Lectures on Algebra

(x) ^ (y). The map is surjective (= onto) if for every y £T there is some x £ S with y = cj>(x). The map <j> is bijective if it is injective as well as surjective. A bijection (resp: an injection, a surjection) is a bijective (resp: injective, surjective) map. The composition of maps <j> : S —> T and ip : T —> £/ is the map ^ 0 : S —» C/ given by (tp / : 5 —> T the inverse -1 : T —> 5 is defined by <j)~1{y) = x <=> <j)(x) = y. The logical symbols A => B, A <= B, and A <=> B stand for A implies B, A is implied by B, and A is true if and only if B is true respectively. §3: G R O U P S A N D FIELDS A group is a set G in which: any two elements x, y can be multiplied to produce a third element xy; this multiplication is associative, i.e., for all x, y, z in G we have (xy)z = x(yz), and so we may omit the parenthesis; there is an identity element 1 in G with lx = xl — x for all i ? G ; and every x £ G has an inverse a: -1 £ G with xx"1 = x~xx = 1. Clearly Sym(S') is a group under composition, with the identity map x — i > x as 1. In particular Sn is a group with \Sn\ = n\, where n! is read as n factorial and stands for the product 1 x 2 x • • • x n. It S is infinite then clearly Sym(S') is infinite. The group G is commutative or abelian if xy = yx for all x, y in G. If G is abelian we may write x + y and 0 and —a: in place of xy and 1 and a; -1 respectively, and when we do so, we may call G an additive abelian group. In the general case when we are writing xy, we may call G a multiplicative group; this time G may or may not be commutative. A field K is an additive abelian group with an associative commutative multiplication defined on it such that the nonzero elements form a commutative multiplicative group and the two operations are connected by the distributive laws saying that for all a;, y, z in K we have x(y + z) = xy + xz and (y + z)x = yx + zx. The rational numbers, the real numbers, and the complex numbers clearly form infinite fields; they are usually denoted by Q, R, and C respectively. To get a finite field, in the eventful year 1830, the 19 year old Galois

4

LECTURE LI: QUADRATIC EQUATIONS

[Gal] noticed that for any prime p, the integers 0 , 1 , . . . ,p — 1 form a field when they are added and multiplied modulo p, i.e., after adding or multiplying the answer is to be divided by p and replaced by the remainder. This gives the Galois Field GF(p). At the Chicago World's Fair of 1893, E. H. Moore [Mol; Mo2] extended Galois' thought by showing that for every power q of p, there is a field GF(q) of q elements. This is unique up to isomorphism, i.e., between any two copies of it there is a one-to-one onto map preserving sums and products; [cf. L5§5(Q32)(T138.2)]. If 1 + 1 + . . . is never zero in a field K then K is said to be of characteristic 0. Otherwise 1 + H h 1 = 0 when 1 is repeated a certain number of times, and the smallest such number is easily seen to be a prime number, which is then called the characteristic of K. For instance the real and complex fields are of characteristic zero, and the Galois Field GF(g) is of characteristic p. Just as a set S is a collection of objects, a subcollection R is called a subset and denoted by R c S; we may also write S D R and call S an overset of R; this is characterized by the implication: x G R =>• x G S. For R\ c S and i?2 C S, their intersection Ri fl R2 is denned to be the set of all x £ S such that x G R\ and x G i?2, and their union Rx U R2 is denned to be the set of all x G S such that x € i?i or x G R%. Similarly for the intersection and union of more than two subsets. For R\ C S and R2 C S we define Ri \ R% to be the set of all x G 5 such that x G R\ but x S- R2; we may call this the complement of R2 in R\. Note that, similar to x ^ y, the symbols x £ R2, Ri <jL R2, and so on, denote the negations of a; G R2,Ri C i?2, and so on. The symbol 0 denotes the empty set. Given any set-theoretic map <j> : S —> T, i.e., given any map 0 of a set S into a set T, for any R C S we put 4>{R) = {4>{x) : x G R}, i.e., <j){R) = the set of all elements of T which can be expressed in the form (x) with x G R; we call <j>{R) the image of R under \ we also put im(<£) = ; finally, for any C / c T w e put <£ -1 ([/) = {x G S : <j>(x) G U). The set consisting of elements x\,...,xe (which may or may not be distinct) is denoted by {x\,..., xe}. However, the group whose only element is 1 may be denoted by 1 instead of the more cumbersome {1}. If H C G are groups under the same operations (and the same identity elements) then we say that H is a subgroup of G or G is an overgroup of H, and we denote this by H < G or G > H; also we write H < G or G > H to mean H < G and H ^ G. If H < G is such that for all h G H and g G G we have ghg~x G H then we say that H is a normal subgroup of G and indicate this by writing H
§3: GROUPS AND FIELDS

5

elements of G which can be expressed in the form xy with y £ H, and we call xH a left coset of H in G; similarly Hx = {yx : y £ H} is called a right coset of H in G; the left cosets of H in G form a partition of G, and so do the right cosets of H in G. Note that a partition of a set 5 is a collection of nonempty subsets of S such that S is their union and any two of them have an empty intersection. For H < G, by G/H we denote the set of all left cosets of H in G; if H < G then G/H becomes a group by defining (xH)(yH) = (xy)H\ we now call G/H the factor group of G by ff. For H < G we put [G : H] = \G/H\ = the number of left cosets of H in G and we call this the index of H in G; this notation is used especially when the index is finite; if G is finite then the index is clearly finite; if G is infinite then the index may be finite or infinite. A finite group G is said to be solvable if there is a chain 1 = Go 5. To introduce A„, we note that any a e Sn, i.e., any permutation on n letters, can be written as a product of transpositions, where by a transposition we mean a permutation which permutes only two letters while leaving all the other letters unchanged. Moreover, the parity, i.e., the evenness or oddness, of the number of transpositions occurring in such a product depends only on a. The permutation a is called even or odd according as the parity of the product is even or odd. Now An is defined to be the group of all even permutations. Clearly \An\ = \Sn\/2 = n!/2 and hence the index [Sn : An] = 2. Therefore An<SnSince An is simple for n > 5, it follows that Sn is unsolvable for n > 5. It only remains to note that the Galois group of the "general" quintic is 55. Therefore the general quintic cannot be solved by radicals. This is Galois' proof. It applies to the general equation of degree n for any n > 5.

6

LECTURE LI: QUADRATIC EQUATIONS

Thus we have the hierarchy of groups in increasing order of complexity: cyclic, abelian, solvable, simple, unsolvable. §4: RINGS A N D IDEALS Generalizing slightly, if in the definition of a group we do not require the existence of the inverse x~l then what we get is called a monoid, and what we get by further dropping the existence of identity is called a semigroup. The terms commutative or abelian semigroup, additive abelian semigroup, multiplicative semigroup, subsemigroup, and oversemigroup are obvious generalizations of the corresponding terms for groups; similarly for a monoid; note that the identity element of a submonoid is required to coincide with the identity element of the monoid. A semigroup H is cancellative means for any x,y, z in it we have the implications which say that: xy = xz or yx = zx =>• y = z. If H is a subsemigroup of a group G then clearly H is cancellative. As an example of an additive abelian group we have the set of all integers, usually denoted by Z, as an example of a submonoid of Z we have the set of all nonnegative integers, usually denoted by N, and as an example of a subsemigroup of Z we have the set of all positive integers, usually denoted by N+. Turning to fields, if K C L are fields under the same operations (and the same zero and identity elements) then we say that K is a subfield of L or L is an overfield of K. As examples, the rational numbers are a subfield of the real numbers which themselves are a subfield of the complex numbers. Again generalizing slightly, a ring R is an additive abelian group which is also a commutative multiplicative monoid such that the two operations are connected by the distributive laws saying that for all x, y, z in R we have x(y + z) = xy + xz and (y + z)x = yx + zx; we call R a null ring if in it we have 1 = 0, i.e., equivalently if \R\ = 1. The concepts of a subring and an overring are defined in an obvious manner; in particular they have the same 0 and 1. A domain is a nonnull ring whose nonzero elements form a cancellative multiplicative monoid. If R is a subring of a field K then R is clearly a domain. In particular the ring of integers, being a subring of the field of rational numbers, is a domain which, as said above, is usually denoted by Z. Dropping the commutativity of multiplication in a field (resp: ring, domain) we get the notion of a skew-field (resp: skew-ring, skew-domain). In the definitions of fields and rings we have spelled out two distributive laws, although they follow from each other, exactly because in case of skew-fields and skew-rings they do not. Moreover, in the definition of a skew-field, by requiring the left-distributive law x(y + z) = xy + xz, but not requiring the right-distributive law (y + z)x = yx + zx, we get the notion of a near-field. Finite near-fields were exhaustively dealt with by L. E. Dickson in 1905. Dickson was the first Ph.D. student of E. H. Moore in the newly opened University of Chicago which had become a vigorous center of mathematical research

U: RINGS AND IDEALS

7

in the westward expansion of the New World. At that time another disciple of E. H. Moore was the Scottish mathematician J. H. M. Wedderburn. Having classified finite fields in 1893, Moore asked his two young followers to study near-fields and skew-fields respectively. Dickson gave a complete list of finite near-fields which was later shown to be exhaustive by the German mathematician H. Zassenhaus as part of his 1935 Hamburg Ph.D. thesis written under the guidance of another famous German mathematician E. Artin. On his part Wedderburn showed that a finite skew-field is necessarily a field. As an additive abelian group, every subgroup / of a ring R is a normal subgroup and so we can form the factor group R/I; note that a typical member of R/I is a residue class (= additive coset) a + I with a £ R. By an ideal in a ring R we mean an additive subgroup I of R such that for all a £ R and x £ I we have ax £ / , and when that is so we make R/I into a ring by defining (a + I)(b + I) = (ab) + I for all a, b in R. We call R/I the residue class ring of R modulo I. Also we call / a maximal ideal or a prime ideal in R according as R/I is a field or a domain. Likewise, we call / a nonzero ideal or a nonunit ideal in R according as I ^ {0} or I ^ R; note that every ideal contains the zero ideal / = {0} and is contained in the unit ideal I = R; moreover, the zero ideal is prime O- R is a domain; likewise, / is the unit ideal <^ R/I is the null ring. For any a £ R we put aR = the set of all multiples az of a with z £ R, and for any I ^ C f l w e put WR — the set of all finite linear combinations a\Z\ + • • • + aeze with a i , . . . ,ae in W and z\,..., ze in R, and we note that these are ideals in R (by convention an empty sum is zero and hence 0 £ WR); they are called ideals generated by a and W respectively; an ideal of the form aR is called a principal ideal in R and a is called a generator of it; in case W = {wi,u>2,... }, we may denote WR by (wi,u>2, • • • )R and call wi,u>2,... its generators. It is easy to see that every ideal in Z is of the form pL with p £ N; moreover: pL is a maximal ideal in Z <=> pL is a nonzero prime ideal in Z «=> p is a prime number. If p is a prime number then we may identify Z/pZ with the Galois field GF(p). The concepts of normal subgroup and ideal can be motivated in terms of homomorphisms thus. For groups G and J, a homomorphism : G —> J is a map such that (p(l) = 1 and (xy) = (f>(x)(f>(y) for all x, y in G; the kernel of , denoted by ker(^), is the set of all x £ G with <j)(x) = 1; note that is injective iff (= if and only if) ker($) = 1 and when that is so we call <j> a monomorphism; if is surjective, i.e., if im() = J, then we call an epimorphism; if is bijective then we call it an isomorphism; if J = G and <j> is an isomorphism then we call an automorphism of G; the set of all automorphisms of G is clearly a subgroup of Sym(G) and we denote it by Aut(G). If G —> J is a group homomorphism then clearly im(0) < J and ker() < G. Conversely, for any H < G we get a group epimorphism G —> G/H with ker(c/>) = H by taking (x) = xH for all x £ G; we call

: G —> J becomes
8

LECTURE LI: QUADRATIC EQUATIONS

4>(x + y) = <j)(x) + 4>(y) for all x, y in G. For rings S and T, a (ring) homomorphism 0 : S —» T is a homomorphism of additive abelian groups such that 4>(1) = 1 and 4>{xy) = cj)(x)4>{y) for all x, y in S; now im(S') is a subring of T and ker(^) is an ideal in S, and the definitions of monomorphism, epimorphism, isomorphism, and automorphism carry over from the group case. Conversely, given any ideal / in a ring 5, for the canonical ring epimorphism <j> : S —> S/I we have ker(0) = / . The set of all ring automorphisms of a ring S is clearly a subgroup of Sym(5') and we denote it by Aut(S'); momentarily letting this Aut be written as Ring-Aut(5) and writing Group-Aut(S) for the automorphism group of the additive abelian group 5 we clearly have Ring-Aut(S) < Group-Aut(5) < Sym(5); it is usually clear from the context which automorphism group is being considered and so it is not necessary to use such a cumbersome notation. Finally for a subring R of a ring S, by an i?-automorphism of S we mean an automorphism
§6: POLYNOMIALS AND RATIONAL

FUNCTIONS

9

generated by U\ U • • • U Ue is denoted by U\ + • • • + Ue. When in the above definitions we want to stress the reference to the ring R, we may say .R-module, .R-submodule, .R-linearly independent, .R-basis, and so on. For modules V and V* over a ring R, by an i?-homomorphism (or .R-linear map) (f>: V —> V* we mean a homomorphism (f> of the underlying additive abelian groups such that for all r £ R and v £V we have ) and im() are submodules of V and V* respectively. Now the meaning of JJ-monomorphism, R-isomorphism, and so on, is obvious. For a skew-ring R, the above description defines a left .R-module V, and an obvious change defines a right .R-module V with a scalar multiplication which to every r £ R and v g V associates vr 6 V. The rest of the above definitions now apply with obvious modifications. By a vector space over a field K we mean a K-module V, and by a subspace of V we mean a .ft'-submodule of V, and so on. Elements of a vector space V, are frequently called vectors. It can be shown that a vector space V always has a basis, and there is a bijection between any two bases. The number of elements in a basis of V is denoted by dim#V, and we call this the dimension of V over K or the /^-dimension of V. Note that then either dim^V € N or dimpcV = oo. It can be shown that any set of generators of V contains a basis of V and hence if V is generated by a finite number of vectors v\,... ,ve then d i m ^ F < e. As said above, any overring of a ring R is an .R-module in an obvious sense. In particular R is an .R-module, and an ideal in R is exactly an .R-submodule of R. In particular, any overring L of a field K is a ff-vector space, and we put [L : K] = dim^L. In particular this applies to an overfield L of K, and then we call [L : K] the field degree of L/K, i.e., of L over K. For a ring or skew-ring R, by R+ we denote the underlying additive abelian group of R, and by Rx we denote the set of all nonzero elements of R; note that K x is a multiplicative group in case of a field or skew-field K. We extend the last notation to any additive abelian monoid (such as an additive abelian group, or a module, or a vector space) V and denote the set of all nonzero elements in it by V x . For an overfield L of a field K, [L : K] usually denotes the field degree rather than the index of K+ in L+. For a homomorphism ) = 1 (resp: im((j)) = 0), where we note that the submonoid {1} (resp: {0}) is denoted by 1 (resp: 0). §6: POLYNOMIALS A N D RATIONAL F U N C T I O N S Given any ring R, we can consider the polynomial ring R[X] in an "indeterminate" X over R, and regard this as an overring of R. A typical polynomial in X

10

LECTURE

LI: QUADRATIC

EQUATIONS

over R, i.e., a typical member of il[X], has a unique (finite) expression A(X) = Y2 A-iX1

with

Ai e R

and i varying over a finite set of nonnegative integers. The degree of A is the largest i with Ai =£ 0; if A = 0, i.e., if A, = 0 for all i, then the degree is taken to be —oo; A is nonconstant means it does not belong to R, i.e., its degree is a positive integer; A is monic means A ^ 0 and Ae = 1 where e is the degree of A. For any other polynomial B(X) = ^2 Bix*

with

Bi&R

the sum C(X) = A{X) + B{X) = J2 CiX1 is given by componentwise addition

and the product = ] T DiXi

D(X) = A(X)B(X) is given by the "Cauchy multiplication rule"

Dk= Y.

AiB

i-

i+j=k

The "formal" X-derivative AX{X) = ^ r Ax(X)

1

=

of i4(X) is denned by putting Y,iAiXi~1.

Clearly A — i > >lx gives an i?-derivation of -R[X], i.e., {A + B)x = Ax + Bx

and

{AB)X = AXB + ABX

with Ax = 0 for all A E R. If i? is a domain then the degree of the product of two nonzero polynomials equals the sum of their degrees, and hence in particular the product is nonzero. It follows that if R is a domain then so is the polynomial ring i?[X]. For an indeterminate X over a field K, by K(X) we denote the field of rational functions in X over K. Members of K{X) are "quotients" of polynomials A(X)/B(X) with nonzero B. The above .ff-derivation of K[X] can be extended to a -^-derivation of K(X) by the "quotient rule" (u/v)x

= (uxv —

uvx)/v2.

