6] standard bitableaux were called standard monomials; however, we will have to consider the ordinary monomials in K[X] so often that we reserve the term monomial for products of the Xij.) In the following [ai,...,a t | & i , . . . ,bt] stands for the determinant of the submatrix (X^: i = 1,..., t, j — 1,..., t). The letter A always denotes a product 61 • • • 6W of minors, and we assume that the sizes \Si (i. e. the number of rows of the submatrix X' of X such that <5j = det(X')) are descending, \S\ > ••• > \6W\. By convention, the empty minor [|] denotes 1. The shape | A| of A is the sequence (|<5i |,..., \SW\). If necessary we may add factors [ | ] at the right hand side of the products, and extend the shape accordingly. A product of minors is also called a bitableau. The choice of this term bitableau is motivated by the graphical description of a product A as a pair of Young tableaux as in Figure 1: Every product of minors is represented by a bitableau and, conversely,
an
n
a
wt
Figure 1: A bitableau
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every bitableau stands for a product of minors if the length of the rows is decreasing from top to bottom, the entries in each row are strictly increasing from the middle to the outmost box, the entries of the left tableau are in {1, . . . ,m} and those of the right tableau are in {1, . . . ,n}. These conditions are always assumed to hold. For formal correctness one should consider the bitableaux as purely combinatorial objects and distinguish them from the ring-theoretic objects represented by them, but since there is no real danger of confusion, we simply identify them. Whether A is a standard bitableau is controlled by a partial order of the minors, namely [01 , . . . , at \ bi , . . . , bt] ;< [ci , . . . , cu | di , . . . , du] t > u and di < Ci, bi < di, i = 1, . . . , u.
A product A = 61 • • • 6W is called a standard bitableau if
in other words, if in each column of the bitableau the indices are non-decreasing from top to bottom. The letter £ is reserved for standard bitableaux. The fundamental straightening law of Doubilet-Rota-Stein says that every element of R has a unique presentation as a K -linear combination of standard bitableaux (for example, see Bruns and Vetter [6]):
THEOREM 2.1.
(a) The standard bitableaux are a K -vector space basis ofK[X] .f ,
(b) If the product 6162 of minors is not a standard bitableau, then it has a representation oi62 = ^Xietrji, Xi € K, xt ^ 0, where e^ is a standard bitableau, ei -< Si , 62 -< f]i (here we must allow that r\i = I ) .
(c) The standard representation of an arbitrary bitableau A, i.e. its representation as a linear combination of standard bitableaux £, can be found by successive application of the straightening relations in (b) . (d) Moreover, at least one £ with |£| = |A| appears with a non-zero coefficient in the standard representation of A. Let BI, . . . , em and /i, . . . , /„ denote the canonical Z-bases of Zm and Zn respectively. Clearly K[X] is a Zm 0 Z"-graded algebra if we give Xij the " vector bidegree" a ® fj . All minors are homogeneous with respect to this grading, and therefore the straightening relations must preserve the multiplicities with which row and column indices occur on the left hand side. The straightening law implies that the ideals It have a AT-basis of standard bitableaux:
COROLLARY 2.2. The standard bitableaux E = 61 • • • Sw such that \Si > t form a K -basis of It-
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All these standard bitableaux are elements of It since S\ € It if |<5i > t. Conversely, every x & It can be written as a JiT-linear combination of products 6M where 6 is a minor of size t and M is a monomial. Properties (b) and (c) of the straightening law imply that the standard bitableaux in the standard presentation of 6M have the required property. We say that an ideal I C R has a standard basis if / is the /^-vector space spanned by the standard bitableaux E € /.
3 THE KNUTH-ROBINSON-SCHENSTED CORRESPONDENCE Let E be a standard bitableau. The Knuth-Robinson-Schensted correspondence (see Fulton [10] or Stanley [15]) sets up a bijective correspondence between standard bitableaux and monomials in the ring .RTfX]. We use the version of KRS given by Herzog and Trung [12].
If one starts from bitableaux, the correspondence is constructed from the deletion algorithm. Let E = (ajj|6jj) be a non-empty standard bitableau. Then one constructs a pair of integers (£,r) and a standard bitableau E' as follows.
(a) One chooses the largest entry £ in the left tableau of E; suppose that {(ii,ji), • ••, (iu,ju)}, ii < • • • < « „ , is the set of indices (i, j) such that (. — a^. (b) Then the boxes at the pivot position (p, q) — (iu,ju) in the left and the right tableau are removed. (c) The entry I = apq of the removed box in the left tableau is the first component of the output, and bpq is stored in s. (d) The second and third components of the output are determined by a "push out" procedure on the right tableau as follows: (i) if p = 1, then r = s is the second component of the output, and the third is the standard bitableau E' that has now been created; (ii) otherwise the entry bpq is moved one row up and pushes out the right most entry & p _u such that 6 p _ifc < bpq whereas 6 p _ifc is stored in s. (iii) one replaces p by p - 1 and goes to step (i). It is now possible to define KRS recursively: One sets KRS([ | ]) = 1, and KRS(E) =
We give an example in Figure 2. The circles in the left tableau mark the pivot position, those in the right mark the chains of "push-outs": In this example we have T/"r>CJ/\^\ __
y
y
y
y
y
y y
It is often more convenient to denote the output by a two row array instead of a monomial by making the row indices of the factors the upper row and the column indices the lower row; in the example
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5
4
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© 2 ® 4
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4 ®
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1 2 © 4
Figure 2: The KRS algorithm In general we set krs(E) = In both rows indices may appear several times; however, the indices Ui in the upper row are non-decreasing from left to right, and if u^ = Uj+i, then i>; > Vj+i for the indices in the lower row, as is easily checked. Conversely, if we are given a monomial, then, by arranging its factors in a suitable order, there is always a unique way to represent it as a two rowed array satisfying the condition just given. The reader may check that one can set up an insertion algorithm exactly inverting the deletion procedure above (what was deleted last, must be inserted first). In combinatorics one most often uses standard bitableaux for the investigation of sequences (or two row arrays). Then insertion is more important than deletion. Since insertion and deletion are inverse operations, one obtains
THEOREM 3.1. The map KRS is a bisection between the set of standard bitableaux on {!,... ,m} x {1,... ,n} and the monomials of K[X}.
This theorem proves half of part (a) of the straightening law: it is enough to check that every element of K[X] can be written as a linear combination of standard bitableaux. Using the straightening law one can now extend KRS to a K-linear automorphism of A^X]: with the standard representation x — £) asS one sets KRS(x) =]T>sKRS(S). The automorphism KRS does not only preserve the total degree, but even the ZTO © Z" degree introduced above: in fact, no column or row index gets lost. Note also that KRS it is not a K-algebra isomorphism: it acts as the identity on polynomials of degree 1 but it is not the identity map . It would be interesting to have some insight on the property of KRS as a linear map like, for instance, its eigenvalues and eigenspaces. REMARK 3.2. We note two important properties of KRS:
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(a) KRS commutes with transposition of the matrix X: Let X' be a n x m matrix of indeterminates, and let T : K[X] -> K[X'] denote the fiT-algebra isomorphism induced by the substitution Xij >->• X'^; then KRS(r(/)) = r(KRS(/)) for all / € ^1-X]- Note that it suffices to prove the equality when / is a standard bitableau. Then the statement follows from [12, Lemma 1.1]. (b) All the powers Sfc of a standard bitableau are again standard, and one has
4 KRS INVARIANTS AND GROBNER BASES
The power of KRS in the study of Grobner bases for determinantal ideals was detected by Sturmfels [16]. He applied Schensted's theorem:
THEOREM 4.1. Let (ti,...,tw) be the shape of the standard bitableau E. Then ti is the length of the longest strictly increasing subsequence in the lower row of krs(S).
If (ujj , . . . , Viq ) is a strictly increasing subsequence of the lower row, then the subsequence (u^ , . . . , uiq ) of the upper row must also be strictly increasing. Therefore
KRS(S) = M • diag[wit , . . . , uiq\ % , . . . , viq]
(*)
where M' is a monomial diag(J) denotes the product of all the indeterminates in the diagonal of the minor 6. Once and for all we now introduce a diagonal term order on the polynomial ring .fiT[^T]. With respect to such a term order the initial monomial in(<5) is diag(<5). There are various choices for a diagonal term order, For example one can take the lexicographic order induced by the total order of the X^j that coincides with the lexicographic order of the (i, j ) . Schensted's theorem implies through its equivalent (*) that for a standard bitableau S 6 It there exists a t-minor S such that in(<5) = diag(<$) KRS(S), and,
in particular, KRS(S) € in(/ t ): if q > t, then we can simply write
» ! , . . . , viq] = M" diag[ufl , . . . , uit | w^ , . . . , u i( ]. Since It has a basis of standard bitableaux, it follows that KRS(/t) C in(/ t ). The K- vector space KRS(/t) has the same Hilbert function as It with respect to total degree since KRS preserves total degree. But in(/ t ) also has the same Hilbert function as It. This implies:
THEOREM 4.2.
The t-minors of X form a Grobner basis of It, and KRS(/<) =
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In fact, the equation KRS(7 f ) = in(7t) has just been observed, and if M € in(/t) is a monomial, then it must be of the form KRS(S) for some standard bitableau S 6 It- But then M is divisible by in(<5) for some t-minor 6. Exactly this condition must be satisfied for the set of ^-minors to form a Grobner basis. It is worth formulating the idea behind the proof of Theorem 4.2 as a lemma:
LEMMA 4.3. (a) Let I be an ideal of K[X] which has a K-basis, say B, of standard bitableaux, and let S be a subset of I. Assume that for all S £ B there exists s E S such that in(s) | KRS(S). Then S is a Grobner basis of I andin(I) = KRS(J). (b) Let I and J be homogeneous ideals such that in(/) = KRS(/) and in(J) = KRS(J). Then in(7) + in(J) = in(/ + J) = KRS(J + J) and in(7) n in(J) = in(/n J) = KRS(/n J).
The proof of part (a) has been explained for the special case of J = It. For (b) one uses KRS(7 + J) = KRS(7) + KRS(J) = in(7) + in(J) C in(7 + J), KRS(7 n J) = KRS(7) n KRS( J) = in(7) n in( J) D in(7 n J),
and concludes equality from the Hilbert function argument. DEFINITION 4.4. Let 7 be an ideal with a standard basis. Then we say that 7 is in-KRS if in(7) = KRS(7); if, in addition, the bitableaux A € 7 form a Grobner basis, then 7 is G-KRS. In slightly different words, an ideal 7 with a standard basis is in-KRS if for each S e 7 there exists x 6 7 with KRS(E) = in(o;); it is G-KRS if x can always be
chosen as a bitableau. As a consequence of Lemma 4.3 one obtains
LEMMA 4.5. Let I and J be ideals with a basis of standard bitableaux.
(a) If I and J are G-KRS, then I + J is also G-KRS. (b) If I and J are in-KRS, then I + J and 7 n J are also in-KRS. In general the property of being G-KRS is not inherited by intersections as we will see below.
The KRS correspondence could be used for the proof of Theorem 4.2 since, by Schensted's theorem, the length of the first row of a standard bitableau is a KRS invariant: DEFINITION 4.6. Let T> be the set of all bitableaux on the matrix X and F : T> -»• N a function on T>. Then we define a function on the set M of monomials, also called F, by F(M) = max{F(A) : A € V, M = in(A)}.
Of course F(M) is well-defined since there are only finitely many A £ T> with M = in(A). We say F is a KRS-invariant if the following conditions are satisfied:
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(a) jP( A) is the minimum of F(E) where E runs through the standard bitableaux in the standard representation of a bitableau A; moreover, if E' appears in the standard representation of xA. for some x € R, then -F(S') > (b) F(E) = F(KRS(E)) for all standard bitableaux E € P. If just condition (a) is satisfied, then we say that we say that F is str-monotone. In order to interpret condition (b) combinatorially we write krs(E) = Then the bitableaux A such that KRS(E) = in(A) correspond bijectively to the decompositions of the lower row of krs(E) into strictly increasing sequences, called inc-decompositions. In fact, if the sequence v^,..., Vit is strictly increasing, then the same holds for u^,..., uit, or equivalently, XUlVl • • • XUtVl is the diagonal product of a i-minor. Note that u ^ , . . . , « $ , is always non-decreasing, and therefore Xmvi • • • XUtVt can only be a diagonal product if v^,..., Vit is strictly increasing. Thus condition (b) requires that F(E), a number associated with the standard bitableau, is encoded in the sequence of integers forming the lower row of krs(E). It is now an easy exercise to show PROPOSITION 4.7. Let F be a KRS-invariant, and let k be an integer. Let Ik(F) be the ideal generated by all bitableaux A such that F(A) > k. Then
(a) Ik(F) has a standard basis formed by all standard bitableaux E such that F(E) > k.
(b) Moreover, Ik(F) is G-KRS. Part (a) follows immediately from str-monotonicity and part (b) follows from 4.3. In general, it does not suffice to take the standard bitableaux in Ik(F) to obtain a Grobner basis of Ik(F); we will discuss an example below. Starting from the ideals Ik (F) and applying 4.5 one can now find new ideals that are G-KRS or at least in-KRS. We have seen that the length of the first row is a KRS-invariant. In order to apply KRS to a wider class of ideals one has to find other (or more general) KRS invariants. One such family of invariants are the functions 74 defined as follows. For an integer s and a sequence si,...,sw of integers one sets w
7t(s) = ( s - £ + ! ) +
and
~ft(si, ...,sw) = ]T Jt(si).
Here we have used the notation (Ar)+ = max(0,A;). One then defines this function for bitableaux A = Si • • • Sw by 7t(A) = 7t(|A|).
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The invariants •jt are of interest since they describe the symbolic powers of the ideals It. Provided the characteristic of the field is 0 or > min(m,n) (we then say K has non-exceptional characteristic), all products Itl •••/{,. have a primary decomposition as intersections of such symbolic powers, and can therefore described in terms of the jt '• (a) A 6 I(tk} «=» 7t(A) > fc;
(b) the standard bitableaux E with 7t(E) > k are a jftT-basis of I\ , (c)
See [6, Section 10] and [9]. The straightening law shows that jt is str-monotone. In [4] we have proved
THEOREM 4.8. The functions jt are KRS-invariants. As a consequence of this theorem and Lemma 4.5 one obtains that all ideals 4 are G-KRS and that all products of ideals of minors are in-KRS if char K = 0 or char .ft" > min(m, n). Furthermore one can then show that the "initial algebras" of the symbolic and ordinary Rees algebras of the ideals It are normal semigroup rings. In particular this implies that these algebras are Cohen-Macaulay. Another object accessible to this approach is the subalgebra At of K[X] generated by the i-minors of X. See [4] for a detailed discussion. EXAMPLES 4.9. (a) We choose m, n > 3. By the above discussion the ideal 7^2) is G-KRS. We want to show that the standard bitableaux in 1% do not form a Grobner basis. The monomial M = ^12^23^21^32 is the initial term of a bitableau of shape (2,2) and hence M 6 in(/2 ). If the standard bitableaux in /|2' were a Grobner basis, then M would be divisible by the initial of a standard bitableaux in 1^. The standard bitableaux in /|2) of degree < 4 have shape (3), (3,1) and (2,2) and clearly their initial term cannot divide M. (b) Suppose that I and J are G-KRS and let E be a standard bitableau in / n J. Then we can find bitableaux AI € 7 and A2 € J such that KRS(E) = in(Ai) = in(A 2 ). In general it can happen that AI $ J and A2 ^ /, and / n J need not be G-KRS. An example is / = 1% ', J = /4 for a matrix of size at least 6x6. In fact, let S = [1345 1236][26145]. (This is the example considered in Section 3). Then £ e J is obvious, and E € / since 72(E) > 2. Since / n J is in-KRS, it follows that KRS(E) = XuX2iX32X45X56XX63 £ in(7 n J). It is however impossible to write this monomial as the initial monomial of a bitableau A in / n J. The bitableaux of degree 6 in /n J are those of shapes (6), (5,1), and (4,2). (By Schensted's theorem, only the last shape would be possible.) Let T be a new indeterminate and define the ideal J in the extended polynomial ring K[X][T\ by
J = Im + Im^T + ••• + hTm~l + (Tm)~
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we assume that m < n. This ideal and its Rees algebra 0°^.0 J* is fundamental for the generic case of MacPherson's graph construction; see [5]. If one "expands" the power Jk into a "polynomial" in T, then the "coefficient" of Tkm~d is
eo + ei + • • • + em - k, ei 4- 2e2 + • • • + mem - d. For a non-increasing sequence si, . . . , sw of non-negative integers let us define k
ak(si,...,sw) - y^Sj j=i
where s» = 0 if i > w. Then we can set at (A) = Q fc (|Al)
for every bitableau A. The straightening law shows that a/t is str-monotone, and it follows easily that
(a) A e J ( k , d ) <=> a f c (A) > d; (b) the standard bitableaux E with afe(S) > d are a ff-basis of J(fc, d)-
THEOREM 4.10. The functions ak are KRS-invariants. An analysis of a^ in terms of inc-decompositions shows that ak(KRS(E)) > d if and only if the lower sequence of krs(E) contains a subsequence of length d that
itself can be decomposed into k increasing subsequences. Thus Theorem 4.10 is just a re-interpretation of Greene's theorem [11]: «*(£) is the maximal length of a subsequence that has an inc-decomposition into k parts. For the "determinantal" consequences of Theorem 4.10 we refer the reader to [5]. The relationship between the KRS invariants 74 and ak is analyzed in Section 6. 5
IDEALS DEFINED BY SHAPE
We say that an ideal I C K[X] is defined by shape if it is generated as an ideal by a set of bitableaux, and, moreover, it depends only on |A| whether a bitableau A belongs to /. In this section we want to characterize the ideals defined by shape in the case in which the characteristic of K is big enough. In particular we will see that all these ideals are in-KRS. The following "balancing lemma" is a crucial argument; it is a simplified version of [6, (10.10)].
LEMMA 5.1. Let TT and p be minors of X, and set u = \p\, v — TT (we include the case TT = 1, in which u — 0). Suppose that u < v and charK = 0 or charK > min(w + l,m — (u + l),n — (u + 1)). Then irp £ Iu+ilv-i.
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The case in which u — 0 is just Laplace expansion. In general the lemma says that a product •np of minors can be expressed as a linear combination of minors that are "more balanced" in size. By repeated application of the lemma we see that irp is even a linear combination of products 6s such that \5\ + \e\ = \K\ + \p\ and 1*1 < N < 1*1 + 1The group GL = GL(m, K) x GL(n, K) operates as a group of linear substitutions on R — K[X] in a natural way: for M 6 GL(m,K) and N € GL(n, K) one substitutes Xij by the corresponding entry of MXN~l. Therefore R is an interesting object for the representation theory of GL, and representation theory offers another approach to the theory of determinantal rings. It is clear that an ideal defined by shape is GL-stable since each element g € GL transforms a minor into a linear combination of minors of the same size. Let a be a shape. Then the ideal 1^ is generated by all bitableaux A with 7t(A) > 7t(
|Aj > a. If even | A|<, > (?k for all k, then we write |A| D a. The ideals 1^ are evidently defined by shape. By definition
Therefore 1^ has a standard basis and is in-KRS. The following theorem should have been contained in [6, Section 11]. THEOREM 5.2. Suppose that char/f = 0. Then the following are equivalent for an ideal I C K[X}: (a) I is defined by shape; (b) I is a sum of ideals of type 1^.
(c) I has a standard basis, and I is stable under the action o/GL on K[X}. Proof. (a)=^(b): / is obviously contained in the sum of all ideals /(I A D where A runs through the generators of I. On the other hand, let S be a (standard) bitableau contained in /d A D, that is, |S| > |A|). If even |S| D |A|, then we can apply Laplace expansion and write S as a linear combination of bitableaux of the the same shape as A. Suppose that |E| ~fi |A|, and let k be the smallest index such that |E|fc < |A|fc. Then there must be an index j < k such that |S|j > |A|j. Now one applies the balancing lemma above, increasing the fc-th row at the expense of the j'-th. After finitely many balancing steps we have written S as a .ftT-linear combination of bitableaux E such that |E D |A|. (b)=>-(a): This is evident, as well as (b)=>(c). (c)=^(b): We have to show that H 6 / for all standard bitableaux E such that E > |S| for some standard bitableau S 6 /.
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Set er = |S|. Suppose first that |H = |E|. By [6, (11.10)] there is a decomposition of GL stable subspaces where I>^ is generated by all standard bitableaux 0 such that |9| > a\ in this decomposition Ma is irreducible and the unique GL-stable complement of /> ' in /(") . Thus we can write
E =x +y where x £ Ma , y £ 1^' . Since E £ /> , we deduce that x ^ 0. For H = S it follows that Ma C I. In fact, / has a unique decomposition as a direct sum of irreducible GL-modules, and these are exactly the MT; since the projection to Ma is non- trivial, it must appear in the decomposition of I. For general H it now follows that x £ /, and therefore all the standard bitableaux in the standard representation of x must belong to /. However, since y £ /> , H must appear in this standard representation. Suppose now that |H| > a. If we find some standard bitableau F of shape |H| such that Tel, then the argument just given shows that E £ I. Since we can connect a and |H| by a chain of shapes with respect to the partial order <, it is enough to consider the case in which |H| is an upper neighbor of a. One obtains the upper neighbors by either inserting a new box below the bottom of the diagram (and thereby increasing 71 by 1) or by removing an "outer corner" box of the Young diagram of shape a and inserting it at an "inner corner" in such a way that the box travels one row up or one column to the right. (At the end of the first row is also an inner corner, provided its length is < min(m,n)). Figure 3 illustrates the three cases.
;t Figure 3: Upper neighbors with respect to < The case in which a new box is added, is trivial. In fact, we fill it with Xmn, and X mn £ is standard of the right shape. We now assume that a box travels one row up, say from row k to row k - 1. With a = (si,..., sw) let p = Sk and q = SK-I- One forms the standard bitableau Q = Oi • • • 6W where
0k = [1,... ,p - 1,q + 111,... ,p - 1,q + 1]
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Brims and Conca
of shape a. Since 0 is a standard bitableau of shape a, it belongs to /, as was shown above. Note that Sk+i < p < q < Sfc_ 2 (where Sk+i = 0 if k = w and Sfc_2 = oo if k = 2). Now we apply the cyclic permutation TT € (pp+l. ..qq+1) to both the rows and columns of the matrix X; the transformation n belongs to GL. Therefore 7r(0) € /. All the factors of 0 except dk-i and Ok are invariant under TT, whereas , . . . , g + l]
(1) (2)
If we straighten this product and multiply its standard presentation with the remaining factors of 0, then we obtain the standard representation of ?r (0). Therefore it is enough that a standard bitableau of shape (q + l,p — 1) appears in the standard representation of it(8k-i)ii(6k)- (Whereas the proof of ((a)=>-(b) is based on "balancing", we now need the "unbalancing" effect of straightening.) Mapping all the indeterminates Xij with i ^ j, i < p or j < p, to 0 and Xu,... , X p _i p _i to 1, one reduces the claim to the assertion that in the standard representation of [111][2,..., r 12,..., r] with r > 3 the minor [1,..., r \ 1,..., r] shows up, and that is immediate from Laplace expansion. The case in which the box travels one column to the right is similar and left to the reader. Essentially it is the case in which a consists of a single column. D One should note that the implications (b)=^(a) and (b)=J>(c) are true over arbitrary fields (actually, over all rings of coefficients), whereas (a)=»(b) needs only that char AT > min(m, n). The implication (c)=>(b) uses the hypothesis that K has characteristic 0 more profoundly: the ideal 7f+1 + (X^,..., X^) satisfies (c) if char .fiT = p > 0, but is not a sum of ideals /^ (provided that X is not just a 1 x 1 matrix).
COROLLARY 5.3. Suppose that K is a field of characteristic 0. Then all GL-stable ideals that have a standard basis are in-KRS. This follows from Lemma 4.5 since the ideals 7^' are in-KRS. Note that there are ideals with a standard basis which are not in-KRS. For instance, let S be standard bitableau with KRS(S) ^ in(S) and set d = degS. Then / = (E) + (Xtj : 1 < i < m, 1 < j < n)d+l has a standard basis and KRS(7) / in(/). As we have seen in the previous section, there are in-KRS ideals that are not G-KRS. The G-KRS ideals among those considered in Corollary 5.3 will be characterized in the next section.
6 BASIC KRS-INVARIANTS In this section we want to show that the functions a^ are basic KRS-invariants, as far as functions F : T> -» N are considered that depend only on shape. First we prove a converse of Proposition 4.7:
PROPOSITION 6.1. Let F : T> -> N be a str-monotone function, and let k be an integer. Let h(F) be the ideal generated by all bitableaux A such that F(A) > k. Then the following hold:
KRS and determinantal ideals
81
(a) Ik(F) has a standard basis, and A e Ijt(F) (if and) only z/F(A) > k; (b) iflk(F)
is G-KRS for all k, then F is a KRS-invariant.
Proof, (a) It follows immediately from the definition of str-monotonicity that Ik (F) is the K-vector space generated by the bitableaux A with F(A) > k, and that the standard bitableaux E € Ik (F) form a jRT-basis. (b) We have to show that F(E) = F(KRS(S)) for every standard bitableau E. First note that F(E) < F(KRS(E)). Set k - KRS(E). Since by assumption Ik(F) is G-KRS we have that KRS(E) = in(A) for some A £ Ik(F) and consequently F(KRS(£)) > k. It follows that for every t one has
KRS(J t (F)) C (M : F(M) >t)C in(It(F)) = KRS(/ t (F)) and hence KRS(J t (F)) = (M : F(M) > t). Now let M be a monomial and set t = F(M). Then E = KRS-1(^) is in It(F) which proves that F(S) > F(KRS(£)). D Now we can show that the KRS-invariance of the functions 7^ follows from that of the otk'PROPOSITION 6.2. For allt,l
Proof. Let E e I j ' be a (standard) bitableau, and suppose that k is the biggest index such that |E|jt > t. Then obviously E e J(k,r + (t - 1)). The verification of the inclusion D is likewise simple. D Since by Greene's theorem the ideals J(k, d) are G-KRS, it follows immediately that the ideals l[r' are G-KRS. In conjunction with Proposition 6.1 we therefore obtain that the jt are KRS-invariants. From hindsight, Theorem 4.8 is an easy consequence of Greene's theorem. The most difficult part of [4] is the proof that the ideals ITt are G-KRS in nonexceptional characteristics. Actually Jtr = J(r, rt) so that the property of being G-KRS is no longer surprising for I£. Not every shape defined, G-KRS ideal is the sum of ideals J(k, d). As an example one can take /^ n Is = /i/s. Another exception occurs for square matrices. For example, if m = n = 4, then 72/4 is G-KRS, but it looses this property for bigger matrices. Nevertheless, the next theorem shows that the functions a>k are truly basic KRS-invariants:
THEOREM 6.3. Let I be a shape defined ideal. If I is G-KRS, then it is the sum of ideals of type J(k, d) n If and, ifm = n, ( J ( k , d) n /r)/£.
Brims and Conca
82 Proof. Let S € 7 be a (standard) bitableau. Suppose that a = |£| = (si,...,s f c , !,...,!)
with Sfc > 2, and st < max(m, n) (equality can only occur if m = n). Set d = otk(a) and £> = deg(S). Then J = 7(fc,d) n if is the smallest ideal of type J ( k , d ) n If containing E. Among the shapes of the elements in the standard basis of J there exists a unique element that is minimal with respect to the partial order defined by the functions 7t (see Section 5). In fact let u = [_d/k\ and r = d-uk. Then the smallest element is
d = (u + 1, . . . , u + 1, u, . . . , u, 1 . . . , 1)
where u 4- 1 appears r times, u appears d — r times, and 1 as often as in a. We have J = /W. Therefore, if a — 0, the ideal J is contained in /, as follows from Theorem 5.2. Now suppose that a > 6. Then it is enough to show that there exists a standard bitableau S of shape a such that |A| < a whenever in(A) = KRS(S). Since we have |A| < a from the KRS invariance of the functions 7t (or a^) it really suffices to find S of shape a such that |A| ^ a for all A with in(<J) = KRS(S). Instead of constructing A directly, we find a monomial M such that KRS-1(M) has the desired shape, but M cannot be written as in(A) where |A| = a. The shape of KRS -1 (M) can be controlled via the 7 or a functions. Let us first consider a = (51,52); we set p = si, q = s2. Then max(m,n) > p > q + 2 > I and min(m,n) > p. It is harmless to assume n — max(m, n). With r — p - q + 2 let
For p = 5 and q = 3 this monomial has the following "picture" :
The reader can check that KRS -1 (M) indeed as shape (p, q). However, it is not possible to decompose M into a product in(#i)in(<52) where \8i\ — p, \52\ = q. In the general case one must multiply M with suitable factors. These are not hard to find. D 7 INITIAL IDEAL OF SYMBOLIC POWERS AND SYMBOLIC POWERS OF THE INITIAL IDEAL
Let S be a polynomial ring and T a term order on S. Given an ideal J of S and an integer b one denotes by «/<& the intersection of all the primary components
KRS and determinantal ideals
83
of height < b in a primary decomposition of J (J
(1)
see [17]. If J has height c, then we define the fc-symbolic power J^ of J to be: •/<*> = (Jk)
Note that J^ = J if and only if J has no embedded primes and all the minimal primes have height c. Now by (1)
On the other hand, since in(J)* C in(J fc ), we have in(J)<*> - (in(J)*)<e C in(J fc )< c . Summing up, there are inclusions
(2) These inclusions are in general strict. In this section we will show that they are equalities when J is the determinantal ideal It and r is a diagonal term order. THEOREM 7.1. In non- exceptional characteristics we have in(J t ( * ) )=in(J*)< c = in(J4)(*) where c is the height of It, i.e. c — (m — t + l)(n — t + 1). The initial ideal in(/t) of It is the square free monomial ideal generated by the diagonals of the f-minors, simply called ^-diagonals in the sequel. Hence it is the Stanley-Reisner ideal of the simplicial complex A t = {A C {1, . . . , m} x {1, . . . , n} : A does not contain t-diagonals}. Denote by Fj the set of the facets of Aj. Then in(J t ) =
PF
where Pp denotes the ideal generated by the Xij with ( i , j ) $ F. The elements of Ft are described in [12] in terms of families of non-intersecting paths. It turns out that Aj is a pure (even shellable) simplicial complex. Since the powers of the ideals PF are P^-primary, it follows that
We start by proving:
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Bruns and Conca
PROPOSITION 7.2. We have
We have already mentioned in Section 4 that If is G-KRS, and in particular the initial ideal in(/t ) of I\ is generated by the monomials M with 7t(M) > k. Now a monomial M = Ili=i xafbi is in Pp if and only if the cardinality of {i : (a;, &;) ^ F} is > k. Equivalently, M is in P£ if and only if the cardinality of {i : (aj,6j) € F} is < deg(M) - k. If we set
wt(M) = max{|A| :AC[l,...,s] and {(oj,6i) : i € 4} G A t } then we have that a monomial M is in PlF€Ft PF if an k. Now Proposition 7.2 follows from:
LEMMA 7.3. Let M be a monomial. Then 7t(M) + wt(M) = deg(M). We reduce this lemma to a combinatorial statement on sequences of integers. Given such a sequence b we define wt(b) to be the cardinality of the longest subsequence of b which does not contain an increasing subsequence of length t, that is,
wt(b) = max{length(c) : c is a subsequence of B and 74 (c) = 0}. Let M — Hi=i xa.ibi be a monomial. We may order the indices such that aj < ai+i for every i and bi+i > 6, whenever Oi = ai+\ . (We have already considered this rearrangement in Section 3.) Then the i-diagonals dividing M correspond to increasing subsequences of length t of the sequence b, and wt(M) = wt(b). Since wt(M) depends only on the sequence b we may assume that o^ = i for every i. Then, by exchanging the role between the Oj's and the 6,'s we may also assume that the bi are distinct integers (see Remark 3.2). Summing up, it suffices to show that:
LEMMA 7.4. One has 7t(6) + wt(b) = length(fr) for every sequence b of distinct integers. Proof. Let P be the tableau obtained from b by the Robinson-Schensted insertion algorithm. We have already discussed the first part of Greene's theorem, namely that the sum a/b(P) of the lengths of the first k rows of P is the length of the longest subsequence of 6 that has a decomposition into k increasing subsequences. But the theorem contains a second (dual) assertion: the sum a*k(P) of the lengths of the first k columns of P is the length of the longest subsequence of 6 that can be decomposed into k decreasing subsequences. It follows that a sequence a has no increasing subsequence of length t if and only if it can be decomposed into t - 1 decreasing subsequences. Then wt(b) is the equal to the maximal length of a subsequence of 6 which can be decomposed into t — 1 decreasing subsequences. Therefore Wt(b) = al_l(P). On the other hand, by Theorem 4.8 we know that 7t(6) is equal to 7t(P) which is the sum of the length of the columns of P of index
> t. Therefore 7^(6) + wt(b) is equal to the number of entries of P which is the D
length of b.
KRS and determinantal ideals
85
We know [4, Thm. 3.5] that in(/tfc) = f|j-=i in(/j fc(t+1 ~ j)) ) (in non-exceptional characteristic) and hence, taking into consideration Proposition 7.2, we have:
3=1
ri
Since the powers of the ideal Pp are Pp-primary we have that (3) is indeed a primary decomposition of in(I*). Hence in(/tfc)
QUESTION 7.6. What is a primary decomposition of the powers of in(Jt)? Is the Rees algebra of in(It) Cohen-Macaulay? Is it normal? 8
COGENERATED IDEALS
As before, let X = (%ij) be an m x n matrix of indeterminates. We consider the set of minors of X equipped with the usual partial order that has been introduced in Section 2. Let <5 = [01,02, ... ,a r |&i,&2, ••• ,br] be a minor of X. One defines 1(6, X) to be the ideal of K[X] generated by all the minors /i such that fj, % S, i.e.
The ideal I ( 6 , X ) is said to be the ideal cogenerated by 6. For general facts about the ideals cogenerated by minors we refer the reader to [6]. We just recall that 1(6, X) is a prime ideal and that it has a standard basis. Namely the set B(6) = {£ : E = a\ • • • crw is a standard bitableau and a\ ^ 6} is a K- vector space basis of 1(6, X). Herzog and Trung have shown in [12, Theorem 2.4] that the natural generators of 1(6, X) (i.e. the minors n such that /j, ^ 6) form a Grobner basis of 1(6, X) with respect to the diagonal term order. Their argument makes use of the KRS correspondence and boils down to the study of the KRS image of the elements of B(6). We will see that this can be rephrased in terms of properties of a suitable 7-function associated to 6. We will henceforth denote its value on the bitableau A by 7,5 (A). We start by defining j s ( p ) f°r a single minor (j,= [ c i , . . . , C k \ d i , . . . , d k ] , namely
7a(//) = max{(i - j + 1)+ : 1 < i < k, 1 < j < r + 1 and (cj < a,- or dt < bj)}
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where, by definition, o r+ i = 6r+1 = oo. Then we extend, by linearity, the 75function to product of minors, that is, if A = /^ • • • nh is a product of minors, then
The standard bitableaux £ such that 7,5 (S) ^ 0 are exactly the elements of B(6). Note that if one takes <J = [1, . . . , i - 1 1, . . . , i — 1] then J(<5, X) — 7t and 7*(/i) = 7t(/i). We may extend, as we have done in Section 4, the definition of the 7,5 -function also to ordinary monomials by setting:
"fi(M) = max{7i(A) : A is a bitableau and in(A) = M}. In terms of the 7,5-function Herzog and Trung [12, Lemma 1.2] proved
LEMMA 8.1. Let £ be a standard bitableau. Then 7«(S) ?* 0 =» 75(KRS(£)) 5* 0 and this implies that
THEOREM 8.2. The ideal 1(5, X) is G-KRS. Note that from 8.2 one has that
There are many natural questions concerning the function 7,5 and related ideals. For instance:
QUESTIONS 8.3. Let J(6,k) = h(ls), that is, the ideal generated by the bitableaux A such that 7,5 (A) > k, and let B(6, k) be the set of the standard bitableaux £ with 7,j(S) > k. (a) Is 7,5 a KRS-invariant? (b) Is B(6, k) a basis of J(6,k)t i.e. is 7,5 str-monotone? (c) Is J(S,k) equal to I(5,X)^7 The inclusion J(6,k) C I(6,X)W holds since the symbolic powers form a nitration and for a single minor /i is not difficult to see that n 6 1(6, X)^ where k = 75 (/z).
(d) Is i REFERENCES
1. S.S. Abhyankar and D.M. Kulkarni. Bisection between indexed monomials and standard bitableaux. Discrete Math. 79 (1990), 1-48.
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2. S.S. Abhyankar and D.M. Kulkarni. Coinsertion and standard bitableattx.) Discrete Math. 85 (1990), 115-166.
3. C. Bae^ica. Rees algebra of ideals generated by pfaffians. (1998), 1769-1778.
Commun. Algebra 26
4. W. Bruns and A. Conca. KRS and powers of determinantal ideals. Compositio Math. Ill (1998), 111-122.
5. W. Bruns and M. Kwiecinski. Generic graph construction ideals and Greene's theorem. Math. Z. 233 (2000), 115-126. 6. W. Bruns and U. Vetter. Determinantal rings. Lect. Notes Math. 1327, Springer 1988.
7. A. Conca. Grobner bases of ideals of minors of a symmetric matrix. 3. Algebra 166 (1994), 406-421.
8. E. De Negri. ASL and Groebner bases theory for Pfaffians bras. Dissertation, Universitat Essen (1996).
and monomial alge-
9. C. De Concini, D. Eisenbud and C. Procesi. Young diagrams and determinantal varieties. Invent, math. 56 (1980), 129-165. 10. W. Fulton. Young tableaux. Cambridge University Press 1997. 11. C. Greene. An extension of Schensted's theorem. Adv. Math. 14 (1974), 254265.
12. J. Herzog and N. V. Trung. Grobner bases and multiplicity of determinantal and pfaffian ideals. Adv. in Math. 96 (1992), 1-37.
13. D.E. Knuth. Permutations, matrices, and generalized Young tableaux. Pacific J. Math. 34 (1970), 709-727. 14. C. Schensted. Longest increasing and decreasing subsequences. Can. J. Math. 13 (1961), 179-191. 15. R.P. Stanley, Enumerative Combinatorics, Vol 2. Cambridge University Press 1999.
16. B. Sturmfels. Grobner bases and Stanley decompositions of determinantal rings. Math. Z. 205 (1990), 137-144. 17. B. Sturmfels, Ngo Viet Trung and W. Vogel. Bounds on degrees of projective schemes. Math. Ann. 302 (1995), 417-432.
The Koszul complex in projective dimension one WINFRIED BRUNS, Universitat Osnabruck, FB Mathematik/Informatik, 49069 Osnabriick, Germany, Winfried.BrunsQmathematik.uni-osnabrueck.de
UDO VETTER, Universitat Oldenburg, FB Mathematik, 26111 Oldenburg, Germany, vetterQmathematik.uni-oldenburg.de
Let R be a noetherian ring and M a finite .R-module. With a linear form \ on M one associates the Koszul complex K(x)- If M is a free module, then the homology of K(x) is well-understood, and in particular it is grade sensitive with respect to Imx-
In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the first nonvanishing Fitting ideal I M has the maximally possible grade r + 1, r = rankM. Then h = grade Imx < r + 1 for all linear forms x on M, and it turns out that Hr.i(K(x)) = 0 for all even t < h and Hr-i(K(x)) = S ( '~ 1)/2 ((7) for all odd i < h where S denotes symmetric power and C — Ext^j(M, R), in other words, C = Cokt/i* for a presentation
Moreover, if h < r, then Hr-/l(K(x)) is neither 0 nor isomorphic to a symmetric power of (7, so that it is justified to say that K(x) is grade sensitive for the modules M under consideration. We furthermore show that the maximally possible value grade Im x —r + 1 can only occur in two extreme cases: (i) r = 1 or (ii) rankF = 1 and r is odd. The note was motivated by a result of Migliore, Nagel, and Peterson (see [MNP], Proposition 5.1). They implicitly prove the result on K(x) for Gorenstein rings R, using local cohomology. Our method allows more general assumptions. (Even the assumption that R is noetherian is superfluous if one uses the correct notion of grade.) It is based on results in [BV1] and has a predecessor in [HM]. The case in which rankF = 1 has been treated in [BV3].
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The situation in which the Fitting ideal IM of M has only grade r is also of interest. For example, it occurs for the Kahler differentials of complete intersections with isolated singularities. While our method also yields results in this case, we have restricted ourselves to the case of grade r + 1 for the sake of clarity. The detailed account of the linear algebra of M and its exterior powers has been given in [BV2], Section 2.
For technical reasons we start with a situation dual to the above one. So let F, GJoe finite free ^-modules of rank m,n and ip : G -> F an .R-homomorphism. Set G = G ® S(F) where S(F) denotes the symmetric algebra of F. Then we may consider ip an S(F) -linear form on G and can define the Koszul antiderivation
with respect to ?/> in the usual way, i.e.
A ... A Xi) —
(-l) J t/>(:Ej):ri A .. .Xj ... A
for xi . . . Xi € G. We use the term Koszul complex also for the complex
associated with a linear map if : R —t G. Suppose that i/>ip = 0 and let
dv:^G^ /\G
be the differential of the Koszul complex associated with tp ® S(F), i. e.
dv(x) = (
for x € A^- Since ip
Koszul complex in protective dimension one
91
bicomplex K.
0
1
,
[
0
i «i
4*
4-
0
0
0
I - ! i -i
"+•
! i
4-
-4*
°^l^ \^"\r^ "L 0———,
«
_^M
-^ ^
-^« -^« 0
where for all integers p, g, and S? means qth symmetric power. The row homology of /C
at Mp'9 is denoted by H™, the column homology by Hp'q . Thus Hp>° is the pth homology module Hp of the Koszul complex associated to
—^-> ]Vp+1 • • •
o ——>. /z — ^-> G ——>. ..-/VPG — ^-* A P+I G where the maps
PROPOSITION 1. Set g = grade 1M, C = Coktfj, and let h = grade Im^*. Assume that r = n — m > 1 and g — r + 1. Then (a) Im
(b) Hi = 0 for 0 < i < min(2, h - 1);
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(c)
if 3 < i < M £ 0 ( 2 ) i/ 3 < » < /i, i = 0 (2);
\0
(d) moreover, for h > 3 £/iere is on ea;ac£ sequence Q -> S^ (C) -> Hh -> Hh h
h
Q-*H -^H
if h £0(2),
if h = 0 ( 2 ) .
Proof. Let M = Cok i/>* . We choose a basis e\ , . . . , em of F* and define the linear map * : G* -> /\m+lG* by *(x) = V*(ei) A ... A *i/>*(em) A a;. Then one obtains a complex 0 -»• F* ^ G" whose dual is the head of the Buchsbaum-Rim complex resolving C = Coktp. It follows that M* = Im\&*, and obviously Im** C IMG. Since ip 6 M*, one has Imtp C IM- This shows (a). We quote some well-known facts about the homology of 1C. Let 0 < p < g. Then H™ =Oforq^Q,p and H™ = SP(C). (See [BV1], Proposition 2.1 for the general statement.) Furthermore Hp>q — 0 for p — q < h by the grade sensitivity of the Koszul complex for ip. Claim (b) on H1 for 0 < i < min(2, ft - 1) is easily proved from the long exact (co)homology sequence. For (c) we modify the complex K. to the complex K, by setting (i) Mp
_ £1,0 ^ _g»-l,l a • • • = jgi'i
if i is even, and -...-
if i is odd, 0 < i < h. In fact, there is an exact sequence
and the extreme terms in this sequence are 0 for all j under consideration. Let now i be even, and j = i/2. Since the map APJ -» M J + l j is injective, the same holds true for its restriction # -> B+l in 5 , and so Hp = Ej'j = 0.
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In the case in which i is odd we can go one further step southwest, and obtain the isomorphism Sj(C) = £^+1 =* &>+1'i for j = (i - l)/2. Since Hh ^ 0, we only have an exact sequence
0 _> Eh-i,h _> gh _^ Hh^ but the arguments above can be applied to Eh~1'1; it is zero if h is even, and isomorphic to S (ft ~ 1)/2 (C) if h is odd.D REMARK 2. Proposition 1 shows that roughly the first half of the symmetric powers Sl(C), i < h, can be interpreted as homologies of a Koszul complex. It is also possible to interpret the other half in a similar way. To this end we consider the column complexes C o , . . . , C/j of /C (not of /C) and set Ch+i — Cok(Ch-i —> Ch)Then we obtain an exact sequence 0 -> C0 -> • • • -»• C ft -*• C ft+ i -> 0 of complexes, and the only nonzero (co)homology can occur along Cft+i and at HP(CP) = SP(C) for p > 0. If one applies arguments similar to those in the proof of Proposition 1 (now proceeding in northwestern direction), then one obtains
fs^C)
i f f c + i = 0(2),
for 0 < i < h. Note that the module C^+1 is resolved by 7?.j, and Hi is just a truncated and shifted version of KQ <8> S*(F). The truncations of ~R0 resolve the exterior powers A/M 9 where Mv = Cok(ip). Thus Clh+1 = f\h~lMv ® Sl(F).As in the proof of Proposition 1 let M = Cok^*. Then it is easy to see that Np = (f\pM)* for all p. In fact, ifj" induces a presentation
f\p~lG* ® F* -> f\pG* ->• /\PM -> 0 for all p, and the exact sequence 0 -> iVp -> /\PG -> /\P~1G <S> F is obtained by dualizing. It follows that ./Vr is free of rank 1 (provided grade IM > 2), and Np — 0 for p > r.
COROLLARY 3. LeiV '• G -* F be as above, and assume thatg = gradelM = H-l. T/ien t/ie following conditions are equivalent.
(1) There is a homomorphism ip : R —> G suc/i i/ioi ^y> = 0 ond the ideal Im
f2J (i) r = 1 or (ii) m = 1 and r is odd.
Proof. The implication (2) => (1) is an easy exercise. (See also the considerations at the end of this note.) For the converse observe that Np = 0 for p > r and that Nr is free of rank 1. So Hr must be cyclic. If r were even, then_H r = 0 by Proposition 1, and Imc^* = R. Thus r must be odd. In this case Hr = S (r ~ 1)/2 (C) by Proposition
1 where C = Coki/i*. So if r > 3, then C must be cyclic, which in turn means m = 1.O
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We now return to our original purpose. Therefore let x be a linear form on M = Cokip*. The induced linear form on G* is denoted by x; note that xV'* = 0. Set ip = x*, and, as above, r — n — m. We want to connect the truncated Koszul complex 0 -> I\TM -> • • • A*-^ ~~$ K~1M • • • -> M -> R -> 0
(1)
with the complex 0 -> fl 4 AT1 -> • • • ATp ^ Afp+1 • • •
(2)
considered above. We have already observed that Np = (/\PM)*.
LEMMA 4. With notation from above, let g = grade 1M > r + I. Then there are maps np : f\pM -» Nr~p, p = 0,... ,r, such that d$fj,p = ±^ p _i9^ and which are isomorphisms for p — 0,... ,r — I and injective for p = r. If, in particular, g = r + 1 > 2, then we obtain the following diagram of maps, the columns of which are exact and with commutative or anticommutative rectangles: 0
0
I
Proof. As in [BV2], p. 17, we choose isomorphisms 7 : /\™F* -^ R, 6 : /\nG* -» and define maps = 0, . . . , r, by
a; € A P G*, J/ G A r " p G*, z = T-^1)- Via the natural isomorphism (/\r~pG*)* ^ /\r~pG we regard i/p as a map f\pG* -> A r " PGf - One has Im VP C A rr ~ p , and it is an easy exercise to show that the diagram
Koszul complex in projective dimension one
95
is commutative or anticommutative (see for example [HM], proof of Theorem 3.1). Consequently the same is true for
PP-I dx, \lmvp
where pp and pp-\ are induced by vp and vf-i. Now let np be the composition of pp and the canonical injection Im vp ->• Nr~p. Then the equation asserted in the proposition obviously holds. In case p < grade IM —1, Imz/ p = Nr~p. This proves the remaining statements.D
If we look at the homology of the truncated Koszul complex (1) associated to x instead of the homology of (2), then a somewhat smoother assertion than Proposition 1 is possible.
THEOREM 5. Let M be module with a finite free resolution of length 1, M = Coktf}* where ip : G —> F is as above, and assume that g = grade IM_ = r + 1. Let X be a linear form on M . Then Imx C IM, and for the homology Hp at /\MP of the truncated Koszul complex (1) associated to x the following holds:
if 0 < i < h, i ^ 0 (2) if 0 < i < h, i = 0 (2), where S°(C) =R/IM. Furthermore there is a x with grade Imx — g if and only if (i) r = 1 or (ii)
m = 1 and r is odd. In this case we have necessarily Im x = IM • Proof. Consider the diagram in Lemma 4. Since p,r is injective, Hr must _be zero if h > 0. Next let h > 1. Then we obtain R/_IM = ff r -i, since H° = H1 = 0. Finally if h > 2, then in addition H2 = 0, so Hr-2 = 0 as stated. The remaining assertions concerning the homology of (1) are contained in Proposition 1. Instead of x we can consider the induced linear form x on G* . Corollary 3 yields the statement about the existence of such a linear form x satisfying grade Im x = 9Assume that such a x exists. If m = 1, then Imx = IM by Proposition 2 in [BV3]. If r — 1, then, under our assumptions, M = Cok«/>* is an ideal in R which must be isomorphic to Imx- Using the Hilbert-Burch Theorem, we have Imx = Q!M with an element a € R. Since grade Imx = 2, a must be a unit.D REMARK 6. (a) It is a noteworthy fact that the homology Hr-i of the truncated Koszul complex (1) associated to x is independent of x for i < h. (b) The Koszul complex associated to a linear form x on a free module of rank r is grade sensitive: its homology vanishes for j > r — grade Im x, and does not vanish at r — grade Im x-
In a sense, this is also true for the linear form x considered in Theorem 5: of course, "vanishes" must be replaced by "vanishes if i is even and is isomorphic to S (i ~ 1)/2 (C) if i is odd". Then Theorem 5 covers alH = 0, . . . , h - 1, but
96
Bruns and Vetter the analogy also persists if i = h < g. In fact, let p be a prime ideal of grade h such that Im* C p. The module Mp is free and Sj(Cv) = SJ'(C% = 0 for all j. Therefore one can apply the grade sensitivity of the Koszul complex for a free module, and H^ can be neither 0 nor isomorphic to a non-vanishing symmetric power of C: otherwise its localization would vanish.
(c) That we have truncated the Koszul complex of x is inessential. In fact, f\rM is torsionfree of rank 1, and /\ r+1 M has rank 0. Hence Hom/{(/\r+1M, f\TM) — 0, and the homology of the full and of the truncated Koszul complexes at f\TM coincide. Let ip : G —* F be as above, r = n — m, and g = grade IM = r + 1. In case r > 1, the existence of a linear form x on G* with grade Im^ = g can be described equivalently and independently of the last theorem.
PROPOSITION 7. With the assumptions on t/> and x from Theorem 5, assume in addition that r > 1. Then grade Im x = 9 is possible if and only if there exists a submodule U of M = Cok ?/>* with the following properties: (1) rank[/ = r- 1;
(2) U is reflexive, orientable, and U9 is a free direct summand of Mv for all primes p of R such that grade p < r. Proof. Let x be a linear form on M such that grade Imx = r + 1. Set U = Ker%. Then (1) and the last property in (2) are obvious. Since Imx is torsionfree and M is reflexive, U must be reflexive. Because Im x has grade > 2, it is orientable. M being orientable, the orientability of U follows from Proposition (2.8) in [B]. Conversely let U be a submodule of M which satisfies (1) and (2). Then / = M/U is torsionfree of rank 1 and therefore an ideal in R which is orientable since
U and M are orientable. Consequently grade/ > 2. So for a prime p in R which contains /, the localization IRf cannot be free. On the other hand I = R is impossible since g = r + 1. In view of the last condition in (2), / must have grade r + LD From Theorem 5 we know that the hypothesis of Proposition 7 can only be satisfied with m = 1 and r odd. The submodule U of M in Proposition 7 has projective dimension r — 1. In fact, the ideal IM = Imx is generated by r + 1 elements and has grade r +1. Therefore it is perfect, i. e. projdim R/I = r +1. This
implies projdim U — r -I via the exact sequence 0 - » [ / - ^ M - » / - » 0 . Dualizing this exact sequence, we obtain an exact sequence 0 -* R -t M* -> U* -> 0. Since M* = f\r~lM has projective dimension r — 1, it follows that U* has projective dimension r - I . The dualization argument just used amounts to interchanging the roles of ip* and x> so that U* plays the same role for x* and if> as U for i/>* and x- If we choose X in a special way, then we can even achieve that U and U* are isomorphic in a skewsymmetric way: for the isomorphism a : U —>• U* one has a* = —a upon the identification of U and U** via the natural isomorphism. As we mentioned at the beginning of the proof of Corollary 3, it is easy to see that
there is a linear form x on G* such that ip* (1) £ Ker x and grade Imx = r + l = n:
Koszul complex in projective dimension one
97
fix a basis zi, . . . ,zn of G* and let if>*(l) = SILi xizi'i the map \ : X^ILi a»z* ^ X)™-! (— l)* a t^n+i-» is an appropriate linear form. Let x be the induced form on M = Cok •;/>*. The submodule U = Ker% has the properties (1) and (2) of Proposition 7 (see the first part of the proof). Consider the commutative diagram 0 ———> Kerx
with exact rows. The isomorphism p is defined by p(zi) = (-l)*z*+1_i where z*,... , z* is the basis of G dual to zi,... ,z n > and pi is induced by p. Since Pi(t/>*(l)) = —x, we obtain a second commutative diagram Kerx ———> U PI
——> 0
PI
with exact rows. (V>*)i is induced by ip* and the first vertical arrow maps 1 to —1. As pi is an isomorphism, the same is true for pi, so U turns out to be self-dual. Moreover p\ is skew-symmetric, i.e. (jp\)* = —pi since the same is true for p. Suppose that R = K\X\,..., Xn] is the polynomial ring in n indeterminates over a field K. If we define tp : Rn —> R by ip(ei) = X^ then the module U is associated with a rank n — 2 vector bundle on Wn~l(K). Such bundles exist however also for odd n; see [V]. The module V associated with the the construction in [V] is self-dual only for n = 4. REFERENCES 1. W. Bruns. The Buchsbaum-Eisenbud structure theorems and alternating syzygies. Commun. Algebra 15 (1987), 873-925.
2. W. Bruns and U. Vetter. Length formulas for the Local Cohomology of Exterior Powers. Math Z. 191 (1986), 145-158. 3. W. Bruns, U. Vetter. Determinantal rings. Lect. Notes Math. 1327, Springer 1988.
4. W. Bruns and U. Vetter. A Remark on Koszul Complexes. Beitr. Algebra Geom. 39 (1998), 249-254.
5. J. Herzog and A. Martsinkovsky. Glueing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with isolated singularities. Comment. Math. Helv. 68 (1993), 365-384 6. J.C. Migliore, U. Nagel, and C. Peterson. Buchsbaum-Rim sheaves and their multiple sections. J. Algebra 219 (1999), 378-420.
7. U. Vetter. Zu einem Satz van G. Trautmann iiber den Rang gewisser kohdrenter analytischer Garben. Arch. Math. 24 (1973), 158-161.
Grobner bases as Characteristic Sets
GIUSEPPA CARRA FERRO Department of Mathematics and Computer Science, University of Catania, Catania, Italy
1 INTRODUCTION The notion of characteristic set was introduced by Ritt [1] in order to study differential polynomials and in particular usual polynomials. This notion is used now in automatic deduction in geometry and it is alternative to the notion of Grobner basis in [2] and [3]. Such notion is based on the concepts of class, ranking, pseudodivision, pseudoremainder and pseudoreduction in the ring of differential polynomials. A characteristic set of a set of differential polynomials is nothing else than a finite set of differential polynomials, such that every differential polynomial is pseudoreduced with respect to the other ones and minimal with respect to a pre-order depending on the ranking and denned on finite subsets of differential polynomials. If a set of polynomials is considered, then a characteristic set of a prime ideal is a regular sequence. When linear partial differential polynomials are considered, then a ranking is nothing else than a term ordering on a free module of finite rank over a polynomial ring and the notions of pseudodivision, pseudoremainder and pseudoreduction almost coincide with the notions of division, remainder and reduction in the Grobner basis theory. These special properties were recently used by Levin [4] in order to define Hilbert polynomials in two variables of algebraic and differential ideals. In this paper it is shown that if the notions of term ordering, division, remainder and reduction are used as in Grobner basis theory and the notions of autoreduced set and ranking pre-order on such sets are used as in Ritt theory, then the notion of characteristic set of an ideal is equal to the notion of reduced Grobner basis. Another approach suggests such proof in [5]. Furthermore the Wu-Ritt-Kolchin algorithm for
99
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Carra Ferro
the construction of an extended characteristic set of a differential polynomial ideal becomes a reduced Grobner basis algorithm. The ranking pre order induces a pre order on the set of all polynomial ideals in K[Xi,...,Xn] and then an equivalence relation. Two ideals are equivalent if they have the same monomial ideal associated to the corresponding reduced Grobner basis. Finally the ranking pre-order induces a linear pre-order on the set of all monomial ideals and it induces a linear pre-order on the Hilbert scheme of all homogeneous ideals in K[Xi,..., Xn] with the same Hilbert polynomial.
2 GROBNER BASES IN POLYNOMIAL RINGS Let NO—{0,1,2,..., n,...}. It is well known that (N$, +) is a commutative monoid. Let X\,..., Xn be n variables. DEFINITION 1 PP(Xi,..., Xn)={ X? • • • X^: (01,..., an) € Ng } is the set of power products in the variables X\,..., Xn.
(PP(X\,...,Xn),.) is a monoid and it is isomorphic to (NQ, +). Let K be a field of characteristic zero. Let A = K[Xi,...,Xn] TA = PP(Xi,..., Xn] be the set of terms of A.
and let
DEFINITION 2 A term ordering a on TA is a total order such that:
(i). 1
Let K* be equal to K \ {0}. Let A* be equal to A \ {0} and let / e A*, f — Y
DEFINITION 3 Let a be a term ordering on TA and let / be an ideal in A.
MM) = (MM) • f e I) and Tff(I) = (Ta(f) : / € / ) . DEFINITION 4 [9]. Let / = (/i,... ,/ r ) be an ideal in A and let a be a term ordering on TA- F={fi,..., fr} is a Grobner ( or standard) basis of / with respect to a iff
Grobner bases as Characteristic Sets
either Mff(I)
101
= (Mff (A, . . . , M ff (/ r )) or Tff(I) = (Tff(fi, • • • , Wr))-
By [10] every ideal / in A has a Grobner basis. DEFINITION 5 Let f,g € A* and let a be a term ordering on TA. / is reduced with respect to g iff no monomial in / is a multiple of Mff(g). More
generally if G={g\, . . . ,gr} C A* f is reduced with respect to G iff it is reduced with respect to every &, i=l,. . . ,r. REMARK 1 Let /, g\,. . . ,gr 6 A*. If / is not reduced with respect to G={gi,. . . ,<7 r }, then there exist gj.,. . . ,qr,r € A such that /=£)i=i,...,r 9»S*+ r and r is reduced with respect to G. r is obtained by the usual reduction procedure in a finite number of steps, because a is a well ordering. In this case / reduces to r with respect to G. DEFINITION 6 Let f , g € A* and let a be a term ordering on TA.
),
]9
is the S — polynomial of / and g and it will be denoted by S ( f , g ) . The following propositions are well known in the literature.
PROPOSITION 1 [10]. F={/l5 . . . , fr} is a Grobner basis of the ideal / in A with respect to the term ordering a iff S(fi,
fj) reduces to zero with respect to F for alii, j — 1, . . . , r.
PROPOSITION 2 [10], [6]. Let / be a nonzero ideal in A and let a be a term ordering on TA- There exists a set F={fi, . . . , fr} C / \ {0} such that:
(i)- fi —ai^i + ^i> with ti £ TA and GJ G -^* f°r alii = 1, . . . , r; (ii).
{ < i , . . . , ir} minimally generates TO-(/);
(iii). ajtj = Mff(fi)
(iv).
for alii = 1, . . . ,r;
Supp(Ri) n T ff (/)=0 for all * = 1, . . . ,r.
F is a Grobner basis of I with respect to a and it is called a reduced Grobner basis of I with respect to a. If #j = a^1 fi for alH = 1, . . . , r, the G— {
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(i). F={fi,..., fr} is a Grobner basis of / with respect to a. (ii). A polynomial / € 7 iff / reduces to 0 with respect to F.
3 AUTOREDUCED SETS
Now the notion of autoreduced set is introduced in order to prove the equivalence between the notions of reduced Grobner basis and characteristic set. DEFINITION 7 Let G={gi,...,gr} C A* and let a be a term ordering on TA- G is autoreduced iff each
PROPOSITION 4 Let a be a term ordering on TA- Every autoreduced set of polynomials is finite. Proof: Let G be an autoreduced set of polynomials in A and let Ta={T(7(g): g 6 G}. By Dickson lemma there exists a finite subset F of G, such that every Ta(g) is a multiple of Ta(f) for some / € F. If F ^ G, then we have a contradiction by definition of autoreduced set. In fact each g € G is reduced with respect to every g' 6 G with g ^ g', i.e. Ta(g) is not a multiple of Ta(sf) for all g1 ± g. A proof of Proposition 4 with different tools is in [5], prop.4.1.36, p.201.
Now a pre-order relation, i.e. a reflexive and transitive relation is introduced on the set of all polynomials in A. This relation can be found in [1] and [11], where only variables are used. It also can be found in [12], [4] and [5], where power products in the variables are used.
DEFINITION 8 Let a be a term ordering on TA and let /, g € A. / h a s lower rank than g if Ta(f)
Only autoreduced sets G={gi,... ,gr} arranged in order of increasing rank will be considered, i.e. autoreduced sets G={gi,... ,gr} with rank(g\) -
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103
REMARK 2 Let f,gi,...,gr € A* and suppose that G= {gi,-.-,gr} is a chain with respect to a. If / is not reduced with respect to G, then by Remark 1 there exist q\,..., g r , r € A such that / = ]Ci=i r 9z9* + r an(^ T is reduced with respect to G. If r is obtained by reducing / in decreasing
order with respect to gr,---,9i, then the polynomials
A similar procedure can be found in [13], p.331 and p.368. DEFINITION 9 Let a be a term ordering on TA. Let JF={/i,... ,/r} and G={#i,..., gs} be autoreduced sets in A. F has lower rank than G and write F - s and rank(fi)=rank(gi) for alH = 1,..., s. If r = s and rank(fi)=rank(gi) for all i = 1,..., s, then F and G have the same rank.
REMARK 3 The ranking relation on the set of all autoreduced subsets of A defined as above is a pre-order relation by its own definition and is called again a ranking pre-order. It is easy also to show that it is linear, i.e. for each pair of autoreduced subsets F and G of A either F ^ G or G -
PROPOSITION 5 Let a be a term ordering on TA- Every nonempty set of autoreduced subsets of the ring of polynomials A has an element of lower rank.
Proof: Let S be a nonempty set of autoreduced subsets of A. By repeating the Ritt's proof in the case of autoreduced subsets associated to a different notion of reduction and rank, there exists a finite decreasing sequence of nonempty subsets of S. Let So=S. Let Si={ F £ S: F ^ 0 and the first
element of F is of lowest possible rank }. Let S2—{F€Si\ F has at least two elements and the second element of F is of lowest possible rank }. More
generally let Sh={ F 6 Sh-i- F has at least h elements and the h-th element of F is of lowest possible rank }. It follows that every autoreduced set in Sh has the first h elements of possible lower rank. If Sh+i is empty, then each autoreduced set F in Sh has minimal rank in S by definition of rank. Suppose that Sh is nonempty for all h G NO- Let G={ fh- fh is the h-th element of an autoreduced set F^ e Sh, h S NO }. G is an autoreduced set in A and then it must be finite by Proposition 4. So Sj must be empty for all j > h + 1 and some h (E NQ.
REMARK 4 The proof of Proposition 5 shows also that if F and G are two autoreduced subsets of A of lower rank in a set 5, then they have the same rank.
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DEFINITION 10 Let a be a term ordering on TA and let / be an ideal in A. An autoreduced subset of / of lowest rank is a characteristic set of /. PROPOSITION 6 Let / be a nonzero ideal in A and let a be a term ordering on TA- The following statements are equivalent:
(i). G={gi,... ,gr} is a characteristic set of /. (ii). An element / of / is reduced with respect to G iff /=0. Proof: Suppose that G is a characteristic set of / and there exists a nonzero element / in I, that is reduced with respect to G. If rank(f) -
PROPOSITION 7 Let / be an ideal of A. G={gi,..., gr} is a characteristic set of / iff G is a reduced Grobner basis of / with respect to a. Proof: Suppose that G is a characteristic set of /. By Proposition 6 / € / iff it reduces to zero with respect to G. By Proposition 3 G is a Grobner basis of I with respect to a. Since G is autoreduced, then it is a reduced Grobner basis of / by Proposition 2. Conversely suppose that G is a reduced Grobner basis of / with respect to a. It follows that G is a autoreduced set by its own definition. So G is a characteristic set of / by Propositions 3 and 6. The above proposition shows once again the existence of a reduced Grobner basis of an ideal / in A with respect to a term ordering a. A similar proposition is suggested in [5], pp.201-202 and in [12], p.217, where a different linear pre-order on the set of all monic Grobner bases of / is used. REMARK 5 By Proposition 1 the well known Buchberger algorithm for the computation of a Grobner basis and its optimizations can be used for the
computation of a characteristic set of an ideal I — ( f i , . . . , f T ) in A. Conversely the Wu-Ritt-Kolchin algorithm for the computation of an extended characteristic set as in [14], pp.178-179 can be used for the computation of a reduced Grobner basis of /.
Grobner bases as Characteristic Sets
105
3.1 Algorithm of Wu-Ritt-Kolchin
Input / = (F), F = {/i,..., fr} ideal in A. Output G = { < ? i , . . . , <7S} a characteristic set of /. G:=0; B : = 0
loop F := F U fl U S; F7 := F; .R := 0; S := 0; while F ^ 0 loop Choose a polynomial / € F' of minimal ranking F':=F' \ {#: 3 is not reduced with respect to /} G:=GU/; end {loop}; for all / e F \ G loop if r:=remainder of / with respect to G and r ^ 0 then jR:=.RUr;
end {if}; end {loop}; for every nonempty subset H C G loop Compute S"=the S-polynomials of H; for every s' € 5' loop if s":=remainder of s' with respect to G and s" 7^ 0 then 5 := S U s" end {loop}; end {loop}; until R = 0; until 5 = 0; return G; end The termination of the algorithm follows from Proposition 5. The correctness of the algorithm follows from Proposition 1 and from the following fact: at each step G C F and then (G) C (F) = I. In the final step each / € F \ G reduces to zero with respect to G and then I=(F) C (G).
4 RANKINGS AND MONOMIAL IDEALS Let K be a field of characteristic zero. Let A = K[Xi,..., Xn] and let a be a term ordering on TAThe ranking pre-order -
106
Ta(f]
Carra Ferro
= Tff(g) and write jRag.
Ra is an equivalence relation on the set A by [12], Lemma 4.24, pp.153-154. There is a biijective map between the quozient set A/Ra and TA, because each polynomial / € A has the same rank as TCT(/). Moreover the ranking pre-order on A induces the total order a on TA by its own definition and by [12], Lemma 4.24, pp.153-154.
Let TT be the canonical map from A onto A/Ra=Tj\- If G={gi,... ,
Proof: Since the ranking pre-order has the reflexive and transitive properties, then it is sufficient to show the antisymmetric property, in order to prove that it is a partial order. I f T a ( G ) ^ Ta(F) and Ta(G) -& «fc, then respectively either Ta(F] XCT Ta(G) or Ta(G] -
DEFINITION 12 Let a be a term ordering on TA and let / and J be ideals in A. I has lower rank than J if the reduced Grobner basis G of I with respect to a has rank lower than the reduced Grobner basis F of J with respect to
Grobner bases as Characteristic Sets
107
REMARK 7 Let cr, 7, J, G and F be as in Definition 12. 7 has the same rank as J if Tff(I)=TlT(G)=Ttr(F)=Ttr(J). In other words the ideals 7 and J have the same rank iff the corresponding monomial ideals TO-(7) and Tff( J) are equal. Since the rankig pre-order on the set of all autoreduced subsets of A is linear, then the ranking pre-order on the set I(A) of all ideals of A is linear, i.e. for each pair of ideals 7 and J in A either 7 -
PROPOSITION 9 Let a be a term ordering on TA. Let F = {/i,..., /r} and let G ~ {g\,..., gs} be respectively reduced Grobner bases of (F) and (G). Let Ta(F}= (T f f (/i),... ,T ff (/ r )} and let Tff(G)= (Ta(9l),.. .,Ta(ga)}. Then:
(G) c (F) =* F -<„ G *=> (F) ^a (G) <=» (T ff (F)) -<„ (Tff(G)). Proof: Let T f f (F)={
If (G) C (F),
then by reduced Grobner bases theory there exists terms mi G TA, such that
Vj = m,jt^ for all j = 1,..., s and some i(j') = 1,..., r. If either z(l) > 1 or mi ^ 1, then t\ - 2 or m^ ^ 1, then £2 ^o- U2 and F -<,„ G. If i(2) = 2 and 7712 = 1, then let ^3 = mst^3\. Let r > s. By repeating the proof as above, if there exists j = 1,..., s such that Vh = m/jt^/jj for all h =
108
Carra Ferro
1 , . . . , j — 1 and either i(j) > j or rrij ^ 1, then tj -<„ Vj and F -
It is easy to find examples of pairs of monomial ideals / and J, such that I -(a J and J g /.
EXAMPLE 1 Let A = K[Xi,X2,X3}. Let a=tdeglex with Xi
If we fix the Hilbert polynomial HP(t), let's consider all monomial ideals I, such that HP(t}=HP(I,t). The Hilbert scheme -H/fp(/,i), n _i as the set of all homogeneous ideals J in A with the same Hilbert polynomial HP(J, t)=HP(I, t) is pre-ordered by the ranking pre-order on the sets of all ideals in A. Such relation defines the equivalence relation Ria as above on the Hilbert scheme #f/p(/,t) !n -i by using the ranking pre-order. It follows that the corresponding quozient set Hfjp(i^),n-i/R^ can be iden-
tified either with a set of autoreduced sets of power products or with a set of monomial ideals. By Proposition 8 such set is totally ordered and well ordered by the ranking -
Grobner bases as Characteristic Sets
109
theory for ^4-submodules of free A-modules of finite rank and the characteristic set theory of Ritt in [1] for systems of linear partial differential equations with constant coefficients.
REFERENCES
1. J F Ritt. Differential Algebra. New York: AMS Coll. Publ. 33, 1950 2. W-T Wu. Basic principles of Mechanical Theorem Proving in Geometries. Journal of Syst. Sci. and Math. Sci. 4(3): 207-235, 1984, Also in Journal of Automated Reasoning, 2(4): 221-252, 1986
3. W-T Wu . Some Recent Advances in Mechanical Theorem-Proving of Geometries. Automated Theorem Proving: After 25 Years, AMS Contemporary Mathematics 29: 235-242, 1984 4. A Levin. A. Computation of Hilbert Polynomials in two variables. J. of Symb. Comp. 28: pp.681-710, 1999
5. M V Kondrateva, A B Levin, A V Mikhalev, E V Pankratiev. Differential and Difference Dimension Polynomials. Dordrecht: Kluwer Academic Publ. MIA 461, 1999 6. D Bayer. The division algorithm and the Hilbert scheme. Ph. D. Thesis. Harvard University, 1982 7. L Robbiano. Term orderings on the polynomial rings. Lect. Not. in Comp. Sci. 204: 513-517, 1985
8. T Mora, L Robbiano. The Grobner Fan of an ideal. J. of Symb. Comp.: 49-74, 1989 9. B Buchberger. Some properties of Grobner bases for polynomial ideals. ACM SIGSAM BULL. 10: 19-24, 1976 10. B Buchberger. A theoretical basis for the reduction of polynomials to canonical form. ACM SIGSAM BULL. 10 (3): 19-29, 1976
11. E Kolchin. Differential Algebra and Algebraic Groups. New York: Academic Press, 1973 12. T Becker, V Weispfenning. Grobner Bases A Computational Approach to Commutative Algebra. New York: Springer-Verlag GTM 141, 1993
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13. D Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag GTM 150, 1995 14. B Mishra. Algorithmic Algebra. New York: Springer Verlag, 1993 15. R Hartshorne. Algebraic Geometry. New York: Springer-Verlag GTM 52, 1977
Threefolds with degenerate secant variety: on a theorem of G. Scorza L. CHIANTINI Dipartimento di Matematica, Universita di Siena, Via del Capitano, 15, 53100 Siena, Italia. [email protected] C. CILIBERTO Dipartimento di Matematica, Universita di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italia. [email protected]
1
INTRODUCTION
In this paper we work over the complex field C.
Let X C Pr be a reduced, irreducible (not necessarily smooth) projective variety of dimension n. Let H be a hyperplane section on X and let ~H C \H\ be the (possibly incomplete) linear system cut out on X by the hyperplanes of Pr . We will assume that X is non-degenerate, i.e. that Ji has dimension r, namely that X spans the whole of P r . Then, by abusing notation, we may, and sometimes will, denote by H both the divisor and the unique hyperplane TTH which cuts it on X. We will also abuse notation by denoting with the same letter a complete linear system of Cartier divisors on a variety and the corresponding line bundle. Let now k be a non-negative integer and let Sk(X) be the k-secant variety of X, i.e. the Zariski closure of the set: {P G Pr : P lies in the span of k + 1 independent points of X} Of course S°(X) = X, Sr(X) = Pr and Sk(X) is empty if k > r + 1. We will also set S (X) — ptyset if k < 0. Notice that, for k > —1, one has: k
:= dim(S*(X)) < min{r,n(k + 1) + k} h
[1.1]
The right hand side of [1.1] is called the expected dimension of S (X) and will be denoted by a^ := a^(X). We will write s^ instead of s^(X) if there is no danger of confusion. Moreover, if k = 1, we will use the shorter notation S(X) instead of S^(X) and similarly for s and a. One says that X is k- defective when strict inequality holds in [1.1]. Furthermore one defines the k- defect of X as the number: 111
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We will use the shorter notation 6k for 6k(X) and we will even drop the subscript k in the case k — 1 if there is no danger of confusion. Of course X is fc-defective if and only if 6k > 0. A variety is said to have a defect, or to be defective, if it has a non-zero fe-defect for some k. The first non trivial example of a defective variety is the Veronese surface in P5, which is 1-defective of defect 1 and the first general result on defective varieties is Severi's classification [S] of 1-defective surfaces which are only cones and the Veronese surface in P5. This old result stimulated the research on the problem of classifying all defective varieties. A problem which is still open in its generality. The theory of defective varieties has been developed by several authors, both classical and modern (see e.g. [A], pro], [Dl], [F], [FR], [PI], [P2], [Scol], [Sco2], [Sco3], [T3], [Z], etc.). A basic result in this theory is that there are no defective varieties of dimension 1. By contrast, as we saw, there are defective surfaces and Palatini in [PI] was the first one who tried to classify them, but his classification contained some serious gaps. A complete classification of defective surfaces, which fills up Palatini's gaps, is contained in Terracini's paper [T3], and has been also worked out in modern times by Dale [Dl], who apparently was unaware of Terracini's paper. For a more recent reference, see [CC2]. As for higher dimensional defective varieties, we are rather far away from a full classification. In Zak's book [Z] one finds several general properties. In particular a smooth, irreducible, non-degenerate, 1-defective variety X c PN of dimension n is such that N + l < (™J 2 ) and if the equality holds then X — V^,n is the Veronese variety of quadrics of Pn, in which case the defect is 1 (see [Z], thm 2.1, pg. 126). Zak has similar, equally beautiful,
theorems also for 1-defective varieties with higher defect (see [Z], chapt. VI). More specifically, one can hope for a complete classification of defective threefolds. This is a subject classically studied by Scorza [Scol], who stated the classification theorem for all 1defective 3-folds (see theorem 1.1 below). He also studied defective 4-folds in [Sco3]. In more recent times the subject has been reconsidered by various authors, e.g. Fujita-Roberts [FR], Fujita [F] and Zak [Z], who essentially looked at the case of smooth threefolds. In the present paper we consider again the case of threefold, but without any smoothness assumption, as Scorza did in [Scol]. We rework here his classification of 1-defective threefolds providing a new proof which substantially simplifies Scorza's involved and sometimes obscure arguments. The main result is the following: THEOREM 1.1 (G. Scorza, [Scol]) An irreducible, non-degenerate, protective 3-fold X C PT is l-defective if and only if r > 6 and X is of one of the following types: (i) X is a cone; (ii) X sits in a 4-dimensional cone over a curve; (Hi) r = 7 and X is contained in a 4-dimensional cone over the Veronese surface Vi 2 in
P5;
(iv) X is the Veronese variety Vz 3 C P9 of quadrics in P3 or a projection of it in Pr, r = 7,S; (v) r — 7 and X is a hyperplane section of the Segre embedding YofP^x P2 in P8.
In section 2 we will discuss the various cases appearing in the statement of theorem 1.1 above, and we will see that they are actually 1-defective and we will compute their defects. In section 3 we will prove more specific statements for 1-defective varieties with a large
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defect. In section 4 we shall give the proof of Scorza's theorem. Let us now talk about the techniques. As well known, the main tool for studying defective varieties is a famous lemma of Terracini's (see[Tl], or, for modern versions, [A], [Dl], [Z]) to the effect that, given a general point P 6 Sk(X), lying in the subspace < PI,..., Pk+i > spanned by PI, . . . , Pk+i general points on X, then the tangent space TSk(x),p *° Sk(X) at P is the span TX,^,...^^ of the tangent spaces TX,P! ,..., Tx,pk+l • Therefore X is fc-defective if and only if:
[1.2]
Let now Z be a subscheme of a variety X and let £ be a linear system of Cartier divisors on X. We will denote by £(— Z) the linear system of all divisors in £ containing Z. If Z is a union of points PI,..., Pm, we write £(-Pi ••• - Pm) for C(-Z). If PI, . . . , Pm are smooth points of X, we let C(-2Pi • • •- 2Pm) be the linear system of all divisors in C(—P\ •••- Pm) having singular points at P\ , . . . , Pm. In case £ = H is, as above, the linear system of hyperplane divisors of X in P r , then T-L(— PI • • • — Pm) is cut out on X by the hyperplanes in Pr containing the subspace < PI, . . . , Pm >, whereas %(-2Pj • • • — 2Pm) is cut out on X by the hyperplanes in Pr containing the subspace Tx,p1,...,pm. These hyperplanes are called m-tangent hyperplanes to X. Hence [1.2] is equivalent to say that X is fc-defective if and only if, for PI, ...,Pk+i general points on X, the system H(—2Pi • • • - %Pk+i) is not empty and: r - S W - 1 = dim(W(-2Pi • • • - 2P*+i)) > min{-l,r - (jfe + l)(n + 1)} In his paper [T3], in which there is the classification of defective surfaces, Terracini implicitely uses a crucial result which can be seen as a natural complement of the aforementioned Terracini's lemma from [Tl]. We are now going to recall it and we refer for its proof to [CC2]. Notice that it could be partly deduced from the results of chapt. V of [Z], to which we refer for the general theory of defective varieties. If X is any variety like above, if PI , . . . , Pjt+i € X are general points and H € H(—"2Pi — • • • — 2Pfc + i) is a general hyperplane section tangent at PI, . . . , Pfc+i, we can consider the contact variety of H, i.e. the union S := 'Splt,,,>pk+l(H) of the irreducible components of Sing(H) containing PI , . . . , Pfc+i . Since P\ , . . . , Pk+i are general points, an obvious monodromy argument shows that S is equidimensional, and we denote by i/k :— i/k(X) its dimension, which we will call the k- singular defect of X. Of course fk < n — 1 and we set i/k = -1 if H(-1P\ - - - - - 2Pk+i) is empty. We will simply denote v\ by v if there is no danger of confusion. We also set /i(S) =r - dim(H(-£)). This is the number of conditions imposed by E to the divisors of ~H containing it. With all this in mind, we can state Terracini's theorem:
THEOREM 1.2 Let X C Pr be a reduced, irreducible, non-degenerate, protective variety of dimension n. If PI, . . . , P/b+i € X are general points, H € H(— 2Pi — • • • — 2P& + i) is a
general (k + I) -tangent hyperplane and S is its contact variety of dimension i>k > 0, then:
k + 1 < ft(S) < (jb + 1)(1 + i/fc) - 6k and therefore: (k + l)vk > Skin particular if X is k-defective then n — 1 > i/*, > 0, i.e. the general (k + l)-tangent hyperplane to X has a contact variety of positive dimension.