Similarly, given any ring R, we can consider the polynomial ring R[X\,...,Xm} in a finite number of "indeterminates" X\,..., Xm over R, and regard this as an

§6: POLYNOMIALS

AND RATIONAL

11

FUNCTIONS

overring of R. A typical polynomial in Xi,..., Xm over R, i.e., a typical member of R[Xi,..., Xm] has a unique (finite) expression = YJAil...imX?...Xi™

A{X1,...,Xm)

with

Ah...im€R

and ( i i , . . . ,im) varying over a finite set of m-tuples of nonnegative integers. The degree of A is the maximum of i\ + • • • + im with Ai...i m ^ 0; if A = 0, i.e., if Ai1.„im = 0 for all i i , . . . ,im, then the degree is taken to be —oo; A is nonconstant means it does not belong to R, i.e., its degree is a positive integer. For any other polynomial B(XU.

..,Xm)

= Y,Bn-^x

---Xt

with

Bix...im

E R

the sum C(X! ,...,Xm)

= A(Xi ,...,Xm)

= J2 di-imXi1

+ B(XU ...,Xm)

•••Xt

is given by componentwise addition ^ii...im

=

-Aii...i m +

Dix...im

and the product D(X1,...,Xm)

= A(X1,...,Xrn)B(X1,...,Xm)

=

Y/Dil...irnXi>...Xl™

is given by the "Cauchy multiplication rule" Uki...km

=

£_J i\+ji=k\,...,im+jm=km

The "formal" partial derivative AXj {Xx,..., is defined by putting AXj{X1,...,Xm)

•™-i1...imBj1...jm.

Xm) =

QA

^->X^)

0f

A{XX

,...,Xm)

^ijAil...imX?...X)'s11Xi'-1X$1\..X%

=

and noting that A >—> AXj gives an R[Xi,...,X,_i,X,+i,..., X m ]-derivation of R[X\,..., Xm]. If J? is a domain then the degree of the product of two nonzero polynomials equals the sum of their degrees, and hence in particular the product is nonzero. It follows that if ii is a domain then so is the polynomial ring R[Xi,..., Xm}. For a finite number of indeterminates X\,..., Xm over a field K, by K(Xi,... ,Xm) we denote the field of rational functions in X\,... ,Xm over K. Note that members of K{X\,... ,Xm) are "quotients" of polynomials A(X\,..., Xm)/B{X\,..., Xm) with nonzero B. Again by the "quotient rule" we can extend the above K[Xi,...,

X,_!, Xj+i,...,

X m ]-derivation of K[Xi,...,

Xm\

to a K(Xi,...,Xj-i,Xj+i,...,Xm)-derivation

of

K(X\,...,Xm).

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EQUATIONS

Given any overring S of the ring R, any finite number of elements x\,..., S can be "substituted" for the indeterminates X\,..., Xm in any A{X\,..., R[Xi,...,Xm] to get A(x\,

. . . , Xm) = y ^ &i-v...imx\

• • • xm

xm in Xm) G

^ &

and we define R[xu...,xm]

= {A(xi,...,xm)

: A{Xi,...,Xm)

G R[Xi,...

,Xm}}

and we note that then R[xi,..., xm] is a subring of S. We call R[x\,..., xm] the subring generated by xi,...,xm over R. The i?-epimorphism R\Xi,..., Xm] —* R\xi,..., xm] and the i?-homomorphism R[Xi,..., Xm] —> S which send every polynomial A(X\,..., Xm) to A{x\,..., xm) are called natural maps or substitution maps. An element x G S is algebraic over R if it satisfies a polynomial equation A(x) = 0 with 0 7^ j4(-f) G -R[X]; otherwise it is transcendental over R; we may express this by saying that x/R is algebraic or transcendental respectively. For W C S, we say that W is algebraic over R if every a; G W is algebraic over R; again we may express this by saying that W/R is algebraic. Finite number of elements (i.e., a finite sequence) x\,..., xm are algebraically dependent over R if they satisfy a polynomial equation A(xi ,...,xm) = 0 with 0 ^ A(Xi,..., Xm) G R[Xi,..., Xm]; otherwise they are algebraically independent over R; we may again express this by saying that (xi,..., xm)/R are algebraically dependent or algebraically independent respectively. Finite number of elements x\,..., xm in 5 form a transcendence basis of S/R (= S over R) if they are algebraically independent over R, and S is algebraic over R[xi,...,xm). For W C S we let R\W] = L1R[XJ1 ,... ,Xje] with union over all finite sequences Xj1,..., Xje in W, and note that this is also a subring of S which we call the subring generated by W over R; note that R[
§7: EUCLIDEAN DOMAINS AND PRINCIPAL IDEAL DOMAINS

13

§7: E U C L I D E A N D O M A I N S A N D PRINCIPAL IDEAL D O M A I N S For any field K, it can be shown that the one variable (= indeterminate) polynomial ring K[X] as well as the many variable polynomial ring K[X\,..., Xm] are UFDs in the following sense. A unit in a ring R is an element u in R such that uu' = 1 for some u' € R; note that then u' is unique and, in accordance with the field case, we may denote it by u - 1 , and for any v £ R we may write v/u for v u _ 1 ; the set of all units in R forms a multiplicative group which we denote by U(R); note that for a field K we have U(K) — Kx. We extend the notation U(R) to denote the multiplicative group of all (two-sided) units in a skew-ring R; similarly, the (two-sided) inverse of a unit u in a skew-ring R may be denoted by u _ 1 ; again note that for a skew-field K we have U(K) = Kx. Elements z, z' in a ring R are associates of each other (in R) if the ideals zR and z'R coincide; note that in case of a domain R, this is equivalent to saying that z' = uz for some unit u in R. An irreducible element in a ring R is a nonzero nonunit element z in R such that z cannot be expressed as z = z'z" where z' and z" are nonzero nonunit elements in R. A ring R is a UFD (= unique factorization domain) if R is a domain and every nonzero element z in R can uniquely be expressed as z = uz\ ...zT where u is a unit in R and irreducible elements in R; here uniqueness means that if z = u'z[... z'r, is any other such factorization then r' — r and there is a permutation . . . , a(r) of 1 , . . . , r such that z',^ is an associate of Zi for 1 < i < r. In case of K[X] or K[X\,..., Xm], the nonzero elements of K are exactly all the units in K[X] or K[X\,... ,Xm] respectively. In case of X p f ] , every nonzero nonunit in K[X] is the associate of a unique nonconstant monic polynomial. Analogous to Z, every nonzero nonunit ideal in K\X] is of the form A(X)K[X] where A(X) is a nonconstant monic polynomial; moreover this ideal is prime <=> it is maximal & A(X) is irreducible. In particular the rings K[X] and Z are PIDs (= Principal Ideal Domains), where by a PID we mean a domain R in which every ideal is principal, i.e., it is of the form xR for some x S R. Moreover both these rings are EDs (= Euclidean Domains), where by an ED we mean a domain R together with a Euclidean Function, i.e., a map <j> : Rx —> N such that for all x,y in Rx we have: y S xR => <j>(x) < 4>{y), and y £ xR => y = qx + r for some q £ R and r G Rx with (r) < 4>{x). If this r (and hence also q) is unique (depending on x and y) then we may call R a special ED, and if there is a subset R of R such that r becomes unique when required to belong to R then we may call R a quasispecial ED relative to the quasispecial subset R. In case of K[X] we take <j>(x) = the X-degree of x, and we note that then K[X] becomes a special ED. In case of Z we take (j)(x) = \x\, i.e., 0 or x < 0, and we note that then Z becomes a quasispecial ED relative to the quasispecial subset N. It can be shown that: ED => PID =• UFD, and: R is a UFD => R[X] is a UFD. Now by induction on m we see that K[X\,..., Xm] is a UFD.
14

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EQUATIONS

§8: ROOT FIELDS A N D SPLITTING FIELDS In view of what we have said above, in case of the univariate (= one variable) polynomial ring K[Y] over a field K, for any irreducible g(Y) G K[Y] the residue class ring K[Y]/g(Y)K[Y] is a field L' which we may regard as an overfield of K, and calling the image of Y under the canonical epimorphism K[Y] —* V to be a.\ we have V = K\a{\ = K(cc\) with g(a\) = 0. We call V a root field of g over K. Given any nonconstant f(Y) £ K\Y] of some degree n, by taking g(Y) to be an irreducible factor of f(Y) in K[Y] we get /(c*i) = 0 and 0 ^ fx{Y) = f{Y)/{Y-al) is of degree n — 1. If n — 1 > 0 then repeating this process we find an element 0:2 in an overfield of K{ax) such that / i ( a 2 ) = 0 and 0 ^ / 2 ( F ) = fi(Y)/(Y - a2) is of degree n — 2. Continuing this process n times we find elements ai,...,an in an overfield of K such that /00=a0(r-ai)...(y-an) where ao is the coefficient of Yn in / ( F ) . The overfield L* = i f [ a i , . . . , a n ] = iiT(ai,..., a n ) of K is called a splitting field of / over K and as notation we put SF(f,K)

= L* =

K(a1,...,an).

For any overfield L of L*, we call L* the splitting field of / over K in L; it can be shown that then for any if-isomorphism <j>: L* —> L** where L** is any overfield of K in L we must have L** = L*, i.e., <j> must be a if-automorphism of L*. To define the concept of a AT-isomorphism, given any overrings S and T of a ring R, a homomorphism (resp: monomorphism, epimorphism, isomorphism) <j> : 5 —> T is called an .R-homomorphism (resp: i?-monomorphism, iJ-epimorphism, il-isomorphism) if (j>(x) = x for all x S R; note that an R-automorphism of S is an .R-isomorphism S —> S. With this definition at hand, it can be shown that any two splitting fields of / over K are -ff-isomorphic, i.e., there is a X-isomorphism of one onto the other. Thus the notation SF(/, K) is unique up to isomorphism. Upon letting f(Y) = a0Yn + axYn-1 + • • • + an with ao,a\,...

,an in K, for the Y—derivative fy(Y) of f(Y) we have fY{Y)

= naoY"-1 + (n - l)aiYn~2

+ •••+ o„_i.

It is easily seen that / is separable (i.e., the roots ai,...,an are distinct) iff f(Y) and / y ( V ) have no nonconstant common factor in ^ [ V ] , By a Galois extension of K we mean an overfield of K which is the splitting field of some nonconstant separable univariate polynomial over K. Assuming / to be separable, by the Galois group Gal(L*,JK') of L* over K we mean the group Aut/f(L*). For any a € Gal(L*,K) we have f(Y) = ao(Y — cr(a\))... (Y — a(an)) and hence ( o i , . . . , a«) i-> (a(ai),..., a(an)) gives a permutation a* of { a i , . . . , an}. Clearly

§S: ROOT FIELDS AND SPLITTING FIELDS

15

a >-> a* gives a monomorphism Gal(L*,if) —> Sn = the permutation group on { a i , . . . , a „ } . We define the image of this monomorphism to be the Galois group of / over K and denote it by Gal(/, K). Thus we have a natural isomorphism Gal(L*,if) —* Gal(/, K) and hence in particular Gal(L*,if) is a finite group. Now Galois himself [cf. L6§6(E6)] characterized Gal(/, K) as the set of all relations preserving permutations of the roots, i.e., the set of all T G Sn such that for every P(Xi,... ,Xn) G K\Xi,...Xn) with P{ai,... ,a„) = 0 we have P ( r ( a i ) , . . . , r(a„)) = 0. For any intermediate field L, i.e., a field L with K C L C L*, it is clear that L* = SF(/,L) and Gal(L*,L) < Gal(L*,if). Likewise, for any H < Gal(L*,if), upon letting fixL.(ff) = {z £ L* : ft(z) = z for all ft G # } it is clear that this is an intermediate field; we call this the fixed field of H. Eventually, following Galois, we shall prove the following [cf. L6§6(E1)]: FUNDAMENTAL THEOREM OF GALOIS THEORY (Tl). The mapping L — i * Gal(L* ,L) = H gives an inclusion reversing bijection of the set A of all fields L between K and L* onto the set T of all subgroups H of Ga\(L*,K), and the inverse bijection is given by H — i > fix/,. (77) = L, where inclusion reversing means: L\ C Li <=> Gal(L*,L x ) D GaI(L*,L 2 ). Moreover iT< Gal(L*,^) <=> L is a Galois extension of K, and when that is so Ga\(L*,K)/H is naturally isomorphic to Gal(L,K); in greater detail, by restricting a € Ga^L*,/^) to L we get <7 G Gal(L,if), and a ^> a gives an epimorphism Gal(L*,if) —> Gal(L, if) with kernel H. Finally for every L e A we have |Gal(L*,L)| = [L* : L}. By an r-cyclic extension of K, where r is a positive integer, we mean a Galois extension L of K such that Gal(L, K) is a cyclic group of order r. The following theorem dates back to 1775 [cf. L6§6(E3)]: LAGRANGE RESOLVENT THEOREM (T2). Assume that K has r distinct rth roots of 1. Then any r-cyclic extension of K is obtained by adjoining an r-th root of some nonzero element a of K, i.e., it is of the form K(a) with 0 ^ ar = a £ K. Conversely, for any 0 ^ a G K, the splitting field of Yr — a over K is an s-cyclic extension of K for some divisor s of r. The above two theorems have the following consequence [cf. L6§6(E3)]: SOLVABILITY THEOREM (T3). Assume that K has n! distinct n!-th roots of 1. Then Gal(/, K) is solvable if and only if the equation / = 0 can be solved by radicals, i.e., if and only if there is a chain of fields K = KQ C ifi C • • • C Kt = K(ai,... ,an) such that for 1 < i < t wehaveifj = Ki-i(ui'r') with 0 ^ Ui e ifj-i and rj G N+. The following theorem dates back to 1665 [cf. L6§6(E4)]:

16

LECTURE

LI: QUADRATIC

EQUATIONS

NEWTON'S SYMMETRIC FUNCTION THEOREM (T4). For the generic n-th degree polynomial F(Y)=Yn

+ X1Yn~1

over K = k(X\,..., Xn), where X\,..., haveGal(F, J ft:) = 5'„.

+ --- + Xn

Xn are indeterminates over a field k, we

We have already noted the following theorem dating back to 1830 [cf. L6§6(E5)]: GALOIS' SYMMETRIC GROUP THEOREM (T5). The group Sn is unsolvable for n > 5. Finally, as a consequence of (T3), (T4), (T5) we get the [cf. L6§6(E5)]: UNSOLVABILITY THEOREM (T6). Assume that the field A; contains n\ distinct n!-th roots of 1. Then, for n > 5, the generic n-th degree polynomial F(Y) over K = k(X\,..., Xn) cannot be solved by radicals. §9: A D V I C E TO T H E R E A D E R We shall now fill in some details of argument which were left out in the above discourse so as not to break the smooth flow of thought. This will be done in a series of Definitions (Dl), (D2), ..., Remarks (Rl), (R2), ..., Examples (XI), (X2), ..., Exercises (El), (E2), ..., and Notes (Nl), (N2), ... . We shall end up with a Concluding Note in the style of Aesop's Fables. The same pattern may be followed in the succeeding lectures. In any lecture, assertions such as Theorems, Corollaries, Claims, and Propositions, will be labelled (Tl), (T2), ..., and formulas and other items will be labelled (1), (2), ..., (Al), (A2), ..., (Bl), (B2), ..., and so on. Sometimes there may be Observations (01), (02), ... . In a lecture, say in Lecture L3, items from another lecture, say Lecture L2, will be cited as L2(D1), L2(T4), and so on. Some details left out in a lecture may very well be covered in a later lecture. The reader may need a certain amount of patience to follow this method of concentric circles, whereby the ideas expounded at a time are explained and expanded in an outer circle. The patience will be rewarded by entertainment and ease of absorption. Also the discerning reader should have no difficulty in separating out the heuristic geometric discourse from the official logical dogma. §10: DEFINITIONS A N D R E M A R K S DEFINITION (Dl). [Divisibility and Prime Ideals]. Let R be a ring. Given elements x, y in R, we say that x divides y (in R) or y is divisible by x (in R) to

§10: DEFINITIONS AND REMARKS

17

mean that y G xR, i.e., y = xz for some z G R; we may express this by writing x\y. Recall that z G R is irreducible (in 7£) means z is a nonzero nonunit in R which is not the product of any two nonzero nonunits in R. Similarly z G R is said to be prime (in R) if z is a nonunit and for all x,y in R we have: 2|(xy) => z|a: or z|j/. Recall that an ideal 7 in R is a prime ideal or a maximal ideal according as R/I is a domain or a field. This is easily seen to be equivalent to the more standard definition, i.e.,

{

an ideal 7 in a ring R is maximal <3> I ^ R and there is no ideal strictly between 7 and 7?

an ideal 7 in a ring R is prime O 7 ^ 7? and for all x, y in R \ I we have xy g' I.

(Ao)

J an element z i n a ring R is prime < I <=$ the principal ideal zR is prime.

Note that a maximal ideal is always prime but not conversely; for instance for a field K, the zero ideal in the univariate polynomial ring K[Y] is prime but not maximal; more interestingly, in the bivariate (two variable) polynomial ring K[X, Y], the ideal generated by X is prime but not maximal since the residue class ring is obviously isomorphic to the univariate polynomial ring 7C[y]. REMARK (Rl). [Division Algorithm]. Given any A = A(X) in the univariate polynomial ring R[X] over a ring R, we call A monic or submonic (in X) if A / 0 and, upon letting a to be the coefficient of Xd in A where d is the degree of A, we have a = 1 or a G U(R) respectively. Assuming A(X) to be submonic, the division algorithm in R[X] says that given any B(X) G R[X] there exist unique q(X) and r(X) in R[X] such that B(X) = q(X)A(X) + r{X) and the degree of r(X) is < d. It follows that the univariate polynomial ring K[X] over a field K is a special ED. Note that the division algorithm in Z says that given any x,y in Z with x 7^ 0 there exist unique q, r in Z such that y = qx + r and 0 < r < \x\. This makes Z a quasispecial ED with N as the quasispecial subset. REMARK (R2). [Long Division or Euclidean Algorithm]. In the ancient Indian literature the following procedure for finding a gcd (= greatest common divisor) of two nonzero elements x\, x to be the euclidean function, let us inductively define qi+2,xi+2 for i = 1, 2 , . . . by saying that if xt $ xi+\R then qi+2 G R and xi+2 G Rx

18

LECTURE LI: QUADRATIC EQUATIONS

are such that Xi = qi+2Xi+i +xi+2 and 4>{xi+2) < (xi+i). This stops after t steps, with (x2) > t £N, when xt+i £ xt+2R- Note that if R is a special ED then the elements qi+2, %i+2 are unique, and if R is a quasispecial ED then they can be chosen uniquely. At any rate, without assuming special or quasispecial, this algorithm enables us to express xt+2 as a linear combination of the elements x 1,0:2, and to express the elements 0:1,0:2 as multiples of 2^+2, i-e., to write o:t+2 = ao:i + bx2, 0:1 = co;t+2, 0:2 = dxt+2, with a, b, c, d in R. Thus the element o: t+2 is a generator of the ideal in R generated by 0:1,0:2, and hence xt+2 is a gcd of 0:1,0:2 in the following sense. For a subset 5 of a ring R (which need not be an ED), by a cd (= common divisor) of S (in R) we mean x £ R such that x\y for all y £ S, and by a gcd of S (in R) we mean a cd x of S such that x is divisible by every cd of S; any gcd of S may be denoted by gcd(S); note that if the ideal in R generated by S is principal then a gcd of S is nothing but a generator of that ideal; moreover, without assuming the ideal SR to be principal, if x £ R is a gcd of S then: y £ R is a gcd of S <=)• x and y are associates of each other. By a cd of a finite or infinite sequence of elements y 1, j/2> • • • in a ring R we mean a cd of S = {2/1,2/2, • • • }, and by a gcd of 2/1,2/2,... we mean a gcd of S = {2/1,2/2, •• • }; again any gcd of 2/1,2/2, • • • may be denoted by gcd(2/1,2/2, • • • )• Although a gcd is determined only up to associates, for euphony we may sometimes call it the gcd. Again assuming R to be an ED, we extend the above procedure of finding a gcd to a finite sequence 2/1,.. •, ys in Rx, with s £ N, by putting ZQ = 0, z\ = 2/1, ^2 = gcd(zi,2/2),23 = gcd(z 2 ,2/3),...,2 s = gcd(z s _i,2/ s ), where the gcd is as found in the above two element procedure, and noting that now clearly zs is a generator of the ideal in R generated by 2/1, • • •, 2/s, and hence zs is a gcd of 2/1, • • •, 2/s • For a finite sequence z/i,..., ys in R we apply this procedure to the sequence obtained by deleting all the zeros from the sequence j / i , . . . , ys, and thereby obtain a generator of the ideal in R generated by j / i , . . . , j/«, which is hence a gcd of j/i,. • •, 2/sREMARK (R3). [ED =» P I D ] . By extending the above method to infinite sequences we can show that ED =>• PID. But this can be shown more easily, although nonconstructively, thus. Given any nonzero ideal 7 in a euclidean domain R with euclidean function (j>, take 0 ^ x £ I having the smallest Rvalue; if there were y £ I \ xR then we could find q £ R and r £ Rx such that y — qx + r and <j>{r) < 4>(x), which would be a contradiction because 0 ^ r £ I. In case R is the univariate polynomial ring K[X] over a field K, for any S C R we put GCD(5) = 0 if S C {0} and otherwise we put GCD(S) = the unique monic generator of the ideal in R generated by S, and for any finite or infinite sequence 2/1,2/2, • • • in R we put GCD(2/i, 2/2, • • •) = GCD({2/i, 2/2, •••})• I n case R is the ring of integers Z, for any S C R we put GCD(S) = the unique nonnegative generator of the ideal in R generated by S, and for any finite or infinite sequence 2/1,2/2, • • • in i? we put GCD(j/i, 2/2, • • •) = GCD({2/i,2/2, •••})• Here GCD stands for the Greatest Common

%10: DEFINITIONS

AND

REMARKS

19

Divisor (in R). Note that in the above two situations, the GCD is a gcd. To make the reference to the ring R explicit we may write gcd^ and GCD^ instead of gcd and GCD respectively. REMARK (R4). [ED => U F D ] . Later on we shall prove PID => UFD [cf. L5§5(Q18.4)]. Here we shall give a direct proof of ED =>• UFD. First let us establish some generalities about UFDs. So let R be a domain. Recall that R is a UFD means (Fl) every nonzero element in R is a product of a unit and irreducible elements and (F2)

the irreducible factors in (Fl) are unique up to order and associates.

It can be shown that if R is a UFD then (F3)

every irreducible element in R is prime.

It can also be shown that if (Fl) and (F3) hold then so does (F2). In other words (Bl)

in a domain R with (Fl) we have: (F2) & (F3).

It is clear that any subset S of a UFD R, and hence any sequence j/i, 2/2, • • • in R, has a gcd; namely if S C {0} then 0 is the only gcd of S, and if S has a nonzero element z then we can write z = uz\x... z|' where u is a unit in R, e i , . . . , e.t are positive integers, and z\,..., zt are irreducibles in R such that no two of them are associates of each other; now upon letting d, to be the largest integer such that zi * divides every element of S, we see that x £ R is a gcd of S iff x = vz^ ... zt* for some unit v. Elements 2/1,2/2, •• • in a UFD are coprime (or relatively prime) means 1 is a gcd of them. To establish some generalities about EDs, recall that R is an ED means it has a euclidean function, i.e., a map (f>: Rx —• N such that for all x,y in Rx we have: (Gl)

y£xR=>

{x) < <j>{y)

and (G2)

y $ xR =>• y = qx + r for some q € R and r € Rx with (j)(r) < <j>(x).

It is easily seen that given any euclidean function <j> : Rx —> N, for all x, y in Rx we have (G3)

x and y are associates => <j>{x) = 4>(y)

and (G4)

y € xR and (j)(x) = (j>(y) => x and y are associates.

20

LECTURE LI: QUADRATIC EQUATIONS

With 0 as above, for all z € Rx by (Gl) we know that 0(z) > 0(1); by induction on (j){z) let us show that z can be expressed as a product of a unit and irreducible elements; for <j>{z) = 0(1) this follows from (G4); so let (z) > 0(1) and assume for all values of <j>(z) smaller than the given one; if z is irreducible then we have nothing to show; otherwise z = z'z" where z' and z" are nonzero nonunits, and by (G4) we get (z') <
every ED satisfies (Fl).

Again with 0 as above, let x, y, z in Rx be such that z is irreducible with xy £ zR and x $ zR; then 1 is a gcd of x, z and hence by (R2) we can write 1 = ax + bz with a, b in R; multiplying this equation by y we get y = axy + byz and hence y G zR. Thus (B3)

every ED satisfies (F3).

By (Bl), (B2), (B3) we see that ED => UFD. REMARK (R5). [Factorization of Integers]. Applying the above Remarks (Rl) to (R4) to the case R = Z we see that: R is an ED, PID, and UFD, with U(R) = {±1}. Every nonzero nonunit element in R is an associate of a unique integer > 1. Every irreducible element in R is an associate of a unique positive irreducible element, i.e., a prime number. Every nonzero element z in the quotient field Q of R has a unique factorization z — ±p\: ... p\ * where e\,..., et are nonzero integers and pi,...,pt are pairwise distinct prime numbers (by convention an empty product is 1); moreover z € R <=> e% > 0 for 1 < i < t. All the nonzero prime ideals in R are maximal, and they are of the form pR where p is a prime number. For any positive integer n the residue class ring S = R/(nR) is clearly a ring of size n; moreover S is a field -^ n is a prime p, and then we call S the Galois Field of p elements and denote it by GF(p). REMARK (R6). [Factorization of Univariate Polynomials]. Applying the above Remarks (Rl) to (R4) to the univariate polynomial ring R = K[Y] over a field K we see that: R is an ED, PID, and UFD, with U(R) = KX. Every nonzero nonunit element in R is an associate of a unique nonconstant monic polynomial. Every irreducible element in R is an associate of a unique nonconstant monic irreducible polynomial. Every nonzero element h = h(Y) in the quotient field K(Y) of R has a unique factorization h = ag*1 ... g\% where a G Kx, e i , . . . , et are nonzero integers, and g\ = gi(Y),... ,gt = gt(Y) are pairwise distinct nonconstant monic irreducible polynomials; moreover h G R •& et > 0 for 1 < i < t. All the nonzero prime ideals in R are maximal, and they are of the form gR where g = g(Y) is a nonconstant irreducible polynomial, which may be chosen to be monic. For any

§i0: DEFINITIONS AND REMARKS

21

nonconstant / = f(Y) G R of some degree n > 0, clearly Kr\(fR) = {0} and hence we may identify K with a subring of the residue class ring L = R/(fR), and then upon letting y to be the image of Y under the residue class map R—>Lwe see that 1) y, • • • > yn~l is vector space basis of L over K and hence [L : K] = n; moreover L is a field <=> / is irreducible. DEFINITION (D2). [Characteristic]. For any ring R there is unique homomorphism Z —•> R. The unique nonnegative generator of its kernel is called the characteristic of R and is denoted by ch(fl). Clearly if R is a domain then ch(R) is either 0 or a prime number. The image of the homomorphism Z —> R is called the prime subring of R. Clearly ch(R) = 0 <=> the prime subring of R is isomorphic to Z; in this case we may identify the prime subring with Z. Also clearly ch(i?) is a prime number p <=> the prime subring is a field; in this case the prime subring is obviously isomorphic to the Galois field GF(p), and we identify the former with the latter. Needless to say that ch(R) = 1 <$ the prime subring is the null ring o R is the null ring. DEFINITION (D3). [Minimal Polynomial]. Let L be an overfield of a field K. Then y £ L is transcendental or algebraic over K according as the kernel of the iiT-homomorphism K[Y] —> L which sends Y to y is zero or nonzero, i.e., according as \K[y] : K) = oo or [K[y] : K] = a positive integer n. In the algebraic case n is the field degree of the field K[y] = K{y) over K, and the unique monic generator of the said kernel is a monic irreducible polynomial of degree n in Y over K; we call this polynomial the minimal polynomial of y/K, i.e., of y over K. Note that for any y G L we always have K[y] C K(y) and hence: y is algebraic over K •& K[y] = K(y) <& [K[y] : K) < oo <^ [K(y) : K) < oo.

DEFINITION (D4). [Least Common Multiple]. In grade school when we learn of cd and gcd we also learn of cm (= common multiple) and 1cm (= least common multiple). To introduce these let it be a ring and let y\,..., ya be a finite sequence of elements in R with s > 0. By a cm oiy\,...,ya (in R) we mean x € R such that x is divisible by j/j for 1 < i < s. By an 1cm of y i , . . . ,ya (in R) we mean a cm of j / i , . . . , ys which divides every cm of j / i , . . . , ys, and we denote it by lcm(j/i,... ,ya). Clearly if \cm(yi,... ,ya) exists then it is unique up to associates. For euphony we may sometimes speak of the 1cm instead of an 1cm. Now assume that it is a UFD. Then we can find a finite number of irreducible elements z\,..., zt in R, no two of which are associates of each other, such that for 1 < i < s we have ?/* = Uiz\iX ... z\H where Ui is a unit in R and en,..., eu are nonnegative integers. For 1 < j < t let lj = max{ejj : 1 < i < s} (with lj = 0 if &ij = 0 for all i), and let y = z± ... ztl. Then clearly y is an 1cm of y\,..., ys. In case R is the univariate polynomial ring K[Y] over a field K (resp: the ring

22

LECTURE LI: QUADRATIC EQUATIONS

of integers Z) we define the LCM (= Least Common Multiple) of y\,... ,ys (in R) to be the unique monic (resp: positive) 1cm of j / i , . . . ,ys, and we denote it by LCM (1/1,..., ys); note that by taking z\,... ,zt to be monic (resp: positive) we get LCM(?/i,..., y3) = y. To make the reference to the ring R explicit we may write lcmij and LCMR instead of 1cm and LCM respectively. Continuing with the assumption that R is a UFD, any element £ in the quotient field of R can be expressed as ( = £/r? with £ G R and r\ G Rx such that the elements £ and 77 are coprime, and we call £/?? a reduced form of £. In case R is the univariate polynomial ring K[Y] over a field K (resp: the ring of integers Z) this becomes unique by requiring 77 to be monic (resp: positive), and so we may call £/?7 the reduced form of (,. REMARK (R7). [Relative Algebraic Closure]. To continue the discussion of (R6) and (D3), let L be an overfield of a field K, and let K' and K" be subfields of L with K c K' c K". It can be shown that then and K / \K" : ^ < 0 ° ° \K" :K'}<0° \ ' -K)
(Hint: if (zi)i
respectively then

J if [K" : .fiT] < 00 then every y G .fiT" is algebraic over K and [the degree of the minimal polynomial of y over K divides \K" : K}. It also follows that (Co)

J if K" is algebraic over K', and .fiT' is algebraic over K, < I then K" is algebraic over .fiT.

Moreover it follows that if K" = K(W) where W C K" is such that (C4)

{ every j / G W is algebraic over .fiT then K" = K\W] and K" is algebraic over if.

Finally it follows that . (Co)

I the set K of all elements of L which are algebraic over K < [is a subfield of L;

we call K the (relative) algebraic closure of K in L; the adjective relative is sometimes used to distinguish this from an (absolute) algebraic closure of K by which we mean an overfield K of K such that K is algebraic over K and every nonconstant univariate polynomial over K has a root in K.