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This theorem explains the difficulty in finding varieties with large defect: as soon as 6k increases, the variety S imposes less and less conditions to the hyperplanes, so that it becomes more and more degenerate. Thus X is swept out by degenerate varieties and, as indicated in [M], this restricts pretty much the possibilities for X. Theorem 1.2 is, in one way or another, our basic tool in this paper, especialy in section 4, where we use it to say that for a 1-defective threefold with S = v = 1 the general contact variety is an irreducible conic. In addition, it turns out that if X is a fc-defective variety
and if PI, . . . , Pfc € X are general points, the projection of X from Tx,p±,...,pk has positive dimensional fibres contained in the contact varieties S. The study of these general tangential projections is crucial in our analysis. Notice finally that the converse to the final part of the statement of 1.2 is false in general: if X is a cone surface, by imposing tangency at a point one gets tangency along a line, but on the other hand no variety is 0-defective. So one is lead, as in [CC2], to the definition of a weakly defective variety:
DEFINITION 1.3 X is a k-weakly defective variety if the general (k + 1)-tangent hyperplane to X has a contact variety of positive dimension. According to Terracini's theorem 1.2, this is a weaker concept than the one of a kdefective variety. It turns out that the classification of weakly defective varieties of dimension smaller than n matters in the classification of defective varieties of dimension n. A full classification of weakly defective surfaces, which includes Terracini's and Dale's classification of defective surfaces, has been recently obtained by Chiantini-Ciliberto [CC2], and we take advantage from these results here.
In conclusion, we notice that, using general tangential projections, one can see that the classification of fc-defective varieties of dimension n matters in the classification of (fc + 1)defective varieties of the same dimension (see proposition 3.6 from [CC2]). Therefore the results from [CC2] and theorem 1.1 above gives us some hope to inductively come to the complete classification of defective threefolds. We have work in progress on this subject and we hope to come back to this in the future.
2
EXAMPLES OF DEFECTIVE THREEFOLDS
In this section we collect examples of defective threefolds. According to the statement of Scorza's theorem 1.1, these, as we will prove, are the only defective threefolds. EXAMPLE 2.1 Every cone X C Pr with vertex V of dimension s over a variety Y C pr-s-i jg i-defective as soon as r > In + 1 - s. Indeed two general tangent spaces to X intersect along the vertex V, thus the defect is S(X) > min{s + l,r + s - In}. More specifically, if Y is also 1-defective, then X is 1-defective as soon a s r > 2 n + l - s - 6(Y) and 6(X) > min{r + s + 6(Y) -2n,s+l + S(Y)}. Notice also that v(X) >s + v(Y) + 1. In particular, if V has dimension n — 2, i.e. if X is a cone over a curve, then the defect is 5(X) = n - l i f r > 2 n + l and is 6(X) = r-n- 2 other wise. Moreover one has v(X) =n-l. One has 6(X) = 1, v(X) = n - 1 if X is a cone over the Veronese surface. Proposition 3.1 and theorem 3.5 below will essentially tell us that the above examples are the only ones for which the defect 6 is maximized.
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EXAMPLE 2.2 Recall from [CC2], example 4.3, that any non-degenerate variety X C Pr of dimension n contained in a (s + m + l)-dimensional cone W over a reduced, irreducible variety Y of dimension m < n, with vertex a linear space V of dimension s is fc-defective as soon as s + 1 < (k + l)(n - m) and r > (m + l)(k + 1) + s + 1. In this case, if in addition r > (k + l)(n + 1) - 1, then Sk(X) > (n - m)(k + 1) - s - 1 and vk(X) >n-m. Thus, for k = 1, we may expect S(X) = n — 1 only if s = n - 2m. Since we must have s + m + 1 > n, the only possibility is m = 1, s — n - 2, i.e. X is a cone over a curve as in example 2.1. Similarly, for k = 1, we may expect 6(X) = n-2, only if m < 2. If m = 2 then X itself has to be a cone over a surface Y, and we have S(X) = v(X] = n - 2 if the surface is not 1-defective. Otherwise, if m = 1, X may lie in a (n + l)-dimenional cone over a curve, in which case one has 6(X) —n-1 but v(X) —n - I. In any event, for n = 3, k = 1, we get only two possibilities for X: (i) X is a cone (see example 2.1);
(ii) X sits in a 4-dimensional cone over a curve in Pr, r > 7.
EXAMPLE 2.3 Let Vd,r be the d-tuple embedding of Pr in P^--, where Nd,r + 1= (d+r). Since quadrics in Pr singular at k + 1 < r points are singular along the P* spanned by them, one has that Vt,r is &-weakly defective. Actually it is fc-defective and the fc-defect is easily computed to be <5& = CJ 1 )In particular V2,n C P" "=+ is a 1-defective n-fold with defect 8 — 1 and also v — 1. Notice that a projection of Vz,n in Pr, with In + I < r < "("2+3? is still 1-defective, in general with the same defects, but a general projection of it in P r , r < 2n, is no longer defective, inasmuch as the secant variety fills up now the entire space.
EXAMPLE 2.4 Let n > 3 and let W be a (n+l)-dimensional cone over the Veronese surface V2,2 in P5, i.e. W C P n+4 is the cone over Va,2 with a vertex V of dimension n — 2 skew with the P5 spanned by Via- Let X be an irreducible, non degenerate variety of dimension n contained in W. We claim that X is 1-defective, with 8(X) — 1 and v(X) = n — 1. Indeed, let P, Q be two general points on X and P',Q' the corresponding projected points on V2,2- The span of TX,P and TX,Q coincides with the span of V,Ty2>2,p' and Tv2 2 ,Q', whose dimension is n + 3. This proves that S(X) = 1. Notice now that TX,P is a hyperplane inside the span fi of V and of Ty2 2 i p<, so it meets Ty2 2) p' along a line and V along a P™~ 3 . Consider the projection of X from Tx,p to a general subspace II of dimension 3 which we may assume to be contained in the P5 spanned by V2,2- The image of V under this projection is the point V = fi n II. The image of V2,2 under the same projection is a quadric cone Z whose vertex is Ty2 2,p' n II, which of course coincides with V. Therefore the image X\ under the aforementioned projection is Z. Now Z is developable, namely a general tangent plane to Xi is tangent along a line. Therefore the general bitangent hyperplane to X is tangent along a divisor, proving that v(X) — n — l. In theorem 3.10 below we will come back on 1-defective varieties of dimension n with 6(X) — n - 2 and v(X) = n - 1 and we will prove that they are only as in the present or in the previous example. EXAMPLE 2.5 The Segre embedding Y of P2 x P2 in P8 is a Severi variety (see [Z]), i.e. it is 1-defective. Actually its 1-secant variety is a hypersurface, namely 6(X) = 1. Furthermore one knows that i/(Y) — 2: a general bitangent hyperplane to Y is tangent along a 2-dimensional quadric (see [Z]). By lemma 3.6 we see that a hyperplane section X C P7 of Y is still 1-defective with 6(X) = v(X] = 1.
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It is interesting to remark that if we project V2,s C P9 from a general secant line to P7 we have a particular hyperplane section of Y. Indeed, the Segre embedding Y of P2 x P2 in P8 can be also interpreted as the image of P4 via the linear system of all quadrics containing two skew lines a, b. By the way, if P, Q are general points of P 4 , a bitangent hyperplane at the image points on X corresponds to the unique quadric in P4 singular along the line L
joining P and Q and also containing a and b. The quadric in question is the union of the two hyperplanes spanned by L and a, resp. by L and b. Hence the quadric is singular along the plane where the two aforementioned hyperplanes intersect, which is the unique plane containing P, Q and meeting a, resp. b, at a point. The plane in question is mapped to a quadric on X along which the bitangent hyperplane touches X. Now, in the rational map / : P4 —> X C P8, the span II of a and b is contracted to a quadric isomorphic to a x 6: this is because a line meeting both a and b is contracted by /. Then a rank 2 quadric consisting of II plus another, general hyperplane II', is mapped to the image of II' via /. On the other hand the restriction of / to II' is nothing but the image of P3 in P7 via the linear system of all quadrics containing two general points, i.e. the points in which a, b intersect II'. This proves our assertion. As a side remark, we notice that the particularity of the hyperplane sections of Y which come from V^,3 by projection from a secant line can be seen to be the following: they have to contain a quadric surface. Remark that if a hyperplane section of Y contains a quadric surface, then in general it contains a 1-dimensional family of such surfaces. An easy count of parameters shows then that such hyperplane sections of X depend on 7-parameters, i.e. they have codimension 1 in the full family of hyperplane sections of X. We leave these
details, which, by the way, we will not need to use later, to the reader.
3
VARIETIES WITH LARGE DEFECT
In this section we consider 1-defective varieties with large defect S := 6^. Our main tool, as in [CC2], §3, is the concept of a tangential projection, which we already used in example 2.4 and which we recall now. Let X C Pr be, as usual, a reduced, irreducible, non-degenerate variety of dimension n and let P be a general point of X. Consider the projection T := TX,P of Pr with centre TX,P- By abusing notation, we will often denote by T := TX,P also the restriction of this projection to X. We will call such a projection a general tangential projection of X. We will set Xi := r(X).
PROPOSITION 3.1 Let X C Pr be a reduced, irreducible, non - degenerate variety of dimension n and let P, Q be general points of X. The tangent space TX,Q to X at Q maps via r := TX,P to the tangent space to Xi at r(Q). Hence one has S(X) = min{n,r — n 1} - dim^i) so that if X is defective then 0 < dim(X1) < n and the converse holds if r > 2ra+ 1. Proof. The first assertion is obvious. Also observe that, since X is non degenerate, then .Xi
cannot be a point, unless n = r-l,'m which case X is certainly not defective. By Terracini's lemma, the dimension of the span of the tangent spaces at P and Q is equal to dim(S(X)). Thus dim(S(X)) — n — 1 = dim(Xi), from which the rest of the claim follows,
o
The previous proposition gives us the upper bound S(X) < min{n — 1, r - n - 2} which may be achieved only in the case X\ is a curve. We will now describe the varieties for which this happens.
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LEMMA 3.2 Let X C Pr be o reduced, irreducible, non - degenerate variety of dimension n. Assume that X\ is a curve. Let H be a general hyperplane section of X. Then X± is also a general tangential projection of H. Proof. Take a general point P € X, which is also a general point of a general hyperplane section H of X through P. The tangent space Tfjtx to H at P is the section of the tangent space TX,P to X at P with the hyperplane TTH cutting out H on X. It follows that the projection H\ of H from Tntx is contained in X\. Since H cannot be degenerate in TT#, then necessarily HI — X\ and the claim follows, o Next we recall a definition.
DEFINITION 3.3 Let Z C pr~l be a pure dimenional variety. One says that it is extendable if there is a variety X C Pr which is not a cone such that Z is a hyperplane section of X. The following result is classical (see [Se], [Sco4], [T2]). A modern proof can be easily deduced from theorems of Badescu, L'vovski and Zak (for a general reference, see [BC]).
THEOREM 3.4 Veronese varieties of dimenion n > 2 are not extendable. Now, by induction on the dimension and by the classification of defective surfaces [CC2], we have the:
THEOREM 3.5 Let X C Pr be a reduced, irreducible, non - degenerate variety of dimension n, with r > n + 3. Assume that Xi is a curve. Then X is either a cone over a curve or a cone over a Veronese surface in P 5 . Proof. For n = 2, the statement coincides with Severi's classification of 1-defective surfaces. For n — 3, we know that a general hyperplane section of X is either a cone over a curve or a Veronese surface. In the former case, X itself must be a cone over a curve with vertex along a line. In the latter case X is a cone over a Veronese surface by theorem 3.4. The case n > 3 follows in a similar way by induction, o Now we go on considering, more specifically, the case of 1-defective varieties. LEMMA 3.6 Let X C Pr be a reduced, irreducible, non - degenerate, l-defective variety of dimension n. A general hyperplane section H has defect 6(H) = S(X) — 1 if r > In + 1,
and defect S(H) — 5(X) otherwise. Moreover v(H) — v(X) - 1. In particular, if 5(X) — 1, r > 2n + l and v(X) > 1 (which happens if Xi is l-weakly defective), then H is not defective but l-weakly defective. Proof. Consider two general points P, Q € H, which are two general points on X. One has TH,P = Tx,p n KH and TH,Q = TX,Q n TTH. Hence TH,p n TH,Q = Tx,p n TX,Q n TTH. Since
one computes 6(X) = dim(Tx,p n TX,Q) + min{l, r - In} and 8(H) = dim(TH,p n TH,Q) + min{l, r - 2n + 1}, the first assertion immediately follows. Notice now that, in the present situation, TX,P,Q = TH,P,Q n 7r#. Hence a general hyperplane section H1 of H which is bitangent at P and Q is cut out on H by a general hyperplane section H" of X bitangent at P and Q. Therefore for the contact varieties we have the relation EP?Q(#") n H C Sp i g(/f'), and actually the equality holds for the genericity of H. This finishes the proof of the lemma, o As a first consequence we have the following:
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THEOREM 3.7 Let X C Pr be a reduced, irreducible, non - degenerate, 1-defective variety of dimension n. Assume r < 2n and set m = 2n — r. Assume o(X) > n — m — 2. Then r > n + 3 and X is either a cone over a curve or a cone over the Veronese surface in P5, in which case r = n + 3.
Proof. The inequality r > n + I is trivial. Assume r = n + 2. Then two general tangent spaces to X are contained in a hyperplane. By standard projective geometry we deduce that either they should be contained in a fixed hyperplane or they should contain a fixed codimension two subspace. Both alternatives lead to a contradiction. A general surface section 5 of X is also 1-defective by lemma 3.6. Indeed taking the first m consecutive hyperplane sections, the defect remains unchanged and it drops at most by 1 in any of the successive n - m - 1 sections. Severi's classification of such surfaces (see [S], [Dl], [T3], [CC2]) tells us that S is either a cone or the Veronese surface V>2,2, whence the conclusion. o Observe that the assumption 6(X) > n — m — 2 always holds when X is a defective threefold.
In view of this theorem, we may and will assume, from now on, that if X C Pr is a 1-defective variety of dimension n, with 5(X) > n — 2, then r > In + 1. Recall that a variety X of dimension n is 0-weakly defective if and only if its Gauss map is not generically finite onto its image: any such a variety is a developable scroll, i.e. it is swept out by a 1-dimensional family of P™-1 along which the tangent space to X is constant.
We will need the following elementary lemma:
LEMMA 3.8 Let C C Pr be a reduced, irreducible, non-degenerate, projective curve, let n < r — 1 be a positive integer and let II C Pr be any fixed subspace of dimension m. Let P € C be a general point; consider the n-osculating subspace T^p to C at P, i.e. the only n-dimensional subspace having with C at P intersection multiplicity n + 1. Then T^'p and II intersect properly in Pr, namely dim(T(l,np n II) = max{n + m — r, —1}. Proof. . The assertion is clear if m = r -1 because C is non-degenerate. Assume m — r - 2 and suppose the assertion is not true. Since C is non-degenerate, this would mean that the hyperplane joining II with the general point P of C would have multiplicity of intersection at least n + 1 > 2 with C at P, a contradiction. Thus we may assume from now on that m < r - 2. Set e := max{n + m - r, -1} and r] := dim(T(?p n II). onsider the projection TT : Pr -» pr-m-i Qf pr £rom j^ ancj jet (~ r — m — 1. The image T of T^p via TT has dimension dim(T) = n — 77 — 1 < n - e - 1 = r — m - 2, so that T is a proper
subspace of p7—"*-1. it has positive dimension, otherwise r? = n — I and we would have a contradiction as in the m = r - 2 case. In conclusion T is a proper subspace of pr-m-1 of positive dimension, which has intersection multiplicity at least n + I with C' at its general point P' = 7r(P). This means that the general point of C" is a point of hyperosculation, a contradiction. The case e = -1 can be dealt with in a similar way. o As a consequence we can caracterize the developable scrolls which are 1-defective:
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PROPOSITION 3.9 Let X € Pr be a reduced, irreducible, non - degenerate, l-defective variety of dimension n which is a developable scroll. Then X is a cone over a developable scroll. Proof. Suppose X is not a cone. Then by the structure theorem of developable scrolls (see [GH], pg. 390), there is a non-degenerate curve C C Pr such that X is the closure of the union of osculating P™"1^ to C. A general tangent space to X is therefore a general osculating Pn to C. By lemma 3.8, two such osculating Pn to C meet properly in P r . By Terracini's lemma this implies the assertion, o Recall now that 1-weakly defective surfaces have been classified by Terracini in [T3] (see also [CC2]). Using this classification, we are able to make one more step with respect to theorem 3.5 in the classification of varieties with large defects. THEOREM 3.10 Let X C Pr be a reduced, irreducible, non - degenerate, l-defective variety
of dimension n > 3, with r >2n+l, 6(X) — n — 1 and f(X) = n — 1. Then either X sits in a cone of dimension n + 1 over a curve, or n = 3, r — 7 and X sits in a cone of dimension 4 over a Veronese surface in P5, or it is a cone over a developable surface. Proof. By lemma 3.6, a general surface section 5 of A" is 1-weakly defective but not defective. By Terracini's classification of such surfaces (see theorem (1.3) of [CC2]), we know that we have only the following possibilities for 5: (i) either S is a developable scroll; (ii) or S sits in a cone W with vertex a point V over a Veronese surface in P5; (iii) or 5 sits in the cone W over a curve with vertex a line L. Assume n = 3. In case (i), the general hyperplane section of A' is a scroll. An easy
count of parameters shows that X contains a 3-dimensional family of lines. Then a classical theorem of B. Segre (see [Seg], [R]) implies that X is either a scroll or a quadric in P 4 . The latter case cannot occur because we are assuming X to be l-defective. In the former, one moment of reflection shows that, since the general hyperplane section S of X is 0-defective, so has to be X. In other words X is also a developable scroll and it must be a cone over a developable scroll by proposition 3.9. Consider now case (ii), in which r = 7. Here we can proceed as in [CC2], §7. The contact varieties S of bitangent hyperplanes to X vary in a 2-dimensional involution T>, whose general element is irreducible and reduced (for the concept of an involution, see [CC2], §5). Indeed E is cut out by a general hyperplane in the irreducible curve F, which in turn is cut out on the general hyperplane section 5 by a quadric cone projecting from V, the vertex of the cone W containing 5, a general conic of the Veronese surface. Hence V is a linear system (see [CC2], thm. 5.10). Notice that deg(E) > 3. Otherwise either S is a plane, or a quadric. Since E varies in ° 2-dimensional linear system, the former case is clearly impossible. In the latter, the gener surface section 5 of X would contain a 2-dimensional system of conies, the ones cut out 1 the surfaces in £>. Hence S would either be the Veronese surface in P5 or a projection of (see [S], [D2]) and therefore r < 6, a contradiction. The previous considerations show that the general surface S € Z> spans a P4 whi we may denote by TS- As S varies in £>, TS cuts the general hyperplane cutting out 5 ^ a 2-dimensional family of P3 all containing the point V. This implies that all the spaces TE contain a fixed line L, which cuts out on a general hyperplane the vertex of the cone containing the corresponding hyperplane section. Furthermore, as we noticed, the general
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hyperplane section of E is a curve F which sits in a quadric cone with vertex at V. Such a curve is projectively normal, which implies that E is projectively normal. Hence any surface through F can be lifted to a hypersurface through E, i.e. the map H°(P 4 ,Zsj^W) ->• H°(P3,Ir,P3(i)) is surjective for every i > 1. In particular this is the case for i = 2, and therefore, if deg(E) > 5, E itself lies in a quadric cone in TS with vertex along L. The same
conclusion takes place also in the cases deg(E) = 3,4: we may leave the easy details to the reader. In conclusion, the projection of E from L is a conic and therefore the projection of X from L is a surface in P5 with a 2-dimensional family of conies, hence, for an argument we already made a few lines above, this projection is the Veronese surface. This concludes the discussion case (ii). Case (iii) can be dealt with, even more easily, with the same ideas. It leads to the case in which X sits in a 4-dimensional cone over a curve. Finally the general case n > 3 can be proved by induction with analogous arguments. We leave the details to the reader, o
4 THE PROOF OF SCORZA'S THEOREM In this section we prove theorem [Scol]. Thus we consider X C Pr a reduced, irreducible, non - degenerate, 1-defective threefold. Notice that by theorems 3.5 and 3.10 we may and will assume r > 7 and S(X) = 1. Furthermore the case v(X) = 2 has been considered in theorem 3.10. Thus we may and will assume from now on 6(X) = v(X) — 1. In order to avoid trivial cases, we will also assume that X is neither a cone, nor contained in a cone over a curve, nor contained in a 4-dimensional cone over the Veronese surface V2,2 in P5. Let P, Q be general points of X. The general hyperplane section H € "H(—2P - 2Q) has a contact curve E := 'SptQ(H). We will prove next that, with our assumptions, E is an irreducible conic. In order to do this, we first need to state the following lemma. LEMMA 4.1 Let 7 be an irreducible, positive dimensional family ofPm's in Pr, i.e. T corresponds to an irreducible, positive dimensional, quasi-projective subvariety of the grassmannian G(m,r). Suppose that m > 2 and that two general subspaces of J- meet along a P m ~ 2 , then only the following cases are possible: (i) the span of the Pm 's of T is a Pk, k < m + 3; (ii) all the Pm 's of T contain the same Pm~*; (iii) the general Pm of jF cuts a fixed Pm along a P™"1. The proof can be found in [Mo]. It is nothing but an exercise in projective geometry and therefore we do not reproduce it here. Now we can prove our: PROPOSITION 4.2 Let X CPr, r > 7, be a reduced, irreducible, non - degenerate, 1defective threefold, with S(X) = v(X) = 1, which is neither a cone, nor contained in a cone over a curve, nor contained in a 4-dimensional cone over the Veronese surface V^,i in P 5 Then the contact curve E of a general bitangent hyperplane to X is an irreducible conic. Proof. . By theorem 1.2, we see that E imposes at most 3 conditions to the hyperplanes in P r . Thus E is a plane curve. If E has degree d > 2, then the projection of X from a general point P € X is not birational, i.e. the general secant to X would be multisecant, which is well known to be a contradiction (see [L], [CCl], pg. 100).
A theorem of G. Scorza
121
If E is a line, then through any pair of points of X there is a line in X joining them, so X is a subspace of P r , a contradiction. Similarly one excludes that E contains a line joining P and Q. Thus the only possibility we are left with is that E is a conic which can be either irreducible or reducible in two distinct lines p and q where p contains P but not Q and q contains Q and not P. We have to exclude the latter possibility. Suppose p stays fixed when H 6 H(-2P - 2Q) and Q move. Then X would be covered by a 2-dimensional family of lines which are pairwise coplanar. Thus they have to pass through the same point, hence X would be a cone, a contradiction. The alternative to the previous situation is that X is covered by a 3-dimensional family of lines. The aforementioned theorem of B. Segre [Seg] implies that X is either a scroll or a quadric in P 4 , but the latter case cannot occur because we are assuming that X is 1-defective. On the other hand, if X is a scroll, the above analysis implies that its planes pairwise meet. By the above lemma 4.1, we have that either X is a cone or it is contained in a cone over a curve, a contradiction again. o Thus with our assumptions, X has a 4-dimensional family of irreducible conies and two general points of X are joined by an element of this family.
COROLLARY 4.3 Same assumptions as in proposition 4.2. Then the general tangential projection Xi C P r ~ 4 of X is a projection of a Veronese surface V2,2> which is not a cone. Proof. We know that a general pair of points in X\ is contained in the projection of a conic. Thus Xi contains a 2-dimensional family of conies. Hence it is the Veronese surface in P5 or one of its projections (recall [D2]). Notice that it cannot be a cone because v(X) = 1. o
We are now ready to finish the proof of theorem 1.1, by proving the following:
PROPOSITION 4.4 Same assumptions as in proposition 4.2. Then X is either the Veronese threefold V2,3 in P9 or a hyperplane section of the Segre embedding Y of P2 x P2 in P8, or a projection of one of these varieties. Proof. Consider a minimal resolution / : Z -> X\ of the singularities of Xi. By corollary 4.3 we have the following cases:
(i) Z is isomorphic to P2, which happens if and only if X\ is the Veronese surface in P5 or an external projection of it to P m , 3 < m < 4. In this case X lies in P r , with 7 < r < 9 ; (ii) Z is P2 blown up at a point V, which happens if and only if either X\ is a projection of the Veronese surface from a point on it, i.e. a cubic scroll in P 4 , or an external projection of it in P3. In this case X lies in P r , with 7 < r < 8; (iii) Z is P1 x P1, which happens if and only if either X\ is a smooth quadric in P3. In this case X lies in P7. In any case, will refer to Hz '•— \f*Oxi(l)\ as to the complete hyperplane system on Z. Notice that the tangential projection induces a rational map T : X -4 Z. By pulling back Hz to X via T, we find a sublinear system of \H\(—2P). Suppose first we are in case (i). Then HZ - |£>P2(2)| on Z ~ P 2 . The complete, 2dimensional, linear system Cz ^ |O P 2(l)j is such that £f 2 ~ Hz and the multiplication map of sections H°(Z, Cz) ® HQ(Z, Cz) -» H°(Z, Hz) is surjective. By pulling back Cz via T we find a 2-dimensional linear system Cp on X. Since the general element in H(—2P) has a double point at P (see, for example, [CC2], proposition (2.3), for an even more general
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result), then the same happens for \H\(—IP) and the above analysis implies that the general element in Cp passes through P and is non-singular there. Notice that, since a general pair of points of X is connected by a rational curve, X, or rather any desingularization X of it, is regular, i.e. Pic°(X) is trivial. Thus, as P varies in X, the linear systems Cp vary, but they are all contained in one and the same linear system C of divisors on X. Since for a general point P € X one has Cp C C(-P), then dim£(-P) > 2 and therefore dim£ > 3. Hence the linear system C defines a rational map <j>: X -» P m , m > 3, and we denote by X1 the image of X via $. Notice that, by numerical reasons, a general conic of the family which sweps out X is sent by (j> to a line. Hence X' is such that two general points on it are connected by a line, hence it is a linear space. This proves that m = 3 and that
that ifi*OY(l) ^ A®B ~ Ox(1). Since h°(X,Ox(l))
= 8 because r - 7, and since
h°(Y, Oy(i)) = 9, we see that X' has to be a hyperplane section of Y and ^ is induced by a linear transformation. o
A theorem of G. Scorza
123
References [A]
B. Adlansvik, Joins and Higher secant varieties, Math. Scand., 61 (1987), 213-222.
[BC]
E. Ballico, C. Ciliberto, On gaussian maps for projective varieties, in Geometry of Complex Varieties, Cetraro (Italy), June 1990, Mediterranean Press, (1993), 35-54.
[Bro]
Bronowski, Surfaces whose prime sections are hyperelliptic, J. London Math. Soc., 8 (1933), 308-312.
[CC1]
L. Chiantini, C. Ciliberto, A few remarks on the lifting problem, Asterisque,218 (1993), 95-109.
[CC2]
L. Chiantini, C. Ciliberto, Weakly defective varieties, pre-print,(1999).
[Dl]
M. Dale, Terracini's lemma and the secant variety of a curve, Proc. London Math. Soc., 49 (3) (1984), 329-339.
[D2]
M. Dale, Severi's theorem on the Veronese surface, J. London Math. Soc., (2) 32 (1985), 419-425.
[F]
T. Fujita, Projective threefolds with small secant varieties, Sci. Papers Collmege Gen. Ed. Univ. Tokyo, 32 (1982), 33-46.
[FR]
T. Fujita, J. Roberts, Varieties with small secant varieties: the extremal case, Amer. J. of Math., 103 (1981) 1, 953-976.
[GH]
Ph. Griffiths, J. Harris Algebraic geometry and local differential geometry, Ann. Sclent. EC. Norm. Sup., IV 12 (1979), 355-432.
[L]
0. A. Laudal, A generalized trisecant lemma, Springer L. N. in Math., bf 687 (1978), 112-149.
[M]
E. Mezzetti, Projective varieties with many degenerate subvarieties, Boll. U.M.I., (7) 8-B (1994), 807-832.
[Mo]
U. Morin, Su sistemi di Sk a due a due incidenti e sulla generazione proiettiva di alcune varieta algebriche, Atti R. 1st. Veneto di Scienze, Lettere ed Arti, 101 (1941-42), 183-196.
[PI]
F. Palatini, Sulle superficie algebriche i cui Sh (h + l)-seganti non riempiono lo spazio ambiente, Atti. Accad. Torino, (1906).
[P2]
F. Palatini, Sulle varieta algebriche per le quali sono di dimensione minore dell'ordinario, senza riempire lo spazio ambiente, una o alcune delle varieta formate da spazi seganti, Atti. Accad. Torino, 44 (1909) 362-374.
[R]
E. Rogora, Varieties with many lines, Manuscripta Math., 82 (1994), 207-226.
[Scol]
G. Scorza, Determinazione delle varieta a tre dimensioni di 5r, (r > 7), i cui 83 tangenti si tagliano a due a due, Rend. Circ. Mat. Palermo, 25 (1907), 193-204.
[Sco2]
G. Scorza, Un problema sui sistemi linear! di curve appartenenti a una superficie algebrica, Rend. R. 1st. Lombardo, (2) 41 (1908), 913-920.
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[Sco3]
G. Scorza, Sulle varieta a quattro dimensioni di Sr (r > 9) i cui £4 tangenti si tagliano a due a due, Rend. Circ. Mat. Palermo, 27 (1909), 148-178.
[Sco4]
G. Scorza, Sopra una certa classe di varieta razionali, Rend. Circ. Mat. Palermo, 28 (1909), 400-401.
[Seg]
B. Segre, Sulle Vn aventi piu di oon~k Sk, note I e II Rend. Accad. Naz. Lincei, (1948) .
[Se]
C. Segre, Sulle varieta normali a tre dimensioni composte di serie semplici razionali di piani, Atti. Accad. Sci. Torino, 21 (1885), 95-115.
[S]
F. Severi, Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e ai suoi punti tripli apparenti, Rend. Circ. Mat. Palermo, 15 (1901), 33-51.
[Tl]
A. Terracini, Sulle Vjt per cui la varieta degli Sh (h + l)-seganti ha dimensione minore dell'ordinario, Rend. Circ. Mat. Palermo, 31 (1911), 392-396.
[T2]
A. Terracini, Alcune question! sugli spazi tangenti e osculatori ad una varieta, Nota I, Atti Accad. Sci. Torino, 49 (1913).
[T3]
A. Terracini, Su due problemi, concernenti la determinazione di alcune classi di superficie, considerati da G. Scorza e F. Palatini, Atti Soc. Natur. e Matem. Modena, V, 6 (1921-22), 3-16.
[Z]
F. Zak, Tangents and secants of algebraic varieties, Transl. Math. Monogr., 127 (1993).
The Unirationality of All Conic Bundles Implies the Unirationality of the Quartic Threefold ALBERTO CONTE, Dipartimento di Matematica, Universita degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italia
Among the many open (uni)rationality problems for algebraic varieties, two are outstanding because of their basic character. To state the first of them, let me remind you that a conic bundle is a 3-dimensional algebraic variety X, defined over any field K, together with a morphism TT : X —> S (where S is a rational surface, in particular S = P2) whose fibers are conies, and that an r-dimensional algebraic variety Y is said to be unirational if there exists a rational dominant map x '• Pr —* Y. Then, our first problem reads as follows: (A) Is every conic bundle unirational?
The answer to (A) is not known: nobody was able to prove that it is positive, but on the other hand no example of non-unirational conic bundle is known. A possible candidate was pointed out by G. FANO in his address to the International Congress of Mathematicians held in Bologna in 1928 ([2]), where he suggested that a hypersurface of degree n V3(n) C P4 containing a line / of multiplicity n — 2 should not be unirational for n > 5 because it should not contain any rational surface (the structure of conic bundle on ¥3 (n) is given by the 2-planes through I, which cut Vs(n) outside I in conies; for n — 3,4 Vs(n) is known to be unirational). However, up to now nobody was able to turn into a proof the above FANO's conjecture. On the other hand, let V r _ 1 (4) C Pr be a quartic hypersurface. It is well known that, for r > 6, the generic V(4) is unirational, whilst nothing is known for r — 4,5 (see [1], [3], [5]). Our second problem is therefore the folllowing
(r = 4): (B) Is the generic quartic threefold unirational?
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Conte and Marchisio
Here too the answer is not known, even if some classes of smooth unirational quartic threefolds (the so called quartics "with separable asymptotics") were constructed (see [9], [4], [5], [6], [7]). In this note, using the result, proved by M. MARCHISIO in the appendix, we shall show that a positive answer to (A) would imply a positive answer to (B), so that, if all conic bundles were unirational then also the generic quartic threefold would be unirational (for a general survey of this kind of problems, see [8]). On the other hand, if the generic quartic threefold were not unirational, as most of the experts think, then our proof would give immediately an example of conic bundle which is not unirational.
To show the above result, let V = Vs(4) C P4 be a quartic threefold and let R £ V (for simplicity we assume K to be algebraically closed and of characteristic zero, but the construction holds for any K). Let TR(V) be the tangent space to V in R and CR(V) the tangent cone to V in R (i.e. CR(V) is the tangent cone in R to the quartic surface VR — V D TR(V), which is singular in R). Now, if V is sufficiently general, we have, if we take R general in V, that CR(V) is a cone over a conic (shortly: CR(V) is a quadric cone), i.e. CR(V) is itself a surface. More precisely there is an open set VQ C V such that if R £ VQ we have that: i) CR(V) is a quadric cone, absolutely irreducible over K(R), ii) if I C CR(V) is a line through R, then it is "at least" a "triple tangent" in R and for almost all such lines lines we have
with P / R (P is called the "fourth" point of intersection) (clearly for every line on CR(V) through R we have either the above situation, or P = R, or / is entirely on V).
Fix now a hyperplane H° C P4 and take a point R £ V which is outside H°. Now take the tangent cone CR(V) as introduced above. Consider on the one hand the intersection QR = CR(V) n H° which is a conic in H° (and more precisely in H° n TR(V)) and on the other hand the intersection WR = V n CR(V). Then we have LEMMA There exists a birational transformation
PR • QR - - -»• WR. Proof: Let P' € QR. Consider the line / = (R,P') spanned by R and P'. This line is on CR(V) by the definition of QR and hence l.V = 3R + P. Now take P = PR(P')-
Unirationality of conic bundles
127
Let now 5° C ¥3 (4) be a rational surface going through the point R £ H° as constructed in the APPENDIX. Now we take the above construction and we get in H° a conic QR(V) spanning a 2-plane in H° (in fact the 2-plane H° r\TR(V) where, as before, TR(V) is the tangent space to V in R). Now we consider the variety
V+ = {(R,P')\R£S°,P'£QR}, which is an irreducible variety defined over KQ and of dimension 3. We have the following diagram
7T
V+
— ->
],
p
V
S°
where ir(R,P') = R and p(R,P') = pn(P') from the LEMMA. Note that the fibers of ir are the conies QR, so that V+ is a conic bundle over the rational surface S°. Therefore, if (A) holds, V+ is unirational, and to show that V = Va(4) is unirational, i.e. that (B) holds, it will be sufficient to show that p is dominant.
To show this, start with a general point P = (p) = (po,...,pm+i) Consider the equations:
(i.e.
consider the polar and second polar of P with respect to V). Intersect these equations with the fixed surface 5° C V. We get a non-empty (and for generic P € V a finite) intersection and taking R in that intersection we get the point P back because by construction P € CR(V) (i.e. P € WR). p is therefore dominant and our claim is proved.
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APPENDIX EVERY QUARTIC 3-FOLD CONTAINS A RATIONAL SURFACE MARINA MARCHISIO Dipartimento di Matematica, Universita degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italia
In [1] Conte and Murre proved that the generic quartic hypersurface V r _i(4) C Pr with r > 5, contains a rational surface. It is possible to prove that also the generic quartic of dimension 3 in P4 contains a rational surface, more precisely we prove the following PROPOSITION Every quartic hypersurface V = V3(4) C P4 contains a ratio-
nal surafce S° and moreover if P £ V is any point of V we can take S° going through it.
Proof: Consider a rational curve C° C V, for istance a line. Let R € C° be a generic point of C°. Fix a hyperplane H° C P4 with R $ H° and consider the variety
X+ = {(R,P')/ReC°,P'£QR}, where QR is as above. X+ is an irreducible surface. We obtain the following diagram
X+ 7T
- -->
p(X+) = S° CV
J,
c° where n(R,P') = R and p(R,P') = pa(P') with
PR • QR = CR(V) nH°-->Xn CR(V). The generic fiber of TT is the isomorphic to the rational curve (7°, so that by M. Noether's theorem we have that X+ is a rational surface and, since p
is dominant, S° too is rational. To obtain So going through a given point P consider again the equations (*) and intersect with the surface generated by the lines on V and then as C° take a line IQ going through an intersection point.
Unirationality of conic bundles
129
References [1]
A Conte, J P Murre. On a Theorem of Morin on the Unirationality of the Quartic Fivefold. Ace. Sc. Torino, Atti Sc. Fis. 132:49-59, 1998.
[2]
G Fano. Sulle varieta algebriche a tre dimensioni aventi tutti i generi nulli. Atti del Congresso Internazionale dei Matematici, Bologna 3-10 settembre 1928, Tomo IV, Zanichelli, Bologna, 1929 pp 115-121.
[3]
J Harris, B Mazur, R Pandharipande. Hypersurfaces of Low Degree. Duke Math. Jour. 95:125-160, 1998.
[4]
V A Iskovskikh, Yu. I. Manin. 3-Dimensional Quartics and Counterexamples to the Luroth Problem. Math. USSR Sb. 15:141-166, 1971.
[5]
M Marchisio. Ipersuperficie quartiche unirazionali. Tesi di Dottorato, Consorzio Universita degli Studi di Genova, Torino e Politecnico di
Torino, Torino 1998. [6] [7]
M Marchisio. Unirational Quartic Hypersurfaces. Bollettino U.M.I., (8) 3-6:301-314, 2000. M Marchisio. A 54 - (114-)dimensional Family of Smooth Unirational Quartic 3 — (4—)folds. To appear.
[8]
A V Pukhlikov. Birational Automorphisms of Higher-Dimensional Algebraic Varieties. Proceedings of ICM 1998, Vol. II, Berlin, 1998 pp 97-107.
[9]
B Segre. Variazione continua e omotopia in geometria algebrica. Ann. Mat. Pura Appl. 50:149-186, 1960.
Gaussian Ideals and the Dedekind-Mertens Lemma ALBERTO CORSO, Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, USA. E-mail: corsoQms.uky.edu SARAH GLAZ, Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, USA. E-mail: glazQuconnvm.uconn.edu
1
INTRODUCTION
Let R be a commutative ring. Throughout the paper the term local refers to a, not necessarily Noetherian, ring with only one maximal ideal. For a polynomial / € R[t], where t is an indeterminate over R, denote by c(f)—the so called content of /—the .R-ideal generated by the coefficients of /. For two polynomials / and g in R[t], the Gaussian ideal of / and g is the R-ideal c(fg). The ideal c(fg) has a number of interesting properties. Of particular interest in this article are the properties of c(fg] that are revealed through its comparison to the ideal c(f)c(g). Two such comparisons received a considerable amount of attention in recent years: c(fg) C c(f}c(g)
c(fg)c(g)n
= c(f)c(g)c(g)n,
(1.1)
(1.2)
for n — deg /.
Containment (1.1) becomes equality when c(f) is an invertible ideal [14, 15]; or more generally when c(f) is a locally principal ideal [36, 38, 5]. A polynomial / in R[t] is called a Gaussian polynomial if containment (1.1) becomes an equality for any polynomial g in R[t\. The focus of a number of recent works was the investigation of the converse, that is the attempt to answer the question: Is the content ideal c(/) of a Gaussian polynomial / an invertible (locally principal) ideal? For a domain R this question was posed as a conjecture by Kaplansky [38]; and it also appears in [36, 2]. In [18, 19], Glaz and Vasconcelos answered the question affirmatively
131
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Corso and Glaz
for a large number of cases where R is a domain. In particular they show that the
answer is affirmative for integrally closed Noetherian domains. Heinzer and Huneke [20] extended the answer to all Noetherian rings provided c(/) contains a non zero divisor. In particular they show that the answer is affirmative for all Noetherian domains. This article surveys both approaches in Sections 2 and 3. Section 4 considers a related topic in both its classical aspects and its more recent manifestations. A ring R is called a Gaussian ring if every polynomial with coefficients in R is a Gaussian polynomial [36, 3]. It is a classical result, proved in [36, 14], that a domain R is Gaussian if and only if it is a Priifer domain. If the Gaussian ring is not a domain the situation is more complicated. We survey some of the results in this direction found in Pahikkala [31], Tsang [36], D.D. Anderson [1, 3], and D.D. Anderson and Camillo [4]. Equality (1.2) is an ancient and beautiful formula which was named by Krull [25], the Dedekind-Mertens Lemma. The ancient versions, which go back to Dedekind [11] and Mertens [26], did not take the form of an equality. See also [24] and [32]. Modern proofs of the present formulation can be found in [28, 37, 16, 6, 5] and a very recent Grobner basis approach is in [8]. A fascinating detailed history of this formula can be found in [21]. Other interesting sources are [3, 13]. There is a natural relationship between questions regarding Gaussian polynomials and questions regarding the Dedekind-Mertens formula. Denote by fJ>R(f)—the so called Dedekind-Mertens number of a polynomial f—the smallest positive integer k such that c(fg)c(g)k~l = c(f)c(g)c(g)k~l for all polynomials g. The Dedekind-Mertens formula says that HR(/) < deg/ + 1. It is shown in [10] that if / and g are polynomials with indeterminate coefficients over a field, then fJ,n(f) = deg /+!. For many polynomials / the Dedekind-Mertens number p-R(f) is smaller than deg/ + 1. For example polynomials / having Dedekind-Mertens number /UR(/) = 1 are precisely the Gaussian polynomials, and there are such of any degree. Denoting by HR(!} the minimal number of generators of an .R-ideal /, we see that the conjecture considered earlier can be rephrased (at least for the local case) as: Does /u#(/) = 1 imply that HR(C(}}) = 1? A number of recent articles considered the more general relation between //«(/) and HR(C(})) (and related notions), for polynomials other than Gaussian. In particular these works considered the question when these two numbers are equal. Section 5 of the current article explores the work done by
Corso, Vasconcelos and Villarreal [10] on generic Gaussian ideals. Section 6 surveys the work done by Heinzer and Huneke [20], and by Corso, Heinzer and Huneke [9] regarding the Dedekind-Mertens numbers. A generalization of the DedekindMertens number appears in a recent paper by Rush [34]. Other aspects of the content of polynomials were considered in [27, 33, 5, 35]. Last but not least, the authors sincerely thank Wolmer Vasconcelos for many helpful comments concerning the material of this survey. 2
GAUSSIAN POLYNOMIALS: A HILBERT FUNCTIONS APPROACH
Kaplansky posed a question regarding the nature of the content ideals of Gaussian polynomials [38, 36, 19, 5] which Glaz and Vasconcelos state in [19] and [18] as the following conjecture:
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CONJECTURE 2.1 Let R be a ring and let f be a Gaussian polynomial over R, then c(f) is an invertible ideal of R.
Without any qualifications on the ring the conjecture is false even with a weaker requirement, that is, the requirement that c(f) be locally principal. EXAMPLE 2.2 ([18]) Let (R,m) be a local Artinian ring with m2 = 0. Clearly any polynomial over R is Gaussian. But there exist such rings where the content of a polynomial is not necessarily even locally principal. Let k be a field, let X and Y be indeterminates over k, and denote by x and y the images of X and Y in k[X, Y\/(X,Y)2. Let R — k[x,y](Xiy), then f ( t ) = x + yt + xt2 is a polynomial with content ( x , y ) . This observation led Glaz and Vasconcelos [19, 18] to restrict their investigation to the case where R is a domain. The most naive approach to test the Gaussian property of a polynomial / is to test the equality c(f)c(g) = c(fg) for judicious choices of g. This yielded the following result:
PROPOSITION 2.3 ([19]) Let R be a domain and let f be a Gaussian polynomial whose content ideal c(f) is generated by two elements, then c(f) is an invertible ideal. Particular cases where c(f) is generated by two elements were treated before: the case where / is linear [38], and the case where / is a binomial [36, 5]. In general, the Gaussian property can be checked locally. We can therefore assume that (R, m) is a local domain. In this case an ideal / is invertible if and only if it is principal, and an investigation into the invertibility of an ideal I turns into an investigation into the minimal number of generators of I, ^R(!). Here the approach via Hilbert functions naturally comes into play. The function of interest
One gives a combinatorial setting for this function as follows. Set / = c(f) and let
R[It] = R + It + I2t2 + . . . c R[t] be the Rees algebra of the ideal /. This function is then the Hilbert function of the special fiber F(I) of the ring R[It], that is the graded 00
ring F(I) = R[It] 0 R/m = (J) Is /mls. Let deg / = n, the Gaussian property of / s=0
implies that HR(C(/)S) = (J,R(c(fs)) <ns + l, and therefore this Hilbert function is itself a polynomial of degree at most one, eos + BI [19]. Moreover, one may assume at this point that the residue field of R is infinite (this is a delicate point since it may require passage from R to R(X), which does not necessarily preserve the Gaussian property of /, but does leave the Hilbert function unchanged). This allows
us to conclude that c(/)s = (a,b)c(f)s~1 for large s, that is c(f) has a reduction J — (a, b) generated by at most 2 elements. We recall that a subideal J of / is a reduction of / if J r+1 = JIT for some positive integer r. The smallest such integer is called the reduction number rj(I) of / with respect to J. A reduction is said to be minimal if it is minimal with respect containment. If the residue field is infinite the
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minimal number of generators is independent of the minimal reduction of /. This
number is called the analytic spread 1(1) of / and it equals the dimension of the special fiber of / (see [29] and [30] for Noetherian rings, and [12] for non Noetherian rings). In conclusion, we have seen that t(c(f)) < ^n(J) < 2. As noted in [19], the case where l(c(f)) — 1 corresponds to e0 = 0; while the case where £(c(f)) — 2 corresponds to CQ ^0. These observations were used to settle the conjecture in the case the characteristic of the residue field of jR is not zero:
THEOREM 2.4 ([19]) Let (R,m) be a local domain with residue field of characteristic p > 0, and let f be a Gaussian polynomial over R, then CQ = 0. In particular, if R is integrally closed then c(/) is an invertible (principal) ideal of R. We now turn our attention to the case where the characteristic of the residue field of R is 0, but R contains the rational numbers. It is shown in [19] that eo < 1. Note that up to this point there was no assumption stating that the ring is Noetherian. In attempting to force CQ to actually be 0 in the case where the characteristic of the residue field is zero, this assumption had to be made in order to employ several 'Rees algebra techniques' which so far are known to hold only
for Noetherian rings. Using these techniques for the case the ring contains the rational numbers, Glaz and Vasconcelos answered the conjecture affirmatively for all integrally closed Noetherian domains:
THEOREM 2.5 ([19]) Let R be a Noetherian integrally closed domain and let f be a Gaussian polynomial, then c(f) is an invertible ideal of R. We remark here that additional information may be obtained via the Hilbert functions approach for Noetherian rings which are not necessarily integrally closed [18, 19], but in view of the more comprehensive Noetherian results of Heinzer and Huneke [20] described in the next section we see no need to elaborate them here. On the other hand the results of Proposition 2.3 and Theorem 2.4 are almost the only results where the state of Conjecture 2.1 is known for rings which are not necessarily Noetherian. Recall that a ring R is coherent if (0: a) and ID J are finitely generated ideals of R for every element a £ R and any two finitely generated ideals 7 and J of R, and a ring R is regular if finitely generated ideals of R have finite projective dimension—a condition that coincides with the classical notion of
regularity for Noetherian rings (see [17]). One can also show that Conjecture 2.1 has an affirmative answer in some cases where the ring is coherent regular. In particular the conjecture is valid if R is a coherent ring of global dimension less than or equal to two.
GAUSSIAN POLYNOMIALS: AN APPROXIMATELY GORENSTEIN RINGS APPROACH Heinzer and Huneke [20] employed a different approach in their attempt to validate Conjecture 2.1. Their response to examples of rings which are not domains over which Gaussian polynomials have content ideals which are not locally principal, such as the ring exhibited in Example 2.2, was to put regularity restrictions on the
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content ideal of the Gaussian polynomial. Specifically they ask that the polynomial be regular, that is a non zero divisor. For such Gaussian polynomials they settled the conjecture affirmatively for all locally Noetherian rings by reducing to the case of approximately Gorenstein rings.
A local Noetherian ring (R, m) is an approximately Gorenstein ring if it satisfies either of the following equivalent conditions: i. For every integer t > 0 there is an ideal / C m * such that R/I is Gorenstein. M. For every integer t > 0 there is an m-primary irreducible ideal / C m * . Approximately Gorenstein rings were defined and investigated in [22], from where it follows that a complete local Noetherian reduced ring of positive Krull dimension is an approximately Gorenstein ring. Let / be a Gaussian polynomial over a ring R, and let / be an ideal of R, then the image of / in (R/I)[t\ is a Gaussian polynomial over R/I [20]. In case the ring is locally both Noetherian and approximately Gorenstein this allowed a reduction to the case where the Krull dimension of the ring is zero and therefore to the following result:
PROPOSITION 3.1 ([20]) Let R be a locally Noetherian and locally approximately Gorenstein ring and let f be a Gaussian polynomial over R. Then c(f) is a locally principal ideal of R. A. polynomial / is Gaussian for polynomials of degree at most n over R if c(f)c(g) = c(fg) for all polynomials g in R[t] with deg
= yR ([20, Theorem 2.1]). From this it follows:
PROPOSITION 3.2 ([20]) Let R be a locally Noetherian and locally approximately Gorenstein ring and let f 6 R[t] be a polynomial o/deg / = n. Then f is a Gaussian polynomial if and only if f is Gaussian for polynomials of degree at most n over R. Let (R, m) be a Noetherian local ring, and denote by (R, in) the m-adic completion of R. One observes ([20]) that / is a Gaussian polynomial (respectively Gaussian for polynomials of degree at most n) over R if and only if / is a Gaussian polynomial (respectively Gaussian for polynomials of degree at most n) over R. Passing from a Noetherian local ring (/?, m) to its m-adic completion and reducing to the case where the Krull dimension is strictly greater than zero allowed Heinzer and Huneke to put all the previous results and observations together and obtain: THEOREM 3.3 ([20]) Let R be a locally Noetherian ring and let f £ R[t] be a regular polynomial of degree n. If f is Gaussian for polynomials of degree at most n, then c(f) is an invertible (equivalently locally principal) ideal of R. In particular Gaussian polynomials over Noetherian domains have invertible content ideals.
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Corso and Glaz GAUSSIAN RINGS
In her 1965 Ph.D. thesis, Tsang defines a Gaussian ring to be a ring over which every polynomial is Gaussian. A substantial part of her thesis is devoted to investigating the nature of Gaussian rings, but she never published her results in a scientific journal. As a consequence some of her results and characterizations were rediscovered later independently by others. In this section we reproduced some of the main results concerning Gaussian rings found in her thesis and place them in the context of some of the later generalizations. It is noted in Tsang's thesis that the Gaussian property can be checked locally, thus her investigation concentrated mainly in the local case. THEOREM 4.1 ([36]) Let (R, m) be a local ring. The following conditions are equivalent: 1. R is Gaussian. 2. For any finitely generated ideal I of R, I/I n (0: /) is a cyclic R-module. 3. For any two generated ideal I of R, I /I n (0: 7) is a cyclic R-module. Several more involved equivalent conditions appear in Section 4 of Tsang's thesis. This theorem allowed her to characterize all Gaussian domains, a result discovered independently by Gilmer [14], which became one of the classical characterizations of Priifer domains.
COROLLARY 4.2 ([36, 14]) Let R be a domain. R is Gaussian if and only if R is a Priifer domain. The prime ideals in a local Gaussian ring (R, m) are linearly ordered, therefore if R is not a domain, its nilradical is its unique minimal prime ideal. It follows that a local Gaussian ring modulo its nilradical is a valuation domain. In particular a reduced local Gaussian ring is a valuation domain. From this discussion we conclude:
COROLLARY 4.3 ([36]) 1. A semi local reduced Gaussian ring is a finite direct sum of Priifer domains. 2. A reduced Noetherian ring is Gaussian if and only if it is a finite direct sum of Dedekind domains. The notion of a Priifer domain can be generalized to rings with zero divisors in a number of ways. One such generalization is a Priifer ring, that is a ring for which any finitely generated ideal containing a non zero divisor is invertible. [31] carried the theme further and showed that R is a Priifer ring if and only if c(f)c(g) = c(fg) for every /, g € R[t] with c(f) and c(g) containing non zero divisors. Next Tsang refined the results of Theorem 4.1 in the case (R, m) is a Noetherian local ring.
THEOREM 4.4 ([36]) Let (R,m) be a local ring satisfying that m is finitely generated and m(p| m*) = 0. The following conditions are equivalent:
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1. R is Gaussian.
2. m/(0: m) is a cyclic R-module. 3. R/(Q: m) is a valuation ring. In particular conditions 1, 2 and 3 are equivalent for a Noetherian local ring (R, m).
As a corollary Tsang concluded that a Noetherian Gaussian ring (R, m) is either a DVR, or a primary ring with m/(0: m) principal, or possesses exactly two prime ideals m and n, with the nilradical of R equal to n and also n = (0: m) ^ 0 and R/n is a DVR not a field. D.D. Anderson [1] carried the theme further and came up with the following characterization of Noetherian Gaussian rings:
THEOREM 4.5 ([!]) Let R be a Noetherian Gaussian ring. Then R is a finite direct sum of indecomposable Gaussian rings of the following two types: 1. Zero dimensional local rings;
2. Rings S in which every maximal ideal has rank one and all but a finitely many of the maximal ideals are invertible. S has a unique minimal prime ty, 5/^3 is a Dedekind domain, ^3mj • • • mn = 0, where {mi,... , mn} is the set of maximal ideals of R which are not invertible.
In addition a ring satisfying the conditions stated in 2 of Theorem 4.5 is an indecomposable Gaussian ring. He also provides a detailed characterization of indecomposable Gaussian rings satisfying 1 of Theorem 4.5. Recently D.D. Anderson and Camillo [4] investigated a condition on a ring R related to the Gaussian property of R. A ring R is called an Armendariz ring if whenever polynomials f = OQ + a^t + ... + antn, g = bo + bit + ,..+ bmtm in R[t] satisfy fg = 0, then atbj — 0 for each i and j. They note that if R is Gaussian, then R is Armendariz which allowed them to give a new characterization of Gaussian rings. PROPOSITION 4.6 ([4]) Let R be a ring. Then R is Gaussian if and only if every homomorphic image of R is Armendariz.
The relation between the Gaussian and Armendariz properties of a ring R allowed D.D. Anderson and Camillo to conclude that the polynomial ring R[t] is Gaussian if and only if R is von Neumann regular; and the power series ring J?|t] is Gaussian if and only if R is von Neumann regular and NO algebraically compact. One notices that under these conditions R[t] (respectively /?[£]) is Gaussian if and only if it is semihereditary. The semihereditary condition, that is the condition under which finitely generated ideals are projective, is another natural generalization of the Priifer condition to rings with zero divisors. Since a semihereditary ring is locally a valuation domain we obtain that a semihereditary ring is Gaussian. One wonders to what extent is the converse true.
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Corso and Glaz GENERIC GAUSSIAN IDEALS
One of the purposes of [10] was to explore the relationship between c(fg) and c(f)c(g], when the polynomials are far from being Gaussian. The approach in this case is more combinatorial. Namely, the authors deal with polynomials / and g whose coefficients are given by two distinct sets of indeterminates. This setting required a different perspective and one had to bring into play new aspects of the theory of Cohen-Macaulay rings, such as o-invariants, Noether normalizations, edge ideals and algebras associated to graphs, and classical determinantal ideals. The path to this analysis started by looking at a 'decayed' Dedekind-Mertens formula. If we multiply both sides of equality (1.2) by c(f)n and rearrange the terms, we obtain
c(fg)[c(f)c(g)]n
= [c(f)c(g)}n+1 ,
(5.1)
where n = deg/.
Equality (5.1) says that the ideal c(fg) is a reduction of the ideal c(f}c(g). By symmetry, the reduction number of c(f)c(g) with respect to c(fg) is less than or equal to min{deg/, deg 3}. In this light, the main result of [10] says:
THEOREM 5.2 ([10]) Le£X = {x0,... ,xn} andY = {y0,... ,ym} be distinct sets of indeterminates and let
f = x0 + xit + ... + xntn
and
g = yo + y\t + . . . + ymtm
be the corresponding generic polynomials over a field k. Set R = fc[X, Y], / = (/)c(s) and J = c(fg) and suppose n < m. Then J is a minimal reduction of I
c
with analytic spread (,(!} = m + n + 1 and reduction number rj(I) = n. Also, the polynomials hq = ^P xtyj are algebraically independent, and k[h0,... ,hm+n] is i+j=q
a Noether normalization of k[xtyj 's\. In particular, Theorem 5.2 implies that the factor c(g)n in the Dedekind-Mertens formula is sharp. The generic form of the ideal c(fg) has also shown two additional interesting features that we describe next. The first feature is an intriguing primary decomposition, in which all components are given by Gorenstein ideals.
THEOREM 5.3 ([10]) Let R be a Noetherian domain and let
f = x0 + xit + .. . + xntn
and
g = yo + yit + ... + ymtm
be generic polynomials of degrees n and m over R. Then c(fg) primary decomposition
has the following
c(fg) = c(f) n C(g) n [c(fg) + c(/)m+1 + c(9r+l] . Furthermore, if R is a Gorenstein ring then the ideal c(fg) + c(f)m+l
a Gorenstein ic
+ c(g)n+l is
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139
The delicate part of the proof of the above result is the determination of the Goren-
steinness of the ideal c(fg) + c(f)m+l + c(g)n+l . For that, one needs to use results on the theory of residual intersections by Huneke and Ulrich [23]. Once we know that the ideal / is Gorenstein, it is not difficult to show that the socle of R/I is generated, for example, by xfy™. Similar but more complicated primary decompositions hold for the Gaussian ideal associated to a triplet, quadruplet, or general n-tuplet of generic polynomials. In these cases, the question of whether the primary
components are Gorenstein is still open. The second feature of c(fg) that we describe here is due to Bruns and Guerrieri [8]. Inspired by the work of [10] on generic Gaussian ideals and motivated by their joint work with Boffi on the Jacobian of a trilinear form [7], Bruns and Guerrieri gave the most recent proof of the Dedekind-Mertens formula. It is a combinatorial proof based on a Grobner basis approach to the ideal c(fg) for polynomials with indeterminate coefficients; in fact they determine the initial ideal of c(fg) with respect to a suitable term order. Namely,
THEOREM 5.4 ([8]) Let k be afield, R = k[Xi,... ,Xc,Yi,... ,Yd] and set i+j=q
with Uij € k, Uij ^ 0 for all i and j . Furthermore, let S denote the set of monomials
Xil---XiuYjl---Yjv
0
Q
Then the set JV of the monomials n £ S generates the initial ideal of the ideal (h-2, . • • ,hc+d) with respect to the reverse-lexicographic term order on R induced by the order
of the indeterminates. In particular, S is mapped to a k-basis of R/I under the natural homomorphism.
As we already mentioned, the first corollary of this result is the Dedekind-Mertens formula. Clearly, it is enough to treat the generic case. Also, to be in the situation to use Theorem 5.4 in the case of / = XQ-\-x\t + . . . + xntn and g = yo + yit + . . . + ymtm one needs to set c = n + 1, d = m + 1, xt = Xi+i, yi = Yi+i and u^- = 1. Then the conclusion readily follows from a 'bi-degree' count. Another corollary of the proof of Theorem 5.4 is the complete classification of the rank one Cohen-Macaulay modules over the determinantal rings /f[T]//2(T), where T is a c x d matrix of indeterminates and /2(T) is the ideal generated by its 2 x 2 minors.
6
THE DEDEKIND-MERTENS NUMBER AND ITS GENERALIZATION
The Dedekind-Mertens formula given in (1.2) is universal in the sense that it continues to hold if the polynomial g has coefficients in any ring S containing R as a
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Corso and Glaz
subring. However, easy examples show that the degrees of the polynomials may be too crude a measure of the relationship between the ideals c(fg) and c(f)c(g). For this reason, Heinzer and Huneke were led to define a new yardstick of comparison [21]. The smallest positive integer k such that c(!d}c(9)k-1 = c(f)c(g)c(g)k-1
for every polynomial g € R[t] is called the Dedekind-Mertens number /AR(/) of the polynomial / e R[t]. We include a subscript 'R' in the notation because HR(}) very much depends upon the coefficient ring R. It is not invariant under base change. Indeed, very little is known of the behaviour of this number even under faithfully flat extensions. The Dedekind-Mertens Lemma says that (J,R(/) < deg/ + 1 and Theorem 5.2 shows that for polynomials with indeterminate coefficients over a field one has Hft(f) = deg/ + 1. At the other extreme, the polynomials / with /WK(/) = 1 are precisely the Gaussian polynomials and Conjecture 2.1 states that c(f) is locally principal. Since deg / + 1 is the maximal number of coefficients of the polynomial /, the thrust of [21] was to relate the Dedekind-Mertens number of / to the minimal number of generators of the ideal c(f). The main result of [21] provides a better upper bound on the Dedekind-Mertens number:
THEOREM 6.1 ([21]) Let R be a commutative ring, let f € R[t] be a polynomial, and let c(f) denote the content ideal of f . If locally at each maximal ideal of R c(f) can be generated by k elements, then the Dedekind-Mertens number /i/j(/) is at most k. In the same paper, Heinzer and Huneke also raised the issue of finding a lower bound on the Dedekind-Mertens number. The inequality p-R(f) < nn(c(f)) is universal, that is, it has nothing to do with the assumptions on the ring R. The conditions under which the reverse inequality holds are considerably less clear. For one, the ring should be local, since the definition of /^R(/) localizes while the number of generators of ideals may change greatly under localization. Question 6.2 ([21]) Let (R, m) be an excellent local domain and let f € R[t]. Is Example 2.2 shows that Question 6.2 has a negative answer in general over zero dimensional rings. The main result of [9] provides the following affirmative answer:
THEOREM 6.3 ([9]) Let (R,m) be a universally catenary, analytically unramified Noetherian local ring. Suppose f E R[t] has Dedekind-Mertens number HR(/) = k. Assume that dim(.R/p) > k for all minimal primes p of R. Then /z/{(c(/)) < k. Therefore, HR(c(f)) =
To prove the above result, the authors introduce a variation of the DedekindMertens number: the polarized Dedekind-Mertens number ptfi(/). For a polynomial / € R[t] they define JiR(f) to be the smallest positive integer k such that C
(f9i)c(gi) • • • c(9i) • • • c(gk) = c(f)c(9l)
• • • c(gk)
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141
for all polynomials gi, . . . ,g& € R[t], where c(gi) indicates the deletion of c(gi). It is clear that for every / € R[t] one has /*«(/) < Mfi(/)- It also holds that /!R(/) < Mfl( c (/))- With the same dimensionality restrictions as in Theorem 6.3, Corso,
Heinzer and Huneke were able to show that jSfl(/) = fJ-n(f)- Finally, using some of the techniques and reductions of [21] and, in particular, the fact that the rings under consideration are approximately Gorenstein, the assertion of Theorem 6.3 follows. [9] also provides a number of examples over one dimensional domains which show that additional assumptions on an excellent local domain are necessary to have the formula nn(f} = /z#(c(/)). Here is one: EXAMPLE 6.4 ([9]) Let A; be a field and let R be the Gorenstein subring fcfs 3 ,s 4 ] of the power series ring fc[s]. Consider the polynomial / = s7 + s6t + s8*2 e R[t]. It is shown that /*«(/) = 2 while /XR(C(/)) = 3.
This example is generalized in [9] _tp one dimensional local Noetherian domains (R, m) such that the integral closure R of R is again local and a finitely generated R-module. The latest development on the topic appears in [34], where Rush introduces the universal Dedekind-Mertens number of a polynomial /. Let M be an ^-module and let / 6 M[t]. The universal Dedekind-Mertens number unn(f) of / with respect to R is the smallest positive integer k such that, for each commutative ^-algebra 5 and each g € S[t], it holds that CR(fg)cft(g)k~l = Cfi(f)cR(g)cfi(g)k~1 as submodules of M g}ft S.
He shows that the counterpart of the questions considered in [21]
and [9] becomes much easier in this setting. He also shows that up,n(f) behaves more predictably that A«R(/): for example, if R is a subring of 5, it follows that
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[25] W. Krull, Idealtheorie, Chelsea, 1948.
[26] F. Mertens, Uber Einen Algebaischen Satz, S.-B. Akad. Wiss. Wein. Abtheilung IlalOl (1892), 1560-1566.
[27] J.L. Mott, B. Nashier and M. Zafrullah, Content of polynomials and invertibility, Comm. Algebra 18 (1990), 1569-1583. [28] D.G. Northcott, A generalization of a theorem on the contents of polynomials, Proc. Cambridge Phil. Soc. 55 (1959), 282-288. [29] D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Phil. Soc. 50 (1954), 145-158. [30] D.G. Northcott and D. Rees, A note on reductions of ideals with an application to generalized hilbert functions, Proc. Cambridge Phil. Soc. 50 (1954), 353-359. [31] J. Pahikkala, Some formulae for multiplying and inverting ideals, Ann. U. Turku. Ser AI 183 (1982), 11.
[32] H. Priifer, Untersuchungen Uber Teilbarkeitseigenschaften in Korpern, J. Reine Angew. Math. 168 (1932), 1-36. [33] D. Rush, Content algebras, Can. Math. Bull. 21 (1978), 329-334.
[34] D. Rush, The Dedekind-Mertens lemma and the contents of polynomials, Proc. Amer. Math. Soc. 128 (2000), 2879-2884. [35] H.T. Tang, Gauss' Lemma, Proc. Amer. Math. Soc. 35 (1972), 372-376. [36] H. Tsang, Gauss's Lemma, Ph.D. Thesis, University of Chicago, 1965. [37] A.I. Uzkov, Additional information concerning the content of the product of polynomials, Math. Notes 16 (1974), 825-827 (English translation of Math. Zametki 16, 395-398). [38] W.V. Vasconcelos, Remarks on content of polynomials, unpublished notes (1970).
Depth Formulas for Certain Graded Modules Associated to a Filtration: A Survey
T. CORTADELLAS Departament d'lgebra i Georaetria, Universitat de Barcelona, Gran Via 585, E-08007, Barcelona, Spain < [email protected] > S. ZARZUELA Departament d'lgebra i Geometria, Universitat de Barcelona, Gran Via 585, E-08007, Barcelona, Spain < [email protected] >
1
INTRODUCTION
The study of the arithmetical properties of Rees algebras of ideals has been one of the major topics in Commutative Algebra during the last decade, and a great amount of results have been obtained on the subject. This is in part due to the much
better knowledge we have of the behaviour of the graded local cohomology of these rings, which allows to determine in many cases their properties. A basic source of information to control the graded local cohomology of the Rees algebra of an ideal is the graded local cohomology of its associated graded ring, which is in principle easier to handle. As a consequence, this straight relation between both graded local cohomologies has been extensively used to characterize the arithmetical properties of the Rees algebra of an ideal in terms of properties of its associated graded ring, the ground ring, and the ideal itself.
Concerning the Cohen-Macaulay property of Rees algebras of ideals in local rings, earlier work based on the above ideas goes back, for instance, to P. Schenzel [1] for parameter ideals, S. Goto and Y. Shimoda [2] for the maximal ideal ideal in a Cohen-Macaulay local ring, and S. Ikeda [3] for the maximal ideal in the general case. Also, without using local cohomology methods, C. Huneke [4] proved that, in a Cohen-Macaulay local ring, if the Rees algebra of an ideal is Cohen-Macaulay °Both authors partially supported by DGICYT PB97-0893
145
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Cortadellas and Zarzuela
then it is so its associated graded ring. These efforts somehow culminated when S. Ikeda and N.V. Trung proved the following criteria for the Rees algebra of an ideal be Cohen-Macaulay.
THEOREM 1.1 (see [5], Theorem 1.1) Let (A,m) be a local ring of dimension d and I c A an ideal of A not contained in the nilradical of A. Let RA(!) be the Rees algebra of I and GA (I) its associated graded ring. Denote by M the maximal homogeneous ideal of RA(!}. Then, RA(!) is Cohen-Macaulay if and only if
(HiM(GA(I)}n = Q for ^ —I, i = 0 , . . . ,d — 1, and
[HdM(GA(I)}n = 0 for n>0. In this case, HiM(G(A}) ^ H^(A) for i = 0 , . . . , d - 1.
This result admits generalizations in several directions. By one side, very often one has to deal with filtrations of ideals that are not adic, that is, filtrations that are not given by the powers of an ideal. Typically, this is the case when taking the nitration of the symbolic powers of an ideal, the filtration obtained by considering the integral closures of the powers of an ideal, or the one given by the Ratliff-Rush closures of the powers of an ideal. So, when noetherian, it is interesting to have a criteria for the Cohen-Macaulayness of the Rees algebras of general filtrations of ideals, similar to the case of adic filtrations. This was done, independently, by S. Goto and K. Nishida [6] and by D. Q. Vit [7], who extended to any noeteherian filtration of ideals Ikeda-Trung's characterization. Similarly, T. Cortadellas [8] has also generalized this criteria to the Rees module associated to a noetherian filtration of modules. In this context, one deals with this kind of filtrations when considering, for instance, the canonical module of the Rees algebra of a filtration of ideals, even if this filtration is adic, see [9]. On the other side, one may fit the investigation of the Cohen-Macaulay property Rees algebras in the more general study of their depth. From this point of view, it is natural to ask if there is any relation between the depth of the Rees algebra of an ideal and the depth of its associated graded ring. S. Huckaba and T. Marley [10] carried on such investigation and, mainly inspired by above cited work of S. Ikeda and N.V Trung, proved very nice formulae relating these depths. Their main results may be summarized in the following way.
THEOREM 1.2 (see [10] Theorem 3.10 and Theorem 3.13) Let (A,m) be a local ring and I C A an ideal of A not contained in the nilradical of A. Let RA(!) be the Rees algebra of I and GA (I) its associated graded ring, and denote by M the maximal homogeneous ideal of RA(!)(i) //depthGU(/) < depthA, then depth^(i) = depthG^C-O + 1-
(ii) Let s = depthG/t(/) and assume [HSM(GA(I))]n (/) > depthG^(/) + 1.
= 0 for any n > 0. Then,
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Later on, T. Marley himself [11] proved that these results are also valid when computing the depth with respect to any homogeneous ideal J containing IR^(I), and T. Cortadellas extended them to noetherian nitrations of ideals [12] and noetherian filtrations of modules [8]. The above results also suggest a similar relation between the Serre property (Sn) of the Rees algebra of an ideal and the corresponding property of its associated graded ring. A first result in this sense was shown by S. Noh and W.V. Vasconcelos ([13, Theorem 2.2]) by proving that if A satisfies (Sk+i) and / C A is an ideal such that height / > k + 1, then the Rees algebra RA(!) has (Sk+i) if and only if the associated graded GA(!) has (S&). This result was extended to ideals with height / > 1 by I.M. Aberbach, C. Huneke and N.V. Trung ([14] Theorem 6.8), and then generalised to noetherian filtrations of modules by T. Cortadellas [12]. In this survey article we want to give in a unified manner an account of all the above results. In this way we shall be able to recover as direct applications or just as examples several results dispersed in the literature and proved with very diffrent methods each one. To do this we shall use the general setting of noetherian filtrations of modules and, in trying to be as self contained as possible, we shall give complete proofs of the most basic and important results. On the contrary, and to be as short as possible, we shall include in the preliminaries, without proofs and refering to the original papers, several technical devices needed throughout the paper, mainly those related with the use of finite graded local cohomology.
2
PRELIMINARIES
Let (A, m) be a d-dimensional local ring. A filtration of ideals X = (In)n>o of A is a decreasing chain of ideals A = IQ 3 Ii D •• • D In--- such that /„I m C. In+m for all n, m > 0. The filtration of ideals determined by the powers of an ideal / is called the /-adic filtration. Let T = (In)n>o be a filtration of ideals of A, Then we denote the Rees ring of A with respect to T by RA(%) — © n >o^«* n ^ ^W and the associated graded ring of A with respect to X by GA(!) = ® n >o In/In+iLet E be an >l-module. A filtration E = (En)n>o of (submodules of) E is a decreasing chain of submodules E = EQ D EI 3 • • • D En • • •. If X = (In)n>o is a filtration of ideals of A and E = (En)n>o a filtration of E, we will say that E is Z-compatible if InEm C En+m for all n, m > 0. Then, we can consider the graded modules R(E) = Q)n>0 En (the Rees module of E) and G(E) = 0 n>0 En/En+l (the associated graded"module of E), respectively over RA(!) and GA(!)Let I be a filtration of ideals and E an I-compatible filtration. We will say that T (respectively E) is noetherian if RA(!) (respectively #(E)) is a noetherian ring (respectively a noetherian /Zyi(I)-module). (See, for example, ([6] Part II, Lemma 2.1) and ([15] Theorem 3.8) for more details about the noetherian property of filtrations.) From now on all rings, modules and filtrations will be assumed noetherian. Then, as in the case of filtrations of ideals ([6] Lemma 2.2), one can see that dimG(E) = dim-E, and dim R(E) = dim E + 1 if h £ p for some p € Assh(E) = {p € Min((0 :A E)) | dim A/p = dim E}, dim R(E) = dim E otherwise. On the other hand, we have
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the following exact sequences of graded RA(
0 -» R(E)+ -> R(E) -» J5 -> 0
(2.1) X)
(2.2)
(often called exact sequences of Huneke). Let 5 = 0 n>0 Sn be a graded ring such that SQ is a local ring with maximal ideal m 0 , and denote by 5+ := @n>0Sn the irrelevant ideal of 5. Let M be a finitely generated graded S-module, J C 5 a homogeneous ideal, and Hj(M) the i-th graded local cohomology module of M with respect to J. If N = mo © S+ is the maximal homogeneous ideal of 5 then, for any i = 1,... ,dimM we denote by a»(M) := maxjn HlN(M)n ^ 0} < oo.
(2-3)
(2.4) and taking components of degree n we get the exact sequences -»•••
(2.5) (2.6)
Throughout this paper we will use the above notations and assumptions. The proofs of the results which relate the depths of the modules E, R(E) and G(E) are mainly based in the use of the exact sequences of Huneke (2.1) and (2.2) to compare the annihilation of the graded local cohomology modules of .R(E), G(E)
and E. An important point for this comparison is determining the values of i for which the local cohomology modules Hj(R(E)) and Hj(G(E)) are finitely graded. Recall that if 5 = © n>0 Sn is a positively graded ring and M = ©nez Mn is a graded S-module one says that M is finitely graded if Mn = 0 for all but finitely many n. For a homogeneous ideal J of 5 we put
gj(M) = supjfc € 1 ff}(M) is finitely graded for all i < k}. Then, a key point for the proofs of the results we obtain is the relation between
gj(R(E)) and gj(G(E)). With respect to this, the formula given by T. Marley ([11] Proposition 3.3) for adic nitrations, and later extended in ([12] Proposition 2.4) for nitrations of ideals, may be similarly proved for nitrations of modules to obtain
gj(R(E)) = 9 y(G(E)) + 1 for any homogeneous ideal J containing
(2.7)
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3
149
DEPTH FORMULAS AND THE COHEN-MACAULAY PROPERTY
Let (A, m) be a local ring, I a filtration of ideals of A, E an A-module and E a noetherian Z-compatible filtration of E. Throughout this section we will use these notations and assumptions. As for the case of nitrations of ideals ([12] Proposition 3.1) we obtain the following result.
THEOREM 3.1 Let J be an homogeneous ideal of RA(!) containing #A(Z)+(1). Then: (i) depthjG(E) < depthj#(E).
(ii) depthjG(E) < depthJr}AE, and
z/depthjG(E) < depthJnAE then depthjfl(E) = depthjG(E) + 1. Proof: Set g := depthG(E), r := depth^(E) and s := depths. (i) Since H}(E)n = 0 for all n ^ 0, from the long exact sequence (2.5) we obtain isomorphisms Hj(R(E)+)n ±2 H j ( R ( E ) ) n for all i and n ^ 0. Furthermore, from the long exact sequence (2.6), we have Hlj(R(E)+)n+l ~ H}(fl(E))n for alH < g and n. Therefore HtJ(R(E))n ±: HiJ(R(E))n+i for n ^ -1 and i < g. On the other hand by (2.7), gj(R(E)) = gj(G(E)) + 1, and hence HiJ(R(E)) = 0 for alii < g which implies depthG(E) < depth-R(E). (ii) Assume the contrary. Then, s < g and so applying (i) we have that r > g > s. Hence, using the long exact sequence (2.5) we obtain injections Hj(E) <-> Hj+1(R(E)+)0 and in particular, that Hj+1(R(E)+)0 ^ 0. Furthermore, from (2.6) and the fact Hj(G(E)) = 0, we have H'j+l(R(E)+)n <-+ HSj+1 (fi(E)) n _i for all n, which implies Hj+l(R(E})-i ^ 0. Moreover, using the isomorphisms Hsj+l(R(E}+}n - Haj+l(R(E)}n (n + 0) we obtain Haj+1(R(E))n ^ Hsj+1R(E)n^ for all n ^ 0 and so HSj+i(R(E)) is not finitely graded since HSj+l(R(E)}-i ^ 0. This implies g j ( R ( E ) ) <s + l< g + l< gj(G(E)) + 1, which contradicts (2.7). It remains to show that r — g + 1 if g < s. In this case one has from (2.5) isomorphisms H$(R(E)+)n ^2 H}(R(E))n for all i < g, and from (2.6) injections
Hlj(R(E)+)n+i <-> HtJ(R(E))n for all n and i < g. Therefore, ff}(fi(E)) n+ i ^> /?}(/?(£))„ for all i < g. Thus, H*j(R(E)) = 0 for i < g since from (2.7) it follows that gj(R(E)) - gj(G(E)) + 1 > g + 1 and so r > g + 1. On the other hand, by (2.3) and (2.4) respectively we obtain F}nA(£) ~ HiJ+l(R(E)+) and #}(G(E)) - HiJ+1(R(E)+(l)} for all i < r-2, which implies H*JnA(E} - F}(G(E)) for i < r - 2. Thus, if r > g + 2 it follows that Hj(G(E)) ±: H9JnA(E) ^ 0. Hence, g — s which contradicts our assumption. D
For J the maximal homogeneous ideal we have as in ([8] Corollary 3.2):
COROLLARY 3.2
(i) depthG(E) < depthtf(E).
(ii) depthG(E) < depthE, and
i/depthG(E) < depth£ then depthjR(E) = depthG(E) + 1.
(Hi) « / a d e t h c E ( G ( E ) ) < °>
then
depth/?(E) > depthG(E) + 1.
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Cortadellas and Zarzuela
Proof: According to Theorem 3.1 it suffices to prove part (iii). Write g = depthG(E) and M. the maximal homogeneous ideal of RA(%)- Since depth -R(E) > depthG(E) it remains to show that H3M(R(E)) - 0. From (2.5) and (2.6) for J = M we obtain injections H9M(R(E)+)n+i
<-4 H^/t(R(E))n
for all n ^ -I. On the other
hand 9M(R(E)) = gM(G(E)) + I > 9 + 1 by (2.7). Thus, H3M(R(E)) is finitely graded. From this fact and the above injections we obtain H_M(R(E))n = 0 for n < — 1. Moreover, for n > 0 the assumption HaM(G(E'))n = 0 implies isomorphisms H9M(R(E))n+l ~ H'M(R(E))n and so H3M(R(E))n = 0 f o r n > 0 since = Oforn»0. D Our next result extends the criteria of Ikeda and Trung (Theorem l.l)for the Cohen-Macaulayness of the Rees algebra in terms of the graded local cohomology modules to the associated graded ring to the general case of nitrations of modules. THEOREM 3.3 Assume that dim R(E) = dim£+l. Then, the following conditions are equivalent:
(i) R(E) is Cohen-Macaulay.