§11: EXAMPLES AND EXERCISES

23

REMARK (R8). [Uniqueness of Splitting Field]. Let L be an overfield of a field K, and let f(Y) be a nonconstant polynomial of some degree n > 0 in Y over K such that / splits completely in L, i.e., f(Y) = ao(Y — a\)... (Y — an) with 0 ^ CLQ S K and c*i,..., an in L; we call L* = K(a\,..., a n ) the splitting field of / over K in L. Let L' be an overfield of a field K' such that there is an isomorphism : K —> K' and / ' splits completely in L' where f'(Y) is obtained by applying (j> to the coefficients of f(Y). Now upon letting a'0 = (ao) we clearly have f'(Y) = a'0(Y - a[) ...(Y — a'n) with a[, •'•• ,a'n in £/. We want to show that there is an isomorphism <j>* : L* —> V* = K'(a[,... ,a'n) such that <j>*(x) = <j){x) for all x £ K. We do this by induction on n. For n = 1 the assertion is obvious because then L* = K and L'* = K'. So let n > 1 and assume for n — 1. Let g(Y) be the minimal polynomial of Qi over K, and let ff'(F) be obtained by applying to the coefficients of g(Y). Then clearly g'(Y) is the minimal polynomial of o^ over K' for some i, and upon relabelling a[,... ,a'n suitably we may assume that i = 1. Now Kx = K{al) and K[ = K'(a[) are root fields of g(Y) and g'(Y) over K and K' respectively, and they are naturally isomorphic to K[Y]/(g(Y)K[Y}) and K'[Y]/(g'(Y)K'[Y]) respectively, and therefore there is an isomorphism (f>\ : K\ —> K[ such that (/>i(ai) = a[ and (x) for all x £ K. Let / i ( y ) = ( y - a 2 ) . • • (Y-an) and /{(y) = (Y-a'2)... (Y-a'n). Then h(Y) is a polynomial of degree n — 1 over /fi, and /{(y) is obtained by applying i to its coefficients. Also clearly L* and L'* are the splitting fields of / i and f[ over i^x and K[ in L and V respectively. Therefore by the induction hypothesis there exists an isomorphism (j>* : L* -> V* such that 0*(a;) = <j>x{x) for all i G i f i . §11: EXAMPLES A N D EXERCISES EXAMPLE (XI). [Special Permutations]. A permutation, say of the integers {1,2,. . . , 6 } , is a rearrangement like (\ 2 3 4 5 6 \ \ 2 4 6 1 5 37 where we send 1 to 2, 2 to 4, 3 to 6, 4 to 1, 5 to 5 and 6 to 3. Let us call the above permutation a. Let us consider another permutation, say r, given by A 2 3 4 5 6\ \ 4 3 6 1 2 5J ' To multiply r by a, we first apply a and then follow it by r. Thus T<7

_ (\ 2 3 4 5 6\ / l 2 3 4 5 6 \ _ A 2 3 4 5 6\ ~ V4 3 6 1 2 5 / V2 4 6 1 5 3y \ 3 1 5 4 2 67

because: er sends 1 to 2 and r sends 2 to 3, a sends 2 to 4 and r sends 4 to 1, a

24

LECTURE LI: QUADRATIC EQUATIONS

sends 3 to 6 and r sends 6 to 5, a sends 4 to 1 and r sends 1 to 4, a sends 5 to 5 and r sends 5 to 2,CTsends 6 to 3 and r sends 3 to 6. Doing it the other way round we get <7T

_ / l 2 3 4 5 6\ / l 2 3 4 5 6^ _ / l 2 3 4 5 6\ ~ V2 4 6 1 5 3 / \ 4 3 6 1 2 5 / ~ \l 6 3 2 4 5 /

which illustrates that in general multiplication is not commutative. The above permutation a can be written as a product of disjoint cycles

where the cyclic permutation (124) is the permutation which sends 1 to 2, 2 to 4, and 4 to 1 (and leaves all the other numbers unchanged) , i.e., in "long form"

and similarly, the cyclic permutation (36) sends 3 to 6 and 6 to 3, and, finally, the cyclic permutation (5) sends 5 to 5. Since (5) leaves all the other numbers unchanged, so indeed (5) is simply the identity permutation (I 2 3 4 5 6\ ~ \1 2 3 4 5 6j •

[b)

The cyclic permutation (124) is a cycle of length 3. A cycle of length 2, such as (36), is called a transposition. The cycles (124), (36) and (5) are disjoint means no two of them have any number in common. EXAMPLE (X2). [General Permutations]. A permutation a of a finite set, say the set of integers { 1 , 2 , . . . , n}, i.e., a £ Sn, is written as / 1

2

•••

n

\

V ( l ) <x(2) • • • a(n)J or more generally as (

%\

\a(ii)

h

•••

in

\

a(i2) ••• cr(in)J

where i\, i2, • • •, in is any labelling of 1,2,..., n. Thus, by nipping the representation of a we get 1=(a(l)a(2).-.a(n)\

\

1

2

•••

n J '

We can chooseCT(1)in n different ways, and then we can choose
%11: EXAMPLES AND EXERCISES

25

\Sn\ = n\, i.e., the symmetric group of degree n has size n\. A permutation on n letters is called a permutation of degree n, to remind us of the fact that an element of the Galois group of a polynomial of degree n is a permutation of the n roots. For a positive integer e < n, an e-cycle or a cycle of length e in Sn is a permutation r for which there is a rearrangement i\,... ,in of 1 , . . . ,n such that i"(ii) = i2,T~(i2) = h,... ,r{ie-i) = ie,r(ie) = i i , r ( i e + i ) = i e + 1 , . . . , r ( i n ) = i n ; we denote this by (iii2 . . . ie), or for clarity by (ii,i2, • • •, *e)- As in the above example, any permutation a of { 1 , 2 , . . . , n} can be written as a product of disjoint cycles, say of lengths e\,.,., e^; note that we include cycles of length 1; also note that every i € { 1 , . . . , n} occurs in exactly one of the ft cycles, and we have e\ + • • • + e^ = n. Now, a cycle of length e can be expressed as a product of e— 1 transpositions, where by a transposition we mean a cycle of length 2; for example (12345) = (12)(23)(34)(45). Therefore, a can be written as a product of e\ + e?. -\ Yeh — h transpositions; the permutation a is called even or odd according as the integer e\ + e^ + • • • + e^ — ft is even or odd. To justify this terminology we have to show that if the integer ei + e2 H 1- eh — ft is even (resp: odd) then any way of writing a as a product of transposition must contain an even (resp: odd) number of them. One way of proving this is described in the following Exercise (El). From the definition of evenness, it follows that the set of all even permutations of 1,2,..., n forms a subgroup of Sn. This subgroup of Sn is called the alternating group of degree n and is denoted by An. It can easily be seen that An < S„ and |5 n /i4 n | = 2 or 1 according as n > 1 or n = 1. EXERCISE (El). Let Sn be the group of all permutations of { 1 , . . . , n} where n is a positive integer. Show that every a G Sn can be expressed as a product of disjoint cycles, and investigate how far this expression is unique; in particular show that the lengths of the cycles are determined by a. Show that a cycle of length e is a product of e — 1 transpositions. Consider the polynomial of degree n(n — 1) in n variables over Z given by

D(X1,...,Xn)=

J]

(Xi-Xj)

i<j in {l,...,n}

and for any a e Sn put D(Xi,...,

Xn)a =

J|

(X^i) -

Xa{j)).

i<j in {l,...,n}

Show that for all a, r in Sn we have

D(x1,...,xn)^

=

(D(xl,...,xnyy

26

LECTURE LI: QUADRATIC EQUATIONS

and for all transpositions 0 we have D(X1,...,Xn)e

=

-D(X1,...,Xn)

and from this deduce that the parity (i.e., the evenness or oddness) of the number of transpositions whose product is a depends only on a and not on the particular product representation. Moreover by defining the signature sgn(o-) to be 1 or —1 according as the said number of transpositions is even or odd we have D{XU .. .,Xny

= sgn(a)D(Xu..

.,Xn).

As we shall discuss later [cf. L5§5(Q32)], the square of the above mentioned polynomial D(Xi,..., Xn) is the modified ^-discriminant of the n-th degree monic polynomial in Y whose roots are Xi,..., Xn, i.e., of the polynomial r i i < i < n ( ^ — ^i). EXERCISE (E2). Show that [S„ : An) = 1 or 2 according as n = 1 or n > 1. Deduce that An < Sn- Generalize by showing that a subgroup of index 2 is always normal. EXERCISE (E3). Show that the symmetric group Sn is generated by the transpositions (12), (13),..., (In); hint: if l,r,s are distinct then (rs) = ( l r ) ( l s ) ( l r ) . Show that An is generated by 3-cycles (123), (124),..., (12n); hint: if 1, r, s are distinct then (ls)(lr) = (Irs), and if l,2,r, s are distinct then (12s)(12s)(12r)(12s) = (Irs). EXERCISE (E4). Show that if H for all x,y m G we have xyx~ly~x £ H. The element xyx~ly~l is called the commutator of x, y and is denoted by [x, y}. The subgroup of a group G generated by the commutators of various pairs of elements of G is called the commutator subgroup of G or the (first) derived subgroup of G, and is denoted by [G,G] or G'. We may rephrase this exercise by saying that for any subgroup H of a group G we have: G' < H <$• H 5 then An is simple and Sn is unsolvable. Hint: for any distinct integers a, b, c, d, e upon letting x = (ode) and y = (cbe) we have xyx~1y~1 = (abc), and hence if H < G < Sn with G/H abelian are groups such that G contains all 3-cycles then so does H; therefore Sn is unsolvable; for simplicity of An [cf. L6§6(E43)]. EXERCISE (E6). For 1 < n < 4, find all subgroups of Sn and show that they are all solvable. Simplify the verification by showing that if H < G are finite groups with G solvable then H is solvable. EXERCISE (E7). Prove the existence and uniqueness of the division algorithm

§12: NOTES

27

equations y = qx + r and B{X) = q(X)A(X) + r(X) asserted in (Rl). Hint: division is repeated subtraction, or equivalently, subtraction is slow division. Thus for the first equation, if y > x > 0 then for r = y — x we have 0 < r < y. Similarly for the second equation, if degxB(X) = e > d = degxA(X) > 0 and e d the coefficients of X and X in B(X) and A(X) are b and a respectively then for r(X) = B{X) - (b/a)Xe-dA(X) we have degxr{X) < e. EXERCISE (E8). Give details of the equations xt+2 = ax\ + bx2, x\ — cxt+2, x2 = dxt+2 asserted in the Euclidean Algorithm described in (R2). Hint: the first step x\ = q$X2 + X3 of the algorithm gives us £3 = x\ — q^X2 and combining this with the second step X2 = QAX^ + x± gives us Xi= X2— 94^3 = -QAXI + (1 + 9394)^2, and so on. EXERCISE (E9). Prove claim (Bl) concerning a UFD made in (R4). Prove that an ED has properties (G3) and (G4) stated in (R4), and prove claims (B2) and (B3) concerning an ED made in (R4). EXERCISE (E10). Concerning algebraic field extensions, fill in the details of argument in (R6) and (D3), and prove assertions (CI) to (C5) of (R7). Hint: recall the "rationalization of surds" method from school math according to which 3 + y / 3 _ (3 + ^ ( 2 - 7 3 ) ^ 3 - V 3 2 +A/3 (2 + \ / 3 ) ( 2 - V3) 1

= 3

^

Generalizing this, given an algebraic element y over a field K with \K(y) : K] = n, we let G(Y) be the minimal polynomial of y over K, and then for any A(Y) G K[Y\ and B{Y) e K[Y] \ (G(Y)K[Y\), by division algorithm and long division we find C(Y),H(Y) in K[Y) with degyC(y) < n such that A(Y) = B{Y)C{Y) + G{Y)H{Y), and this gives

proving that 1,«/,..., yn~1 is a if-vector-space basis of K(y) = K[y\. §12: N O T E S NOTE (Nl). [Derivations]. In calculus derivatives are defined in terms of limits. In algebra we define derivatives in terms of formal rules of manipulation. In greater detail, by a derivation of a ring S with values in an 5-module V we mean a homomorphism D : S —> V of the underlying additive groups such that for all x, y in S we have D(xy) = xD(y) + yD(x). For any s e S w e put (sD)(x) = sD(x) for all x € S, and for any other derivation of D' of S with values in V we put (D + D'){x) = D(x) + D'{x) for all x £ S. This makes the set of all derivations of S

28

LECTURE LI: QUADRATIC EQUATIONS

with values in V into an S-module which we denote by Der(5, V). If D(x) = 0 for all a: in a subring R of S then we say that D is an .R-derivation of S or a derivation of S/R. The submodule of all derivations of S/R with values in V is denoted by Derfl(5, V). By a derivation of S we mean a member of Der(S,S), and by an Rderivation of S we mean a member of Der/{(5, S). Now, referring to the section on Polynomials and Rational Functions, for any ring R the formal derivative A i-> Ax is an i?-derivation of iipf], and if R is a field K then this induces a ^-derivation of K(X); likewise, for 1 < j < m, the formal partial derivative A ^ Axd is an R[X\,..., Xj-i, Xj+i,..., X m ]-derivation of R[Xi,..., Xm], and if R is a field K then this induces a K(Xi,..., Xj-i,Xj+i,..., X m )-derivation of K(X1,..., Xm). Note that in the said section, in the defining equation Ax(X) = ^2iAiXi^1 for i = 0 we take iAiXl~l = 0, and in the defining equation of Ax{X\,..., Xm) for ij — 0 we take the corresponding term to be zero. This avoids negative powers. EXERCISE ( E l l ) . Let D be a derivation of a ring S. Show that for all x in the prime subring of S we have D{x) = 0. Show that if S is a field then for any u, v in S with D ^ O w e have (the quotient rule): D(u/v) = (vD(u) - uD(v))/v2. Deduce that if 5 is a domain then D has a unique extension to the quotient field K of S, i.e., there is a unique derivation D' of K such that D'(z) = D(z) for all z € S. EXERCISE (E12). Let A(XU ...,Xm)e R[XU. ..,Xm] where Xu...,Xmbe indeterminates over a ring R, and let x\,..., xm be elements in an overring S of R. Show that for any D in Der^(5, S) we have (the chain rule): D(A(xi,...,xm))=

Yl

AXj(xi,...,Xm)D(xj).

l<j<m

More generally show that for any D in Der(S', S) we have (the more general chain rule): D(A(xi,...,xm))=AD(xi,...,xm)+

^2

AXd(xi,...,xm)D(xj)

l<j<m

where AD(XI, . . . , Xm) 6 R[X\,..., cients of A(Xi,..., Xm).

Xm] is obtained by applying D to the coeffi-

NOTE (N2). [Very Long Proofs]. As said before, CT = the Classification Theorem of finite simple groups has a proof of about ten thousand pages. Actually, the 1980 proof had some gaps which were repaired (?) only in 2004. Now a proof which is that long is not very digestible. In doing mathematics it is good to make clear what your assumptions are, and to base your arguments on results whose proofs you have checked. But results like CT can be useful guides in the search for truth. For applications of CT see Kleidman-Liebeck's book [KLi].

§ 13: CONCL UDING NO TE

29

§13: C O N C L U D I N G N O T E Algebra means what you do not know, call it x. The experimental data will produce an equation in x. Solve it. The value of x so obtained will tell you what you wanted to know.

Lecture L2: Curves and Surfaces §1: MULTIVARIABLE W O R D PROBLEMS In the last lecture I switched from calling the unknown quantity x to calling it Y. I did this to make room for bivariate (= two variable) equations because a word problem may simultaneously have two unknown quantities. For instance, let us again go to the garden and pluck some roses and some gardenias. Then let us go to two more gardens and pluck the same number of roses as well as gardenias as we did from the first garden. From yet one more garden, we pluck only roses, as many as from the first garden, but do not pluck any gardenias. Counting up all the flowers we see that we have collected a total of fifty which is a nice number to make a bouquet. Needing also a garland, we go to four more gardens and from each pluck the same number of roses as well as gardenias as from the first garden, and make a garland of the sixty flowers so collected. How many roses and how many gardenias did we pluck from the first garden? Calling X the number of roses we plucked from the first garden and Y the number of gardenias which were plucked there, we get the two simultaneous equations: AX + 3Y - 50 = 0

and

AX + AY - 60 = 0.

Subtracting the first equation from the second we get Y = 10 and substituting this in the first or the second equation we obtain X = 5. Now the word problem may be more complicated, such as going to as many shops as the number of roses first picked and from each of them buy the same number of roses, and so on. This could lead to equations such as: Y2 + 7XY + 3Y + 2X - 9 = 0

and

AY3 + 2XY2 + 2Y - 11 = 0

or even three variable equations such as Y3 + XZ-5

= 0

and

Y2 + AX2 - 4 = 0.