(ii) H*M(G(E))n =Qifn?-l,i = Q,...,timE-land H%mB(G(E))n = 0 if n > 0. In this case, HtM(G(E)) ~ H*m(E) for i = 0,..., dimE -1. Proof: (i) => (ii} Assume that H1M(R(E))
= 0 for i < dimE. Using the exact
sequence (2.5) one obtains that HtM(R(E)+)n = 0 f o r n / 0 and HiM(R(E)+)Q H^~l(E). Since on the other hand, the exact sequence (2.6) gives isomorphisms
H'M(G(E))n - H%l(R(E)+)n+l for i < dimE - 1, we have H*M(G(E))n = 0 for
n ± -1 and i < dimE - 1. It remains to show that ^ m£ (G(E)) n = 0 forn > 0. In this case we have again for the exact sequences (2.5) and (2.6) 0 —> HEG(En — > Hdm+lREnl — > H™+lREn — > 0 for all n ^ -1. Furthermore, H'^nE+l(R(E))n = 0 for n » 0 and so the epimorm£+1 (#(E)) n+1 -»• H^mE+1(R(E))n,n ? - 1, implies that #£,mB P hism//^ 0 for n > 0 which proves the assertion. (ii) => (i) We want to see that H*M(R(E)) = 0 for all i < dimE. From (2.5) and (2.6) we get isomorphisms ff^(.R(E)) n+ i ~ HlM(R(E))n for all n ^ -1 and i = 0 , . . . , dimE - 1, isomorphisms H$nE(R(E))n+l - H^nE(R(E))n for n > 0, and monomorphisms H<$"E(R(E))n+i ^ H$"E(R(E))n for all n ^ -I. On the other hand, by (2.7) gM(R(E)) = gM(G(E)) + 1 > dimE+1; that is, the modules H1M(R(E)) are finitely graded for all i < dimE. Together with the above monomorphisms and isomorphisms this proves that ^^(^(E)) = 0 for all i < dim E. D In particular, if we assume E Cohen-Macaulay we have the following simple formulation:
COROLLARY 3.4 Assume that dim R(E) = dimE 4- 1 and E is Cohen-Macaulay. Then: R(E)
is Cohen-Macaulay & G(E) is Cohen-Macaulay and o(G(E)) < 0.
Depth Formulas for Certain Graded Modules
4
151
APPLICATIONS AND EXAMPLES
In this section we are going to describe some consequences and applications of the above theorems. We start with the case of a regular local ring.
COROLLARY 4.1 Let A be a regular local ring of dimension d and X a noetherian filtration of A with ht/i ^ 0. Then, depthjR(J) = depthG(l) + 1. Proof: If G(I) is not Cohen-Macaulay the assertion follows from Corollary 3.2(ii). Otherwise, we use the known fact that in a regular local ring R(X) is CohenMacaulay if and only if G(T) does (see [17] Theorem 5). D EXAMPLE 4.2 Let A be a regular local ring and p a prime ideal of A. Consider
the symbolic Rees algebra and associated graded ring. If we assume Rs(p) is noetherian, then depth Rs(f>) = depth G«(p) + 1. We may also consider the case of a positively graded algebra.
COROLLARY 4.3 Let S — ® n>0 Sn be a noetherian graded ring with SQ a local ring. Consider the filtration In := 0 m > n >S TO - If o,(S) < 0 then depth R(X) >
depth S + 1 . In particular, if S is Cohen-Macaulay then R(X)
is Cohen-Macaulay.
Proof: Applying Corollary 3.2(iii) to the filtration IM of S_M, where M is the maximal homogeneous ideal of S and taking into account that Gs(X) = 5. D
If S in the above result is a Rees algebra we recover the following result of Goto and Nishida ([6] Part II, Corollary 6.4).
EXAMPLE 4.4 Let A be a local ring and I a noetherian filtration of A with ht /: >
0. Consider the filtration of R(X) defined by X'n = ©ro>n R(X)m = © m > n Im. If R(X) is Cohen-Macaulay, then R(X') is also CM since d(R(F)) = -1 < 0 (see [6] Part II, Lemma 3. 3). Let (A, m) be a local ring, X a filtration of ideals of A, E an A-module, and E an I-compatible filtration of E. For / 6 Z we will denote by *E the filtration of the module EI defined by ; E n = En+[, where En+i := E if n + I < 0. Then, 'E is a an X -compatible filtration of EI. Put a(E) := max{n E = E0 = EI — • • • — En}. Observe that G( ; E) = G(E)(l). Thus, if / < 0 we have that G('E) and G(E) are isomorphic as /?(I)-modules. This allows to prove the following results. PROPOSITION 4.5 Let r>Q. Then, we have the following:
(i) depthG(E) < depthR(~rE). (ii) depthG(E) < depth£ ^> depth.R(-rE) = depthG(E) + 1.
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Cortadellas and Zarzuela
(Hi) r < -adepthG(E)(G(E)) =* depth/?(~rE) > depthG(E) 4- 1. (iv)
Assume that dimfi(E) = dimE + 1 and E is Cohen-Macaulay. Then:
R(~rE) is Cohen-Macaulay <£• G(E)
is Cohen-Macaulay and r < -a(G(E)).
Proof: Since G(~ r E) = G(E)(-r) it follows that depthG(~ r E) = depthG(E) and aj(G(~ r E)) = aj(G(E)) + r. Parts (i), (ii) and (iii) are obtained from Corollary 3.2(iii) whereas (iv) follows from Corollary 3.4. D Let / be an ideal of A. Then, the fractionary ideal (1, t)r of R(I) coincides with R(-TI) = A 0 At @ • • • Itr+l 0 / 2 r+ 2 0 • • •. Often we will denote (1, t)r instead R(~rZ). Prom the above proposition one has in this case: PROPOSITION 4.6 Let r > 0. Then: (i) depthG,i(/) < depth(l,£) r .
(ii) depthG^/) < depthA =>• depth(l,i) r = depthG^(J) + I. (iii) r < -adepthGA({)(GA(I))
=» depth(M) r > depthG^/) + 1.
(iv) Assume that dimRA(!) = d+ I and A is Cohen-Macaulay. Then: (l,i) r is Cohen-Macaulay <$ GA(!) is Cohen-Macaulay andr < —O,(GA(I})-
As a consequence we may recover the following results obtained by Morey, Noh, Vasconcelos [18] and Herzog, Simis, Vasconcelos [19], by using very different techniques:
COROLLARY 4.7 ([18] Proposition 1) Let A be a Cohen-Macaulay local ring and I an ideal of positive height. 7/grade(G/i(/)+) > 0 andGA(I) is not Cohen-Macaulay, then:
+ 1, ifr = 1, r depth(l,i) > i depthG , , , _ A,(/) K v r , ., ' - ' depthG / j(/) ifr Proof: By Proposition 4.6, for all r > 0 we have that depth(l, t)r - depthG(J) + 1.
(The assumptions ht (/) > 0 and grade(G(/)+) > 0 are not necessary.) D
COROLLARY 4.8 ([19] Theorem 2.6) Let A be a Cohen-Macaulay local ring and I an ideal with grade(J) > 2. Assume RA(!) Cohen-Macaulay. Then, (l,t)r is Cohen-Macaulay if and only if r < —a(GA(I)) ~ 1Proof: It follows from Corollary 3.4 and Proposition 4.6(iv). D We will recall now the definition and some basic properties of canonical modules for graded rings. For details see [20]. Let S = ® n > 0 5 n be a graded ring over a local ring (So, trig). Put d = dim S. If SQ is complete, the canonical module of S is denned by KS = Horns(H^(S), E), where N is the maximal homogeneous ideal of S and E is the injective envelope (in the category of graded S-modules) of the residue field S/N. If S0 is not complete, a module S is said to be a canonical module of S
Depth Formulas for Certain Graded Modules
153
if KS ®s0 S0 i= Ks»s So' wnere So is the mo-completion of SQ. If 5 has a canonical module, it is a finitely generated graded 5 -module and unique up to isomorphisms. Moreover, by graded local duality a(5) = — min{n\ [Ks]n ^ 0}. Assume that A has a canonical module KA and let Z be a noetherian filtration of A with dim R(l) = d + I. Put KR(T) and KG(x) respectively the canonical modules of R(T) and G(I). Trung, Viet and Zarzuela [9] proved that if R(T) is Cohen-Macaulay, then there exists a filtration K = (Kn)n>o of KA which is Icompatible and noetherian such that KK{I) — © n >i Kn and KG(i) — Kn-i/Kn. Thus, if KQ = K I , KR(I) and ^Qj(Z) can be considered as the Rees module and the associated graded module of a filtration of K\ . Then, we may apply the previous results to obtain information about the depths of KG(I) and KR(Z)-
PROPOSITION 4.9 Let (A,m) be a d-dimensional local ring and I - (/ n ) n >o a
noetherian filtration of ideals of A with dim RA(%) = d+1. Assume that RA(%) is Cohen-Macaulay aru/ a(GU(I)) < ~2. Then:
depth KGA(x) = depth/Oi.
Proof: Let K be the filtration of KA for which KR(I) ^ R(lK)(-l) and KG(X) ^ G(K)(-1). By assumption a(G(I}) < 2 and so o(K) > 1. Therefore KG(X) ~ G( 1 K)(— 2). The assertion is then a consequence of Corollary 3.2 and the Cohen-Macaulayness of /?( 1 K). 0 Goto [21] studied the graded local cohomology modules for the associated graded ring G(m) when (A, m) is a Buchsbaum ring of maximal embedding dimension. In this case, we may then obtain the following corollary.
COROLLARY 4.10 Let (A,m) be a Buchsbaum ring of dimension 3 and maximal embedding dimension. Assume depthyl = 2. Then, depthKGA^m) = depth/d. Proof: By ([21] Proposition (1.1)) one has: a(G(m)) < 2, depth(?(m) = depthA = 2, F^(G(m)) n = Hl(A) for n = -1 and #* (C?(m)) n = 0 for all n ^ -1. Then, one uses Theorem 3.3 and Proposition 4.9. D EXAMPLE 4.11 Let d > 3 be an integer. In [22] Ikeda constructs a Buchsbaum ring (A,m) of dimension d and depth A = 2 such that -R/t(m) is a CohenMacaulay ring. Moreover, the reduction number of m is less or equal than 1, so A has maximal embedding dimension. Then, applying ([21] Theorem (1.1)) we have that a(G*,4(m)) < 1 - d < -2. Thus, for these rings one has that m) = depth/d. 5
SERRE's CONDITION
The techniques and results in Section 3 allow also to study the Serre properties for the blow up modules.
THEOREM 5.1 Let (A, m) be a local ring, I a noetherian filtration of A, E an A-module and E a filtration of E I-compatible. Assume that R(E) is a finitely generated R(X)-module and dim/?(E) = dimE + 1. Then, if E fulfils Sk+i the following conditions are equivalent:
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(i) R(E) fulfils 5,fe+i(ii) G(E) fulfils Sk and R(Ef) dimEp < k.
is Cohen-Macaulay, for all prime ideal p D I\ with
Proof: Write R = R(I), G = G(Z) and I = R(I)+(l). (i) => (ii) Assume that R(E) fulfils Sk+i and let Q be a prime ideal of G. We want to show that depthG(E)<2 > min{dimG(E)Q,fc}. First notice that it suffices to consider homogeneous prime ideals. Let P be an homogeneous prime ideal of R such that Q = P/I and p = P n A. Localizing at p we may suppose p = m since depthG(E)g = depthG(Z j G(Ep)g. If P ^ R+ we have depthG(E) Q = deptLR(E)P - 1 > min{dim R(E)P, k +1} - 1 = min{dim R(E)P l,k} min{dimG(E)g,A;}, since R(E) satisfies Sk+i and G(E)g ~ R(E)P/xR(E)P for some element x regular in R(E)P. Assume now that P D R+. Then P = M, the maximal homogeneous ideal of R. With this assumption, by Corollary 3.2,
depthG(E) < depthE and if depthG(E) < depthE then depthG(E) = deptLR(E) 1; therefore, in this case, depthG(E) = depthJ?(E)-l > min{dim.R(E),A;-t-l}-l = min{dimG(E),fc} since R(E) satisfies Sk+i and dimfl(E) = dimG(E) + 1. Next suppose that depthG(E) = depthE. Then, depthG(E) = depthE > min{dimE,fc+ 1} = min{dimG(E),fc + 1} > min{dimG(E),A;}, since E fulfils Sk+i. Let now p be a prime ideal of A with p D /i and dim Ep < k. We need to show that R(IP) is a Cohen-Macaulay ring. Considering P = (p,R+) € Spec./? it is suffices to prove that R(E)P is Cohen-Macaulay, since this is the localization of R(Xf) in the maximal homogeneous ideal of R ( I f ) . Since R(E) satisfies Sk+i one has depth J R(E)p > min{dimR(E) P ,k + I } , and the assertion will be clear if dim.R(E)p < k + 1. But dimfl(E) P
Depth Formulas for Certain Graded Modules
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(i) R(I) satisfies Sk+i • (ii) G(I) satisfies Sk and 7?(7p) is Cohen-Macaulay, for all prime ideal p 2 I with
ht (p) < k. Also, if A is regular we may formulate the following:
COROLLARY 5.3 Let A be a regular local ring and I a noetherian filtration with ht/i > 0. Then, R(I) satisfies Sk+i if and only if G(I) satisfies SkProof: By Theorem 5.1 it suffices to show that R(If) is Cohen-Macaulay for all p D 7j with htp < k in case that G(X) fulfils Sk- Suppose that G(T) satisfies Sk and let p be a prime ideal of A with p D /i and ht p < k. Av is a regular local ring and then, by Corollary 4.1, R(Xf) is Cohen-Macaulay if and only if G(ZP) is Cohen-Macaulay. Moreover, G(X P ) is Cohen-Macaulay if and only if G(Z)>p is Cohen-Macaulay for P = (p, G(X)+). Thus, it suffices to show that the property Sk for G(I) implies the Cohen-Macaulayness of G(X) P . But depthG(I)F > min{htP,fc} = htP and the assertion is clear. D We may determine the values of r for which (l,t) r satisfies the property SkNamely, we have the following.
PROPOSITION 5.4 Let A be a local ring and I an ideal with ht (/) > 0. Assume that A and RA(!) fulfils Serre's condition Sk+i- Then, for r > 0 (1, t)r fulfils Sk+i <£> r < min{-a(G(/ p )), V prime p D / uritA ht (p) < k}. Proof: Write R(~rl) = (l,i) r . Then, the assertion follows from Theorem 5.1 and a similar argument as in the proof of the above Corollary. D Let 7 be an ideal of A. Following Herzog, Simis and Vasconcelos [19] we say that the canonical module KR^ of R(I) has the expected form if KR(I) — At ® • • • © Atr ® Itr+l ® • • • for some r > 0. The above results allows us to recover the following result of Herrmann, Hyry and Korb [23]. COROLLARY 5.5 ([23] Lemma 5.1) Let A be a local ring and let I C A be an ideal with ht (I) > 0 such that the canonical module -Kfl(j) has the expected form. Let k be an integer such that A and RA(!) fulfil Sk- Then KRA(J) fulfils Sk, too.
Proof: Write KR(J) ^ fl(-( r - 1 >/)(-l) where r = -(o(G(/)) + 1). According to Proposition 5.4, it is enough to prove that -a(G(/)) - 2 < -a(G(/p) for all p D 7 with ht(p) < fc. But K R ( I f ) ±: (^fl(/)) P (see [6, (2.10)]) and so we have that ) < a(G(J)). D
References [1] P. Schenzel. Standard system of parameters and their blow-up rings, J. Reine Angew. Math., 344: 201-220, 1983. [2] S. Goto and Y. Shimoda. On the Rees algebras of Cohen-Macaulay local rings, Commutative algebra. In: Lecture Notes in Pure and Appl. Math. New York: Dekker, 1982, pp 201-231.
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[3] S. Ikeda. The Cohen-Macaulayness of the Rees algebras of local rings, Nagoya Math. J., 89: 47-63, 1983. [4] C. Huneke. On the associated graded ring of an ideal, Illinois J. Math., 26: 121-137, 1982. [5] N. V. Trung and S. Ikeda. When is the Rees algebra Cohen-Macaulay?, Comm. Algebra., 17(12):2893-2922, 1989.
[6] S. Goto and K. Nishida. The Cohen-Macaulay and Gorenstein Rees algebras associated to nitrations.: Memoirs of the Amer. Math. Soc. 526, 1994. [7] D. Q. Viet. A note on the Cohen Macaulayness of Rees algebras of nitrations, Comm. Algebra 21(1): 221-229, 1993.
[8] T. Cortadellas. On the Depth of the Canonical Modules of Graded Rings Associated to an Ideal, Comm. Algebra 28(2):601-612, 2000. [9] N. V. Trung, D. Q. Viet and S. Zarzuela. When is the Rees algebra Gorenstein?, J. Algebra 175(1), 137-156, 1995.
[10]
S. Huckaba and T. Marley. Depth formulas for certain graded rings associated to an ideal, Nagoya. Math. J. 133: 57-69, 1994.
[11]
T. Marley. Finitely graded local cohomology and the depths of graded algebras, Proc. Amer. Math. Soc 123: 3601-3607, 1995.
[12]
T. Cortadellas. Depth Formulas for the Rees Algebras of Filtrations, Comm. Algebra 24(2): 705-715, 1996.
[13]
S. Noh and W. Vasconcelos. The 82 Closure of a Rees Algebra, Results in Math. 23: 149-162, 1993.
[14]
I. M. Aberbach, C. Huneke and N. V. Trung. Reduction numbers, BrianonSkoda theorems and the depth of Rees rings, Compositio Math. 97(3): 403-434, 1995.
[15]
W. Bishop, J. W. Petro, L. J. Ratliff, Jr and D. E. Rush. Note on noetherian nitrations, Comm. Algebra 17(2): 471-485, 1989.
[16]
M. P. Brodmann, R. Y. Sharp. Local Cohomology: An algebraic introduction with geometric application. Cambridge studies in advanced mathematics, 60. Cambridge University Press, Cambridge, (1998).
[17]
J. Lipman. Cohen-Macaulayness in graded Algebras, Mathematical Research Letters 1: 149-157, 1994.
[18]
S. Morey, S. Noh and W. V. Vasconcelos. Symbolic powers, Serre conditions and Cohen-Macaulay Rees algebras, Manuscripta math. 36: 113-124, 1995.
[19]
J. Herzog, A. Simis and W. V. Vasconcelos. On the canonical module of the Rees algebra and the associated graded ring of an ideal, J. Algebra 105: 285302, 1987.
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[20] M. Herrmann, S. Ikeda and U. Orbanz, Equimultiplicity and Blowing Up, Springer-Verlag, Berlin-Heidelberg, (1988).
[21] S. Goto. Buchsbaum rings of maximal embedding dimension, J. Algebra 76: 383-399, 1982. [22] S. Ikeda, The Cohen-Macaulayness of the Rees algebras of local rings, Nagoya Math. J. 89: 47-63, 1983. [23] M. Herrmann, E. Hyry and T. Korb. On Rees algebras with a Gorenstein Veronese subring, J. Algebra 200(1): 279-311, 1988.
Extremal Betti numbers of lexsegment ideals MARILENA CRUPI, Dipartimento di Matematica, Universita di Messina, Contrada Papardo, salita Sperone 31, 98166 Messina (Italy); e-mail: [email protected] ROSANNA UTANO, Dipartimento di Matematica, Universita di Messina, Contrada Papardo, salita Sperone 31, 98166 Messina (Italy); e-mail: [email protected]
ABSTRACT: The behaviour of extremal Betti numbers of a lexsegment ideal is
here studied. It is shown that a lexsegment ideal can have at most two extremal Betti numbers. Moreover, the possible sequences of extremal Betti numbers for a lexsegment ideal are characterized.
I
INTRODUCTION
Let R = K[XI, . . . , xn] be the polynomial ring in n variables over a field K. Let / be a graded ideal of R and let
F.
0 —> Fr —* . . . —>• Fo — » / —> 0
be a minimal graded free resolution of / over R, where Ft = (SjR^j)13'' • The numbers fa are called the graded Bettinumbers of 7. One has /?,-j = dim# Tor/*(/
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free resolution in the form of the following picture
where the outside corners of the dashed line give the positions of the extremal Betti numbers. Let / be a lexsegment ideal of R. We want to determine its extremal Betti numbers. In Section 1 we discuss extremal Betti numbers for stable monomial ideals. In Section 2 we prove that any lexsegment ideal can have at most two extremal Betti numbers and we determine their possible values. Moreover, given two pairs (k\,ii) and (^2,^2) of positive integers, we determine conditions for the existence of a lexsegment ideal / with the extremal Betti numbers (3klkl+(.1 and We like to thank Jiirgen Herzog for some useful comments.
2
EXTREMAL BETTI NUMBERS OF STABLE IDEALS
Let K be a field and R — K[x\,..., xn] be the polynomial ring in n variables over K. The support of a monomial u of R is supp (u) = {i : a;,- divides u}. Let m(u) denote the maximal integer belonging to supp ( u ) . For every monomial ideal I C R, we denote by G(I) the unique minimal set of monomial generators of I. Recall that a monomial ideal I of R is strongly stable if:
for all u 6 G(I) one has (xju)/Xi
6 / for all i € supp (u) and all j < i.
A stable ideal is defined by the following combinatorial property:
for all u £ G(I) one has ( x j u ) / x m ( u ) G / for all j < m(u). A set £ of monomials is called a lexsegment if for all u 6 £ and all v >ie:r u with deg v = deg u, it follows that v 6 £ where >iex denotes the lexicographic order ([!]). Then a monomial ideal / C R is called a lexsegment ideal if all homogeneous components of / are spanned by lexsegments. Every lexsegment ideal of R is obviously a strongly stable ideal and each strongly stable ideal is stable.
LEMMA 2.1 Suppose I C R is stable. If Pn+j ^ 0, then /3kk+j ^ 0 for k = 0 , . . . , i.
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161
Proof. The graded Betti numbers of a stable ideal / are given by Eliahou and Kervaire ([2]): V—^
where G(/)j = {u £ G(7) : deg u - j}. It follows that /?i,- + j(/) ^ 0 <=» there exists u 6 G(I)j
with m(w) > i + 1.
(*)
Thus if Pn+j(I) ^ 0 then there exists « € G(/)j with m(w) > z + 1. Therefore m(w) > k + I for k = 0, . . . , i and hence Pkk+j ^ 0. The lemma says that all linear strands of a stable ideal begin in homological degree 0. COROLLARY 2.2 Let I C R be stable. The following conditions are equivalent: (a.) fikk+e.
(b)
zs
extremal.
(1) pkk+t ^ 0 (2) /3kk+j = 0forj>l (3) Pn+l = 0 for i > k.
PROPOSITION 2.3 Let I C R be a stable ideal. equivalent:
The following conditions are
(a) Pkk+i is extremal. (b) k + l= max{ra(u) : u 6 G(I)t} and m(u) < k for all j > i and all u £ G(I)j . Proof, (a) => (b): (*) and Corollary 2.2 (b), (1) imply max{ra(u) : u £ G(I}t} >
k +1. Suppose i + 1 := max{m(u) : u £ G(I)t] > k + 1. Then by (*), /3a+i ^ 0 for i > k, contradicting Corollary 2.2, (b), (3). So we have max{m(u) : u 6 G(I)t} = k + 1. Suppose that m(u) > k + I for some u 6 G(I)j
with j > (.. Then by (*),
fikk+j 7^ 0, contradicting Corollary 2.2 (b), (2). (b) => (a): Since k + 1 = max {m(u} : u G G ( I ) t ] , fak+t ^ 0 and /?,-,•+/ = 0 for all i > k. Moreover since m(w) < k for all j > t and for all u £ G ( I ) j , we have that /?fcfc+ j = 0. Hence, by Corollary 2.2, /3kk+i is an extremal Betti number. From Proposition 2.3, we have the following
COROLLARY 2.4 Let I be a stable monomial ideal and /3kk+i an extremal Betti number of I . Then (3kk+e = \ { u & G ( I ) e : m ( u ) = * + l } | COROLLARY 2.5 Let I be a stable monomial ideal, let t = max{j : G(I)j ^ 0} and m = max{m(u) : u G G(I}}. Then I has exactly one extremal Betti number if
and only if m = max{m(u) : u G G ( I } i } .
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Proof. The condition is equivalent to the fact that no linear strand is longer than the highest.
REMARK 2.6 Let / be a stable monomial ideal and hk+i an extremal Betti number of /. Then 1 < Pkk+t < CJl^1), because there are exactly (k^^1) monomials of degree t with m(w) = k + 1.
3
EXTREMAL BETTI NUMBERS OF LEXSEGMENT IDEALS
In this section we determine the extremal Betti numbers of lexsegment ideals. The following lemma will be crucial in the proof of our main result.
LEMMA 3.1 Let u, v G K \ X I , ..., xn] be two monomials in a lexsegment £ in degree t such that m(u) = m(v) < n and u > v. Then there exists a monomial w G £ such that: (i) u > w > v;
(ii) m(w) = m(u) + 1. Proof, u and v can be written as follows:
u = Xi^i,, • • • xit,
1 < ii < j'2 . . . < it < n,
Since u > v there exists k < t — 1 such that z
'l = jl i »2 = jl, • • • , ik = jk ,ik + l < jk + l •
Hence we may choose
In fact w > v since i/,+1 < jk+i, and u > w since it < it + I .
THEOREM 3.2 Let I c R = K [ x l } ...,xn] be a lexsegment ideal. Then I has at most two extremal Betti numbers.
Proof. Let ki — max{m(u) - 1 : u £ G(I)} and ti — max{j : there exists
u G G(I)j
with
m(u) = ki + I } .
Extremal Betti numbers of lexsegment ideals
163
By Proposition 2.3 fiklki+tl is an extremal Betti number of / and & l f c ,+/i = \iu € G(7k
:m u
( ) = *i + Ill-
Let G(I}>il = {u £ G(7) : degw > £1). If G(I)>il = 0, then ftklkl+tl is the only extremal Betti number. Suppose G(I)>tl ^ 0. Let ki + I = max{m(u) : u € G(7)>/,}. It is 1 < ki < ATI. Let £2 > i\ be the maximal degree such that there exists u with m(u) — &2 + 1. Then fik?k2+t? is an extremal Betti number. By Lemma 3.1 u is the unique monomial in G(I)t2 with the property m(w) = A:2 + 1. Therefore Pk2k2+ti = 1- This monomial w is the smallest monomial of the lexsegment £ which spans 7{2. In fact, let u £ G(7)^ 2 be the unique element with m(u) = A;2 + 1, u = Xj-j • • •a;i, 2 _iX,- < , zj < . . . < i^, i^ = A:2 + 1 < n. Suppose u is not the smallest element in £. Then w = a;,-, • • -Xi( _iXit^+i £ £. For a lexsegment ideal we have that if u e G(7)^ 2 , w < u, w £ 7^2 then u> £ G(I)t2. This is a contradiction, since m(w) > ?TI(U). We claim that G(7)>^ 2 = 0. Suppose that G(7)^ 3 ^ 0 for some £3 > £ 2 . Let £3 be the smallest such number and let v £ G(7)£ 3 . Then v < ux^3~il = KJ, • • -Xit x^3~il. The next element in the lexicographic order following ux^~tl is w — Xil • • • a;,- <2 _ia;^~^j +1 . Since 7 is a lexsegment ideal it follows that w £ 7. But m(w) > & 2 + 1, contradiction. EXAMPLE 3.3 Let / C k[xi,...,xn] be the ideal / = , , 0:3). The Betti diagram of / is
2
3 4 5
ii
25
26
16
6
1
6 2 2 1
15
20 1 4 1
15 1 —
6 -
1 — —
3
5 2
—
J has 3 extremal Betti numbers, however J is not a lexsegment ideal, but only strongly stable.
PROPOSITION 3.4 Given two pairs of positive integers (ki,t\), (A: 2 ,£ 2 ) such that 1 < A;2 < k\ and 2 < li < £ 2 , then there exists a lexsegment ideal I C k[xi,.. .xn] with extremal Betti numbers /3i,1itl+tl and fik^kv+t? if and only if ki + 1 = n. Proof. Suppose @klkl+t>i is an extremal Betti numbers and ki + I < n. Let u — X{1 .. . X i t G ^i with i^, = ki + 1. Then u is the lexicographic smallest monomial in /£,, because otherwise v = Xil ... X{( - \ X j t -|_i 6 Iil with m(v) > k\-\-\, a contradiction. Since /?^ 2 ^ 2+ £ 2 is an extremal Betti number of / then G(/)^ 2 ^ 0 and A;2 + 1 = max{m(u) : u € G(/)>^}. Then 7<2 \ Rt^tjtl ^ 0. The lexicographic smallest monomial in Re2-[1Iz1 is ux^~il. Since 7^2 is spanned by a lexsegment it follows that the next monomial v — Xil • • •£; / i _ 1 x^ ! ~^ I + 1 following ux^>~il in
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the lexicographic order belongs to 0(1)^, a contradiction since m(v) = itl -f 1 = &! +2 > fci + 1 > A 2 + 1. Conversely, if ki + I = n we construct a lexsegment ideal / having generators in degrees i\ and £2- Choose all monomials in the lexicographic order from x^ up to the largest monomial u having m(u) — n as monomial generators in degree t\ . Let
The smallest element in degree £2 that belongs to R^-^I^ is x{lx^ /1+1. The next element in the lexicographic order is it) = xllx2*~ 1+ . Hence we choose _ — l l
2
i
1*2
3i • • • i - !
2
COROLLARY 3.5 // a graded ideal I C k[x\, . . .,xn] has more than two extremal Betti numbers, then the generic initial ideal, Gin(I), cannot be a lexsegment ideal. Proof. By [3], Corollary 1, or [4], Theorem 2.3, the extremal Betti numbers of an ideal /, as well as their positions are preserved passing to the generic initial ideal, then by Theorem 3.2 Gin(I) cannot be a lexsegment ideal.
PROPOSITION 3.6 Let I C K[xi,...,xn] be a lexsegment ideal. Let ti be the smallest integer such that Pkiki+ti 2S an extremal Betti number of I for some fci. // either (1) ATI + 1 < n or
(2) k\ -\- I = n and the lexicographic smallest monomial in Ii1 is of the form u = xjl • • -Xjt^k-.ixn-ixkn, k > 0
then fik-iki+ti is the only extremal Betti number of I. Proof. If /EI + 1 < n, by using the same arguments of the previous proposition, Pkiki+ti is the only extremal Betti number of /. Now let k\ + 1 = n and let u — Xjt • • • Xjti-k-iXn-iXn the smallest monomial in Itl. Suppose G(I)t3 ^ 0 for £2 > ^i- Hence '/^ \ R^-iJt, ^ 0. The lexicographic smallest monomial in Ri2-il It, isux^~il — Xj1 • • • Zj / i _ t _ 1 x n _ia;*" t "^ 2 ~ f l . The next element in the lexicographic order following ux^~il isv = Xj1 - - - a ; ^ _t-i ; E n + ^ 2 ~^ 1 + 1 Since / is a lexsegment ideal it follows that v £ G(/)^ 2 , a contradiction since m(v] = n.
REFERENCES 1. W. Bruns, J. Herzog. Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Rev. Ed., Cambridge 1996. 2. S. Eliahou, M. Kervaire. Minimal resolutions of some monomial ideals, J.
Alg., 129: 1-25, 1990. 3. D. Bayer, H. Charalambous, S. Popescu. Extremal Betti numbers and Applications to Monomial Ideals, J. Alg., 221: 497-512, 1999. 4. A. Aramova, J. Herzog. Almost regular sequences and Betti numbers, Preprint 1998. 5. D. Eisenbud. Commutative algebra, with a view towards algebraic geometry, Graduate Texts Math., 150, Springer 1995.
Local Monomialization STEVEN DALE CUTKOSKY 1 , Department of Mathematics, University of Missouri - Columbia, MO 65211, USA, [email protected].
1
INTRODUCTION
Consider a local homomorphism
RCS
(1.1)
of local rings essentially of finite type over a field k. The structure of such an extension is extremely complicated, even when R and S are regular. The two simplest examples of an extension (1.1) are:
Monomial Mappings. R -> S is a monomial mapping if R has regular parameters (j/i, . . . , y m ), an etale extension of S has regular parameters (xi, ... ,xn) and there is a matrix (ay ) of natural numbers such that
m
—
Urn —
Monoidal Transforms. Suppose that R, S have a common quotient field. 5 is a monoidal transform of R if there exists a regular prime P in R, 0 7^ y 6 P and
where m is a prime ideal that contracts to the maximal ideal of R. If P is the maximal ideal of R, S is called a quadratic transform. 1
Partially supported by NSF
165
166
Cutkosky
If R -4 5 is a monoidal transform and the residue fields of R and S are isomorphic to a subfield k of R, then R has regular parameters (3/1, ... ,2/n), S has regular parameters (xi, ... , x n ), and there exists r
J/r =
X\XT
yr+i = xr+i
yn =
%n
Suppose that V is a valuation ring of the quotient field K of S, such that
V dominates S. Then we can ask if there are sequences of monoidal transforms R -> R' and S -> 5' such that V dominates 5', 5' dominates R, and #' -> 5' has an especially good structure.
2
K
->
S" C V
t fl
->
t S
(1-2)
VALUATIONS AND LOCAL UNIFORMIZATION
A basic problem is to prove Local Uniformization (resolution of singularities) along an arbitrary valuation. This was first proved in characteristic zero by Zariski in 1940. THEOREM 2.1 (Local Uniformization, Zariski [21]) Suppose that K is a field of algebraic functions over a field k of characteristic zero, and V is a valuation ring of K (containing k). Then there exists a regular local ring essentially of finite type over k, with quotient field K , such that V dominates R (R C V andmynR = ran).
The proof is by a case by case analysis of the different types of valuations which can occur. Even in a function field of dimension two, the variety of valuations is very rich. We will give an example of a valuation of a rational function field k(x, y) with value group Q (Example 3, Section 15, Chapter VI [24]; Section 6, second case [20]). Choose constants 0 -£ Ci £ k. Define m = x,u2 = y
Set Ri = k[ui,Ui+i](Ui,ui+i)- Each map Ri -> Ri+\ is a product of quadratic transforms (blowups of points). V =Ufli
Local Monomialization
167
is a valuation ring of k ( x , y ) . A valuation v of k(x,y) such that K = {/€*(*,!,) | !/(/)> 0}
is determined by the conditions
If we normalize v(u\] ti be 1, then -.
The value group of v is thus Q. The above example is a non discrete rational rank 1 valuation. This is the most difficult case to analyze. In general, these non Noetherian valuations can be represented by "formal series"
where 7;? € R, and the sums are transfinite [13]. If / 6 fc(zi,... ,xn), then v(f) is the order of/(zi ( £ ) , . . . ,xn(t)). In dimension two and three, Zariski [20], [23] uses local uniformization to obtain resolution of singularities of projective varieties (in characteristic zero). In Zariski's 1939 proof of resolution of surface singularities [20], he presents the following algorithm for resolving surface singularities. Suppose that 5 is a projective surface over a field k of characteristic zero. Let F be the normalization of S. F has only finitely many singular points. Let F± be the normalization of the blowup of these singular points. We can iterate to construct a sequence 5 «- F 4- Fi <- F2 <- • • • The main result of [21] is:
THEOREM 2.2 Fn is nonsingular for some n.
If not, there exists a sequence of points qi € Fi, such that g, maps to
are not regular for all i. V = Ug 0 fli
is a valuation ring. The valuation could be quite complicated! However, it can be deduced from local uniformization in dimension 2 that Ri must be nonsingular for i sufficiently large.
A complete variety over a field k is an integral finite type fc-scheme which satisfies the existence part of the valuative criterion for properness (c.f. Chapter 0 [12]). Complete and separated is equivalent to proper. In arbitrary dimension, Zariski proves resolution by a complete variety.
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Cutkosky
THEOREM 2.3 (Zariski) If X is a proper variety of characteristic zero, then there exists a complete nonsingular k-variety Y, and a birational morphism Y —I X. The Theorem follows from Local Uniformization, Theorem 2.1, and the compactness of the Zariski Riemann manifold [22]. Resolution of singularities in all dimensions and characteristic zero has been proven by Hironaka [12]. Resolution in dimension < 3 and positive characteristic has been proven by Abhyankar [3]. 3
LOCAL MONOMIALIZATION
It is possible to obtain a diagram (1.2) making R' -* S' a monomial mapping whenever the quotient field of 5 is a finite extension of the quotient field of R, and the characteristic of k is 0.
THEOREM 3.1 (Monomialization) (Theorem 1.1 [9]) Suppose that R C S are regular local rings, essentially of finite type over a field k of characteristic zero, such that the quotient field K of S is a finite extension of the quotient field J of R. Let V be a valuation ring of K which dominates S. Then there exist sequences of monoidal transforms R —> R' and S —> S' such that V dominates S' , S' dominates R' and R' —> S" is a monomial mapping. R' t
-»•
S'CV t
R
-+ S
The standard theorems on resolution of singularities allow one to easily find R' and 5" which have regular parameters (xi, . . . ,xn) and (3/1, ... ,yn) such that Xl
=
tf".....^1"*!
:
(3.2)
where the 8i are units. However to obtain a monomial mapping, we must have the essential condition Det(ay) / 0. The difficulty of the problem is to achieve this condition. This Theorem is false over a field of characteristic p > 0. x
= yp + yP+i
gives a simple counterexample. Taking a pth root of 1 +y is not etale. However, we can ask if it is possible (in positive characteristic) to obtain (3.2) with det(fly-) ^ 0. This would be sufficient to prove Theorem 3.3 below in positive characteristic. ^,From Theorem 3.1, we obtain an affirmative answer to a conjecture of Abhyankar. The conjecture is that simultaneous resolution from above along a valuation (proved in 2 dimensional function fields by Abhyankar [1]) is true for arbitrary function fields. The statement of the conjecture is as follows.
Local Monomialization
169
Suppose that we are given a diagram
L -»
K t V t
R where K/L is a finite extension of algebraic function fields over a ground field k, V is a valuation ring of K, and R is a regular local ring, essentially of finite type over k, with quotient field K. The conjecture asks if there exist local rings S* and R*, which are essentially of finite type over k, with respective quotient fields L and K, making a commutative diagram L -*• K t t 5* ->• R* t
R such that 5* is normal, R* is a obtained from R by a finite sequence of monoidal transforms, and R* is a localization of the integral closure of S* in K. The conjecture is a key step in extending Abhyankar's proof [1] of Local Uniformization in dimension 2, characteristic > 0 to higher dimensions. Abhyankar [2] has shown that the conjecture is false if we ask for 5* to be regular. The conjecture is true in characteristic zero and all dimensions.
THEOREM 3.3 (Theorem 1.1 [10]) Simultaneous resolution from above is true in any dimension over fields k of characteristic zero. 4
MONOMIALIZATION OF MORPHISMS
In this section we will suppose that $ : X —¥ Y is a morphism of nonsingular proper &-varieties, where k is a field of characteristic 0. The structure of $ is extremely complicated. However, we can hope to perform sequences of monoidal transforms Xi —> X and YI —> Y and obtain a morphism $ : X\ —> Y\ which has a relatively simple structure. ^From Theorem 3.1, we know that the strongest result that is
possible is to obtain a $ which is monomial like.
DEFINITION 4.1 $ is locally monomial if for every p € X there exist regular parameters ( j / j , . . . ,ym) in OY,$(P), and an etale cover U of an affine neighborhood of p, uniformizing parameters (xi,... ,xn) on U and a matrix ay- such that ,i. — — 3/1
r al i 1 x
• • •x r ani n
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DEFINITION 4.2 A morphism * : Xi -* Y\ is a global monomialization of $ if there are sequences of monoidal transforms a : X\ —> X and /3 : Y\ —I Y , and a morphism $ : Xi —>• Y\ such that the diagram
X,
*
4, X
Y1
4*
Y
commutes, and $ is a locally monomial morphism. Recall that there exist morphisms over a field k of positive characteristic which do not have a global monomialization. If <£ : X —> C is a morphism from a projective variety to a curve, the existence of a global monomialization follows immediately from resolution of singularities. In the case of a morphism of complex projective surfaces, a proof of the existence of a global monomialization follows from results of Akbulut and King (Chapter 7 of
[6]). An important case when the existence of a global monomialization is still open if for birational morphsisms of nonsingular, characteristic 0, varieties of dimension > 3. Birational maps are known to have a simple structure, since recently, Wlodarczyk [18], [19] and Abramovich, Karu, Matsuki and Wlodarczyk [5] have proven the "Weak Factorization Conjecture" for birational morphisms. They have shown that a birational morphism of nonsingular projective varieties, over a field of characteristic zero, can be factored by alternating sequences of blowups and blowdowns.
The construction of a monomialization by complete varieties follows from Theorem 3.1. THEOREM 4.3 (Theorem 1.2 [9]) Let k be a field of characteristic zero, $ : X -> Y a generically finite morphism of nonsingular proper k-varieties. Then there are birational morphisms of nonsingular complete k-varieties a : X\ -> X and 0 : Y\ —> Y , and a locally monomial morphism \f : X\ —» Y\ such that the diagram
4, X
; *
Y
commutes and a and J3 are locally products of blowups of nonsingular subvarieties. That is, for every z £ X\, there, exist affine neighborhoods V\ of z, V of x = a(z), such that a : V\ —» V is a finite product of monoidal transforms, and there exist affine neighborhoods W\ of ^(z), W of y = a($(z)), such that j3 : Wi —> W is a finite product of monoidal transforms.
Recall that a monoidal transform of a nonsingular fc-scheme S is the map T —t S induced by an open subset T of Proj(®£"), where I is the ideal sheaf of a nonsingular subvariety of S. A key step in the local proof of monomialization is to define a kind of multiplicity, which measures how far the situation is from a specific form which is close to being monomial. In the local valuation theoretic proof, we make use of special products of monoidal transforms defined by Zariski called Perron transforms. Under appropriate
Local Monomialization
171
application of Perron transforms our multiplicity does not increase, and we can in fact make the multiplicity decrease, by an appropriate algorithm. An essential difficulty globally is that our multiplicity can increase after a permissible monoidal transform. This is a significant difference from resolution of singularities, where a foundational result is that the multiplicity of an ideal does not go up under permissible blowups. Recently, we have been able to overcome this in the case of a mapping from a 3-fold to a surface. We have proven that a global monomialization exists for a morphism from a 3-fold to a surface [11].
THEOREM 4.4 (Theorem 1 [11]) Let $ : X -> 5 be a proper dominant morphism from a 3-fold X to a surface S, over an algebraically closed field k of characteristic zero. Then has a global monomialization. our proof, we can in fact put the mapping in a more combinatorial form. Toroidal embeddings and morphisms are defined in [14] and [6].
THEOREM 4.5 (Theorem 2 [11]) Let $ : (Ux C X) -> (Us C S) be a dominant proper morphism of toroidal embeddings from a 3-fold X to a surface S. Then
toroidal morphism. We will give a brief overview of the proof of Theorem 4.4. Step 1. First construct a diagram
X' 4 X
^ *
5' I S
where the vertical maps are products of monoidal transforms, such that sing($') has simple normal crossings in X and $'(sing $') has simple normal crossings in S'. Then for all p £ X', q = <J?'(p), there exist regular parameters (u,v) in Os',q, (x,y,z) in Ox>,p such that one of the following holds: 1.
u = xa,v =p(x) +xbF
where x jfF, F has no terms which are monomials in x. 2.
u = (xayb}m,v = p(xayb) + xcydF
where (a, b) = I , x jfF, y jfF, xcydF has no terms which are monomials in xayb.
3. u = (xaybzc)m,v = p(xaybzc) + xdyezfF where ( a , b , c ) — I , x jf F, y K F, z jf F, xdyezfF has no terms which are monomials in xaybzc.
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Cutkosky
Step 2. This is the difficult step. We construct a commutative diagram
X" I
X'
*" >
*
5
so that everywhere we have one of the forms: 1. u = xa, v = p(x) + xby,
2. u = (xayb)m,v = p(xayb) + x°yd, 3. u = (xayb)m,v = p(xayb) + xcydz, 4. u = (zVz c ) m ,i; = p(xaybzc) + xdyezf with . f a
rank <
.
6
c 1
_
. > = 2.
\ d e f J
Step 3. We construct a commutative diagram Vtlt
SL
$
I v" j\
Oil
J
4*" _
c' o
such that the vertical maps are products of monoidal transforms, to get $" locally monomial.
References [1] ABHYANKAR, S., Local uniformi/ation on algebraic surfaces over ground fields of characteristic p ^ 0, Annals of Math, 63 (1956), 491-526.
[2] ABHYANKAR, S., Simultaneous resolution for algebraic surfaces, Amer. J. Math 78 (1956), 761-790.
[3] ABHYANKAR. S., Resolution of singularities of embedded algebraic surfaces, second edition, Springer Verlag, 1998.
[4] ABRAMOVICH D., KARU, K., Weak semistable reduction in characteristic 0, preprint.
[5] ABRAMOVICH, D., KARU, K., MATSUKI, K., WLODARCZYK, J., Torification and Factorization of Birational Maps, preprint.
[6] AKBULUT, S. AND KING, H., Topology of algebraic sets, MSRI publications 25, Springer-Verlag Berlin.
[7] CHRISTENSEN, C., Strong domination/ weak factorization of three dimensional regular local rings, Journal of the Indian Math Soc., 45 (1981), 21-47.
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[8] CUTKOSKY, S.D., Local Factorization of Birational Maps, Advances in Math. 132, (1997), 167-315. [9] CUTKOSKY, S.D., Local Factorizaton and Monomialization of Morphisms, Asterisque, 1999.
[10] CUTKOSKY, S.D., Simultaneous resolution of singularities, Proc. American Math. Soc. 128, (2000), 1905-1910.
[11] CUTKOSKY, S.D. Monomialization of a proper morphism from a 3 fold to a surface, in preparation. [12] HIRONAKA, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math, 79 (1964), 109-326.
[13] KAPLANSKY, I., Maximal fields with valuations I, Duke Math. J, 9 (1942).
[14] KEMPF, G., KNUDSEN, F., MUMFORD, D., SAINT-DONAT, B., Toroidal embeddings I, LNM 339, Springer Verlag (1973). [15] SALLY, J., Regular overrings of regular local rings, Trans. Amer. Math. Soc.
171 (1972) 291-300. [16] SHANNON, D.L., Monodial transforms, Amer. J. Math, 45 (1973), 284-320. [17] WLODARCZYK J., Decomposition of birational toric maps in blowups and blowdowns. Trans. Amer. Math. Soc. 349 (1997), 373-411.
[18] WLODARCZYK J., Birational cobordism and factorization of birational maps, preprint. [19] WLODARCZYK J., Combinatorial structures on toroidal varieties and a proof of the weak factorization theorem, preprint. [20] ZARISKI, O., The reduction of the singularities of an algebraic surface, Annals of Math., 40 (1939) 639-689. [21] ZARISKI, O., Local uniformization of algebraic varieties, Annals of Math., 41, (1940), 852-896.
[22] ZARISKI, O., The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc., 45, (1944), 683-691. [23] ZARISKI, O., Reduction of the singularities of algebraic three dimensional varieties, Annals of Math., 45 (1944) 472-542.
[24] ZARISKI, O. AND SAMUEL, P., Commutative Algebra II, Van Nostrand, Princeton (1960).
Isomorphism of Complexes and Lifts STEVEN DALE CUTKOSKY1, Department of Mathematics, University of Missouri - Columbia, MO 65211, USA, [email protected], partially supported by NSF. HEMA SRINIVASAN2, Department of Mathematics, University of Missouri Columbia, MO 65211, USA, [email protected], partially supported by NSF.
1
INTRODUCTION
R denotes a commutative Noetherian local ring with maximal ideal m. Let F be an exact complex of free R modules. Let H C R be an ideal of R. Suppose C is a complex of free R-modules and 4> : F
Partially supported by NSF Partially supported by NSF
175
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Cutkosky and Srinivasan
of isomorphism of singularities and has been well studied. In the first section, we will concentrate on this case and list the known results Samuel [8], Mather [6], Elkik [4], Cutkosky and Srinivasan [3] and their modifications. We also discuss the problem of finding the largest such H, up to radical of course. In the next section, we concentrate on the simple but strongest case Q = {!}. By a previous theorem of the authors, for any complex F of length I, if K — /(/i), there is a t such that any isomorphism $ : F <8> R/K1 -> C ® R/K* can be lifted to an isomorphism of complexes F and C which agree with (j> modulo H where H CT/K. In theorem 4.3 we prove that when the complex F is of length one, then \/I(fi) is the largest possible ideal. Whether this is true for complexes of length 2, is still open. We also list some cases where this is true. In the context of isomorphism of singularities, when R is a power series ring over a field and q is the group of k algebra automorphisms of R, this means that two singularities R/I and R/J are isomorphic if their truncated resolutions are isomorphic modulo a proper ideal H. By a theorem of Peskine and Szpiro, [7], given any exact complex F of length /, there is a positive integer t such that any complex C that is isomorphic to it modulo m1 is also exact. However, it will not in general be isomorphic. Indeed, theorem 4.5 demonstrates this. 2
NOTATIONS
R is a commutative ring and Q is a subgroup of the group of automorphisms of R. Let a € Aut(R). (j) : M —> N is a a homomorphism of two R-modules M and N if (j> is additive and <j)(rx) — o-(r)<j>(x) for any r € R and a; 6 M.
DEFINITION 2.1 We will say that a Q isomorphism <j> : F
Two complexes F and C are said to be Q isomorphic modulo an ideal H if they are Q isomorphic after tensoring with R/H. When Q = {!}, we will suppress it. Given a complex F, it would be nice to find an ideal H such that any Q isomorphism from F to a complex C modulo H can be lifted in the above sense, to a Q isomorphism of F and C. The next question would be to find the largest such ideal up to radical for any given complex F.
3
Q ISOMORPHISMS
In this section, R will be a powerseries ring or a polynomial ring in xi, • • • ,Xd over a field k and F is a free complex with FQ = R. Let Q be the group of k-algebra
Isomorphism of Complexes and Lifts
177
automorphisms of R. In this context, to say that all Q isomorphisms of F -» C modulo H can be lifted to Q isomorphisms is to say that F is to say that F is determined by H. If FI -> F0 — R is a map represented by / = (01, a 2 , • • • , an), then
denotes the n x d Jacobian matrix
THEOREM 3.1 [8, 6, 9]: Let I = I , so that F = FI 4 F0 = R. Let K = In(W(fi) + I(fi)). LetH = K3. F&R/H = C ® R/H for some complex C implies that there exists a Q isomorphism aofF and C which is' the identity modulo K. THEOREM 3.2 [4, 3] Let I = 2 so that F = F2 4 FI 4 F0 = R. Let I ( f l ) be an ideal of height h. Let
be the Jacobian ideal of I. Then there exists positive integers t > s such that for any Q isomorphism
When the characteristic of k is zero, a converse to theorem 1 is true. That is,
THEOREM 3.4 [6, 3] Let F = FI 4 F0 = R be an exact complex. Suppose that H C R is a proper ideal such that any complex C which agrees with F modulo H is Q isomorphic to it. Then H C
Note that K = I unless /i = (ai , • • • , an) is a complete intersection in Rp for some prime ideal P. When R is equidimensional, the above theorem says that the only ideals I whose presentations have a nontrivial finite determination are the complete intersections. In fact, if F = FI -> R is the presentation of R/I, then if F equals C modulo J3 , then they are isomorphic by Nakayama's lemma! Thus there is no special advantage in looking for a Q isomorphism in this case. __ However, a similar converse to theorem 2 is not true. That is, \/j(I) is not the largest ideal to finitely determine a finite presentation of R/I up to a Q equivalence.
THEOREM 3.5 Let R = k[[xij, 1 < i < n, 1 < j < n - I]} be a power series ring and m be the maximal ideal. Let I = In-i(X) be the ideal of the (n - 1) x (n - 1) minors of the generic matrix X — (xij) and F be the resolution of R/I. If C is
any complex such that F ® R/mn is Q isomorphic to C ® R/mn, then there is a Q isomorphism between F and C. We have \/j(I) = \/In-^(X) ^ m if n > 3. In particular, when n > 3, any complex C which is isomorphic to F modulo m" is Q isomorphic to it.
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Cutkosky and Srinivasan
Proof. Let F = Rn~l 4 Rn 4 R and C = Rn~l ^ Rn 4 R. Let a € Q be such that F ® R/mn and C <8> R/mn are a isomorphic. Replacing Xy by
REMARK 3.6 // j/y- = Zy- + ay /or any ay m m2 and g = f\n~lY, then the corresponding complex C will be isomorphic to F modulo m™ ana* will therefore be Q isomorphic to F. This is in sharp contrast to the the situation when Q — { I } . We will see in Theorem 4.5 that F will be isomorphic to C if and only if it is isomorphic to C modulo a suitable power of /(/) whose radical is not the maximal ideal unless n=2 and F is the Koszul complex.
This brings us to the question:
Question: Let 1=2 so that F = F2 -4 FI -4 R. What is the biggest ideal K such that for some integer t, and any complex C any Q isomorphism (f> : F <8> R/Kf -> C ® R/K1 can be lifted to a Q isomorphism of complexes F and C which agree with 4> modulo H such that K C H C \fK1
4
R-MODULE ISOMORPHISMS
In this section R is any commutative noetherian local domain with maximal ideal m. Let F be the complex
F
/ 1 .• ft ip ~^ J( ri—1 771 •'~? ~; ri—i z? r1 —7 A .TO i? — v> • • • —_v> r\
We will give some results when the subgroup of automorphisms of R in question is the trivial group. Thus the isomorphisms here are .R-module isomorphisms. This is the strongest conclusion and hence the ideal K will be the smallest. By a theorem of Peskine and Szpiro, [7] given any exact complex F of length /, there is a positive integer t such that any complex C that is isomorphic to it modulo m1 is also exact. A recent result of C.Huneke and D. Eisenbud, shows that if R is Cohen-Macaulay then even if F is an infinite complex, there is a positive integer t such that any complex C which is isomorphic to F modulo m4 is exact. However,
in either case, F and C will not in general be isomorphic. Indeed, theorem 4.5 in this section demonstrates this. To begin with if /(/i) = R, then any homomorphism gi such that /i = g\ mod m2 will result in an isomorphism. This is naturally the strongest possible. So, consider the case that /(/i) ^ R- From now on, all R-module homomorphisms / will have /(/) ^L R, If /(/) = R: the statements of these theorems will still be true if we replace /(/) by /(/) fl m.
Isomorphism of Complexes and Lifts
179
THEOREM 4.1 [3] Let F : Fl -4 F0 be such that rank of f = rank of F0. Suppose that R is complete with respect to I(fi ) . Then there exist positive integers t, s with t > s, such that for any isomorphism <j> : F ® R/I(fiY -»• C ® R/I(fi)1 where C is a complex there is an isomorphism 4> '• F -> C SMC/J In fact, v^(/i) is indeed an ideal that will do for complexes of any length. The following is a consequence of theorem 5.8 in [3].
THEOREM 4.2 lei R be a complete local domain. Let F : P; A P<_! '41 Fi-2 -» • • • -> F! 4 P0 6e an ea;ac£ complex of free R-modules. Then there exist positive integers t,s with t > s, such that for any isomorphism <j> : F ® R/I(fi)t -> C
A converse to Theorem 4.1 is true.
THEOREM 4.3 Let f : PI —> F) be an R-module homomorphism such that rank(f) = rank(Fo). Suppose H is any ideal such that for any R-module homomorphism g : FI —>• PO, any isomorphism
0 xl
0
where the first Q i s r - t - l x r - t - l zero matrix and xl is the x times the t + 1 x t + 1 identity matrix. Then rank of g is at least r. But the first r xr minor of g is b = yp + xy\ + • • • + xi + • • • + xt+lz, where yj € 7 r _j(/). It follows that b is not in \ / I ( f ) = \/ann M. So, there cannot be an isomorphism <[> such that
Fl
4. 4>i
4 F0
4- <^o
commutes. But / = g mod /f . This contradicts the assumption that all isomorphisms modulo H can be lifted. So, H C \//(/). Using the similar construction t ( x ) , we prove:
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Cutkosky and Srinivasan
THEOREM 4.4 Suppose that F is a complex of length I where rank(fi) — r = rank(Fi) and I ( f \ ) is of generic height. Then again, if H is any ideal so that isomorphisms <j> : F (g> R/H —)• C ® R/H give rise to isomorphisms of F and C, then H C Proof. Just as in the previous theorem, if H is not a subset of y/(/i)> x G H and a t maximal with respect to IT~t C -\/I(/i). Let g\ = /i + t(x) and C be the Buchsbaum-Rim complex. Then F <8> R/H - C® R/H and hence they are isomorphic, which is impossible. Hence we conclude that H C \//(/i)Now, when M = R/I, where 7 is a height two perfect ideal, its resolution is special. Thus,
THEOREM 4.5 Suppose F : F2 4 Fl A- R is exact. Suppose further that cokernel of f i = R/I is Cohen-Macaulay and of homological dimension 2. That is, fa is injective and F* is exact. Suppose H is an ideal such that any isomorphism : F eg) R/H —> C 0 R/H gives rise to an isomorphism of F and C. Then H C The proof uses the same construction as in 4.3 except we apply it to fa. To give a sketch, let F be a minimal prime of I(fa) and x be not in P. Let t be such that Ir2-t C P but /r 2 -t-i is not a subset of P. Define t(x) as in the proof of theorem 4.3 and let g2 = fa + t(x) Then if gi = /\r2~1g?, the corresponding complex C would be isomorphic to F modulo (x). But, F and C cannot be isomorphic because coker(fa)* ^ coker(g2)* .
REMARK 4.6 Comparing theorems 4.5 and 3.5, we see that when Q= {!}, the largest ideal, up to radical, to determine a resolution F of a Cohen-Macaulay, codimension two cyclic module M , is V ann M , where as it is the maximal ideal when Q is the group of k- algebra automorphisms of R.
References [1] BUCHSBAUM, DAVID A, AND RIM, D.S. A generalized Koszul Complex II. Depth and Multiplicity, Trans.of AMS., Ill (1964), 197-224
[2] CUTKOSKY, S.D., AND SRINIVASAN, HEMA An Intrinsic Criterion for isomorphism of singularities, American J. Math., 115 (1993), 789-821. [3] CUTKOSKY, S.D., AND SRINIVASAN, HEMA Equivalence and Finite Determination of Mappings, J. Algebra, 188 (1997), 16-57. [4] ELKIK, R Solutions d'equations a coefficients dans un annequ henselian, Ann.Sci. Ecol.Norm. Sup. (4) (1973)
[5] HIRONAKA, H Formal Line bundles along Exceptional loci, in "Algebraic Geomety" pp.201-218, Oxford Univ.Press, Bombay, 1969 [6] MATHER, J Stability of C°° mappings, III, Finitely determined map-germs, Publ. Math. I.H.E.S., 35 (1969), 127-156
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[7] PESKINE,C AND SZPIRO, F Dimension projective finite et cohomologie locale, Publ. Math. I.E. E.S., 42(1972) 47-119. [8] SAMUEL, P.Singularities des varieties algebriques, Bull.Soc.Math.Prance, 79 (1951), 121-129 [9] TOUGERON, L Ideaux de fonctions differentiables, I Ann.Inst.Fourier (/grenoble) 18, (1968), 177-240
Weil divisors on rational normal scrolls RITA FERRARO Dipartimento di Matematica Universita di Roma "Tor Vergata" Viale della Ricerca Scientifica, 00133 Roma. Con il supporto dell'Istituto Nazionale Alta Matematica.
1
INTRODUCTION
Let X C P™ be a rational normal scroll of degree / and dimension r = n — / + 1. X is the image of a projective bundle X over P1 of rank r — I through the birational morphism j defined by the tautological line bundle O^(l). Depending on X, the scroll X may be smooth (in this case j is an isomorphism) or singular. The aim of this note is to study Weil divisors on a singular rational normal scroll. Let V be the vertex of X, we have that codim(V, X) > 2. When codim(y, X) = 2 the scroll X is a cone over a rational normal curve of degree /; in this case let E — j~1(V) be the exceptional divisor in X. The paper is divided into four sections. In section 2 we set up the notation and recall some standard facts about Weil divisors on normal varieties: the strict image map j# : CaCl(X) -» Cl(X) (defined in Prop 2.1), is used to describe the group Cl(.X') of Weil divisors on X modulo linear equivalence. It turns out that there are two different cases: (i) when codim(V,X) > 2, then j# is an isomorphism; (ii) when codim(V, X) = 2, then j# is surjective and ker(j#) is generated by E. Since the group Cs,C\(X) of Cartier divisors on X modulo linear equivalence is well known, this provides an explicit description of C\(X) (Cor. 2.2). Furthermore we introduce the basic theme of the next section: the relation between the strict image, the scheme theoretic image and the direct image of the ideal sheaf of a divisor on X (Lemma 2.5). The goal of section 3 is to describe the sheaves corresponding to Weil divisors on X. It is known (see Prop. 3.3) that the group C\(X) is in bijection with the set T>iv(X) of divisorial sheaves on X (i.e. coherent sheaves which are reflexive of rank one), and induces on it a natural group structure. We describe explicitly the group T>iv(X) via the direct image morphism j, : Pic(X) -> Coh(X) of sheaves. In particular, we will be able to say when the direct image jf^F of an invertible sheaf T G Pic(X) is reflexive. The analysis naturally splits in the two cases above mentioned. When codim(V, X) > 2, we will prove with standard tecniques that 183
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Ferraro
j * P i c ( X ) - Div(X) (Cor.
3.10). In particular (Prop. 3.9)
the divisorial sheaf
OX(D) of a Well divisor D on X is
where D is the proper transform of D in X. When codim(F, X) = 2, this is no longer true. To overcome this problem we introduce the concept of integral total transform D* C X of a Weil divisor D C X (see Def. 3.13). We prove (Th. 3.17) that T>iv(X) consists of the direct images of those line bundles T € Pic(X) such that the degree deg^g) of f restricted to the exceptional divisor E is < /, where / is the degree of the scroll X. In particular we will prove the projection formula:
Ox(D)~j.(Ox(D')). In section 4 we will study the intersection of Weil divisors on X in the critical case, i.e. codim(F, X) = 2. According to [6]., Prop. 2.4. an effective Weil divisor D on X C Pra is itself a closed subscheme of X of pure codimension 1 with no embedded components. Given two effective divisors D and D' on X with no common components, we can consider the scheme-theoretic intersection Y = D n D' C P™ and ask, for example, for its degree. Note that when codim(y, X) > 2 one can compute the degree of a "complete intersection" of / (1 < I < r - 1) divisors DI, . . . , DI using the natural intersection form on Div(X) inherited from X , via the isomorphism j#. When codim(F, X) — 2, the unique linear theory of intersection which can be defined in X is the generalization to higher dimension of the theory developed by Mumford in [7] for a normal surface. According to this theory, the intersection number of two divisors D, D' on X is a rational number defined as the intersection number of the corresponding rational total transforms in X (see Def. 3.12), which is in general a rational number and does not represent the degree of the scheme theoretic intersection Y = D n D'. In Th. 4.6 we will use the integral total transform to find the minimal reflexive resolution of Oy as an Ox-module, which allows us to compute the degree of Y (Prop. 4.11). The last section is devoted to examples and applications. In particular in Ex. 5.1 we show that every divisor of degree > n on a rational normal cone of degree n — 1 in P™ has maximal arithmetic genus. In Ex. 5.2 we show that, when codim(F, X) = 2, every effective divisor D C X and every scheme theoretic intersection Y = DnD' C X of two effective divisors with no common components are arithmetically CohenMacaulay schemes in P". In Ex. 5.4 we will compute the arithmetic genus of Y. Much of this material was motivated by the subject of my doctoral thesis: Classification of curves of maximal genus in P5, since these curves lie on (possibly singular) rational normal three-folds. In that context it is necessary to compute the degree of the scheme theoretic intersection of two divisors on a rational normal cone X over a twisted cubic in P5, and to develope some linkage tecniques on X. Moreover, for the linkage problem, it is necessary to know the divisorial sheaves on X . Thanks to my advisor Giro Ciliberto.
2
PRELIMINARIES
In this section we recall some basic facts about rational normal scrolls and we describe the group Cl(X) in terms of CaCl(X). For more details about rational normal scrolls the reader may consult for example [8] or [3].
WeiJ divisors on rational normal scrolls
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A rational normal scroll X C P™ is the image of a projective bundle ?r : P(£) -» P1 over P1 of rank r — 1 through the morphism j defined by the tautological bundle Op(£)(l), where £ = Opi(ai) © • • • ©C?pi(a r ) with 0 < ai < • • • < ar and J2ai — /• If ai = • • • = a; = 0, 1 < / < r, X is singular and the vertex V of X has dimension I - 1. Let us denote P(£) = X. The morphism j : X -» X is a rational resolution of singularities, i.e. X is normal and arithmetically Cohen-Macaulay and Kj^O^ = 0 for j > 0. We will call j : X -» X the canonical resolution of .X". It is a general fact that Weil divisors on a normal scheme do not depend on closed subsets of codimension > 2. We refer to [5], Prop. II, 6.5. for this basic fact which we will continuously use in this note. In Th. 3.1 we will see the equivalent result in terms of sheaves. Let Xs = X \ V be the smooth part of X; since X is normal by [5], Prop. II, 6.5. there is an isomorphism Cl(X) -> Cl(Xs) defined by D — J^niDi -» ^n;(D; D Xs) , where Di are prime divisors on X; in other words, given a prime divisor DS on Xs there is an unique prime divisor D = DS (the closure of DS in X with the induced scheme structure), which extends DS on all X. Since J|j-ix s 'IS an isomorphism, we can rewrite [5], Prop. II, 6.5. in the following way:
PROPOSITION 2.1 Let X be a singular rational normal scroll, let V be its vertex and let Xs be its smooth part. Let j : X -4 X be the canonical resolution. Then:
1. there is a surjective homomorphism j# : CaCl(X) -» Cl(X) defined by C = Y^iniCi —> ^nij(Ci <~}j~1Xs), where we ignore those Ci <~}j~1Xs which are empty;
2. if codim(V, X) > 2, then j# : Pic(X) -> Cl(X) is an isomorphism; 3. i/codim(V, X) — 2 and E is the exceptional divisor of j, then there is an exact sequence: 0 —> Z -> CaCl(X) -4 Cl(X) —>• 0 where the first map is defined by
Given a Cartier divisor C on X, the Weil divisor j#(C) on X will be called the strict image of C through j. It is well known that Pic(^T) = Z[H] ® Z[jR], where [H] — [Ox (1)] is the hyperplane class and [R] = [?r*Opi(l)] is the class of the fibre of the map TT : X —> P1 . Let H = j#H and R = j#R be the strict images of H and R respectively (i.e. respectively an hyperplane section and a divisor in the ruling of X). Then as a consequence of Prop. 2.1 we have the following: COROLLARY 2.2 Let X C Pra be a singular rational normal scroll of degree f and let j : X —> X be its canonical resolution. Then
1. ifcodim(V,X)
> 2, Cl(X) ^
2. if codim(F, X)=2,E~H-fR
and Cl(X) S Z[R].
Proof. 1) Follows immediately from Prop. 2.1 2). When codim(F, X) = 2 an hyperplane section H passing trough V splits in the union of / fibers R, therefore we have H ~ fR, i.e. by Prop. 2.1 3) E ~ H - fR. D
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REMARK 2.3 The Weil divisor R on X is not Cartier, since it is not locally principal in a neighborhood of the vertex V . The proof is similar to the classical example of a quadric cone ([5], Ex. II, 6.5.2).
However, besides the strict image, there are other natural ways of "pushing down" a divisor on X. Namely one can consider the scheme theoretic image. If C C X is a subscheme of X, the scheme-theoretic image j*(C) of C through j is the unique closed subscheme of X with the following property: the morphism j o i, where i : C t-^> X is the canonical injection of C in X, factors trough j*(C), and if D is any other closed subscheme of X through which joi factors, then j» (C) <->• X factors through D also (see [2], §6.10 pp. 324-325 and [5] Ex. II, 3.11 d)). According to [6], Prop. 2.4. an effective Weil divisor D on X C Pn, i.e. a formal sum Y^niDi with r?,j € Z+ and Di irreducible, corresponds to a closed subscheme of X of pure codimension 1 with no embedded components. Therefore we can talk about the scheme theoretic image j* (C) of an effective divisor C. Given a prime divisor D on X , the proper transform D of D on X is the closure j~1(DC\Xs) with the induced reduced scheme structure. The proper transform of any Weil divisor in X is then defined by linearity. If D is effective it follows that D is the scheme-theoretic closure of Dy-iXs m X, that is the smallest closed subscheme of X containing D\J-\XS as subscheme (see [2] Def. 6.10.2 pg. 324); this implies that D is the scheme-theoretic image of D\j-iXs through the canonical injection k : Dy-iXs <-» X. REMARKS 2.4 1. For any Weil divisor D on X , from the definition of j# and of D it follows that j#(D) = D. 2. Let codim(V, X) > 2 and let D be a divisor on X . The unique Cartier divisor
C on X such that j#(C) = D is the proper transform D. Indeed, since C and D coincide outside a subset of codimension > 2, they must be equal.
3. If codim(V, X) = 2 we have j#(D + nE) = D for every n £ Z. In fact 4. If D is effective, then j * ( D ) = D. Let us denote by k : D r\j~[Xs "-> X and i : DnXs <-» X the canonical injections and by h : Dr\j~~lXs —t DnX$ the isomorphism j\f>nj-iXs ', we then have jok = ioh. By [2] Prop. 6.10.3 pg.324
(transitivity of the scheme-theoretic images) we have thatjt(kt(Dr\j~lXs))
—
it(ht(D nj~1X$))- Since, as we noted before, kt(D r\j~1X$) — D and (for the same reason) i f ( D n Xs) — D, this proves what we claimed.
In the following Lemma we prove that for an arbitrary divisor C on X the scheme theoretic image j*(C) is defined by the ideal sheaf j*Zc<x '-> j*Ox = Ox. As it will be more clear in the next section, the strict image j#(C) is defined \yy jflc,xyv (Prop. 3.3 4)). LEMMA 2.5 Let C be an effective Cartier divisor on X and let jt(C) be its schemetheoretic image in X . Then jt(Xc,x) — Ij,(c)\x-
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Proof. Let us consider the following diagram of sheaves on X: f\
,
T"
i
-»•
/O
C?x
v
->
/^
Oj-
v
{c)
->•
Since j*Ox = Ox, the morphism j* : Ox -» j*Ox is an isomorphism; therefore, by the Snake's Lemma, we have to prove that the morphism Oj,(c) ~^ j*Oc is injective. This follows because otherwise ker(j,* (C) ) <-»• C'j.(c) would define a subscheme C"
of j,(C) such that the morphism Oj,(c) ~* j*Oc factors in C'j.(c) ~* ^C" —^ J*Oc, but this cannot happen by universal property of j*(C). D 3
DIVISORIAL SHEAVES
We consider here the problem of describing the group Div(X) of divisorial sheaves on a singular rational normal scroll X in terms of the Picard group Pic(J^T) of the
canonical resolution X. It is known (see Prop. 3.3 below) that T>iv(X) is naturally isomorphic to the group Cl(X) of Weil divisors modulo linear equivalence, The analysis naturally splits in two cases: codim(V, X) > 2 and codim(V, X) = 2. First we deal with the first case, which can be treated in a more or less standard way.
The result (Cor. 3.10) is that the natural map j* : Pic(X) —>• Coh(X) is in fact an isomorphism j» : Pic(X) —» Viv(X). This is no longer true when codim(F,X) — 2. To overcome this problem we will introduce the concept of integral total transform of a Weil divisor D on X. But first let use give basic definitions and properties of
divisorial sheaves on a normal scheme; for details and for a more general point of view the reader may consult [6], §2. Let X be a normal scheme. We recall that a coherent sheaf T on X is reflexive if the natural map T -> ^-"vv is an isomorphism, where ,FV denotes the dual sheaf Hom(Jr, Ox)- The following Theorem, which says that reflexive sheaves depend only on subsets of codimension 1, is a basic fact which we will use in this section.
THEOREM 3.1 Let X be a normal scheme and let Y C X be a closed subset of codimension > 2. Then the restriction map induces an equivalence of categories from the category IZefl(X) of reflexive sheaves on X to the category Tiefl(X \ Y) of reflexive sheaves on X \ Y.
Proof. [6], Th. 1.12. We recall from this proof the way to extend a reflexive sheaf Q on X \ Y to a reflexive sheaf T on X. It consists in taking a coherent extension T§ of Q in X (which exists by a general result on extensions of coherent sheaves), and then to put T = ^b V V - The double dual of any coherent sheaf is in fact always reflexive. D
DEFINITION 3.2 Let X be a normal scheme. Let D be a Weil divisor on X. If K(X) denotes the function field of X ([5], pg. 91 and pg. 141), then the sheaf Ox(D) defined for every open set U C X as
= {f e K ( X ) \ d i v f + D > 0 is called the divisorial sheaf of X.
on U}.
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The following Proposition (see [6], Prop. 2.8.) describes the equivalence between reflexive sheaves and divisorial sheaves. In point 4) is defined the group structure on Viv(X) induced by C\(X).
PROPOSITION 3.3 Let X be a normal scheme.
1. For any Weil divisor D the sheaf Ox(D) is reflexive and locally free of rank one at every generic point and at every point of codimension 1. 2. Conversely, every reflexive sheaf which is locally free of rank one at every generic point and at every point of codimension 1 is isomorphic to Ox(D) for some Weil divisor D. 3. If DI and D^ are Weil divisor on X , DI ~ D^ if and only if Ox(Di) = OX(DZ) as Ox -modules,
4. If D, Dl, D2 are Weil divisors on X, OX(-D) = OX(D)V and Ox(Dl + We come back now to the case of a singular rational normal scroll X. Our goal is to explicitly describe T>iv(X) with its group structure and the idea is to compare T>iv(X) with P'ic(X), via the direct image map j» of sheaves. There is a natural surjective map, induced by the strict image map j# : Pic(X) —> C l ( X ) , which we call again j# : j# : Pic(X)
0X(C)
->•
-Div(X)
H+ UOX(C))^
which is bijective when codim(F, X) > 2. The sheaf j*(Ox(C))v^ must be the divisorial sheaf Ox (j# (C)) by Th. 3.1 since it is reflexive and isomorphic to Ox (j#(C)) outside a subset of codimension > 2 (in the open set Xs)- We are then going to compare j# : Pic(X) -> T>iv(X) with jf : Pic(X) -> Coh(X) or, in other words, we are going to check when j*((9^(C ( )) is a reflexive sheaf. Of course we will have, according to Prop. 2.1, two separate cases: a) codim(Vr, X) > 2 and b) codim(V, X) = 2. First we briefly describe j»(Pic(X )). We refer for details to [10] or [4]. Let us denote:
the invertible sheaf associated to the class [aH + bR] in Pic(X) and let us consider on X their direct images:
Ox(a,b):=j.0x(a,b), with a, 6 6 Z. The cohomology of Ox(a, b) can be explicitly calculated using the Leray spectral sequence, which, since Rt-ir:tOx(a, b) — 0 for every a, 6 £ Z and 0 < i < r — 2, simplifies as follows:
jj (a,
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189
Therefore for i < r — 1 and a > 0 we obtain:
|/|=o
j€l
which are zero of course if a < 0 and for 1 < i < r — 1; while for j = r — l,r we obtain:
which can be computed also by Serre duality using (3.4). For a > 0 and 6 > — 1 using (3.4) we compute: (3.5)
For b < — 1, the dimension h°(X,Ox(a,b))
depends on the type of the scroll, i.e.
on the integers aj, . . . , ar. We recall from [10] that we have the vanishing tfj,0x(a,b)
=0
(3.6)
for i > 0 and for all a € Z and 6 > — 1, which implies, via the degenerate Leray spectral sequence associated to j: hi(0x(a,b))=hi(0x(a,b)}
(3.7)
for i > 0. Moreover in [10] it is proved that the dualizing sheaf MX of X is: wx=j*Ox(Kx) = Ox(-r,f-2).
(3.8)
As we will see (Cor. 3.10 and Th. 3.17), ujx is always a divisorial sheaf, so it makes sense to talk about the canonical divisor KX ~ —rH + (f — 2)7? on X.
Let us consider the case when codim(V, X) > 2. PROPOSITION 3.9 Let codim(V,X) > 2, let D ~ aH + bR be a Weil divisor on X and let D ~ aH + bR be its proper transform on X . Then
Proof. It is sufficient to consider the local situation. Let U be an open set containing the vertex V and let U' = rl(U). Call V = j~l(V). Since codim(V',X) > 2, then H°(U',0X(D)) = H°(U'\V',0X(D)); moreover H°(U' \ V',OX(D)) * H°(U\ V, OX(D)) S H°(U, OX(D}) where the last isomorphism follows from Th. 3.1. D COROLLARY 3.10 Let codim(V,X) > 1, then Ox(a,b) is a reflexive sheaf for every a, b € Z and
Moreover the natural group structure on T>iv(X) inherited from Cl(X) is given by
< Ox(a, 6), Ox (a1, b') >^ Ox((a + a'), (6 + b')) and Ox(a, 6)v = Ox(-a, -b).
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Proof. The first assertions follows directly from Prop. 3.9. The description of the group structure on Viv(X) follows from Prop. 3.9 and part 4) of Prop. 3.3. D
REMARK 3.11 Note that Ox((a +a'),(b +V)) S (Ox(a,b)
When the codimension of V is 2 we know (Remarks 2.4 3)) that the map j# : Pic(X) -> Cl(X) is not injective and therefore also the map j# : Pic(X) -> Piw(.X') is not injective. It turns out that the study of T>iv(X) becomes very simple if we introduce, for any Well divisor D on X, a Cartier divisor D* on X, which we will call the integral total transform of D, which plays, in some sense, the role of the proper transform in case codim(Vr, X) > 2. The problem of defining, for a Well divisor D trough V on X, the total transform X has been considered and solved by Mumford in [7] on a normal surface, with the goal of developing the bilinear intersection theory on normal surfaces (see also [9]). Mumford's theory can be generalized on rational normal cones by defining the total transform j*D of a divisor D on X as:
DEFINITION 3.12 Let codim(V,X) = 2 and let D be a Weil divisor on X. Then the (rational) total transform of D in X is: j*D = D + qE,
where E is the exceptional divisor on X and q is a rational number uniquely determined by the equation: (D + qE) • E • HT~2 = 0. If D ~ aH + bR, then we find that q = 4. If D is effective, since D does not contain E, we have that b = D • E • Hr~2 > 0, i.e. q > 0. Mumford's total transform j*D is in general a Q-divisor and it is integral if and only if D is Cartier. As we will see, it is more convenient for our purposes to consider, if D is effective, the round-up \j*D~\ of j*D, i.e. the smallest integral divisor on X containig j*D. So let us give the following definition:
DEFINITION 3.13 Let codim(V,X) = 2. Let D be an effective
Weil divisor on X,
we define the integral total transform D* of D as:
D* = \j*D]=D+\q}E where D ~ aH + bR is the proper transform of D, q = 4 is the same number appearing in Def. 3.12 and \q~\ is the round-up of q, i.e. the smallest integer > q. We define the total transform of -D as (-£>)* = -D*.
NOTE 3.14 It is a simple computation to get the following equivalent expression of D*. Let D ~ dR be effective, i.e. d > 0, and divide d - 1 = kf + h (k > —1 and 0 < h < f). Then: D* ~(k+l)H-(f-h-
l)R.
(3.15)
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191
The relations between these coefficients and the ones in Def. 3.13 are k+1 — a+ \q] and / - h - 1 = f\q\ - b. From formula (3.15) it is clear that D* is uniquely determined by the class of linear equivalence of D ~ dR, i.e. by the degree d.
PROPOSITION 3.16 Let codim(V,X) = 2. Let D be an effective Weil divisor on X. Then the integral total transform D* — D + \q]E is the biggest Cartier divisor C on X such that j » ( C ) = j#(C) = D. More precisely:
j.(D + aE) = j#(D + aE} = D if and only if 0 < a < \q]; in this case j<J-f)+aE\x ~ ^D\X-
Proof. In Remarks 2.4 we have seen that j#(D) = j»(D) = D and j#(D+mE) = D for every m € Z. Therefore it is enough to prove that j*(D*) = D and that D is a proper subscheme of j, (D* +mE) for every integer m > 0. Proving the first equality is equivalent to prove that j*(D*) does not have embedded components. Let us fix a divisor D' ~(f -h-
l)R on X; then F - D + D' ~ (k + l)H is a Cartier divisor on X cut out by a hypersurface F of degree k + 1. Let us consider on X the divisor j*F ~ (k + l)H. We have that D* + D' — j*F, in fact they are linearly equivalent and coincide outside E. Therefore the scheme-theoretic union F of the schemetheoretic images jtD' and j«D* is contained in j»(j*F} — F, which is equal to the scheme theoretic union of the strict images_j#D' = D' and j#D* = D. Therefore we must have F = F. This implies that F cannot have embedded components, since it is cut out by a hypersurface on the arithmetically Cohen-Macaulay scheme
X. Since the ideal sheaf of F is given by the product of the ideal sheaves of j*(D*) and D', an embedded components of j*(-D*) would be an embedded components
of F, but this is not possible. On the other side j*(D* + mE) for m > 0 can not be equal to D; in fact D* + mE ~ (k + 1 + m)H - ((m + I)/ - h - l)R is not contained in j*F since j*F - D* - mE ~ -mE + (f - h - l)R is not effective. The isomorphism between the ideals follows from Lemma 2.5.