Of course, we need not always pick flowers. We could shoot arrows instead or count the number of elementary particles as they collide in a cyclotron. Or we may compare distances between stars, or what have you. At any rate, such word problems frequently lead to several polynomial equations in several variables. We solve them, or try to solve them, and interpret the found values of the variables as measurements of the physical quantities we started with. To solve these equations we could take recourse to the geometric method initiated by Descartes around 1630. For instance the first two linear equations represent two lines in the (X, F)-plane, and they intersect in the point (5,10). Turning to the second set of equations and plotting them as curves in the (X, Y)-plane, the first equation, which is of degree 2, is a hyperbola and the second equation, which 30

%1: MULT1VARIABLE WORD PROBLEMS

31

is of degree 3, is a cubic curve. By Bezout's theorem they intersect in 2 x 3 = 6 points, giving us six solutions of the two simultaneous equations. The third set of trivariate (= three variable) equations represent surfaces in the (X, Y, 2T)-space. Their degrees being 3 and 2, again by Bezout's theorem they intersect in a space curve of degree 3 x 2 = 6. Note that the second equation is a vertical elliptical cylinder. So what is this theorem of Bezout? Well, Descartes' analytic geometry eventually evolved into algebraic geometry and Bezout's theorem, proved around 1770, was the first substantial theorem of that new discipline. It says that in the plane, a curve of degree m and a curve of degree n always meet in mn points. Likewise, in three-space, a surface of degree m and a surface of degree n always meet in a curve of degree mn. Moreover, in three-space 3 surfaces of degrees di,d2,d^ will meet in ^1(^2^3 points. All this, as we shall see, provided things are counted properly! But learning to count properly may take a while. Quite generally, our word problem may lead to several polynomial equations in several variables. Generalizing the idea of a curve in the plane and a surface in 3-space, for any positive integer N, the points in A^-space satisfying an A'-variate (= N variable) polynomial equation F(Xi,... ,XN) = 0 are said to form a hypersurface, and the degree of F is called the degree of that hypersurface. Again as a curve in the plane is 1-dimensional and a surface in 3-space is 2-dimensional, so a hypersurface in N-space is (N — l)-dimensional, the intersection of two of them is (N — 2)-dimensional, the intersection of three of them is (N — 3)-dimensional, and so on; all this only "in general." If our word problem leads to N hypersurfaces Fi(X\,..., XN) = 0 for 1 < i; < N, then Bezout says that they meet in d\.. .dry points where di is the degree of Fi. If for some M < N we have only the first M hypersurfaces F\ = 0 , . . . , FM = 0 then Bezout says that they meet in an (N — M)-dimensional variety of degree d\... &M- Again provided everything is counted properly. We also have to ensure that the hypersurfaces are "mutually independent"; for instance the equation F% = 0 should not be a consequence of the equations F\ = 0 and F2 = 0, as it would be say if F3 = Fi + F2. Also we must attach precise meanings to a variety, its dimension, and its degree. Very briefly, a variety is the geometric locus consisting of the common solutions of a finite number of polynomial equations in a finite number of variables, its dimension is the "degree of freedom" of a point moving on it and is definable in terms of suitable transcendence degrees, its degree is the number of points in which it intersects a complementary dimensional linear subspace and is definable in terms of suitable field degrees, and so on. To get ready for this proper counting, let us start by considering an algebraic curve C in the (X, F)-plane over a field k. Now C is given by a bivariate polynomial equation f(X, Y) — 0 where

f(X,Y) = a0(X)Yn+ J2 di(X)Yn~i l
32

LECTURE L2: CURVES AND SURFACES

with Oi(X) G fe[X] and 0 ^ a 0 (X) G k[X]. To discuss points in the (X, Y)-plane let us introduce cartesian products. The cartesian product 5 x T of sets S and T consists of all pairs (a, /3) with a £ S and (3 G T. Similarly the cartesian product S\ x • • • x Sm of sets Si,... ,Sm consists of all m-tuples ( 7 1 , . . . , 7 m ) with 74 G Si. If 5i = • • • = Sm — S then S\ x • • • x Sm is denoted by Sm. Now for a typical point (a,/?) in the (X, Y)-plane A;2 we have a £ k and /? G k; this point is on C if / ( a , /3) = 0. Assuming / to be irreducible in k[X,Y], and projecting vertically onto the X-axis, above each point X = a, in general there lie n points ( a , / 3 j ) , . . . , (a,@n) on the curve C. The degree of C, i.e., the (X, F)-degree of / , could be greater than the covering degree n, i.e., the Y- degree of / . To ensure that in general / ( a , Y) has n roots 0i,..., /3n in k, we need to assume that the field k is algebraically closed, which means every nonconstant univariate polynomial over k has a root in k. For instance the complex field C is algebraically closed. This used to be called the fundamental theorem of algebra, though it may be argued that it is neither fundamental nor a theorem of algebra. At any rate, most of its proofs involve analysis and topology. Many of them were given by Gauss between 1800 and 1840. Then around 1880, Kronecker constructed an algebraic closure of any field in a purely algebraic manner. Here by an algebraic closure of a field K we mean an algebraically closed overfield K of K which is algebraic over K. Sometimes we may call K an absolute algebraic closure of K to distinguish it from the (relative) algebraic closure of K in an overfield L of K by which we mean the subfield of L consisting of all elements of L which are algebraic over K. Indeed, over any field K the method of first constructing a root field of a univariate irreducible polynomial and then using it to construct a splitting field SF(F, K) of any nonconstant univariate F(Y) G K\Y] described in Lecture LI is due to Kronecker. In turn this may be used to construct an algebraic closure K of K. Namely, let us label all the distinct nonconstant monic polynomials in K[Y] as Fi(Y),F2(Y),..., and then after letting Ki+i - SF (FuKi) with Kx = K, let us putK = l)?L1Ki. This is ok if K is countable, i.e., if K is either finite or there is a bijection of N+ onto K, because in both these cases the nonconstant monic polynomials in K\Y] can be labelled as .Fi, F2, But if K is uncountable then what to do? Well, let us try making "longer" sequences. At the end of all positive integers let us put a symbol u>, thus getting the "sequence" 1,2,..., w, where u > i for all i G N+. Then start again with LJ + 1,W + 2,.. .,LJ2+OJ,... ,LJ2 + 2LJ, .. . , 2 w 2 , . . . ,3w 2 ,... and then 3 4 w u u w w , . . . , w , . . . ,w ,w ' + l,w + 2,... ,u) +CJ, ... ,w w ^, and so on to kingdom come! But will the kingdom ever come? Nobody knows! In other words, it is not known if in this "constructive" manner we will ever get beyond a countable set. So we take recourse to the axiom of choice or the well ordering principle or Zorn's Lemma. Let us proceed to describe these three postulates and show their mutual equivalence. The axiom of choice says that given any set of nonempty subsets of a set, it is

§J: MULT1VARIABLE WORD PROBLEMS

33

possible to pick one element from each subset. To reformulate this in terms of maps, for any set T let V{T) = the power set of T, i.e., V(T) = the set of all subsets of T, and let Vx (T) = the restricted power set of T, i.e., V* (T) = V(T) \ {0}. Then the axiom of choice says that given any map 4> : S —> VX(T) from a set S to the restricted power set Vx (T) of a set T, there exists a map ip : S —> T such that for all a; € 5 we have VK^) € x < z, and for all x, y in S we have x < y and y < x •& x = y; we write x < y to mean x < y and x ^ y. A poset (= partially ordered set) is a set with a partial order on it. A lo (= linear order = order) on a set S1 is a partial order on S such that for all x, y in S we have either x < y or y < x. A loset (= linearly ordered set) is a set with a linear order on it. A chain in a poset S is a nonempty subset of 5* which is a loset under the induced partial order. A maximal element of a poset S is an element y of S such that for all x € S we have: y < x => y = x. An upper bound of a nonempty subset R of a poset S is an element y of S such that x < y for all x £ R, and a smallest element of R is an element z of R such that z < x for all x € R. A wo (= well order) on a set S is a po on S such that every nonempty subset R of S has a smallest element; the said smallest element is clearly unique, i.e., it is determined by R; we may denote it by min(fl). A woset (= well ordered set) is a set with a well order on it. For any x =fi y in a woset 5, we have x < y or y < x according as the smallest element of {x, y} is x or y. Therefore every woset is a loset. The well ordering principle says that every set has a well order. A poset has the Zorn property if every chain in it has an upper bound. Zorn's Lemma says that if a nonempty poset has the Zorn property then it has a maximal element. As said above, out of the three postulates, the axiom of choice is self-evident. Out of the remaining two, the well ordering principle is surprising but Zorn's Lemma is more versatile. At any rate, as we shall see in a moment, the three are equivalent. After establishing the equivalence, we shall complete the above proof of the existence of an algebraic closure first by using the well ordering principle and then by Zorn's Lemma. We shall also show that any two algebraic closures K and if of a field K are if-isomorphic, i.e., there is a if-isomorphism K —> K . Consequently we may sometimes speak of the algebraic closure of K, rather than an algebraic closure of K. Now bijections define cardinals or cardinality. Two sets have the same cardinality if there is a bijection between them. For instance, all triads, i.e., sets of size three, define the cardinal 3, or as they used to say in school "new math," the threeness of three. The cardinality of a set S is denoted by |£|. It can be shown that cardinals are linearly ordered by defining l^l < \T\ to mean that there is an injection from 5 to T. The assertion that cardinals are partially ordered under this

34

LECTURE L2: CURVES AND SURFACES

binary relation, is known as the Schroeder-Bernstein Theorem; we can reformulate it by saying that if there are injections S —> T and T —> S then there is a bijection S —> T. To see that this partial order is a linear order, what we need to prove is that given any sets S and T, either there exists an injection from S to T, or there exists an injection from T to S. As bijections between (unordered) sets define cardinals, order preserving maps between well ordered sets define ordinals in the following manner. A bijection

x e R. Let us denote the ordinal of a woset S by ||5||, and for any wosets S and T let us define | | 5 | | < ||T|| to mean that there is an order preserving bijection from S onto a lower segment of T. By using Zorn's Lemma it can be shown that under this relationship, any set of ordinals is well ordered. An ordinal is a limiting or nonlimiting ordinal according as it does not or does have an immediate predecessor, i.e., according as the set of all smaller ordinals does not or does have a maximal element. For instance w, 2u>,..., u)w are limiting ordinals, whereas 1,2,..., w + 1, u> + 2 , . . . , ww + 1 are nonlimiting ordinals. The idea of polynomials in UJ is relevant for constructing the algebraic closure of the meromorphic series field k((X)) in characteristic p ^ 0. So let us proceed to introduce meromorphic series. §2: P O W E R SERIES A N D M E R O M O R P H I C SERIES Writing a polynomial in an indeterminate X over any ring R in ascending powers of X and letting them go to infinity on the right hand side we get a power series in X, and letting them have a finite number of terms on the left, i.e., terms with negative exponents, we get a meromorphic series. We denote the power series ring by -R[[X]], and we denote the meromorphic series ring by R{(X)). We regard R[X] as a subring of i?[[X]], and we regard i?[[-X"]] as a subring of R((X)). More precisely, a meromorphic series A(X) in X over R, i.e., a member of R((X)), has a unique expression

A(X) = YlAiXi

with

AieR

such that Supp(A) is a well ordered subset of Z where the support of A = or the X-support of A = A(X), is defined by putting Supp(A) = Supp x A = {i e Z : At ^ 0}

A(X),

§2: POWER SERIES AND MEROMORPHIC SERIES

35

and we note that: Supp(A) = 0 •& A = 0. We define the order of A, or the X-order of A, by putting ord(,4) = ordxA = min Supp(A) i.e., ,, ,. [ the smallest element of Supp(A) oral A) = < \oo

if A ^ 0 ifA = 0.

As in the case of polynomials, for any other meromorphic series B(X) = J2 BiX*

with

Bi e R

C{X) = A(X) + B(X) =

J2°ixi

the sum

is given by componentwise addition Ci = Ai + Bi and the product

D(X) = A(X)B{X) = ] £ A** iGZ

is given by the "Cauchy multiplication rule"

Dk = 5 3 AiBj. i+j=k

It is this rule which makes us require Supp(A) to be well ordered, because otherwise there may be infinitely many pairs (i,j) with Ai ^ 0 ^ Bj and i + j = k. At any rate, in an infinite summation, by convention we disregard all the zero terms, and so the summation makes sense provided there are only finitely many nonzero terms. The power series ring is defined by putting R[[X}} = {A = A{X) 6 R((X))

: ord(A) > 0}.

As in the polynomial case, the "formal" X-derivative Ax{X) = R((X)) is defined by putting AX{X)

=

dx

' of A{X) in

Y,iAiXi~1

and again this gives an /?-derivation of R((X)). If R is a domain then the order of the product of two nonzero meromorphic series equals the sum of their orders, and hence in particular the product is nonzero. It follows that if R is a domain then so is the meromorphic series ring R((X)) as well as the power series ring i?[[X]].

LECTURE LS: CURVES AND SURFACES

36

For finitely many variables (= indeterminates) Xi,... ,Xm over a ring R, we consider only the power series ring H[[Jfi,..., Xm]], which we regard as an overring of R. A power series in Xi,..., Xm over R, i.e., a member of R[[Xi,..., Xm]}, has a unique expression

A(Xu...,Xm)=

A

h-imX[1...Xi^

E

with

Ah...imeR.

(ii,...,im)eN m

The order of A, or the (Xi,..., X m )-order of A, is the minimum of %i + • • • + im with j4jj.,.jm ^ 0; if A = 0, i.e., if Aiu^im = 0 for all z i , . . . , i m , then the order is taken to be oo; we denote the order of A by ord(A) or ord^x1,...,xm)AFor a n y other power series B(Xu...,Xm)=

Yl

Bil...imX?...X%

with

Bh...imGR

«m)6Nm

(»i

the sum C ( X i , . . . , Xm) = A ( X i , . . . , X m ) + B ( X i , . . . , X m )

=

cv.. im xf i ...x£

E (n,...,i m )eN m

is given by componentwise addition <~'ii...im

— - ^ i i ...im. >

£>ii...iTn

and the product D(X\,...

,Xm)

= A(X\,...

=

,Xm)B(Xi,...

E

,Xm) i

Dh.,.imX ^...Xi-

(«l,...,»m)€N"'

is given by the "Cauchy multiplication rule" i'fei...fc tn =

2^i

•Ai1...imBj1,..jm.

»l+jl=fcl>...,*m+jm = fcm

The "formal" partial derivative AXj {X1,...,Xm)= in R[[X\,..., X m ]] is defined by putting AXj (Xi,...,Xm)=

(Xu^..,xm)

h A i i • • -im X%\ • • • X j ' - 1

22 (»i,...,«m)eN

aA

X%

of

j

^ ^ ^ _^

^

X

j+~1 • • • X%m

m

with the understanding that for ij = 0 the corresponding term in the above summation is zero, and we note that A — i > Ax, gives an R[[Xi,..., X , _ i , X , - + 1 , . . . , Xm]}derivation of R[[Xi,..., Xm]]. If R is a domain then the order of the product of two nonzero power series equals the sum of their orders, and hence in particular the product is nonzero. It follows that if R is a domain then so is the power series ring i?[[-Xi,..., Xm}\. For a finite number of variables Xi,..., Xm over a field K, by

37

%2: POWER SERIES AND MEROMORPHIC SERIES

K((Xi,..., Xm)) we denote the field of meromorphic functions in X\,..., Xm over K. Members of the meromorphic function field K((Xi,... ,Xm)) are "quotients" of power series A{X\,..., Xm)/B{X\,..., Xm) with nonzero B. By the "quotient rule" we extend the above X[[-Xi,... ,Xj~i,Xj+i,...,Xm]]-derivation of the domain K[[Xi,..., Xm}} to a K((Xi,..., Xj-i, Xj+i,..., X m ))-derivation of the field

K((Xu...,Xm)). Formally speaking, for any ring R, we may regard R((X)) as the set of all functions (= maps) from Z to R with well ordered support. At any rate, R((X)) c i? z , where for any sets S and T, by ST we denote the set of all functions :T —> 5 . In case S has a special element e, like the zero element in a ring, by the support of (j> we mean the set of alH € T such that 4>(t) ^ e, and we denote it by Supp((/>). By (5'T)finite let us denote the set of all 4> £ ST such that Supp(0) is finite, and in case T is a poset, by (5 r ) we iiord let us denote the set of all € ST such that Supp(i/>) is well ordered. Note that then R{{X)) = (#z)weiiord, R[[X]] = Rn, and R[X] = (i?N)flniteIn case of a finite number of variables Xi,..., Xm we have -R[[X 1 , . . . , X m ]] = -ft" and R[X\,... ,Xm] = (RN )flnite- In case of R((X)), the function which sends 1 £ Z to 1 S R and every other element of Z to 0 £ R corresponds to X. In case of R[[Xi,..., Xm]}, the function which sends ( 0 , . . . , 0 , 1 , 0 . . . , 0) e N m with 1 in the i-ih spot to 1 € R and every other element of N m to 0 6 R corresponds to Xj. Similarly for R[[X}} and R[X]. To speak formally about quotient fields, we first introduce the concept of an equivalence relation. An equivalence relation on a set S is a binary relation x ~ y which holds between certain pairs (x, y) 6 S2 and which is reflexive: for all x € S we have x ~ x, symmetric: for all x, y in S we have x ~ y => y ~ x, and transitive; for all x, y, z in S we have x ~ y and y ~ z =^ £ ~ z. By putting all the elements which are equivalent to each other in one box, the set S gets divided into a disjoint partition, i.e., a family (= set) of nonempty subsets any two of which have an empty intersection. This family of boxes = equivalence classes is denoted by S/ ~ and is called the quotient of S by ~; the map which sends every x G S to the box containing it is called the natural surjection S —> S/ ~ . Now the quotient field F of a domain E is defined to be the set of all equivalence classes of pairs (u, v) £ E2 with v ^ 0 under the equivalence relation: (u, v) ~ (u',v') <$• uv' = vu'. The equivalence class containing (u, v) is denoted by u/v or ^, and we add and multiply the equivalence classes by the rules ~L + ^ = "i"a+u2t'i and i£ x *a = ajsa. Also we "identify" any u e £ with f e F . This m a k e s F a field, and £ a subring of F . We may write QF(F) for F. Given two finite sets of variables X\,..., Xm and Zi,...,Zd over a ring i?, and given any power series P^(Zi,..., Zj) "devoid" of constant term, i.e.,

pW(Zll...,Zd)=

J; 0'i,.-.,jd)eN<»

^

^

-

^

with

P^.,d£R

38

LECTURE L2: CURVES AND SURFACES

such that

PQ...O

=

®> ^n a n y P o w e r series

A(X1,...,Xm)

=

Ail...imXi^...Xi-

J2

with

Au...imeR

im)eN m

(tl

we can "substitute" P ( i ) for Xt for 1 < I < m to get A(p\Z1,...,Zd)

A(pW(Z1,...,Zd),...,p(m\Z1,...,Zd))

= (JI

jd)eNd

This makes sense because it does not involve infinite sums of coefficients. Indeed, for any i = (ii,...,im) € N m , upon writing

p^(z1,...,zdr...p^mHz1,...,zdy-= with P-j

Yl

p

it.iA'---zdd

• G R, clearly we have P

h-jd

=

°

whenever

jx +

\- j d < ix -\

+ im.

Thus A i-> A ( P ) gives the P-homomorphism 0 : R[[Xi,..., Xm]] —> P [ [ Z i , . . . , Zd]], which we call the substitution homomorphism. Note that by taking d = 0 we get R[[Zi,..., Zd]] = R and by taking pW = 0 for 1 < I < m we get the P-epimorphism R[[Xi,..., Xm}} - • P given by A(XU. . . , X m ) >-> A{0,..., 0) = 4>... 0 . T n e P ° w e r series P ^ ^ Z i , . . . , Zd),..., P ^ m ' ( Z i , . . . , Zd) (which are assumed devoid of constant terms) are said to be analytically independent or analytically dependent over P according as ker(0) = 0 or ker() ^ 0. If P ( 1 ) ( Z i , . . . , Zd),..., P ( m ) ( Z i , . . . , Zd) are power series (devoid of constant terms) over a field K such that the substitution map <j> : K[[Xi,.,. ,Xm}] —> K[[Zi,... ,Zd}] is injective, then <j> uniquely extends to an injective K-homomorphism K((X\,..., Xm)) —> K((Zi,..., Zd)) which we again call the substitution map. Considering the geometric series identity (i)

(1-X)(1

+ X + X2 + ...) = 1

in the univariate power series ring P[[X]] over a ring P , and taking any power series P ( Z i , . . . , Zd) in a finite number of variables Z\,..., Zd over P with P ( 0 , . . . , 0) = 0 and substituting it for X we get (l-P(Z1)...,Zd))P'(Z1)...,Zd) = l where

P'(Z1,...,Zd)=

J2 0
P{Zl,...,Zd)i£R[[Zu...,Zd)}.

§3: VALUATIONS

39

If the element Q{Z\,..., Zd) £ R[[Z\, • • •, Zd]] is such that Q(0, • •., 0) is a unit in R then letting P(ZU ...,Zd) = l - Q ( 0 , . . . , 0 ) - 1 Q ( Z 1 , . . . , Zd) we get P ( 0 , . . . , 0) = 0 and Q(Zi,...,Zd) = Q(0,... ,0)(1 - P(ZU... ,Zd)) and therefore by letting Q'(ZU..., Zd) = Q ( 0 , . . . , O ) " 1 ^ ^ ! , . . . , Zd) we get Q(Zlt..., Z d ) Q ' ( Z i , . . . , Zd) = 1. Thus we have shown that if Q ( 0 , . . . , 0) is a unit in R then Q(Z\,..., Zd) is a unit in R[[Zi,..., Zd]]. Conversely if Q(Zi,..., Zd) is a unit in R[[Zi,..., Zd}} then for some Q*(Z1:. ..,Zd) in R[[Z1:..., Zd}\ we have Q(ZU ..., Zd)Q*{Zu. ..,Zd) = 1, and putting Zx = • • • = Zd = 0 we get Q ( 0 , . . . ,0)Q*(0,... ,0) = 1, and hence Q ( 0 , . . . , 0) is a unit in R. Recalling that U(R) is the group of all units in R, we conclude that (u(R[[Z1,...,Zd}])=

{Q(Z1,...,Zd)eR[[Z1,...,Zd}}:

\

Q(0,...,0)£U(R)}.

Applying the above consideration to the power series ring ivT[[Xi,... ,Xm}] in a finite number of variables X\,... ,Xm over a field K, we see that the units in . ^ [ [ X i , . . . , Xm]} are exactly those power series whose order is zero. It follows that in the univariate case, the meromorphic series field K((X)) is a concrete representation of the quotient field of X[[X]], i.e., there is no discrepancy in the two meanings of

K((X)). §3: VALUATIONS Again considering the power series ring if [[Xi,... ,Xm]} in a finite number of variables over a field K, we can uniquely extend the order function from the domain K\[Xi,..., Xm]] to its quotient field K((X\,... ,Xm)) by defining the order, or the (Xi,..., X m )-order, of u/v for any u, v in K[[X\,..., Xm]\ with v ^ 0 by putting oid(u/v) = ovd(x1,...,xm)u/v

= ord(u) — ord(v)

with the convention that ord(0) = oo. Now for all x,y in K((Xi,..., (VI)

Xm)) we have

ord(xy) = ord(x) + ord(j/)

and (V2)

ord(o; + y) > min(ord(x),ord(2/))

with obvious conventions about oo. In particular, for all x in K((X\,..., have (V3)

Xm)) we

ord(:r) = o o « i = 0.

The second inequality also holds for all x,y in i?[[-X"i,... , X m ] | for any ring R. To make this transparent it is best to express every power series as a sum of

40

LECTURE

L2: CURVES AND

SURFACES

homogeneous polynomials by "collecting terms of like degree." In greater detail, a polynomial H(X1,...,Xm) = ^fr<1...imX*l...X^

with

Hh...im£R

is homogeneous of degree j if for all ( i i , . . . , im) £ Supp(H) we have i\ -\ \-im = j ; note that the zero polynomial is regarded as homogeneous of every degree. Now a power series

A(X1,...,Xm)=

A

Y,

n-imX^...X^

with

4..,mefi

(ti,...,tm)€Nm

can uniquely be written as A(X\,.. .,Xm)

=

2_^

Aj(Xi,...,Xm)

0<j
where the polynomial

Aj(X1,...,Xm)=

J2

Ail...imX?...X%

is homogeneous of degree j . Given any other power series B(X\,...

,Xm)

=

2_^ Bj(X\,...

,Xm)

0<j
where Bj (X\,..., Xm) e R[Xi,..., Xm] is homogeneous of degree j , assume A ± 0 ^ B and let ord(A) = a and ord(B) = b. Then A(Xi,...,

Xm) — Aa(Xi,...,

Xm) + terms of degree > a

B(Xi,...,

Xm) = Bb(Xi,...,

Xm) + terms of degree > b

and

with Aa{Xu.

..,Xm)^0^

Bb(Xu...,

Xm).

It follows that A(Xi,.

..,Xm)

+ B(Xi,..

.,Xm)

f

Aa(Xi,..., Xm) + terms of degree > a if a < b Xm) + terms of degree > b if a > 6 = < Bb(Xi,..., Aa(Xi,.. .,Xm) + Ba(Xi,..., Xm) + terms of degree > a if a = b

§3: VALUATIONS

41

and hence ord(A + B) > min(a, b) with equality in case of a ^ b. Also note that if R is a domain then A(Xi,...,

Xm)B(Xi,...,

= Aa (Xi,...,

Xm)

Xm)Bb(Xi,...,

Xm) + terms of degree > a + b

and 0 ^ Aa(X\,... ,Xm)Bb(Xi,... ,Xm) G R[X\,... ,Xm] is homogeneous of degree a + b, and hence ord(AB) = a + b. To generalize the idea of meromorphic series, let G be an ordered abelian group, i.e., G is an additive abelian group which is also an ordered set such that for all x, y, x', y' in G we have: x < y and x' < y' => x + x' < y + y'. For instance G = Z or Q or R. Or G could be the set M'd' of lexicographically ordered d-tuples of real numbers r = (n,. • • ,rd.),s = (s\,..., Sd), • • •, where lexicographic order means: r < s •*=> either n — Si for 1 < i < d, or for some j with 1 < j < d we have T\ = Si for 1 < i < j and rj < Sj. Or G could be a subgroup of any of these. Inspired by properties (VI) to (V3), we define a valuation of a field K to be a map v : K —* G\J {oo} such that for all x,y m K we have (VI) to (V3) with ord replaced by v, with the conventions about oo that: for all g G G we have g < oo and g + oo = oo, and also 00 + 00 = 00. We call G the assigned value group of v, and by the value group of v we mean the subgroup of G given by :xGKx}.

Gv = {v(x) By the valuation ring of v we mean the ring Rv = {xGK

: v(x) > 0}

and we note that this is clearly a subdomain (= subring which is a domain) of K which is its quotient field. We say that v is trivial over a subfield k of K, or that v is a valuation of K/k, to mean that x G kx =>• v(x) — 0 and we note that this is equivalent to saying that k C Rv, i.e., k is a subfield of Rv. Now the ord function is obviously a valuation v of K{(X\,... ,Xm))/K with GV = G = Z

and

K[[XU . . . ,Xm]] C Rv.

Note that in case of m = 1 the above displayed inclusion is an equality. To construct a valuation having a given ordered abelian group G as its value group, and at the same time to generalize the idea of the univariate meromorphic series field K((X)) over any given field K, let K((X))G

= {A&KG

: Supp(A) is well ordered}

where the support of A G KG, denoted by Supp(j4), is defined by putting Su P P (A) = {g G G : A(g) ± 0}.

42

LECTURE L2: CURVES AND SURFACES

For A, B in

the sum is componentwise, i.e., for any g G G we have

K((X))Q,

(A + B)(g) = A(g) + B(g) and the product is by the "Cauchy rule" which says that for all g G G we have

(AB)(g) = J2

A

0)B(j)

i+j=g

and the well ordering enters in showing that (V4)

\{(h3)£G2

:i+j

= g with i eSupp(A) and j GSupp(B)}

1 is a finite set. We also have (V5)

Supp(yl) and Supp(B) are well ordered => Supp(AB) is well ordered.

Note that A = 0 means A(g) = 0 for all g G G. Also A = 1 means A(0) = 1 and A(g) = 0 for all other g G G. For any A G i f ( ( X ) ) G we define ,. ,, fmin SuppfA) ord(A) = < \oo

ifA^O ifA = 0

and we note that then for all x,y in K((X))G we have (VI) to (V3). Mimicking the proof of (ii) using (i) in §2, we can show that (V6)

0 ^ €

K((X))G

=* AA' = 1 for some A' €

K{{X))G.

Thus K((X))G is a field, and ord is a valuation of K{{X))a/K with value group G. Concretely speaking, a typical member of K((X))G can be written as

A{X) = Y,AiXi where we are writing Ai for the previous A(i). Assuming G ^ 0 and designating a suitable nonzero element of G as 1, the function which sends I £ G to 1 £ K and sends every other element of G to 0 S R now corresponds to X. For instance, if G = ZM or QM or R ^ , then designate ( 1 , 0 , . . . , 0) to be 1 G G. We also put K[[X]]G = {A(X) G A"((X)) G : oid(A) > 0} and we note that iC[[X]]c is the valuation ring of the ord valuation of K((X))GTaking Q to be the exponent group, i.e., taking Q for G, for any n G A^+ we put Qn = {j G Q : nj G Z} and for any prime p we put Qn

= {j G Q : njp r G Z for some r G N + }

§4: ADVICE TO THE READER

43

where r may depend on j , and we let ^(PO)newt = U ne N + {A(X) G K((X))Q

: Supp(A) C Q„}

and in case K is of characteristic p we let K((X))gnewt

= UneN+{A(X)

These are clearly subfields of K((X))q, of the domains

K[[X]] newt

G X((X)) Q : Supp(A) C Q n , p }. and they are the respective quotient fields

—

{A(X) G tf((A-))„ewt : ord(A) > 0} and K[[X}]enewt

= {A(X) G K((X))gnewt

: ord(A) > 0}.

We shall give several proofs of the following theorem which dates back to 1660. NEWTON'S THEOREM (Tl). Let k be an algebraically closed field of characteristic 0. Then k((X))newt is an algebraic closure of k((X)). We shall also prove the following generalization of Newton's Theorem. GENERALIZED NEWTON'S THEOREM (T2). Let k be an algebraically closed field. Then k((X))q is an algebraically closed overfield of k((X)). If k is of characteristic p > 0, then k((X))gnev^ is an algebraically closed overfield of k((X)), which is strictly bigger than the algebraic closure of k((X)); more precisely, k((X))gnewt contains elements whose supports are such that their ordinals are not polynomials in u, but the ordinal of the support of any element which is algebraic over k((X)) is a polynomial in w. §4: A D V I C E TO T H E R E A D E R The next section on Zorn's Lemma and Well Ordering, spread over seven pages, is rather heavy logical stuff. This inspires me to verbatim reproduce the following lines of advice from the preface of the Second Part of the great and famous Algebra Text-Book of George Chrystal written in 1889 [Chr]: "I may here give a word of advice to young students reading my second volume. The matter is arranged to facilitate reference and to secure brevity and logical sequence; but it by no means follows that the volume should be read straight through at a first reading. Such an attempt would probably sicken the reader both of the author and the subject. Every mathematical book that is worth anything must be read "backwards and forwards," if I may use the expression. I would modify the advice of a great French mathematician and say, "Go on, but often return to strengthen your faith." When you come on a hard or dreary passage, pass it over;

44

LECTURE L2: CURVES AND SURFACES

and come back to it after you have seen its importance or found the need for it further on." So, in a first reading, without loss of understanding, the reader may prefer to immediately proceed to the section after next where I shall give a Utilitarian Summary of the logical stuff. §5: ZORN'S LEMMA A N D WELL O R D E R I N G We want to show the equivalence of the three postulates: Axiom of Choice, Well Ordering Principle, and Zorn's Lemma. Let us abbreviate these as (AC), (WO), and (ZL), and restate them. Let us also include an Alternate Version of Zorn's Lemma which we abbreviate as (ZL*). (AC). Given any map (f> : S —• Vx (T) from a set S to the restricted power set P (T) of a set T, there exists a map ip : S —* T such that for all x £ S we have ip(x) £ 4>{x). (WO). Every set has a well order. (ZL). If a nonempty poset has the Zorn property then it has a maximal element. Recall that a poset has the Zorn property means every chain in it has an upper bound. (ZL*). If a nonempty poset has the weak Zorn property then it has a maximal element, where a poset has the weak Zorn property means every woset in it (= well ordered subset of it) has an upper bound. x

First let us prove the following: LEMMA (T3). Let P be a set of subsets of a poset S. Assume that every member of P is well ordered (in the partial order induced from S). Also assume that P is a loset (in the "lower segment" partial order on P according to which A < B in P means A is a lower segment of B). Let Q = 1)R€PR. Then Q is well ordered (in the partial order induced from S). PROOF. Given any nonempty subset Q' of Q, we can take R € P such that R n Q' is nonempty. Let x = min(i? n Q'). We claim that then x = min(Q'). So let there be given any y £ Q''. If y £ R then obviously x < y (in the partial order of S). If y £ R then y € Ri for some i?i £ P; since y £ Ri \ R, Ri cannot be a lower segment of R; since P is a loset, R must be a lower segment of the woset R\; therefore, since y £ R\\R and x £ R, we must have x < y. Now we are ready to prove the: EQUIVALENCE THEOREM (T4). (AC) =*> (ZL*) => (ZL) =• (WO) =>(AC).