D
THEOREM 3.17 Let codim(V,X) = 2. A sheaf Ox(a,b) with a,be1 is reflexive if and only if b = degOx(a,b),E < f . Proof. The divisors on X of type dR with d > 0 are proper transforms of divisors on X, therefore the sheaves Ox(0, -d) are ideal sheaves on X by Prop. 3.16, and therefore they are reflexive. The divisors ~ H - (f - h - l)R with 0 < h < f are total transforms of divisors ~ (h + l)R on X; by Prop. 3.16 Ox(—l,f - h - I) with 0 < h < f are ideal sheaves on X, therefore they are reflexive by Th. 3.1. By the projection formula we have that Ox (a + a',b) = Ox(a,b) ® Ox(a',Q) for every a,a',b £ Z. Since the tensor product of a reflexive sheaf with an invertible sheaf is reflexive, we obtain that, the sheaves Ox(a, b) are reflexive for every a € Z and b < f . It remains to prove that the sheaves Ox(Q,d) are not reflexive for d > f . Let us suppose they are reflexive and divide d — I = kf + h as usual, with k > -1 and 0 < h< f; then the sheaves Ox(k, h+l) and Ox(0, d) with h
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As a consequence of Th. 3.17 we have that we can write the divisorial sheaf associated to a Weil divisor D in X in more than one form. To be more precise, let d > 0 and divide d = kf + h + 1 > 0, with k > -1 and 0 < h < f . Let
D ~ dR and let us suppose that D is not Cartier, i.e. d ^ 0 mod / (otherwise the representation is unique and it is Ox(D) = Ox(k + 1,0)), then:
Let D ~ — dR (now D may be Cartier), then
0X(D) - Ox(-(k + I),/ - ft - 1) s 0 x (-fc, -(/i + 1)) - . - . - Ox (0, -d). In the next, Corollary we esplicity describe the group Div(X), fixing a particular
form in which we write a divisorial sheaf. In this way it is evident the bijection between Viv(X) and C1(X) S 1.
COROLLARY 3.18
Let codim(V,X) = 2. Then: Viv(X) = {Ox(a,b)\a,b£Z,0
The natural group structure on T>iv(X) is given by: < Ox(a,b),Ox(a' ,b') >>->•
' + [ - } , b + b' mod /) andOx(a,b)v S Ox(-a,-b), where [ • ] denotes the integral part. Moreover for a fixed b : 0 < 6 < /, the set T>ivb{Ox(a>, b} \ a € It] is the set of divisorial sheaves of Weil divisors D ~ dR with d — b mod /. Proof. By Th. 3.17, for every d € Z, we can write the divisorial sheaf associated to D ~ dR in the form Ox(a, b) with a € Z and 0 < 6 < /. D
This particular choice is convenient if we want to compute h°(Ox(D)), in fact by (3.7) we know how to compute it.
COROLLARY 3.19 Let codim(V,J>0 = 2 - Let D ~ dR be an effective divisor; divide d = kf + h + I with k > — 1 and 0 < h < f . Let \D\ be the complete linear system of D. Then:
/ — 1 (i.e.
D is not Cartier), or
if D is Cartier.
Proof. Cor. 3.18,
(3.7), (3.5).
D
We conclude with the projection formula:
COROLLARY 3.20 (Projection formula) Let codim(V,X) = 2. Let D be a Weil divisor on X and let D* be its integral total transform. Then:
Proof. By Th. 3.17 j*O^(D*) is a reflexive sheaf; since it is the divisorial sheaf associated to D in the open set Xs, then by Th. 3.1 it is the divisorial sheaf of D an all X. D
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193
INTERSECTION OF WEIL DIVISORS
As we have already noted, an effective Weil divisor D on X C P™ is a closed subscheme of X of pure codimension 1 with no embedded components ([6], Prop. 2.4.). For this reason we can regard D C P" as a projective scheme. Given two effective divisors D and D' on X with no common components, we can consider the scheme-theoretic intersection D n D' C Pn and ask for its degree. For degree of a scheme Y C P™ we mean the lenght h0(OyL) of the zero-dimensional scheme YL which represents a generic (codim y)-dimensional linear section of Y. As we will see, this problem has an immediate solution when codim(F, X) > 2, via the isomorphism Cl(X) = CaCl(X). In this section we show how to use the integral total transform to compute this degree in case codim(Vr, X) — 2. If codim(V, A') > 2, by Cor. 2.2 we can define an intersection form on Div(X): I : Div(X)r
-> H->
Z Di-Dy-Dr
exactly as in X; i.e. 7 is determined by the rule:
Hr = d
Hr~l-R=l
Hr~'2-R2 = 0.
The intersection form / determines the degree of the intersection scheme Y = DI H • • • n DI C Pn of I (I < r) effective divisors which intersect properly. If codim(V, X) = 2, the linear theory of intersection developed by Mumford in [7] in the case of a normal surface can be generalized to our case by D1---Dr=j*D1---j*Dr
where j*Di is the Mumford's total transform of Di. Given two effective Weil divisors D and D' with no common components, the intersection number D • D' • Hr~2 does not represent in this case the degree of the intersection scheme Y = D n D1 C P™.
To compute this degree we will find the minimal resolution of Oy as Ox-module (Th. 4.7); other applications of this resolution are described in the next section. First we need to prove some properties of the integral total transform.
LEMMA 4.1 Let codim(y, X) = 2 and let D be an effective Then:
Weil divisor on X.
Pa(D}=Pa(D").
(4.2)
Proof. Let D ~ dR and divide d — 1 = kf + h with k > -1 and 0 < h < /; since D* ~ (k + \)H — (f — h — l)R, with / — h — 1 > 0, from the exact sequence 0 -> lD,\x ~^ @x ~* OD* -* 0 by (3.6) we obtain the exact sequence
0 -> j*lD.\x ~* J*°x ~* J*®D' -> 0. Since j*ZD.\x ~ %D\X by Prop. 3.16 and j*Ox = Ox, we get j*Oo' = OD- By (3.7) we obtain Pa(D)
= 1 - X(0D) = 1 - x(0D-) = Pa(D').
a In the next lemma we analize the behaviour of the total transform with respect to the sum of Weil divisors. First we need the following definition:
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DEFINITION 4.3 With the notation of Def. 3.13 define: e := IYI -
By definition e is a rational number in the interval [0, 1[ of the kind e = ^j- with I = 1, 2, . . . , / . With the notation of (3.15) we have:
f-h-l ——J—— We note that given D ~ dR on X, e is uniquely determined by the class of linear equivalence of D, i.e. by d. £=
LEMMA 4.4 Let codim(V, X) = 2 and let DI ~ d\R and DI ~ d^R be two effective divisor on X . Then: (4.5)
'
where [ • } denotes the integral part, Proof. By definition 3.13 of integral total transform we have:
since the proper transform of Z?i +£>2 isDi+D?. We find the two cases [( [?i 1 + [92! if 0 < ei + e2 < 1 and \qi + q2] = \qi~\ + fel - 1 if 1 < e: + e2 < 2. D
THEOREM 4.6 Let codim(V,*) = 2 and let Y C X C Pn be a "complete intersection" of two effective divisors D\ and D^ on X. The following sequence is exact:
0 -> j.Ojt(-(Dl + £> 2 )*) -> j.O^-D^Qj.Oxi-DS) -> IY{X -* 0
(4.7)
and therefore it is a "reflexive resolution" of OY as a Ox -module. Proof. First let fa : Ool+D-2 -* Oot for i = 1,2 be the projection and let
0 -> 0Dl+D,
2
0Dl ® 0D,
l
^ 2 0 Dl no 2 -* 0.
(4.8)
For any effective Weil divisor D on X we have already seen the resolution of OD'
Q^jtOx(-D*)-+Ox
->0c->0
(4.9)
in the proof of Lemma 4.1. The mapping cone (see [1] pg. 432, pg. 657) between the resolution (of type 4.9) of OD^+D-Z an^ of OD-I © Oo2 gives a resolution of
j.Ojf (-(A + Z? 2 )*) -> Ox e j.Ojf (-DI) ® j.Ojt (-^) -> 0x ® Ox. In this resolution we can suppress redundant terms and obtain the required resolution:
j.Ox(-(Di + D 2 )*) -> j.Ojf (-Z?I) ® J.Ojf (-D2*) ^ OxD
Well divisors on rational normal scrolls
195
NOTE 4.10 The resolution 4.7 allows us to find a resolution, in general not minimal, of Oy as Op*, -module. Indeed, as shown in [10], a minimal resolution of Ox (a, b) as Opn-module is given by the Eagon-Northcott type complex Cb (a) for 6 > — 1. Since each of the terms in 4.7 is a sheaf Ox (a, b) with 6 > 0, then a suitable mapping cone between the complexes Cb(a)'s gives us the required resolution in Pn.
PROPOSITION 4.11 Let codim(V,X) = 2 and let D and D' be two effective divisors on X with no common components. Then the degree of the "complete intersection" scheme Y = D n D1 is given by: r 2
-'
e'-l) + l
if {f. + e'} = I
Proof. Let us call XL , YL , DL and D'L general (r - 2)-dimensional linear sections of X, Y, D and D' respectively, and let us call XL the canonical resolution of the rational normal surface XL. The resolution of OyL as an OXL -module is by Th. 4.6:
By the proof of Lemma 4.1 we have that x(J*@xL(—D*)) = pa(D*) for every effective divisor Don XL . Let [e + e'] =0; by Lemma 4.4 (DL + D'L)* = D*L + D'L*. Looking at the resolution of OYL , by adjunction formula we compute:
= DI-D'L* = D* -D" -
,
where K%L ~ -2H + (f - 2) ft is the canonical divisor of XL. Let [c + e'} — 1; by lemma 4.4 we have that (DL + D'L)* = D*L+D'L*-E. By an analogous computation we find:
h°(0YL)
= Di-D'L'-(D'L+D'L*)-E-±(KxL-E)-E =
D" • £ " • • Hr~2 + f ( e + e' - 1) + 1 D
We note that when [e + e'} = 1, the quantity f(e + e' - 1) is bigger than or equal to zero, therefore in this case deg(D n D') is strictly bigger than the number D* • D'* • Hr"2. The scheme theoretic intersection Y contains the vertex V as a component with a certain multiplicity > 0, we call this number the integral intersection multiplicity m(D,D';V) of D and D' in V, i.e.
m(D, D'; V) = deg(£> n D') - D • D' • H7""2. If D ~ aH + bR and D' ~ a'H + b'R are the proper transforms of D and D', then by Prop. 4.11 we explicitly compute:
f(f^
if[c + e'] = 0
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Ferraro
NOTE 4.13 The intersection multiplicity of two effective divisors D and D' with no
common components through the singular locus V on a normal surface X , is defined in the linear intersection theory of Mumford as the rational number: i(D,D';V) — j*D • j* D' — D • D' . If X is a rational normal cone, using the same notations as above, we find that the linear intersection multiplicity i(D,D';V) is exactly ^j-.
5
EXAMPLES AND APPLICATIONS
In this section we show some applications of the previous results. In particular in Ex. 5.4 we use Th. 4.6 to compute the arithmetic genus of the scheme theoretic intersection Y of two effective divisors on a rational normal cone. EXAMPLE 5.1 In this Example we show that every effective non degenerate divisor on a rational normal cone X of degree n — l in P™ is a curve of maximal arithmetic
genus pa(C) - G(n,d). Let C ~ dR with d > n — 1, let us divide d — 1 = m(n — 1) + S with m > 1,
0 < o < n - 2; then G(n,d) = (™)(n - 1) + m6. By Lemma 4.1 we know that Pa(C) — pa(C*), where C* ~ (m + l)H - (n - 2 - S)R. By adjunction formula on X we then compute pa(C*) — G(n,d).
EXAMPLE 5.2 Let codim(V, X) = 2, then every effective divisor D and every "complete intersection" Y of two divisors D, D' on X C P" is arithmetically CohenMacaulay. In the case of one divisor we know by (4.9), (3.7) and (3.4) that
for 1 < i < r — 1 and every k. Since X is aritmetically Cohen-Macaulay from the exact sequence 0 -4 IX\P" ~> %D\fn ~^ %D\X -» 0 we conclude hl(XD\y,n(k)) — 0 for 1 < i < r - 1 and every k. Looking at the resolution (4.7) and using (3.7) and (3.4) we compute: h l ( X y \ x ( k ) ) = 0 f o r l < i < r — 2 and every fc; as in the previous case, since X is aritmetically Cohen-Macaulay we conclude ht(IY\p«(k)) — 0 for 1 < i < r — 2 and every k.
EXAMPLE 5.3 If codim(V,X) > 2 and Y is a "complete intersection" of I (I < I < r — 1 ) divisors D, ~ a,iH ~ b{R with hi > 0, then the resolution of Oy as an Ox -module, is a Koszul complex (see [10], Ex. 3.6.): 0
-4
Ox(-(ai + - - - + o / ) , 6 i + • • • + &;) -» ...
From this resolution using (3.7) and (3.4) we conclude that Y is arithmetically Cohen-Macaulay iff bi + • • • 6; < /.
EXAMPLE 5.4 Let codim(7,^) = 2 and r > 3. Resolution 4.7 can be used to compute the arithmetic genus of the intersection scheme Y of two effective divisors D and D' with no common components.
Well divisors on rational normal scrolls
197
Let us suppose that Y is non degenerate, i.e. d, d' > f , then using (3.7), (3.4) and (3.5) we compute: Pa(Y)
= h0(Ox(Kx + (D + D')*})-h0(Ox(Kx + D*}-h°(Ox(Kx+D'*}
where a_ t = [f ] and /3_! = /[f ] -d'; Ol = [fj and ft = / f f | -d;
and /?o = /[ i' +f ^c l — (^' + d). With these notations the degree of Y is:
References [I] D Eisenbud. Commutative Algebra with a View toward Algebraic Geometry. GTM 150, Springer Verlag, New York, 1994. [2] A Grothendieck, J Dieudonne. Elements de Geometric Algebrique. Die Grundlehren der mathematischen Weissenschaften, Band 166. New York: Springer-Verlag, 1971. [3] D Eisenbud, J Harris. On varieties of minimal degree ( a centennial account), Proceedings of the AMS Summer Institute in Algebraic Geometry, Bowdoin, 1985. Proceedings of Symposia in pure Math., 46, AMS, 1987. [4] R Ferraro: Curve di genere massimo in P5 and Explicit Resolutions of Double Point Singularities of Surfaces. PhD dissertation, Universita di Roma "Tor Vergata", 1998.
[5] R Hartshorne. Algebraic Geometry. GTM 52, Springer Verlag, New York, 1977. [6] R Hartshorne. Generalized divisors on Gorenstein schemes. K-Theory Journal,
8, 1994, pp. 287-339. [7] D Mumford. The topology of normal singularities of an algebraic surface and
a criterion for simplicity. Publ. Math. I.H.E.S. 9, 1961, pp. 229-246. [8] M Reid. Chapters on Algebraic Surfaces in Complex Algebraic Geometry. Lectures of a summer programm Park City, UT, 1993. Kollar J., lAS/Park City Math. Series 3, AMS, Providence, RI, 1997, pp. 5-159. [9] F Sakai. Weil divisors on normal surfaces. Duke Math. J., 51, 4, 1984, pp. 877-887.
[10] FO Schreyer. Syzygies of canonical curves and special linear series. Math. Ann., 275, 1979, pp. 105-137. [II] J. P. Serre. Prolongement de faiseaux analytique coherent. Ann. Inst. Fourier, 16, 1966, pp. 363-374.
Families of Wronskian Correspondences LETTERIO GATTO1 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 TORINO - Italy and Departamento de Matematica, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire S/N , Cidade Universitaria, Pe -Brazil, email: [email protected] and [email protected] Alia mia terra, net giorno del mio ritorno
1
INTRODUCTION
Let TT : X—>S be a proper flat family of smooth curves. The purpose of this paper is to detect the fibers of the family carrying special wronskian correspondences, to be defined below. This is a way to produce cycle classes in Mg, the moduli space of smooth project!ve curves of genus g , corresponding to loci of curves possessing certain special linear series. The original motivation of this research was to look for divisors in Mg whose
closure in Mg, the Deligne-Mumford compactification of Mg by stable curves ([!]), had an easily computable class in the group Pic(Mg)
smooth curves. Nevertheless, recently, the author and E. Esteves ([4]) used a wronskian correspondence on a family of_stable hyperelliptic curves to recover a relation in
Pic(Hg) <8> Q, where Hg C Mg is the stable hyperelliptic locus, whose first proof was based on a rather technical argument ([5]). Hence, this paper is also intended to be a reference for the quoted [4] and the forthcoming [6].
Let C be a general smooth projective complex curve of genus g > 3, and let a, b be two positive integers such that a + b = g. The simplest and most natural example of wronskian correspondence is the cycle class in Al(C x C) associated to the locus 1 1991 Mathematics Subject Classification: 14H10, 1415, 14H55, 14H99. Key words and Phrases: Partial Jets, Families of Wronskian Correspondences, special Weierstrass points, moduli spaces of curves, divisors classes in M 9 . Work partially supported by MURST (Italy), Progetto Nazionale "Geometria Algebrica, Algebra Commutativa e Aspetti Computazionali" (coordinatore Claudio Pedrini) and CNPq (Brasil), contrato n. 451574/98-0 and processo n. 300680/99-6(NV).
199
200
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Wr (a, b) of points (P, Q) such that aP + bQ is part of a canonical divisor. Applying the formulas of Sect. 5 to the canonical linear series, such a cycle class is given by:
(1) in A1 (C x C), where A is the class of the diagonal of C x C, and K^ is the pull-back of the canonical bundle on C via the projection pt. Notice that JCxC[Wr(a, 6)] • A = (g — 1 )g(g + 1), the total weight of Weierstrass points on C, as it should be. The locus Wr(a, b) admits the following simple interpretation: If (P, Q) € Wr(a, b) and the curve C has general moduli, then Q is a ramification point (Section 2) of the linear system Vap = \K — aP\. Clearly, if P is general, the linear system has no special ramification points. Hence one may look for all the pairs (P, Q) € C x C such that Q is a special ramification point with respect to Vap. Let us call wtya(2) such a locus, whose associated cycle class is what we call a special wronskian correspondence. The way to determine such a class relies on the following fact: there is a a multiwronskian section (Section 4) of type (a, 6), u A . . . A Da~lu A w A ... A Db~lu of the line bundle (pf tf)®2*?11 ® (p^)®*^ that vanishes along Wr(a, b) and along the diagonal with multiplicity ab, so that, actually, Wr(a, b) is the zero locus of a section W a> 6 of the bundle:
It follows that Q is a special ramification point of the system Vap if and only if the partial derivative of W0,6 with respect to the second factor vanishes. In other words:
wtya(2) = Z(D2(Wa,b)), where, in the right hand side, one has the zero scheme of a section of the partial jet bundle (Section 3) :
After checking that the expected and the actual codimension of the locus coincide, one may compute the degree of [wtvaP(2)}.
CxC
JCxC
= 2a(g - 1) • [(a + 1)(62 + b+l)(g-l)- b(g2 + g-ab+ 1)],
(2)
where the computations have been performed by using the fundamental exact sequence (Proposition 3.1) for partial jets bundles. If g = 3, a = 2 and 6 = 1 one gets fCxC[wtva(2)} = 56, which is exactly twice the number of the bitangents of a smooth plane quartic. One can continue the game and look for the class of the
locus wtVa(3) = {(P,Q) € C x C\ wtVap(Q) > 3}. For general C this locus is empty, but for special C this locus may be even positive dimensional: this occur for instance when C possesses Weierstrass points of type g - 1, i.e. such that h°(C, (g - 1)P) > 1. This implies that D|(w A ... A D9~2u/\u) does not vanishes in the expected codimension in X x s X. We plan to discuss this case in more detail in [6]. In the last section, we study in detail, as a guiding example, the case a = g — 2 (hence b — 2), proving that (Theorem 6.2) each irreducible component of tu(3), the
Families of Wronskian Correspondences
201
locus of curves possessing a special wronskian correspondence of type (g — 2,2), has the expected codimension 1 in Mg. If W is an irreducible component of w(3) and m\v is the number of pairs (P, Q) such that wtvig_2}P (Q) > 3 on the general curve belonging to W, then (Corollary 6.3) states that in Pic(Mg), the Picard group of the moduli functor of complex smooth protective curves of genus g, the following equality holds:
W] = 6(9a5 - 78g4 + 331g3 - 780#2 + 838# - 244)A.
(20)
w It should be remarked that the purpose of this paper is to emphasize the techniques used. But clearly new questions, which do not seem to be easy (at least to me),
arise. For example: is w(3) irreducible? If not, what are its irreducible components and what are the multiplicities mn/7 If it is irreducible, what is the number M of pairs (P, Q) such that wtv
PRELIMINARIES
In all the paper we work over the complex field (or any algebraically closed field of characteristic zero). The basic references for the entire paper are [1] for the notion of stable curve, [7] for the Picard Group of the moduli functor of smooth curves and [8] for special linear series. All such foundational material is covered in [9] as well. For the basic notation of intersection theory we refer to [10]. We refer to such a book also for the notion of correspondence. If TT : X—>S is a proper flat family of smooth projective complex curves, we recall that a point P € X is said to be a ramification point for a given relative linear system (V, £) of rank r and degree d (Section 4) if and only if the vector bundle map over X, Dr : x*V—>Jr£, defined by Dr(Q, v) = Drv(Q) drops rank at P or, which is the same, if the section Ww = A r+1 £> r of the bundle A r+1 (J r £) ® A r+1 (7r*F v )
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Gatto
vanishes at P. The class of the ramification locus of the grd in A1 (X) is given by (see [11], [12]): „/„
i
1\
(3)
where Kv is the relative dualizing sheaf of the family. If 5 = Spec(C) one has the Brill-Segre formula ([12]):
c 3
Z(W)] = (r + l ) d - ( 0 - l ) r ( r + l).
(4)
PARTIAL JETS
Let TT : X—>S be a proper flat family of stable curves of genus g parametrized by a scheme of finite type over C. A complex stable curve of genus g is a connected projective curve having only nodes and such that each smooth rational component must intersect the rest of the curve in at least three points (see [1] for details). The title partial jets of this section means that we want to learn how to perform partial derivatives of sections of line bundles on families which are fc-fold fibered products of our given TT : X—>5. We restrict our attention to the case of only two factors. The general case work with obvious modifications and is discussed in [13]. We assume known the construction of jets bundles for families of stable curves as in [2] or [14]. Let TT : X——)S be a stable curve over 5. As it is well known the sheaf of relative differentials £llxis comes equipped with a universal derivation dx/s '• Ox—>H^. ,s. There is a natural map 72 : £llx,s—^K^, which restricts to the identity over smooth points, where K* is the relative dualizing line bundle of the family TT (see [1], p. 76-77). Let d* := 72. o dx/s : Ox—^ be the composition. Adopting a terminology used by R. F. Lax ([15]), d^ will be said to be the exterior derivative along the fibers of TT. Now let X x5 X be the fiber product over 5 of X by itself, fitting in the following cartesian diagram: XxsX
P2
——————>• X
X
By the universal property of dn, it turns out that p*d^ is the exterior derivation along the fibers of PJ. We set, d^i = p^d^ and dn^ = Pid*- Clearly, d^ti : Oxxsx—>KPj = P*KT,, (i ^ j). Let / e Ox*sx(U) where U trivializes Kpj via a local generator CTJ. Define 9,(/) by the equality: dn,i(f)
= 9if ' O,.
(5)
Clearly dif e Oxxsx(U], so that the higher order partial derivatives can be recursively computed. Now, let £ € Pic(X xs X / S ) , i.e. a line bundle over X *s X. By simplicity of exposition, we shall assume that h°(£) > 1, Let U =
203
Families of Wronskian Correspondences
{(Ua;tpa,aa)} be an open covering of X xs X trivializing simultaneously £ and, say, Kpi , for a given i e {1, 2}. If A <E H°(X x s X, £), then we have: A^ = £a^a, where la £ Oxxsx(Ua). Consider now the collection Z?*(A) = {Df(X)\Ua} = {(Uc,;ia,dila, • • • ,3fO}. The reader can easily check that on triple overlapping the set of transition functions between the data (U;d^a)o
PROPOSITION 3.1 Let X^ be the k-fold fibered product of X over S and let £ € Pic(XJ/S). Then for each i £{!,..., k}, each q > 0, the following exact sequence holds:
^Jqi-lC-^Q. 4
(6)
MULTIWRONSKIAN SECTIONS
Let (V, £) be a grd on a stable curve ?r : X —>S over S, i.e. V is a locally free subsheaf of rank r + 1 of 7r*£, where £ € Picd(X/S) and the induced map Vs := V <8> k(s) t-» H°(A's,£|;i, ) is injective. Let K^ be the relative dualizing sheaf of the family and let a, b be positive integers such that a + b = r + I. Construct the fiber product p : X xs X —>S. Notice that for each i € {1,2}, p; o ?r = p, where Pi : X xs X —>X is the projection onto the i-th factor. Construct Ja~lC and Jb~l£, the jet bundles of order a - 1 and b - I of £, relatively to the projection IT. Such a jet bundle is constructed according to [2]. Form the rank r + 1 vector bundle pi Ja~l£ ffi p^Jb~l £. Then we have the natural evaluation map:
(7)
S defined by: ((P^.A) * ( ( D a - l X ) ( P 1 ) , ( D " - l X ) ( P 2 ) ) , where Da~l\ £ a l b l b l H°(X,J - £) (resp. D ~ \ £ H°(X,J ~ C)). Let Ui,U2 C X be open sets such
that H°(Ui, £) = Ox(Ui) • ^ and H°(Ui, K,) = Ox(Ut) • a. If A is an Os-basis of V, one has Aj^ = £j • ipi. It is not difficult to check that the local representation of Da~l ® Db~l on the open set Ui x s [72 is given by: )
a a(b-l)\T )\ ) i2 ' ' ' ' ' -2 /
(8)
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Gatto
where the superscript T denotes the transpose of a matrix and the derivatives are taken in the sense explained in Section 3, according to [2], i.e. dv^ = £•l'<jj. However, in the diagram (7), we want rather to consider the exterior product of the map Da~l © D*-1, which is: A r+lpa-l 0 D 6 - 1 )
. A r + 1 (p*V)——» A r+1 (p* ja-1 £ & p» J&-1 £ )
(g)
whose local representation, in the open set U\ xg [72 is, by construction, given by the determinant of the matrix (8). Moreover, since A r+1 pTv is a line bundle, r+1 (pJ Ja~l£® A r+i (£a-i 0£>&-i) may be identified with a section of the line bundle A b l r+1 v p^J ~ C)
I
(10) DEFINITION 4.1 Tfte section A^-D0-1 ©D 6 " 1 ) is said to be the multiwronskian section of type (0,6) associated to the given grd.
5 5.1
FAMILIES OF WRONSKIAN CORRESPONDENCES Relative Wronskian Correspondences
The set up is as in Section 4 with the exception that from now on we shall only work with families of smooth curves. The multiwronskian section:
vanishes along the diagonal A, as suggested by the notation itself, which is a Cartier divisor. If L : X xs
Wrv(a,b)(S) = i*(Z(W+i(Da-1 ®Db~1))), where the closure is taken in X x 5 X.
DEFINITION 5.1 The class \Wrv(a,b)](S) 6 Al(XxsX)
oftheschemeWrv(a,b)(S)
is said to be a wronskian correspondence over 5. If S is a point, Spec(C), then it
is simply called a wronskian correspondence. In other words, a wronskian correspondence over S is a family of wronskian correspondences, one for each fiber of the family TT. This explains the title of the paper. Geometrically, a wronskian correspondence of type (a, 6) is the cycle class associated to the closure in C x C of the locus of pairs of distinct points (Pi,P2) such that the divisor a PI + bP-2 imposes less than r + l conditions on the linear system of hyperplanes of P(V) cutting the image of C through the morphism induced by the given grd.
EXAMPLE 5.2 Let C be an hyperelliptic curve and let i: C—>C the hyperelliptic involution. An important example of wronskian correspondence is the class of the locus of points (P, t(P)) in Al(Cx.C). Off the diagonal, it coincides with [Z(A2(D°© D°)], where A 2 (£>° © D°) <E HQ(C,pl£ ® p^C) and (H°(C,£),C) is the g\. The
Families of Wronskian Correspondences
205
study of such apparently innocuous (and well known) wronskian correspondence plays an important role in [4], where some theorems of the paper [5] are revisited. In particular in [4] it is studied the wronskian correspondence associated to an admissible relative g\ on a family of stable hyperelliptic curves. 5.2
Vanishing Along the Diagonal.
Given a family of wronskian correspondences, by its very definition it is clear that Wry (a, b) is the zero locus of a section of the line bundle:
where m is the order of vanishing of Z(f\r+l(Da~~l © D6""1)) along A, so that, at the level of classes one has:
[Wrv(a,b)] = \Z(f\r+l(Da-1 @ L^1))] - mA = a(a-l)\ „ ;
fb(b-l)\ r. ^ Ki +
^
/
(11)
where we set Li — c\(p\C) and KI = ci(plK^). For future use, set also L — ci(£) and K = ci(Kn). Our next goal is to determine the integer m. The most direct way to find it would be to write a multiwronskian section of type (a, b) and to apply the formalism of the partial jet bundles to compute the multiplicity with which the sections vanish along the diagonal. This method is illustrated in an example in [6]. Such a direct approach, although quite simple, may be combinatorially unpleasant.
For this reason, we shall use a more geometric argument.
LEMMA 5.3 The following equality holds:
[ W r v ( a , b ) ] ( S ) - & = [Z(y/,)].
(12)
Proof.
The cycle class [Wry (a, b)](S) is associated to the locus of (Pi,P-2) £ X Xs X, such that dim(Vs(—aPi — bP^)) > 0, where s = p(Pi). Intersecting with the diagonal A amounts to look for pairs such that PI = P2, i.e. for pairs (P, P), such that dim(Vs(-aP - bP)) > 0, i.e., for points P such that dim(Vs((-a - b)P)) > 0, i.e. exactly to Z(Wn}. The equality stated is hence proven.
QED PROPOSITION 5.4 The multiwronskian section of type (a,b] vanishes along A with the multiplicity m = ab. Proof.
Let 6 : X «-> X xs X be the diagonal embedding defined by pi o <5 = p2 o 5 = id*. For i — 1,2, we have the following identifications:
Li-k = 6*(Ll) = ( P i ° 5 ) * ( L ) = L Ki • A = S'(Ki) = (Pi o 5)*(K) = K A 2 = -KtA = -6*(Ki) = -K p*Ci(V) = (po5)*ci(V) = n*ci(V).
(13) (14) (15) (16)
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Gatto
Now, use expression (11) for [Wry(a,6)] in equality (12), using the identifications (13), (14), (15), (16) arid the fact that a + b-r + l. One has: , ,.T fa(a-l) b(b-l); \ _. (a + b)L + ( ~~—- + v +m)K-
Hence, equating the coefficients of K one has: a(a-l) _ 6(6-1) ____ _ _ + m = _ r(r _ _+_1) + implying that m = ab, as desired.
QED REMARK 5.5 The integer m = ab found above is exactly what is classically known as the valence of the correspondence ([10], p. 309).
6
SPECIAL WRONSKIAN CORRESPONDENCES OF TYPE (j-2, 2)
Let TT : X —>S be a family of smooth curves. If 5 = Spec(C) = {pt}, it is simply a smooth projective curve of genus g. If C is a curve and P € C let Vap — H°(C,Kc(-aP)). Define:
Vw(k)(S) = {(P,Q) £XxsX: wtV(g_2)p(Q] > k}
(17)
where we say that the weight of Q, wtvaP(Q), with respect to Vap, is > k (and we write wtvaP(Q) > k) if (Dk~LW)(Q) = 0, W being the wronskian associated to the linear system Vap • For sake of brevity, from now on we shall omit the reference to the base 5, the latter being clear from the contest. For example, if C is a general curve (hence it has no Weierstrass point of type g — 2, see below), Vw(T) is the locus of points in C x C defined by all pairs (P, Q) such that (g - 2)P + 2Q is contained in a canonical divisor. The purpose of this section is to find an expression in Pic(Mg), the Picard group of the moduli functor of smooth curves of genus g > 4 of the locus:
io(3) := wtV(!i_2)(3) = {[C] e Mg/C possesses a pairs (P,Q) such that Q is a ramification point of V( 9 _2)P of weight at least 3}, We start by equipping to (3) with a structure of closed subscheme of Mg. To this purpose denote by M° the moduli space of smooth curves without automorphism. The complement of M° in Mg has codimension 2. There is a universal curve TT : C° —>M°. In C° XMO C° we may consider the special wronskian correspondence [Vw(3)°] associated to the locus Vu>(3)° which is described, away from the diagonal, as the zero locus of the section D'2(/\9(Dg~'2 © D)) of the bundle J| (L) where we set, for brevity,
L = (p^)®""2"*'" ® (P*M®Z ® p* A9 Ev.
(18)
Families of Wronskian Correspondences
207
As previously discussed such a section induces a section D 2 (W 9 _2,2) of the bundle J|(L(-2(g- 2)A)). Then Vw(3) is scheme theoretically defined as Z(P 2 (W g - 2 ,2)The map p is proper, and hence w(3)° = p(Vw(S)°). One then set w(3) = w(3)°, the closure being taken in Mg.
This defines w(3) scheme theoretically and it is clear that each irreducible component of w(3), if non-empty, has codimension at most 1, since each irreducible
component of Vw(3) = p~1(w(3)) has codimension at most 3 and the fibers of p are 2-dimensional. The main result of this paper is that each irreducible component of Vw(3) has the expected codimension 1 in Mg. We start with a proposition that probably belong to the folklore of the subject and is certainly known to specialists. PROPOSITION 6.1 Let C be a general smooth complex curve of genus g. Then Vw(2) is finite. Proof.
Let pi : C x C—>C be the projection onto the i-th factor. Consider the restriction of pi to the locus Vui(l) C C x C. Then pi : Vw(l)—>C is a flat morphism of relative dimension 0 of degree ^ wtyp (Q) = 4g — 2. The locus of points Q such that wtvP(Q) > 2, Z(Z? 2 (>V 9 _2,2)), is exactly the locus where the differential of pi vanishes. But each irreducible component of Vw(l) has only finitely many ramification points, so that the general (P,Q) € Vw(l) is such that wtvP(Q) = 1-
QED The proposition implies that the locus Vw(2) has the expected dimension 0 in CxC.
Hence, the total weight of such points in C x C is simply: = /
02 (J21(L(-2(.9 - 2) A))) = 2(g - l)(g - 2)(552 - I2g - 3). (19)
JCxC
It is worth remarking that the above formula, for g = 3, yields 24, the total weight of Weierstrass points on a curve of genus 3, as it should be. We may now prove our main theorem.
THEOREM 6.2 Each irreducible component of the locus Vu>(3) has the expected codimension 3. The morphism p : Vw(3)—>w(3) is generically finite and, hence, w(3) has the expected codimension 1 in Mg. Proof.
Let [C] be a general element of w(3). Then, for each P &C, w(P)A.. .AL>»-3a;(P) ^ 0 (or, put otherwise, the Brill-Nother matrix ([8], p. 154) associated to the divisor (g - 2)P has maximal rank). This is because if tj(P) A ... A D9~3ui(P) - 0 then [C] would belong to the locus -D 9 _2 of the curves possessing a Weierstrass point P such that h°(C, (g - 2)P) = 2. But such locus has codimension 2 in Mg ([16]), so that the general element of w(3) does not satisfy such a condition which would
invalidate all the arguments below. To prove the claim it is sufficient working with a general family of smooth curves parametrized by a complete base which does not contain curves belonging to Dg-2- This is possible by the above remarks. Let TT : X—>S be one such and
208
Gatto
let p : X xs X —>S be the fiber product of the family with itself over S and let us denote by pi the projections, as usual. By Proposition 6.1, the restriction of the map p to Vw(2) —>S is a flat morphism of relative dimension 0 and of degree N(g) (formula 19). Notice that such a result is compatible with the fact that the Weierstrass points of weight at least 2 occur in codimension 1 in S. Now, if a fiber Xs possesses a pair (P, Q) € Viu(3), the differential of the restriction of pi to Vu>(2) vanishes at (P, Q). Since p = -n o p\, one has dp = dx o dp\ so that dp vanishes along Vw(3) that is then contained in the ramification locus of the map Vu>(2) —>S. Hence Vw(3) is zero dimensional in X x§ X, implying that w(3) is actually a divisor in 5, concluding the proof. QED The theorem above ensures that the computations below make sense.
COROLLARY 6.3 The following equality holds in in Pic(Mg):
=
p,[Vt«(3)] =
w
= 6(9g5 - 78g4 + 33103 - 78002 + 8380 - 244)A.
(20)
where the sum is over all the irreducible components W 's of w(3) and raw M the number of pairs (P, Q) such that wtv(g_y)P (Q) > 3 on the general curve belonging to W. Proof.
It is sufficient to evaluate both members of the above expression on a family TT : X —>S of smooth curves parametrized by a complete smooth base. The first equality is due to the definition of the push-down of cycles, and from the definition of iu(3)(S). The second equality is only a matter of computation. In fact, we know that: Since the expected codimension of the zero locus of the section D%(
\Dl(u A . . . A D3~3iJ A w A Du)} = c3 (j|(L <8> Ox*(-1(g - 2) A)))
(21)
where L is as in formula (18). Using the fundamental exact sequences (Prop. 3.1): 0 -> L(-2( S - 2)A) ® (p'2K*)®2 -> J22(L(-2(S - 2)A)) -+ 4(L(-2(g - 2)A)) -y 0,
and 0— »L(-2( 5 - 2) A) ®p*J^-4J21(L(-2(5 - 2)A))— >L(-2(j - 2)A)— >0, one can express the third Chern class above as:
c 3 (J|(L(-2( f l -2)A)))
- ( c 1 ( L ) - 2 ( S - 2 ) A ) . ( c 1 ( L ) - 2 ( ( / - 2 ) A + /ir 2 )2X2), (22)
Families of Wronskian Correspondences
209
where, again, we do not distinguish the divisor A from its first Chern class. The rest of the proof follows from a tedious but essentially trivial computation, using the following push-down rules:
where KI - K*(ci(Kn)'2), A = cj.(E) and KI = 12A ([17]).
QED REMARK. We do not know if w(3) is irreducible. If it were so, we would have a formula for the class [u>(3)], namely the right hand side of (20) divided for the number (deg(p\Vwm)) of pairs (P, Q) such that wtv(g_2}P (Q) > 3 on the general curve belonging to w(3), which is still a unknown datum.
REFERENCES
[I] P. Deligne, D. Mumford, On the irreducibility of the space of curves of given genus, Publ. Math. I.H.E.S., 36 (1969), 75-109.
[2] L. Gatto, On the closure in Mg of Smooth Curves having a Special Weierstrass Point., Math. Scandinavica, to appear. [3] L. Gatto, F. Ponza, Derivatives of Wronskians with applications to families of special Weierstrass points, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2233-
2255.
[4] E. Esteves, L. Gatto, A Geometric Interpretation and a new proof of a relation by Cornalba and Harris, 2000 (alg-geom/0008176)
[5] M. Cornalba, J. Harris, Divisor classes associated to families of stable varieties, with application to the moduli space of curves, Ann. Scient. EC. Norm. Sup., 21 (1988), 455-475. [6] E. Esteves, L. Gatto, Special Wronskian Correspondences on families of Stable Curves, forthcoming.
[7] J. Harris, D. Mumford,On the Kodaira Dimension of the Moduli Space of Curves , Invent. Math. 67, (1982), 23-86. [8] E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of Algebraic Curves, Vol 1, Springer Verlag, 1984.
[9] J. Harris, I. Morrison, Moduli of Curves , GTM 187, Springer-Verlag, 1998. [10] W. Fulton, Intersection Theory, Springer-Verlag, (1984).
[II] H. Popp, Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Mathematics 620, (1977). [12] D. Laksov, A. Thorup, Weierstrass points and gap sequences for families of curves, Ark. Mat. 32 (1994), 393-422.
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[13] L. Gatto, The Recife Notes on Intersection Theory on Moduli Space of Curves, Monografias de Matematica, 61, Institute de Matematica Pura e Aplicada, Rio de Janeiro, (2000). [14] E. Esteves, Wronski Algebra systems on families of singular curves, Ann. Scient. EC. Norm. Sup. (4),29, 107-134 (1996).
[15] R.F. Lax, Weierstrass Points of the Universal Curve, Math. Ann. 42, (1975), 35-42. [16] E. Arbarello, Weierstrass Points and Moduli of Curves, Compositio Mathematica, 29-3 (1974), 325-342. [17] D. Mumford, Stability ofProjective Varieties, L'Ens. Math. 23, (1977), 39-110.
A Note on Hodge's Postulation Formula for Schubert varieties SUDHIR R. GHORPADE1
Department of Mathematics, Indian Institute of
Technology Bombay, Powai, Mumbai 400076, India.
1
srgQmath.iitb.ernet.in
INTRODUCTION
Given a projective algebraic variety X in the JV-dimensional projective space F^ over an algebraically closed field k, we can associate to X its Hilbert function, viz., the function h : N -> N defined on the set N of nonnegative integers by
h(i) = d\mk Re,
for I € N
where R denotes the homogeneous coordinate ring of X, i.e., R is the polynomial ring over k in N + 1 variables, quotiented by the vanishing ideal of X, and Rt denotes its £-th homogeneous component. It is well-known that h(t) is a polynomial function, for all large enough values of I , and this polynomial, called the Hilbert polynomial of X, contains a lot of useful geometric information about X. However, the Hilbert polynomial is not always easy to compute explicitly in substantial examples of geometric interest. Classically, the known cases seem only to include simple examples such as hypersurfaces, rational normal curves, and more generally, complete intersections. Recently, Hilbert polynomials of determinantal varieties of various kinds, have been explicitly determined [1, 3, 6, 8]. However some of these formulae are quite complicated. Schubert varieties in Grassmannians form an important class of projective varieties, which have been considered at least since the last century. A nice, short
and explicit formula for the Hilbert polynomial of such a variety, known as the postulation formula, was obtained by Hodge [10] in 1943. We feel that this formula deserves to be better known, especially since it arises in a number of seemingly different contexts, particularly in Combinatorics. Partly motivated by this desire, 'Partially supported by a 'Career Award' grant from AICTE, New Delhi and an IRCC grant from IIT Bombay.
211
212
Ghorpade
we describe in this note the formula of Hodge, derive a slightly simpler description of it and give an alternate proof of the formula. This will also provide us with an opportunity to briefly introduce combinatorics of nonintersecting lattice paths, which, in the same vein, deserves to be better known among the algebraists. The basic facts needed about Schubert varieties in Grassmannians are described in the next section. We review some combinatorial facts about lattice paths in Section 3. Finally, a proof of the postulation formula is given in Section 4.
2
SCHUBERT VARIETIES IN GRASSMANNIANS
Let k be a field, which, for simplicity, we assume to be algebraically closed. Let V be a vector space of dimension n over k. Given any integer d with 1 < d < n, the
Grassmannian Gj(V) is defined to be the space of d-dimensional linear subspaces of V. This has a canonical embedding
Gd(V) <-» P(A d K) ~ P J p 1 known as the Plucker embedding, and via this embedding Gd(V) has the structure of a projective algebraic variety. Briefly, the Plucker embedding may be described
as follows. Fix a basis {ei,...,en} of V. Then ea := eai A . . . Ae a d , where a varies over the indexing set I ( d , n) = {a = ( c c i , . . . , ad) € 1d : I < aj < . . . < ak < n}
form a basis of A.dV. The image of a subspace W £ Gd(V) is the point of f ( A d V ) corresponding to the one-dimensional subspace of A.dV spanned by wi A ... A Wd, where {wi,..., Wd} is (any) basis of W. If we write w\ A . . . A Wd =
2_J
pa(W)ea,
where pa(W) € k, for a £ I(d,n)
ael(d,n)
then p(W) = (pa(W))a€l(d n) is called the Plucker coordinate of W. Schubert variety Given any a =€ I(d,n), the corresponding Schubert variety fia is defined by Oa = {W &Gd(V) :dim(WnAi) > t for t = l , . . . , d } , where Ai = span{ei,... ,eai} for 1 < i < d. We have a partial order, called the Bruhat order, on the indexing set I(d, n), which is defined by
/ ? < / ? ' <=$> (3j < P'J
for all j = 1,..., d,
where /? = ( j 3 i , . . . , (3d) and /?' = (0{,--.,0d)i are arbitrary elements of I(d,n). Using this, we can describe fia in terms of the Plucker coordinates as follows. na = f t { p = (p7) £ Gd(V) :p/3=Q for all /3 £ I(d,n) with /? ^ a}. Hence, fiQ is also a projective algebraic variety. In fact, fia is irreducible and dim f l n =
Hodge's Postulation Formula for Schubert varieties
213
Note that Gd(V) is a particular case of fla with a = (n — d+l,n — d + 2, ...,n). In particular, dimGd(V) = d(n - d). For proofs of the basic results mentioned above as well as for a more leisurely discussion, we refer to Hodge and Pedoe's book [11] or the expository article of Kleiman and Laksov [13]. We shall now turn our attention to the structure of the homogeneous coordinate ring of Schubert varieties. Let P = {P7 : 7 6 I(d,n)} be a family of independent indeterminates over k and k[P] be the ring of polynomials in these indeterminates
with coefficients in k. Fix a € I(d,n), and let I(fla) denote the (vanishing) ideal of fi a . This is a homogeneous ideal of k[P] and thus the residue class ring Ra = k[P]/I(£la) has a natural graded ring structure. For I 6 N, let Rf denote the £-th graded component of Ra. Also let p~, denote the image of P7 in Ra. By a standard monomial on fla we mean an element of Ra of the form p^p^ • • -p 7 «, where 7 1 , . . . , -ye are elements of I(d, n} with the property that 71 < ... < je < a. The number of terms, i.e., (, is called the degree of this monomial. Now we have the following basic result.
THEOREM 1 The standard monomials on f l a form a vector space basis of Ra. More precisely, for each I £ N, the standard monomials on fla of degree I form a vector space basis of Rf.
Observe that a standard monomial on f l a of degree £ is completely determined by a f. x d rectangular array (ay) of integers with the property that 1 < an < ... < aid for 1 < i < I and a^r < ... < a(r < ar for 1 < r < d. (1)
We shall refer to such integral arrays as rectangular standard tableaux bounded by a of size I x d. The following corollary is an immediate consequence of the above theorem.
COROLLARY 2 The Hilbert function ha of fi a is given by
ha((.) = ^(rectangular standard tableaux bounded by a of size I x d), for I € N.
REMARK 3 The above theorem describing an explicit basis for the homogeneous coordinate ring of Schubert varieties is sometimes called Hodge Basis Theorem. A proof in characteristic zero is given in Hodge and Pedoe [11, Ch. XIV, Sec. 9]. For a characteristic free proof, see, for example, the paper [16] by Musili or the book of Arbarello, Cornalba, Griffiths and Harris [2, p. 74]. The Brandeis lecture notes of Seshadri [17] is also a nice readable reference for a proof of this theorem as well as for extensions to the context of Schubert varieties in G/P, where G is a nice algebraic group and P a parabolic subgroup.
3
NONINTERSECTING LATTICE PATHS
Let A = (a, a') and B = (b,b') be points with integer coordinates, i.e., in Z 2 . By a lattice path from A to B we mean a finite sequence L = (Po, PI, ... , Pm) of points in Z2 with PO = A, Pm = B and
Pi-Pi-i =(1,0) or (0,1) f o r i = l , . . . , r o .
Ghorpade
214
Note that the lattice path L is determined by its point set L = {Pj : 0 < j < m} by simply arranging the elements of this set in a lexicographic order.
In more intuitive terms, a lattice path consists of vertical or horizontal steps of length 1. For example, a lattice path from A = (1,1) to B = (5,6) may be depicted as follows.
B
FIGURE 1 It is clear that a lattice path from A to B would require b - a vertical steps and b' — a' horizontal steps, and it would be determined once we choose which of the total b — a + b' — a' (ordered) steps are vertical. Thus, # (lattice paths from A to B) =
b - a + b' - a' b —a
(2)
Note that this number is positive if and only if b > a and b' > a'.
Given any two d-tuples A — (Ai,..., Ad) and 2 = (Bi,..., Bd) of points in Z 2 , by a d-path, or simply, a path, from A to IS we mean a d-tuple £ = (L\,..., Ld) where Lr is a lattice path from Ar to Br, for 1 < r < d. We call L to be nonintersecting if no two of the paths L\,..., L,i have a point in common; otherwise, we call it intersecting. For counting the number of nonintersecting lattice paths, we
have the following beautiful result. THEOREM 4 Let Ar = (ar,a'r) and Br = (br,b'r), r = l,...,d, be points in satisfying and
a{ < . . .
a'd,
b'.
(3)
Let A = (Ai,..., Ad) and "B = (B^,..., B(i)- Then the number of nonintersecting paths from A to T> is equal to the binomial determinant
det
(4)
In other words, this is the determinant of a d x d matrix whose (i,j)-th entry is the number of lattice paths from Ai to Bj.
Hodge's Postulation Formula for Schubert varieties
215
The idea behind the proof is quite simple and elegant. Let us illustrate in the case d - 2. So let A = (Ai,A2) and 3 = (Bi,B^), with Ai,Bi as in Theorem 4. Prom (2) it is clear that # (lattice paths from A to B) = (h ~ ?' + ^ ~ ^\ (^ ~ f
V
DI - 01
yv
+
^ ~ 4) .
&2 - 02
/
(5)
If a path £ = (Li, £,2) from .A to CB, where Lj = (P 0> . . . , Ps) and L2 = (Qo,-- • ,Qt) is intersecting, then we can find least i > 0 such that PJ = Qj for some j < t. Now switch the paths at Pi — Qj, i.e., consider L( — (Po, . . . , P^Qj+i, • • • ,Qt) and L2 = (Qo,...,Qj,Pi+i,...,Pa)- In this way we get a path from (Ai,A 2 ) to (B-2,Bi). Conversely, by (3), any path from (yli,^) to (B^,Bi) must intersect, and a similar switching yields an intersecting path from A to 23. Thus one obtains a bijection which shows that M.f
4. <-• ^ t # (intersecting paths from .An t.to
/& 2 - Oi + 62 - tt'
V
&2 - 01
/ V
oi - o2
The last equation together with (5) implies the formula (4) in the case d — 2. In the general case, one considers permutations
By a similar switching trick, one can obtain an involution on the set of pairs (
and further, the RHS above can be rewritten as sgn(cr) ]T 1 =
sgn(a)
+ a € Sd, C € T, -C intersecting
Thanks to the sign-reversing involution , the terms in the last summation cancel each other, whereas by (3), a path £ 6 fa can only be nonintersecting when a is the identity permutation. Thus the right hand side is the number of nonintersecting paths from A to 23, and we have the desired formula. REMARK 5 The above result is due to Gessel and Viennot [5] though the ideas can perhaps be traced to the works of Karlin-McGregor [12] and Lindstorm [15]. In [5] the endpoints A», Bi are of a more restrictive nature, but the arguments in the proof remain similar.
4
HODGE'S POSTULATION FORMULA
Fix positive integers d,n with d < n and a = ( a i , . . . , a d ) € I(d,n). Let fia denote, as before, the Schubert variety corresponding to a in the Grassmannian
216
Ghorpade
of d-dimensional linear subspaces of an n-dimensional vector space over k. The postulation formula of Hodge [11, Thm. Ill, p. 387], in our notation, can be stated as follows.
THEOREM 6 The Hilbert function of HQ is given by
In particular, the Hilbert polynomial of fiQ equals the Hilbert function for all nonnegative integral values of the parameter I.
An alternate description of this formula is provided by the following lemma. LEMMA 7 Given any I £ N, we have
Proof:
If (bij) is any d x d matrix, then by a cyclic permutation of the rows (or alternatively, by successive row interchanges) we can transform it to the d x d matrix whose (i,j)-th entry is bd-i+i,j. This will change the sign of det(&ij) by (_l)d(d-i)/2_ We can also do a similar operation on the columns. Combining these two, we see that det (fejj) = det (bd-i+i,d-j+i)- In particular, we have
Now for the d x d matrix corresponding to the determinant on the right hand side of the above equation, we successively make the n — 1 elementary column operations Cn - Cn-\,Cn-i - Cn-2, - . . , C-2 - Ci. Then in view of the Pascal triangle identity
t-l the (i,j)-ih entry changes from ( a ' + /T/^/~ 1 ) to ( Qi+ /J^"/~ 2 ), for j > 2, while the determinant is unaltered. Next, we make the n — 2 elementary column operations Cn - Cn-i, C*n-i - Cn-2, • • • , C3 - C-2, and continue in this way till in the end we just make a single column operation Cn — Cn-\ • In this process the j-th column of the original matrix is altered j — I times, and its entries eventually become
-(j -l)\
(a.i+1- i
By taking the transpose of the resulting matrix, we have the desired formula.
D
We will now link the rectangular standard tableaux appearing in Corollary 2 to
nonintersecting paths. The idea to associate such a path to a rectangular tableaux (aij) is quite simple. Start at (1,1). Move vertically till (l,an). Go one step to
Hodge's Postulation Formula for Schubert varieties
217
the right, and then move vertically till (2,a 2 i). Continue in this way till we reach
(£, a(i). Take a final step to the right and then move vertically to end at (£+1, ai). Next, start at (0,2) and use the second column to obtain the second lattice path, and so on. To recover the tableaux from a nonintersecting d-path, it suffices to look at the northeast corners in each path. More formally, we have the following.
LEMMAS Lette N. Define Ar = (2-r,r) and Br = (£-r + 2,ar), for 1 < r < d, and let A = (Ai,...,Ad) and "B = (Bi,...,Bd). Then the set of rectangular standard tableaux bounded by a of size I x d is in one-to-one correspondence with the set of nonintersecting paths between A and 1$. Proof:
Let (a^) be a rectangular standard tableaux bounded by a of size t X d.
As a convention, we set, OQJ — j and a,t+\j — aj for 1 < j < d. By (1), it is clear that QOJ < aij < ... < atj < ae+i j for each j — 1,..., d. For 1 < r < d, define Lr to be the lattice path from Ar to Br whose point set is given by
LT = {(i - r + 1, j) : 1 < i < I + 1 and a;_i r < j < a^}
It is easy to see that Lr is well-defined. Moreover, if for 1 < r, s < d, r ^ s, the lattice paths Lr and Ls intersect, then we have (i\ — r 4- 1, ji) = (i% — s + 1,^2)
for appropriate ii,iz,ji,J2, and if, without loss of generality, we have r < s, then *i < *2 — 1 and hence ji = j\ < Oj ir < a j 2 _ j r < aj 2 _i s , which is a contradiction.
Thus £ = ( I / i , . . . , Ld) is a nonintersecting d-path from A to ®. Conversely, let £ = (Li,..., Ld) be a nonintersecting d-path from A to 33. Given 1 < r < d, the fact that Lr is a lattice path implies that for 1 < i < (., we have max{j 6 N : (i - r + 1, j) £ L,.} = mm{j € N : (i - r + 2, j) e Lr};
let air be this common value. In this way we get a I x d array (a^) of integers. Since Lr is a lattice path from Ar to Br, we have 1 < a,r < a^+i r < ar for i < i. Further, if air > a i r + 1 for some r < d, then it is clear that the truncated paths L'r and IY+I obtained respectively by moving along Lr and Lr+i till (i - r + l,aj r ) and (i — r + 2 , a j r + i ) , must intersect. This contradicts the assumption that £ is nonintersecting. It follows that the array (a^) satisfies the properties in (1).
In this way we get a map from the set of nonintersecting c?-paths from A to 3 to the set of rectangular standard tableaux bounded by a of size I x d, and vice-versa.
It is easy to see that the composites, either way, are identity maps.
D
Proof of Theorem 6: Follows from Corollary 2, Theorem 4 and Lemmas 7 and 8. d REMARK 9 It may be remarked that the above proof of Hodge's postulation formula is structurally analogous to the proof given in [3] and [7] for determining the Hilbert function and Hilbert series of certain determinantal ideals. The correspondence between the standard bitableaux appearing there and nonintersecting lattice paths (with a given number of turns) was somewhat more intricate, i.e., first one had to use a version of the Robinson-Schensted-Knuth correspondence and then something like Viennot's light-and-shadow procedure. Here, as Lemma 8 shows, we can use a more direct correspondence. On the other hand, it does not appear easy to find a 'good formula' for the Hilbert, series of fi a . Connection with lattice paths has also been exploited in [9] to prove a formula for the multiplicity of
218
Ghorpade
determinantal ideals. Hodge's postulation formula, and particularly the unusual fact that the Hilbert function ha(t) is a polynomial for all values of I was in fact, the starting point of the investigations by Musili [16], which led to certain vanishing theorems for Schubert varieties in Grassmannians. These results have now been vastly generalized (see [14] for a survey) while a far-reaching generalization of Hodge's postulation formula itself has been obtained by Fulton and Lascoux [4].
REFERENCES 1. S. S. Abhyankar. Enumerative Combinatorics of Young Tableaux. New York:
Marcel Dekker, 1988. 2. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris. Geometry of Algebraic Curves, Vol. I. New York: Springer-Verlag, 1985. 3. A. Conca, J. Herzog. On the Hilbert function of determinantal rings and their canonical module. Proc. Arner. Math. Soc., 122:677-681, 1994. 4. W. Fulton, A. Lascoux. A Fieri formula in the Grothendieck ring of a flag bundle. Duke Math. J., 76:711-729, 1994. 5. I. Gessel, G. Viennot. Binomial determinants, paths and hook length formulae. Adv. Math., 58:300-321, 1985. 6. S. R. Ghorpade. Abhyankar's work on Young tableaux and some recent developments. In: C. Bajaj ed. Algebraic Geometry and Its Applications. New York: Springer-Verlag, 1994, pp. 215-249. 7. S. R. Ghorpade. Young bitableaux, lattice paths and Hilbert functions. J. Statist. Plann. Inference, 54:55-66, 1996.
8. S. R. Ghorpade. On the enumeration of indexed monomials and the computation of Hilbert functions of ladder determinantal varieties. In: C. Martinez, M. Noy and 0. Serra ed. FPSAC'99 Actes/Proceedings. Barcelona: Universitat Politecnica de Catalunya, 1999, pp. 225-232. 9. J. Herzog, N. V. Trung. Grobner bases and multiplicity of determinantal and pfaffian ideals. Adv. Math., 96:1-37, 1992.
10. W. V. D. Hodge. Some enumerative results in the theory of forms. Proc. Camb. Phil. Soc. 39:22-30, 1943.
11. W. V. D. Hodge, D. Pedoe. Methods of Algebraic Geometry, Vol. II. Cambridge: Cambridge Univ. Press, 1952. 12. S. Karlin, G. McGregor. Coincidence probabilities. Pacific J. Math., 9:11411164, 1958. 13. S. L. Kleiman, D. Laksov. Schubert Calculus. Amer. Math. Monthly, 79:10611082, 1972.
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14. V. Lakshmibai, C. S. Seshadri. Standard monomial theory. In: S. Ramanan, C. Musili and N. Mohan Kumar ed. Proceedings of the Hyderabad Conference on Algebraic Groups, Madras: Manoj Prakashan, 1991, pp. 279-322.
15. B. Lindstorm. On the vector representation of induced matroids. Bull. London Math. Soc., 5:85-90, 1973. 16. C. Musili. Postulation formula for Schubert varieties. J. Indian Math. Soc., 36:143-171, 1972. 17. C. S. Seshadri. Introduction to Standard Monomial Theory. Lecture Notes
No. 4, Brandeis University, Waltham, MA, 1985.
SUBINTEGRAL EXTENSIONS AND UNIMODULAR ROWS JOSEPH GUBELADZE1 A. Razraadze Mathematical Institute, Alexidze St. 1, 380093 Tbilisi, Georgia, gubelQrmi.acnet.ge
To the memory of Emzar Rtveliashvili
1 STATEMENT OF THE RESULT Recently there has been a revival in the study of the orbit sets of unimodular rows under the natural action of elementary matrices. This development is related to the nice relationship between van der Kallen's group structures on the mentioned orbit sets [9] [10] and the Euler class groups of commutative noetherian Q-algebras of Bhatwadekar-Sridharan [BhSr], an invariant where the obstructions for splitting off free summands from top rank projective modules live. On the other hand subintegral ring extensions have played an important role in the study of projective modules. Recall that a (commutative) ring extension A C B is called elementary subintegral if B — A[x] for some x € B such that x2 , x3 € A. The extension A C B is called subintegral if B is a filtered union of elementary subintegral extensions. The link between projective modules and subintegral extensions is established by R. Swan's following THEOREM 1 ([7]). Let A C B be a subintegral extension and P be a finitely generated projective A-module. Then:
(1) IfB®AP&Qi@Q2®---®Qn «/ienP«Pi®P 2 e---®Fn where B®APi faQt.
(2) If B ®A P has the form free® rank 1 then the same is true of P. Many other relevant results on low dimensional .ftT-theoretic objects can be found in [6]. Here we prove the following 1
Supported by TMR grant ERB FMRX CT-97-0107
221
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Gubeladze
THEOREM 2. Let Ac B be a subintegral extension and u € Um n (yl), n > 3 such thatu~En(B) (1,0,...0). Thenu~En(A) (1,0,...0). Actually, this is contained in [5] but in an implicit form - the result is derived in the course of the proof of Theorem 3.3.5 there. Here we make the claim more explicit because of its independent interest and outline its connections with other objects.
COROLLARY 3. Let A C B be as in Theorem 2 and u,v <E Vmn(A). Assume the rings are Noetherian and 2 < dim A < In—4. Then u ~En(B) v implies u ~En(A) vIn fact, it is proved in [10] that in the mentioned range the orbit set (with respect to multiplication on the right) \Jmn(A)/En(A) (and, hence, Umn(B)/En(B) too) carries a group structure. Therefore, Theorem 2 applies once we notice that the induced mapping Um n (yl) —> Um n (B) is a group homomorphism and that (1,0... ,0) represents the neutral element - direct consequences of the very definition of the group structures via universal weak Mennicke symbols of van der Kallen [10]. QUESTION 4. Is the same statement on equivalence of unimodular rows valid without any restriction on the Krull dimension dim A?
REMARK 5. It is shown in [6] that a subintegral ring extension of noetherian rings A C B implies an isomorphism of Chow-groups Chj(B) ->• Chi(A). Now the Euler class group can to some extent be thought of as the Chow group of 'oriented' zero cycles [2]. It is therefore interesting to find out the relationship between two Euler classes S(A) and £(B).
2 PROOF OF THEOREM 2 The proof is based on the following THEOREM 6 ([3]). Assume M is an affine simplicial monoid and R is a noetherian ring of Krull dimension dimJ? = d. Then En(R[M]) acts transitively on Umn(J?[M]) for any n > max(3, d + 2).
Recall that a monoid is called affine if it is a finitely generated submonoid of a free abelian group Z r , r e N. An affine monoid M is called simplicial if the cone C(M) it spans in W is simplicial, i. e. C(M) is spanned (over H+) by linearly independent vectors. Actually, we need the following special case:
COROLLARY 7. For a monoid M as in Theorem 6 the action Umn(Z[M]) x En(I*[M}) -»• Umn(Z[M]) is transitive whenever n > 3. We also need the following Milnor patching for unimodular rows. This is Proposition 9.1 (a) in [4], but [4] only contains the proof of Proposition 9.1(b).
LEMMA 8. Let m > 2 be a natural number and
Subintegral extensions and unimodular rows
223
be a pull-back diagram of commutative rings where either 7 or 5 is a surjection. Assume an element U\ € Um m (Ai) is such that its image U € Um m (A) is in the same Em(A)-orbit as (1,0, ... ,0). Then there is an element W € Um m (Ai) such that Ui ~e m (A 1 ) W and W has a preimage in Um m (A').
Proof. Let an elementary matrix e € Em(A.) be such that Ue = (1, 0, . . . , 0). Case 1. 7 is surjective. There is a lifting e2 6 Em(A.z) of e"1. The first row of 62, say C/2 is mapped to U under 7. Therefore, there exists [/' € (A')™ which is mapped to Ui and f/2 respectively. It only remains to show that U' € Um m (A'). Observe that the entries of U' generate an ideal of A' which contains 1 + a for some a € Ker(a) (in our situation a is surjective). If we knew that this ideal also contains a + b for some b € Ker(/3) then the desired unimodularity would follow from the equalities 1 = l+ab= l+a-a((l+a)-(o+6)). (One uses the equality Ker(a)nKer(/3) =0.) There exist c\ , . . . , cm € A2 such that +•••+ (P(a)cm)Xm = /3(o),
where (Ai, . . . , A m ) = t/2. Because of the inclusions
/8(o)ci, . . . , P(a)cm 6 Ker(7) these elements have preimages in A. It follows that the ideal of A, generated by the entries of U, contains an element of the type a + b for some b € Ker(/3). Case 2. 8 is surjective. In this situation there is a lifting si € Em(\i) of s. The elements Uie\ € Um m (Ai) and (1,0,... ,0) € Um m (A 2 ) have the same images in Um TO (A). Then the same arguments as in Case 1 show that f/i£i admits a preimage in Um m (A'). D Now we turn to the proof of Theorem 2. Step 1. Without loss of generality we can assume that B = A[x] with x2,x3 6 A. Put 5 = AU{a;} and consider the polynomial algebra Z[{£s}s]. The homomorphism g : %[{ts}s] -> B, sending ts to s, is surjective. Consider the pull-back diagram
D
——— > A
It follows from Lemma & that we are done once the transitivity of the action Um n (£>) x En(D) -> Um n (D) is established. Step 2. Let N denote the multiplicative monoid consisting of 1 € Z and those monomials in the ts which are divisible by t2x. We claim that N is a filtered union of affine simplicial monoids. Consider any finite subset
T ={*,„... ,t.k}c{t. \s<=A}. It is enough to show that the monoid MT of the monomials, which only involve indeterminates from T U {tx} and are divisible by t%., is a filtered union of affine
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Gubeladze
simplicial monoids. We can think of the monoid of all monomials in the tSi and tx as the free commutative monoid Z++1 so that tx corresponds to (0, ... , 0, 1). Consider the simplex A& — conv(£ S l ,... ,tSh,tx) and its facet Afc_i = conv(£ S l ) ... , t S k ) . The subset A f c \A/t_i is a filtered union of rational simplices, say UjA^. ('Rational' means 'with rational vertices'). The classical Gordan Lemma (see, for instance, §4 in [7]) guarantees that the submonoids M W) = (m € MT
m + 1 and K+m n A (j) + 0} U {1} C MT
are all finitely generated and, hence, affine simplicial. (Here K + m refers to the ray in K*+1 spanned by m.) The claim follows because MT is a filtered union of the ' Step 3. The Z-submodule of 1[{ts}s\ spanned by N\ {1} is an ideal of Z[{t,}5]Denote this ideal by 7. It is clear that any monomial in I has image (under g) in A - one just uses the fact that any natural number > 2 is a non-negative integral linear combination of 2 and 3. This means that / is also an ideal of D. Consider the pull-back diagram D
I
1
———> D/I
The action Umn(Z[Ar]) x En(Z[N}) -> Umn(Z[iV] is transitive by Corollary 7 and Step 2. Therefore, if the action Um n (£»//) x En(D/I) -> Umn (£>//) is also transitive then by Lemma 8 we get the transitivity of the action Um n (D) x En(D) —> Um n (jD) and this completes the proof by Step 1. It is easily observed that (£>//)red = Z[{£3},4]. Since the property we are looking at is invariant modulo nilpotents we are done by Suslin's theorem [8] (applied to filtered union of polynomial rings). REMARK 9. As mentioned, the special case of Theorem 6 for free commutative monoids is due to Suslin [8] (the case of a trivial monoid being the Bass-Vaserstein classical results on surjective KI stabilizations, see Ch. 5, §4 in [1] and [11]). In [4] we have extended the transitivity result to all submonoids M C Qr which
satisfy the following condition: the cone C(M) finite polyhedral and admits a sequence
spanned by M in Rr is pointed,
0 = C0 C Ci C • • • C Ck = C(M)
of finite polyhedral cones for some k 6 N such that the closures of the sets Ci\Ci-i C K r , i € [i,k] are all simplicial cones.
In [4] the mentioned monoids are called monoids of ^-simplicial growth. This result suffices to cover all rank 3 affine monoids without nontrivial invertibles and
it extends the class of affine simplicial monoids considerably. (Actually one can also invoke monoids with nontrivial units, as it is done in [4] with use of Suslin's relevant result on Laurent polynomial rings.) However, despite several attacks using different approaches [4] [5] the general case of affine monoids remains open. The simplest case of a monoid for which the transitivity of the action Umn(.R[M]) x En(R[M]) -> \Jmn(R[M]) is not known in
Subintegral extensions and unimodular rows
225
the range n > max(3,dim/Z + 2) is the octahedral monoid Z+(±1,0,0,1) + Z + (0,±1,0,1)+Z + (0,0,±1,1)CZ 4 , - a discrete cone over the standard octahedron conv ((±1,0,0), (0, ±1,0), (0,0, ±1)).
REFERENCES 1. H. Bass, Algebraic K-Theory, W. A. Benjamin, Inc., 1968.
2. S. M. Bhatwadekar and Raja Sridharan, Euler class group of a noetherian ring, Compositio Math., to appear
3. J. Gubeladze, The elemenatry action on unimodular rows over a monoid ring, J. Algebra 148 (1992), 135-161. 4. J. Gubeladze, The elementary action on unimodular rows over a monoid ring II, J. Algebra 155 (1993), 171-194.
5. J. Gubeladze, Geometric and algebraic representations of commutatice cancellative monoids, Proc. A. Razmadze Math. Inst. 113 (1995), 31-81. 6. F. Ischebek, Subintegral ring extensions and some .K"-theoretical functors, J. Algebra 121 (1989), 323-338.
7. R. G. Swan, Gubeladze's proof of Anderson's conjecture, Contemp. Math. 124 (1992), 215-250. 8. A. A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11 (1977), 221-238. 9. W. van der Kallen, A group structure on certain orbit sets of unimodular rows, J. Algebra 82 (1983), 363-397. 10. W. van der Kallen, A module structure on certain orbit sets of unimodular rows, J. Pure Applied Alg. 57 (1989), 281-316.
11. L. Vaserstein, On the stabilization of the general linear group over a ring, Math.
USSR Sb. 8 (1969), 383-400.
On commutative FGS'-Rings with ascending condition on annihilators CHEIKH THIECOUMBA GUEYE, Departement de Mathematiques et Informatique, Faculte des Sciences et Techniques, Universite Cheikh Anta Diop, Dakar, Senegal MAMADOU SANGHARE, Departement de Mathematiques et Informatique, Faculte des Sciences et Techniques, Universite Cheikh Anta Diop, Dakar, Senegal
INTRODUCTION Let R be a commutative ring. An .R-module M is said to satisfy property (S) if every surjective endomorphism of M is an automorphism. R is called a FGS-nng if every ./?-module with property (S) is finitely generated. In [5] W.V.Vasconcelos proved that for a commutative ring R, every finitely generated .R-module satisfies property (S). In general, the converse is not true.
The purpose of this note is to characterize commutative FG5-ring with ascending condition on annihilators.
1. STATEMENT AND PROOF OF THE MAIN RESULTS
1.1 J(R) is a nil ideal
J(R)
is the Jacobson radical of R.
PROPOSITION 1 Let R be a commutative FGS-nng. Then every prime ideal of R is maximal. Moreover the set of all prime ideals of R is finite. Proof. Let be P a prime ideal of R and let B be the classical quotient field of the integral domain R/P. It's clear that B, considered as a R/P-module satisfies property (5). Since R/P is also a FGS-ring, then B is a finitely generated R/P module, therefore B — R/P and P is maximal. 227
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Gueye and Sanghare
Let now L be the set of all prime ideals of R. For every P £ L, R/P is a simple
.R-module, furthermore if P, P' £ L and P ^ P', we have HomR(R/P,R/P') = {0}, hence the .R-module M = 0P6L R/P has property (5), then M is a finitely generated /?-module and this fact implies that the set L is finite. According to proposition 1, if R is a commutative FGS-ring, then its Jacobson radical J(R) is a nilideal and R is a semi-local ring. In particular, idempotents can be lifted modulo J(R), so that R is commutative semi-perfect ring, i.e., a finite direct product of local rings RI , R2, . . .,Rn.
PROPOSITION 2 A product of rings Ri(l < i < n) is an FGS-nng if and only if every Ri, (1 < i < n) is an FGS-ring. Proof. Let R = rii=i,...,n Ri- If -R is a FGS-ring, it is clear that Rt is a .FGS-ring as a homomorph range of R. To prove the converse, assume that Ri is a .FGS-ring for any i, (1 < i < n). Let M be a R- module, M is also a .Rj-module. Therefore if M satisfies property (S) then M as an .Rj-module is finitely generated for any * > ( ! < * < • n)- Since R = fliLi Rii therefore M is finitely generated as an .R-module. PROPOSITION 3 A commutative artinian ring with principal ideals is a commutative FGS-ring. Proof. R is a direct product of finitely many local artinian ring Ri(l < i < n) with principal ideals. Therefore by proposition 2 we can suppose that R is a local artinian principal ideal ring. Let M an .R-module,then by [1] M = ®ie/ Mi where Mi are cyclic /?-modules . Suppose that M is not finitely generated. Then there exists an infinite countable subset K of / such that / = K [_) L with K |~| L = 0 and Mi = Aci = ACJ for all i,j € K. Thus we may assume that K =. Let us put now G = (0ng Acn), D = (0igL M;)
Z =
$:G
—>
C
a
!-»
$(z) =
c
n n
n>0
a
ncn-l
where C_i = 0.
n>0
Then the surjective map defined as follows
h:M = C®D
->
M
m = z +x
1-4
h(m) — §(z) + x.
is not injective. Therefore M does not verify property (S). If / is a subset of a ring R, we denote by arm/? (I) the set of all elements of R that annihilate 7.
PROPOSITION 4 Let R be a commutative FGS-ring such that annihilators of subsets of R verifie the ascending chain condition (a.c.c.). Then the Jacobson radical J(R) of R is nilpotent. Proof. Since J(R) is a nilideal, then J(R) is T-nilpotent ([4] proposition 2-1-6). Let n be the least integer such that anrifi(Jn(R)) = anrif{(Jn+1(R)). Suppose that «7(jR) is not nilpotent, then there exist an element x\ in J(R) such that J(R) x x\ ^ 0, which implies Jn+l(R] x x\ ^ 0. We can find x2 £ J(R) such that
Commutative FGS -Rings
229
Jn(R) x x1 x x2 ^ 0 and Jn+l(R) x a:1 x x2 ^ 0 etc... We have then put in a pominent position a sequence (x n ) n gjv in J(R) such that x n x xn-\ x . . . x xi is nonzero for any integer n > 1. This contradicts the T-nilpotency. Therefore J(R) is nilpotent. Let R be a ring, / a subset of #, and annR(I) denotes the set of annihilators of I'mR. PROPOSITION 5 Let R a commutative FGS-rmg such that annihilators of subsets of R verify the ascending chain condition. Then R is artinian. Proof. Without loss of generality, we may assume that R is a local ring with Jacobson radical J2(R) - 0. Then J ( R ) / J ' 2 ( R } is an R/J(R) vector space. Assume that dimR/J(fi)J(R)IJ'2(R} > 2. Then by [2] there exists an ^-module M which is not finitely generated and which verifies the property (S), which is absurd. Therefore dimRij^J(R)/J'2(R} = 1, this implies that J(R) is principal and R is artinian. Moreover R is an principal ideals ring. 1.2 Characterization theorem
THEOREM A commutative ring is an FGS-ring satisfying the ascending chain condition on annihilators if and only if it is an artinian principal ideal ring. Proof. => proposition 4.
4= proposition 3.
References [1] Cohen, I.S. and Kaplansky, I.Ring for which every module is a direct sum of cyclic modules, Maths Zeitshr.Bd,54(H25) (1951).
[2] Kadi,A,M et Sanghare, M : Une caracterisation des anneaux arteniens ideaux principaux.Lect. Notes in Math.N01328, Springer-Verlag 245-254 (1988). [3] Laradi, A. On Duo Rings, pure semi-simplicity and finite representation type, commucations in algebra 25(12),3947-3952 (1997).
[4] Renault, G. Algebre non commutative, Gauthier Villars (1975). [5] Vasconcelos W.V, On finitely generated flat module, Trans.Amer.Math.Soc. 138 (1969), 505-512.
COHEN-MACAULAY F-INJECTIVE HOMOMORPHISMS MITSUYASU HASHIMOTO Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602 JAPAN
I
INTRODUCTION
Throughout this article, p denotes a prime number. Some ring theoretic properties of a noetherian ring R of characteristic p are characterized by module theoretic properties of the Frobenius map FjS. : R^ -4 R given by a H> ap , where we conventionally use the symbol R^ for R in order to indicate that R is an /?'e'-algebra via Fjj. A classical result of Kunz [18] tells us that R is regular if and only if there exists
some e > 1 such that Fj| is flat. More recently, since M. Hochster and C. Huneke invented the notion of tight closures [14], F-regularity and its variations [14, 13], F-rationality [8] and its strong form [25], F-injectivity, F-purity [8] and so on have been studied intensively, see [16]. The relationship between characteristic zero singularities and the singularities listed above has been clarified via modulo p reduction, see [24, 11]. It has been known that some properties of morphisms in characteristic p > 0 are characterized by properties of the Radu-Andre homomorphisms. Let / : A —>• B be a homomorphism of noetherian commutative rings of characteristic p. We define the eth Radu-Andre homomorphism of / to be the map from B^ ®A(ei A to B which sends &( e )
231
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Hashimoto
between F-finite noetherian commutative rings of positive characteristic is Cohen-
Macaulay F-injective (CMFI, for short) if / is flat Cohen-Macaulay and all fibers are 'geometrically F-injective,' see Definition 5.4. It is not so difficult to prove that a Cohen-Macaulay flat homomorphism / : A —> B between F-finite Cohen-Macaulay rings of characteristic p is CMFI if and only if / is reduced and the cokernel of <&e(A, B) is a maximal Cohen-Macaulay B^'-modulefor any (or equivalently, some)
e > 1, see Proposition 5.5. Utilizing this fact, we prove the following. THEOREM 5.8 Let f : (A, m) -» (B, n) be a flat local homomorphism of F-finite noetherian local rings of characteristic p. If the closed fiber B/mB is geometrically CMFI over A/m, then f is CMFI. The key to the proof is Kawasaki's Macaulayfication theorem [17]. Utilizing the study of CMFI homomorphisms, we prove the following. THEOREM 6.4 Let f : (A, m) -> (B, n) be a flat local homomorphism of excellent local rings of characteristic p. Assume that k& := A/m and kg '•= B/n are both F-finite. If A is F-rational, the generic fiber B
the F-finiteness of k^ or kg, but assuming that / is regular. Note also that this is a natural generalization of a theorem of R. Fedder and K.-i. Watanabe [8] (see Corollary 6.6). As we have a regular alteration for algebras essentially of finite type over a field [4], if B/mB is geometrically F-rational and A is essentially of finite type over a perfect field of characteristic p in the theorem, then we need not to assume that B ®A K is F-rational, see Corollary 6.8. Note that Corollary 6.8 for characteristic zero rational singularities was proved by R. Elkik more than twenty years ago [6, Theoreme 5]. After the Messina Conference, the author got to be aware of Enescu's paper [7]. He independently proved Theorem 6.4 without F-finiteness assumptions. Acknowledgement: The author is grateful to Professor Kei-ichi Watanabe for valuable suggestions. Special thanks are also due to Professor Kazuhiko Kurano for pointing out an essential error in the draft. The author is grateful to Professor Melvin Hochster for kindly sending him the preprint [7]. The author would like to express his thanks to Professor Melvin Hochster and Professor Florin Enescu for their encouragements. This paper is a revised version of the preprint distributed at the conference, and in particular, many references in [7] that the author was not aware of are added. 2
NOTATION AND TERMINOLOGIES
Throughout this article, a ring means a commutative ring with identity, and R, A and B stand for rings. For a ring R and p 6 Spec/?, we denote the field Rp/$Rp by K(P). For an Rmodule M, we denote /c(p) ®R M by M(p). For an /J-linear map / : M —> N, we denote l s ( p ) ® / by /(p) : M(p)
Cohen-Macaulay F-injective homomorphisms
233
Let if : M —>• N be an /J-linear map of _R-modules. We say that if is pure if
lw <S> V '• W <S> M ->• W <S> N is injective for any .R-module M/. Let / : A —> -B be a ring homomorphism. We say that / is pure (resp. split) if
/ as an A-linear map is pure (resp. a splittable monomorphism). Let k be a field and R a noetherian fc-algebra. We say that R is geometrically regular (resp. geometrically reduced) over k if for any finite algebraic field extension
k' of k, k' (gife R is regular (reduced). A homomorphism of rings / : A —> B is said to be regular (resp. reduced), if both A and B are noetherian, / is flat, and B(P) is geometrically regular (resp. geometrically reduced) over /c(P) for any P G Spec(A).