§5: ZORN'S LEMMA AND WELL ORDERING

45

PROOF OF (AC) =>• (ZL*). Assume (AC), and let 5 be any nonempty poset in which every woset has an upper bound. Suppose if possible that S has no maximal element. By taking <j> to be the identity map VX(S) —> PX(S) in (AC), we get a map V : VX(S) - • S such that for all 0 ^ R C S we have i>{R) £ R. Let W(S) be the set of all nonempty well ordered subsets of 5, and for every R £ W(S) let R* be the set of all upper bounds of R which do not belong to R; by assumption R has an upper bound and S has no maximal element, and hence we can take an element in 5 which is strictly bigger than that upper bound thereby getting an upper bound not in R; thus R* is nonempty and so tp(R*) £ R*. For every y £ S let L(y) = {x £ S :x (S). Let V{S) be the set of all R £ W(S) such that: (i) min(i?) = z and (ii) for all z ^= y £ Rv/e have il>{L{y,R)*) = y. Clearly {z} € V(S) and hence V(S) ^ 0. We claim that V(S) is a loset (in the "lower segment" partial order). To see this, given any Ri and R2 in V(S) let Ro = {y £ Ri D R2 : L(y,Rx) = L(y,R2)}. Then z £ R0 and hence RQ T^ 0. Clearly RQ is a lower segment of Ri as well as R2. Suppose if possible that Ro ^ Ri for 1 < i < 2; then upon letting yt = min(i?j \ Ro) we get i?o = L(yt, Ri) for 1 < i < 2; consequently j/i = V(i(2/i,i?i)*) = ^((-Ro)*) = ip{L{y2,R2)*) = 2/2, and hence yi = y2 £ Ro which is a contradiction. Therefore either Ro = -Ri or Ro = R2, and hence either JJi is lower segment of R2 or i?2 is lower segment of R\. This proves the claim. Now, upon letting T = L)R^V(S)^-: by the above lemma we get T £ W(S) and from this it follows that T £ V(S). Consequently, upon letting f = T U {ip(T*)}, we see that f £ V(S) with f <£ T which is a contradiction. Therefore S1 must have a maximal element. Thus (AC) => (ZL*). PROOF OF (ZL*) =* (ZL). Obvious. PROOF OF (ZL) => (WO). Assume (ZL), and given any set S let W(S) be the set of all well ordered subsets of S. Note that W(S) contains a subset of S as many times as the ways it can be well ordered. Put the "lower segment" partial order on W(S), i.e., Ri < R2 in S •$=> R\ is order isomorphic to a lower segment of R2. Given any chain P in W(S), upon letting T = Dn^pR and upon putting the partial order on T induced by the partial orders of the various R £ P, we see that T £ W(S) and R < T for all R £ T, i.e., T is an upper bound of P. Therefore by (ZL), the set W(S) has a maximal element T. If T ^ S then by taking y £ S\T and letting T = T U {y} and putting the same partial order on T as T with the understanding that x < y for all £ £ T, we get T £ W(S) with T < T which is a contradiction. Therefore f = S. Thus (ZL) => (WO). PROOF OF (WO) =^(AC). Assume (WO), and put some well order on T; now given any 4> : S —> V*(T) as in the statement of (AC), for every x £ S let il)(x) = min(0(i)). Thus (WO) =*• (AC).

46

LECTURE LZ: CURVES AND SURFACES

REMARK (Rl). The above theorem indirectly tells us that (ZL) «• (ZL*). Of this the implication (ZL*) =>• (ZL) is obvious. Here is a direct proof of the implication (ZL) => (ZL*). Given a poset S, by a weakly cofinal subset S we mean a subset R such that there is no y € S for which we have x < y for all x € R. Let W(S') be the set of all subsets of 5 which are well ordered (in the partial order induced from S). In the "lower segment" partial order, clearly W(S) has the Zorn property because the union of any chain is clearly an upper bound for it. Therefore, assuming S to be nonempty and noting that any singleton subset {y} with y £ S obviously belongs to W(S), by (ZL) we get a maximal element T in W(S). If T is not weakly cofinal in S then by taking y € S\T such that x < y for all s e T and letting f = T U {y} we get f € W(S) with T < f which is a contradiction. Therefore T is weakly cofinal in S. It follows that if S has no maximal element then T has no upper bound in 5. Thus (ZL) =» (ZL*). REMARK (R2). In (Rl) we have shown that: (ZL) => every poset has a weakly cofinal well ordered subset. REMARK (R3). Given a poset 5, by a cofinal subset of S we mean a subset R such that for every x E S we have x < y for some y £ R. As a variation of (Rl) and (R2), we can show that: (ZL) => every loset has a cofinal well ordered subset, and we can use this implication to prove the implication (ZL) => (ZL*). REMARK (R4). [Existence of Algebraic Closure by Well Ordering]. The usual mathematical induction is based on the fact that the positive integers are well ordered. Similarly transfinite induction is based on well ordered indexing sets, and so in effect on the well ordering principle (WO). For instance we can prove the existence of an algebraic closure of a field K, which need not be countable, thus. Let the set S of all nonconstant monic polynomials in K[Y] be indexed by a well ordered indexing set I and write them as Fi(Y) with i £ J; for instance we could take I = S and well order it by the well ordering principle. Let K\ = K with 1 = min(i'). For any 1 ^ j e J, having denned Ki for all i < j in 7, put Kj = SF(Fj,UiSi with i<jKi). Now K = Ujg/Xj is an algebraic closure of K. If this sketch of a proof is not very convincing, because of the problems such as where to take the two unions, the reader is referred to the original source of this proof which is the 1910 Crelle Journal paper of Steinitz reprinted as a book [Ste] by Chelsea. In Remark (R6) we shall give a (perhaps more convincing) proof using Zorn's Lemma. But first in Remark (R5) we shall use Zorn's Lemma to show the existence of prime ideals and maximal ideals. REMARK (R5). [Existence of Maximal Ideals by Zorn's Lemma]. Let I be an ideal in a ring R. Recall that I is a maximal ideal if the residue class ring R/I

§5: ZORN'S LEMMA AND WELL ORDERING

47

is a field, or equivalently if I ^ R and there is no ideal between I and R other than these two. Assuming I ^ R, let 5 be the set of all ideals in R which contain I but are different from R; in the partial order given by inclusion, the union of any chain is an upper bound for it, and hence by Zorn's Lemma S has a maximal element M. Obviously M is a maximal ideal in R. REMARK (R6). [Existence of Algebraic Closure by Zorn's Lemma]. Alternatively, we can use Zorn's Lemma (ZL) to prove the existence of an algebraic closure of a field K thus. Let I be the set of all nonconstant monic polynomials in K[Y}. For any F = F(Y) G I, let D(F) be the Y-degree of F, let aF>1,..., aF KF given by (j)F(XF]j) = aFj for 1 < j < D(F); since the image of F is a field, its kernel MF is a maximal ideal in PF. Now consider the polynomial ring Qj in the infinitely many variables {XFj : F £ I and 1 < j < D(F)} over K. Note that the polynomial ring R[{Xi : I G L}] in an infinite number of variables {Xi : I G L) over a ring R is constructed in an obvious manner, and it equals the union of the polynomial rings U-R^X/j : h G H}} taken over all finite subsets H of L; thus each polynomial involves only a finite number of variables; any two polynomials together involve only a finite number of variables and so they can be added and multiplied; finally, if W = {xi G S : I G L} is any subset of any overring S of R then Xi H-> xi gives us an i?-homomorphism R[{Xi : I G L}] —> S (called the substitution map) whose image is the subring R[W] of S. In our case, for every F G / , the ring PF is a subring of the ring Qj. Moreover, we have the partition {(F,j) : F G / and 1 < j < D(F)} = UFei{(F>J) • 1 < J < D(F)}. The following Lemma can be proved easily: LEMMA (T5). If L = U f f e f i if is a partition of a nonempty set L (where ft is a set of pairwise disjoint nonempty subsets of L), if RL = R[{Xi : I G L}] is the polynomial ring in indeterminates {Xi : I G L} over a field R, if for each H G ft we are given a nonunit ideal Jfj in the polynomial ring RH = R[{Xh : h G H}], and if JL is the ideal in RL generated by all the ideals JH as H varies over O, then JL is a nonunit ideal in RL; [cf. L6§6(E44)]. In our situation, by (T5) we see that the ideal in Qj generated by all the ideals MF as F varies over I is a nonunit ideal in Qi, and hence by (R5) there exists a maximal ideal Mi in Qi such that for every F G I we have Mp C M/. Clearly Mi C\K = 0 and hence we may identify K with a subfield of the field K = QI/MI. For every F e / w e must have Pp D Mi = Mp and hence we may identify Kp with a subfield of K. Now K contains a splitting field of F over K for every F G I. Also K = K[{apj : F G I and 1 < j < D(F)}} and hence K is algebraic over K.

48

LECTURE LS: CURVES AND SURFACES

Therefore K is an algebraic closure of K. REMARK (R7). [Uniqueness of Algebraic Closure by Zorn's Lemma]. Let L be an overfield of a field K, and let 7 be a set of nonconstant polynomials in Y with coefficients in K such that every / = f(Y) £ I splits completely in L, i.e., upon letting n(f) to be the F-degree of / and F = F(Y) to be the monic associate of / , we have (Wl)

F{Y) =

(Y-aftl)...(Y-aiMf))

with a / , i , . . . , <Xf,n(f) m J->. We call (W2)

L* = K({af,j

: / G / and 1 < j < n(/)})

the splitting field of I over K in L; note that if 7 = 0 then L* = K; also note that if I is the set of all nonconstant polynomials in Y with coefficients in K then L* is an (absolute) algebraic closure of K as well as the (relative) algebraic closure of K in L. Let V be an overfield of a field K' such that there is an isomorphism : K -> K' and every / ' = f'(Y) £ V splits completely in V where / ' c K'[Y] is obtained by applying (j> to the coefficients of the various members of / . Let V* be the splitting field of V over K' in L'. We want to show that there is an isomorphism cj>* : L* —> V* such that 4>*{x) = (j>(x) for all x G K. For any J c I let J' C / ' be obtained by applying (j) to the coefficients of the various members of J, and let Lj and L'j, be the splitting fields of J and J' over K and K' in L and L' respectively. Let A be the set of all pairs (J,<j>j) where J d I and cf>j : Lj —> L'j, is an isomorphism such that 4>j(x) = j0{x) =
J s = [J J

(W3)

(./,<MeB

it can be shown that

{

there is a unique isomorphism jB : LjB —> L ^ such that for all (J,(j)j) £ B and x £ Lj we have (j>jB{x) = j{x)

and moreover (W5)

{JB^JB)

is a n upper bound of-B in A

Therefore by Zorn's Lemma, A has maximal elements. Finally by L1(R8) we see that (W6)

(Ji, 4>Ji) is maximal in A => J\ = I.

§5: ZORN'S LEMMA AND WELL ORDERING

49

REMARK (R8). [Existence of Vector Space Bases and Transcendence Bases by Zorn's Lemma], We shall now show that any vector space has a basis and any two of its bases have the same cardinality (which is called the dimension of the vector space). At the same time we shall show that any overneld has a transcendence basis and any two of its transcendence bases have the same cardinality (which is called the transcendence degree of the overneld). So let if be a field and let L be either a vector space over K or an overneld of K. Let H ^ b e a subset of L. Just for a few minutes, in the vector space case (resp: overfield case): let us call W independent if every finite subset of W is linearly (resp: algebraically) independent over K, and let us call W generating if L coincides with KW (resp: if L is algebraic over K{W)); in both cases call W a basis if it is independent and generating; given any other subset U of L, let us say that U is dependent on W if every u e U belongs to KW (resp: is algebraic over K(W)). It is easy to show that I W is a basis < ^

(W7)

<=> it is a minimal generating set •£> it is a maximal independent set

where the minimality means that W is a generating set but there is no generating set W with W' C W and W' ^ W, and the maximality means that W is an independent set but there is no independent set W' with W C W' and W ^ W'. It is also easy to see that (W8)

W is generating =>• there exists a basis W' with W' C W.

Obviously L is generating and hence by taking W = L in (W8) we get the existence of a basis. To prove equicardinality, assuming IF to be a basis and given any other basis U, let A be the set of all triples {U',W',^') where U' C U, W' c W, and bijection <j>' :U' —> W', are such that (W \ W') U U' is a basis. Now A becomes a poset by declaring (U',W',(j)') < (U",W",(j)") to mean that U' C U", W' c W", (j)"(U') = W, and ^"(u) = 4>'(u) for all u € U'. By taking U' = 9 = W and ' to be the obvious map, we see that A is nonempty. Moreover, given any chain B in A, by taking

UB=

(J (U',w,4>')£B

U' and

WB =

(J

W

(U',w,4>')eB

it can be shown that ,TTT„. (W9)

I there is a unique bijection &B • UB —> WB such that < \for all ({/', W, ') G B and u G V we have B{u) = '(u)

and moreover (W10)

(UB, WB, 4>B) is an upper bound of B in A.

50

LECTURE

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SURFACES

Therefore by Zorn's Lemma, A has maximal elements. Finally it can be shown that (Wll)

([/*, W*,*) is maximal in A => U* = U and W* = W.

This completes the proof. REMARK (R9). [Linear Order on Cardinals by Zorn's Lemma]. For the cardinals | 5 | and \T\ of sets S and T, we recall that | 5 | = \T\ means there is a bijection S —> T, and | 5 | < \T\ means there is an injection S —> T. To prove that this is a partial order, i.e., to prove the Schroeder-Bernstein Theorem, given any injections (/>: S —> T and ip :T —> S we have to find a bijection S —> T. Let f = ip(p and R = ip(T). Then clearly: (*) / : S —> S is an injection and f(S) C i? C S; we claim that (*) implies the existence of a bijection g : S —> R. Obviously ip induces a bijection h : ii —> T, and hence the claim yields the bijection hg : S —* T. To prove the claim, for any x G 5 let /°(a;) = x and / n + 1 (a;) = /(/"(a;)) for all n € N; also let /°°(a;) = {/"(a;) : n G N}, and for any V C S let /°°(V) = U x e v/°°(a;). Let R' = f°°(R \ f(S)), and note that then R' c R. Since / ( 5 ) C R and i?' C R, we get a map g : 5 —> R by putting g(a;) = a; or /(a;) according as a; G R' or a: G 5 \ R'. It can be shown that then (W12)

g is a bijection.

To prove that < is a linear order, given any sets U and W, we want to show that either there exists a bijection of U onto a subset of W or there exists a bijection of W onto a subset of U. To do this let A be the set of all triples (U',W',4>') where U' C U, W C W, and 0' : U' —> W is a bijection. Now A becomes a poset by declaring (U',W',(p') < (U",W","(U') = W, and "{u) = '(u) for all u G C/'. By taking U' = 0 = W and 0' to be the obvious map, we see that A is nonempty. Moreover, given any chain B in A, by taking

UB=

(J (U\W',')eB

tf'

and

WB =

[J

W

(W ,W\4>')€B

it can be shown that (W13)

I there is a unique bijection cf>B '• UB —» W^B such that < [for all ([/', W ,
and moreover (W14)

(C/B, V^BI ^ B ) is an upper bound of B in A.

Therefore by Zorn's Lemma, A has maximal elements. Finally it can be shown that (W15) (U*,W*,*) is maximal in A => either U* = U or W* = W. This completes the proof.

§5: ZORN'S

LEMMA

AND WELL

ORDERING

51

REMARK (RIO). [Well Order on Ordinals by Zorn's Lemma]. For the ordinals | | 5 | | and ||T|| of well ordered sets S and T, we recall that | | 5 | | = ||T|| means there is an order preserving bijection S —> T, and | | 5 | | < ||T|| means there is an order preserving bijection of S onto a lower segment of T. To prove that this is a partial order, given any order preserving bijections <j>: S —• T" and ip : T —> S', where T" and S' are lower segments of T and S respectively, we have to find an order preserving bijection S —* T. Let R = ip(T') and for every x € S let f(x) = ip( R is an order preserving bijection; we claim that (*) implies R= S and / : S —> S is the identity map of S. Since <j> and ip are injections, the claim yields the conclusion that T' = T and hence (p : S —> T is an order preserving bijection. The above claim is clearly subsumed under the following stronger claim which can be proved easily: if / ' : S —> T" and / " : S —> T" are order preserving bijections (W16)

< of a well ordered set S onto lower segments T" and T" of a well ordered set T then we must have T" = 7" and / " = / ' .

To prove that < is a linear order, given any well ordered sets U and W, we want to show that either there exists an order preserving bijection of U onto a lower segment of W or there exists an order preserving bijection of W onto a lower segment of U. To do this let A be the set of all triples ({/', W, (/>') where U' is a lower segment of U, W is a lower segment of W, and <j>' : U' —» W is an order preserving bijection. Now A becomes a poset by declaring (U',W',') < (U",W",(j)") to mean that U' is a lower segment of U", W is a lower segment of W", "{u) = <j>'{u) for all u €U'. By taking U' = 0 = W and
(J

B=

{U',W',')€B

U

'

and

WB=

U

W

'

(U',W',')€B

it can be shown that [ there is a unique order preserving bijection B • UB -* WB (W17) < [such that for all ([/', W, <j>') £ B and u G U' we have 0B(u) = cj>'{u) and moreover (W18)

(f/s, WB, 4>B) is an upper bound of B in A

Therefore by Zorn's Lemma, A has maximal elements. Finally it can be shown that (W19)

(U*, W*, (/>*) is maximal in A =>• either [/* = U or M^* == W.

This completes the proof that < is a linear order. Let us now set up an order preserving bijection between a well ordered set and its lower segments. Namely,

52

LECTURE L2: CURVES AND SURFACES

given any element y in a well ordered set C, clearly {x G C : x < y} is a lower segment of C different from C; conversely given any lower segment L of C different from C, by taking y to be the smallest element of C\L, we get L = {x € C : x < y}; it follows that under the above defined partial order < on well ordered sets, y i—• {x e C : x < y} gives an order preserving bijection (W20)

of any well ordered set C onto the set of all lower segments of C other than C.