For a noetherian ring /?, we denote the set of maximal ideals and minimal primes of R by Max(_R) and Min(_R), respectively. We say that a i , . . . , as are parameters of R if h t ( a i , . . . , as) > s (i.e., ( a i , . . . , a s ) = R or h t ( a i , . . . , a 5 ) = s). We denote the set R \ (\JP€u'm(R) P) by -^°- Note tnat « £ -R° if and only if a is a parameter of R (of length one). Let R be a ring of characteristic p. We denote the Frobenius map R —> R (a H->. aP) by FR. So F| is the map given by FeR(a) = a?° for e > 0. The symbol _R' r ' stands for the same ring R for any r ^TL. We consider that R is an .R(e)-algebra via Fjj for e > 0. For an -R-module M, the .R-module M, viewed
as an _R( r )-module, is denoted by M^. For a ring homomorphism
We sometimes identify (R^)(r > with #( r + r '). So (F£)W is identified with the Frobenius map from _R( r + e ) to R^ for r e Z and e > 0. It is sometimes necessary to emphasize that an element a G R is viewed as an element of R^. In this case, we use the notation a^. Finally, R^, M(°), A<°) and a<°) are simply denoted by R, M, A and a.
Hence we have F^a^) = ap for a 6 R and e > 0. For an ideal I of R and e > 0, J^ 6 - 1 J? is the ideal generated by the all p e th powers of elements of /, which is denoted by /^, where q = pe. A ring R of characteristic p is said to be F-finite if R is a finite J?'^-module. If R is noetherian and F-finite, then it is excellent [19]. Note that a complete noetherian local ring of characteristic p with the F-finite residue field is F-finite. Let R be a noetherian ring of characteristic p, and I an ideal of R. M. Hochster and C. Huneke [14] defined the tight closure of /, which we denote by /*. By definition, /* := {x e R 3c e R° 3e0 > 1 Ve > e 0 cxq € / M (q := pe)}.
Note that /* is an ideal of R containing /. An ideal I of R is called tightly closed
if 1= I*. R. Fedder and K.-i. Watanabe [8] defined that a noetherian commutative ring R of characteristic p is F'-rational if all parameter ideals of R are tightly closed. The following is known. THEOREM 2.1 Let R be a noetherian ring of characteristic p. The following hold: 1 If R is F-rational, then R is normal.
2 If R is F-rational and is a homomorphic image of a Cohen-Macaulay ring, then R is Cohen-Macaulay.
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Hashimoto
3 // (R, m) is an equi-dimensional local ring with at least one tightly dosed m-
primary parameter ideal and is a homomorphic image of a Cohen-Macaulay ring, then R is F-rational.
4 The following are equivalent: a R is Cohen-Macaulay F-rational;
b For any m € Max(J?), Rm is Cohen-Macaulay F-rational. c For any multiplicatively closed subset S of R, R$ is Cohen-Macaulay F-
rational. 5 If (R,m) is local, x 6 m is a non-zerodivisor of R, and R/Rx is Cohen-Macaulay F-rational, then R is Cohen-Macaulay F-rational. 6 // (R, m) is excellent local, then R is F-rational if and only if its completion R is F-rational.
7 If R is locally excellent F-rational, then R is Cohen-Macaulay. For the proof, see [15, Thoerem 4.2] and [25, Proposition 0.10].
3
QUASI-SPLIT MAPS INTO MAXIMAL COHEN-MACAULAY MODULES
Let R be a noetherian ring. An ideal / of R is called a parameter ideal if / is generated by parameters of R. An _R-linear map
R. A ring homomorphism A —)• B is said to be quasi-split if it is quasi-split as an A-linear map. Let if : M —> JV and if> : N —>- L be /2-linear maps. If f and i[> are quasi-split, then tjj o (p is quasi-split. If 4> ° V is quasi-split, then tp is quasi-split. We say that an A-module M is maximal Cohen-Macaulay if M is /^-finite, and depth fim Mm = dim_R m for any m 6 Max(_R). If M is a maximal Cohen-Macaulay jR-module, then for any multiplicatively closed subset S of R, we have that MS is a maximal Cohen-Macaulay .Rs-module.
LEMMA 3.1 Let R be a Cohen-Macaulay ring, N a maximal Cohen-Macaulay Rmodule, M an R-finite module, and if> : M —\ N an R-linear map. Then the following are equivalent. 1 if is quasi-split.
2 For any m G Max/?, there exists an mRm-primary parameter ideal J of Rm such that if®l:M® Rm/J —> N
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COROLLARY 3.2 Let f : (A,m) —S- (S,n) fee a Cohen-Macaulay local homomorphism of Cohen-Macaulay local rings [3]. Lei / : M —> N be an A-linear map between A-finite modules. Then the following are equivalent. 1 M and N are maximal Cohen-Macaulay A-modules, and / is quasi-split.
2 B ®A M and B ®A N are maximal Cohen-Macaulay B-modules, and IB <E> / : B ®A M -} B ®A N is quasi-split. Note that if B is Cohen-Macaulay and / is flat, then the local homomorphism / : (-A,m) —>• (B, n) is a Cohen-Macaulay homomorphism. We use the corollary only in this restricted case. The following is also trivial.
LEMMA 3.3 Let (A,m) be a Cohen-Macaulay local ring, and f : M —» JV an Alinear map between maximal Cohen-Macaulay A-modules. Then the following are equivalent: 1 f is quasi-split;
2 fm is quasi- split for any m £ Max(^4);
3
LEMMA 3.4 Let f : (A, m) —>• (B, n) fee a flat local homomorphism between CohenMacaulay local rings. Let N be a maximal Cohen-Macaulay B-module which is A-flat, and M a finite B-module. Let f : M —> N be a B-linear map. Then the following are equivalent.
1
4
n
RADU-ANDRE HOMOMORPHISMS
In this section, all rings are assumed to have characteristic p. Let / : A —>• B be a homomorphism of rings. We define the Radu-Andre homomorphism of /, denoted by $ e (/) or $e(A, B), to be the map
B(e] ®A(f) A -» B
(& (e) ®a^bp°(fa)}.
It is a ring homomorphism for e > 0. The (— e)-shift Qe(A, B)(~e^ of this map
is denoted by Ve(A,B). By definition, we immediately have the following.
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LEMMA 4.1 Let f : A -> B, g : B -> C , and h : A —>• A be ring homomorphisms , and set B :=• A ®A B. Let e and e1 be non-negative integers. Then the following hold:
1 The composite map
C<«> ® A( .> A = <:<•> ®B(., BW ®A(e} Al agrees with <&e(A,C).
2 Lei a 6e i/ze canonical isomorphism
Then we have
3 The composite map
A ®A (B ®A A(-^) = B ®A (A ® agrees with the composite map
A ®A (B ®A A(-°>) = l^^A'
4 The composite map
B^ ® A(S) A = (B^ ® A(B , A) ®A ^ * ' agrees with the map
B^ ®A(., A = (B (e) ®A(.) A( e )) ®AM A S 5 T/ie composite map
agrees with FA.
6 The composite map
agrees with Fg.
(
-^
)
g ®A A = B
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7 The composite map
(e) ®AM A agrees with (^ ( e) 8 A ( e ) ^) ( " e ) Note that B^ —» B^ e ^ <E> A (e> .A —> B induces a sequence of homeomorphisms of affine schemes. In particular, if B is local, then so is B^ ® A (e) A. If B is local and B^ (E> A (
( x [ e , . . . , X j ) of B^ is again a system of parameters both for B^ (g> A (
LEMMA 4.2 Lei / : A —> 5 6e a homomorphism of noetherian commutative rings of characteristic p. If both A and B are F '-finite, or if A is Nagata and f is essentially of finite type, then for any e > 1, B^®A(e)A is noetherian and$e(A,B) is finite. Proof.
Let e > 0 and set C := B (e) <S> A(e ) A.
Assume that both A and B are F-finite. As A is F-finite, we have
IBM ® -P1! : -B(e) -> B( e ) ® A( .) A = C is finite for any e. It follows immediately that C is Noetherian. Since Fjj is finite, $ e ( J /l, B) is finite by Lemma 4.1, 6. Assume that ^4 is Nagata and / is essentially of finite type. As A —)• C is essentially of finite type, C is also noetherian and Nagata. As A —>- B is essentially of finite type, the /1-algebra map $ e (j4, 5) : C —>• B is also essentially of finite type. As it is also a S^ e )-algebra map and hence is integral, we have that $e(A,B) is finite. n
The following was proved by Radu [22] and Andre [2].
THEOREM 4.3 Let f : A —> B be a homomorphism between noetherian commutative rings of characteristic p. Then the following are equivalent.
1 There exists some e > 1 such that $e(A, B) is flat. 2 For any e > I , we have that
3 / is regular.
5
COHEN-MACAULAY F-INJECTIVE HOMOMORPHISMS
Let (R, m) be a local ring of characteristic p. We say that R is F-injective [8] if the -linear map
is injective for any i E ZL The following is an immediate consequence of the local duality theorem.
LEMMA 5.1 Let (R, m) be an F-finite Cohen-Macaulay local ring. Then the following are equivalent:
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1 R is F-injective; 2 The Frobemus map
UIA ^ Horri£(1) (R,uAm)is surjective; 3 The Frobenius map F^ : R^ —> _R is quasi-split; 3' 27?e Frobenius map FR : R^ —> /J is quasi-split; 3" T/ze Frobenius map FR : R^ -^ R is quasi-split for e > 0; where R denotes the m-adic completion of R, andw^ denotes the canonical module of A for a complete noetherian local ring A. Proof. The only non-trivial implication is 3'o3. As R is excellent, <3>i (R, R) is flat by Theorem 4.3. It is also finite, since both R and R are F-finite. As $i(R/m, R/mR) is obviously an isomorphism, $i(/?, R) <S> R/m is an isomorphism by Lemma 4.1, 4. It follows that <3>i(-R, -R) is an isomorphism. By Lemma 4.1, 6, the assertion follows. D Thus, the Cohen-Macaulay _F-injective (CMFI, for short) property of an _F-finite local ring localizes and passes to the completion. An F-finite ring R is said to be CMFI if each localization at a prime of R is a CMFI local ring. Note that an F-finite noetherian ring R of characteristic p is CMFI if and only if R is Cohen-Macaulay and FR quasi-splits.
LEMMA 5.2 Let f : A -> B be a faithfully flat homomorphism of F-finite rings. If B is CMFI, then so is A. Proof. Clearly, A is Cohen-Macaulay. As FB is quasi-split, we have l B (i) <S> FA is quasi-split by Lemma 4.1, 6. By Corollary 3.2, FA is also quasi-split. D DEFINITION 5.3 Let k be a field of characteristic p, and R a fc-algebra. We say that R is geometrically CMFI over k if for any finite algebraic extension k' of k, k' ®k R is a CMFI ring. DEFINITION 5.4 Let / : A —>• B be a homomorphism of noetherian rings of characteristic p. We say that / is CMFI if / is flat and B(P) is geometrically CMFI over K(P) for any P e Spec A. PROPOSITION 5.5 Let f : (A,m) ->• (5,n) be a flat local homomorphism of Ffinite Cohen-Macaulay local rings. Then the following are equivalent. 1 .B(m) is geometrically CMFI over A/m. 2 There exists some e > 1 such that B(m) ®A/m (A/m}^~e^ is F-injective. 3 For any e > 1, <5>e(A, B) quasi-splits and A-pure.
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4 For any homomorphism g : A —> K essentially of finite type, the base change
f®!K :K^B®AK is CMFI. 5 f is CMFI. Proof.
1=^2 Trivial. 2=>3 Assume that B(m)®A/m (A/rn^-^ is F-injective. Set C :- B^®A(e) A, so that C(m) is F-injective. Then F^,, quasi-splits, and hence $e(A(m), B(m)) also quasi-splits by Lemma 4.1, 7. By 2,4 of the same lemma, <&U(A, 5)(m) quasi-splits for any u > 0. As B is A-flat,
Theorem 4.3. Hence, FB(L) quasi-splits by Lemma 4.1, 6. is trivial.
D
COROLLARY 5.6 Let f : A ->• B and g : B -> C be CMFI homomorphisms of F-finite noetherian rings of characteristic p. Then, g o / is CMFI. Proof. We may assume that A is a field. Then, A, B, and C are Cohen-Macaulay. Now the result follows immediately from Lemma 4.1, 1. D
COROLLARY 5.7 Let f : B -> C be a CMFI homomorphism of F -finite noetherian rings of characteristic p. If B is CMFI, then C is CMFI. Proof.
Let A := Fp in Corollary 5.6. The assertion follows immediately.
P
THEOREM 5.8 Let f : (A,m) -» (5,n) be a flat local homomorphism of F-finite noetherian local rings of characteristic p. If the closed fiber B/mB is geometrically CMFI over A/m, then f is CMFI. Proof. As the completion B —>• B is regular, we have that B/mB is geometrically CMFI over A/m by Corollary 5.7. By Lemma 5.2, it suffices to show that the composite A -> B -» B is CMFI. As the completion A —> A is regular, it suffices to prove that / : A —» B is CMFI by Corollary 5.6. As B/(mA)B = B/mB is geometrically CMFI over A/m, we may assume that A has a dualizing complex, replacing / by /. Let P e Spec A, and K a finite algebraic extension field of «(P), and p e Spec(B <SU K). We want to prove that (B ®A K)v is CMFI. Let A' be the integral closure ofA/P in A'. As A/P is F-finite (hence is Nagata), we have that A' is finite over A/ P.
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By Kawasaki's Macaulayfication theorem [17, Theorem 1.1], there is a proper birational morphism X —> Spec A' such that X is Cohen-Macaulay. Let g be the composite morphism X —> Spec A' —> Spec .A, / : X' = Spec B x Spec A X ->- Spec B the first projection, and h' : X' —>• X the second projection.
Note that Spec(B (g)^ A') is the generic fiber of h'. Let £' be the point of X' corresponding to p. As g is proper, g' is also proper, and hence there is a specialization x' € X' of £' such that 5'(a;') is the closed point of Spec B.
Set x := h'(x'), B" := OX',x', A" := Ox,x, f" to be the associated map A" -> B", and m" to be the maximal ideal of A". Note that g(x) is the closed point of Spec A As X is Cohen-Macaulay, A" is Cohen-Macaulay. As h' is flat, /" : A" -> B" is flat. Note that A" and B" are F-finite. Note also that A"/m" is an essentially
of finite type field extension of A/m. By Proposition 5.5, B/mB <S>A/m A"/m" is geometrically CMFI over A"/m", and hence its localization B"/m"B" is also geometrically CMFI over A"/m". By Proposition 5.5, we have that /" is a CMFI homomorphism. Hence B"®A" K is CMFI. As £' is a generalization of x1, we have that (B ®A K)y is a localization of B" ®A,, K, and hence it is CMFI. n
LEMMA 5.9 Let (A,m) be a noetherian commutative F-finite local ring of characteristic p, and t G m a non-zerodivisor. If A/tA is CMFI, then so is A. Proof. Let R be the DVR F P [T]( T ), and / : R -» A be the map given by T ^ t. As both R and A are F-finite and both R and / are CMFI (note that the residue field ¥p of R is perfect), A is CMFI. O
LEMMA 5.10 Let A be an F-finite noetherian ring of characteristic p. Then, the CMFI locus of A is Zariski open in Spec A Proof.
This is nothing but the intersection of the Cohen-Macaulay locus of A and
the maximal Cohen-Macaulay locus of CokerFyj.
6
D
AN APPLICATION TO F-RATIONAL RINGS
In this section, all rings are assumed to be noetherian commutative rings of characteristic p.
DEFINITION 6.1 ([25, Definition 1.2]) Let I? be a ring. We say that R is strongly F-rational if for any c G J?°, there exists some CQ > 0 such that cFj| : R^ —>• R (a' r ) >->• cap ) quasi-splits for any e > CQ. REMARK 6.2 Velez [25] proved that, quite generally a strongly F-rational ring is F-rational, and conversely, a Cohen-Macaulay local F-rational ring is strongly Frational. He also proved that an F-rational algebra of finite type over an excellent local ring is strongly F-rational.
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241
It follows that an F-finite F-rational ring R is also strongly F-rational. Note that R is Cohen-Macaulay (Theorem 2.1, 7). Hence, the non-quasi-split loci of
(cFj|)(~ e ) for e > 1 form a decreasing sequence of closed subsets of the noetherian space Spec R, and the problem is reduced to the F-finite local case, which is done by Velez as above.
DEFINITION 6.3 Let k be a field of characteristic p, and A a fc-algebra. We say that A is geometrically F-rational over k if k' <S>fc A is F-rational for any finite algebraic extension field k' of k. A ring homomorphism / : A —> B is called F-rational if B(P) is geometrically F-rational over re(-P) for any P £ Spec A
THEOREM 6.4 Let f : (A,m) -» (B,n) be a flat local homomorphism of excellent local rings of characteristic p. Assume that k^ := A/m and kg := B/n are both F -finite. If A is F -rational, the generic fiber B®A K is F -rational, and the dosed fiber B/mB is geometrically CMFI over kA, then B is F -rational, where K is the field of fractions of (the normal domain) A.
Proof.
By Velez' theorem of smooth base change [25, Thoerem 3.1], the composite
map t/> : A —> B —> B satisfies the assumption of the theorem, where B denotes
the n-adic completion of B. Note that the m-adic completion A of A is also Frational (Theorem 2.1, 6). As the closed fiber of the induced map / : A —> B is B/(mA)B = B/mB, and the generic fiber of / is a localization of that of A ->• B, the assumption is also satisfied by /. It suffices to prove that B is F-rational by Theorem 2.1, 6. So replacing / by /, we may assume that both A and B are
F-finite. As a CMFI ring is reduced Cohen-Macaulay, the CMFI homomorphism / is reduced Cohen-Macaulay. As A is a Cohen-Macaulay normal domain, B is reduced and Cohen-Macaulay. By [25, Theorem 3.5], we have that the non-F-rational locus of Spec B is Zariski closed. Let / be the radical ideal of B defining the closed subset. If In A = 0, then there is a minimal prime P of I such that PnA — 0. As Bp, which is not F-rational, is a localization of the F-rational ring B ®A K, this is a contradiction. Hence, we have / n A ^ 0. So we can take c G (/ n A) \ {0}. By the definition of /, B[l/c] is Frational. Replacing c by its appropriate power, we may assume that c is a parameter test element of B by [25, Theorem 3.9] and the reduced Cohen-Macaulay property of B. In other words, we may assume that c € A° fl (f}j(J '-B J * ) } , where the
intersection is taken over all parameter ideals J of B. As A is strongly F-rational and c € A° , there is a positive integer e such that cF| : A^ —5- A quasi-splits as an A( e )-linear map. It suffices to show that cF# : p(e) _^ B quasi-splits for such an e, because then
<S> IB/J • B/J -» B is injective for any parameter ideal J of B, and hence we have x 6 J* =>• xq G (J[• x 6 J
and the proof is complete.
(q:= pe),
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By Lemma 3.2, 1 B («) ig> (cFjj) quasi-splits. On the other hand, 3>e(A,B) quasisplits by Proposition 5.5. By Lemma 4.1, 6, cFg quasi-splits. n
COROLLARY 6.5 Let f : A —> B and g : B —}• C be F-rational homomorphisms between F-finite noetherian rings of characteristic p. Then the composite g o / is F-rational. Proof. We may assume that A = k is a field. Let K be a finite algebraic extension of k. As K ®fc B is F-rational and IK <E> g is F-rational, K ®fc C is F-rational by the theorem. D
COROLLARY 6.6 ([8, Proposition 2.13]) Let (A,m) be an F-finite noetherian local ring of characteristic p, and t £ m a non-zerodivisor of A. If A[\/t] is F-rational and A/At is CMFI, then A is F-rational. Proof. Let R := Fp[T](T), and / : R -> A be the map given by T ^ t. As t is a non-zerodivisor, / is flat. The generic fiber A[l/i] is F-rational, and the closed fiber A/At is geometrically CMFI over ¥p. By the theorem, we have that A is F-rational. n REMARK 6.7 The author does not know if aflat local homomorphism / : (A, m) —>• (B, n) of F-finite noetherian local rings of characteristic p with B/mB geometrically F-rational over A/m is F-rational. This is affirmative if A is essentially of finite type over a field of characteristic p. As Spec A' has a regular alteration for any integral domain A1 which is finite over A [4], this is a slight modification of [9, (IV.7.9.8)]. Hence, we have the following.
COROLLARY 6.8 Let k be a perfect field of characteristic p, (A, m) a local kalgebra essentially of finite type, f : (A,m) —>• (B,n) a flat local homomorphism of local rings. If A is F-rational, B is noetherian and F-finite, and B/mB is geometrically F-rational over A/m, then B is F-rational. Let R be a noetherian F-finite ring of characteristic p. We say that R is strongly F-regular if for any c 6 R°, cF^ : R^ —$• R splits as an /?( e )-linear map for some
e > 0. We can give a slight simplification of the proof of a part of a lemma by G. Lyubeznik and K. E. Smith [20, Lemma 4.1].
LEMMA 6.9 Let f : R —t S be a map of F-finite rings. If f is regular and R is strongly F-regular, then S is strongly F -regular. Proof. Take c G R° such that R[l/c] is regular. As / is regular, S[l/c] is As R is strongly F-regular, there exists some e > 1 such that cFJ. splits. Theorem 3.3], it suffices to show that cFJ splits as an S^-linear map. 15(<0 <8> (cFj|) splits. On the other hand, as / is regular, $e(R,S) splits as linear map by Theorem 4.3. Hence, cFJ splits.
regular. By [13, Clearly, an S^n
REMARK 6.10 Even if (5, n) is essentially of finite type over a perfect field, t € n a non-zerodivisor, and S/tS is strongly F-regular, S may not be strongly F-regular, see [23]. See also [1].
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References
[1] I Aberbach, M Katzman, B MacCrimmon. Weak F-regularity deforms in QGorenstein rings. J Algebra 204:281-285, 1998. [2] M Andre. Homomorphismes reguliers en characteristique p. C R Acad Sci Paris
316:643-646, 1993. [3] LL Avramov, H-B Foxby. Cohen-Macaulay properties of ring homomorphisms.
Adv Math 133:54-95, 1998. [4] AJ de Jong. Smoothness, semi-stability and alterations. Publ IHES 83:51-93, 1996.
[5] T Dumitrescu. Reducedness, formal smoothness and approximation in characteristic p. Comm Algebra 23:1787-1795, 1995. [6] R Elkik. Singularites rationnelles et deformations. Invent Math 47:139-147,
1978. [7] F Enescu. On the behavior off-rational rings under flat base change, preprint, 1999. [8] R Fedder and K-i Watanabe. A characterization of F-regularity in terms of -F-purity. In: M Hochster, C Huneke, JD Sally, eds. Commutative Algebra. Proceedings of the microprogram held in Berkeley, California, 1987. New YorkBerlin: Springer, 1989, pp 227-245. [9] A Grothendieck. Elements de Geometrie Algebrique, IV (Seconde Partie). Inst. Hautes Etudes Sci. Publ. Math. 24, 1965.
[10] A Grothendieck. Seminarie de Geometrie Algebrique 1, Revetements etale et groupe fondamental. Lecture Notes in Mathematics, 224. Berlin-New York:
Springer, 1971.
[11] N Hara. A characterization of rational singularities in terms of injectivity of
Frobenius maps. Amer J Math 120:981-996, 1998. [12] Auslander-Buchweitz Approximations of Equivariant Modules, to appear in London Math Soc Lecture Note Ser, Cambridge: Cambridge. [13] M Hochster, C Huneke. Tight closure and strong F-regularity. Mem Soc Math
France (N.S.) 38:119-133, 1989. [14] M Hochster, C Huneke. Tight closure, invariant theory, and the BriangonSkoda theorem. J Amer Math Soc 3:31-116, 1990. [15] M Hochster, C Huneke. F-regularity, test elements, and smooth base change.
Trans Amer Math Soc 346:1-62, 1994. [16] C Huneke. Tight closure and its applications. CBMS Regional Conference Series in Mathematics 88. Providence, RI: AMS, 1996.
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[17] T Kawasaki. On Macaulayfication of Noetherian schemes, to appear in Trans Amer Math Soc.
[18] E Kunz. Characterizations of regular local rings of characteristic p. Amer J
Math 91:772-784, 1969. [19] E Kunz. On Noetherian rings of characteristic p. Amer J Math 98:999-1013, 1976. [20] G Lyubeznik, KE Smith. Strong and weak F-regularity are equivalent for graded rings. Amer J Math 121:1279-1290, 1999. [21] H Matsumura. Commutative Ring Theory, 1st paperback ed. Cambridge: Cambridge, 1989. [22] N Radu. Une classe d'anneaux noetheriens. Rev Roumanie Math Pures Appl
37:79-82, 1992. [23] A Singh. F-regularity does not deform. Amer J Math 121:919-929, 1999. [24] KE Smith. Vanishing, singularities and effective bounds in prime characteristic local algebra. In: J Kollar, ed. Algebraic Geometry, Santa Cruz 1995. Proc
Symp Pure Math 62 (1). Providence, RI: AMS, 1997, pp 289-325. [25] JD Velez. Openness of the F-rational locus and smooth base change. J Algebra
172:425-453, 1995. [26] K-i Watanabe. F-regular and F-pure normal graded rings. J Pure Appl Algebra
71: 341-350, 1991.
Valuations of ideals, evolutions and the vanishing of cohomology groups
REINHOLD HUBL, NWF I - Mathematik, Universitat Regensburg, 93040 Regensburg, Germany; email: [email protected] ANTON RECHENAUER, NWF I - Mathematik, Universitat Regensburg, 93040 Regensburg, Germany; email: [email protected]
The concept of evolutions has been introduced by B. Mazur [13] in connection with Galois-deformations and the work of Wiles [18] resp. Taylor and Wiles [17] on modular elliptic curves. Given a local ring k and a local algebra R/k, essentially of finite type, an evolution of R/k is a local fc-algebra S, also essentially of finite type over k, together with a surjective rmghomomorphism 5 —+ R of fc-algebras such that the induced map £lls,k(g>sR —> ^Jj/j. of Kahler differentials is an isomorphism. The algebra R/k is called evolutionarily stable, if any such evolution s : S —* R is trivial (i.e. if any such e is an isomorphism). In connection with evolutions, B. Mazur raised the following Question. Suppose k is a field of characteristic 0 or a complete discrete valuation ring of mixed characteristics. Is it true, that any reduced flat local algebra R/k, essentially of finite type, is evolutionarily stable? The question was of interest in connection with Wiles' work on Fermat's last theorem. A positive answer would provide an alternative approach to a crucial part in its proof. It has been studied by Eisenbud and Mazur [1], who also showed that the corresponding question in positive characteristics has a negative answer. The first example of a curve singularity in characteristic 2 that is not evolutionarily stable actually was already provided by E. Kunz. Eisenbud and Mazur also obtain positive answers to the above question in several cases. Some more positive answers have been obtained by Huneke and Ribbe [5] and by the first one of the present authors [8], though the question in general is still wide open.
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In this note we try to shed some light on this problem from a birational point of view, following ideas and techniques introduced by J. Lipman ([10], [11]). This approach raises some problems and questions that seem to be of interest in their own right. Special thanks go to J. Lipman for helpful conversations and discussions on the techniques used in [11] and [12]. During the preparation of this paper the first author was partially supported by a Heisenberg-Stipendium of the DFG, and the second author was partially supported
by a scholarship of the University of Regensburg. The following criterion for an algebra to be evolutionarily stable is essentially due to Eisenbud and Mazur ([I], §1):
Theorem 1. Let k be a noetherian local ring, let R/k be a local algebra, essentially of finite type, and write R = P/I for some smooth local algebra (P, m)/k essentially of finite type. Then the following are equivalent: (1) R/k is evolutionarily stable. (2) // / G / with 6(f) & I for all 8 <E Derk(P), then f £ m • I.
(3) If f €l with dR/kf e / • Op/j., then / e m - / . Proof. The equivalence of (1) and (2) is essentially due to Eisenbud and Mazur, [1], §1, see also [8], (1.1), and the equivalence of (2) and (3) is obvious as flp/j. is a finitely generated and free P-module by [9], (8.1), and as
by the universal property of Kahler differentials. We note in particular that the conditions (2) and (3) of thm. 1 are independent of the special choice of a presentation R — P/I of R as & homomorphic image of a smooth local algebra (P, m)/k .
From now on k will be a field of characteristic 0. Let R/k be a local algebra, essentially of finite type, and let R = P/I be a presentation of R/k with a local algebra (P, m)/k, smooth and essentially of finite type.
Theorem 2. In the above situation consider the following conditions: (1) R/k is evolutionarily stable. (2) // / 6 / such that v(f) > v ( I ) for each Rees-valuation v of I , then
/em-/.
(3) If f £ I with /" £ /"+1 for some positive integer n, then f G m • I . i) We always have (2) <=> (3) =» (1) ii) If I is a radical ideal, then all three conditions are equivalent. Remark 3. The integral closure / of an ideal / C R is defined to be the set of all elements x G R satisfying an equation of type xn + aix"-1 + • - • + a n _ i a ; + an = 0
with at € /' for all t £{!,..., n}.
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cf. [19], app. 4. Note that I always is an ideal with 7 C 7 and with 1 = 1. An ideal I C R is called integrally closed, if 7 = 7 , and it is called normal if In = 7" for all positive integers n. If R is an excellent domain, Rees [15] has shown, that
for each non-trivial ideal I C R there exists a unique finite set
T(r):={Vl,...,vm} of discrete rank 1 valuations Vj on K = Q(R), the field of fractions of R, such that for all x G R and positive integers n the following are equivalent: i) x 6 7" . ii) Vj(x) >n-Vj(I):=n- min{vj(r) : r e 7 \ {0}} for all j € {1, - . . , m). and such that T(7) is minimal with this property. These valuations are called the Rees-valuations of 7. They can be obtained as follows: Let 7£(7) := R[It] be the Rees-ring of 7 and let 'R.(I) be its integral closure. Then 7£(7) is a graded normal domain and 77£(7) is a homogeneous ideal of pure height 1 in it. Thus, if Min(7ft(7)) = {*plt . . . ,
homogeneous localization of 1i(I) in ^3,- , is a discrete valuation ring, contained in K , and the corresponding valuations Vj of K (j — 1, . . . , m) are precisely the Rees-valuations of the ideal 7 . For the proof of thm. 2 we need the following easy lemma:
Lemma 4- Let V/k be a discrete valuation ring, essentially of finite type and let TT 6 V be a regular parameter. Then there exists a derivation 8 € Derfc(V) with
«(») = 1 For this derivation 8 we have
v(6(x)) = v(x) - 1
for all x £ mv \ {0}
where v is the normed valuation defined by V .
Proof of the lemma. As Oy/j. is free as a K -module by [9], (7.2), and as
it suffices to show that dv/kif £ my -fi^/j. • This is however obvious by the exactness of the sequence
0 —»• mvmr =ir-I —>• fLl with / = V/mv (cf. [9], (6.5)). Thus we find a 8 e Der^V) with %) = 1. For any x e m^ \ {0} write x = irn • £ with a unit s e V* and some n > 0. Then
and therefore v(8(x)) — v(x) — 1 as Q C K.
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Proof of thm. 2. The assertions of this theorem are essentially contained in [8],
though the result is not stated there in this form. Therefore a complete proof is included here. First we show the equivalence of (2) and (3): ___ n+1 Clearly if /" G 7 for some positive integer n, then /" G 7n+1 , hence w ( / )>!L±l. ,(/)>„(/)
for each Rees-valuation v of /. Thus (2) implies (3). Conversely suppose that condition (3) is satisfied, let T(7) := {vi,...,vm} be the set of Rees-valuations of 7, and let < p 1 ) . . . , < p m G Spec(7£(7)) be the prime ideals of 7£(7) of height 1 determining these valuations, and let py be the corresponding maximal ideals of the discrete valuation rings 1^(1)^ .) (i.e. ) for j = 1, . . . , m . If / G 7 with Vj(f)
> «/(/) for
all j 6 {!,.. . ,m}, then (*)
/ 6 pr I • K(I)(V.)
for all J G {!,..., m}
On the other hand (**)
1
If we assume that ft (£ ^3 j for some j G {!,..., m} , then we have that
and with p^ := tyj -"R.(I)jt n72.(7)/^ t - ) we conclude from (**):
contradicting (*). Thus ft g
conclude that dy/kf G 7 • ^yik, hence 6(f) G 7 for all 6 G Der^V), implying that v(<5(/)) > v ( I ) for all such 8 , As / G 7 • V C my (v being a Rees-valuation of 7), we conclude from lemma 4 that v(f) > v(I) + 1, hence / G m • 7 by (2). Thm. 1 now implies that R/k is evolutionarily stable. If 7 is a radical ideal, then the equivalence of (1) and (3) is an immediate consequence of [8], (1.2) in connection with thm. 1.
Valuations of ideals
249
Remark 5. Condition (3) of thm. 2 may be viewed as a condition on the associated graded ring
G := grj(P) = 0 In/In+l of P with respect to /: If a homogeneous element / + I2 6 G\ of G of degree 1 is nilpotent in G , then we ask whether / cannot be part of some set of minimal generators of /. If / is a radical ideal, then R = P/I is evolutionary stable as a fc-algebra if and only if the answer to this question is positive. In [8] in particularly this condition was studied. There it was shown that (3) is satisfied for some classes of radical ideals, such as ideals generated by rf -sequences, Cohen-Macaulay ideals of minimal multiplicity and Gorenstein ideals of almost minimal multiplicity. In each of these cases it was shown that some stronger condition is satisfied, mostly involving the fibre cone Fm(I) := 0 7"/m-7 n of I . Obviously the above question ngN
has a positive answer if Fm(I} is reduced (or even a domain), for in this case any / £ / that is part of a minimal set of generators of I satisfies /" ^ m • 7" for all n £ N , hence in particular its leading term in G is not nilpotent. This condition can be verified for certain classes of ideals, cf. [8], §3, [6]. However in [8], §4 it is also shown that Fm (I) is not reduced in general. Under some suitable depth condition it can be proved that G has no nilpotent elements of degree 1 at all, cf. [8], §2, however this also will not be the case in general. It seems to be very hard to verify (3) of thm. 2 directly as fairly little is known about the structure, in particular about the nilpotent elements of gr/(P).
Remark 6. Following Samuel [16] we define the 7-adic order Vi(x) of an element x € P \ {0} in a local domain P to be the largest nonnegative integer n with x £ In , and we define the 7-adic limit order Vj (x) of x to be V
:=
lim ! 2 H
We note that this limit always exists and is a rational number. For a > 0 we define ideals Ia and I~ by Ia := {x € P \ {0} : Vi(x) > a} U {0}
and 7- : = { * € P \ { 0 } : V F ( a O > a } U { 0 }
Then under the assumptions of thm. 2 for a radical ideal I C R the following conditions are equivalent: (1) R = P/I is evolutionarily stable as a fc-algebra.
(2) Ia C m • I for all a > 1 . (3) /f C m • I In fact, Rees [15] has shown that
and from this description of the 7-adic limit order and thm. 2 the equivalence of the above statements follows easily.
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From now on let (R, m) be a regular local ring of dimension d, essentially of finite type over k, let 7 C R be a radical ideal and let T( J) be the set of its Reesvaluations. Furthermore we denote by K — Q(R) the field of fractions of R. By an R -valuation we mean a discrete rank 1 valuation v on K such that v(r) > 0 for all r g R\ {0} and such that its valuation ring V is essentially of finite type over k (or equivalently over R). However we do not require that v is an m-valuation in the sense of [6], i.e. that v(x] > 0 for all x £ m \ {0} . Set V/ := {v : v is an ^-valuation with v(I) > 0} For subsets A, B C Vi (not necessarily distinct or disjoint) we define
ZA,B(!) :={r&R\ {0} : v(r) > v(I) + 1 for all v € A and v(r) > v(I) + v(m) for all v e B} U {0} Note that the ideals 0,4^(7) are always integrally closed as they may be obtained as the (not necessarily finite) intersection of valuation ideals, cf. [19], app. 4.
Theorem 7. In the above situation assume in addition that (R, m) contains a field k of characteristic 0 over which R is essentially of finite type, and over which R/m is finite, and assume that I is a radical ideal. Then the following are equivalent: (1) R/I is evolutionary stable as a k-algebra.
(2) aA,B(I) Cm-1 for all A,BCVi with T(I) C A. (3) There exist subsets A, B C Vi with T(I) C A such that aA,B(I) Cm-1 Proof. Assume that we have (1), i.e. that R/I is evolutionary stable as a kalgebra, let A, B C V/ with T(7) C A, and let A0 - T(I), B0 = 0. Then clearly
&A,B(I) C aAo
QAi,Bi(7) C aAis(I) C m • 7 and thus we may assume that A = B = V/. Let m — (xi,...,xn) with regular parameters xi,... ,xn. As R/m is finite and separable over k, we conclude from [9], (6.5) that ,- .
/^~2
Tu/m
^>
f~il
/
LT-l
f*) 1
:= ii^/j./ITl • i'^j/j.
which in turn implies that £llR/k — Rdxi + - • - + Rdxn . Let v 6 V/ be any valuation, let V be its valuation ring and let TT be a regular parameter of V. By the above description of £llR/k we have , T , J _ r- _max V —— &v/k) = VdXl + ••• + Vdxn C
0,« m
1 . Q^
Valuations of ideals
251
Thus if / e / with d B /fc/ € 7 • fi}j /jb , then
(*)
dv/kf € ,r max { 0 Xm)-i} . / . &v/k = T m«{»W.»(0+«(»)-i} . nj, /fc
As w £ V/ we have that / • V C mv , hence / £ mv - Thus by lemma 4 there exists a 6 £ Derfc(V) with v(S(f)) = v(f) — 1, and from this and (*) we get (as in the proof of thm. 2) f £ T max{»(/)+l,w(/)+w(m)} . y
Since v G V/ was arbitrary, this implies that / e 0^(7), hence / € m • 7. By thm. 1 we conclude that R/I is evolutionarily stable as a fc-algebra. Corollary 8 (cf. [8], 3.11). In the situation of thm. 7 assume that m • I is integrally closed. Then R/I is evolutionarily stable as a k -algebra.
Proof. Let A - T(I) C V/ and let B = T(m • I). Note that T(m • 7) C V/. Then
OA,B(/) C a0 ]B (7) = m • 7 = m • 7 and the claim follows.
Remark 9. If R is two-dimensional, then any product of integrally closed ideals is integrally closed again. In higher dimensions this is not true any more, cf. [3] or [8], §4. In [4] conditions on ideals I C R that force m • 7 to be integrally closed are studied. If 7 is a normal ideal (i.e. all powers of 7 are integrally closed), and if its fibre cone Fm(I) — 07"/m • 7" has no embedded components, then m • 7" is integrally closed for all positive integers n . This is the case for instance
if dim(TJ) = 3 and 7 C R is a normal height 2 prime having reduction number at most 2. In general however the structure of Fm(I) is not well understood. For instance it seems to be not known whether the fibre cone Fm (7) is equidimensional if 7 C R is a prime ideal. ___ Other examples of ideals I C R satisfying m • 7 = m • 7 are given in [7]. An ideal / C R is integrally closed if and only if it is contracted for every birational proper map IT : X —> Spec(7?), and to check integral closedness, it suffices to consider one such w for which X is normal and J • Ox is invertible (cf. [10]). Such maps also can be used to study the ideals 0,4,5(7). Corollary 10. In the situation of thm. 7 let Y - Spec(R) and let it : X —f Y be a birational projective morphism such that X is regular and such that I • Ox defines a divisor. Write I -Ox ~ Ox(^ —a,jDj) with positive integers a,j and pairwise
i=i
distinct prime divisors Dj . Then the following are equivalent:
(1) H°(X, Ox(i: -(a, + 1) • Dj)) Cm- 1 j=i (2) 7?/7 is evolutionarily stable as a k -algebra.
Remark 11. i) Note that H°(X,OX) = R and H](X,OX)
= 0 for / > 0. Thus we
can think of H°(X, Ox( £ -(a; + 1) • Dj)) as an ideal of R.
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ii) As 7 • Ox is an invertible ideal of Ox and as X is a regular scheme, we have t that 7 • Ox - OX(D) for some Weil-divisor D = £ -o,-7^ (cf. [2], II. 6.11). If
Wj is the discrete normalized valuation on K = Q(R) determined by Dj (i.e. by OX,DJ ), then o,j = Wj(I), i.e. aj > 0 for j £ {1,..., t}, and
{r € J? \ {0} : uy(r) > a,- + 1 for all j 6 {1,... ,t}} U {0}
Proof of Corollary 10. Let g : Z :— Proj(72.(7)) —* V be the normalized blow-up of 7. As 7 • Ox is invertible and X is regular, the universal properties of blowing up and of normalization imply that TT : X —> Y factors as X
—^
Z
Y for some projective birational map / : X —>• Z. Thus if Wj denotes the normalized 7Z-valuations defined by OX,DJ for j £ {!,...,<}, then T(7) C {wi,..., wt] ='. A, and by rem. 11 ii) (1) is equivalent to
«U,0(-0 C m • 7 Hence the corollary follows from thm. 7.
Corollary 12. In the situation of thm. 7 let Y = Spec(R) and let ir : X —>Y be a birational projective morphism such that X is regular and such that I • Ox defines t a reduced divisor, i.e. I • Ox = Ox($3 ~Dj) with pairwise distinct prime divisors 3=1
Dj . Then the following conditions are equivalent:
(1) 7 2 C m - 7 (2) 72/7 is evolutionarily stable as a k -algebra.
Proof. From [12], (1.4) resp. [10], (6.2) we conclude
H°(X,0X(J: -2 • Dj)) = H°(X,I2 -0X) = J5 j=i and cor. 12 is immediate from cor. 10.
Remark 13. Condition (1) of cor. 12 is satisfied in particular if 7 is a normal ideal. Hence if 7 C R is a normal ideal, defining a reduced divisor in some resolution of singularities TT : X —*• Y, then R/I is evolutionarily stable as a ^-algebra. This
however is also an easy consequence of [6] (4.1) in connection with [8], (2.1).
Valuations of ideals
253
Using thm. 7 we even can replace condition (1) of cor. 10 by a (seemingly) stronger one:
Corollary 14. In the situation of thm. 7 let Y = Spec(K) and let ir : X —> Y be a birational projective morphism such that X is regular and such that both I • Ox and m • Ox are invertible. Write
E
-&,-•£>,•)
j=6+l
with positive integers a j , b j , with S < t and with pairwise distinct prime divisors DI, ..., Dt. Then the following conditions are equivalent:
(1) H°(X, CM E -(a} + l).Dj+
E
3=1
3=6+1
-(a,- + 6,-) • D,-)) C m • I.
(2) R/I is evolutionarily stable. Remark 15. If v is a prime divisor of X of the first kind which is nonnegative on R with v(m) > 0, then also v(I) > 0. Hence any prime divisor in the support of m • Ox is also in the support of I • Ox • Proof. The proof of this result is similar to the proof of Cor. 10. It remains to note that TT factors both through the normalized blow-up of m and I of Y. Thus, if Wj denotes the normalized valuation of K associated to Dj , and if we set A = {wi,...,wt} and B — {ws+i,..., Wt} we will have that T(I) C A and A, B C Vi. Hence thm. 7 completes the proof. Now we fix a birational projective morphism ir : X —> Y = Spec(fi) such that X is a normal k-variety (not necessarily smooth), and such that both m • Ox and / • Ox are invertible ideals, and we set £ :— m • Ox • Then £ is a line bundle on X, generated by its global sections. In fact if m = (ri,...,rj) with a regular system of parameters r i , . . . , r<j of R, then these elements, viewed as elements of H°(X,£), generate £. Furthermore each of these elements r,- defines a morphism d
ri : C~l —> Ox , given on local sections by e >—^ e(r,-). Thus setting M := ® L~l, i=l
the d-fold direct sum of C~l, the regular parameters r i , . . . , r<j define a canonical morphism
a : M —>OX d given on local sections by < r ( e i , . . . , e<j) = E £i(ri). As £ is generated by r\,..., r^, 1=1 the morphism cr is surjective, and it therefore defines a Koszul complex
K*(M,
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Hiibl and Rechenauer
is a direct sum of copies of £~s. As tn* • 7* • Ox is locally free as an Ox -module, we also get that the complexes
(£J|t, ff;it) := V(M, a) ®Ox m' • I1 • Ox are exact sequences of sheaves of locally free Ox -modules for all non-negative integers s,t. Proposition 16. In the above situation for a pair s, t of non-negative integers the
following conditions are equivalent:
(1) m» •/< = t n - m ' - 1 - / * . (2) H°(X,o-lit):H°(X,IClit)—>
H°(X,IC0,it) is surjective.
Proof. We have that H°(X,IC°it) - H\X, m' • 7* • Ox) = m* • /* is the integral closure of m • / by [10], (6.2), and that
H°(X,ICl!t) = (H°(X, m-1 - 7< • Ox))' = is the d-fold direct sum of copies of m*"1 • 7* (using [10], (6.2) again). Furthermore
d 1=1
i.e. Im.(H°(X, 0"Jit)) = m • m
s-1
• 7* , and from this the claim follows.
Corollary 17. Assume that for some pair s, t of positive integers
(*,,«) Then
Hl(X, I* • £—*) = H\X, 7* • C'~3) = ••• = Hd~l(X, f • C°-d} = 0 _____ _______ m' • 7* = m - m ' - 1 -7*
In particular if (* it i ) is true, then m-7 is integrally closed and R/I is evolutionary stable as a k -algebra. If (*s,i) for all s > 1 .
zs
true for all s > 1, iAen m* - 7 is integrally closed
Proof (cf. [12], §5). The Koszul complex £J t decomposes into short exact sequences
(Ri)
o — % — K;I( — ft,-_i— ^ o /d\
with Tlo = /C°_ t and K^ - K-ds>t - 7* • C'~d . As £j >t = (7* • £"-'') V i / is a direct sum of copies of 7* • £'""' , the long exact cohomology sequences associated to and the assumptions imply that the composition of connecting morphisms
is injective. As Hd~l(X,Tld-i) - Hd~l(X,tCd>t) - 0 by our assumption, we conclude that Hl(X,1ti) = 0. Hence the long exact cohomology sequence associated to (Ri) implies that
H\X,
Valuations of ideals
255
Remark 18. i) In general m • / is not integrally closed, even if we assume that I is a radical ideal (or even a prime ideal), cf. the examples in [8], (3.12) and [8], §4. Thus the vanishing required in cor. 17 will definitely not hold in general, not even in case s = t = 1. In each of the examples given in [8] however condition (1) of cor. 14 is satisfied. ii) If in the situation of cor. 17 the scheme X arises from Y by blowing up I • m and normalizing, then 7 • m • Ox is a TT-ample invertible sheaf on X, hence also ample (as Y is affine). Thus for all sufficiently large s,t the conditions (*i,t) will be satisfied. Hence for all s, t sufficiently large, we will have
m5 •/* = m - m " - 1 • 7<
Assume now in addition to the above assumptions, that X is regular, hence that 7 • Ox and m • Ox define divisors on X. As above we write
with positive integers aj , bj , with 8 < t , and with pairwise distinct prime divisors DI, . . . ,Dt, supported on the exceptional locus of TT . With these notations let
(ai + l)-Di + E -(a,- + 6,-) ' A) and note that J C m • / • Ox = £?(1 .
Proposition 19. If the inclusion L : J <—* K° x admits a lifting e : J —> K,\ l then
R/I is evolutionarily stable. Proof. If L admits a lifting of the desired kind, then
H°(X, J) C lm(H°(X, a\>}}) = m • 7 and the proposition follows from cor. 14.
Remark 20. More generally the ^-algebra R/I is evolutionarily stable if and only if we have
However this inclusion does not necessarily imply that the inclusion i : J <—>• K,° l admits a lifting £ : J —»• K\ l . Theorem 21. Suppose that for each I g {1, . . . , d — 1} we have
H!(X,Ox(EDi+
E (/+!)&,-• A)) = 0
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Hubl and Rechenauer
Then R/I is evolutionarily stable. Proof. We will show that the inclusion
t : J —y m • / • Ox
admits a lifting e : J -+ K,\^ . Let C* := J~l ® AC^ , and let
Then it remains to show that the canonical inclusion L : Ox —»• .A/" admits a lifting e : Ox —»• C11 . For this it suffices to proof that H\X,Cl) = Eomox(Ox,Cl) -^U Hom Ox (Ox» := H°(X,M) is surjective. As C" is an exact sequence of locally free Ox -modules, decompose C* into short exact sequences
(Ri)
we
may
0 —>Ki —>& —>Hi-i —> 0
(with 72.Q = A/" and Tid-i = Cd). We conclude from the long exact cohomology
sequence associated to (Ri) that it suffices to show
H1(X,K1) = Q Note that
i=s+i
J
and therefore the assumptions imply the H!(X, Cl+1) = 0 for all / € {1, . . . , d— 1} . Thus the long exact cohomology sequences associated to (Ri) for i > 2 imply that the composition of the connecting morphisms
is injective. As
Hd-l(X,Ud-i)
= Hd~l(X,Cd) = 0
by our assumptions (cf. above), the claim follows. The above theorem raises the following
T Vanishing Question. In the above situation let E Di be the reduced exeptional i=i T
divisor of w , and let E := E p,-£>; be an effective
divisor supported on this excep-
i= l
tional locus (i.e. p,- > 0 for all i e {!,... ,T] ). Under which conditions on the p,is it true that
H'(X,0X(E))
=0
for all I e {1, . . . , d- 1}
Valuations of ideals
257
Remark 22. i) The vanishing required in thm. 20 only depends on the prime divisors of X , dominating / (but not m ) and on m , but not on the multiplicities of I at these prime divisors. ii) If E = 0, i.e. if Pi = 0 for all i=l,...,T, then H'(X, OX(E)) = 0 for all / > 1 . This holds even true if we only assume that (R, m) has rational singularities. For arbitrary pi we however do not expect the above cohomology groups to vanish. In fact, if / C R is a radical ideal such that m • / is not integrally closed (cf. [8], T §§ 3, 4), and if C := m • Ox , then / • C~s = £ aj,»Dj =: E, with (however not J=i necessarily non-negative) integers o^, , and cor. 17 implies that for one of these E, the above vanishing will not hold. iii) As far as evolutions are concerned, we may in addition assume that the T
reduced exceptional divisor ^ Dj is a strictly normal crossing divisor. j=i iv) If in the above situation Ox(—E) is generated by global sections, then
Hl(X,ux <S> OX(-E)) = 0
for all/ > 1
(where u>x is the canonical divisor of X) by a result of S. D. Cutkosky (cf. [11], app. A. 2). This vanishing is related to the above vanishing question via formal duality, does however not imply it. These relations will be studied more closely in [14]. So far we do not know whether in this situation and for this E also the above vanishing holds. As (R, m) is regular and local, the canonical divisor u>x of X is supported on the exceptional locus of TT , and therefore Cutkosky 's result produces divisors supported
on the exceptional locus for which vanishing holds. v) The lifting property of prop. 19 seems to be much stronger than the property of R/I being evolutionarily stable as a & -algebra. Thus we suspect that it might not be satisfied very often. Therefore it would be interesting to find other conditions that force H°(X,J) to be contained in Im(.ff 0 (X,<7i )1 )). Remark 23. Instead of looking at the Koszul complex arising from a (minimal) set of generators of m, we also might look at / = («i, . . . , s n ). Setting C — I • Ox , ._
/ _.
\ n
M. — ( C~l J
__
and looking at the map a : M. —* Ox induced by si, . . . , sn , we
obtain as above a Koszul complex f C ' ( M ) and complexes
,
0x
Similarly to prop. 16 the surjectivity of H°(X,cfl
m' • I* • Ox t)
is equivalent to
ms -P = I-ms -I*- 1 which again is implied by the vanishing of certain cohomology groups, cf. cor. 17. Instead of looking at I we also might look at a reduction J — (si,..., s\) C I of /, and we obtain conditions for the equality ms • I* = J • m* • It~1 to hold. This gives theorems of Brianc.on-Skoda type, cf. [12], §5.
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Eisenbud, D. and B. Mazur: Evolutions, symbolic squares, and Fitting ideals. Jour, reine angew. Math. 488 (1997), 189-201. Hartshorne, R.: Algebraic Geometry, Springer, New York, Heidelberg, Berlin, 1977. Huneke, C.: The primary components and integral closures of ideals in 3-dimensional regular local rings. Math. Ann. 275 (1986), 617-635. Huneke, C. and R. Hiibl: Fibre cones and the integral closure of ideals. Preprint. Huneke, C. and J. Ribbe: Symbolic squares in regular local rings. Math. Z. 229 (1998), 31-44. Hiibl, R. and I. Swanson: Discrete valuations centered on local domains. To appear in Jour. Pure AppL Algebra. Hiibl, R. and I. Swanson: Normal cones of monomial primes. Preprint. Hiibl, R.: Evolutions and valuations associated to an ideal. Jour, reine
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On integral schemes of vector fields ERNST KUNZ, Fakultat fur Mathematik, Universitat Regensburg, 93040 Regensburg, Germany; email: [email protected]
ABSTRACT: In algebraic differential calculus analoga of facts from analysis, differential topology and differential geometry are proved which then can be applied to commutative algebra and algebraic geometry. The purpose of this article is to show how the wellknown theorem of Frobenius about integral manifolds of vector fields translates into algebra (see Theorem 4.1 and Corollary 4.2 below.)
1 BASIC FACTS AND NOTATIONS In the following R/RQ is always a commutative algebra, QlRiRa its module of
Kahler differentials, DeiRoR = Homft(fik, flo , R) its derivation module and §(fl]j;flo) the symmetric algebra of £llR,Ra • We consider Der# 0 R as a, Lie algebra with respect to the Lie bracket [ , ]. If the universally finite module of differentials of R/Ro exists it will be denoted by fijj/^. If in this case R is noetherian we also have Der#0.R = HornR(fj!jj, fio , R). We denote by d the universal (universally finite) derivation of R/RQ. For p G Spec/? the residue field of the local ring Rf is denoted by fc(p). Let X := SpecJ?. Then T(X/R§) := Spec§(fi]j, R() ) is called the tangent bundle of X/RQ, and the canonical morphism TT : T(X/Ro) —>• X corresponding to the natural injection R —> §(^Jj/^ 0 ) is called the projection of the tangent bundle onto its base space X. The formal properties of tangent bundles in the theory of schemes are described in EGA IV.16.5. A survey on results about the structure of tangent bundles is given in [Ku2J. A section V : X —> T(X/Ro), i.e. a morphism with TT o V — idx, is called a vector field on X/Ro. It is of the form V — Spec 9? with a unique .R-homomorphism §(Qjj, fl(j ) —+ R. By the universal property of the symmetric algebra these
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For p e X the fiber 7r~ 1 (p) at p is the subscheme SpecS(QL flo )p/p§(n] l/ , Ho )p = Spec §fc( p )(fc(p) ®B ^/J/H O )- I*8 &(p)-rational points correspond by the universal property of the symmetric algebra to the _Rp-linear maps £ : fi)j ,R —»• fc(p), hence to the elements of Tf (X/Ro) := DerH 0 (/Z p , fc(p)). This fc(p)-vector space is called the (Zariski-) tangent space of X/Ro at p, its elements are the tangent vectors at p. For a vector field V on X/Ro the point V(p) € ?r~ 1 (p) corresponds to the canonical
composition Q1R/R —>• J2 —»• &(p), where I is the linear form associated with V . If we consider V as an element of Der/e0/?, then V(p) € Tp(X/Ro)
is the canonical image
of V by the composed map
We call V(p) the tangent vector of I/ at p. LEMMA 1.1. Assame fAai /or p € X the module of differentials Q^ ,R is free of finite rank. Then for Vi, . . . , Vm € Derfi0.R the following statements are equivalent: a
) ^i(p)i • • • i Kn(p) are linearly independent over Ar(p).
b) (Vi, . . . , Vm} is part of a basis of the R^-module DerH0.Rp. PROOF: Since ^lRp/Ro is free, we have Der flo (^ p , fc(p)) S Hom Rp (Q^ /fio , fc(p)) S
Hom^Cfi^/^, ^p)/pHom fip (n^ /fio , H p ) S Der Ro /2p/pDer Ro fip, and the claim follows from Nakayama's lemma. In case m = 1 condition a) is of course equivalent with: a') Vv(Ry) contains a' unit of Rp. Now let X C X be a closed subscheme, i.e. X — Spec/?// with an ideal I C R, which we call the defining ideal of X . An X-morphism V : X —»• T(X/Ro) is called a vector field on X/Ro along X. It corresponds to an /2-homomorphism §(^/^j0) —* R/I, hence to an element of DeTRo(R,R/I). Therefore we identify the vector fields along X with the elements of this derivation module. Set R :— R/I, let e : R ^- R be the canonical epimorphism, and W : T(X /Ro) —* X the canonical projection.
Similarly as above for V 6 Der #„(./?, R) and p 6 X the tangent vector V(p) of V at p is by definition the canonical image of V in Tp(X/Ro). The canonical epimorphism §(fi/{/# 0 ) —>• ^(^TJ/D ) induces a natural commutative diagram
T(X/R0)
-i-+ T(X/R0)
(1) X <——————> X
where i is a closed immersion. For each vector field V : X —•> T(X / RQ) its composition with /, is a vector field on X/J? along X , called the tangential vector field along X
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induced by V . In terms of derivations it is the image of V by the canonical map a : DeiR0R —> DeiRo(R, R) (V >—*• V o e). This map is injective, and we shall identify Deift0R with its image by a. For p G X we have an injective functorial map
Therefore Tv(X/Ro)
T,(X/Ro).
can and will be identified in the following with its image in
Thus F(p) = a(
Similarly for a vector field V : X —* T(X/Ro) its composition with X ^ X defines a vector field along X called the restriction of V along X . In terms of derivations the restriction of vector fields is given by the canonical map
(R,R) At the points p £ X the tangent vectors V(f) and /?(V)(p) in Ty(X/Ro)
agree.
Since / • Derfi0.R C Der/{0(.R, I) the functorial exact sequence (2)
0 -> Der^CTZ, /) -> Der^U
induces an exact sequence of 7?-modules
(3)
0 -* DeiRa(R, /)// • DeiRoR -> DeiRoR/I • DeiRoR -^ De?Ro(R, R)
If fl^j, „ is a finitely generated projective _R-module, then /? is an isomorphism, since
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For a submodule D C DetRoR we may ask for conditions whether /3(D) C , i.e. whether for each vector field from D its restriction along X is tangential toT.
LEMMA 1.2. The following assertions are equivalent: a) J3(D] C Der^R
b) I is D-invariant, i.e. V(I) C / for each V 6 D. PROOF: a)^b) If /3(D) C Derfio#, then for any V £ D there is a V e DerfioS such
that e o V = Vo e. Then e(V(l)) = F(g(/)) = 0, hence V(I) G /. b)—»-a) If / is D-invariant, then each V G D clearly induces a map V : R —* R with e- o V = V o e, and V is a derivation of
DEFINITION 1.3. A closed subscheme X C X is called an integral scheme of D if
/3(D) = DeTRoR. In other words: X is an integral scheme of D if and only if each tangential vector field on X/Ro along X is the restriction of some vector field from D. Given p € X we may also ask whether the initial value problem has a solution, i.e. whether there exists an integral scheme X for D with p 6 X, and whether it is unique.
REMARKS 1.4 a) If X is an integral scheme of a direct summand D of Der/{0/? where QIR/RO is finitely generated and projective, then the exact sequence (3) shows that J3 induces an isomorphism of /^-modules D/ID ^ De?R0R. b) If D is the /?-submodule of Der#0.R generated by a single vector field V, then X C X is an integral scheme of D, if and only if the restriction of V along X is tangential and generates Der #„-/?. Assume now that fi/j/# 0 is a finitely presentable module, for example R/Ro is
essentially of finite type and R0 noetherian. Then for each multiplicatively closed subset S C R we have
Moreover the canonical sequence (3) localizes. Thus if X is an integral scheme of D and p G X, then Xv :— SpecRf is an integral scheme of Dv in Xv := SpecRy. Integral schemes of vector fields need not exist, not even in the local and onedimensional situation, see example 2.3 below. If they do there are some uniqueness assertions which we will discuss in the next section.
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2 UNIQUENESS OF INTEGRAL SCHEMES OF VECTOR FIELDS Let D C DeiR0R be a submodule and I ^ R a D-invariant ideal. By Zorn's lemma / is contained in a maximal such ideal. If R := R/I, then D := /?(D) C Der/{0J? by 1.2, and I is a maximal D-invariant ideal if and only if R is D-simple, i.e. (0) is the only proper D-invariant ideal of R. We call R/Ro differentially simple, if it is D-simple for D = DerH 0 /2. The following facts are known:
I) If Q C R and R is noetherian, the maximal D-invariant ideals / ^ R are prime, since all associated primes of D-invariant ideals are so (see [S], Thm.l or [SS], 1.3, for example). Since sums of D-invariant ideals are D-invariant, in a local ring there can be only one maximal D-invariant ideal ^ R. II) If K is a field of characteristik 0 and A an affine /^-algebra, then the following assertions are equivalent: a) A/K is differentially simple. b) A is a regular domain. In fact a) implies that A is a domain, by what was mentioned in I). If A has singular primes, then the minimal such primes are differentially invariant (see [S], Thm.3 and [Ki] for generalizations). Thus A must be regular. Conversely, if A is a regular domain, Q— is a projective module. By [K^], Prop, the only invariant prime is the zero ideal, and there are no other invariant proper ideals by I) above. III) In the local c.ase the same arguments show that for p € Spec^l the following assertions are equivalent: a') A p is differentially simple. b') Ap is regular. From these facts we obtain PROPOSITION 2.1. Let A be an affine algebra over a field K of characteristic 0 and set X := SpecA. For a submodule D C DetffA let X C X be an integral scheme of D defined by an ideal I C A and let $ 6 X.
a) p is a regular point of X if and only if Ip is the unique maximal Dp -invariant ideal
of A,. b) If Dp is a free direct summand in Der^j4 p of rank p and p € X is regular, then the irreducible component of X which contains p has dimension p.
c) X is irreducible and regular if and only if I is a maximal D-invariant ideal of A. PROOF: a) and c) are immediate from II) above. Under the assumption of b) let A := A/I. Then Dp//D p = Der^p by 1.4a), and this is a free ^-module of rank p. For L := Q(AV) the derivation module Der/cL is a vector space of dimension p, which implies that L/K has transcendence degree p. The assertion of b) follows.
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COROLLARY 2.2. Under the assumptions of the proposition let X and Z be integral schemes of D and p 6 X a point which is regular on both X and Z. Then the irreducible components of X and Z which contain p agree. PROOF: If J is the denning ideal of Z, then /„ '= Jp by 2.la), hence Ip n A = Jp n A EXAMPLE 2.3. Under the assumptions of 2.1 let X be a regular scheme and V a
vector field on X such that F(p) ^ 0 for a p e X, and let D := (V}. Then Dp is a free direct summand of DerjfAp (1.1). If X is an integral scheme of D such that p is a regular point of X, then the component of X containing p is a curve, uniquely defined by V and called the integral curve of V through p. Consider a polynomial algebra K [ X , Y] in variables X, Y over K and the vector field V = ^ + (l + Y)^r. For .ft := tf[X, Y] (Jfl y) let R be the completion of/?. Then V^ can be looked at as an element of Der# J? and of Der^/?. The ideal p € Spec./?
generated by Y—exp X + l is the maximal D-invariant ideal in R where D — (V). We have R/f ^ K[[X]], and the kernel of the Jsf-homomorphism (p : K[X, Y] -+ K[[X}] with
REMARK 2.4. It was shown by Alok Kumar Maloo ([M], Thm.5) that for a noetherian local ring R, a finitely generated submoduie D C Derg/?, which is a Lie subalgebr,a, and a maximal D-invariant ideal I ^ R the R//-module D/ID is always free. He also gives an example of a l-dimensional noetherian local ring /? containing a field K of characteristic 0, but not essentially of finite type over K, for which Der#/? is free, R/K is differentially simple, but R is not regular. We turn now to positive results about the existence of integral schemes of vector fields. A crucial assumption is that D is a Lie subalgebra of the derivation module. In the next section we collect some facts about this condition.
3 SUBALGEBRAS IN THE LIE ALGEBRA OF VECTOR FIELDS Let D C Der#0/? be an /?-submodule.
LEMMA 3.1. If D is generated by vector fields Vi,...,Vp such that [Vi,Vj] = 0 for i,j= 1,.. .,p, then D is a Lie subalgebra o/Der/{„/?•
PROOF: Let V = £ a.-VJ, V = £ 6,-V; (a; A € R). The claim follows from the >=i j=i formula (4)
[V, V'}
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which is easy to verify.
LEMMA 3.2. For V i , . . . , V p € DerRoR with [Vi,Vj}
= 0 (i,j = l,...,p),
a i , . . . , ap 6 N and a, b £ R we have (Vr'o...oV^O(«*)=£(£)'--^ Pi-0
PROOF by induction on |o = a\ 4- • • • + ap. For a — \ this is the product formula for derivations. In the induction step it suffices to apply V\ to both sides of the formula, since the derivations Vi commute. Then the claim follows.
PROPOSITION 3.3. Suppose R is local and DerRoR has a basis (Wi,. ..,Wn) such that [Wi, Wj] — 0 for i,j=l,...,n. If D is a Lie subalgebra of Der#0.R and a direct summand, then D too has a basis (Vi, . . . , Vj>) with [Vi, Vj] = 0 for i,j = 1, . . . ,p.
PROOF: Since R is local and D C DeiR0R a direct summand, D has a basis ( d i , . . . , d p ) . Write n
vpWfi
(avti € R, v- l , . . . , p , /•« = !,..-,")
Possibly after renumbering the W/j we may assume that [av/t]VilJL=ii__ip is an invertible p matrix. Let [bvfl] be its inverse and Vv := ^ by^d^ (v = 1, . . .,p). Then (Vi, . . ., Vp) 11=1 is a basis of D and n
(5)
p
Vv = XI E fr "/i<wW/» = ^" + Z c"^^ P=I #1=1 P>P
with certain cvp £ R. Since D is a Lie subalgebra of Der/{0/? we have
o-\
On the other hand we can write n [Vl/,Vl,} = Y,SrWr
(sr£R)
r=l
Comparing these two expressions by means of (5) we find that ra = sa for cr = 1, . . . , p. Now apply (4) and (5) for calculating \VV, VM] and make use of [Wv, W^} = 0 (v, // = 1 , . . . , n) . We obtain sa = 0 for a = 1 , . . . , p, hence [ Vu , VM] = 0 for v, /j, = 1 , . . . , p.
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COROLLARY 3.4. Assume RQ is noetherian and R/Ro is essentially of finite type and smooth. For a direct summand D ofDeTR0R the following assertions are equivalent: a) D is a Lie subalgebra o/Der/j 0 /?.
b) For all m e M&xR there exists a basis (Vi,...,Vp) of Dm such that [Vi,Vj] - 0
PROOF: The smoothness of R/Ro implies that &1R/R has locally at each m a basis (dxi, . . -,dxn) with xi,...,xn G Rm. Let (W\, . . ., Wn) be the dual basis of DerR0Rm, Wi = gf- (i = l,...,n). Then [Wi,Wj] = 0 for i,j = l,...,n. That D is a Lie subalgebra of Derfl 0 /Z is a local property. The claim of the corollary follows from 3.1 and 3.3. LEMMA 3.5. Let R/Ro be an algebra withQ C RO, and let R[[ti,.. .,tp]] be the power series algebra in indeterminates t\,.. .,tp over R. For V\,..., Vp € DeiR0R assume the [Vi,Vj] = 0 (i,j = l,...,p). Then there exists a unique RQ~homomorphism ij> : R —+ R[[ti,..., tp}] with the following properties:
a) i/}(r)\t-0 — r for all r € R. b) For k = 1,... .,p the following diagram commutes
R vk
R
sir
,... ,tp]]
PROOF: If if} exists, then «/> ° Vi o Vj - £- o -JL. 0 ^'for i,j - l,...,p. For r € R assume that t/>(r) = J^ pv^...Uftv^ • ... • tp" (pVi...Vf € R). Then
p =
"
/"I *>\
fltI'p C/tp
where v = (y\, . . . ,
j_
v\ = v\ +
t=o
v\
vp and v\ = v\\ • . . . • vp\
As for the existence of tb we define it by the formula K)
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267
Since the Vi commute it follows from 3.2 that t/> is an .Ro-homomorphism. Obviously V"( r )|t=o = r- Further
The canonical pairing
defines the non-degenerate scalar product
between differentials and vector fields. It can be extended to a non-degenerate bilinear form
where (wt A • • • A w p , Vi A • • • A Vp) = det[{wj, V j ) } i j = i t . . . t p
for W I , . . . , W P € ^fl//? 0 J Vi,...,!^, G Der#0.R. Here for w € ®IR/RO
an
d Vo,V"i the
following formula is valid
(6)
(du, V0 A Vi) = K 0 ({w, Vi}) - ^({w, V 0 )) - (w, [V0, ^])
LEMMA 3.6. Let 0)j/ flo be finitely generated and projective. For a direct summand D C Der flo # set DL := {w £ VR/Ro \(u,V) = 0 for all V ED}. Then the following statements are equivalent: a) D is a Lie subalgebra o/Der# 0 /?.
b) (dw, Vo A Vi) = 0 for all w <E D1- and V0, Vi € D. This is an immediate consequence of (6). If the universally finite module of differentials ^Jj//j 0 exists what was said in connection with lemma 3.6 holds true if is replaced by fik
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4 EXISTENCE OF INTEGRAL SCHEMES OF VECTOR FIELDS Let K be a field of characteristic 0 and A an affine /^-algebra. Assume D C DeiifA is a direct summand. Then for each m 6 Maxyl the ylm-module Dm is a direct summand of Der^^lm, and Dm :— Am ®A D is one in Der^A m . In example 2.3 we have seen that there need not exist an integral scheme of Dm in Specj4m even if Am is a regular local ring and Dm a Lie subalgebra of Der#A m though there was such a
scheme for Dm. The algebraic version of the theorem of Frobenius which we are going to show states that this fact is always true.
Set R := Am. Since R/mR is a finite extension field of K there is a unique Cohen subfield 6 of R which contains K . If m is a regular point of X , then R = l[[Xi , . . . , Xn}} is a power series algebra over f, and Der/c/? = Dere/? is a free fl-module with the basis (j^-, • • • , ~QJ£-)- Therefore we may denote 6 by K. We also write D for Dm and X for Spec/2. THEOREM 4.1. For R = K [ [ X i , . ..,Xn}} as above assume D C VerKR is a direct . summand of rank p and
F := DL := {w € frR/K
( w > V) = Q for all V- € D}
Then the following statements are equivalent:
a) F has a basis consisting of exact differentials. b) D is a Lie subalgebra o/Der#.R. c) There are non-units Ylt...,YneR with R = K[[Yi, ..., Yn}] and D = 0 R-^-. !=1
'
PROOF: a)-+b) If F = Rdfi ® • • • 0 Rdfn-p (/; 6 R), then d(rdfi) = dr A dft for r 6 R, i — 1, . . . , n — p. It suffices to apply 3.6 to u G F with dw = LU' A 77 where w' G &R/K, rjeF. Then for V0, Vi € D .
and 3.6 shows that D is a Lie subalgebra of Derjf R. b)—»-c) Since D is a direct summand of Der# R it has by 3.3 a basis (Vi, . . . , Vp) with [Vi, Vj] = 0 (i, j = 1, . . .,p). Then Nakayama's lemma implies that after renumbering the Xi, if necessary, ( V i , . . . , Vp, j^-— , . . . , •$%-) is a basis of DeixRBy 3.5 there is a A'-homomorphism ip : R —» R[[ii, . . ., tp}} with ^(r)| t=0 — r for r € R and ij> o Vk = gf- o $ for k — I , . . . , p. Let
be the composition of tj) with the canonical map
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The last ring is isomorphic to /^[[
The matrix [Vi(Xk)]i,k=i,.,.,p is invertible since (Vi, ...,VP, j^ — , . . ., •§%-) is a basis of Der/f R, hence so is its image by i/>, which is [^-ip(^k)]i,k=].,...,p as ^- o ip =
$ o Vi (i — l,...,p). The derivatives gy-^pG) for i ^ k and k > p belong to the maximal ideal ms of 5 and the ^-V'(-X'fc) are units of 5 for k = p + 1, . . . , n. The coefficient matrix of the system (7) is therefore invertible, hence the i/; (dXi) (i = 1, . . . , n) form a basis of QlSiK. If m.R denotes the maximal ideal of R we have a canonical commutative diagram
where x and 77 are the maps induced by ij). Since rj is an isomorphism, so is x, and it follows that $ is an isomorphism. With Yk := W\tk) (k = l,...,p) and Yk := ^~l(Xk) (* = p+1, . . . , n) we have R = A'[[yi, . . . , yn]], and statement c) follows as
l c)->a) Since F = Dx we obtain F = ® »=P+I
COROLLARY 4.2. If under the assumptions of 4-1 one of the conditions a)-c) holds, then X :— Spec^Z/(y p +i, . . . , Y n ) is
a p x g-matrix with entries FI/P € P-
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THEOREM 4.3. There exist power series u\,..., uq £ K[[ti,.. .,tp}] satisfying
(8)
jfiL = Fri(ti,...,tp,u1,...,uq)>Vi(Q) = 0
(i/=l,...,p;t =!,...,?)
if and only if the "integrability conditions"
dF.i
+
A t
8Fvi_dF,i
dt, ^" 'dx7~~dt^ are satisfied for i/,/j, = 1,.. .,p; i = 1 , . . . , q. The Uf (i — 1 , . . . , q) are then unique. Of course in case p = 1, the case of ordinary differential equations, the integrability conditions are vacuous, and the initial value problem has always a unique solution.
PROOF of 4.3. Set t = (ti,...,tp) and u = (ui,...,uq). Assume the system has a solution. Then w,(0) = 0 and ff-(O) = F^(0) (i - 1,.. .,q; v = l,...,p). Further
dF
x , v->
= - ( vi« . « ) + a-l E 1
hence ^f^(O) = f^(0) + E If^C 0 )' ^(°)- BY induction we see that the values (7 = 1
at 0 of the partial derivatives of the u, can be expressed by the Fvp and their partial derivatives at 0, which implies the uniqueness of the solution. Moreover it follows from (9) that the integrability conditions are necessary for the existence of a solution. Consider now the derivations
and the differentials p := dXp - ^ Fvpdtv € &1
which generate a submodule F C &lPiK- Then D := (Vi, . . . , Vp} is a direct summand of rank p in Der^fP, and from
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we see that F — DL . Further q
q
[K, Vy =^2{[-^,Ft,pg^-] + [F!/pg^-,-^-}}+ p=l
£ [ P,r=l
p
8fV,,x
9
shows that [K, V^] = 0 for i/, ^ = 1, . . .,p if and only if the integrability conditions are satisfied. Then D is a Lie subalgebra of Der/fP, and 4.1 implies the existence of
non-units Ylt . . . , Yp+q € P such that P - K[[Yi, ..., Yp+?]], D = 0 P-£r and
Set P := P/(ti, . . . ,tp) = K [ [ X i , . . . , Xq]]. Since the functorial homomorphism P ®p &IPIK —*• &p/K maps F bijectively onto &p/K , the matrix [ ajc +; ]j,p=i,...,? is invertible and
p = A: [pi,...,
v-l
which is equivalent with the equations (8).
REFERENCES EGA Grothendieck, A. and J. Dieudonne. Elements de geometric algebrique, Publ. math. (1960-67) [Ki]
Kunz, E. Differentially closed prime ideals. Communications in Algebra 26
(1998) 3759-3763 [K2J
— '. On the tangent bundle of a scheme. Univ. Jagellonicae acta math. 37 (1999) 9-24
[M]
Maloo, A.K. Differential simplicity and the module of derivations. J. pure appl. algebra 115 (1997) 81-85
[SS]
Scheja, G. and U.Storch. Fortsetzung von Derivationen. J. Algebra 54 (1978) 353-356
[S]
Seidenberg, A. Differential ideals in rings of finitely generated type. Amer. J. Math. 87 (1967) 22-42
Roberts rings and Dutta multiplicities KAZUHIKO KURANO Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan
1 INTRODUCTION This is a survey on the theory of Roberts rings and Dutta multiplicities. Proofs of all statements except for the final one will be omitted. In 1983, using Frobenius maps, S. Dutta [4] defined the limit multiplicity Xco(lF.) (that we refer as the Dutta multiplicity) for a bounded finite free complex F. with homology of finite length over a local ring A of characteristic positive, and showed
that it was a rational number. In 1989, P. Roberts [24] proved that, for a local ring A of positive characteristic, if a complex F. is not exact and its length (as a complex) equals the dimension of A, then its Dutta multiplicity is positive. The fact played an essential role in his proof of the new intersection theorem in the mixed-characteristic case. After that, the theory of Dutta multiplicities were established for complexes not only over a ring of characteristic positive but also over a homomorphic image of an arbitrary regular local ring [12]. It is not known whether the positivity above (Conjecture 5.2) is true or not in the case where A is mixed-characteristic. We see that the positivity of Dutta multiplicity is deeply related to Serre's positivity conjecture of intersection multiplicities (Remark 5.3). In 1985, P. Roberts [21] proved the vanishing theorem of intersection multiplicities for a local ring that satisfies T~A([^.]) = [Spec-A^m^, where TA is the RiemannRoch map for Spec A (Fulton [6]). We refer such a ring as a Roberts ring [16]. Over a Roberts ring, the Dutta multiplicity of a complex coincides with the Euler characteristic, that is, the alternating sum of length of homology modules. Therefore the theory of Roberts rings are deeply related to that of Dutta multiplicities. In fact, P. Roberts proved the above vanishing theorem over Roberts rings by proving a corresponding vanishing of Dutta multiplicities (Theorem 5.1 (3)). Complete intersections are Roberts rings, but there is an example of a Gorenstein non-Roberts ring. Normal Roberts rings are Q-Gorenstein. Quotient singularities and Galois extensions of regular local rings are Roberts rings (see Section 6). For rings of characteristic positive, we can describe Dutta multiplicities and characterize Roberts rings by Frobenius maps as in Section 2. In the general case, we can define and characterize Dutta multiplicites and Roberts rings using the 273
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singular Riemann-Roch theory (Fulton [6]) or Adams operations (Gillet-Soule [8]) as in Section 3. For local domains with field of fractions of characteristic 0, we can characterize Dutta multiplicities and Roberts rings in terms of Galois extensions as in Section 4. In Section 5, we shall explain the origin and some applications of
the theory of Roberts rings. In Section 6 we shall give basic properties on Roberts rings. Here we shall explain terminologies that will used later. Throughout the paper, we assume that all rings are commutative with 1. For a ring A, Go(^4) denotes the Grothendieck group of finitely generated Amodules. For an yl-module M, [M] denotes the element in GO (A) corresponding to
M. If a ring homomorphism g : A —)• B is finite (i.e., B is a finitely generated A-module), we have the induced homomorphism g* : GO(5) —>• Go(A) of additive groups defined by g* ([M]) = [gM] for a B-module M, where gM is an A-module M whose A-module structure is given through g.
A# (A) = ®jA;(A) is the Chow group of the affine scheme Spec A. For a prime ideal p of A of dim^l/p = i, [SpecA/p] denotes the cycle in Ai(A) corresponding to the closed subscheme SpecA/p.
For an additive group N, NQ denotes N ®z (Q>. A bounded complex of finitely generated free modules is called a perfect complex. The support of a bounded complex F. is defined to be the union of the supports of the homology modules, and denoted by Supp(F.). Let (A, m) be a Noetherian local ring. Let F. be a perfect A-complex with support in {m} of the following form:
F. : 0 -> Fs -» • • • ->• F0 -» 0 If Fs is not 0, we say that the complex F. is of length s. We define the Euler characteristic xw. '• GO (A) —» Z (or xw. '• Go(A)q —>• Q) to be
l)'M# t (F. ®A M)), where Ht(V. ®A M) is the t-th homology module of the complex F. ®A M and f-A(Ht(^-®A M}} denotes its length as an A-module. We sometimes denote simply by
2 THE CASE OF POSITIVE CHARACTERISTIC In the section, we assume that (^4, m) is a d-dimensional complete local ring containing a field of characteristic p > 0 with perfect residue class field A/m unless otherwise specified. In the case, the notion of Roberts rings and Dutta multiplicities are defined in an elementary way, i.e., we do not need the singular Riemann-Roch theorem to define them. Let / : A —> A be the Frobenius map, i.e., f ( x ) = xp . Since / is finite, we have the induced map /* : GO(^.)Q —>• Go(^4)- We put Li G 0
(A)0 = {c e G 0 (A) Q | /
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for i = 0 , . . . , d, that is, L;Go (A)^ is the eigen space of /* e End) of eigen value p*. Then,
GO (A)ty = ©j = o Lj-Go (-A)^
(1)
is satisfied. (It will be proved elementarily. Using the theory of Adams operations instead of Frobenius maps, an equality like (I) was proved by Gillet-Soule [8] for a homomorphic image of an arbitrary regular ring.) Put (2)
DEFINITION 2.1 Let F. be a perfect yl-complex with support in {m}. The rational number xr.(
We say that A is a Roberts ring if g,- = 0 for i = 0, 1, . . . , d — 1. By the equation (2), we have (/* ) e (M) = I/-4 = Pde1d + P[d-l}eqd-i + ' • • + Peqi + 90
for each e. Since
XT.([f- A]) = x(Fe (¥.)), we have
X=o(F.) = ».(9.) = l™
r».(M) = lim
X(*"(F.)),
(3)
where Fe(F.) is the perfect complex whose entries are p e -th power of those of F.. (S. Dutta [4] defined the multiplicity of the form as above.) Assume that A is a Cohen-Macaulay ring and / is an m-primary ideal of finite projective dimension. Let F. be the minimal free resolution of A/ 1. Then, by the equality (3), the Dutta multiplicity Xoo(IF-) coincides with the Hilbert-Kunz multiplicity € H K ( I , A ) , that was introduced by Kunz [11] and Monsky [20]. Note that A is a Roberts ring iff [/A] — pd[A] in Go(-A)
3 THE GENERAL CASE In the section, we assume that (A, m} is a
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of Q- vector spaces TA '• GO (A)^ -> ^(A)^1 . Put
TA ({A}) = rd + rd- 1 + • • • + T0 ,
(n e A,-
These TJ'S enjoy interesting properties as follows:
PROPOSITION 3.1
With notation as above, the following are satisfied:
(1) If A is a Cohen-Macaulay ring, then
is satisfied (by the definition of TA and Example 18.1.2 in [6]), where WA denotes the canonical module of A. (2) If A is a Gorenstein ring, then we have r^-i = 0 for each odd i. (3) If A is a complete intersection, then we have r, = 0 for i < d (by the definition ofrA and Corollary 18.1.2 m [6]). (4) Td is equal to [Spec A] d (Theorem 18.3 (5) in [6]), where
p6Spec A dim A/p=d
In particular, we have TJ ^ 0.
(5) Assume that A is normal. Then, by definition, Ad-i(A) coincides with the divisor class group Cl(-A) of A. Let C!(U>A) G C1(A) be the isomorphism class containing UJA- Then, we have T^-I — cl(o;/i)/2 in Ad-i (Lemma 3.5 m [12]). If A is a ring as in Section 2, then it is known that r/i(L,-Go (A)^) = A2(we can prove it using Theorem 18.3 (1) and Example 18.3.12 in [6]). Therefore, in the case, we obtain TA(
DEFINITION 3.2 Let F. be a perfect A-complex with support in {m}. The rational number XiF.(rJ 1 ( r d)) is called the Dutta multiplicity of the complex F. and denoted by Xoo(lF-)We say that A is a Roberts ring if Ti = 0 for i = 0 , 1 , . . . , d — 1. 'Let A be a homomorphic image of a regular local ring 5. Then, TA may depend not only on A but also on S by its construction (Fulton [6]). However, in many cases (e.g., the case where A is a complete local ring or A is essentially of finite type over a field or Z (Section 4 in [16])), it is proved that TA does not depend on the choice of 5. The author knows no example that TA actually depends on the choice of a regular local ring 5".
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By Proposition 3.1 (3), a complete intersection is a Roberts ring. By Proposition 3.1 (5), we can easily find examples of normal Cohen-Macaulay non- Roberts rings. We shall give some examples of normal Gorenstein non-Roberts rings later (Remark 3.4 and Section 6 (Rl)). By definition, we know that the Dutta multiplicity %oo(F.) coincides with the Euler characteristic Xr.([A]) if .A is a Roberts ring. Here we give some sufficient conditions for Xoo(F.) = X PROPOSITION 3.3
// one of the following conditions is satisfied, then we have
(1) A is a Roberts ring. (2) F. is liftable to a regular local ring, i.e., there exists a regular local ring S and an S-free complex G. such that A is a homomorphic image of S and
F. = G. ®s A is satisfied (Theorem 1.2 in [15]). (3) A is equi- dimensional with d = dirndl < 2 (Proposition 3.4 in [12]). (4) A is Gorenstein (or Q- Gorenstein) with d = dim A < 3 (Proposition 3.4 in
[12]). If IK. is a Koszul complex with respect to a system of parameters for A, then K. is liftable to a regular local ring. Hence the Dutta multiplicity of K. coincides with the usual multiplicity of A with respect to the parameter ideal. As in [12], the Dutta multiplicity of a complex enjoys properties like the usual multiplicity. REMARK 3.4 Using an example of a negative intersection multiplicity due to Dutta-Hochster-MacLaughlin [5] [10], there exists a 3-dimensional normal CohenMacaulay ring (A, m) and a perfect A-complex F. with support {m} such that Xoo(F.) ^ Xr.(IXl) (p295 in [26]). Miller-Singh [19] constructed 5-dimensional normal Gorenstein local ring (A,m) and F. such that Xoo(F.) ^ X By Proposition 3.1 (4), we have Xoo(F.) = XV.(TA 1([Spec A ] d ) ) . By the local Riemann-Roch formula (Example 18.3.12 in [6]), xw. = c^i(^-)TA is satisfied, where ch(F.) : A, (yl)nj —> Q is the localized Chern character defined by MacPherson's graph construction [6]. Therefore, we have
Xoo(F.) = ch(F.) n [Spec,4]d = chd(F.) n [Spec A]d = chd(F.) n TA([A]), where ch,-(F.) : A»(.A)
= e!'ch;(F.) proved in Theorem 3.1 in [17]. THEOREM 3.5 Let (A,m) be a homomorphic image of a regular local ring and put dimj4 = d. Let F. be a perfect A-complex with support in {m}. Let ipe denote the e-th Adams operation.
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(1) For any k > 2, we have lim - \- lim
e->oo
There exist rational numbers ai, . . . , a^+i (depending only on d) such that
d+i
THEOREM 3.6 Let A be a d-dimensional Noetherian local ring that is a homomorphic image of a regular local ring S. Put k = dim 5 — dim A. Then the following conditions are equivalent:
(1) TA([A}} £ Ad(A)iQ, i.e., A is a Roberts ring.
(2) Letting G. be a finite S-free resolution of A, i?*([&.]) = tk[G.] is satisfied in KS0pecA(SpecS)9 for some t > 2. (3) With notation as above, ^*([G.]) = tk[G] is satisfied in KQPSC A (Spec S) 1.
Here, K 0 pec (SpecS1^ is the Grothendieck group of perfect S-complexes with support in Spec A (see Gillet-Soule [8], Soule [28]).
4 THE CASE OF DOMAINS In the section, we assume that (A, m) is a cf-dimensional local domain that has a subring 5 satisfying the following two conditions; (1) 5 is a Nagata regular local ring, (2) A is a localization of a module-finite ring extension of S. It is easy to see that, if A is a complete local domain or a local ring of an algebraic variety, then there exists a subring S satisfying the above two conditions.
Let L be a finite algebraic field extension of Q(S) containing Q(A), where Q( ) denotes the field of fractions. Let BL be the integral closure of A in L. Since A is a Nagata domain, BL is a finitely generated yl-module with field of fractions L. Then, we can characterize Roberts rings and Dutta multiplicities in terms of BL as follows2:
THEOREM 4.1
The following conditions are equivalent:
(1) TA([A}) G Ad(A)q, that is, A is a Roberts ring. (2) For some finite algebraic field extension L ofQ(A) such that L is normal over Q(S), [BL] = rankyt BL • [A] m Go(A)q$ is satisfied. (3) For any finite algebraic field extension L of Q(A) such that L is normal over Q(S), [BL] = rankA BL • [A] in GO(^)Q is satisfied. 2
They are generalizations of Theorem 1.1 in [15] and Theorem 1.1 in [16]. Proofs are much the same, however.
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THEOREM 4.2 Let L be a finite-dimensional normal extension ofQ(S) containing Q(A). For a perfect A-complex F. with support in {m}, we have
xw.([BL}) _ E.-(-i)^(g.-(F.®x^)) [L:Q(A)}
[L:Q(A)}
where [L : Q(A)} denotes the dimension of the Q(A)-vector space L. Then, we immediately obtain the following corollary:
COROLLARY 4.3 Let A be a Noetherian local normal domain such that A has a Noether normalization 3 S with Q(A)/Q(S) Galois. Then, A is a Roberts ring. Corollary 4.3 says that, if a Noetherian local normal domain A is not a Roberts ring, then A never have a Noether normalization S with Q(A)/Q(S) Galois. REMARK 4.4 Combining Theorem 4.1 and Definition 2.1, we have the following: Suppose that (A, m) is a d-dimensional complete local domain with perfect coefficient field of characteristic p > 0. Take a Noether normalization S of A. Then, the following are equivalent: (1) A is a Roberts ring. (2) Let / : A —> A be the Frobenius map. (By our assumption, / is finite.) Then,
\jA] = rankA } A - [ A ] in G0(A)9 is satisfied. (3) For some (any) finite-dimensional normal extension L of Q(S) containing
Q(A), (BL\ = rank A BL • [A] in GO(/!)CQ> is satisfied, where BL is the integral closure of A in L. generated A-module since A is complete.)
(BL is a finitely
5 THE ORIGIN OF ROBERTS RINGS AND THEIR APPLICATIONS In 1985, P. Roberts [21] proved the following: Assume that A is a Roberts ring. Let M and N be finitely generated A-modules satisfying the following three conditions; (1) pdAM < oo, pdAN < oo, (2) £A(M ®A N) < oo, (3) dimM + dim// < dirndl, where pd^ denotes the projective dimension as an A-module. Then, £,.(-l)%i(Torf (M, JV)) - Q is satisfied. (It was conjectured by Serre [27] when A is a regular local ring, and it was independently solved by P. Roberts [21] and Gillet-Soule [7].) 3
We say that regular subring S of A is a Noether normalization if the inclusion S <—>• A is finite.
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The above was the beginning of the theory of Roberts rings. P. Roberts proved the above vanishing theorem by proving a corresponding equality on Dutta multiplicities (Theorem 5.1 (3)) as below.
Before stating Theorem 5.1, we give a short remark. Let F. : 0 -> Fs -» • • • ->• F0 ->• 0
be a perfect ^4-complex. Then, the A-dual Fv : 0 -» F0V ->• • • • -» Fsv ->. 0
of F. is defined. Here, we regard .F/ as the part in degree —i of the complex Fv . It is easy to see that the support of F. coincides with that of Fv . Let F. and G. be perfect A-complexes. Then, F. ®^ G. is a perfect A-complex with support SuppF. D SuppG..
THEOREM 5.1 Let A be a d-dimensional local ring that is a homomorphic image of a regular local ring. Let F. and G. be perfect A-complexes.
(1) Assume that the support o/F. is in {m}. Then,
is satisfied (Example 18.1.2 in [6]). (2) Assume that the support o/F. is in {m} and A contains a field of characteristic p > 0. Let Fe(W.) be the perfect A-complex whose entries are pe-th power of those o/F.. (It is easy to see Supp(F e (F.)) = Supp(F.)J Then,
is satisfied for each e > 0. (Using Proposition 2.6 in [12], we may assume that A is a complete local ring with perfect coefficient field of characteristic p > 0. Then, by the equation (3), it will be proved.)
(3) Assume that SuppF. D SuppG. is in {m}. 7/dimSuppF. +dimSuppG. < d, then Xoo(F. «uG.) = 0 is satisfied (P. Roberts [21]). (4) Assume that SuppF. n SuppG. is in {m}. //dim SuppF. + dim SuppG. < d, then
Xoo(Fv ®A G.) = (~l) d - d i m is satisfied (C- Y. Chan [3]).
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(5) Assume that (A, m) contains a field. Let W. be a perfect A-complex of length d with Supp(F.) = {m}. (In particular, the support is not empty, and F. is not exact.)
Then
Xoo(F.) > 0 is satisfied [1 7]. If one replace the Dutta multiplicity Xco( ) by the Euler characteristic x( ) in Theorem 5.1, then all of (1), (2), (3), (4), (5) are false (see 13.3 in Roberts [26]). However, if A is a Roberts ring, then the Dutta multiplicity of a complex coincides with the Euler characteristic. That is to say, if A is a Roberts ring, then Theorem 5.1 is still true for the Euler characteristic. By Theorem 5.1 (1) and (2), we have the following:
(SI) Let (A, m) be a d-dimensional Roberts ring and F. be a perfect A-complex with support in {m}. Then,
is satisfied. (S2) Let (A,m) be a d-dimensional Roberts ring of characteristic p > 0. For a perfect A-complex F. with support in {m},
is satisfied for any e. They were conjectured by Szpiro (Conjecture C2 in [29]) without the assumption that A is a Roberts ring. (There exists an example (13.3 in Roberts [26]) of a nonRoberts ring A and a perfect A-complex F. with support {m} such that both (SI) and (S2) are not satisfied.) Let M and N be finitely generated ^-modules satisfying the following two conditions; (1) pd^ M < oo, pd A N < co, (2) IA(M ®A N) < oo. Let F. (resp. G.) be the minimal A-free resolutions of M (resp. N). Then, it is easy to see
By Theorem 5.1 (3), ^.(-^^(Torf (M, N ) ) vanishes if A is a Roberts ring with dimM + dim JV < dim A. That is a vanishing theorem of P. Roberts [24]. Assume dimM + dim AT < dim A. By Theorem 5.1 (4), we have
if A is a Roberts ring. That is a theorem due to C-Y. Chan [3].
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Theorem 5.1 (5) was proved by P. Roberts [24] when (A, m) is a complete equalcharacteristic Noetherian local ring such that the residue class field is perfect of positive characteristic. Using the fact, he proved the new intersection theorem in the mixed-characteristic case. Recently, it has been solved in [17] when A contains
a field. Here, it is natural to ask:
CONJECTURE 5.2 Let (A, m) be a homomorphic image of a regular local ring. Put d = dim A Let F. be a perfect A-complex of length d with Supp(F.) = {m}Then *oo(F.) > 0. Conjecture 5.2 is open even if A is a Gorenstein ring. As in Remark 3.6 and 3.7 in [12], Conjecture 5.2 is true if A is a Cohen-Macaulay ring of dimension less than or equal to 3. If A is a Cohen-Macaulay Roberts ring in Conjecture 5.2, then we have
by the depth sensitivity [1]. Conjecture 5.2 is true if A has a test module [15]. REMARK 5.3 As in Theorem 1.2 in [15], the conjecture as above is deeply related to the Serre's positivity conjecture as below (Conjecture 5.4). In fact, if the complex in Conjecture 5.2 is liftable (see Proposition 3.3 (2)) to a regular local ring, then we can prove that the Dutta multiplicity of F. is nonnegative by Gabber's non-negativity theorem [2]. Conversely, if Conjecture 5.2 is ture, so is Conjecture 5.4 in the case where either M or TV is a (not necessary maximal) Cohen-Macaulay module. CONJECTURE 5.4 Let (S, n) be a regular local ring. Let M and N be a finitely generated 5-modules such that dim M + dim N = dim S and 0 < £s(M®s N) < oo. Then, £;(-l)%(Torf (M, N)) > 0 is satisfied.
It is well-known that the following conjecture (that is called Small CohenMacaulay Modules Conjecture [9]) implies the Serre's positivity conjecture. CONJECTURE 5.5 If A is a complete local ring, then A has a maximal CohenMacaulay module M, i.e., M is a finitely generated A-module with depth M = dim A By Theorem 1.3 in [15], Small Cohen-Macaulay Modules Conjecture also implies Conjecture 5.2 using the theory of test modules. By Theorem 4.2, we know that Conjecture 5.2 is equivalent to the following one:
CONJECTURE 5.6 Let (A, m) be a d-dimensional complete local normal domain and assume that A has a Noether normalization S with Q(A)/Q(S) Galois. Let
F. : 0 -» Fd -> • • • ->• F0 ->• 0 be a perfect A-complex with Supp(F.) = {m}. Then x(F.) = EiM^'M0.
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6 BASIC PROPERTIES ON ROBERTS RINGS
In the section, we shall give basic properties on Roberts rings. For proofs, we refer the reader to [16]. However we shall give a proof to (Rll), because it was not proved anywhere else.
Throughout the section, (A,m) is a homomorphic image of a regular local ring. (Rl) If .A is a complete intersection, then A is a Roberts ring. However, a Gorenstein ring is not necessary a Roberts ring. Here, we give a concrete example.
Let t, m, n be integers such that 1 < t < m < n. Let R be the polynomial I < i < m; 1 < j' < n] over a field k divided by the ideal /t(zjj) generated by all t by t minors of the m by n matrix (xij). Using a criterion ring k[xij
(Theorem 1.3 in [14]) below, we conclude that, for t = 2, A = R(Ri) 'ls a Roberts ring if and only if R is a complete intersection as in Section 3 in [14].
Since any localization of a Roberts ring is a Roberts ring again (see (R4)), we know that, for any t, A = -R(fi,) is a Roberts ring if and only if A is a complete intersection. Therefore, R is a Roberts ring if and only if t — 1 or t = m = n. Hence, if 1 < t < m = n, then A is a Gorenstein non-Roberts ring.
THEOREM 6.1 Let R = ®n>0Rn = R0[Ri] be a Noethenan graded ring over a field RQ. Assume that X = Proj(R) is smooth over RQ. Then, the local ring A = R(RI) is a Roberts ring if and only if
In particular, if X is an abelian variety, then A = R(RI) is a Roberts ring. Here, CH*(^) denotes the Chow ring of the smooth projective variety X, GI( ) is the first Chern class, fl^ is the tangent sheaf of X, and td(fi^) is its
Todd class (see [6]).
(R2) If A is a normal Roberts ring, then it is Q-Gorenstein, i.e., the isomorphism class C\(UA) containing the canonical module WA of A is a torsion in the divisor class group C\(A). Assume that A is a 2-dimensional normal domain. Then A is a Roberts ring if and only if A is Q-Gorenstein. (R3) If dim A < I , then A is a Roberts ring.
If A is a Roberts ring, then A is equi-dimensional, i.e., dim>l = dimA/P is satisfied for any minimal prime ideal P of A. (R4) Let S —>• T be a flat extension of regular local rings that may not be a local homomorphism. Let / be an ideal of S. If S/I is a Roberts ring, then so is T/IT.
In particular, if A is a Roberts ring, then so are its completion A, its henselization h A and its localization Ap for any prime ideal P of A.
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(R5) Assume that A is an excellent henselian local ring. If its completion A is a
Roberts ring, then so is A. It is natual to ask the following: QUESTION 6.2 Suppose that A is a homomorphic image of a regular local ring. When the completion A is a, Roberts ring, is A so?
It is proved that the above question is equivalent to the following one: QUESTION 6.3 Suppose that A is a homomorphic image of a regular local ring. Is the natural map Go(-4)((j —> Go(^4)nj injective? The author does not know whether they are true or not. If they are true,
then we know that the Riemann-Roch map TA : Go(A)
image of 5. (R6) Let T be a finite-dimensional regular (not necessary local) ring and R = T/I be a homomorphic image of T. Then, the set
{P € SpecT | P D / and TRp/Tp((RP}) £ AdlmRp(RP)q} is an open set of Spec R, where r/j p /y p : Go(-Rp)iQ> —>• A*(.Rp)
(R7)
Let a; be a non-zero-divisor of A. If A is a Roberts ring, then so is A/xA. (The converse is not true in general.)
(R8) Let / be an ideal of A contained in the 0-th local cohomology group H^(A) with respect to the maximal ideal m of A. Then, A is a Roberts ring if and only if so is A/1.
(R9) Let xi, ... , xs be a filter regular sequence of A. If A is a Roberts ring, then so is A / ( X I , ... ,xs). (RIO) Assume that there exists a regular local ring T containing A such that the inclusion i : A —>• T is finite. Then, A is a Roberts ring. In particular, an invariant subring of a regular local ring with respect to a finite group is a Roberts ring if the regular local ring is finite over the invariant subring. Furthermore, let H be a simplicial submonoid of NQ. (Here, NO denotes the set of non-negative integers. We say that a submonoid H of NQ is simplicial if there is a positive integer n such that n • NQ C H.) Let &[Ng] (resp. k[H]) be the semi-group ring of NQ (resp. H) over a field k. Then, since the inclusion k[H](H\{o}) ^ ^[No](H d \{o}) is finite, we know that &[#](.H"\{o}) is a Roberts
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(Rll) Let F = {Ffc}fcgg be a filtration of ideals of A, i.e., F^'s are ideals of A such that (1) Fk D Fk+i for k e Z, (2) F0 = A D m D FI, (3) F;Fj C Fi+J- for i, j e Z. Put G = ®fc> 0 F fc /F fc+ i, J?' = >1[{FA<* | A; £ Z}] C A^.i" 1 ]. Assume that /?' is Noetherian. Then we have the following: (1) There exists a graded homomorphism p : A*(A)(jj —> A*(G) such is a Roberts ring, then so is any local ring ofG. (3) Assume that both A and G are normal domains. If A is Q-Gorenstein, then so is G. Proof. Since G = R' /t~lR' , we know that G is Noetherian. It is well-known that dimG = d and dim/?' = d+ I , where d = dim A In the same way as Chapter 5 in [6], we shall construct the map p as below.
Let g : R' -» G be the projection and <j> : R' -> -R'^"1)"1] = A[t,t~l\ the
localization. Then, we have the following exact sequence
A»(G) Q A A,(#)Q A A*(A[t,r1}) — »• 0
(4)
since Spec_ff'\SpecG = Spec Ajt,^ 1 ] (Proposition 1.8 in [6]). Since SpecG —> Spec R' is an effective Cartier divisor, we have the map i' : A* (R')*Q —> A* (G)nj (2.3 and Proposition 6.1 in [6]). By the definition of r , for a prime ideal P of R', we have r([Spec R'/P]) = [SpecR'/P + (t'1)] (resp. 0) if r1 £ P (resp. t"1 e P), where [Spec_R'/P + (t~1)] is an element in the Chow group of R' defined in 1.5 in [6]. Since i'g* — 0 by the definition of r, we obtain the map ^ : At(A[t,t~l]) —> A*(G)<jj such that ip<j>* = i' by the exactness of the sequence (4). On the other hand, the flat map £ : A —> ylf^t" 1 ] induces the isomorphism Put p = ip£* : Af(A)
A*(G)nj. By definition, /> is a graded homomorphism, i.e., /?(Aj (j4)nj) C AJ(G)Q for each i. In order to prove /?(r^ ([>!])) = T"G([G]), it is suffucient to show the following three equalities:
(i) t,( (iii) i](r Here, ^ and <j> are smooth morphism. Therefore, by Theorem 18.3 (4), we obtain (i) and (ii) since ^[t.t-^/A — A[f,^ - 1 ] and ^^[t.t- 1 ]/^' — 0- We have (iii) by Corollary 18.1.12 and Example 18.3.12 in [6]. We have completed the proof of (1).
Put
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where T; 6 Ai(A)iQ and r[ € Aj(G)
Next assume that both A and G are normal domains. Then, by Proposition 3.1 (5), we have rd^l = cl(w A )/2 £ Ad_l(A},Q = Cl(A)® and r'd_l = cl(w G )/2 6 Ad_i(G)q) = C1(G)(Q>. Therefore, if C[(UIA) is a torsion in Cl(A), then so is
cl(wG) in C1(G).
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[8] H. Gillet and C. Soule, Intersection theory using Adams operations, Invent.
Math. 90 (1987), 243-278. [9] M. Hochster, Topics in the homological theory of modules over local rings, C. B. M. S. Regional Conference Series in Math. 24, Amer. Math. Soc., Providence, RI, 1975. [10] M. Levin, Localization on singular varieties, Invent. Math. 79 (1985), 253-291.
[11] E. Kunz, On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), 999-1013. [12] K. Kurano, An appoarch to the characteristic free Dutta multiplicities, J. Math.
Soc. Japan 45 (1993), 369-390. [13] K. KURANO, On the vanishing and the positivity of intersection multiplicities over local rings with small non complete intersection loci, Nagoya J. Math. 136
(1994), 133-155.
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[14] K. Kurano, A remark on the Riemann-Roch formula on affine schemes associated with Noetherian local rings, Tohoku Math. J. 48 (1996), 121-138.
[15] K. Kurano, Test modules to calculate Dutta multiplicities, preprint. [16] K. Kurano, On Roberts rings, preprint. [17] K. Kurano and P. C. Roberts, Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic 0, to appear in Trans.
Amer. Math. Soc.. [18] K. Kurano and P. C. Roberts, The positivity of intersection multiplicities and symbolic powers of prime ideals, to appear in Compositio Math.. [19] C. M. Miller and A. K. Singh, Intersection multiplicities over Gorenstein rings,
[20] P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), 43-49. [21] P. C. Roberts, The vanishing of intersection multiplicities and perfect com-
plexes, Bull. Amer. Math. Soc. 13 (1985), 127-130. [22] P. C. Roberts, MacRae invariant and the first local Chern character, Trans.
Amer. Math. Soc. 300 (1987), 583-591. [23] P. G. Roberts, Local Chern characters and intersection multiplicities, Algebraic geometry, Bowdoin, 1985, 389-400, Proc. Sympos. Math., 46, Amer. Math. Soc., Providence, RI, 1987.
[24] P. C. Roberts, Intersection theorems, Commutative algebra, 417-436, Math. Sci. Res. Inst. Publ, 15, Springer, New York, Berlin, 1989. [25] P. C. Roberts. Local Chern classes, multiplicities, and perfect complexes, Mem. Soc. Math. France No. 38 (1989), 145-161.
[26] P. C. Roberts. Multiplicities and Chern classes in local algebra, Cambridge University Press (1998). [27] J-P. Serre, Algebre locale, Multiplicites, Lecture Notes in Math. 11, SpringerVerlag, Berlin, New York, 1965. [28] C. Soule, Lectures on Arakelov Geometry, Cambridge studies in advanced math. 33, Cambridge University Press 1992. [29] L. Szpiro, Sur latheorie des complexes parfaits, Commutative Algebra: Durham
1981. 83-90, London Math. Soc. Lecture Note Ser., 72, Cambridge Univ. Press, 1982.
Hilbert functions of squarefree Veronese subrings HIROYUKI NISHIDA, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan HIDEFUMI OHSUGI, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan TAKAYUKI HIBI, Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
ABSTRACT. Let 2 < q < d/2 and U(dq) the g-th squarefree Veronese subring of the polynomial ring K[ti, t%, . . . ,td] with each degi; — 1 over a field K, i.e., 7i^ is the affine semigroup ring generated by all squarefree monomials of degree q belonging to K[ti,t2,... ,td]. A combinatorial study of the ft-vector of TZj will be presented. First, it will be proved that, for all d > 5, the ft-vector h(7i^ ) = (ft 0 , f t i , . . . , ft s ) with s = [2d/$] of the third squarefree Veronese subring 7td
satisfies (i) ft; < ft s _; for all 0 < i < [s/2] and (ii) fto < fti < • • • < ft[s/2]-
Second, for the ft-vector h(7cf ) = (fto, f t i , . . . ,ft s ) o f f l / , where s = (d — 1) — [(d — !)/
I
INTRODUCTION
Let K be a field and A a homogeneous K -algebra, i.e., A is a graded K -algebra A = AQ (J) AI 0 Ay (J) • • • such that (i) AO = K, (ii) A is generated by A\ over K, and (iii) dim^ AI < oo. The Hilbert function of A is
H(A,n] — dim^ An,
n = 0,1,2,....
Let d = dim A, the Krull dimension -of A. We then define the sequence of integers
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Nishida, Ohsugi and Hibi
called the h-vector of A, by the formula
n=0
i=0
Thus, in particular, ho = 1 and hi = dimjf AI — d. It is known that hi = 0 for all sufficiently large i. We write h(A) = (ho, hi, . . . , hs) if hi = 0 for all i > s. We refer the reader to [1, Chapter 4] for historical background and fundamental information about ft-vectors of homogeneous A'-algebras. An intriguing open problem on h- vectors is to find a complete characterization of the A- vectors of Cohen-Macaulay integral domains. It is shown (Stanley [8]) that the ft-vector h(A) = (ho, hi,... ,hs) with hs ^ 0 of a Cohen-Macaulay integral domain A satisfies the inequalities h0 + hi + ••• + hi < hs + hs-i + • • • + hs_i,
0 < i < [s/2].
The third author [3] conjectured that the h-vector h(A) = (ho, hi,... ,hs) with hs ^ 0 of a Cohen-Macaulay integral domain A satisfies the inequalities
hi
(1.0) (1.0)
Unfortunately, it turned out (e.g., [5]) that the above conjecture is false in general. However, it is still possible to ask if the conjecture is true for a "reasonable" class of Cohen-Macaulay integral domains.
CONJECTURE 1.1 (a) Let A be a homogeneous affine semigroup ring and suppose that A is Cohen-Macaulay. Then the h- vector h(A) — (ho, hi, . . . ,hs) with hs ^ 0 of A satisfies the inequalities (1) and (2). (b) Let A be a homogeneous affine semigroup ring and suppose that A is Gorenstein. Then the h-vector of A is unimodal. (A finite sequence ( a o , a i , . . . ,as) of nonnegative integers is called unimodal if do < a\ < • • • < a,j > Oj+i > • • • > < * « for some 0 < j < s.) Since the ft-vector h(A) = (ho, hi, . . . , ft s ) with hs ^ 0 of a Gorenstein ring A is symmetric, i.e., hi — hs-i for all i, the conjecture (b) follows from (a). Even though both (a) and (b) of Conjecture 1.1 are presumably false, it would be of great interest to find a class of Cohen-Macaulay homogeneous affine semigroup rings with h- vectors satisfying the inequalities (1) and (2) as well as to find a class of Gorenstein homogeneous affine semigroup rings with unimodal /z-vectors. Let 2 < q < d/2. Let K[ti,t2, • • • , £<*] denote the polynomial ring with each degti = I over a field K. The q-th squarefree Veronese subring of K[ti,tz, • • • ,td] is the affine semigroup ring 72.^ generated by all squarefree monomials of degree q belonging to K[ti,t2, • • • ,td\- The subring 7?^ is normal and Cohen-Macaulay. In addition, it follows easily from [10, Corollary 9.6] that, for all d > 3, the /i-vector of c?\ the second squarefree Veronese subring TLd satisfies the inequalities (1) and (2). Now, the first result of the present paper says that, for all d > 5, the ft- vector /o\ of the third squarefree Veronese subring Tld satisfies the inequalities (1) and (2).
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291
THEOREM 1.2 Letd>5 and h(U(*}} = (h0, hlt... , hs) with hs ^ 0 the h-vector of the third squarefree Veronese subring fld . Then s = [2d/3] and hi
LEMMA 1.3 Let h(K{dq}) = (h0, hlt h2, . . .) denote the h-vector of Tl(q} . Then
^ = D-1) k=0
where At 'q with i G TL is the nonnegative integers satisfying the equality (1 + A + A 2 + • • • + A"- 1 )" = Y, Ai'q xii£E
On the other hand, Lemma 1.3 produces the following Theorem 1.4 which together with [6, Theorem 4.4] yields an elementary proof to the well-known fact [2] that 71d with 2 < q < d/2 is Gorenstein if and only if d = 1q.
THEOREM 1.4 Let 2 < q < d/2 and h(R.} = (h0, hlt . . . , hs) with hs ± 0 the h-vector ofH(^ . Then s = (d - 1) - [(d - l ) / q ] . Moreover, hs = h0 (= 1) if and only if q divides d. In case q divides d, one has ft 5 _i = hi (= (d) — d) if and only It is now reasonable to try to prove (but not yet proved) that the ft-vector
)
=
(ho, hi,... j h z q - z ) of the Gorenstein ring TC^q' is unimodal. Via the theory of toric varieties, it follows from [9, Lemma 2.2] that hq > /z ? +i > • • • > /Z2g-2- Thus the question is how to prove the inequality hq_i > hq, i.e.,
k=0
(-!)* /^-)-^
(L4)
/
We do not know if the above inequality (3) can be shown by using fundamental and standard techniques in the frame of elementary combinatorics. A large part of the present research appears in the master's thesis (unpublished) of the first author entitled "Hilbert functions of squarefree Veronese algebras," Department of Mathematics, Graduate School of Science, Osaka University, February,
1999. The second author is supported by JSPS Research Fellowship for Young Scientists.
292
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Nishida, Ohsugi and Hibi
THIRD SQUAREPREE VERONESE SUBRINGS (•q\
The main purpose of the present section is to show that the h- vector h(7id ) =
(ho, hi, . . . ,hs) with hs ^ 0 of the third squarefree Veronese subring TCd ' satisfies hi < ft s _j, 0 < i < [s/2], and ho < hi < • • • < /i[ s /2]First of all, a combinatorial formula to compute the h- vector of the g-th squarefree Veronese subring 13,^' is discussed. Consult, e.g., [4] for further information. Let q > 1, d > 1 and let A^q with i G "Z, denote the nonnegative integers satisfying the equality
A + A 2 + • • • + A' - 1 " Thus, in particular, At'q = 0 unless 0 < i < d(q — 1).
LEMMA 2.1 (a) Let H(H(f ,n) denote the Hilbert function ofK(^.
Then
^? ) ,n) =k=o D
(b) Let h(n(dq]) = (h0,h1,h2,...) denote the h-vector of K^ . Then
Proof, (a) It follows from, e.g., [2, Lemma 2.1] that H(Ti^ , n) is equal to the number of sequences ( 0 1 , 0 2 , . . . , a<j) € Zd such that (i) each 0 < Oj- < n and (ii) ^^j Oj = ng. Then a simple combinatorial argument yields Hin(ti n\
I d ' /
fc _ V(-D (^ f"9 - *(" + 1) + 1 + (d - 1) - 1 1
I /f I \ \ / \
Z^ k>0
d-
as required.
(b) What we must prove is the equality 77 ( f i
_
fa\ — i\"
|
// _
1 \
'
for all 0 < k < q. Note that dim72.j = d. Now,
»=o d
/ ,\
oo n=0
n=0 j=0
n=0
I
Hilbert functions of squarefree Veronese subrings
293
together with 00
1
n=0
n=0 j=0
yields the desired equality (4). We now turn to the discussion of the h-vector of the third squarefree Veronese subring Tid . Let d > 5 and let h(7id ) = (ho, hi, h % , . . . , hs) with hs ^ 0 denote /o\ the h-vector of the third squarefree Veronese subring TCd '. Lemma 2.1 (b) says that
'd 2 for i= 1 , 3 , 4 , . . . .
The list of the h-vectors ofR^1 with 5 < d < 19 drawn below guarantees that the inequalities (1) and (2) are correct for h(Hd ') with 5 < d < 19.
rf = 6; (1,14,36,14,1) d=7; (1,28,133,119,21) d=8; (1,48,364,568,202,8) d=9; (1,75,834,2005,1230,147,1) d = 10; (1,110,1695,5830,5565,1352,55) d=ll; (1,154, 3157,14773, 20438,8437,869,11) d = 12; (1,208,5500,33748,64285,40612,7930,352,1) d= 13; (1,273,9087,71071,179335,161811,51610,4992,91) d= 14; (1,350,14378,140140,454545,557662,265174,45278,2184,14) d = 15; (1,440,21945,261690,1065090,1712532,1141140,305160,
28650,665,1) d= 16; (1,544,32488,466752,2337296,4786112,4273272,1656496, 258314,13328,136) d= 17; (1,663,46852,800462,4851392,12364882,14298547,7611172,
1789114,165087,4556,17) d= 18; (1,798,66045,1326884,9598914,29883960,43579977,30604896, 10157058,1480784,80580,1122,1) d = 19; (1, 950,91257,2135030,18217200,68198866,122755200,110282061, 49293030,10480685,955415, 30039,190) (List of h-vectors of ft^ with 5 < d < 19)
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We now come to the first main result of the present paper.
THEOREM 2.2 Let d > 5 and h ( H ) = (h0, hi,... , hs) with hs ^ 0 the h-vector of the third squarefree Veronese subring 71d . Then s = [2d/3] and hi
0
ho < hi < • • • < /i[ s /2]-
Proof. Since A$? = 0 if i > [2d/3]; A$d/3] + 0; (^ = 0 if i > ((d + l)/2], and since [2cf/3] > [(d + l)/2], it follows that hf = 0 if i > [2d/3] and /i[2d/3] ^ 0. Hence s = [2cf/3], as required. /o\
(First Step) It will be proved that the ft-vector h(R.d ) = (ho, hi, h^, . . . , /i[2d/s]) /o\
of 7td satisfies the inequalities hi < h[2d/3]-i f°r all 0 < z < [d/3]. Since the sequences Ad,3
^T-O
Ad,3
i^1!
Ad,3
i • • • i ^2d
and
are symmetric and unimodal (e.g., [7, p. 503]), it follows that Af' < Aj' with (i + j)/2 < d and that (^) < (
lii < j
Let i ^ 2. First, let d = U with / > 7. If 2 ^ z < £, then hi = A3^'3 -
U(^} and hu.i = Al_i} - Sf^/^J. Now, (3i + 3(2^ - z))/2 = 3£ = d and ((2z - 1) + (2(2^ - z) - l))/2 = 2£ - 1 > 3^/2. Thus ht < hU-i- Second,
let d - U + 1 with £ > 1. If 2 ^ i < I , then hi = A^+l'3 - (U + l ) ( f ^ l ) and hu-i = Af(^i} - (U + l)( 2(2 3 /_+)_ 1 ). Now, (3z + 3(2£ - i))/2 = 3£ < d and ((2z - 1) + (2(2£ - i) - l))/2 = 2£ - 1 > (U + l)/2. Thus hi < hU-i. Third, let d = U + 2 with t > 6. l i l ^ i
A (M+1) _,- = Af$ll}_i} - (U + 2)( 2((2 4 £ +!. ) _ 1 ). Now, (3i + 3((2£ + 1) - i))/2 = 3£ + 3/2 < d and ((2i - 1) + (2((2£ + 1) - i) - l))/2 = 2* > (3* + 2)/2. Thus hi < hu-iLet i = 2. First, if d = U > 21, then ft 2 = >if'3 - 3£(33£) + (320 and /i 2£ _ 2 =
Af^6 - 3£(4^5) = A3^6 (since U < M - 5). Hence ft 2 < /i 2 f _ 2 . Second, if 1 > 22, then A 2 = Af+1'3 - (3£+ 1)(%+1) + (%+1) and ft 2 £ _ 2 = Af+_l6'3 l)(«ls) = ^-e'3 (since 3£+ 1 < 4£ - 5). Hence ^2 < ft 2 £- 2 . Third, if
d - U + 2 > 20. Then ft 2 = Af+2'3 - (3£ + 2)( 3 ^+ 2 ) + (3£+2) and ^w-s'3 - (3* + 2) (4^3) = ^£-33 (since 3£ + 2 < 4£ - 3). Hence h2 < /i 2£ _ 2 . /Q \
(Second Step) In order to show that the /z-vector h(7td ) = ( h o , hi, h?,... , ofli3
satisfies ft 0 < hi < /i2 < • • • < /Z[d/3], what we must prove is the inequalities
^N
Hilbert functions of squarefree Veronese subrings
295
and A d,3 ^d,3 \ j\ ^•3i+3 ~ AZi _ "I
®
\ jl } ~ I
•
\ )'
o / • ^ \j /ol 1 — —t i l ~
/o o\ ("•*)
We know the inequalities (6) by Corollary 2.4 below, while to see the inequalities (5) for d > 20 can be easily done by the routine computation with (3) < (d^1) and
LEMMA 2.3 Let d, i, k 6 Z with i > -1, d > 5, 3« + 3 < d and I < k < d. xd-1,3
Ad-l,3
k
>
k
~3
~
/
d
d
\_(
(k-(i + 2 ) j
(k-(i + 4)
Proof. Since
(1 + A + A 2 ) d = (1 + A + A 2 )(l + A + A 2 )^ 1 , it follows that Ad,3 _ /(d-1,3 _,
Ad-1,3 , ^d-1,3
In general, Ak~ ' — Ak~_^ > 0 since k — 3/2 < d — 1. Hence (7) is true if i > 1, d> 5, 3i + 3 < dand 1 < k < 2. If ( k , i ) is one of (1,-1), (1,0), (2,-l) and (2,0), then (7) is true for all d > 5 since (d - l)(d - 2) > 2. Moreover, (k_,d+^) - (k_di+4)) < 0 unless k < d/1 + i + 3. If k = d > 5, then k < d/2 + i + 3 only for d < 12. The possible pairs (d, i) £ Z2 with d > 5 satisfying [d/2] - 2 < i < [d/3] - 1 are (5, 0), (6,1), (7,1) and (9, 2). The direct computation of (l + A + A 2 )''- 1 says that (7) is true if (d, i, k) is one of (5, 0, 5), ( 6 , 1 , 6 ) , (7,1,7) and (9,2,9). These observations guarantees that (7) is true in case k = 1, or k = 2, or k — d. In particular, if d = 5, then (7) is true for all i and k with i = -1,0 and k = 1,2,5. Again, the direct computation of (1 + A + A 2 ) 4 says that (7) is true if (d, i, k) is one of (5,-1,3), (5,0,3), (5,-1,4) and (5,0,4). We now prove the inequality (7) by using induction on d. First of all,
where c^iAd-i,3 A
Ad~2'3
^ —\ k ^k-3 n _ {Ad-2,3 _ Ad-2,3
U
— (-H-k-l
A
(k-l}
and (Ad-l,3 _
(•"•k
Ad-l,3}_
A
k-3 I
,(
d
\_(
Ufc-(i+2)/
= (Ad~2'3 - Adk-_23'3) + C' + D',
U
d
296
Nishida, Ohsugi and Hibi
where d
~2'3
d 23 ry -— i(A&_2 ~' /1
d
~2'3
Ad~2'3 \
~~ A(k-2)-3> ~
3 3 One has Adft~l' > Ad^ft~—2'£3j — O, since (k - 2) - 3/2 < d - 2. Moreover,' AKd~2'3 —— > ^f"o' —^ —— ' ' —— K —O if k — 3/2 < d — 2, i.e., if k < d. Work with induction on d. Recall that we already know that the inequality (7) is true if k = 1, or k = 2, or k = d. Let d > 3 and 2 < k < d. Since 3z + 3 < d, either i > 0 or 3z + 3 < d. If i > -1 and 3? + 3 < d (together with 2 < k < d), then (7 > 0 and D > 0. If i > 0 and 3z + 3 < of (together with 3 < & < d), then C' > 0 and D' > 0. In each of these two cases, the required inequality (7) arises. v
COROLLARY 2.4 Let d, i, k 6 Z wzi/i z > 1, 3z + 3 < d and I < k < d. Then Ad,3
Ad,3
.
,1
d
\
,[
d
Proof. The technique appearing in the proof of Lemma 2.3 together with ~ ^-3) "~ (^(fc-(i + 2)) ~ r f (fc-(i+4))) d l 3 d 1 3 d A — O -j- iJ -I- (Ak_2 (k-l)-3> \\(k-2)-((i-2) + 2)J (A;'
— c -i- n -i- (A ~ >
A->
\ — <(
}
d (\(k-2)-((i-2)+4)J!> \\
where
C = (Adk~1'3 - At1,'3) - ((d - I)(47i2)) - (d- 1)(47J4))); D = (At\'3 - Ad(-^}_3) - ((d-l)^-_\i+t) - (d- IK^:1^))), and
where
yields the inequality (8), as desired.
3
GORENSTEIN SQUAREFREE VERONESE SUBRINGS
Let 2 < q < d/1 as before. It is known [2] that the squarefree Veronese subring Tij is Gorenstein if and only if d = 2q. The purpose of the present section is to give an elementary proof to this fact by using the following Theorem 3.1 together with [6, Theorem 4.4].
Hilbert functions ofsquarefree Veronese subrings
297
THEOREM 3.1 Let 1 < q < d/2. Let h(n(dq}) = (h0,h1}... ,hs) with hs ^ 0 denote the h-vector ofTl^. (a) If d — 1 = jq + r with j > 1 and 0 < r < q, then hd-j-i = A '^_^'q'. (b) ([2, p. 635]) One has s = (d-l)-[(d- l ) / q } . (c) Suppose that d = jq with j > 2. Then hs^\ — A^ — d. Proof, (a) Since
one has U\
.Hn — k
Jtr"(d-j-l)(q-k)-k
M—"> ,
a
Z_^,\
^l
% f c / U
\f,
First, let j > g. Since q — r—l—jk < 0 for all fc > 0, it follows that h^-j-i = A 'qr_1, as desired. Second, let j < q. Then
=
((1 + A + • • • + A^ 1 ) - AJ'(1 + A + • • • + A^-1 + A + ' ' ' + A«- 1 d -*A J '*l + A
Now, in case k > 0, the coefficient of A 9 ~ r ~ 1 ~ j r f c in
(1 + A + • • • + Xq-l)d-k(\ + A + • • • + A^-y is equal to the coefficient of A 9 ~ r "" 1 ~ jfe in (l + A + . - . + A 9 -*- 1 )* since g — r— 1— jk < q — j — 1 < q — 1,
g — r— 1— jk < q — k — 1.
Hence
l
_
~
-jrf.9
A
q-r-l
k=0
(b) Let s = (d-l)-[(
298
Nishida, Ohsugi and Hibi
i> ck. Since c0 > GI > • • • > c g _i , it follows that ht = Ef=o(~ 1 )' S (k)Ai(q-k)-k
~° if i > CQ = d — d/q. In other words, ft, = 0 if i > (d — 1) — [(d — !)/ 1 and 0 < r < g, then s = d — j — 1. The result (a) says
that h, = A*'™-¥'9} • Since 0
(c) Let 2. Then s — d — j and
k=0
v
k
-
(d-j-l)(q-k)-k-
^ ~ i
k-Q
'
v
k
Let j > q. Then /M-J-I = Ad'i = A^ - d. Let j = q. Then — d. Let j < q. The similar technique appearing in the proof of (a) yields
A? =
k
COROLLARY 3.2 Let 2 < g < d/2 and h(R,^) = (h0,hi,... ,hs), with s = (d — 1) — [(d — l)/g] the h-vector ofTCq. Then hi — ft s _; for all 0 < i < s if and only if d — 2g. Proof. First, suppose that h{ = hs-{ for all 0 < i < s. In particular, hs = I and hs_i (= hi] = (d) —
Then Theorem 3.1 (c) says that hs-i = A^ - d. Since A^ = ( d ) , if follows that j = 2, as desired.
Second, if d = 2g, then s — 2g — 2 and
snce (i(q -k)-k) + ((2q - 2 - i ) ( q - k) - k) = 1q(q - k - I ) .
Hence Lemma 2.1 (b) guarantees that ht = ft 2 g-2-j for all 0 < i < q — l, as required. Corollary 3.2 together with [6, Theorem 4.4] yields an elementary proof to the well-known fact [2] that the squarefree Veronese subring "R^ with 2 < q < d/2 is Gorenstein if and only if d = 2g.
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References [1] W. Bruns and J. Herzog, "Cohen-Macaulay Rings," Cambridge University Press, Cambridge, New York, Sydney, 1993.
[2] E. De Negri and T. Hibi, Gorenstein algebras of Veronese type, J. Algebra 193 (1997), 629-639. [3] T. Hibi, Flawless O-sequences and Hilbert functions of Cohen-Macaulay integral domains, J. Pure and Appl. Algebra 60 (1989), 245 - 251. [4] M. Katzman, The structure of path algebras, preprint. [5] G. Niesi and L. Robbiano, Disproving Hibi's conjecture by CoCoA or projective curves with bad Hilbert functions, in "Computational Algebraic Geometry" (F. Eyssette and A. Galligo, Eds.), Birkhauser, Boston / Basel / Berlin, 1993, pp. 195-201.
[6] R. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), 57 - 83. [7] R. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in "Graph Theory and Its Applications: East and West," The Annals of the New York Academy of Sciences, Volume 576, 1989, pp. 500 535.
[8] R. Stanley, On the Hilbert function of a Cohen-Macaulay domain, J. Pure and Appl. Algebra 73 (1991), 307 - 314. [9] R. Stanley, A monotonicity property of ft-vectors and ft*-vectors, Europ. J.
Combin. 14 (1993), 251 - 258. [10]
B. Sturmfels, "Grobner Bases and Convex Polytopes," Amer. Math. Soc., idence, RI, 1995.
Prov-
On the conductor of a surface at a point whose projectivized tangent cone is a generic union of lines FERRUCCIO ORECCHIA 1 , Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Complesso Universitario di Monte S. Angelo - Via Cintia , 80126 Napoli, Italy, e-mail: [email protected].
ABSTRACT. Let (A, m) be the local ring at a singular point x of a surface V with regular normalization A. Let e(A) = e and emdim(A) = r + 1 denote the multiplicity and embedding dimension of A and b denote the conductor of A in A. Assume that \/b = m, Spec(G(A)) is reduced and the projective tangent cone Proj(G(A)) to V at x is a union of e lines in generic position. In this paper we show
that with a few exceptions b = m*, where s = Min{n 6 N (e — l)(n + 1) <( n ^ r )}-
1
INTRODUCTION
In [I] the conductor of the local ring at a multiple point x, of an algebraic reduced curve C was computed under the assumption that the projectivized tangent cone Proj(G(A)) is reduced and consists of points in generic position. In this case it was shown that the conductor is a power of the maximal ideal of the local ring. In [4] the conductor of the local ring of an algebraic reduced variety X at a general point
x of a multiple irreducible subvariety Y of codimension one was computed under the assumption that the tangent cone to X at x of Y consists of linear varieties in generic position. In this case it was shown that the conductor is a power of the prime defining Y. No result is known about the computation of the conductor of a non-normal variety whose singular locus has codimension > 1. In this paper we want to start filling this gap in the case of surfaces. In fact let (A,m) be a reduced non-normal local ring of multiplicity e — e(A] and Hilbert function H(A,n). Let '1991 Mathematics Subject Classification. 13A30. Key words and phrases. Surfaces, tangent cones, conductor. Work partially supported by MURST
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r + 1 = emdim(A) = H(A, 1) be the embedding dimension of A. Let b be the conductor of A in its normalization (A, 3) (where 3 is the Jacobson radical of A) Suppose that \/b = m and that the tangent cone Spec(G(A)) is reduced. First we show that the natural homomorphism G(A) —»• G(A) is injective so that G(A) can be identified to a subring ofG(A). Then we prove that if G(A) is reduced and G(Zn] is the conductor ofG(A) in G(A), for some integer n, then m" = 3™ is the conductor of A in A. Let now A be the local ring of a surface X at a singular point x. Suppose that A is regular. Assume that Spec(G(A)) is reduced and Proj(G(A)) C W consists of e skew lines {li,...,le}. We say that Proj(G(A)) = {li,...,le} is in generic e-position if the Hilbert function of A ( that is of the coordinate ring of the lines G(A)} is maximal i.e.
H(A,n) = Min{l
,en + e}
Proj(G(A)) is in generic e — 1-position if any e — 1 lines of Proj(G(A)) are in generic position. By using the previous result on the conductor of G(A) and the computation of the conductor of reduced rings given in [1] we prove the main result
of the paper:
let Proj(G(A)) be in generic e — 1, e-position and s = Min{n £ N | (e - l)(n + 1) <
Then b = ms if and only if e ^ [CT)/( S + *)J + *• Throughout the paper all ring are supposed to be commutative, with identity and noetherian.
Let S be a semilocal ring. By G(S) we denote the associated graded ring (Bn>o(3"/3 n+1 ) with respect to the Jacobson radical 3 — mi H • • • fl rn e . If £ G S1, x ^ 0, x G 3™ —3™ +1 ,n G N, we say that x has degree n and the image x* G 3n/3n+1
of a; in G(S) is said to be the initial form of x. If o is an ideal of S, by G(a) we denote the ideal of G(S) generated by all the inital forms of the elements of a.
If (A,m) is local with maximal ideal m and residue field k = A/m, H(A,n) = ofo'mfc(m n /tn n+1 ), n G N, denotes the Hilbert function of A and e(A) the multiplicity of A at m. The embedding dimension emdim(A) of A is given by H(A, 1). If R = 0 n>0 Rn is a standard graded finitely generated algebra over a field k, of maximal Homogeneous ideal n, H(R,n) = dim^Rn — H(Rn,n) denotes the Hilbert function of R and emdim(R) — H(R, 1) = emdim(Rn) the embedding dimension of R. The multiplicity of R is e(R) = e(Rn). One has e(A) = e(G(A)) and emdim(A) = emdim(G(A)). If B is any ring B denotes the normalization of B. If A is a subring of B AnriA(B/A) is the conductor of A in B (that is the largest ideal of A and B}.
2
THE CONDUCTOR OF REDUCED ASSOCIATED GRADED RINGS
In this section we assume that (A, m) is a reduced equidimensional local ring with finite normalization (A, 3) (where j is the Jacobson radical of A). We assume also
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that the conductor b of A in A has radical \/b = m (this is equivalent to saying that Ay is regular for any prime p of codimension one) . In this case there is an integer h such that 3A C A. We assume also that G(A) is reduced.
PROPOSITION 2.1 The natural homomorphism G(A) -> G(A) is injective. Proof. G(A) ->• G(A) is injective if and only if m"/m"+1 -)• 3"/3"+1 is injective for any n, but this means 3"+1 fl m" = m n+1 . Then it is enough to prove that 3n n A = m" for any n > 0. We prove first, by induction, that 3mn ft m = m™ +1 for every n > 0. It is obvious that 3 n m = m. Suppose 3m""1 n m = m". If x G 3m" n m, x G Cftn™"1 n m — m™. Now, since by assumption \/b = m we have 3* C m for some integer h. Then xh € 3/!m/m C mhn+1 i.e. the leading form of x in G(A) is nilpotent. But G(A) is reduced, so x G m"+1. As a consequence we have that 3" n m""1 = m". In fact if x £ 3" n m""1 we have xh = xxh~l G the leading form of x is nilpotent. So x G m". Thus we can conclude that
n THEOREM 2.2 Lei b fee £fce conductor of A m A. and G the conductor of G(A) in G(A) then: (a) G(b) C G;
(b) ifG = G(T) for some integer n then b = m" = 3". Proof, (a) Let b* G G(A) of degree m and x* G G(b) of degree n. We have bx G 3m+" n,4 = mm+n (see Proposition 2.1) i.e. b*x* G G(/l) which proves the inclusion. (b) If G(T) = 0i>n(3V3!'+1) !
m + 3
2+1
is
the conductor of G(A) in G(A) then tf C
for any z > n and so
3" = mn + 3n+1 = m" for any i > n. But there exists an integer h such that 3h C A. Hence 3" C A
and 3" is an ideal of A. So 3" C b and G(3n) C G(b). Thus G(3n) = G(b). This implies that 3" = b. Finally 3n = 3" n A = m" by Proposition 2.1. D In the following we will need also a general result on the computation of the conductor. Let R be a reduced ring and p;, i = 1, . . . ,ra, be the minimal primes of R. Set .R; = -R/pf. The canonical homomorphisms TT; : R —>• /?z- induce an embedding
Thus we can identify _R with a subring of j?'. Furthermore R' is integral over R.
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PROPOSITION 2.4 The conductor of R in R1 is the ideal
pj =
n(p«+n ') n
where (\j is the image of the ideal pj under the homomorphism R —> Ri ( for any j ) . Furthermore if the rings Ri are normal then R' is the normalization of R. Proof. See [1], Proposition 2.5.
3
D
LINES IN GENERIC POSITION AND CONDUCTOR OF SURFACES
In this section we assume that k is an algebraically closed field of characteristic zero.
THEOREM 3.1 Let R be the homogeneous coordinate ring of e lines {li,... ,le} C P£, r > 3. Then H(R,n) < Min{(n+r),en + e}. Furthermore let Xj = aijS + bijt, j G { 0 , . . . , r } , be a parametric representation of the line /,, i € {!,..., s}, and let (aio,bio,... ,asr,bsr) G A 9 , q = 2e(r + 1) be the q-tuple of all the coefficients of the representations of the lines lj. Then the set U C A| of the q-tuples (aio, • • • , asr, bio, • • • , bsr) for which H(R, n) = Mm{( n + r ), en + e} is open and nonempty (and consists of skew lines). Proof. All claims are proved in Proposition 1.3 and 1.5 of [5] except for the crucial fact that the set U is non-empty. This is proved in Theorem 0.1 of [2]. D
DEFINITION 3.2 A set of lines X = {k U • • • U le} C P£, r > 3 is in generic position if its homogeneous coordinate ring R has Hilbert function H(R, n) = Mm{( n + r ),en + e}. X is in generic t-position, t < e, if every i-subset of X is in generic position (then generic e-position is generic position).
THEOREM 3.3 Let R be the coordinate ring of a set of lines X - { l i U - - - U / e } C P£, r > 3 and set s = Min{n G N
(e — l)(n + 1) <(™j!~ r )}. Then X is in generic
position if and only if H(R, s) = (e + l)s and H(R, s — l j = ( 5 ~* +r ) that is X is not contained in any hypersurface of degree < s and X is contained in ( s+r ) _ ( e + l)s linearly independent hyper surf aces. Proof. See Theorem 1.6 of [5].
D
THEOREM 3.4 Let R be the homogeneous coordinate ring of e lines { I j , . . . ,le} C P£, r > 3 in generic e — l,e position. Let emdimR = r + 1, n be the maximal homogeneous ideal of R and n,, i — I,... ,e, be the maximal homogeneous ideals of ~R. Set s = Min{n € N (e - l)(n + 1) <("+ r )} and let c be the conductor of R in R. Then:
(a) c C n' C n-=i n|
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(b) c = n« = nU«? if and only if e ? [(
Proof. Let p;, i = 1, . . . ,n, be the minimal primes of R = k[xo, . . . ,xr}. Then by assumption Ri = R/fy ;= k{[X, Y], kf = k and, by Proposition 2.4, we can identify R with the ring []"=! h[X,Y]. Hence p£=i ni = ®i=i(x,YYki[x,Y]- ( a ) Since the
lines {/i U • • -Ule} — {li} C P£ are in generic position we have that the ideal PWj $j
is generated by forms of degree > s (see Theorem 3.3) and the same happens to its image in Rt. Hence p)j?« q,- C (X, Y) s fc;[^, Y]
(b) The condition e ^ [(s^r)/(s + l)j + l is equivalent to saying that (e-l)(s + 1) > ( 5 + r ) or e(s + 1) < (s+r). Since by assuption we have (e -!)(«+ 1) <( s + r ) e ^ L( S | r )/( s + !)J + !
is
equivalent to e(s + 1) < ( s + r ).
Fix an integer i, 1 < i < e. Let (/i, . . . , /„) be the elements of degree s of a minimal set of generators of the ideal Q^. pj. By the minimality (/i, . . . , /«) are linearly independent modulo pj and then their images
(71,...,7jc(~]qjC(X,Y)ski[X,Y] are linearly independent forms. But u = H(R,s) — H(R/^]-,i pj, s). Then if e(s + l) < C+ r ) we have u= e(s + I) - (e - l ) ( s + l ) = s + 1 and ( 7 i , . . - , 7 u ) =
If e(s + 1) > ( s + r ) we have fs _i_ r \
u=
\
r
J
- (s - l)(e + 1) < s(e + 1) - (s - l)(e + 1) = e + 1
and
n REMARK 3.5 The numbers e of lines in generic e,e — I position for which e — [( s + r )/(s + 1)J + 1 (s - Min{n 6 N (e - l)(n + 1) <(™+ r )}) are sparse. For example if e < 20, these numbers are 4,10,19 in IP3, 9 in F^/IO in P6, 19 in P9. The following two examples were checked with CoCoA [3], EXAMPLE 3.6 Let R be the homogeneous coordinate ring of the following 4 skew lines li : w — x = y — z = Q, l^:w-\-x = y + z = Q l3:w-2x = y-2z = G, 14 : 2w - x = 2y + z = 0 in generic e — 1, e position.The conductor of R in R is the ideal (y2z, w2x, w3, y3,x3, z3,wx2, yz2,xy, wy, wz, xz) which is not a power of the maximal homogeneous ideal (x,y,z,w) of R.
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EXAMPLE 3.7 Let R be the homogeneous coordinate ring of the following e = 5 skew lines
/i : w — x+y = y—z+x = 0, li : Qw+x+2y = y+7z — x = Q: 13 : w — lx = y — lz = 0 /4 ! £11) — ox ^^ 2*y -}- oz = U, 45 ! o£f — 2*x — oy i ^z — U
which are in generic e — l,e position. Then by Theorem 3.4, (b), the conductor of R in R is n3 where n is the maximal homogeneous ideal of R.
THEOREM 3.8 Let A be the local ring at a singular non-normal point x of an equidimensional surface X with regular normalization A. Assume that the conductor b of A in A has radical vb = m (that is x is not contained in any multiple curve of X) and that the tangent cone Spec(A) is reduced and consists of e = e(A) planes.Let Proj(G(A)) be in generic e — l,e- position and set s = Min{n £ N | (e — l ) ( n + (a) b C m"
(b) b = ms if and only if e ± [ ( s + r ) / ( s + 1)J + 1. Proof. First we prove that G(A) is the normalization of G(A). The natural splitting 3e/3e+1 =_§)en(me/me_+1) induces the isomorphism G(A) = rii=i G (^)m,- But since k = A/mi and Amt is regular we have G(A)mi = ki[X, Y], where k{ = k. Then we can identify G(A) with the ring Y[t=i ^*[-^i^]- Now by assumption G(A) is the coordinate ring of e planes. Thus if p,- are the minimal primes of G(A)
we have G(,4)/p; ^_k[X,Y]. Then by 2.3 and Proposition 2.4 we have G(A]_ = Hi-i ki[X,Y] — G(^4)Let G be the conductor of G(A) in its normalization G(A).
(a) By Theorem 2.2 and Theorem 3.4, (a), we have G(b) CG= G(m)s = G(ms). Hence b C ms
(b) If b = ms by Theorem 2.2, (a), G(b) = G(m s ) = G_(m)s C G. But G C G(m)s by Theorem 3.4. Hence G = G(m) s = G(m)sG(A) = (G(m)G(A))s Pl^=i G(nii)s, where G(mz-) are the maximal homogeneous ideals of G(A), and e ^ L ( S r r ) / ( s + !)J + ! by Theorem 3.4, (b).
Viceversa if e ^ [ ( s + r ) / ( s + 1)J + 1 G = DU G(m f ) s = G(f^ =1 mj) = G(3'), by Theorem 3.4, (b) and the claim follows from Theorem 2.2, (b).
D
EXAMPLE 3.9 Let A be the local ring at the origin x = 0, y = 0, z = 0, w = 0 of the surface of A4
xz — yw = z + z — w = yz -\-yz- xw = y z — x -\- y = 0 . Then it is easily shown that
G(A) = k[x, y, z, w\/(xz - yw, z2 - w'2, yz - xw, x2 - y2) and then the projectivized tangent cone Proj(G(A]} is the union of the two skew lines /i : x — y = z — w = 0, 1% : x + y^z + w — 0. Since two skew lines are_always in generic position by Theorem 3.8, (b), we get that the conductor of A in A is the maximal ideal of A ( s = I ) .
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References [1] F. Orecchia. Points in generic position and conductors of curves with ordinary singularities. J. London Math. Soc., 24, 1981, 85-96. [2] R. Hartshorne, A. Hirschowitz. Droites en position generale dans 1'espace projectif, Springer Verlag, Lecture Notes in Mathematics, 961, 1982, 169-188.
[3] D. Capani, G. Niesi and L. Robbiano. CoCoA (Computational Commutative Algebra), 1999, available at http://lancelot.dima.unige.it . [4] F. Orecchia. Points in generic position and conductor of varieties with ordinary multiple subvarieties of codimension one J. Pure Appl. Algebra , 142, no. 1,
1999, 49-61. [5] F. Orecchia. The ideal generation conjecture for s general rational curves, to appear in Journal of Pure and Applied Algebra.
Lifting problem for codimension two subvarieties in P n+2 : border cases MARGHERITA ROGGERO, Dipartimento di Matematica, Universita di Torino, via Carlo Alberto 10, 10123 Torino, Italy, e-mail: [email protected]
1
INTRODUCTION
Let X be an integral non-degenerate projective variety of dimension n and degree d in p r a + 2 j defined over an algebraically closed field K of characteristic zero. Consider the subvariety Y — X n H of X, where H = Pn+1 is a general hyperplane in P"+2. The "lifting problem" is the problem of finding conditions on d, n and s such that any hypersurface G of degree s in P n+1 containing Y can be lifted to a hypersurface in P"+2 containing X. For n = 1, Laudal's "generalized trisecant lemma" states that any curve in P2 containing the set of points Y can be lifted to a surface in P3 containing X, if d > s2 + 1 (see [5], [4], [10]). More recently, many results have been proved generalizing Laudal's theorem to higher dimension varieties (see [2], [7], [11] ), which use different techniques of algebra and algebraic geometry. In the present paper we
will turn our attention to algebraic techniques, mainly to "Strano's method" (see
[9], [10], [3], [11]). Consider TX and Iy, the ideal sheaves of X and Y, and suppose that there are hypersurfaces of degree s containing Y that do not lift to hypersurfaces containing X. Then the restriction map H°XX(S)-^H°IY(S) is n°t surjective, that is ker(H1Ix(s - l)^-H1Ix(s)) is non zero. In [10] Strano proved that the same property also holds for the hyperplane section Y of X and a suitable integer s' < s (see Theorem 4.3). This result gives rise to a reduction both in dimension and in degree, so that the lifting problem for a codimension 2 variety in P™+ 2 turns into a problem about its general plane section, which is a set of points in uniform position in P 2 . In §3 and §4 we consider properties of ker(H1Ix(s - l)->H1Ix(s)), and, more generally, of elements of H12x(s — a) which vanish if multiplied by suitable degree Written with the support of the University Ministry funds. 1991 Mathematics Subject Classification: 14M07
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a forms and we prove that such elements are closely related to hypersurfaces in
p™+ 2 containing X and also to hypersurfaces in Pn+1 = H containing a general hyperplane section Y = X D H. Using those results and the theory of generic initial ideals, we are able to obtain in §5 and §6 some new cases of the following conjecture:
CONJECTURE 1.1. ([7]) Let X be a codimension 2, integral, non-degenerate variety in P n+2 , Y its general hyperplane section and s the minimal degree of hypersurfaces containing Y. If deg(X) > s2 - (n - l)s +
"
\
L
+1 /
then, every hypersurface of degree s in P™+ 1 containing Y lifts to a hypersurface in p n + 2 containing X . Case n = 1 is Laudal's lemma; cases n = 2 and n = 3, besides some partial
results and weaker bounds, are proved in [7], [8], [11], [12]. We obtain the following new cases: - n = 4 (Theorem 6.1). - n = 5, with the additional hypothesis h°Xz(s — 1) ^ 1 on the generic plane
section Z of X (Theorem 6.2) - any n, with the additional hypothesis h°Xz(s — 1) = 0 on the generic plane section Z of X (Theorem 5.1). Most of our notations and techniques are taken from a paper by Green ([3]);
our improvements mainly concern a stronger use of Strano's method and of initial ideal theory, applied to the variety X and to its sections with linear spaces of every codimension, not only to sets of points, plane sections of X . 2
NOTATIONS AND DEFINITIONS.
Let X be a closed subscheme in PN . We will denote by A = 0 At the graded ring in JV + 1 variables K[XQ, . . . , XJY]- Without any further notice, elements of A will always be homogeneous elements; we will often denote by the same symbol an
element a G A and the hypersurface in P^ defined by the equation a = 0. We will denote by Ix C A and Xx C OpN the ideal and the ideal sheaf of X , so that Ix = 0 H°Xx(i). We will denote by W 1x the graded A-module 0 HlXx(i] and by h^Xx(i) the dimension of the A'— vector space
DEFINITION 2.1. Let V be a non zero subspace of A f . Define N%_i(XyV}
the
set of elements a G HlXx(m — i) which vanish if multiplied by every element of V.lfV = Ai, then we will write /C_,-PO instead of N^_t(X, At). Observe that ~Nr^n_l(X) is the degree m — I component of the socle of Ji1 x and we will call generalized socle any vector space N™_i(X, V). We will denote by n^_{(X, V) the dimension of N^(X, V). Let's recall the following, well known, definitions (see for example [3]). DEFINITION 2.2. The ideal sheaf Xx of X mPN is m-regular if mXx(m-q) = 0 for every q > 0. The ideal Ix of X is m-regular if the ideal sheaf Xx is m-regular.
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311
We will denote by:
- ao(Ix)= min{m such that h°Xx(m) ^ 0} - cei(Ix)= min{m such that h°Ix(m) > h°Opn+2(m — o?o)} - r(/fc)= min{m such that I^ is TO- regular }. DEFINITION 2.3. Let Z be a set of d points in P2. The Hilbert function of Z is the function hz defined on the set of integers Z:
If t < 0, then hz(t) = 0 and if t » 0, then hz(t) - d. DEFINITION 2.4. A set of points Z C P 2 is in uniform position if every subset of d' points Z' C Z has Hilbert function:
hzl(t)=min{hz(t),
LEMMA 2.5. Let Z be a set of d points in uniform position in P 2 . Then: (i) dm = 0 ifm < a0(Iz); (ii) dm = TO + 1 - ao(Iz) i f o t o ( I z ) <m < ui(Iz); (Hi) dm+i > dm + 2 if oti(Iz) < m < r(Iz] and the equality holds if and only if Iz has no minimal generators in degree m + 1;
(iv) dm — m+l if m > r(Iz).
/ \ \—^°° / i 1 / \ v^^1"! ( i -i i \ i (v) Em =0 ( T O +! - dm) = E m = o ( m + * ~ d™) ~ d'
(m) n£_i(Z) = s m+2 - 3m+2 - dm+2 + 2d m+ i - dm. Proof: See for example [3] Theorem 2.30, Proposition 4.12, 4.14 and 4.32.
3
GENERALIZED SOCLE.
The aim of the present section is to state close relations between some non vanishing generalized socles and hypersurfaces containing X of suitable degree.
REMARK 3.1. From the cohomology exact sequence associated to
(I) it follows that (Ul x)m = Hllx(m) ~ H°Ox(m)/H0OpN(m). So, every element <j 6 HlXx(m) is the image of some global section a' G Ox(m). Observe that such a global section cr' can be defined by fixing local equations ^-fc on the open sets UXl — {xj ^ 0}, where the Fj are polynomials of degree TO + k. We will say that -£ are local equations also for cr G HlZx(m). On the other hand, local equations ^f define a global section cr' G Ox (m) if and
only if TJ- and -^ glue as functions on Ux- <^UXj CtX , for every 0 < i, j < N , that is,
if and only if each degree m + 2fc polynomial x^Fj —x'jFi vanishes on UXl r\UXj f~\X.
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Thus, if or e N%+k(X, V) and y £ V, then ay - Q and so a'y is cut on X by a degree m + k hypersurface F of PN and — is a local equation for a' on Uy O X. Suppose 3/1,3/2 G V', cr'j/! = FI and a'y^ = F%. So cr'yiy^, as a function over X, is equal to y\F^ and also to 3/2-Fi, so that 3/1^2 ~~ 2/2-^1 belongs to the degree m + 2k component of Ix • On the other hand, let 2/1,3/2 forms in A and suppose that the element a of HlXx(m) is given by local equations ^ on the open sets W y j n X , 2 = 1,2. Then, 2/1-^2 — 3/2-^1 £ ^X if no irreducible component of X has support contained in
the hypersurfaces y1 = Q and 3/2 = 0, while, in general, we can simply say that (2/1-^2 — y^Fi)yihy2h belongs to /x, for some positive integer h. The following two results concern the dimension of the vector space spanned by the polynomials y\F^ — y-^Fi of the previous remark.
PROPOSITION 3.2. Let X be a closed subscheme ofPN. Suppose Nm (X) ^ 0 and denote by v the number of minimal generators of Ix in degrees < m + 2. Then: (a) H°Ix(m + 2) > N and v > 2. (b) IfN>3, then v > 3. (c) If X is a set of points in P 2 in uniform position, then v > 3 or X is a complete intersection (p,q), p + q = m + 3. Proof: Let x be a linear form such that the corresponding hyperplane does not contain the support of any (isolated or embedded) component of X, so that Ux r\X is a dense subset of X: we can suppose x = (a) If cr is a non-zero element of N'm (X], then the ideal of X contains at least the degree m + 2 polynomials x0F; — XfFo, i = I , . . .,N (see Remark 3.1). It remains to prove that they are linearly independent: if not, there is a linear form y £ ( x i , . . . , Xjv) (suppose y — xi) such that xoFi — xiFo = 0. But, in this case, FQ = xoG and an inverse image a1 £ H°Ox(m) of a is cut by the hypersurface G on the dense subset IAX H X of X, so that a is zero, against our assumption. Now we prove the three statements on v all together.
Let X' C PN be the subscheme defined by all generators of Ix in degrees < m + 2. The same local equations ^-, which define cr, also define a non zero element of HlXxi(m) (see Remark 3.1). So, Ix> has at least 2 minimal generators, since, if not, Ti^-, should be zero. Thus, v > 2. If N > 3, we have H^, = 0 if Ix> is generated by less than 3 elements. So v > 3. Finally, suppose X be a set of points in uniform position in P 2 and suppose that X' has just two minimal generators of degrees p, q < m + 2. By uniform position property, X' is a set of pq points (see [6] Proposition 1). Since the same local equations that define a also define a non zero element of Nm ( X ' ) , we can easily see, by the minimal free resolution of Ix>, that p+q = m+3. Again, by the minimal free resolution of Ix1 and by the uniform position property of X, it follows:
pq > deg(X) > hx(m + 2) = h'x(m + 2) = d e g ( X ' ) = pq and so X = X1.
THEOREM 3.3. Let X be an integral, non-degenerate, subvariety of PN and let V C Am be a vector space of dimension r > 2; such that N™+1(X, V) ^ 0.
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Then If, moreover, r > 3, then H°Ix(m + 2) > r. Proof: Let's choose a non zero element
Fir -i — Oj.fr- lr + / ^ bijFjr.
3=1
By a suitable linear change on V, we can suppose that xi corresponds to an eigenvector of the matrix (6»j), that is Fir-i = aF r _i r + bF\r. Thus: r_i
- bFr) — ( X T - \ — bxr)(Fi + aFr) = 0
and then, again,
In fact, in the proof we have just used the fact that the complementary in X of a hyperplane section is a dense open subset of X . On the other hand, if some component of X is contained on a hyperplane and the vector space V is strictly contained in A\, then all the elements y{Fj — yjFi could be zero, as the following example shows. EXAMPLE 3.5. Consider a set of simple points X C P^ and let t be the minimal degree of hypersurfaces containing X . Suppose P = (I, 0, . . . , 0) 6 X . If m is any integer < t — 2, (even m < 0), there is a global section a1 G H°Ox(rn) whose value is 0 at every point of X - P and is ^ 0 at P; moreover a1 is not defined by a hypersurface, since there are no hypersurfaces of degree m containing X — P. So, its image a G Hllx(m) does not vanish. However, such an element
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GENERALIZED SOCLE AND HYPERPLANE SECTIONS.
Let X C PN and A be as in §2. If a; is a general element in AI , we will denote by Y the section of X with the hyperplane H = {x = 0} as a subscheme of H = P^"1. So, Iy will be the ideal of Y in B = A\(x) and 2y will be the ideal sheaf of Y in OpN-l
.
Let's consider the vector space N^_i(X, V) in the special case when V is spanned by re; by the cohomology sequence of:
Q^Ix(m - l)-+Ix(m) A IY (m)->0.
(2)
we can deduce the two following exact sequences:
Q->H°Ix(m-l)->H0Zx(m)
^ H°IY(m) 4 N^X^^Q
(3)
and
).
(4)
(Here and in what follows, we denote by the same symbol TT every map induced by the natural restriction from X to Y). It is easy to see, by (3), that the elements of N™_±(X, x) correspond to degree m hypersurfaces in p-^-1 containing Y which don't lift to hypersurfaces of P^ containing X, that is, to so called sporadic zeros of degree m of X (see [11]). Observe that the section Y is a general section of X; nevertheless by semicontinuity, if n™_1(X,x) = r, then we also have n™_1(X, x') = r, for every general x' € AI. The vector spaces N™_1(X,x') depend on x' and their common part Nm_1(X) is (in general) a proper subset of them. The following result gives a good description of the non-liftingjiypersurfaces containing Y, which correspond to elements of N™_l(X, (x, x ' ) ) or Nm_l(X).
PROPOSITION 4.1. Let X and Y = X n {x = 0} be as above and let F be the hypersurface containing Y which corresponds to a non zero element a £E N™_l(X,x). Then: < r € N^^X^x,^}) tfand only if x' F £ Tr(H°Zx(m + 1)) Proof: By the exact sequence (3) in degrees m and m + I , it follows: o- G N%_1(X,(x,x'}) if and only if i>(x' F) = x'^(F) = x' a = 0 if and only if
As a consequence of the above result, we see that
COROLLARY 4.2. Let X and Y = X n {x = 0} be as above. Suppose that:
_
(ii) IY has no minimal generators in degree m + 1. Then, N™+l(X, x) = 0 so that H°Ix(m +!)->• H°IY(m + 1) is surjective.
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315
Proof: Let's consider the following commutative diagram:
<j)X
(m+l))
\-<j)Y
-»
H°(XY(m+l))
\-4>
^
N^(X, x)
By (ii) (fiy is surjective and by (i) (ft is the zero map. Thus:
^m) = 0 as required. The following result is one of the main tools in this paper. It was first proved by Strano (see [10]) and restated by Green in [3], where a particular case of the sequence (5) was also introduced (cfr. [3] Proposition 4.31 and 4.37).
THEOREM 4.3. (Strano) Let X be an integral, non degenerate variety in PN and Y = X Pi {x = 0} be its general hyperplane section. Then, the following sequence is exact:
o^^^_ 1 (x ) xH^_ j _ 1 (x ) x < + 1 )4^_,.(x, a: < )AlC-,-(y).
(5)
REMARK 4.4. Suppose N^^X^) ^ 0. Observe that /C-i( y ) C W™-i( y ,2/) for every linear form y <E BI . So, if the map TT is injective, we can see, by the exact sequence (5) with i = 1, that every sporadic zero of degree m of X restricts to a sporadic zero of degree m of Y . Moreover, if TT is not injective, then N^_t(X,xl)
^ 0, for some i > 1 (recall that
1
H Ix(t) = 0 if t < 0). Thus, by (5), /C_;(Y) ^ 0 for a suitable i and then, again, ]V^/_ 1 (Y) ^ 0 for some m' < m. In any case, a sporadic zero of X induces a sporadic zero of Y, in the same or in a lower degree, depending on the injectivity of TT. The following results, that we will use very often in the proof of the main results, precisely concern properties of the map TT.
LEMMA 4.5. Let X and Y - X n {x = 0} be as above. (i) If h°Ix(rn) + n™_i(X,x) < N - 1 (or h°Ix(m) + n^.^X.z) = N and h°Ix(m) > 1), then TT : W^^^X) -» l\C-i(^) zs injective. (ii) Suppose, moreover, dim(X) > 2. 7//i 0 Iy(m)+n^-i( y ) < ^-2 (°r h°IY(m)+n^_1(Y) = N-l andh°IY(m) > 2), then, TT : N^.^X, x) -* TC-i(Y) »« injective. Proof: If codim(X) = 1, then Hl x = 0 and there is nothing to prove. So, suppose codim(X) > 2 and N > 3. _
(i) If /? is any non zero element in the kernel of IT : Nm_l(X) -)• Nm_1(Y), then, by (5), /3 — ax, where a is a suitable element of hlXx(m — 2). So, we have aA!X = axAi = J3A1 = 0, that is aAl C N^^X^x). Since n^^X^) < N- 1 -h°Xx(m), a belongs to N^(X, V) where V C Al and:
dim(V) > dim(Ai) - n^^X^) = N + I - n^.^X.x) >h°Ix(m) + 2,
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(or dim(V) > h°Ix(m) + 1 > 3) against what Theorem 3.3 states. (ii) First, we want to stress that, in the hypothesis dim(X] > 2, also the general hyperplane section Y of X is integral and non degenerate. Suppose TT : N™_l(X, x) —>• Nm_1(Y) not injective and let /? G N^_1(X, x) be a non-zero element in its kernel. By Theorem 4.3, there is a suitable strictly positive integer p such that /? = axp and 7r(a) is a non-zero element of Nm_p_1(Y). So, we have AC_ 2 (7) ^ 0. In fact, if J\C-200 = 0, then, JC-P-i(Y) = A^.p-i^) ^ 0, and so , ~Nr^_t_i(Y) ^ 0, for a suitable integer t, 2 < t < p. By Theorem 3.3 it follows h°2Y(m) > h°XY(m — t + 1) > dim(Bi) = N against our hypotheses.
Thus, we can choose anon zero element T 6 Nm_2(Y). Since, rB\ C Nm_1(Y), then T vanishes if multiplied by a vector space V C B\ such that:
dim(V) > d i m ( B 1 ) - n ™ _ 1 ( y ) = 7 V - n ^ _ 1 ( Y ) > /i°Jy(m)+2 (or e?zm(7) > h°IY(m) + 1 > 3) against Theorem 3.3. PROPOSITION 4.6. Lef ^ anrf Y - X n {x = 0} 6e as a6oue. Suppose that: (i) l^-p-i(Y) = 0 for every p>l Then TT : 7 V _ 1 ( ^ ) -> //™_ 1 (Y) is noi zero, so t t a t T - i ( ^ ) ^ 0 . Proof: Suppose that Nm_1(X) —>- Nm_1(Y) is the zero-map and take any non zero element a € Nm_l(X). Since 7r(cr) =
Since, by hypothesis, N^^X^) we have:
- /C-iPO>
then
^ € ^C-p-iW-
In fact
.
= (r^-MOa; = (rxP-1A1)A1 = (rxP'l}A2 = ••• = r(Ap+1) So rAp C A r m _ 1 (^) and, having supposed that TT is the zero map, we find T}YBp = (rAp)\Y = ^(rAp+i] = 0 that is T]y would be a non zero element of Nm_p_1(Y), against (i). 5
THE MAIN RESULT
In this section, we will prove the following result, which is an important special case of Conjecture 1.1. THEOREM 5.1. Let X be a codimension 2, integral, non- degenerate subvariety of degree d in P"+ 2 J Y be a general hyperplane section of X and Z be a general plane section ofY. Ifh°Ix(s) = 0, h°IY(s) ^ 0 and h°Iz(s - 1) = 0, then: d<s2-(n~l)s+ ( " } + 1. \
•"
/
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317
First of all, we observe that codimension 2 subvarieties X in P™+ 2 that satisfy the hypotheses of the previous result, exist for every n > 2 and s > n.
EXAMPLE 5.2. (Chang) Let X be a codimension 2 subvariety o/P n + 2 defined by the ^.-resolution:
= 0, h Iy\sj = 1, h Xz(s)
=n
and h 1z\s — 1) = 0 (see [1] and
[11])To prove Theorem 5.1, we will follow the standard technique of cutting the variety X by general linear spaces, until we reach a set of points in uniform position Z in P 2 . Using results on generalized socle, we can determine upper bounds for the Hilbert function of Z at some strategic levels, which give us the required bound for the degree d = deg(Z] = d e g ( X ) . Now we introduce some new notations we will use in the following sections. Let X be an integral, non degenerate, codimension 2 subvariety of p n + 2 and let Yfc = XDLk+2 be sections of X with general linear subspaces Lfc+2 = Pk+2 C P ra + 2 such that Ln+2 D £n+i D • • • D £3 D £2- Of these, specific cases are: Yn = X, Yn-i =Y,Yi = C a smooth, irreducible curve in P3 and YQ = Z a set of points in uniform position in P 2 . Let's denote by An+2 the graded ring K[XQ,XI, .. .,xn+2\ — H°Opn+2. We can suppose L/t+2 defined by a;n+2 = £n+i = • • • = a:&+3 = 0 so that A ~^~2 = K[XQ,XI, .. .,Xk+2\ = H°O-pk+?. If h > k, the ring Ak+2 is a quotient of Ah+2. We will often use the same symbol for an element in Ah+2 and for its class in the quotient Ak+2. We consider in Ak+2 the reverse lexicographic order on monomials induced by XQ > Xi > • • • > X / , + 2-
Note that we have chosen general linear sections of X and so XQ, x\,..., X/.+2 are general coordinates for Ak+2. Thus, the initial ideal in(I) of an ideal / C Ak+2 will be the generic initial ideal gin(I) of I (see [3]). Moreover, if h > k and a is any element of Ah+2, then the class in Ak+2 of its initial monomial in(a) is the initial monomial of the class of a. From now on, in the present section, we suppose that the hypotheses of Theorem 5.1 hold. First of all, observe that every section Y/~ has a sporadic zero in degree s' < s (see Remark 4.4) and that the inequality is not allowed by H°Xz(s — 1) = 0. So HOl Yk(s] > H°IYk+1(s) + I for every k, 0 < k < n - 1, so that H°Iz(s) > n. lfH°Iz(s) > n + l, or H°Iz(s) = n and H°Iz(s+l) > 2n + 3, easy calculations on the first difference of the Hilbert function show that the statement of Theorem 5.1 holds (see [7] and [11]). The same is true if the ideal Iz is s + 1—regular. Then, in order to state Theorem 5.1, we can restrict to the following case:
h°IYk(s - 1) = 0 for every k, 0 < k < n I,OT_. in\ _ „ _ fe |-or everv ^ o < k < n and prove that Iz is s + 1—regular. In the following lemmas we compute the dimension of the ideals and generalized socles of the varieties Y^ at some strategic levels around s.
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LEMMA 5.3. In the previous notations and hypotheses:
ft) nss~[_p(Yk,xk+i} - 0 andnss_1(Yk,xk+2) = 1 for every 1 < k < n. (ii) the restriction map TT : N I _ l ( Y k + i , X f . + 3 ) —>• Ns_1(Yk) and NI^(Yk,xk+2) = J^.^Y*) for every l
is an isomorphism
Proof: (i) Since H°XYk_1(s — 1) = 0, Yk has no sporadic zeros in degree < s — 1, then N*~i_ (Yk,xit+2) must be zero (see Remark 4.4). Moreover Yk has sporadic zeros in degrees < s, that is nss_1(Yk, Xk+z) ^ 0, and, by our hypotheses, (ii) First of all, we want to prove that TT is non the zero map. If TT = 0 in the case k = n — 1, then, N S_^(Y) ^ 0. If a is some non-zero element in NS_2(Y), then, aA™~1 C nj_ 1 (y) which is a 1-dimensional vector space. So, a € N*~2(Y, V), dim(V) > n+ 1 and, by Proposition 3.3, h°IY(s) >n+l, against our hypothesis.
Now, the (i) part allows us to use lemma 4.6 and conclude by induction. Finally, note that in our hypotheses, Iz has no minimal generators in degree s + I (see also lemma 2.5 (iii)) and then (iii) follows from lemma 4.2 (where X = C,
Y = Z e m = s). NOTATIONS 5.4. Under the previous hypotheses, we will denote by: Fn+2 a generator of H°XY(s); o- n _i a generator of N^^Y, xn+1) = N^^Y); Fi, i = 0, . . . , n + 1, polynomials such that ^ are local equations for
LEMMA 5.5. In the previous hypotheses and notations, the following properties are true for every k, 0 < k < n — 1: (i) H°XYk(s) is generated by (classes in Ak+2 of) Fn+z, . . . , -Ffc+3,' (ii) H°IYk(s + 1) is generated by Fij, 0 < i < j < n + 2; (iii) n*+l(Yk,xk+2) = 0 andnss+l(Yk,xk+2) = 1. Proof: If k = n — I , (i) is true by construction. Let k < n — 1 and suppose (i) true
for k+l. Since the polynomials x,Fj —XjF, G H0Xy(s+ 1) , then their classes also belong to H°XYk(s + 1) and in the quotient ring Ak+2 we have xtFj — XjFi = XiFj for every j > k + 2. But all the varieties Yk are irreducible and non-degenerate and so Fj € H°IYh (s} for every j = k + 3, . . . , n + 2. These are n - k = h°IYk (s) polynomials and so for (i) it remains to prove that they are linearly independent. F^+2, . . . , Ffe+s are linearly independent by induction (see exact sequence (3) where X = Yk+1, Y = Yk and H°XYk+i(s - 1) = 0). If there is a linear relation Fk+2 — \k+3Fk+3 + • • • + A ra+2 ^«+2 in Ak+2, then we have Ffc +2 = A fe+3 F fe+ 3 + - • •+A n+2 -Fn+2 + Zfc + 2 G in Ak+3; then ak+i, the restriction to Yfc+i of
Lifting problem for codimension two subvarieties in p n + 2
319
Suppose that the statement holds for some integer k — I . We know that the polynomials Fij, as elements of the ring Ak+2 , belong to H0Iyk(s + 1) and, as elements of the ring Ak+l , generate H°XYk_1(s + I): thus the restriction map:
TT : H°IYk(s + 1) A H^Y^S + 1) is surjective. Let F be any element of H°XY,.(S + 1); there exists an element G = "^,0-ijFij such that (j>(F) =
__
n+ 2 a Fi
F = G + xk+2( Y^ hfi) = X ^ i + i=k+3
as required. Finally (iii) is an easy consequence of Lemma 5.3, the exact sequence (5) and what just proved.
LEMMA 5.6. Under the previous hypotheses and notations, the following properties are true: (i) JV/_!_p(X, xn+2) = N^l_p(X, xn+2) = 0 for every p>l.
(iv) If (Tn generates Ns_i(X]
and if ^- are its local equations, then the classes
of the Fi m An+l can be the elements Fi in Notation 5.4. (v) H°Ix(s + 1) is a vector space of dimension ("+ 2 K"+ 1 ) _j_ ^ generated by
Fij - XiFj - XjFi, 0 < i < j < n + 2. Proof: First of all, observe that (by Lemma 5.3 (i) and Lemma 5.5 (iii)) we have:
n'.±\(Y,xn+1) = nss+11(Y,xn+1) = nl_1(Y,xn+1) = n^.^Y) = 1. So, ^/^.^y.zn+i) = JV/_ p _i(y,i n +i) = 0 for every p > 1, because h°IY(s) = 1 < n (see Theorem 3.3). Then, from the exact sequence (5) and the irreducibility of Yn, it follows: n't±\(X,
xn+2) = ns+_l(X, xn+2) = • • • = 0
and
nss_2(X, xn+2) = nss_3(X, xn+2) - • • • - 0 that is (i). Consider the exact sequence (5) with m = s + 1 and i = 1:
0 -j. N^X.x^) -> N',^(X,xn+2) H- N;+1(X,xn+,) -+ NSS+1(Y) and recall that nss_l(X, a; n +2) = 1 (see Lemma 5. 3 (i)) and nss+1(Y) = 0 (see Lemma 5.5 (iii)). So we obtain n$+l(X, xn+2) = nss±\(X, xn+2) - I . Moreover, by (5) with m = s + 1 and i = 2 and by part (i) it follows:
320
so that naat.\(X,xn+2) < 1 and nss+1(X, xn+z) = 0, which is (ii). The equality (iii) is an easy consequence of the previous, since #/_!(*, Zn+2 K+ 2 c N;+1(x,xn+2) = o. In order to prove (iv) it is sufficient to chose the restriction of an to Y as crn-i in Notation 5.4, so that local equations for
h°XYk(s + 1) = h°IYk_,(s + 1) + h°XYk(s) for k = 1, . . . , n (where the equalities hold because nss+1(Yk, £^+2) = 0).
Proof (of the main theorem) : As we have stated at the beginning of the present section, we can limit ourselves to the situation: H°Iz(s - 1) = 0, H°Xz(s) = n H°Iz(s + 1) - In + 2 and prove that lz is s + 1-regular. In degree s, Iz is generated by Fn+2, Fn+i, . . . , F3 and its generic initial ideal Jz = gin(Iz) by the monomials xf),xs0~1xi, . . . , xs0~n+lx™~1 , where XSQ = in(Fn+2)
and < x'0, ^"^i, . . . , x^^x^'1 >= in < Fn+2,Fn+1, ...,F3>. To simplify notations, we suppose that the hyperplanes x3, . . . , xn+i have been chosen so that: Fn+2 = xs0 +
where (.) denotes the sum of monomials whose orders are strictly lower than the
order of the preceding monomials. Note that we can do that without changing Z, which actually is a general plane section of X.
In degree s + 1, Jz is generated by the 2n + 2 monomials: -r,(vs /_» s-n + l n-l\ s + 1 s-n n + 1 Ts-n + l n-l\ J •"'l^O' ' ' ' ' 0 1 / ' '2^ J -Oi • • • i •'•0 1 h^O ' 0 1
Then, in degree s + 1, Iz is generated by the In + 2 polynomials: 2, x0Fr
where r is a suitable integer 3 < r < n + 2 and xs0~nx"+1 = in(F0r- Fir+i +brFi3). In fact, if in(F0r — Fir+i + brFi3) ^ xs0~nx™+1, then it should be zero, because no monomial of lower degree belong to Jz', if these elements were zero for every r, then there would be the same number n — I of independent syzygies in degree s + 1 for both Iz and Jz and Iz would be s-regular: see [3] Proposition 2.28). The ideal Ix is generated in degree s + I by the polynomials Fij (Lemma 5.6) and so its generic initial ideal Jx contains the following monomials:
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321
_ tft(f{j. • / r?.i j. \i j -i1 < |-9 3 «** s-n-J+j n •t' n I + l-j "'i — ^v,• )t' <\- ^y7 < \ n/1_ ^^ ii j _y? > / o. +I +I XQ = m(F 0n+2 ) and XQ~"X" = m(F0r - Fir+i + fer-Fis)n+ n 1
Those are ( M + ) + l = ft°J^-(s + 1) independent monomials and so they are a basis of Jx in degree s+ 1. Thus, the corresponding polynomials Fij form a basis S of Ix in degree s + 1:
By direct computations, we can see that Jx has ("+ H"+ K"+ ) independent syzygies in degree s + 2. Furthermore, there are exactly the same number of linear relations between the FJJ given by:
XiFjk - XjFik + xkFij = 0, 0 < i < j < k < n + 2. Claim: The above relations correspond to independent syzygies Sijk for IxFirst of all observe that we can suppose that
is zero. If not Iz and Jz have one minimal generator in degree s + 1 and 1% is s + 1— regular as required (see Lemma 2.5 and [3] Corollary 2.29). Moreover, we can suppose that _Fo, F I , F% belong to the ideal generated by #3, . . . , x n +2- To get this it is enough to replace F; by F, — x;G, where G £ An is such that the hypersurface G — 0 cuts out on Z the principal divisor locally given
b
y {ft' * = 0,1,2}.
Now, we can write the Fij, which don't belong to B, as linear combinations of element of B as follows:
ckFl3 + dk(F0r - Flr+l + brF13
where the (.) contain combinations of Fij with i > 2 and 3 < f c < n + l , f c ^ r . Let's consider the free resolution:
El -A E0 -A IX —> 0 where EQ is the free module on the elements e,-j such that f ( e i j ) = Fij for every Fij £ B and EI is the free module of first syzygies. Let's denote by Sijk the syzygy that we obtain from XiFjk — XjFik + xi-Fij — 0, using (6) if it is necessary. Suppose that the Sijk are linearly dependent: ^,-A ctijkSijk — 0, ctijk £ K. Then, 0 = g(Y^ijkankSijk] = Y J i j k a i J k 9 ( S i j k ) — T^HijCij where all the coefficients Hij, that are linear forms, must be zero and so the coefficients of every Xk in Hij must be zero. We will compute a few of those coefficients in terms of the otijk and of the constant coefficients that appear in (6). If j = I or 2, and k > 3, the coefficient of XQ in Hjk is aojk'- it follows ceoik = a02k = 0 for every k > 3. Then the coefficient of x\ in Hy, is a^k and so, ai2k = 0 for every k > 3. Let's now consider indices i, j, k such that Fij £ B and k > 3 (we don't suppose j < k); by what just proved, the coefficient of Xk in Hij is sum of ctijk and a linear
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Roggero
combination of some aotfc; if we prove that aotk — 0 for every t, we also obtain that ctijk = 0. Let's use descent on k. If k = n + 2, the coefficient of xt in -ffon+2 is Qotn+2 = 0 (note that Fon+2 does not appear in (6)) and then a;j n +2 = 0 for every pair of indices ij such that Fij € B. Now, let k > 3, k ^ r, be such that aijk+i = 0 for every ij such that F;J G #. The coefficient of Xj in -ffi^+i is aijfc+i + aojk = aoj/c = 0; then a;jfc = 0 for every ij such that Fij € S. Finally, if k = r and ajjv+i = 0 for every ij such that Fij € B, the coefficient of £j in Hir+i is aijv+i — 5^dto:oj£ = — ^dtaojt = 0; on the other hand the coefficient of Xj in Hor is aojr + ^^t^ojt = <^ojr — 0, and from this a,-jr = 0 for every zj such that F,-j G £>We have just proved that all the aijk are zero, unless possibly 0:012Let's suppose 0:012 ^ 0. The coefficient of XQ in _ffj^ is aoi^Pojk = 0; so pojfc must be zero, for every j, k and then PI 2 = 0. By the factoriality of the polynomial ring An+2 we obtain FI = x^G, and
Thus 12 is s + 1-regular too and the bound on the degree d of X holds.
6
LIFTING PROBLEM IN P6 AND P7.
In the present section we will prove Conjecture 1.1 for n = 4 and, with an additional hypothesis, for n — 5. We will follow the notations introduced in the previous sections for general sections of X and corresponding rings and ideals: in particular, X is a codimension 2, integral, non-degenerate subvariety of P 6 or P7, Y a general hyperplane section of X, C an irreducible curve section of Y with a general P3 and Z a set of points in uniform position, general plane section of C.
THEOREM 6.1. Let X be a codimension 2, integral, non-degenerate subvariety of P6 and let Y be its general hyperplane section. If h°Ix(s) = 0 and ft°Jy(s) ^ 0, then: d=deg(X) < s 2 - 3 s + 7. Proof: It si sufficient to prove the statement for the minimal integer s satisfying
the hypothesis; in fact, if /i°Iy(s — 1) ^ 0, then s — I > 2, since Y is not a complete intersection, and (s — I) 2 — 3(s — 1) + 7 < s2 — 3s + 7. Then, let's suppose
If h°Iz(s - 1) — 0 the statement is the case n = 4 of Theorem 5.1. More generally, easy computations involving the Hilbert function prove that the bound on d holds if 1% is s + 1-regular. So, let's suppose h°Iz(s — 1) ^ 0 and r ( I z ) > s + 2.
Lifting problem for codimension two subvarieties in Pn+2
323
In such hypotheses we have Ns_2_a(Z)) ^ 0 for some a > 0 and then there are at least 3 minimal generators in degrees < s — a (see Proposition 3.2 (c)). If h°Xz(s — 2) ^ 0, then the number a above is I and so there are at least 3 minimal generators in degree < s — 1. The Hilbert function of Iz gives deg(X) < deg(Z] < s2 — 4s + 6 which is an even better bound. In a similar way we conclude if one of the following cases happens: h°Xz(s - 2) = 0 and h°Xz(s - 1) > 2, h°Iz(s - 1) = 1 and h°Xz(s) > 6
h°Xz(s - 1) = 1, h°Xz(s) = 5 and h°Xz(s + 1) > 12. We want to prove that those are all the possible cases. By Proposition 2.5 it remains just one case to examine: h°Z z (s-l) = l, h°I z (s)=5, h°Iz(s + l) = ll. In this hypotheses we also have h°Xc(s — 1) = 0 and N*_i(C, £3) = 0 (see[3], Lemma 4. 37). Using the exact sequence (3) for all the varieties Y& we obtain:
5 = h°lz(s) = h°Xc(s) = nl_ 1 (y 4 ) + nj-i^a) + "I-i^a).
(6)
Observe that the last three summands cannot be zero, since nss_l(Y4J ^ 0 and ns,~\~aa(Yi) - 0 for every a > 0 and i = 2,3,4, because h°XYl(s - 1) = 0. Moreover, nss_3(C, ^3) = 0 and nss_2(C,X3) = 1 ([3] Lemma 4.37); from this and the exact sequence (4) it follows nss_3_i(Yk, Xk+i) = 0 for every i > 0 and k= 1,2,3,4. Using again (4) with i = I and j = 2:
0 = NiY^N^Y^N^Y^N^C)
=0
and with i = 2 and j = 2:
o = ;v;_3( we find n*_ 1 (Y 2 ) = nss_2(Y2) < nss_2(C) < nss_2(C) = 1, that is n^_ 1 (Y 2 ) = 1. Moreover, from Proposition 4.5 (ii) (with Jf = Y^} it follows that ns$ nJ_ 1 (Ys) < nj_ 1 (y3). Thus, there are just two possibilities for the summands in (6): 5 = 2 + 2 + 1 or 5= 1 + 3 + 1 . _ Case 2 + 2 + 1 must be rejected, since it implies that N^_1(Y3) = A^.^Ys) and that the map N s_l(Y3)-^N s_l(Y2) is not injective, against Proposition 4.5 (i) (with X = Y3). Finally, let's consider case 1 + 3 + 1. _ The kernel Nf_2(Y3, x5) of A^jL^Ys, x5}-^~NSs_1(Y2} has dimension > 2 and then, by (4), A r g_ 2 (y 2 ) should have dimension > 2. But rzj_ 1 (y 2 ) = 1 and then there is __s
__s _ -j.
some non zero element 7 £ ^-2(^2) which also belongs to N ^^(Y^), against the assumption h°Xc(s — 1) = 0.
THEOREM 6.2. Let X be a codimension 2, integral, non- degenerate subvariety of P7. I f h ° X x ( s ) = 0, h°IY(s) ^ 0 and h°Iz(s - 1) ^ 1, i/ien:
d=deg(X) < s2 - 4s + 11.
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Proof: We may suppose that s is the minimal integer satisfying the hypothesis. If h°Xz(s — 1) = 0 the statement is the case n = 5 of Theorem 5.1. If h0Xy3(s — 1) ^ 0, we get the bound on d by replacing s — I instead of s in Theorem 6.1 and, if h°!Y3(s - 1) = 0 but h°IY2(s - 1) ^ 0, we get the bound by replacing s — 1 instead of s in [12], Theorem 3.11. . Furthermore, easy computations involving the Hilbert function prove that the bound on d holds if Iz is s + 1-regular. So, from now on, we will assume that /i°Iy2(s — 1) = 0, that r(Iz) > s + 2 and that there are at least 3 minimal generators of Iz in degrees < s + 1 . The Hilbert function of Iz allows us to prove the bound if one of the following conditions holds:
h°Iz(s - 2) = 0 and h°Xz(s - 1) > 4 or h°Iz(s - 1) = a < 3 and h°Iz(s) > a + 6 or h°Xz(s - 1) = a < 3, h°Xz(s) = a + 5 and h°Xz(s + 1) > a + 13. Claim: No other cases are allowed.
Suppose h°Xz(s-2) = 0, h°Iz(s-l) = a, h°Iz(s) = a + 5, h°Iz(s+l) = a+12, h°Xz(s + 2) = a + 21, with 2 < a < 3 (lower values are not allowed by Lemma 2.5). First consider case: a = 3 By the exact sequence (3) we get nss~^(Y2, £4) + ns.T.\(C, x3) = 3 and, by Lemma
2.5, ns~-l(z] - 0, n',-l(z) = l and nl-l-k(z) = ° for every k > °Thus n^I^C, £3)= "sIslC'i^s) < 1 and so 0*12(^2,^4) > 2, which is impossible by Proposition 4.5 (ii) (where we put X = Y?, Y — C and m = s — 1). Now consider case a = 2. By the hypotheses it follows b = h°Ic(s — 1) < 1 and ns,T_\(C, £3) = 2 — 6. From this we obtain NS_3(C) = 0. In fact if 7 is any non zero element of S
~N S~3(C), then •yV = 0 with dim(V) > 4 - (2 - 6) - b + 2 and this implies h°Ic(s - 1) > b + 1 (see Theorem 3.3). The ideal Iz has two minimal generators in degree s — 1, one in degree s and just one syzygy in degree s+1; other generators and syzygies have degrees > s + 3. By the minimal free resolution of Iz we get:
From this, using repeatedly the exact sequence (5), we can get: nlll(C, x3) + n'+l(C, x3) + nJ.^C, x3) + n'-_\(C, x3) < 4.
Suppose h.°Xc(s - 1) = 0 that is nss1\(C, x3) = 2. In this case we have: nss^\(C, x3) + nss+1(C, x3) + n^^C, x3) < 2. Moreover, since there are no minimal generators of Iz in degree s + 1 and s + 2, if nss+1(C, x3) = 0 then nss^l(C, x3) is zero too, and if ra*_ 1 (C 1 , £3) = 0 then the other two are also zero; in any case nss~^(C, £3) = 0. Thus: nss-i(C) < 3, n°,±l(C) < 4 and nss+\(C) = 0 (if not, any non-zero element in n^^tC) = 0 should be annihilated both by a plane and an independent conic surface and so h°Xc(s — 1) ^ 0, against our hypothesis)
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and from this, by (5) again: tf,±a(C) < 2 and 2 = nssll(C,x3) < n^C^s) + nss±{(C) = nJ^(C) < ns+_l(C) + 1. If nss+l(C) = 2 then h°Ic(s - 1) ^ 0, which is impossible. So, let nssti(C) = 2 and nsst\(C) = 3. Moreover, n's±\(C) = n^^C1,^) +
nss+l(C)
= 1 and then nss+1(C,x3) = 0 and n|_j(C) = n^^C,^) = 1.
If there exists some element a £ N"S_3(C) = NSa_3(C) such that dim(aA\) < 3,
then h°Ic(s — 1) ^ 0. This happens for instance if there is a linear form x such that x7V^_ 3 (C') = 0. Let {cr, r} be a basis for NS_3(C); we can suppose CTXQ = 0 and TX\ = 0. Then (crxi,
case, NS_2(C) ^ 0 and so NS_3(C) ^ 0 against what we have proved above.
Finally, let's suppose h°Xc(s — 1) = 1.
Since nss-l(Y2,x4) = I and T^I^C) = 0 , then Tfl'^C) = N°:%(C,x3) = I . From a free resolution of Ic it follows that Ic ha just one 2— syzygy in degree s + 2 and at least 4 1— syzygies in degree s + I (and none in lower degrees). On the other hand, gin(Ic) has at most 5 1— syzygies in degree s + 1 and such number of 5 is reached if and only if N^_l(C, x) = 0. So, if Ic is not s + 1— regular, then Ic has 4 1 -syzygies in degree s + 1 and ^V/_ 1 (C I ) = 0 that is h°Ic(s) — 8. Then Tv'I^C) = ~Nl-3(C) = 0. Using repeatedly (5) we also get A^_ 3 (Y;) = 0 and n*_ 2 (^-) < 1 for every i > 1.
In particular nss_2(Y2) = ns$_2(Y2) = 1 and then nss_1(Y2) = 0, n*_i(>3) < 1, "5-1(^4) < 2 > «s-i( y s) < 3 and fr°m tnis h°Ic(s} < 0 + l + 2 + 3 = 6 which is a contradiction. REMARK 6.3. The condition h°Iz(s — 1) ^ 1 in the previous result is in fact a restriction on the statement, which might be furtherly investigated. In any event it cannot be escluded at the moment that h°Xz(s — 1) = 1 yelds a contrexample.
References [1] M. Chang, Characterization of arithmetically Buchsbaum subschemes of codimension 2 m Pn, J. Differential Geometry 31 (1990), 323-341 [2] L. Chiantini, C. Ciliberto, A few remarks on the lifting problem, Asterisque
218 (1993), 95-109. [3] M. Green, Generic initial ideals , Summer School on Comm. Alg. (1996) 11-85.
[4] L. Gruson, C. Peskine, Section plane d'une courbe gauche: postulation, In: Enumerative Geometry and Classical Algebraic Geometry (Nice, 1981), Progress in Math. 24 (Birkhauser, Boston 1992), 33-35. [5] O. Laudal, A generalized trisecant lemma, In: Algebraic Geometry Proceedings, LNM 687 (Springer 1978), 112-149.
[6] R. Maggioni, A. Ragusa, The Hilbert function of generic plane sections of curves o/P 3 , Invent. Math. (1988) 253-258.
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[7] E. Mezzetti, Differential-geometric methods for lifting problem and linear systems on plane curves J. Algebraic Geometry 3 (1994) 375-398. [8] E. Mezzetti, I. Raspanti, A Laudal type theorem for surfaces in P 4 , Rend. Sem. Mat. Univ. Politec. Torino 48 (1990), 529-537.
[9] R. Strano, A characterization of complete intersection curves in P3, Proc. AMS 104 (1988) 711-715 [10] R. Strano, On generalized Laudal's lemma, In: Complex Projective Geometry (Trieste, 1989), London Math. Soc. Lecture Note Series 179 (1992), 284-293.
[11] A. Tortora, On the lifting problem, in codimension 2 Le Matematiche Vol LII (1997) Fasc. I pp. 41-51 [12] M. Valenzano, Bounds on the degree of integral varieties of codimension two in projective space To appear in Journal of Pure and Applied Algebra
Rank Two Bundles and Reflexive Sheaves on P3 and Corresponding Curves: an Overview MARGHERITA ROGGERO Dipartimento di Matematica, Universita di Torino, via Carlo Alberto 10, 10123 Torino, Italy, e-mail: [email protected] PAOLO VALABREGA Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail: [email protected]
MARIO VALENZANO Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail: [email protected]
1
INTRODUCTION
Rank two vector bundles and reflexive sheaves on projective n-space or, more generally, on a smooth algebraic variety, have been investigated since the seventies, with quite a few interesting results covering many areas of research. However the present survey is focused only on the interaction between rank two vector bundles and reflexive sheaves on a three-dimensional projective space on the one hand, and algebraic curves embedded in P3 on the other hand. Section 2 below is a "dictionary" with notations, definitions and technical tools, so that the reader can look it up when necessary; Section 3 outlines a picture of the matter also from a historical point of view, and lists the related research; Section 4 contains some results about rank two bundles and reflexive sheaves only; Section 5 discusses the first and second relevant sections of a reflexive sheaf in connection with the problem of finding the minimal degree of a surface containing a space curve; and finally Section 6 points to Gherardelli-type theorems. 2
DICTIONARY
(2.1) Ground field k: We work over an algebraically closed field k of characteristic 0. (2.2) Projective space P3: We consider the projective space of dimension 3 over Written with the support of the University Ministry funds. 1991 Mathematics Subject Classification: 14F05, 14H50
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the ground field k.
(2.3) Notations: Given a coherent sheaf F on P3 we use the following notations: F(n) for F ® 0 p 3 ( n ) , #!>(n) for £P(P 3 ,F(n)), /z!>(n) for dim A /P(P 3 , F(n)), where 0 < i: < 3 and n G Z. (2.4) Curve in P3: Curve will always mean a closed subscheme of P3 of pure dimension 1, i.e. with no embedded or isolated point, locally Cohen-Macaulay and generically locally complete intersection, so our curves are possibly reducible or even non-reduced. (2.5) Subcanonical curve: A curve C is called a-subcanonical if its dualizing sheaf MC is isomorphic to Oc(a) for some integer a. (2.6) The integers e(C) and e'(C) associated to a curve C: If C is a curve in P3, then e — e(C) is the speciality index of C, i.e. the greatest integer such that h°<jJc(—e) — hlOc(e) ^ 0, while e' = e'(C) is the greatest integer such that u>c(—e') has a section generating it almost everywhere. In general we have e' < e, but for an integral curve e' = e. We observe also that, if C is a-subcanonical, then a = e'. (2.7) Rank two vector bundle on P3: It is a locally free sheaf on P3 of rank two, i.e. whose fiber at every (closed) point is a ^-vector space of dimension 2. (2.8) Rank two reflexive sheaf on P3: It is a rank two coherent sheaf F such that the natural map from F to its double dual F vv is an isomorphism; we recall that the dual F v of F is defined as Hom(F, 0ps) and the rank of a coherent sheaf F is by definition the rank of the stalk of F at the generic point. Such a sheaf is locally free except at finitely many points; moreover any vector bundle is a reflexive sheaf, so one can consider reflexive sheaves as a generalization of vector bundles. (2.9) Split vector bundle: A rank two vector bundle (or reflexive sheaf) on P3 is called split if it is isomorphic to the direct sum of two line bundles. Throughout this paper the bundles or reflexive sheaves are often implicitly assumed to be non-split: this is due to the fact that many theorems are either trivially false or trivially true for split bundles as the reader can by himself easily understand. (2.10) Scheme of zeros (or zero locus) of a section of a reflexive sheaf: If s is a nonzero global section of F, then it gives rise to a map
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F is not free; in fact cs(F) = h°Sxt1(F,wF3j, so a reflexive sheaf is a vector bundle if and only if 03 = 0. (2.12) Normalized sheaf: A sheaf is called normalized if its first Chern class c\ is either 0 or — 1. For every sheaf F we get a normalized sheaf if we twist by ro, where m = — %J- if c\ is even or m = —^^- if c\ is odd. (2.13) Serre's correspondence: (see [41, Theorem 4.1]) Fix an integer c\. Then there is a one-to-one correspondence between: (i) pairs (F,s) where F is a rank two reflexive sheaf on P3 with c i ( F ) = GI, and s 6 H°F is a global section whose zero locus has codimension 2, and (ii) pairs (C, £), where C is a locally Cohen-Macaulay curve in P3, generically locally complete intersection, and £ £ H°wc(4 — ci) is a global section which generates the sheaf u>c (4 — cj) except at finitely many points. Furthermore, under this correspondence, c^(F) = d and cs(F) = 2pa — 2 + d(4 — GI), where d and pa are the degree and arithmetic genus of C. Moreover the sheaf F and the ideal sheaf of the curve C are linked by the fundamental exact sequence »0.
(1)
(2.14) Reflexive sheaf canonically associated to a curve: Given a space curve C, let £ be a section generating uc(—e') almost everywhere, where e! — e'(C) is the number defined in 2.6; such a section gives rise to a reflexive sheaf F , called the sheaf "canonically associated" to C; its first Chern class is c\ = e' -j- 4. (2.15) First and second relevant sections: The least integer a = a(F)
such
that H°F(a) ^ 0 is called the level of the first relevant section of the reflexive sheaf F . In fact every nonzero section of F(a) gives rise to a scheme of zeros of codimension two, i.e. a curve, called minimal curve of F. The level of the second relevant section is the least integer /? = /3(F) > a such that h°F(/3) > h°O-p3(j3—a), so this is the first level where there are sections which are not multiple of a same section at level a. (2.16) Stability: A rank two reflexive sheaf or bundle F on P3 is stable (resp. semistable), in the sense of Mumford and Takemoto, if for every invertible subsheaf L of F it holds a(L) < c-4jp- (resp. Cl(L) < £i^i). Notice that stability implies semistability and if F has odd first Chern class the two concepts coincide. If semistability condition is not verified we say that the sheaf is non-stable. If we consider a normalized reflexive sheaf, then it is stable if and only if a > 0, semistable if and only if a + c\ > 0, non-stable if and only if a + GI < 0; in this last case the sheaf is non-stable of order r = —a — c\ . (2.17) Null correlation bundle: It is characterized by c\ — 0, c-i = 1; its twist by 1 is the vector bundle canonically associated to a pair of skew lines (see [59] or
M). (2.18) Arithmetically Buchsbaum sheaf and curve: They are defined by the property of having zero multiplication on the 1-cohomology module (and are shortly referred to as a.B.).
3
HISTORICAL OUTLINE. SERRE'S CORRESPONDENCE. LIST OF RELATED RESEARCH.
The correspondence between rank two vector bundles on three-space and subcanoni-
cal curves (see 2.5) goes back, although not explicitly, to Serre (see his famous paper
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[74] on projective modules in Seminaire Dubreil-Pisot, 1960-61) and Horrocks ([48], 1968). Subcanonical curves, at least the smooth irreducible ones, had been investigated since the beginning of the forties by the Italian mathematician Giuseppe Gherardelli, with a geometric approach and of course no use of bundles. He proved two main results, known as first and second theorem of Gherardelli. They deal with
irreducible curves free from singularities in P3 (projective space over the complex field) and state what follows (see [33] and [34]): I. If all surfaces of any degree n cut out on C complete linear series and all surfaces of a given degree cut out the canonical series, then C is simple complete intersection of two surfaces. II. If C is residual of a subcanonical curve, then C is simple complete intersection of three surfaces. Conversely all such curves can be so obtained. Therefore, in modern language, Theorem I states that a subcanonical arithmetically normal curve is complete intersection of two surfaces (the converse being well known), while Theorem II concerns liaison and states that a curve linked to a subcanonical curve is defined, as a scheme, by three equations, and conversely.
The theorems above gave both rise, in the eighties, to problems and results on subcanonical curves (and rank two vector bundles). To our knowledge no mathematician investigated subcanonical curves (either in connection with bundles or for themselves) until the mid-seventies, when Barth and Van de Ven considered two-codimensional subvarieties of P" zero loci of sections of rank two bundles ([7]). But it was in 1975 that Ferrand ([32]) revived Serre's construction, in order to investigate some multiple structures on curves which came
out to be subcanonical as schemes of zeros of sections of rank two bundles. It is also worth recalling that in 1978 the first theorem of Gherardelli was rediscovered by Griffiths and Harris, who gave two new proofs of it both with a classical geometric approach and using the bundles ([36]). As to the rank two vector bundles on P3 (or P", n > 4), it is due to Barth, Elencwajg and Maruyama ([4], [5], [52], [53]) a careful study of stable bundles (and reflexive sheaves), of their Chern classes, cohomology and moduli. In particular Barth and Elencwajg showed that a stable rank two vector bundle in characteristic 0 has a spectrum, a set of c% whole numbers whose knowledge gives information on the intermediate cohomology of the bundle. Serre's point of view is actually investigated and furtherly developed by Hartshorne between the end of the seventies and the beginning of the eighties ([39], [41], [42]). The correspondence between rank two vector bundles and subcanonical curves is presented in [39] and later extended in [41] to rank two reflexive sheaves and more general curves (see 2.13). It is worth remarking that, while vector bundles correspond to subcanonical curves, reflexive sheaves correspond to locally CohenMacaulay generically locally complete intersection curves, which means almost all non-pathological curves. We recall that the key link between a vector bundle or reflexive sheaf F and the scheme of zeros C of a nonzero section s of F(n) is given by the following exact sequence
O-J-Ops -* F(n) -»X c (ci + 2n) -> 0 (see 2. 13 and 2.11).
(2)
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Through the exact sequence above, properties can be transferred from bundles or reflexive sheaves to curves and conversely. We wish to emphasize the following fact. A priori, working with curves and working with bundles or reflexive sheaves should be equivalent. Actually the techniques employed may be very different and sometimes it might come out to be much easier to deal with curves and then translate into the language of sheaves, or conversely. It may also happen that both techniques are useful to obtain a com-
mon result. For instance Gherardelli theorems and their extensions can be proved for smooth irreducible curves on the basis of geometric arguments involving only curves, but the results can be used to obtain properties of bundles; on the other hand, through the bundle machinery, the extension to arbitrary subcanonical curves becomes very easy. Hartshorne's paper [41] is not focused only on Serre's correspondence; in fact it extends the construction of the spectrum due to Barth and Elencwajg to a stable reflexive sheaf. The existence of the spectrum in the less studied case of non-stable
reflexive sheaves is later proved by Sauer ([72]). However Hartshorne's main concern is the investigation of the level a of the first relevant section of a bundle or a reflexive sheaf (see 2.15). It must be remarked that a nonzero section at this level gives rise for sure to a scheme of zeros which is a
curve, called minimal curve of F. Hartshorne in [39] conjectures that a is not larger that a certain function of the Chern classes, when F is a bundle; such a conjecture is investigated and finally proved for a reflexive sheaf in [41] (see Section 5). These results on a give rise to papers by Nollet, Roggero and Valabrega concerning, instead of the first nonzero section, the second relevant section ([70], [71], [58]). Thanks to the exact sequence above, such a section is related to the smallest degree of a surface containing a projective curve. As far as Gherardelli theorems are concerned, his second theorem is proved and extended in 1979 by Rao ([61]), while his first theorem is investigated and improved by Chiantini and Valabrega in 1984 and furtherly in 1987 ([18], [19]). Many more papers in the eighties and nineties investigate properties of rank two reflexive sheaves, from many points of view. In connection with Buchsbaum curves we must quote Ellia-Fiorentini and Chang ([26], [16]), while Miro-Roig proves the existence of sheaves with given Chern classes ([55], [56]) and Hirschowitz of sheaves with seminatural cohomology ([47]). Since we want to offer a survey which specially emphasizes those results on rank two vector bundles and reflexive sheaves which are relevant for curves, we shall pay attention to the following list of related research: 1. General properties of rank two bundles and reflexive sheaves: spectrum, computation, vanishing and connection of 1-cohomology, stability, non-stability and plane restriction, Chern classes and theorems of existence, seminatural cohomology. 2. First and second relevant sections of a reflexive sheaf in connection with the smallest degree of a surface containing a curve. 3. Gherardelli-type theorems on subcanonical curves, also in connection with liaison and arithmetically Buchsbaum curves. We wish to recall that in 1978 Hartshorne, after the Oxford conference on algebraic vector bundles on projective spaces, and the discovery of connections with some partial differential equations of the mathematical physics, made a list of 26 problems in the field ([40]). Among them only a few concern rank two vector bundles on P3 and the present survey will point out answers to some questions of the
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list.
4
GENERAL PROPERTIES OF RANK TWO BUNDLES AND REFLEXIVE SHEAVES.
Stable bundles have been investigated in many papers in the seventies and eighties:
Barth, Maruyama, Ein, Harthshorne, Vogelaar and Miro-Roig should be quoted
([4], [52], [22], [55]). First of all we list results about restrictions to planes and lines. The following key result is due to Barth ([4, Theorem 3]): Let F be a rank two stable bundle on P3 and let H be a general plane; then the restriction FH of F to H is a stable rank two bundle, unless F is a null correlation bundle, in which case FH is only semistable. We remark that the same restriction theorem is true in positive characteristic, with the further exception of the Frobenius pullback of a null correlation bundle ([21]). The following extension of the above theorem is due to Maruyama ([54]) and also to Ein, Hartshorne and Vogelaar ([22]):
If F is a semistable reflexive sheaf, then its general plane restriction is semistable. It is worth remarking that, using the restriction property, it can be proved the following inequality for a stable reflexive sheaf:
c\ — 4c2 < 0, (hence c2 > 0 when the sheaf is normalized). We observe that a(F#) can be either equal to or smaller than c*(F); moreover ct(Fn) is the same for a general plane H. If, for a special H, the restriction Fg is non-stable, then the order of instability (that is the largest integer r such that H°FH(-r - cj ^ 0) cannot exceed c2 - GI - 1 ([57]). If L is a general line, FL is a split bundle OL(O] ® OL(CI — a) (see [37] or [59]), where, by the theorem of Grauert and Miilich ([35]), a — 0 when F is a stable normalized reflexive sheaf. A jumping line for a stable normalized bundle F is a special line such that a ^ 0. Chang has bounds on the maximal order a of a jumping line, depending on Chern classes ([12]):
a < 2c2 + 2 + ^ c i - Y (4 + 3ci)c 2 + 5 + —ci. When F is non-stable, it is known that, for a general H, a(Fn) — <*(F), while for special H a(Ffj) can be smaller than a(F); also in this case there is a bound on the order of instability ([57]). We recall here that a stable normalized bundle has a spectrum ([5]), which is a set of K = c2 integers ki whose knowledge gives dimensionally the 1-cohomology in the range n < — 1 and the 2-cohomology in the range n > —3 — ci, as follows:
A 1 J F(n) = ft°(®Opi(Jfe,- + n + l ) )
and h2F(n) = hl(®O^(ki + n + I)).
The existence of the spectrum has been extended to stable reflexive sheaves by Hartshorne ([41]). As to non-stable reflexive sheaves, Sauer ([72]) proves that the
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spectrum, a set of K = c2 + c\r + r1 integers, r being the order of instability, gives dimensionally the 1-cohomology below r—1 and the 2-cohomology above — r—ci — 3. Now we list a few properties of the spectrum. The spectrum of a normalized bundle, stable or not, has a symmetry property ([41], [72]): A weaker property, involving also 03 holds for a reflexive sheaf ([41], [72]):
c3 = -'<. The following connection property holds for a normalized stable reflexive sheaf ([41]): If k > 0 occurs in the spectrum, then 0 , 1 , . . . , k also occur, and if k < 0 occurs, then — 1, — 2 , . .., k also occur. For a non-stable sheaf a similar properties holds ([72]). Inequalities involving Chern classes and vanishing theorems for the intermediate cohomology of a rank two normalized reflexive sheaf F can be obtained using the spectrum ([41], [42], [72]). As examples in the stable case (see [41]):
(i) hlF(n] - 0 for n < -1 - i(ci + c 2 ) and h2F(n) = 0 for n > c2 - ci - 3; ( i i ) c 3 < C 2 - ( c 2 - 2 ) ( C l + l). Many papers deal with the classification of rank two bundles and reflexive sheaves having small Chern classes (for instance ci = 0, c2 = 2) or narrow assigned spectrum, also in connection with moduli spaces ([13], [14], [15], [27], [45],
[46], [64]). All the results listed below are related to the existence of rank two bundles or sheaves whose Chern classes or spectrum or cohomology are assigned. A natural cohomology rank two vector bundle has, by definition, at most one nonzero H1 F(n), 0 < i < 3 for each n. Hirschowitz ([47]) shows that, for c\ — 0 and c2 > 0 or c\ — — 1 and even c2 > 6, there exists a stable rank two vector bundle with Chern classes GI, c2 and natural cohomology. This result is related to the theory of instanton bundles because a mathematical instanton bundle is characterized by ci = 0 and #^(-2) = 0 (see also [44]). A seminatural cohomology reflexive sheaf has by definition at most one nonzero Hl F(n), for 0 < i < 3 and for each n > —2—^-. Hirschowitz proves that seminatural cohomology sheaves with given Chern classes exist, with few exceptions, under the following conditions: Cic 2 = c3 (mod 2)
and
0 < c3 < 4c2 — c\ — 4
and so proves the existence of maximal rank curves which are zero loci of sections of these sheaves. In this context see also [9]. As to the existence of general reflexive sheaves, Miro-Roig gives necessary and sufficient conditions on the Chern classes both in the stable and in the non-stable case ([55], [56]). As a consequence the following result is obtained: For all (ci, 02,03) € Z3 such that Cic 2 = csfmod 2) and €3 > 0, there exists a rank two reflexive sheaf (stable or not) F on P3 with Chern classes c;(F) — c,-,
i=l,2,3. When a.B. reflexive sheaves (see 2.18) are concerned there are results by Ballico,
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Bolondi and Miro-Roig ([3]) giving lower bounds for the third Chern class and discovering gaps in the existence range of the Chern classes. Partial results concerning the existence of reflexive sheaves with given spectrum are obtained by Hartshorne and Rao ([45]) and Rao ([64]). It is well known that any finite length module is the first cohomology module (also called Rao module) of a reflexive sheaf, but this is not true if we deal with bundles. Full characterizations of the Rao module of a rank two bundle, as requested by problem 10 of Hartshorne's list [40], are known. Decker in [20] shows the following result:
M is the Rao module of the bundle F if and only if there is a minimal free resolution 0 —> £-4 —> £3 —>• LI —>• L\ —> LQ —> M —> 0 such that: i) rank(Li) = 2 rank(L0) + 2, ii) there is an isomorphism <3> : L\(c\) —> LI, some integer c\, such that a o <3> o The above theorem completes the necessary condition for M to be the Rao
module of F given by Rao in [62], giving an answer to Problem 10 of Hartshorne's list [40]. A related theorem of [67] shows that the dimensional knowledge of the intermediate cohomology gives the three Chern classes of a reflexive sheaf. One of the most meaningful results on the Rao module of a vector bundle is due to Buraggina ([10]) and it states that the Rao module of a bundle is connected, i.e. a zero component cannot exist between two nonzero components. So a final answer is given to a long investigated conjecture. We do not get involved in details, but we want to recall that many papers and results concerning vector bundle and reflexive sheaves make use of the so called monads or, more generally, of locally free resolutions ([49], [6], [63], [45], [51]). We also recall that a few papers concern the vanishing of the intermediate cohomology of a stable reflexive sheaf and of its general plane restriction ([25], [29], [30]). 5
FIRST AND SECOND RELEVANT SECTIONS. MINIMAL DEGREE OF A SURFACE CONTAINING A SPACE CURVE.
In [41] and [42] special attention is paid to the smallest integer a such that H°F(a) ^
0, F being either a bundle or a reflexive sheaf. Such a number has an interesting property: all nonzero sections of F(a) give rise to two-codimensional schemes of zeros, so curves and not surfaces. These are called minimal curves of F. Minimal curves of reflexive sheaves and liaison are closely related. Exact sequence (1) and (2.13) show that among all curves corresponding to F the minimal ones have smallest degree and genus; moreover all curves arising from sections of F(n), any n, belong to the same even class of liaison, having the same (up to shifts) Rao module as F. In addition, Lazarsfeld and Rao show ([50]) that a general integral curve is minimal in its biliaison class, so being the minimal curve of the reflexive sheaf corresponding to it in the natural way (see 2.14). Such an integral curve is directly linked to the minimal curve in the "residual" biliaison class. Buraggina ([11]) extends to reflexive sheaves the concept of biliaison and shows that any (non-arithmetically Cohen-Macaulay) minimal curve in a biliaison class
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corresponds to the sheaf whose CB is smallest; this means that, if a biliaison class contains a vector bundle, then it is a minimal element in the class. This is equivalent to the following property: if in a biliaison class there is a subcanonical curve, then the minimal curve is for sure subcanonical. In order to consider minimal curves of a reflexive sheaf, it is useful to obtain information about a. The following upper bound for a, depending upon the Chern classes of F, is proposed as a conjecture in [39] and fully proved in [42]: Let F be a reflexive sheaf such that c% > 0 (for instance F semistable) and ci = 0 or -1. Then h°F(t) ^ 0, provided that t > v/3c2 + 1 - 2 if Cl = 0 and
t > ^302 + 5 - | i f c i = -I. (This is a full answer to Problem 9 of Hartshorne's list [40]). Therefore an upper bound for a (when F is normalized) is a function of 02 of the same growth as \/3c2- We observe that the above integer t is the first one such that the Euler-Poincare characteristic of F becomes positive for good.
The proof requires techniques involving unstable surfaces and reduction steps. The general plane restriction of a stable bundle is stable (see Section 4), but the restriction to a special plane may be non-stable (such planes are called unstable planes). An unstable surface for a reflexive sheaf F can be defined in an analogous way, but using the dual of the restriction, because the restriction of F to some surface could be a non-reflexive sheaf: An unstable surface for a normalized rank two reflexive sheaf F, unstable of order > r, is a surface X such that there exists an injective map Ox —> Fx(~r)> where F^ denotes the dual of the restriction of F to X. An equation / = 0 defining an unstable surface X can be obtained looking at an annihilator / of minimal degree d of an element in H2F(m)*, some m, where * denotes the dual vector space. Such a surface X gives rise to a reduction step
0 -> F' ->• F -» Iz,x(-r) -> 0 where F' is a rank two reflexive sheaf, Z is a closed subscheme of X of dimension < 1 and F' has Chern classes ci(F') = c1(F)-da,nd c2(F') = c 2 ( F ) - cl(F)d- dr - k, k being the degree of the curve part of Z, defined as the coefficient of the linear part of the Hilbert polynomial of Z. A suitable annihilator giving rise to an unstable surface can be obtained glueing annihilators of plane restrictions. The theorem above is also proved in [43], but the approach is quite different because rank three reflexive sheaves are involved; the paper contains also a slight improvement, because the third Chern class c3 appears in the inequalities. The above results are applied in the same paper to find the maximum genus g of an
irreducible non-singular curve C of degree d in P3 which is not contained in any surface of degree < k. This is in fact a partial application (with bundles replaced by reflexive sheaves) of what Hartshorne suggests in his problem list ([40, Problem
12] ), where a suitable use of rank two vector bundles is assumed to be of some help in order to obtain such a bound on the genus. It is also interesting to find upper bounds for the level /? of the second relevant section, which has the following two properties: 1) zero loci of a general section of F(t) are curves if and only if either t = a or t > /?, while F ( t ) has no nonzero section for t < a, and the zero locus is a surface ifa
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2) if C is minimal for F, then, because of the exact sequence (1), a + /? + c\ is the smallest degree of a surface containing C. An estimate of such minimal degree, depending on the degree of C, can be obtained if an upper bound for a + /? + GI is obtained as a function of 03 + ci a + a 2 . Moreover, if t > /? is large enough, the curve scheme of zeros of a section of F(t) lies on two independent surfaces of degrees t + a + ci and t + j3 + cj.. If C is an arbitrary curve, nothing better than the following result can be obtained in general ([18]): // C has degree d, the smallest degree s of a surface containing C is d. Better results can be obtained introducing other conditions on the curve (e.g. smoothness) or other parameters beside the degree d, like the genus or the index of speciality. Through exact sequence (1) sheaves can be used to discover bounds for /? depending upon a, ci and eg, or, in the language of curves, upon d and e'(C) . Precisely in [70], [71] and [58] bounds both for /? and for s are obtained. We quote the following:
(i) P < smallest integer t such that x(F(t + a + ci)) > x(Op^(t + ci)) for good; (ii) s < vd, if e1 + 4 < 2yd (in this event F is necessarily non-stable);
(Hi) s < ^/3d + i — |, if C is not minimal for a semistable F. We wish to emphasize that reducible and non-reduced curves are included. Observe that the integer t of (i) has the same order as \/6d, if e'(C) > —3 (and this is true for all stable curves according to [1]); therefore C lies on a surface of degree < \/£>d. Otherwise C is minimal for a non-stable sheaf and certainly non-reduced, and the bound is d. Bound (i) is reached by the skew union of q lines in general position. In fact q lines in general position, which by the way form a (—l)-subcanonical curve, have maximal rank, i.e. the restriction map H°Opz(n) —> H°Oc(n) has maximal rank for every n. Therefore the smallest degree s cannot be smaller then the least integer n such that h0Opa(n) > (n+ l)s. More generally the bound is reached by minimal curves of reflexive sheaves having seminatural cohomology ([71], [58]). In this context we quote the main result of [69]: If en reaches the upper bound of [41], then /? = a.
A related question is the following: when is the scheme of zeros of a section of F(n) smooth and/or irreducible? If -F is a bundle it is known that a smooth irreducible curve is obtained for n large enough ([39]), while if F is a reflexive sheaf n large enough ensures only that the zero locus is integral. This is proved in [65] and is related to an algebraic version of Bertini theorems.
6
GHERARDELLI-TYPE THEOREMS.
The first theorem of Gherardelli concerns irreducible subcanonical curves in complex projective space, which are free from singularities, and complete series on them. It states (see [33]):
Theorem I (Gherardelli). Let C be an irreducible curve such that (i) all surfaces of any degree n cut out on C complete linear series; (ii) the surfaces of a give degree r cut out the canonical series.
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Then C is simple complete intersection of two surfaces. (In particular C is free from singularities).
Therefore, if we associate subcanonical curves to bundles, it states that a rank two bundle splits if its 1-cohomology module vanishes; or, which is equivalent, a subcanonical arithmetically normal curve is complete intersection. This is in fact the statement presented as Gherardelli theorem in Ellia [23] and actually it is simply a translation into the language of schemes of Horrocks' general splitting criterion for a vector bundle on P n (see [49] or [59]), later extended by Evans and Griffith ([28]). But this is not the spirit of the theorem, which comes out of essentially geometric arguments, typical of classical Italian algebraic geometry, like Brill-Noether reciprocity theorem and the number of conditions imposed to a linear series by a plane section of C or the study of surfaces of a given degree containing the curve. Therefore it is far away from the spirit of Hartshorne as well as of Ferrand
and Serre. Both the spirit of classical Italian geometry and of Serre are revived in a paper by Griffiths and Harris ([36]). They give a proof based on Riemann-Roch and a careful study of the second difference of the sequence h°Oc(n) and a second proof which makes a strong use of vector bundles. Actually after the forties no other mathematician, in Italy or everywhere else, investigated subcanonical curves in this spirit, although with the aid of bundles, until the beginning of the eighties. It goes back to 1981 Sernesi's paper [73], where homological techniques (but not bundle techniques) are used to show that a resolution of ®n£zH°Oc(n), C being an a-subcanonical curve, is self dual, and that H1Xc(n) vanishes for good if it vanishes for n = a + I (see also [2], where old Petri's results are rediscovered). But it is after Hartshorne's lectures on rank two bundles and curves in Kyoto (1981) and Cortona (1982) that Gherardelli theorem is revisited in Italy. In 1984 Chiantini and Valabrega give the following improved statement: An a-subcanonical curve C is complete intersection if and only if the surfaces of degree 1 + f when a is even, of degree ^4p when a is odd, cut out a complete series on C. This means obviously that a rank two normalized bundle F splits if H1F(—1) = 0. The proof is not based on bundles but on the bound for the genus of the space curves of given degree proved by Gruson and Peskine ([38]), hence on a result obtained with algebraic tools. The bundle machinery is useful only to the purpose of including, through the twists, also reducible and non-reduced curves, with any singularities. In fact, if C is any subcanonical curve, then it is zero locus of some section of a rank two vector bundle F, where F and Xc are linked by the exact sequence (1). But (see [41]) F(n), n large enough, gives rise to a subcanonical smooth irreducible curve C". Since Gherardelli-type conditions mean that HlF(n) vanishes, for some or all n, the following chain of implications can be easily seen: Gherardelli-type conditions on C ^ vanishing of 1-cohomology of F ^ Gherardelli-type conditions on C"
338
Roggero, Valabrega, and Valenzano C' is a complete intersection F splits C is a complete intersection.
The paper also shows that the following reasonable conjecture is not true: C is complete intersection if the surfaces of degree a cut out a complete series (which is of course the canonical series). Such a condition only forces deg C to be smaller than a suitable power of a. It is the use of bundles that suggests the falsehood of the conjecture: in fact the completeness of the series cut out by all surfaces of degree a is not invariant under twisting of the bundle. The improved version of the first Gerardelli theorem of Chiantini and Valabrega has also been proved in 1988 by Popescu ([60]), avoiding the Gruson-Peskine theorem and using bundle techniques. A further proof is due to Ellia ([24]) and it is based on the spectrum. Chiantini and Valabrega prove also the following result (see [19], and [68] for an improvement): Let C be an a-subcanonical not complete intersection curve lying on a surface of degree s and not less, and assume that t > s is the smallest degree of a new surface containing C; then no series can be complete among those cut out by surfaces of degree n, s — 2 < n < i — 2. It must be remarked that a key tool for the proof is the above quoted result by
Sernesi. The above results have been generalized in [66] by Roggero. Among others we quote the following statement: A curve C in P 3 is a-subcanonical if and only if uic(—a) has a section which generates it almost everywhere and hlOc(n) = h°Oc(a — n) for some n < a/2. In [26] another Gherardelli-type theorem concerning a.B. curves (see 2.18) is proved using cohomological techniques on reflexive sheaves. The homogeneous coordinate ring of an a.B. curve is by definition a Buchsbaum ring; this is equivalent to say that the Rao module of the curve has trivial multiplication. In [26] Ellia and
Fiorentini proves what follows: A non-split rank two vector bundle on P 3 has zero multiplication on its first cohomology module if and only if it is a null correlation bundle. [26] gives also a curve version of the result above: Let C be an a.B. integral curve in P3. // C is subcanonical, then C is either a complete intersection or the scheme of zeros of a section of (a twist of) a null correlation bundle. The theorem on curves is simply an application of the theorem on bundles. It is however remarkable that the hypothesis "integral" is not necessary in their proof, because it is only a question of bundles. Chang gives a slightly more general result ([16]): If X is an a.B. subcanonical two-codimensional subvariety ofPn, then X is either a complete intersection or the zero locus of a section of a twist of a null correlation bundle on P 3 . The proof is cohomological and bundle-based but different from [26].
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The second theorem of Gherardelli is stated as follows (see [34]): Theorem II (Gherardelli). Let L^v be the system of all curves in P3 which are
simple complete intersections of two surfaces F1*, Fv; if C is residual, with respect to the given system, of a subcanonical curve (for instance another complete intersection Lpq or an elliptic curve ... ), then C is simple complete intersection of three surfaces. Conversely all such curves can be so obtained. Therefore, in modern language, it concerns liaison and states that a curve directly linked to a subcanonical curve is defined, as a scheme, by three equations, and conversely. As before the proof is based on geometric arguments (base points of linear systems, adjoint curves, . . . ). It was forgotten as well for many years, until 1979, when Rao ([61]) gave a new proof in the general context of liaison and rank two bundles (but without a proper quotation of Gherardelli, perhaps because it was an old paper written in Italian and not known). Such a proof is strongly based on resolutions of ideal sheaves and on homological techniques and disregards geometric
arguments (in the sense of Italian geometry). So Rao's theorem is in some sense the counterpart of the second theorem of Gherardelli, from the point of view of the proof, because homological bundle techniques produce a geometric result. Actually some of the many papers on liaison among curves deal also with subcanonical curves and curves directly linked to them, or even with self-linked subcanonical curves. For instance we recall the following characterization ([17]): Let C be a subcanonical curve corresponding through the section s to the rank two vector bundle F; then there is a surjective map u : ®!=i ^P 3 (~ a ») ~~^ F, ai being an integer, all i, and s is member of a system of 4 generators of F if and only if C is directly linked to another subcanonical curve. It is also worth to be quoted the following result by Beorchia and Ellia ([8]): A smooth subcanonical curve C in P3 is self-linked if and only ifC is a complete intersection. The above result has recently been generalized by Franco, Kleiman and Lascu
([31]).
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[70] M. Roggero, P. Valabrega, On the second section of a rank 2 reflexive sheaf on P3, J. Algebra 180 (1996), 67-86 [71] M. Roggero, P. Valabrega, On the smallest degree of a surface containing a space curve, Boll. U.M.I. 1-B (1998), 123-138 [72] T. Sauer, Nonstable reflexive
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J.P. Serre, Sur les modules projectifs, Seminaire Dubreil-Pisot 1960/61, Seer. Math. Paris (1961)
On the structure of Ext groups of strongly stable ideals ENRICO SBARRA 1 , Universitat GH-Essen, FB6 Mathematik/Informatik, 45117
Essen,
Germany, Enrico. SbarraOuni-essen.de
ABSTRACT: We consider special classes of strongly stable ideals such as squarefree lexicographic ideals generated in one degree and principal strongly stable ideals. We prove a structure theorem for the Ext groups of such ideals. By means of earlier results on lexicographic ideals we also prove a-i upper bound theorem for the sheaf cohomology of some families of coherent sheaves on PJ-.
1
INTRODUCTION
The main goal of this paper is to study the behaviour of the Ext groups of special classes of ideals which are strongly stable. A monomial ideal / of R = K[Xi,... ,Xn] is strongly stable (resp. squarefree strongly stable) iff given any monomial (resp. any squarefree monomial) u € / and i 6 suppu the monomial jr^u is again in / for any j < i (resp. j < i and j (jt suppw). These ideals turn out to have combinatorial properties which arising the interest of many mathematicians became the subject of their enquiry. As a special class of strongly stable ideals one recognizes lexicographic ideals. Because of their particular nature, one can expect to find easy descriptions of some of their invariants, for instance that of their Betti numbers in terms of their Hilbert functions (see for example [3], [5] and [2]). Our starting point is the characterization given in [8], Proposition 6.6, of Ext'R(R/J, R), where J is a lexicographic ideal and i < n. In particular, it is shown that such Ext groups are cyclic and that one can easily check whether or not they vanish. This is also interesting because such Ext groups can be viewed as Matlis dual of the local cohomology modules H^(R/J), where m is the graded maximal ideal of R, and therefore from their structure one can deduce information about these modules, such as their Hilbert series. Our aim is on one hand to find analogous char-
acterizations of the Ext groups for other classes of strongly stable ideals and, on the other hand, to use the aforementioned Proposition 6.6 in order to prove an Upper Bound Theorem for families of sheaves 7 on P^-, where J is a graded ideal of K[Xo, • • • , Xn], with a given Euler characteristic. The present paper is organized in three sections, in each of which we consider the lexicographical term order induced by Xi > X% > . . . > Xn. In the first section we study squarefree lexicographic ideals generated in one degree. Recall that a squarefree lexicographic segment of degree d Supported by Istituto Nazionale Alta Matematica Francesco Sever! (INdAM), Rome.
345
346
Sbarra
is a subset £ of the squarefree monomials of Rj with the property that if m 6 £ is a squarefree monomial, then n & C for every squarefree monomial of Rd lexicographically greater than m. A squarefree monomial ideal J is a squarefree lexicographic ideal iff for every squarefree monomial u € J and all squarefree monomials v of the same degree with v > u, then v £ J. We prove a
structure theorem for Exi'R(R/J,R), with i < n (see Proposition 2.3). Unfortunately, unlike the non-squarefree case, the assumption that the ideal is generated in one degree is restrictive. Section 2 deals with principal strongly stable ideals. A strongly stable ideal I is called principal iff it is the smallest strongly stable ideal containing a given monomial, which is referred to as the principal generator of I. The main results on the Ext groups of these ideals are contained in Proposition 3.3 and Proposition 3.4. The final section is dedicated to the proof of the Upper Bound Theorem (cf. Theorem 4.3).
2 SQUAREFREE LEX-IDEALS Our idea is to use the description of the Eliahou-Kervaire resolutions of squarefree stable ideals as given in [2], Theorem 2.1, in order to pass to the duals and compute the Ext groups. It is easier though to prove a reduction argument and to work only with the first two maps of such a resolution.
LEMMA 2.1. Let A == K[Xi, . . . , X r ] be a polynomial ring and let J be the ideal of A generated by all of the squarefree monomials of Ad- Then, for i / r — d + 1, Ext'A(A/J, A) = 0. Proof. Since J is squarefree, it defines a simplicial complex A. The corresponding Stanley-Reisner ring /f [A] is Cohen-Macaulay of dimension d — I . In fact A can be viewed as the d — 1-skeleton of the simplex generated by {!,... ,r}, which is Cohen-Macaulay. Thus, if n denotes the graded maximal ideal of A, H£~l(K[&\) 7^ 0 and this is the only non-vanishing local cohomology module. The conclusion follows from the Local Duality Theorem. D Given a squarefree monomial v of degree d, we denote by £(v) the set of all squarefree monomials of degree d which are lexicographically greater than or equal to v. LEMMA 2.2. Let A = K[Xi, . . . , X r ] be a polynomial ring and I a squarefree lexicographic ideal of A generated by jC(v), for some v 6 Ad- Suppose Xi $• suppv. Then, I = Xi J + I' A, where I' is the ideal generated by jC(v) in A' = K [X^, . . . , Xr], and, for 0 < z ' < r — d + 1 ,
Proof. The first assertion is clear, since J is the ideal of A generated by all squarefree monomials of A' of degree d— I . Since J is generated by monomials in the variables Xi, . . . , Xr and I' A C J, it is easy to see that (Xi) n I' A = ( X i ) J n I'A. Therefore,
On the other hand, from the short exact sequence
0 —)• ((Xi) + I1 A) 1 1 — »• Afl -4 A/((Xi) + I'A) —* 0 we derive the long exact sequence in homology
) -4 Ext*A(A/((X1) + I' A), A) In view of the previous lemma one can deduce that, for 0 < i < r — d + 1,
and the conclusion follows from Rees' Lemma.
D
Ext groups of strongly stable ideals
347
Let v any squarefree monomial of S = K\Xi, . . . ,Xm]. Grouping together consecutive variables, we may write ft
Vj
with the standard convention that the empty product is equal to 1. This notation will be useful in what follows. Note that, if I is a squarefree lex-ideal of 5 generated in degree d, then proj dim 5/7 < m-d+1.
PROPOSITION 2.3. Let S = K[Xi, . . . ,Xm] be a polynomial ring and I be a squarefree lexicographic ideal generated by £(v) for some v £ Sj- Then, for any j < m — d + 1, Ext3s(S/I, S) is cyclic and, if Uj == Xj • . . . • Xj+Vj-i,
Ex4(5//,5) ~ ————~————r£> + j - 1). ( X 1 } . . . ,*_,•_!, Uj ) f£ Proof. We shall prove that, for any j < m — d + 1,
where we let 7y)
=
(XVi+i,XVl+V2+2, . . . ,XVl+...+Vj_1+j--i, u'j)
and u'j
is set
to be
rii=jJ -^"i+.-.+tfj-i+ii which is equivalent to the assertion, upon re-indexing the variables. The case j > I can be reduced to the case j = 1, as suggested by the previous lemma. In fact, if Si denotes the polynomial ring K[X{, . . . ,Xm] for any i < m, since C(v) = «i-C(^-), one has
Extj(5//,5) ~ Ex4
"'
,SVl
[ X i t . . . ,XV1]
and, by Lemma 2.2,
Ex4(5/7, 5) ~ Ex4-'
lp,**l ——
, 501+1 {X,, . . . , XVl](Vl + I).
It remains to be shown the case j = 1. For this purpose one can use, as observed before, the EliahouKervaire resolution for squarefree stable ideals, adapt the main ideas which led to Proposition 6.6 in [8] and achieve the desired conclusion. D REMARK 2.4. By means of Theorem 3.4 in [8], Proposition 2.3 provides an easy way of computing an explicit upper bound for the Hilbert series of the local cohomology modules of 7s.'[A] when A is pure and K has characteristic 0. REMARK 2.5. The formulae of Proposition 2.3 and Proposition 6.6 in [8] are formally very similar. In fact, if J C R = K[Xi,.,. ,Xn] is a (non-squarefree) lexicographic ideal generated in degree if by the monomials u > v, and v = X"1 • . . . • • -X%n, one has Ext^j(7?/J, 7?) ~ R/'
REMARK 2.6. In view of the Local Duality Theorem and Proposition 2.3, for any A C { I , • • . , n — 1}, one can exhibit examples of a graded 7E-module M such that H^(M) = 0 iff i £ A. Indeed it is enough to set M = R/J, where J is a squarefree lexicographic ideal generated in one degree by £(v), and choose a monomial v € R such that supp v = {j: n — j £ A}.
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Sbarra
We conclude this section with an immediate consequence of Proposition 2.3.
COROLLARY 2.7. We consider an arbitrary squarefree ideal J of R generated in one degree. Let i > 0 and H'm(R/J) ^ 0. Then H*m(R/J) is isomorphic as an R-module to M(— Y^h
where aj < 0, ^ • dj = c and there exists h G {n — i, . . . , n — i + vn-j — 1} such that a^ = 0.
3 PRINCIPAL STRONGLY STABLE IDEALS Let / be a principal strongly stable ideal. It is clear that / is generated in one degree and that the minimal set of generators of /, denoted here by S(v) , is given by those monomials of R which can be obtained by exchanging variables in the support of v with greater ones. Let C(v) denote the set of monomials u with deg« = degt; and u > v. In general S(v) C C(v), while clearly S(X? ) = £(Xf) and S(Xb) = £(Xb). It is interesting to find necessary and sufficient conditions for S(v) to be a lex-segment. LEMMA 3.1. Let v be a monomial of R. Then S(v] = £(v) iff v = XfX^
or v = X f X j X b .
Proof. We start proving the "if part. Let [Xi, . . . ,Xi+h]c denote the set of monomials of Rc in the variables Xi, . . . ,Xi+h and let v = X f X ^ . The assertion is trivial if 6 = 0. Thus, we may assume 6 > 1 and write £(v) as the (disjoint) union of the sets X°+1[Xi, . . . ,Xn]b-i and Xa[X-2, . . . , Xn}b. Both of these sets are clearly contained in S(v). Analogously, let v — X"XjX% and observe that the case b = 0 is again trivial. Thus, one may write C(v) •=. X°+l [X%, . . . , Xn]b U XfXj[Xj+i, . . . ,Xn]b-i U XfXj[Xj+i, . . . ,Xn]b, and deduce the conclusion from the previous case. Conversely, we may present v in the following way: v = XfX^1 • . - . . • X^h , where a > 0, h > I , ij+i > ij > 1 for j = 1, . . .h — 1 and Vij > 0 for j = 1, . . . , h. Observe that, if ft = 1 and i^ > 1 or if ft = 2, v,^ = 1 and 22 < n then S(v) is strictly contained in £(v). Thus we may assume that
h > 1 and, therefore, that d = deg v > Wj h . We set w = X^ 'h X^h . Clearly w does not belong to $(v), since the exponent of the last variable is greater than that of v. In order to test when w > v, it is sufficient to check the inequality Xl '' ''"~1 Xih > X"^ • . . . • X;^'1 , which is always verified if h > 2, or if h = 2 and vtt > 1. This exhausts also the last possible case and the proof is completed. D
The following proposition is the analogue, for principal strongly stable ideals, of the result on the structure of Borel-fixed ideals proven in [7], Chapter VI, Proposition 1. PROPOSITION 3.2. Let I be a principal strongly stable ideal generated byS(X"), for some monomial X" g R. Then
Proof. The assertion is clear if the support of X" consists of only one element. Let / be the largest integer in the support of X". It is sufficient to prove that I — J(Xi,... , Xi)1", where J is the principal strongly stable ideal with principal generator X» = ^-. Observe that S ( X l / ) - S(X»}X"' \JB, where B is the set of monomials of S(X") which are not divided by X\l. Moreover, if w is any monomial of S(X"}, since S(X") is strongly stable, then wXf> e S(X^)X^' C S(X"). Accordingly, with the same notation as in the proof of the previous lemma, w[X\,... , Xi]^, £ S(X1') and, therefore, S(X") D 5(X")[X 1) ... , */]„,. On the other hand, B C <S(^' i )([^i,... ,X{\V, \ [Xi\vt). Thus, S(XV) = S(X^}X^ U B C S(X^)[Xi,... , Xt}Vl and we are done. D
Ext groups of strongly stable ideals
349
Let A = K[X\ , . . . , Xn] and B = K[Xi , . . . , Xm] be two polynomial rings over the same field K and let n > m. Let I be an ideal of B and denote by I A its extension to A. Recall that, since
A is flat over 5, one has, for any z, Ext'A(A/IA, A) ~ Exi*B(B/I, B) ®B A, which is isomorphic to Ext'B(B/I, B}[Xm+i , . . . , Xn}. This is to say that, in this situation, one may study the Ext groups of R/I in the smaller ring without loss of information. In particular, one has Ext'A(A/IA, A) = 0 for i > m. Let v be any monomial of R such that the principal strongly stable ideal generated by S(v) is not a lex-ideal. Write v as Xf X?' •. . • •Xi^h , with the notation as in the proof of Lemma 3.1, let v' = v/X^hh and consider polynomial rings 5 = K[Xlt. . . , Xih] and T = K[Xi,. . . , X i h _J. Furthermore, let J be the ideal of T with principal generator v1 and / the ideal of S with principal
generator v. We know from the proof of the proposition that / = JS • (X±, . . . ,Xih)v'i>. Since JS/I has finite length, for any z > 0, there is a short exact sequence
0 _,. JS/I — ». H°(Xl_iXih)(S/I) and, for any i > 0, an isomorphism H*,x
x
—» H°(Xi_iXth)(S/JS)
)(•$*/ -0 — HJX
x
JS/JS).
— »- 0. Thus, for any i < «'/,,
Exi^R/IR, R) ~ Ext's(S//, S) ®s R ~ Ext!s(S/JS, S)®s R~ Ex4(T/J, T) 0x505^
Moreover, the above short exact sequence shows that H ?x
x
(1)
JS/I), and therefore Ext^1 (S/I,S)
does not vanish. On the other hand, if i^-i < i < ih or i > i/,, Ext*R(fi///Z, /?) = 0 while for i < Zft_i the problem is reduced to that of a principal strongly stable ideal in a ring with less variables. Using induction the next proposition follows immediately.
PROPOSITION 3.3. Let I be a principal strongly stable ideal of R with principal generator v. Then, ) = 0 iff j'
The next proposition completes the results of this section.
PROPOSITION 3.4. Let I be a principal strongly stable ideal of R with minimal system of generators S(v) jL C(v). Let v = X%X?; • ...• X^h with a>0,h>l, ij+i > ij > 1 for j = 1, . . . , h - I and vfj > 0 for j=l,...,h. '
Set c = \ l lfVii ^ 1 and, for any c < j < h, let Rtj = K[Xi, ... , X{i]. ^2 if Vi1 — 1 Furthermore, let Iic be the ideal of Ric generated by ^(X^X^1 ) ifv^ > 1 or by S(XfXilX^) Vi1 = I. Finally, for any j = c + 1, . . . , h, let
;=c+i Then, for any j < ia,
, R) ~ ExtJHic (RiJIic,Ric) ®R,C R and, for j — c + 1,... , h,
,R)-
if
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Sbarra
Proof. For j < ic the assertion follows immediately from what was said before Proposition 3.3. Note that, by virtue of Lemma 3.1, /,-c is a lex-ideal and, thus, its Ext groups can be computed as in Proposition 6.6 in [8]. Furthermore, since Xii is a non-zerodivisor of Rij/Ii:i_lRi1, then
^.....x.^C^-Ay-i^) = °-
Thus
> Extfl,.^/^,^) - (li^Rij/IijY
and the conclusion
for j = c + 1, . . . , h follows from (1).
D
4 THE UPPER BOUND THEOREM We first quote the statement of [8], Theorem 5.4, which we shall use in the proof of Theorem 4.3.
THEOREM 4.1. Let R = K[Xi,... ,Xn] a polynomial ring with graded maximal ideal m = (Xi,... ,Xn) and let I C R an homogeneous ideal. If Ilex denotes the lexicographic ideal associated to I, then, for any i,j, dim* H^(RlI)i < dim* H' As a consequence, by means of Proposition 6.6 in [8], one deduces the following lemma. LEMMA 4.2. Let I be a graded ideal with a given Hilbert polynomial
with ai > ...a; > 0 and let Vf = \{aj : n — ay — 1 = i}\. Finally, let 2!rl'^''~n~'7~I)
Then H(H'm(R/I)J)
otherwise
< 6"/, for all j and for all i > 0.
Proof.v Without loss of generality one may assume Ilex to be generated in degree d. Thus /'ea7 n n\\ i\L\A^ r{v \ • ... • A 'voA n j) and
H(H'm(R/I)J)
< H(K(Xn.i,... ,X B ]/(XlV), E
v
" - J - i - !)•
Let us consider S = J-Cf^,-,... ,Xn]/(^,''i). It is easy to see that H ( S , v f ) = Y^l'=i (n~li+k) E^Li (nn-2ik)' Therefore,
which implies the desired conclusion.
D
Let 5 be the polynomial ring /{'[.XTo, • • • >Xn] with graded maximal ideal n. To any 5-graded module M one can associate a sheaf M on the n-dimensional projective space Proj S = P^. Let O be the structure sheaf of PJ-. Given an arbitrary sheaf JF of O-modules, one defines a graded 5-module Tt(F) associated to T to be r*(J") = ®j-r(PJ r ,^'(j)), where J"(j) denotes the twisted sheaf F ®o S(j).
We denote by D n ( ) the functor limHomgfn", ). It is known that Dn is naturally equivalent to F, (~) as functors from the category of all graded S-modules and homogeneous S-homomorphisms to itself. In particular, for any j and for any graded 5-module M, T t ( M ) j = F(PJ-, M(j}) ~ Dn(M)j. Furthermore, one can check that fl"*(P^,M) = ®jHt(Vff,M(j)) has a natural graded structure as an 5-module. Observe that, given a family of graded ideals with a fixed Hilbert polynomial P, there exists a unique saturated lexicographic ideal which belongs to the family and that this ideal can be described explicitly in terms of P.
Ext groups of strongly stable ideals
351
THEOREM 4.3. Let J be the family of ideals of S with a given Hilbert polynomial P and let L be the saturated lexicographic ideal of the family. Then, for any i and any I 6 J , the Hilbert function of Hl(y"jf, I) admits a sharp upper bound depending only on P, which is reached for I — L.
Proof. Let I 6 J and let us consider the short exact sequence 0 —^ / —> S —> S/ 1 —¥ 0. From the corresponding long exact sequence in cohomology it is easy to see that (a) H ° ( I ) = 0, (b) H^~1(S/I) ~ Hln(I) for any i < n + 1 and (c) there exists a short exact sequence 0 —^ H%(S/I) —> HZ+1(I) —> H2+1(S) —?• 0. Note that the sequence 0 — > H°(I) — »• I _+ Dn(I) — > ffi(7) —> 0 th
is exact and the i right derived functor R'Dn( ) of Dn is naturally equivalent to H'n+1( ), for any i > 0 (cf. [4], Theorem 2.2.4). Since the Qth local cohomology module of J is 0 and H®(S/I), one obtains a short exact sequence 0 — »• 7 — )• £>„(/) — >• #S(S/7) — >• 0.
Thus, since H°(S/I) ~ ^, !>„(/) ~ / : n°°, and Rf Dn(I) ~ ff* +1 (/) for any i > 0. By virtue of Theorem 4.1, H(I : n°°, j) < H(L,j) for any / € J and for any j. In fact,
< H(H°(S/I**),j) Accordingly,
H(Dn(I),j)
for any j.
Let now 0 < i < n. Since -ff£+1(/) ~ Hln(S/I), one has, again by Theorem 4.1, H(RiDn(l),j)
H(Hln(S/I),j)
< H(H'n(S/L),j).
(2) =
Therefore, applying Lemma 4.2,
H(RiDn(l),j)
for any j.
On the other hand, since H(H^+1 (S), j) = H(S(-n-l),-j)
(3)
= H(S,-n-j-l) = ("+1_~n"_7ri1"'1) =:f;
J+
j!
(4)
_ .n + l,L , / -j'-l \
—
W
n,j
^ V-n-j-1/'-
Recall the functorial isomorphism
ir(Pl, M(j}) S H?Dn(M)j, for any i, j, provided by the Serre-Grothendieck Correspondence Theorem, which implies in particular that, for any i, tfj(P£.,M) ~ R{Dn(M) as graded 5-modules (cf. [4], Theorem 20.3.15). In view of the last observation, the conclusion follows from (2), (3) and (4). It is clear that the bounds depend only on the Hilbert polynomial P and are reached iff / = L. d We conclude with the following remark. Theorem 4.1 has been generalized as follows (cf. Theorem 5.1 in [9]): Let F = ®ri=lRe.i be a free and graded ^-module and let M be a family of graded submodules of F with a given Hilbert function. If L denotes the lexicographic submodule of the family, then, for any M 6 M and for any i,j,
dim* H'm(F/M)i < dim*
352._
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By virtue of this result one may generalize the above theorem for families of quasi-coherent sheaves on Vff with a given Euler characteristic. In this case the bounds are dependent upon the Hilbert polynomial only in case i > 0, while they depend upon the whole Hilbert function for i = 0.
REFERENCES 1. A. Aramova, J. Herzog, T. Hibi. Shifting operations and graded Betti numbers. To appear in: Journal of Algebraic Combinatorics.
2. A. Aramova, J. Herzog, T. Hibi. Squarefree lexsegment ideals. Mathematische Zeitschrift 228
(1998) 353-378. 3. A. Bigatti. Upper bounds for the Betti numbers of a given Hilbert function. Communications ^n Algebra 21(7) (1993), 2317-2334.
4. M. Broadmann, R. Sharp. Local Cohomology. Cambridge University Press, Cambridge, 1998. 5. H. Hulett. Maximum Betti numbers of homogeneous ideals with a given Hilbert function. Com-
munications in Algebra 21(7)
(1993), 2335-2350.
6. G. Kalai. The diameter of graphs of convex polytopes and /-vector theory, in: Applied Geometry and Discrete Mathematics. (P. Gritzmann and B. Sturmfels, Eds) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 4, Amer. Math. Soc., 1991, 387-411.
7. K. Pardue. Nonstandard Borel-fixed ideals. Thesis, Brandeis University, 1994.
8. E. Sbarra. Upper bounds for local cohomology for rings with given Hilbert function. Preprint 1999.
9. E. Sbarra. Upper bounds for local cohomology modules of rings with a given Hilbert function. Thesis, University of Essen, 2000.
An introduction to tight closure KAREN E. SMITH Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, [email protected]
1
INTRODUCTION
This is an expanded version of my lecture at the Conference in Commutative Algebra and Algebraic Geometry in Messina Italy in June 1999. The purpose of the talk was to give a brief introduction to the subject of tight closure, aimed at commutative algebraists who have not before studied this topic. The first part focused mainly on the definition and basic properties, with the second part focusing on some applications to algebraic geometry, particularly to global generation of adjoint linear series. These lecture notes follow even more closely a series of two lectures I gave in Kashikojima, Japan, at the Twentieth Annual Japanese Conference in Commutative Algebra the previous fall, and were distributed also in conjunction with that conference. I wish to thank the organizers of both conferences, Professors Restuccia and Herzog for the European conference, and Professors Hashimoto and Yoshida, for the Japanese conference. Both events were a smashing success. Special thanks are due also to Rosanna Utano, for help editing the tex file. Tight closure was introduced by Mel Hochster and Craig Huneke in 1986 [18]. Today it is still a subject of very active research, with an ever increasing list of applications. Applications include areas like the study of Cohen-Macaulayness. For example, the famous Hochster-Roberts theorem on the Cohen-Macaulayness of rings of invariants has a simple tight closure proof [18]. Also, the existence of big Cohen-Macaulay algebras for rings containing a field was proved with ideas from tight closure [20], and the existence of " arithmetic Macaulayfications" in some cases was discovered with tight closure [3], [29]. Tight closure has provided us with greater insight into integral closure, and into the homological theorems that grew out of Serre's work on multiplicities. For example, it gives us simple proofs of the Briangon-Skoda Theorem, the Syzygy Theorem of Evans and Griffith and of the monomial conjecture (in equi-characteristic) [18]. Tight closure provided the inspiration for results on the simplicity of rings of differential operators on certain 353
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Smith
rings of invariants [45], and it has produced "uniform" Artin-Rees theorems [22]. There are also numerous applications to and connections with algebraic geometry, such as in the study of singularities [46], [38], [11], of vanishing theorems [12], [25], [14], and of adjoint linear series [41], [44]. In Section 3, I will summarize some of these applications to algebraic geometry, although of course, there will not be enough time to do any of them any justice. Let us begin with our first task: to introduce the definition of tight closure before tackling its main properties in the next section. Tight closure is a closure operation performed on ideals in a commutative, Noetherian ring containing a field (that is, of "equi-characteristic"). The tight closure of an ideal J is an ideal /* containing /. The definition is based on reduction to characteristic p, where the Frobenius (or p — th power map) is then used. To keep things as simple as possible, we treat only the characteristic p case here.
DEFINITION 1.1 Let R be a Noetherian domain of prime characteristic p, and let
/ be an ideal with generators (yi , . . . , yr). An element z is defined to be in the tight closure /* if there exists a non-zero element c of R such that
for all e » 0.
Loosely speaking, the tight closure consists of all elements that are "almost" in / as far as the Frobenius map is concerned. Indeed, if we take the pe — th root of (*) above, we see that l
c
'^z e m1^
for e ^> Q. As e goes to infinity, l/pe goes to zero, so in some sense cl/p° goes to 1 (this idea can be made precise with valuations). So z is "almost in" /, at least after applying the Frobenius map.
It is not important to restrict to the case where R is a domain; we can define tight closure in an arbitrary Noetherian ring of characteristic p by requiring that c is not in any minimal prime. However, because most theorems about tight closure reduce to the domain case, we treat only the domain case in this lecture. Example. Let R be the hypersurface ring
k[x,y,z] (x3 + y3 - z3)' where k is any field whose characteristic is not 3. Then
(x,y)* = (x,y,z2). Indeed, if k has characteristic p, we can write
An introduction to tight closure
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where r = 1 or 2 and q = pe. Expanding this expression as
it is easy to see that each monomial xmyn appearing in the sum has either m > q or n > q unless both m and n equal q — 1 (which only happens in the case where q — 1 mod 3). So we can take c = x (or y), and conclude that c(z2)q G (xq,yq) for all g = pe. Thus z 2 £ ( x , y)*. A similar argument can be used to show that z is not in (x,y)*. Because this works for all p (except p — 3), we declare that z 2 , but not z, is in the tight closure of ( x , y ) also in characteristic zero. So ( x , y ) * = (x,y,z2) in every characteristic p > 0 except p = 3.
2
MAIN PROPERTIES OF TIGHT CLOSURE
The definition of tight closure takes some getting used to. Fortunately, one can understand many applications of tight closure if one simply accepts the following properties of tight closure as axioms: Main Properties of Tight Closure
(1) If R is regular, then all ideals of R are tightly closed. (2) // R c-> S is an integral extension, then IS ft R C /* for all ideals I of R.
(3) If R is local, with system of parameters xi,... ,xa, then ( x i , . . . , X i ) : Xi+i C (xi,..., x^* ("Colon Capturing"). (4) If n denotes the minimal number of generators of I, then If C /* C /, where for any ideal J, J denotes the integral closure of J. (5) IfR-^-Sis
any ring map, PS C (IS)* ("Persistence").
For the remainder of this section, we will discuss these five main properties, their proofs and main consequences. Some of the five require some mild hypotheses; precise statements will be given. All of them are true in any equicharacteristic ring (although Property 2 is not interesting in characteristic zero). All of them are quite elementary to prove, at least in the main settings, with the exception of Property 5 which requires a new idea. We will stick to the prime characteristic case, and simply remark that " by reduction to characteristic p", one can prove the characteristic zero case without essential difficulty. Note that one important property is omitted from the list. Any decent closure operation ought to commute with localization, but amazingly, we still do not know that tight closure does.
Open Problem // U is a multiplicative system in a ring R, is 1
} = (IR[U~1})*1
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It is easy to see that one direction holds, namely, PRlU*1] C Indeed, if z G /*, then we have the equations czp° = aiey^ + • • • + areyf in R. Expanding to R[U~l], the same equations show that |- is in the tight closure of IR[U~1}. This is a very special case of Property 5 above. On the other hand, the other direction is not known in any non-trivial case (see, however, [2], [43]). The localization problem is probably the biggest open problem in tight closure theory. It is remarkable that the theory is so powerful while such a basic question remains unsolved. The power is derived from the five main properties above, which we now discuss. Property One: All ideals are tightly closed in a regular ring.
It is easy to see why all ideals are tightly closed in a regular ring. For example, consider the special case where (R, ra) is local domain and the Frobenius map is finite. This is not a very restrictive assumption from our point of view, because we are usually interested in the local case anyway; also the Frobenius map is finite in
a large class of interesting rings— for example, for any algebra essentially of finite type over a perfect field or for any complete local ring with a perfect residue field. We have a descending chain of subrings of R
R D Rp D Rp2 D Rp3 D . . . Because R is regular, the ring R is a, free module considered over each one of the subrings Rp . Indeed, the Frobenius map is flat for any regular ring, but because we have assumed that R is local and the Frobenius map is finite, we actually get that R is free over Rp . This means that, for any non-zero c, we can find an Rp -linear splitting
so long as e is large enough that c is not in the expansion of the maximal ideal of Rp° to R (that is, c ^ m^'J, where ra^ denotes the ideal of R generated by the pe — th powers of the generators of m).
Now if we have an ideal I = ( y i , • • . ,yr) of R and an element z 6 /*, then we can find equations p ° = ° for all large e. Applying the Rp -linear map <j> above, we see that
where each coefficient (a;) is some element of Rp . By taking the pe — th root of this equation, we see that z is an _R-linear combination of y\ , . . . , yr. Thus z £ I, and I* — I for all ideals of R. This completes the proof that all ideals are tightly closed in a regular ring, at least in the special case we considered. The general case (of prime characteristic) is not much harder. The point is that flatness of Frobenius in a regular ring. See [18]. Property Two: Elements mapped to / after integral extension are in I*.
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We now prove Property 2: if R c—>• S is an integral extension of domains of prime characteristic, and I is an ideal of R, then IS fl R C /* . The following lemma will be useful also in the proof of Property 3.
Key Lemma // R c—> S is a module finite extension of domains, and d is any fixed non-zero element of S , then there is an R-linear map, S —>• R sending d to a non-zero element of R. The point in the proof of the Lemma is that after tensoring with the fraction field, K , of R, we have an inclusion K '—} K ®R S, where the latter is simply a finite dimensional vector space over K. So of course there is a A'-linear splitting K®S —>• K sending 1
write
z = aij/i + • • • + aryr 1
where a,- G S and the j/,-'s generate /. Because this expression involves only finitely many elements from S there is no loss of generality in assuming S is module finite over R. Now, raising this equation to the pe — th power, we have
Using the lemma, we find an /J-linear map S —> R sending 1 to some non-zero element c 6 R. Applying <j> to this equation, we have
This is an equation now in R, showing that z £ /* . Property 2 is proved. Essentially the same argument shows the stronger property: if R M- 5 is an integral extension of prime characteristic domains and / is an ideal of R, then (IS)* D f l C / * . Property 2, unlike the other four properties, is interesting only in prime characteristic. For example, if R is any normal domain containing Q, then R splits off of every finite integral extension S (using the trace map). In this case, IS f~) R = I for every ideal of R and every integral extension S.
Property 2 can be phrased in terms of the absolute integral closure. For any domain R, the absolute integral closure R+ of R is the integral closure of R in an algebraic closure of its fraction field. In other words, R+ is the direct limit of all finite integral extensions of R. Property 2 can be stated: IR+ CtR C /* for all ideals I of R. This leads to the following interesting problem.
Open Problem Let R be a domain of prime characteristic. Is IR+ D R = I* for all ideals R? In addition to providing a very nice characterization of tight closure, an affirmative answer to this question would immediately solve the localization problem.
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Indeed, it is easy to check that the closure operation defined by expansion to the absolute integral closure and contraction back to R commutes with localization. There is no non-trivial class of rings in which this open problem has been solved. However, we do have the following result.
THEOREM 2.1 [38] Let R be a locally excellent1 domain of prime characteristic. Then I* = IR+ D R for all parameter ideals I of R. A "parameter ideal" is any ideal / generated by n-elements where n is the height of /; if R is local, an ideal is a parameter ideal if and only if it is generated by part of a system of parameters. As we see from the theorem, tight closure commutes with localization for parameter ideals. However, this does not follow from the theorem because this fact is used in its proof. See instead [2].
The proof of this theorem is somewhat involved, so we do not sketch it here; see [38]. The result has been generalized to a larger class of ideals, including ideals
generated monomials in the parameters, by Aberbach [1]. Property Three: Colon Capturing.
Property 3, the colon capturing property of tight closure, is particularly instrumental in applications of tight closure to problems about Cohen-Macaulayness. Of course, if R is a Cohen-Macaulay local ring with system of parameters X i , . . . , X d , then by definition, (xi,.. . , X i ) : xi+i C ( x i , . . . , X i )
for each i = l,2,...,d — 1. Colon capturing says that, even for rings that are not Cohen-Macaulay, the colon ideal ( x i , . . . , X { ) : Xi+i is at least contained in (xi,.. .,£;)*. Loosely speaking, tight closure captures the failure of a ring to be Cohen-Macaulay. We now prove the colon capturing property of tight closure: if .R is a local domain (satisfying some mild hypothesis to be made soon precise) and x-i,... ,Xd is a system of parameters for R, then
( x i , . . . , X j ) : Xi+i C ( x i , . . . , X i ) * for each i = 1,... ,d — 1. Let us first assume that R is complete. In this case, we can express R as a module finite extension of the power series subring fc[[a;i,.. .,£<*]], where k is afield isomorphic to the residue field of R. Suppose that z 6 ( x i , . . . , X { ) :R z;+i. Consider the ring A contained in R obtained by adjoining the element z to the power series ring &[[:EI, . . . , Xd]]- Observe
that the ring A is Cohen-Macaulay; in fact, A is a hypersurface ring because its dimension is d and its embedding dimension is d + 1 (or d, if z happens to be in power series ring already). Now, because z £ (xi,..., x,) :R Xi+i, we can write
zx{+i = a1x1 + • • - + aiXi 1
Virtually all rings the commutative algebraist on the street is likely to run across are locally excellent, but see [34] for a definition.
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for some elements a,- in R. Raising this equation to the pe — th power, we have zyr
1
1" —— ft T -I^i+1 — «i ^i T
. • .
T* -rI- ft«,- -i,
Because the inclusion A °-» R is a module finite extension, we can use the Key Lemma to find an >i-linear map R —> A sending 1 to some non-zero element c G A. This yields equations
where the (a? ) are just some elements of A. In other words, cz G (x^ ,... , x f ) :A xi+l
in the ring A. But A is Cohen-Macaulay, and x\ ,..., xpd is a system of parameters for A, so we see
for all e. This shows that z G (xi,..., a;,-)* in -R (also in A, but it is R we care about). Thus ( x i , . . . , X i ) :R xi+i C (xi,. . ., Xi)*, and the proof of the colon capturing property is complete— at least for complete local domains.
Inspecting the proof, we see that we have not used the completeness of R in a crucial way: what we need is that R the domain is a finite extension of a regular subring. So this proof also works for algebras essentially of finite type over a field (the required regular subring is supplied by Noether normalization) and in many other settings. In fact, colon capturing holds for any ring module finite and torsion free over a regular ring. See [18] and [23] for different proofs and more general
statements. The philosophy of colon capturing holds for other ideals involving parameters. For example, if / and J are any ideals generated by monomials in a system of parameters {XQ, ..., xd], one can compute / : J formally as if the EJ'S are the indeterminates of a polynomial ring. Then the actual colon / : J is contained in the tight closure of the 'formal' colon ideal. Furthermore, even more is true: we have /* : J is contained in the tight closure of the formal colon ideal. Essentially
the same proof gives these stronger results with very small effort. For an explicit example, let XQ, ..., xd be a system of parameters in a domain R. Then (XQ, . • • , X^J
. (XQXl
. . . Xd)
C
(XQ
, • • • , Xd
)
and even One reason for tight closure's effectiveness is that these sorts of manipulations can often help us prove a general statement about parameters if we already have an
argument for a regular sequence. Some Consequences of the First Three Properties.
It follows immediately from the colon capturing property that if R is a local ring in which all ideals are tightly closed, then R must be Cohen-Macaulay. Indeed, if all parameter ideals are tightly closed, then colon capturing implies that R is Cohen-Macaulay. This leads us to define two important new classes of rings.
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DEFINITION 2.2 A ring R is weakly F-regular if all ideals are tightly closed. A ring R is F-rational if all parameter ideals are tightly closed. So far we have seen the following implications: Regular =>• weakly F-regular ==>• F-rational =>• Cohen-Macaulay. The first implication is Property 1, while the last implication is Property 3. The reason the adjective "weakly" modifies "F-regular" above goes back to the localization problem. Unfortunately, we do not know whether the property that all ideals are tightly closed is preserved under localization. The term "F-regular" is reserved for rings R in which all ideals are tightly closed not just in R, but also in every localization of R. That is, we have the following special case of the localization problem: Open Problem // R is weakly F-regular, and U C R is any multiplicative system, is the localization R[U~l] also weakly F-regular? This problem is much easier than the localization problem itself. Indeed, it has been shown in a number of cases. For example, Hochster and Huneke showed the answer is yes when R is Gorenstein [19], [21]. This was later generalized to the Q-Gorenstein case, and even to the case where there are only isolated non Q-Gorenstein points, by MacCrimmon [33]. Using this, it is possible to see that weakly F-regular is equivalent to F-regular in dimensions three and less. (These statements require some mild assumption on R, such as excellence). Recently, an affirmative answer was given also for finitely generated N-graded algebras over a field [32]. By contrast, the full localization problem has not been solved in any of these cases. The problem is reminiscent of an analogous problem in commutative algebra that looked quite difficult in the mid-century: is the localization of a regular ring still regular? With Serre's introduction of homological algebra to commutative algebra, the problem suddenly became quite easy. Perhaps a similar revelation is necessary in tight closure theory. Returning to the applications of the first three properties, we now prove the following easy, but important, theorem. THEOREM 2.3 [18] Let R C S be an inclusion of rings that splits as a map of R-modules. If S is (weakly) F-regular, then R is (weakly) F-regular. The proof is simple. Suppose that / is an ideal of R and that z G /*. This means that for some non-zero c, czp° G 7^°] where 7^ 1 denotes the ideal generated by the pe — th powers of the generators of /. Expanding to S, we have czp G (IS)& \ so that z G ( I S ) * . But all ideals of 5 are tightly closed, and so z G 75. Now applying the splitting S -)• R (which sends 1 to 1 72-linearly), we see that z G 7 in R as well. This completes the proof. The importance of this Theorem lies in the following corollaries. COROLLARY 2.4 Any ring (containing a field) which is a direct summand of a regular ring is Cohen-Macaulay.
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The proof is obvious: a regular ring is F-regular by Property 1, so any direct summand is also F-regular. By Property 3, this summand is Cohen-Macaulay. COROLLARY 2.5 (The Hochster-Roberts Theorem) The ring of invariants of a linearly reductive group acting linearly on a regular ring is Cohen-Macaulay.
This is essentially a special case of the previous corollary because the so-called Reynold's operator gives us a splitting of RG from R.
We emphasize that both the Theorem and its corollaries make sense and are true in characteristic zero. Thus even though there are very fewer linearly reductive groups in prime characteristic, the Hochster-Roberts Theorem for reductive groups over the complex numbers has been proved here by reduction to characteristic p. To be fair, we have not proved Properties 1 and 3 in characteristic zero (nor even given a precise definition of tight closure in characteristic zero). However, if one accepts the existence of a closure operation in characteristic zero satisfying Properties 1 and
3, then we have proved that the Hochster-Roberts Theorem follows.
We now mention one of the crown jewels of tight closure theory.
THEOREM 2.6 [20] Let R be an excellent local domain of prime characteristic. Then the absolute integral closure R~*~ of R is a Cohen-Macaulay R-module. We can see that this must be true as follows. Let xi,.. .,Xd be a system of parameters. Suppose z 6 (xi,... ,X{) : %i+i. By the colon capturing property, z G (xi,...,£;)*. But for parameter ideals, tight closure is the same as the contraction of the expansion to R+ (see the discussion of Property 2). Thus z 6 (xi,..., Xi)R+r\R. This holds for all i, so x\,..., x^ is a regular sequence on R+, and the Theorem is "proved". Unfortunately, this is not an honest proof because the proof that /* = IR+ n R for parameter ideals / uses the Cohen-Macaulayness of R+. Property 4: Relationship to integral closure.
Property 4 is really two statements. First, the tight closure is contained in the integral closure for any ideal /. Second, the integral closure of /^ (where /* is the least number of generators of /) is contained in the tight closure /*. The point in proving both statements is the following alternative definition of the integral closure J of an ideal J in a domain R: an element z € J if and only if there exists a non-zero c in R such that czn 6 Jn for all (equivalently, for infinitely many) n 3> 0. (This can be easily proved equivalent to the more standard definition of integral closure by recalling another characterization of integral closure: J consists of all elements z such that z 6 JV for all discrete valuation rings V lying between R and its fraction field. ) Note that in particular, /* = / for all principal ideals /. Now, with this definition of the integral closure, it is immediately clear that the tight closure of any ideal is contained in the integral closure. Indeed, since the pe — th power of the generators of / are contained in the pe — th power of /, we have
czp° e i&C] c ip° for all e. So any z in /* is in /.
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For the second statement, suppose that z £ I P . This means that there exists a, non-zero c such that for all n, czn G /^ n . If t / i , . . . , j / ^ generate /, then I^n is generated by monomials of degree /j,n in the y,. But if y^y^2 • • -y^ is such a monomial, at least one a,- must be greater than or equal to n. So
for all n. In particular, this holds for n = pe, for all e, and we conclude that z <E I*. The proof that P C /* is complete. The statement that IP C /* is sometimes called the Briangon-Skoda Theorem. The original Briangon-Skoda Theorem stated that for any ideal / in a ring of convergent complex power series, the integral closure of the //-th power of/ is contained in I, where p, is the minimal number of generators of / [4] . This statement was later generalized by Lipman and Sathaye to more general regular local rings and then by
Lipman and Tessier to certain ideals in the the so-called 'pseudo-rational' local rings (for a ring essentially of finite type over a field of characteristic zero, pseudo-rational is equivalent to rational singularities) [30], [31]. Tight closure gives an extremely simple proof of the Briangon-Skoda theorem for any regular ring containing a field: IP C I* C /, where the first inclusion follows from Property 4 and the second by Property 1. But better still, tight closure explains what happens in a non-regular ring as well. The original motivating problem for the Briangon-Skoda theorem is said to be due to J. Mather: if / is a germ of an analytic function vanishing at the origin in C" , find a uniform k (depending only on n} such that fk is in the ideal generated by the partial derivatives of/. The Briangon-Skoda theorem tell us that we can take k = n. Indeed, it is easy to check t h a t / e 7J = (, • • •, )- So fn e J> C 7f C J/. It is remarkable how easy the tight closure proof is for this problem that once seemed very difficult. Before moving on to Property 5, we consider one more comparison of tight and integral closure. Let J be an m-primary ideal in a local domain of dimension d. Recall the Hilbert-Samuel function defined by HS(n) = lengths/I" . This function is eventually a polynomial in n, and its normalized leading coefficient
dl lim —rHS(n)
n-i-co n
is called the Hilbert-Samuel multiplicity of /. Analogously, when R is of characteristic p, we can define the Hilbert-Kunz function
This function has polynomial growth in pe , and its leading coefficient lim —1
n-»oo (pe)
is called the Hilbert-Kunz multiplicity of /.
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As is well known, the integral closure of 7 is the largest ideal containing 7 having the same Hilbert-Samuel multiplicity (assuming the completion of R is equidimensional). What is also fairly straightforward to prove is that the tight closure of / is the largest ideal containing I having the same Hilbert-Kunz multiplicity (assuming the completion of R is reduced and equidimensional) [18]. In this sense, tight closure is a natural analog of integral closure. Hilbert-Kunz functions are interesting and mysterious, with important connections to tight closure theory and surprising interactions with number theory. Much has been proved about them by Paul Monsky, among others; see, for example, [36]. and to the bibliography of [23] for more references on this topic. Property Five: Persistence of Tight Closure.
The persistence property states: whenever R —> S is a map of rings containing a field, 7*5 C (75)* . In other words, any element in the tight closure of an ideal 7 of 7? will "persist" in being in the tight closure of 7 after expansion to any Ti-algebra. Before discussing the precise hypothesis necessary, let us consider what is involved in proving such a statement. Suppose z E 7* where 7 is an ideal in domain 7?. Thus there exists a non-zero c such that
for all large e. Expanding to S, of course, the same relationship holds in S (using the same letters to denote the images of c, z, and 7 in S). This would seem to say
that the image of z is in (IS)* , which is what we need to show. The problem is that c may be in the kernel of the map 7? -> 5. Thus we need to find a c that "witnesses" z G 7* but is not in this kernel. Unlike the first four properties, Property 5 does not follow immediately from the definition. The new idea we need is the idea of a test element. DEFINITION 2.7 An element c in a prime characteristic ring 7? is said to be in the test ideal of 7? if, for all ideals 7 and all elements z € 7* , we have czp £ 7^ ^ for all e. An element c is a test element if it is in the test ideal but not in any minimal prime of 7?.
Note that the definition of the test ideal requires that czp° £ 7^ for all e, not just for all sufficiently large e. We could also define the asymptotic test ideal as above but require only that czp € I® J for e 2> 0. An interesting fact is that the aymptotic test ideal is a 73-module— that is, it is a submodule of the module 7? under the action of the ring of all Z- linear differential operators on 7?. See [39]. It is not at all obvious that there exists a non-zero test ideal for a ring 7?. Fortunately, however, it is not very difficult to prove the following.
THEOREM 2.8 [19] Let R be a ring of prime characteristic, and assume that the Frobenius map of R is finite. If c is an element of R such that the localization Rc is regular, then c has a power which is a test element. That is, the test ideal contains an ideal defining the non-regular locus of Spec R.
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In a later paper, Hochster and Huneke prove this without the assumption that the Frobenius map is finite, imposing the weaker and more technical hypothesis of being finitely generated over an excellent local ring. Although the theorem stated above for rings in which Frobenius is finite is quite easy to prove, the proof in the more general setting is difficult and technical; see [21]. Note that in any ring R, the element 1 is a test element if and only if R is weakly F-regular. We expect that much more is true:
Conjecture. The test ideal defines precisely the non~F-regular locus in Spec R. The conjecture is proved in some cases, such as for (excellent local) Gorenstein rings [21] and for rings Kf-graded over a field [32]. Having introduced the idea of a test element, we resume our discussion of persistence. First of all, we should say that Property 5 is not known to hold in the generality we've stated; some mild hypothesis on R is needed. The problem is in finding test elements for R. Let us now sketch the proof of persistence. Let R —> S be a map of domains.2 As we remarked above, persistence is trivially true when (j> is injective, so factoring <j> as a surjection followed by an injection, we might as well assume <j> is surjective. Now factor ^ as a sequence of surjections
R -> R/Pi -» R/P2 - > • • • • - > R/(ker(f>) - S, where PI C PI C • • • C (ker <j>) is a saturated chain of prime ideals contained in
the kernel of <j>. By considering each map separately, we see that we might as well assume that the kernel of the map R —> 5 has height one. Now if R is normal, then the non-regular locus of R is defined by an ideal J of height two or more. But as we mentioned above, this means that the test ideal has height two or more, 3 so we can find a c which is a test element but not in the kernel of <j>. The proof is complete in the case R is normal. Finally, it is not difficult to reduce the problem to the case where R is normal, using Property 2. What happens is the normalization R of R maps to an integral extension S of S, namely the domain S obtained by killing a prime of R lying over the kernel of
practice, all rings we run across will satisfy this hypothesis.
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can be checked, it satisfies Property 4 as well. On the other hand, since we expect /* = IR+ n R, this is perhaps not very convincing. A more interesting question is whether we can define a closure operation for rings that do not contain a field (that is, in 'mixed characteristic') which satisfies Properties 1 through 5. If so, many theorems that can now be proved only for rings containing a field, such as the homological conjectures that grew out of Serre's work on multiplicities, would suddenly admit "tight closure" proofs. The only serious attempts at defining such a closure operation in mixed characteristic are due to Mel Hochster, but so far none has proved successful; see, for example, [16]. I hope it is clear from section one that the main ideas in tight closure theory are remarkably simple and elegant, and also that they have far-reaching consequences. In section two, we will look more closely at applications of tight closure to algebraic geometry.
3
THREE APPLICATIONS OF TIGHT CLOSURE
At the beginning of the part one, we mentioned that tight closure is applicable to a wide range of problems in commutative algebra and related fields. Now we will discuss in greater detail how tight closure has increased our insight in three areas of algebraic geometry: adjoint linear systems (Fujita's Freeness Conjecture), vanishing theorems for cohomology (Kodaira Vanishing), and singularities. We will mainly discuss the first of these, giving a tight closure proof of Fujita's freeness conjecture
for globally generated line bundles, but we point out connections with the other two topics as they arise. In all three areas, characteristic p methods are used to prove characteristic zero theorems. The unifying theme for the tight closure approach to these three problems is the action of the Frobenius operator on local cohomology. Reduction to Characteristic p.
Reduction to characteristic p is easiest to understand by example. Say we want to study the affine scheme associated to the ring
(x3 + y3 + z3)' We instead consider the "fibration" Spec
ff'fj
(x3 + y3 + z6)
-»SpecZ.
The fiber over a closed point (p) £ Spec 2, is the characteristic p scheme
Z/(p)[x,y,z] SP
(l3 + y3 + Z3) '
whereas the fiber over the generic point (0) 6 Spec TL is the original scheme
Qfr.y.*] gccc "Pec (a:3 + y3 + z3)'
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For the sorts of questions we are interested in here (which are ultimately cohomological) the following philosophy holds: what is true for the generic fiber is true for a Zariski dense set of closed fibers, and conversely, what is true for a Zariski dense set of closed fibers is true for the generic fiber. So in order to study the ring X lip fnr aa "crpnprir" e rincr ring "&I(P){ s s
' '—^-,
(xd + jr + z6)
is obtained from Spec /xs^'j^3\ by the flat base change Q —> C. Again, from the point of view of the types of questions we will consider, we might as well study Sx3+y3+z3\' and nence (Js+ a.ffj for a "generic" p. The philosophy holds for any scheme of finite type over a field of characteristic zero. For example, if we are interested in the ring rPU ,, ^
R=
(irx3 + VTfy3 + z3) '
we set A = Z[TT, \/l7] and consider the fibration
A[x,y,z] Spec ———— __—-——— —> Spec A. 3 (nx + VlJy3 + z3)
Again, we might as well study the prime characteristic ring -.— , , fe^, , \T{X
-f"VJ-'y
~TZ
3N
, where
)
/j, is a generic maximal ideal in A. Each A//J, is a finite field, so these closed fibers are all " characteristic p models" , for varying p, of the original ring R. In general, if
where A; is a field of characteristic zero, we let A = Zfcoefficients of the Ff's] C k and set
A
A[xi,...,xn]
~ TWi——FT' V - ^ l j • • • i r) r
Then the map
Spec RA -> Spec A (or the map A -^ RA) will be called a family of models for Spec R (or R). The generic fiber is the original scheme Spec R (after extending the field if necessary) and a generic (or typical) closed fiber is a characteristic p model of Spec R. We will prove theorems about R by establishing the same statement for a generic characteristic p model of R, that is, "for all large p." The idea of a family of models can be used to define concepts in characteristic zero which seemingly only make sense in prime characteristic. For example, we can define F-regularity and F-rationality for finitely generated algebras over a field in this way.
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DEFINITION 3.1 Let R be a finitely generated algebra over a field of characteristic zero. Then R is said to have F-regular type if R admits a family of models A —> RA in which a Zariski dense set of closed fibers are F-regular. (This does not depend on the choice of the the family of models.) Similarly, we can define weakly F-regular type, F-rational type, or F-split type for any finitely generated algebra over a field of characteristic zero. (In characteristic p, F-split means that the Frobenius map splits, that is, Rp C R splits as a Rp-module map.) There is a subtlety in the meaning of F-regularity for algebras of characteristic zero. As we've said in part one, the operation of tight closure can be defined for any ring containing a field, so it makes sense to define a finitely generated algebra over a field of characteristic zero to be weakly F-regular if all ideals are tightly closed. This is a priori different from the condition of weakly F-regular type. We expect that these notions are equivalent, but this remains unsolved. See [17]. The notions of F-rational type and F-regular type turn out to be intimately connected with the singularities that come up in the minimal model program. The
first theorem in this direction explains the name " F-rational". THEOREM 3.2 [40], [11] A finitely generated algebra over a field of characteristic zero has F-rational type if and only if it has rational singularities. The concept of rational singularities is very important in birational algebraic
geometry. Recall that by definition, a ring R has rational singularities if and only if it is normal and it admits a desingularization X for which Hl(X, Ox] — 0 for all » > 0.) We will not dwell on this theorem here, rather refering the the papers [40] and [11] in the bibliography. Later, we will later mention some ideas in the proof. Now we move on the application of tight closure to Fujita's freeness conjecture, where many related ideas appear. Application of Tight Closure to Adjoint Linear Series.
Let X be a smooth projective variety of dimension d, and let £ be an ample invertible sheaf on X. We are interested in the adjoint line bundles aix <8> £n,
for n > 0. Because £ is ample, we know that for large n, this adjoint bundle is globally generated. Fujita's freeness conjecture provides an effective version of this statement. Fujita's Freeness Conjecture. With X and C as above, the sheaf wx <8> £d+1 is globally generated.
The conjecture is known in characteristic zero in dimension up to four [37], [7], [26]. See [28] for a survey. In arbitrary characteristic, the best that is known is given by the following theorem.
THEOREM 3.3 [41] If X is a smooth projective variety of dimension d and C is a globally generated ample line bundle on X, then uix ® /C d+1 is globally generated.
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See [44] for a recent improvement of this result. Our next task is to prove this theorem, that is, to establish Fujita's Freeness Conjecture for globally generated line bundles. This will give a good overview of some of the methods that can be used in applying tight closure to algebro-geometric questions.
If X has characteristic zero, the first step is to reduce to the characteristic p case using the standard technique we described. So it is enough to prove the theorem in the case that X has prime characteristic. A good way to study an ample line bundle on a projective variety X is to build the section ring This is a finitely generated, N-graded ring whose associated projective scheme recovers X . Its dimension is d+1. Assuming that X is irreducible, every section ring S will be a domain. If X is smooth, then 5 has (at worst) an isolated singularity at the unique homogeneous maximal ideal m. The invertible sheaf £" corresponds to the graded S-module S(n), the S-module S with degrees shifted by n. Fujita's Freeness Conjecture is equivalent to the following more commutative algebraic statement. Fujita's Freeness Conjecture in terms of local cohomology // (S, m) is a section ring with an isolated non-smooth point, then H^'1(S) has the following property: there exists an integer N such that for all 77 G H^1 (S) of degree less than N, r] has a non-zero S-multiple of degree —d— I.
The proof of the equivalence of this statement with Fujita's Conjecture is not difficult. This is essentially the dual statement (using Matlis duality for S or Serre duality for X). Details can be found in [41]. To prove Fujita's Conjecture, we will tackle this local cohomological conjecture. First we describe a convenient way to think about elements in the local cohomology module ff£ +1 (S). Let XQ, X i , . . . , x,j be a system of parameters for S of degree one. Such a system of parameters exists by our assumption that L is globally generated (after enlarging the ground field if necessary) . The local cohomology module H^l(S) can be computed as the cokernel of the following map
where x denotes the product XQXI ... x
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We represent elements of H^~1(S) by fractions [^] , with the square bracket
reminding us of the equivalence relation on fractions. If the degree of r\ is —n, we see that —n = deg z — t(d+l). It is easy to see that if z 6 (XQ, x\, . . . , x*d), then 77 = [^-] must be zero, by thinking about the image of the map above. Unfortunately, the converse is false. However, we have the following interesting observation.
LEMMA 3.4 // 77 = [£] = 0, then z
,.
The Lemma is easily proved: if 77 = [^] = 0, then this means that for some integer s, we have 7
z
TS 7
where now x* z € (XQ+S , • • . , x+s). Thus
so by colon capturing, z £ (x*0, . . . , xfd)* . The Lemma is proved. The Frobenius Action on Local Cohomology.
The Frobenius action on local cohomology is the main idea in the proof of Fujita's Freeness Conjecture for globally generated line bundles, and in the proof of the equivalence of rational singularities with F-rationality. It is also the central point in the relationship between tight closure and the Kodaira Vanishing theorem. The idea of using the Frobenius action on local cohomology to study tight closure
first appears in the work of Richard Fedder and Kei-ichi Watanabe [9] . The Frobenius action is easy to understand. Indeed, Frobenius acts in a natural way on each module Sx, ...Xtr in the Cech complex denning the local cohomology modules; it simply raises fractions to their p — th powers. This action obviously commutes with the boundary maps, so that it induces a natural action on the local cohomology modules. In particular, the Frobenius action on H^l(S) is given by
Using this, it makes sense to define tight closure for submodules of H^l(S) by mimicking the definition for ideals. For example, we can define the tight closure of the zero submodule in
0* = {77 £ H^l(S] | there exists c / 0 with erf' = 0 for all e » 0}. The tight closure of zero in H^"l(S] is an important gadget. One can show that it is the unique maximal proper submodule of H^f1 (S) stable under the action of Frobenius [40]. Returning to the proof of Fujita's Freeness Conjecture, we observe the following two facts.
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(1) 7 ? = [ | r ] e O *
(2) Any test element c kills 0*. These two facts are straightforward to prove using nearly the same argument as in the proof of the Lemma above. Now the proof can be summarized in two main steps. First we show that if T] 6 H^f1(S) does not have a multiple of degree — d — 1, then 77 is in 0*. Next we show that 0* vanishes in all sufficiently small degrees. Obviously, upon completion of these two steps, the proof is complete.
Step One: if r] 6 H^'1(S) does not have a multiple of degree —d— I , then rj is in 0* . The main point is colon capturing. Assume on the contrary, that an element r] = [ jf] of degree — n has no non-zero multiple of degree — d — 1. This means that
every element of 5 of degree n — d — I must kill 77. In particular,
By the Lemma, this means that
or in other words, (Z. (X (vt, . . . , X • vt\* Z7 fc . ((XVn Q , . . . , Xf dJ \) 0 d)
H
~d~1
Now we use colon capturing. We manipulate the parameters XQ, . . . , xj formally as if they are the indeterminants of a polynomial ring, in which case the colon ideal (ignoring the *) would be easily computed to be
Colon capturing says that the actual colon ideal is contained in the tight closure of this "formal" colon ideal, that is,
But note that the degree z is (d + l)t — n (because i] = [JV] has degree -n = deg z-(d+ l ) t ) . Thus
A moment's thought reveals that this forces
Indeed, if
e (4, we see immediately that because the degree of c is fixed, the degrees of the generators of [(xo, • • •,«d) des2+1 ]'- 9 -' are much larger than the degree of czq , so that czq must
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in fact be in the ideal (XQ, . . . , ar^)^ for large q = pe . But by Fact (1) above, then
we see that and the proof of step one is complete.
Step two: 0* vanishes in sufficiently
small degrees.
The point is to consider the test elements of S. Because X is smooth, the section ring S has an isolated singularity. This means that the denning ideal of the nonregular locus of 5 is m-primary. As we mentioned in Lecture 1, this implies that the test ideal of S (of all elements that "witness" all tight closure relations) contains an m-primary ideal. But according to Fact 2 above, the test ideal of S annihilates 0*, so that 0* is killed by an m-primary ideal. This says that 0* has finite length, so of course, it must vanish eventually in all degrees sufficiently small. This completes the proof of step two, and thus the proof of Fujita's Freeness Conjecture for globally generated line bundles. Experts will notice that the argument above does not really require that X be smooth. We used smoothness only in Step 2, to conclude that 0* is finite length. But 0* is of finite length more generally, and is in fact equivalent to the variety X being F-rational (or F-rational type in characteristic zero). Thus Fujita's Freeness Conjecture holds for any globally generated ample line bundle on a projective F-
rational (type) variety. We should remark that Fujita's Freeness Conjecture for globally generated line bundles can also be proved, in characteristic zero, using the Kodaira vanishing theorem. As far as I know, however, tight closure provides the only proof in prime characteristic. Interestingly, the Frobenius action on local cohomology seems to act as a substitute for Kodaira Vanishing. There is a good reason for this: it turns out that Kodaira vanishing theorem is equivalent to a statement about the action on Frobenius on local cohomology modules. Tight Closure and Kodaira Vanishing.
Recall the classical Kodaira Vanishing Theorem: Kodaira Vanishing If X is a smooth protective variety of characteristic zero, and jC is any ample invertible sheaf on X, then Ht(X,C~1} = 0 for all i < dimX. The Kodaira Vanishing Theorem is false in characteristic p, although it can be proved by reduction to characteristic p [6]. See also [8]. Let S = (£>n>oH°(X,£n) be the section ring of the pair (X,£). Unwinding definitions using the point of view that local cohomology can be computed from the Cech complex of the Ox -algebra ®£ n , Kodaira Vanishing is seen to be equivalent to 11^(3) vanishes in negative degree for all i with I < i < dim S. Because S has (at worst) an isolated non-Cohen-Macaulay point at m, we know that each Hlm(S) is supported at m, and hence must vanish in degrees sufficiently small. So we could state the Kodaira Vanishing Theorem as follows: the Frobenius
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action on a dense set of characteristic p models for S is injective in negative degrees
on H*m(S), for I
where D is the sum of the degrees of the X{ 's. This theorem is equivalent to the Kodaira Vanishing Theorem. Just as Kodaira Vanishing can fail in prime characteristic, so can this tight closure statement. However, the statement holds when S is a generic characteristic p model for a section ring of characteristic zero, that is, "for large p." By allowing the possibility that we have a full system of parameters in the above version of the Kodaira Vanishing Theorem, we get the strong Kodaira Vanishing Theorem. In fact, if XQ, x\, . . . , xj is a full system of parameters for a section S as above, we get a more precise statement. Strong Kodaira Vanishing [25] [11]. Let S be an N-graded ring over a field of characteristic zero, and let XQ, xi, . . . ,xj be a full system of (homogeneous) parameters for S, with degXi ^> 0. Then d (XQ, . . . , Xd)* = Y^(X°> • • • , £ ; , • • • , Xd)* + S>D j=0
where D is the sum of the degrees of the X{ 's. The reason we get equality here is that S>rj is contained in (XQ, . . . , x,i)* , as can be verified with the Brianoon-Skoda theorem (Property 4). It is possible to say precisely how large the degrees of the a;,-'s must be in the statements of Kodaira and strong Kodaira vanishing in terms of tight closure. In
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both theorems, each Xf should have degree larger than a, where a is the a-invariant of S. By definition (due to Goto and Watanabe), the a-invariant is the largest integer n such that H^mS(S) is non-zero in degree n. The strong form of Kodaira Vanishing is conjectured in [25], where the idea of the "monomial property of a d+ sequence" due to Goto and Yamagishi is used. It is proved in [25] for rings of dimension two, from which it is shown that the Kodaira Vanishing Theorem follows for any normal surface of dimension two. In full generality, however, the statement was not known until Nobuo Kara proved the injectivity of the Frobenius action on the negatively graded part of local cohomology [11]. Hara has since greatly generalized his work; see [13]. Tight Closure and Singularities.
Finally, we summarize some more connections between tight closure and singularities in algebraic geometry. Let X be a normal variety of characteristic zero. Assume that X is Q-Gorenstein, that is, that the reflexive sheaf u>x represents a torsion element KX in the (local) class group of X. In other words, the Weil divisor class KX is assumed to have a multiple which is locally principal. Consider a desingularization X —>• X of X, where the exceptional divisor is a simple normal crossings divisor with components EI, ... ,En. Write
for some unique rational numbers a;. To understand this expression, suppose that rKx is locally principal, so that it makes sense to pull it back; then compare to rKx. The difference is some divisor supported on the exceptional set, hence of the form Y^i-i miEi- Dividing by r, we arrive at the above expression, where 'equality' means numerical equivalence of Q-divisors. See [27]. In general, the a,-'s can be any rational number, although if X is smooth, we can easily see that each a, will be a positive integer. This leads us to the following restricted class of singularities.
DEFINITION 3.5 The variety X has log-terminal singularities if all a; > -1, and has log-canonical singularities if all a,- > 1. (This is independent of the choice of desingularization.) The relationship to tight closure is is evidenced by the following theorem.
THEOREM 3.6 Let X be a normal Q-Gorenstein variety of characteristic zero. X has F-regular type if and only if X has log-terminal singularities. This theorem follows immediately from the equivalence of rational singularities and F-rational type discussed earlier, using the "canonical cover trick". Indeed, assuming X is local, set
y = Spec {Ox 0 Ox(Kx) ® OX(ZKX) ® . ..Ox((r - 1}KX)}
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where r is such that Ox (rKx) is isomorphic to Ox via a fixed isomorphism (so that we can define a ring structure on Ox ®OX(KX)®OX(2KX)® • • • Ox((r-l)Kx)}. The natural map Y —^ X is called the canonical cover of X. It is easy to check that when X is Cohen-Macaulay, the canonical cover Y is Gorenstein, and that the map is etale in codimension one. With these properties, it is not hard to show the following two facts: (1) (Kawamata) Y has rational singularities if and only if X has log-terminal singularities. (2) (K.-i. Watanabe) Y has F-rational type if and only if X has F-regular type. Thus the equivalence of F-regular type with log-terminal singularities follows from the equivalence of F-rational type with rational singularities. There are some subtleties involved in the argument using the canonical cover. Watanabe's argument shows F-rationality for Y is equivalent to strong F-regularity for X. Strong F-regularity is a technical condition conjectured to be equivalent to weak F-regularity (when both are defined), introduced because it, unlike weak F-regularity, is easily shown to pass to localizations [19]. However, in the case of Q-Gorenstein rings, weak and strong F-regularity turn out to be equivalent [33]. The first proof that F-regular type Q-Gorenstein singularities are log-terminal is due to Kei-ichi Watanabe and uses a different argument [46]. This different argument also produces the following nice result. THEOREM 3.7 [46] Let X be a variety satisfying the conditions above. If X is of F-split type, then X has log-terminal singularities. (Recall, a local ring of characteristic p is F-split if the inclusion Rp '-^ R splits as a map of Rp modules.) A very interesting open problem that has deep connections with number theory is the following. Open Problem. If X has log-canonical singularities, does X have F-split type? Further Reading on Tight Closure.
The original tight closure paper of Hochster and Huneke [18] is still an excellent introduction to the subject. There are also a number of expository articles on tight closure. Craig Huneke's book Tight Closure and its Applications [23] is an good place for a beginning commutative algebra student to learn the subject; it contains several applications more or less disjoint from the ones discussed in detail here. It also contains an appendix by Mel Hochster [17] discussing tight closure in characteristic zero. Another nice survey is [15], which contains a list of open problems; although the article is now seven years old, many of these problems remain open. A more recent view is provided by the expository article [5]. The article [42] is a survey written for algebraic geometers. Huneke's "Tight Closure and Geometry" is another nice read for algebraists [24]. All these sources, but especially [23], contain long bibliographies to direct the reader to numerous research articles on tight closure.
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References [1] I. Aberbach. Tight closure in F-rational rings. Nagoya Math. J. 135, 1994, 43-54. [2] I. Aberbach, M. Hochster and C. Huneke. Localization of tight closure and modules of finite phantom projective dimension. J. Reine angew., 434, 1993, 67-114. [3] I. Aberbach, C. Huneke and K.E. Smith. A Tight Closure Approach to Arithmetic Macaulayfication. Illinois Journal of Math, 40, 1996, 310-329. [4] J. Briangon, H. Skoda. Sur la cloture integrale d'un ideal de germes de fonctions holomorphes en un point de Cn. C. R. Acad. Sci. Paris Ser. A, 278,
1974, 949-951. [5] W. Bruns. Tight Closure. Bulletin Amer. Math. Soc., 33, 1996, 447-458. [6] P. Deligne, L. Illusie. Relevements modulo p2 et decomposition du complexe de de Rham. Inventiones Math., 89, 1987, 247-270. [7] L. Ein, R. Lazarsfeld. Global generation of pluricanonical and adjoint linear series on smooth projective three-folds. Jour, of Amer. Math. Soc., 6, 1993,
875-903. [8] H. Esnault, E. Viehweg. Lectures on vanishing theorems. Birkhauser DMV
series 20, 1992. [9] R. Fedder, K. Watanabe. A characterization of F-regularity in terms of Fpurity. in Commutative Algebra in MSRI Publications No. 15, Springer-
Verlag, New York, 1989, 227-245. [10] S. Goto, K. Yamagishi. The theory of unconditioned strong d-sequences and modules of finite local cohomology. preprint [11] N. Hara. A Frobenius characterization of rational singularities. American
Journal of Math, 120, 1999, 981-996. [12] N. Hara. Classification of two-dimensional F-regular and F-pure singularities.
Adv. Math., 133, 1998, 33-53. [13] N. Hara. Geometric interpretation of tight closure and test modules . preprint,
1999. [14] N. Hara. A characteristic p proof of Wahl's vanishing theorem for rational surface singularities. Arch. Math (Basel) volume 73, 1999, 4, 256-261. [15] Melvin Hochster. Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure (in Commutative algebra: syzygies, multiplicities,
and birational algebra). Contemp. Math., 159, 1994, 173-196. [16] M. Hochster. Solid closure (in Commutative algebra: syzygies, multiplicities, and birational algebra). Contemp. Math., 159, 1994, 103-172.
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[17] M. Hochster. The notion of tight closure in equal characteristic zero. Appendix to 'Tight Closure and Its Applications', by C. Huneke, GBMS lecture notes, 88, 1996. [18] M. Hochster, C. Huneke. Tight closure, invariant theory, and the Briancon-
Skoda theorem. Jour. Amer. Math. Soc., 3, 1990, 31-116. [19] M. Hochster, C. Huneke. Tight closure and strong F-regularity. Memoires de la Societe Mathematique de France, 38, 1989, 119-133. [20] M. Hochster, C. Huneke. Infinite integral extensions and big Cohen-Macaulay algebras. Annals of Math., 135, 1992, 53-89. [21] M. Hochster, G. Huneke. F-regularity, test elements and smooth base change. Trans. Amer. Math. Soc., 346, 1994, 1-62. [22] C. Huneke. Uniform bounds in Noetherian rings. Invent. Math., 107, 1992, 203-223. [23] C. Huneke. Tight Closure and its Applications. CBMS Lecture Notes in Mathematics, 88, American Math. Soc., Providence, RI, 1996. [24] C. Huneke. Tight Closure and Geometry. Lecture notes from Commutative algebra summer school in Barcelona, preprint. [25] C. Huneke, K.E. Smith. Tight Closure and the Kodaira vanishing theorem. J. reine angew. Math., 484, 1997, 127-152. [26] Y. Kawamata. On Fujita's freeness conjecture for threefolds and fourfolds. Math. Ann., 308 (3), 1997, 491-505. [27] Y. Kawamata., K. Matsuda , K. Matsuki. Introduction to the minimal model program. Advanced Studies in Pure Math., Algebraic Geometry, Sendai 1985, 10, 1987, 283-360. [28] Kollar. Singularities of Pairs, in Algebraic geometry—Santa Cruz 1995, 289325, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997. [29] K. Kurano. On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal, With an appendix by Kikumichi Yamagishi. J. Algebra,
190 (2),1997,405-434. [30] J. Lipman, A. Sathaye. Jacobian ideals and a theorem of Briangon-Skoda. Michigan Math. J., 28, 1981, 199-222. [31] J. Lipman, B. Teissier. Pseudo-rational local rings and a theorem of BriangonSkoda about integral closures of ideals. Michigan Math. J., 28, 1981, 97-116. [32] G. Lyubeznik, K.E. Smith. Weak and Strong F-regularity are equivalent for graded rings. Amer. J. Math. 121(6), 1999, 1279-1290. [33] B. MacCrimmon. Strong F-regularity and boundedness questions in tight closure, PhD. Thesis. University of Michigan, 1996.
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[34] H. Matsumura. Commutative Ring Theory. Cambridge University Press, Cambridge, 1986. [35] V.B. Mehta, V. Srinivas. A characterization of rational singularities. Asian J. Math. 1, 1997, 2, 249-271. [36] P. Monsky. The Hilbert-Kunz function. Math. Ann., 263, 1983, 43-49. [37] I. Reider. Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. of Math., 127, 1988, 309-316. [38] K.E. Smith. Tight closure of parameter ideals. Invent. Math., 115, 1994, 4160. [39] K.E. Smith. The D-module structure of F-split rings. Math. Research Letters, 2, 1995, 377-386. [40] K.E. Smith. F-rational rings have rational singularities. Amer. Jour. Math.,
119 (1), 1997, 159-180. [41] K.E. Smith. Fujita's conjecture in terms of local cohomology. Jour. Algebraic Geom., 6 (3), 1997, 417-429. [42] K.E. Smith. Vanishing, singularities and effective bounds via prime characteristic local algebra, in Algebraic geometry—Santa Cruz 1995, 289-325, Proc. Sympos. Pure Math., 62, Part 1 (Please also see the erratum at http://www.math.lsa.umich.edu/ kesmith) Amer. Math. Soc., Providence,
RI, 1997, 289-325. [43] K.E. Smith. Tight closure commutes with localization in binomial rings. Proc. Amer. Math. Soc. 2000 (to appear). [44] K.E. Smith. A tight closure proof of Fujita's Freeness Conjecture for globally generated line bundles. Math. Annalen 317 Issue 2, 2000, 285-293. [45] K.E. Smith, M. Van den Bergh. Simplicity of rings of differential operators in prime characteristic. Proc. London Math. Soc., 75 (3), 1997, 32-62. [46] Kei-ichi Watanabe. F-purity and F-regularity vs. Log-canonical singularities, preprint.
Eisenbud-Goto inequality for Stanley-Reisner rings NAOKI TERAI, Faculty of Culture and Education, Saga University, aga 840-8502, Japan, [email protected]
ABSTRACT: We give an upper bound for the regularity of the sum of squarefree monomial ideals. Then we prove the Eisenbud-Goto inequality : reg /A < deg&[A] — codim&[A] + 1 for a pure and strongly connected simplicial complex A. Finally, we classify the simplicial complexes for which the upper bound is attained.
INTRODUCTION As the theory of minimal free resolutions is developed, the regularity has been recognized to be an important invariant. As for the regularity of monomial ideals, many interesting estimates are obtained in [Ho-Tr]. They give the upper bound for regularity in terms of the arithmeticdegree or the degrees of minimal generators of monomial ideals. While their method is algebraic, a combinatorial and topological method is adopted in [Fr-Te], based on Hochster's formula and Alexander duality. We give the same bound for the regularity using the arithmetic-degree. This paper is a continuation of [Fr-Te]. As for general homogeneous ideals, the following Eisenbud-Goto conjecure has been giving one of motivation to study:
CONJECTURE 0.1 ([Ei-Go, Introduction]). Let A = k[xi,x2, • • . ,xn] be the polynomial ring in n-variables over a field k, and P a prime ideal of A which contains no linear form. Then reg P < degA/P - codimA/P + I .
Now it is actively studied in algebraic geometry and in commutative algebra. See, e.g., [Kw] and [Mi-Vo]. In this paper we prove a monomial ideal analogue of the above conjecture, which is suggested by Eisenbud, i.e.,
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THEOREM 0.2 (Eisenbud 's Conjecture). Let k be a field. Let A be a pure and strongly connected simplicial complex. Then reg /A < deg k[A] — codim&[A] + 1. We announced this result in [Fr-Te] and gave a sketch of the proof there. Here we give a rearranged and detailed proof, which is necessary for the later argument. As an application to the original Eisenbud-Goto Conjecture, we have:
COROLLARY 0.3 Let A — k[xi,xz, • • • , xn] be the polynomial ring in n-variables over a field k, and P a prime ideal of A which contains no linear form. Assume A/inP is reduced, where inP is the initial ideal of P with respect to some term order. Then reg P < deg A/P - codimA/P + 1. In §4 we classify the simplicial complexes for which the Eisenbud-Goto bound is attained :
THEOREM 0.4. Let k be afield. Let A be a (d— 1) -dimensional pure and strongly connected complex. We put r = reg/A. Then reg/A = deg k[A] — codim&[A] + 1.
if and only if A satisfies the following condition: (1) A is a (d — 1)-tree which is not the (d — l)-simplex if r=2. (2) A = A'(v —>• w) for some (d — l]-tree A' and for some separated v, w (E V(A')
if r=3. (3)A S dA(r) * A(d- r + 1) + ((d- l)-branches) i f r > 4. See §1 and §4 for undefined terminology. The author would like to appreciate D. Eisenbud for informing the conjecture and giving him helpful suggestions.
1
PRELIMINARIES
We first fix notation. Let N(resp.Z) denote the set of nonnegative integers (resp. integers). Let | S \ denote the cardinality of a set S. We recall some notation on simplicial complexes and Stanley-Reisner rings. We refer the reader to, e.g., [Br-He], [Hi], [Hoc] and [St] for the detailed information about combinatorial and algebraic background.
A (abstract) simplicial complex A on the vertex set V = { x i , X 2 , • • • ,xn} is a collection of subsets of V such that (i) {x,} £ A for every 1 < i < n and (ii) F G A, G C F => G £ A. The vertex set of A is denoted by V(A). Each element F of A is called a face of A. We call P G A an i-face if F \= i + 1 and we call a maximal
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face a facet. Let F be a face but not a facet. We call F free if there is a unique facet G such that F C G. If {x,} is free, we call Xi free. We define the dimension of F 6 A to be dimF = F \ — 1 and the dimension of A to be dim A = max{dim F F G A}. We say that A is pure if all facets have the same dimension. In a pure (d — l)-dimensional complex A, we call (d — 2)- face a subfacet. We say that a pure complex A is strongly connected if for any two facets F and G, there exists a sequence of facets p' p> . . ., r pm — rF _— rQ,r\, —/7 u-
such that Fi-i Pi Fj is a subfacet for i = 1,2,...,m. Let /, = /»(A), 0 < 2 < d — 1, denote the number of i-faces in A. We define /_i = 1. We call /(A) = (/o, / i , . . . , /rf-i) the f-vector of A. Define the h-vector h(A) = (h0,hi,...,hd) of A by d '
d J
^
i=0
'
/ -<
l
i=0
Let Hi(A;k) denote the z'-th reduced simplicial homology group of A with the coefficient field k. Let A = k [a; 1,35 2 , . . . ,xn] be the polynomial ring in n-variables over a field k. Define /A to be the ideal of A which is generated by square-free monomials Xi1X{2 • • • X i r , I < ii < i-2 < • • • < ir < n, with {z'i, i % , . . . ,ir} $ A. We say that the quotient algebra At [A] := A/I& is the Stanley-Reisner ring of A over k. Next we summarize basic facts on the Hilbert series. Let k be a field and R a homogeneous fc-algebra. By a homogeneous A;-algebra R we mean a noetherian graded ring R — ® i>0 Ri generated by RI with R0 = k. Let M be a graded Rmodule with dim^ M,- < oo for all i 6 Z, where dim/^ M,- denotes the dimension of Mi as a k-vector space. The Hilbert series of M is defined by
It is well known that the Hilbert series F(R,t) of R can be written in the form
F(R,t) =
hit + • • • + hsts
(1-t) dimR
where ho(= 1), hi,... ,hs are integers with deg R := ho + hi + • • • + hs > 1, which is called the degree of R. The vector h(R) = (h0, hi,..., hs) is called the h-vector of R. We consider k[A] as the graded algebra A;[A] = (J)i>o ^IA]i with degarj- = 1 for 1 < j < n. The Hilbert series F(fc[A],i) of a Stanley-Reisner ring A;[A] can be written as follows:
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where dimA = d — 1, (/o, /i, . • • , f d - i ) is the /-vector of A, and (ho, hi, . . . , h^) is the ft- vector of A. It is easy to see deg&[A] = fd-i- On the other hand, the arithmetic degree of k[A] is defined to be the number of facets in A, which is denoted by a-deg&[A]. See, e.g., [Ho-Tr] for the definition of the arithmetic degree of a general ring R.
Let A be the polynomial ring k[xi, x%, . . . , xn] over a field k. Let M(£ 0) be a finitely generated graded ^-module and let
(-j)pa>i
-+ M
jez be a graded minimal free resolution of M over A. The length h of this resolution is called the protective dimension of M and denoted by h = pdM . We call f)i(M) = Ylij^z Pi,j(M) the z'-th Betti number of M over A. We define the CastelnuovoMumford regularity reg M of M by
reg M = max {j - i /?,-,j(M) ^ 0}. See, e.g., [Ei] for further information on regularity. We define the initial degree indeg M of M by indeg M = min {i Mi ^ 0} = min {jf
/? OJ (M) ^ 0}.
Let I be a natural number. We say that M satisfies (N;) condition if /3;^+ s (M) = 0 for i < /, s ^ indeg M. We denote the number of generators of M by fJ,(M) = /?o(M). The following two theorem are a starting point for our study.
THEOREM 1.1 (Hochster's formula on the Betti numbers [Hoc, Theorem 5.1]).
where
AF = {G&A GCF}.
It is easy to see:
COROLLARY 1.2. reg /A = max {i + 2 | #"i(AF; fc) ^ 0 for some _F C K}.
If F is a face of A, then we define a subcomplex link&F by
l i n k A F = { G < E A F D G = 0,-FU G e A}.
Eisenbud-Goto inequality for Stanley-Reisner rings
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THEOREM 1.3 (Hochster's formula on the local cohomology modules (cf. [St, Theorem 4.1])). ^ F(H'm(k(A}), *) = £ F6A
where Htrn(k[A\) denote the i-th local cohomology module of k[A] with respect to the graded maximal ideal m.
COROLLARY 1.4. reg /A = max {i + 2 Hi(link^F; k) ^ 0 for some F € A}. Next we recall the definition of Alexander dual complexes. For a simplicial complex A on the vertex set V, we define an Alexander dual complex A* as follows:
A* = {F C V : V\F
(£ A}.
THEOREM 1.5 [Te, Corollary 0.3]. Let k be a field. Let A be a simplicial complex. Then reg JA = Pd
2. REGULARITY OF THE SUM OF IDEALS In this section we give a upper bound for the sums of sqare-free monomial ideals. In the rest of the paper we always assume that k is a fixed field. First we prove the following proposition. It seems to be known, but we cannot find it in literature.
PROPOSITION 2.1. Let I be a monomial ideal in the polynomial ring A — k [ x i , x % , . . . , xn] and m a monomial in A. Then
p
The following proof is simplified by suggestion of Eisenbud.
Proof.
If we show that
then the mapping cone guarantees that (m)) < p d A / / + l .
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by [Ei, Exercise A.3.30]. We have
=
(m)/(lcm(m, mi), . . . ,lcm(m, S A/(m'1,...,m't)®A(m),
here I = (mi , . . . , m t ), m^ =
cm
(^' m ') _ Hence, we have only to show
Now we have (A/I)m = Am/(m\, . . ., m't)Am. Hence we have
pd A/I>pd (A/I)m =pdAm/(m'1,...,m't)Am = pd A/(m(, . . .,m't). We are done.
qed
For the regularity of the sum of square-free monomial ideals, we have the following conjecture:
CONJECTURE 2.2. Let Aj(^ 0) be a simplicial complex for i = 1,2. Then we have reg(/ Al + /A 2 ) < reg / Al + reg JA2 - 1. If /A! and /A2 are complete intersections, then the above inequality holds. The next theorem gives a weaker upper bound.
THEOREM 2.3. Let A,-(^ 0) be a simplicial complex for i = 1,2. Then we have reg(/ Al + / A2 ) < min{reg / Al + a-degfc[A2], reg /Aa + a-d
Proof. Only in this proof, we define a simplicial complex A by only the condition (ii) of the definition of a simplicial complex. We do not require the condition (i). Then we have (A*)* = A. And Theorem 1.5 also holds under this definition. By the above proposition we have
pd A/(IAl +/A 2 ) < pd Since / Al + /As = ^A!nA 2 , we have
reg (/ ( A l nA 2 )*) < reg JA; + a.-degk[A*2] by Theorem 1.5 and //(/A Z ) = a-degfcfAj]. Since we have /( A l nA 2 )* /AJ H /A*, then we have reg (/A; n /A-) < reg JA. + a-
=
Eisenbud-Goto inequality for Stanley-Reisner rings
385
Similarly we have reg (/A; f~l / A j) < reg /A. + a-deg£[A*]. Consider the exact sequence
0 -»• A/(I±. n /A;) -> A//A; 8 A//A; -> M/A; + JA;) -J- 0. By [Ei, Corollary 20.19], we have
reg >!/(/A; + /A;) < max{reg A/(I&* n /A; ) - 1, re Hence reg >i/(/A; + /A;) < minjreg /A- + a-degfc[A;] - l,re We obtained the desired result.
REMARK. Since the inequality reg 7A < a-degfc[A] holds (cf. [Ho-Tr] and [Fr-Te]), Theorem 2.3 is weaker than Conjecture 2.2.
3. EISENBUD-GOTO INEQUALITY In this section we prove Eisenbud-Goto inequality for Stanley-Reisner rings of pure and strongly connected simplicial complexes. First we prove a lemma which is necessary for inductive argument.
LEMMA 3.1. Let A be a pure and strongly connected simplicial complex. Then there exists a facet F G A such that
A' := {H € A 1 H C G for some facet G(^ F) e A} is pure and strongly connected. Proof. We define a graph GA corresponding to A as follows: The vertex set V(GA) consists of {yp F is a facet of A}. The edge set E(G&) is defined by: {yF,yo} G E(G&) if and only if F n G is a subfacet. If A is pure and strongly connected, GA is connected. It is well known that there exists a vertex yp G V(G A ) such that GV(G A )\{F} is connected. Then A' is pure and strongly connected, qed Now we prove the main result in this section. THEOREM 3.2(cf. [Fr-Te, Theorem 4.1]). Let A be a pure and strongly connected simplicial complex. Then we have reg/A < deg A; [A] — codim&[A] + 1.
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Proof. Let V be the vertex set of A. Put | V |= n and dirnArfA] = d. We prove the theorem by induction on the number fd-i of facets in A. First if codim Ar[A] < 1, then k[A] is a hypersurface. In this case the theorem is clear.
Suppose codim A; [A] > 2 and f^-i > 2. By the above lemma, there exists a facet F € A such that A' := {H 6 A | H C G for some facet G(^ F) e A} is pure and strongly connected. Denote by V' the vertex set of A' and by f'd_1 the number of facets in A'. There are two cases. Case 1 V ^ V. Put V\V = {v}. For W C V with v <£ W we have Aw = A'w. On the other hand, for W C V with v £ W, Hi(Aw;k) S Hi(A'w\{v}; k) for i>l. Since reg /A = max{e + 2 | //j(An/; A;) ^ 0 for some W C V}, we have reg/A
=
re:
Case 2 F = I/'. We have reg /A = pd fe[A*] . Now we see that fc[A*] = fc[(A')*]/(m), where m = rL,eK\,F ^j- By Proposition 2.1, we have reg /A
<
reg /A' + 1
< fd-i-(n-d) + 2 = fd-i-(n-d) + l. qed
COROLLARY 3.3. Let A be a simplicial complex such that codim&[A] > 2. Assume /A satisfies (N2) condition. Then we have pdk[A] < H(IA) - indeg/A + 1. Proof. If JA satisfies (N2) condition, then A:[A*] satisfies (82) condition by [Ya, Corollary 3.7] and then A* is pure and strongly connected. If A* is pure, then deg&[A*j = //(/A)- If codimfe[A] > 2, then indeg/A = codimfc[A*]. We are done by Theorems 1.5 and 2.4. qed
Proof of Corollary 0.3. Put JA = in/3. Then by [Ka-St, Theorem 1], A is pure and strongly connected. By Theorem 3.2, we have regP
<
reg/A
< deg A;[A] - codimA;[A] + 1 = deg A/P - codimA/P + 1. qed
Eisenbud-Goto inequality for Stanley-Reisner rings
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4. EQUALITY CASE
In this section, we classify pure and strongly conneceted simplicial complexes A which satisfy reg/A = deg A;[A] — codimfcfA] + 1, and give some characterization for such complexes. First we introduce some notation. Put [TO] = { 1 , 2 , . . . , ™ } . We denote the elementary (m- l)-simplex by A(m) = 2^ and put A(0) = {0}. We put <9A(m) = 2^ \ {[m]}j which is the boundary complex of A(m). Let A; be a (d— l)-dimensionalpure simplicial complex for i = 1,2. If A i f l A 2 = 2F for some F with dim F = d — 2, we write AI UF A 2 for AI U A 2 . We sometimes write AI U* A 2 for AI \Jp A 2 if we do not need to express F explicitly. We define a (d — l)-tree inductively as follows. (1)A(cO is a (cf-l)-tree. (2)if T is a (d - l)-tree, then so is T U* A(d). If TI , T 2 , . . . , Tm are (d — l)-trees, we abbreviate A U* TI U* T2 U* • • • U* T m as A + ((d — l)-branches). Let A be a (d — l)-dimensional pure and strongly connected complex. Take v, w 6 ^(A). We say v and w are separated in A if {v, w} (£ A and that there exists no subfacet F in A with {v} U F, {w} U F 6 A. If v and w are separated in A, We denote A(v —» w) for the abstract simplicial complex which is obtained by substitution of w for every v in A. The vertex set of A(v —» w) is V(A) \ {v}. By Lemma 3.1 we know that every (d — l)-dimensional pure and strongly connected simplicial complex can be constructed from the (d— l)-dimensional elementary simplex A (of) by a succession
A(d)
= AI ->• A 2 - » • • • - > • Afd_,
of either of the following operations : (1)Ai+i = A, UF' 2 F , where x £ V(Ai), F' is a subfacet of A,- and F = F' U {x}. (2)Aj+i = (A, UF' 2F)(x -> y), where x ^ V(A;), F' is a subfacet of A,- and y £ V'(Aj) such that a; and t/ are separated and F = F' U {x}. Let A; be a simplicial complex for i = 1,2 such that V(Ai) D V(A 2 ) = 0- We define the simplicial join AI * A2 of A j and A2 by
A! * A 2 = {FU G | F € AI, G 6 A 2 }.
LEMMA 4.1. Let A be a (d—1)-dimensional pure and strongly connected complex. We assume reg/A = deg k[A] - codimA;[A] + 1 = 3.
Then A can be expressed as follows: A = A'(x -> y) * A(d - s) + ((d - l)-branches)
for some (s — l)-tree A' and for some separated x, y 6 V(A') with Hi(A'(x —)•
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Proof. We may assume A has no branches. Then A can be expressed as A = A'(x —>• y), where A' is a (d — l)-tree and x, y £ V(A') are the only free vertices in A'. Let F be the facet with x € F and G the facet with y 6 G in A'. Let GA' be the graph introduce in the proof of Lemma 3.1. Since A' is a (d— 1)tree with only two free vertices x, y, then GA; is a line with the end points yp and 2/G • Hence there exists a sequence of facets
F = F0,Fi,...,Fm=G such that Fi-i fl Fi is a subfacet for f = 1, 2, . . . , m. Then FQ, F I , . . . , Fm are all facets in A'. We put W — F n G. If we have z £ F; and 2: ^ Fi+i, then z ^ Fi+2, since A' is a (cf — l)-tree. Then we have W C F, for i = 0,1,2, ... ,m. Then we have A' = AI *2 V K , and A'(x -> y) = Ai(a; -> j/) * 2 H/ , where AI is an (s - l)-tree and s = d— W \. It is easy to check that Ai(or —> y) is contractible to the circle S1.
qed
THEOREM 4.2. Let A be a (d — I) -dimensional pure and strongly connected
complex. We put r = reg/A . Then
reg/A = deg A; [A] — codimfe[A] + 1. if and only if A satisfies the following condition: (1) A is a (d — I) -tree which is not the (d — l)-simplex if r=2. (2) A = A'(t> —> to) /or some (d — I) -tree A' ancf for some separated v, w £ j/ r=5. S dA(r) * A(d - r + 1) + ((d - l)-branches) ifr>4. Proof. First we assume that A satisfies r = deg k[A] — codimfc[A] + 1. We use induction on r. If r = 2, then A is a [d — l)-tree by [Frj. If r = 3, then by the procedure to construct pure and strongly connected complexes, (3) is easy to check. We assume r = 4. We prove the statement by induction on dim A. We may assume A has no branches. Then A is of the form
A = (A'UF- 2F)(x->y) where A' is pure and strongly connected and F — F' U {x} is a facet of A and y € V(A') such that x and y are separated in A' Up> 2F. We have degfc[A'] codimfcfA'j + 1 = 3 and hence from the proof of Theorem 3.2 we get reg/A< = 3. By the assumption of induction and the previous lemma, A' is of the form A' = A"(w —> w) * A(d — s) + ((d — l)-branches) for some (s — l)-tree A" and for some separated
v,w € V(A") with HI(&"(V -> w);k) ^ 0. If x <£ V(A"(v -> to) * A(d - s)) or if F' £ A."(v —>• w) * A(d — s), then the branch part can be contractible to a 1-dimensional subcomplex, then we have ^(A^;^) = 0 for each X C V'(A). Contradiction. Since A has no branches, we have A' = A"(i> —>• w) * A(d — s) and x € V(A"(v -> w) * A(d - s)) and F' € A"(v -^ u;) * A(d - s). Case 1. We assume F' fl V(A(d — s)) ^ 0. In this case A is a cone . Hence we are done by induction. Case 2. We assume F' n V(A(cf - s)) = 0. In this case d - s < 1. Then we have d — soi~d=s + l. Then A and its subcomplexes of the form AX for X C V'(A)
Eisenbud-Goto inequality for Stanley-Reisner rings
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are contractible or contractible to a 1-dimensional complex, unless d — s + I and A"(v —>«;) = <9A(3). Here we omit a detail. Only case we must consider is s = 1,
d=1 and A = <9A(4). In this case r = deg k[A] - codimfc[A] + 1 = 4 . If r > 5, we prove the statement by induction on dim A. We may assume A has no branches. Then A is of the form
where A' is pure and strongly connected and F = F' U {x} is a facet of A and y G V(A') such that x and y are separated. We have deg &[A'] — codim&[A'] + 1 = r — l and hence from the proof of Theorem 3.2 we get reg/A' = r—l. By the assumption of induction, A' is of the form A' = <9A(r - 1) * A(d -r + 2) + ((d- l)-branches). If or £ V(dA(r - 1) * A(d - r + 2)) or if F' <£ <9A(r - 1) * A(d - r + 2), then the branch part can be contractible to a 1-dimensional subcomplex, then we have // r _2(A;t; k) = 0 for each X C V(A). Contradiction. Since A has no branches, we have A' = <3A(r - 1) * A(d - r + 2) and x e V(dA(r - 1) * A(d - r + 2)) and F' <E <9A(r- 1)* A ( d - r + 2). Case 1. We assume F' D V(A(d - r + 2)) 7^ 0. In this case A is a cone. Hence we are done by induction.
Case 2. We assume F'rW(A(d-r + 2)) = 0. In this case d-r + 2 < 1. Then we have d = r — 1 or d = r — 2. If d = r — 2, then reg / A < d + l = r — 1. Contradiction. Hence we have d — r — 1. In this case, for F(^ 0), dimlinkA-F 1
COROLLARY 4.3. Let A be a (d— I) -dimensional pure and strongly connected complex on the vertex set [n]. Assume r := reg/A ^ 4. Then the following conditions are equivalent: (1) reg/A = deg fc[A] — codim&[A] + 1.
(2) A = <9A(r) * A(d - r + 1) + ((d - l)-branches). (3) k[A] is Cohen-Macaulay with h-vector (1, n — d, 1, . . . , 1(= /z r -i)).
/or i = j = 0
J
/? M + J -(*[A])=<
„_,_!
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Proof. (!)=> (2) follows by Theorem 4.2. (2)=> (3) is easy to show, since A is shellable. (2) =£-(4). It is easy to see that /?,-i,-+J-(fc[A]) = 0 unless j = 0, l,or,r— I by Hochster's formula. We see that
\W\=i+r-l
for i = 1,2, . . . ,n — d We can compute /?,-)t-+1(Ar[A]) by the Hilbert series of A: [A]. (3)=> (5) follows from [St, Theorem 6.4]. '(4)=> (3), (5)=> (3), and, (3)=> (1) are trivial. qed COROLLARY 4.4. Lei A be a (d — 1)- dimensional pure and strongly connected complex on the vertex set [n]. Assume reg/A = 3 and A; [A] satisfies (83) condition. Then the following conditions are equivalent: (1) reg/A = deg A; [A] - codimfc[A] + 1.
(2) A = A(/-gon) * A(d - 2) + ((d - l)-branches) /or some / > 3, where A(/-gon) is the boundary complex of the l-gon. (3) A; [A] is Cohen-Macaulay with h-vector ( I , n — d, 1).
(4) fori = j = Q
otherwise
f 0, F(ITm(k[Ay, t) = I t-'+'-Kn-flt-'+i+t-"
fori^d , ._ ,
Proof. Note that A; [A] satisfies (82) if and only if (a) A is pure and (b)linkA-F is
connected for every F 6 A with dimlinkA-f > 1- Then (!)=>• (2) follows by Lemma 4.1. The rest is similar to the proof of the above corollary.
qed
REMARK. A Cohen-Macaulay homogeneous ring R with ft-vector h(R) = (1, fti, 1 , 1 , . . . , 1) is called a stretched Cohen-Macalay ring (cf. [Oo]). These corollaries also give the classification of stretched Cohen-Macaulay Stanley-Reisner rings.
References [Br-He] W. Brans and J. Herzog, "Cohen-Macaulay Rings," Cambridge University Press, Cambridge / New York / Sydney, 1993.
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[Ea-Re] J. A. Eagon and V. Reiner , Resolutions of Stanley-Reisner rings and Alexander duality, Journal of Pure and Applied Algebra 130 (1998) 265-275. [Ei]
D. Eisenbud, "Commutative Algebra with a view toward Algebraic Geometry," Springer-Verlag, New York, 1995.
[Ei-Go] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicities, J. Alg. 88 (1984) 89-133. [Fr]
R Froberg , On Stanley-Reisner rings,, in "Topics in algebra " Banach Center Publications , No. 26, PWN-Polish Scientific Publishers, Warsaw, 1990, pp.5770.
[Fr-Te] A. Friibis-Kriiger and N. Terai, Bounds for the regularity of monomial ideals, Mathematiche (Catania) 53 (1998),83-97. [Hi]
T. Hibi, "Algebraic Combinatorics on Convex Polytopes," Carslaw Publications, Glebe, N.S.W., Australia, 1992.
[Ho-Tr] L.T. Hoa and N.V. Trung, On the Castelnuovo-Mumford regularity and the
arithmetic degree of monomial ideals, Math. Z. 229 (1998) 519-537. [Hoc]
M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in "Ring Theory II (B. R. McDonald and R. Morris, eds.)," Lect. Notes in Pure and Appl. Math., No. 26, Dekker, New York, 1977, pp.171 - 223.
[Ka-St] M. Kalkbrener and B. Sturmfels, Initial complexes of prime ideals, Ad-
vances in Math. 116 (1995) 365-376. [Kw]
S. Kwak, Castelnuovo regularity for smooth subvarieties of dimension 3 and 4, J. Algebraic Geometry 7 (1998), 626-642.
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[Oo]
A. Ooishi, Castelnuovo's regularity of graded rings and modules, Hiroshima Math. J 12 (1982), 627-644.
[St] R. P. Stanley, "Combinatorics and Commutative Algebra, Second Edition,"
[Te] N. Terai, Generalization of Eagon-Reiner theorem and h-vectors of graded rings, Preprint. [Ya] K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree N™graded modules, Preprint.