Finally let us show that under the above defined partial order < on well ordered sets, any set D of well ordered sets is well ordered. Namely, given any nonempty subset E of D, take C e E, and let E' = {C e E : C < C}. Then by (W20) we see that E' has a smallest element C", and clearly C" is a smallest element of E. §6: UTILITARIAN S U M M A R Y In the material of the previous section through (R3) we have shown the equivalence of the axiom of choice, Zorn's Lemma, and the well ordering principle. In (R4) we have shown that the well ordering principle implies the existence of algebraic closure. In (R5) we have shown that Zorn's Lemma implies that, given any ideal / of any ring R with I ^= R, there exists a maximal ideal M of R with I c M. In (R6), (R7), and (R8) we have shown that Zorn's Lemma implies the existence and (up to isomorphism) uniqueness of algebraic closure, where uniqueness means: given any isomorphism : K —> K' of fields and given any algebraic closures L* and L'* of K and K' respectively, can be extended to an isomorphism of L* onto L'*, i.e., there exists an isomorphism cf>* : L* —• V* such that <j)*{x) = cf)(x) for all x £ K; note that if J ' is the set of all members of if'[y] obtained by applying <j) to the coefficients of the members of a set J of nonconstant members of AT[y], and if Lj and L'j, are the splitting fields of J and J' over K and K' in L* and L'* respectively, then we automatically have (j)*(Lj) = L'j,\ in particular this shows that <j> can be extended to an isomorphism of Lj onto L'j,. In (R8) we have shown that Zorn's lemma implies the existence of vector spaces bases and transcendence bases. In (R9) we have shown that Zorn's Lemma implies that any set of cardinals is linearly ordered. Finally in (RIO) we have shown that Zorn's Lemma implies that any set of ordinals is well ordered. Without loss of understanding, all this may be taken for granted. As an exception, the material of (R8) through display (W8), i.e., the first 21 lines in the discussion of the existence of vector space bases and transcendence bases, should be read. §7: DEFINITIONS A N D EXERCISES DEFINITION (Dl). [Real and Complex Numbers]. Real numbers may be

§7: DEFINITIONS AND EXERCISES

53

defined as equivalence classes of Cauchy sequences of rational numbers, and the complex number field C may then be defined as the splitting field of the quadratic polynomial Y2 + 1 over the real number field R. By analogy with N+, by Q+ we denote the set of all positive rationals; moreover, by Q - > a n d Qo- we denote the set of all nonnegative rationals, negative rationals, and nonpositive rationals respectively; as usual, the absolute value of any r G Q is denoted by \r\, i.e., \r\ = r or — r according as r G Qo+ or r G Q_; by context, this will not be confused with the size | 5 | of a set 5. A sequence x = (xi)i Ne and j > Nc we have \xi —Xj\ < e. This is equivalent to the Cauchy sequence x' = (a^)i in symbols x ~ x', if for every e G Q+ there exists Me G N + such that for all i > Me we have lij — a^| < e. Now R may be defined to be the quotient CQ/ ~ of the set CQ of all Cauchy sequences in Q by the equivalence relation ~ . We "identify" Q with a subset of R by sending any q £ Q to the equivalence class of the Cauchy sequence {Qi)i 0}, G 0 + = G+ U {0}, G_ = {g G G : g < 0}, and Go- = G _ U { 0 } ; also we define the absolute value of any g G G by putting \g\ = g or —g according as g G Go+ or g G G_. In particular this defines the sets R+, Ro+, R~, Ro~, and defines the absolute value \r\ of any r G R. Note that now Z + = N+ and Zo + = N. An ordered abelian group G is archimedean means for all x,y in G + we have nx > y for some n G N+. Clearly R is an archimedean ordered field, i.e., an ordered field whose underlying additive ordered abelian group is archimedean. A sequence x = (zi)i Ne and j > Ne we have |#,- — Xj\ < e; the sequence x is convergent means it converges to a limit £ & G, i.e., for every e G G+ there exists M £ G iV+ such that for all i > M€ we have |£ — Xi\ < e; we indicate this by some standard notation such as xt —• £ as i —• oo or limj-Kjoajj = £. An ordered abelian group is complete means in it every Cauchy sequence is convergent. Clearly R is a complete field, i.e., an ordered field whose

54

LECTURE

L2: CURVES AND

SURFACES

underlying additive group is complete as an ordered abelian group. In the next several Definitions and Exercises we extend the above construction of the reals to embed any ordered abelian group in a complete ordered abelian group. DEFINITION (D3). [Torsion Subgroups and Divisible groups]. By the subgroup of a group G generated by elements x\,X2,--. in G we mean the smallest subgroup of G which contains these elements. The order of an element in a group is the order of the subgroup generated by it. The subgroup of an additive abelian group G generated by all of its elements of finite order is called the torsion subgroup of G; if this is zero then G is said to be torsion free. An additive abelian group G is divisible means for every g e G and n € Z x there is h e G with nh = g. EXERCISE (El). Show that for any elements a, 6 in a torsion free additive abelian group and any nonzero integer n we have: na = nb =>• a = b. Show that any ordered abelian group is torsion free, and hence any ordered field is of characteristic zero. Show that for all x,y in an ordered field we have \xy\ = \x\\y\. Show that the usual order on Q is the only order on it which makes it an ordered field. EXERCISE (E2). Show that for all x, y in an ordered abelian group G we have \x + y\ < \x\ + \y\, and from this deduce that any convergent sequence in G is Cauchy and has a unique limit. HINT: Divide the argument into two cases according as there does or does not exist a sequence of positive elements (ei)i<j £ and Xi —> - f in G => (xi) is Cauchy and £' = £. Upon letting y\ = £ — Xi and r] = £ — £', it suffices to show that: j/j —> 0 and yi —> r\ in G =3> (y$) is Cauchy and r\ = 0. In the first case, we claim that for any e > 0 in G we have e; + e/ < e for some I G N+; namely, since e» —> 0, for some m £ N + we must have e m < e; again since ej —> 0, for some n S N + we must have en < e - e m ; it suffices to take I = m o r n according as e m < en or em > e„. Now y, —> 0 =>• there exists N £ N+ such that for allz > N we have \yt\ < £/, and hence for &\\ i > N and j > n we have \yi — j/j| < e, which shows that (j/j) is Cauchy. Since y, -+ r], by choosing N large enough we can also arrange that for all i > N we have \r] — yi\ < e/, and this yields \v\ < \Vi\ + \v — Vi\ < e; + e; < e, which shows that 77 = 0. In the second case, y* —> 0 =>• there exists J V £ N + such that for all i > N we have y, = 0, because otherwise we can find an infinite sequence of integers 0 < j i < J2 < • • • such that y^ ^ 0 for all i and upon letting e* = (y^ | this would produce a sequence of positive elements {ei)i
§7: DEFINITIONS AND EXERCISES

55

free additive abelian group G, let G* — G x Z x / ~ where the equivalence relation ~ is given by: (g,n) ~ (g',n') •£> n'g = ng', and embed G in G* by identifying every g £ G with the equivalence class containing (g, 1). Define addition in G* by taking equivalence classes in the proposed equation (g, n) + (h, m) = (mg + nh, nm). Note that then G* is a divisible additive abelian group such that for every g~ £ G* we have rig £ G for some n £ Z x . We call G* the rational completion of G. Note that if G is divisible then G* = G. Now assume that G is an ordered abelian group. Note that then the order on G uniquely extends to an order on G* so that G* becomes an ordered abelian group. G is said to be orderwise complete if every Cauchy sequence in G is convergent and has a limit in G. Moreover, G is said to be an orderwise completion of a subgroup H if G is orderwise complete and every element of G is the limit of a sequence all of whose members belong to H. Let G = CQ/ ~ where the equivalence relation ~ on the set CG of all Cauchy sequences in G is defined as in (Dl) with G replacing Q. As in (Dl) and (D2), G is made into an ordered abelian group and G is embedded into it; see (E4) below. Clearly G is an orderwise completion of G, and we call it the ordered completion of G. Also we call G* the real completion of G. Note that G* is archimedean o G is archimedean <=> G is archimedean. Also note that if G is complete then G = G. Finally note that if G is divisible and complete then &=G. In particular if K is an ordered field then its underlying additive group is divisible and hence K* = K; as in (Dl) and (D2) we make this into an ordered field; it is a complete ordered field, i.e., an ordered field whose underlying additive group is orderwise complete. EXERCISE (E3). In (D2) show that the induced relation < on the equivalence classes of Cauchy sequences in Q is a linear order. EXERCISE (E4). In (D4) show that the induced relation < on the equivalence classes of Cauchy sequences in an ordered abelian group is a linear order. EXERCISE (E5). Let G be any nonzero archimedean ordered abelian group. Show that given any g > 0 in G and x > 0 in R, there exists a unique order monomorphism (i.e., a group monomorphism which is order preserving) 0 : G —> M such that 4>(g) = x, and there exists a unique order isomorphism (i.e., a group isomorphism which is order preserving) ijj : G* —> M. such that ijj(g) — x. Moreover, for these maps we always have (h) = tp(h) for all h £ G. EXERCISE (E6). Show that x K+ X2 gives a surjection M -> M 0+ . HINT. The usual method of finding the decimal expansion 1.14142 . . . of \/2 can be explained in terms of decimal expansions of integers by saying that l 2 < 2 < 2 2 ,

LECTURE L2: CURVES AND SURFACES

56

142 < 2 x 102 < 15 2 , 141 2 < 2 x 104 < 1422, 14142 < 2 x 106 < 14152, 141422 < 2 x 108 < 141432, and so on. More generally let n > 1 and d > 1 be any integers, and let y > 0 and i > 0 be any integers. Then clearly there is a unique integer Xi such that xf < yndx < (xi + l)d; Xi is nothing but the n-adic expansion of the largest integer < y 1 / , d n\ Obviously the sequence (xi/nl) is Cauchy and for its limit x in R+ we have xd = y. Any positive rational can be written in the form y/zd where y and z are positive integers, and then we get x/z G R + with (x/z)d = y/zd. Finally, any rj £ R+ can be written as the limit of a sequence (r]j) in Q + and then taking £,• G R + with £d = rjj we get a Cauchy sequence (£,) for whose limit £ € R+ we have £d = 77. EXERCISE (E7). Show that the identity map is the only field automorphism of R. Hint: by (E6) any field automorphism of R is order preserving, and since it must send 1 to 1, by (E5) it must be the identity map. EXERCISE (E8). In contrast with (E7), assuming the fact that C is an algebraic closure of R, show that C has uncountably many field automorphisms. DEFINITION (D5). [Rational and Real Ranks]. In view of the first sentence of (El), any torsion free divisible additive abelian group may clearly be regarded as a Q-vector-space. The Q-vector-space dimension of the rational completion of a torsion free additive abelian group G is called the rational rank of G and is denoted by r(G). Alternatively, r(G) may be characterized as the cardinal of a maximal (= nonenlargeable) Z-linearly independent subset H of G, where independent means that for any finite number of distinct elements x\,...,Xd in H and any integers n\,...,

rid we have: n\Xi + • • • + n^Xd = 0 => n\ = • • • = rid = 0.

Now let G be an ordered abelian group. By a segment of G we mean a nonempty subset H of G such that: g e G and h e H with \g\ < \h\ => g € H. By an isolated subgroup of G we mean a subgroup which is a segment. The next Exercise says that the set S(G) of all nonzero isolated subgroups of G is linearly ordered by inclusion. The order-type of S(G) is called the real rank of G and is denoted by p{G). Two linearly ordered sets are said to have the same order-type if there is an order preserving bijection between them. An ordinal is the order-type of a well ordered set. Examples of nonordinal order-types are the order-types of Z or Q or Q + . We write p{G) € N or p(G) = 00 according as S(G) is finite or infinite. As a comparison, recall that the dimension dim/cV of a vector space V over a field K is the common cardinality of a vector space basis of V/K, and we write dimj<-V 6 N or d i m ^ y = 00 according as the said basis is finite or infinite. Similarly the transcendence degree trdeg/c-^ of an overfield L of a field K is the common cardinality of a transcendence basis of L/K, and we write trdeg/fL € N or trdeg^L = 00 according as the said basis is finite or infinite. In the section on Valuations, for any nonnegative integer d, we introduced the

§7: DEFINITIONS AND EXERCISES

57

ordered abelian group R' d ' as the lexicographically ordered set of all d-tuples of real numbers. Upon replacing K by any ordered abelian group G we generalize this to G' d '. As a further generalization, given any well ordered set T we introduce the ordered abelian group GlT! as the set GT of all functions / : T -> G with componentwise addition and the order defined by setting: / < g •$=> / ^ g and f(i) < g(i) where i is the smallest j £ T with f(j) ^ g(j). EXERCISE (E9). Show that for any isolated subgroups H\,H2 of an ordered abelian group G we have either Hi C H2 or H2 C Hi. EXERCISE (E10). Let G be a nonzero archimedean ordered abelian group; by (E5) this is equivalent to saying that, up to order isomorphism, G is a nonzero subgroup of R; for instance G = K. For any nonempty well ordered set T, describe the nonzero isolated subgroups of C?'T' and show that they are in an order reversing bijective correspondence with T; hint: « * - » { / £ GT : f(j) = 0 for all j < i) is the desired bijection from T to the nonzero isolated subgroups. In particular, for any positive integer d, the ordered abelian group G ^ has exactly d nonzero isolated subgroups. EXERCISE ( E l l ) . Let G be any ordered abelian group. Show that r{G) = 0 <=> p(G) = 0 o G = 0. Show that G is archimedean <=> p(G) < 1. Show that r(G) < 00 =4> p(G) < r(G). Show that r(G) = p(G) = d e N+ => there exists an order monomorphism G —> Q' d ' and an order isomorphism G —» Q ^ . See the Hint in (E12) below. EXERCISE (E12). Show that, given any ordered abelian group G with p(G) = d 6 N+, there exists an order monomorphism G —* R^ and there exists an order isomorphism G* —» M.W. Hint: make induction on d starting with (E5), and use the fact that if H is any nonzero isolated subgroup of G then in a natural manner G/H can be made into an ordered abelian group with p(H) + p(G/H) = d. DEFINITION (D6). [Dedekind Cuts]. Instead of using Cauchy sequences to prove (E5), we can use Dedekind Cuts. So let G be a nonzero divisible archimedean ordered abelian group. A Dedekind cut of G is a pair (L, U) of nonempty subsets of G with U = G\L such that for a l l / e I and u £ U we have I < u and there is no u' e U with U = {u e G : u' < u}. For any t G G we get a Dedekind cut (Lt,Ut) with Lt = {I £ G : I < i) and Ut = {u S G : t < u}. Let DG be the set of all Dedekind cuts of G. It can be shown that G is complete O « H (L t , Rt) gives a surjection G —> DG. Indeed, Do may be defined to be the completion of G. At any rate, for proving (E5), given any h £ G let 9(h) be the real number which corresponds to the Dedekind cut (L, U) of Q where L = {m/n £ Q with m £ Z and n £ N+ : nh < mg} and U = Q\L, and take <j>(h) = x6(h).

58

LECTURE L2: CURVES AND SURFACES

EXERCISE (E13). Prove the geometric series identity (i) in the univariate power series ring i?[[X]] over any ring R, and from it deduce description (ii) of units in the multivariate power series ring i?[[i?i,... ,Zd]]- Prove claims (V4) to (V6) and thereby show that, for any field K and any ordered abelian group G, K((X))Q is a field with a valuation whose value group is G. [cf. L6§6(E45)]. EXERCISE (E14). Complete the existence proof for vector space bases and transcendence bases by establishing claims (W7) to (Wll) made in (R8). Complete the proof that cardinals are linearly ordered by establishing claims (W12) to (W15) made in (R9). Complete the proof that ordinals are well ordered by establishing claims (W16) to (W20) made in (RIO). DEFINITION (D7). [Approximate Roots]. In the Hint to (E6) we showed how to use n-adic expansions of positive integers to find successive approximations to the d-th root of a positive integer. Mixing a generalization of this with a generalization of the completing the square method of solving quadratic equations leads us to the concept of approximate roots of polynomials. So consider a monic polynomial

F = F{Y) = YN + J2

A YN l

i ~

l
of degree N > 0 in Y with coefficients Ai in a ring R. It N is a unit in R then we can generalize the completing the square idea to completing the iV-th power by writing

F(Y) = (Y - Ai/N)N

+ ]T

AHiiY-Ai/N)"-'

2
with A\ e R, i.e., by killing the coefficient of YN~X. To generalize this further let D > 0 be an integer which divides N. Instead of assuming N to be a unit in R, assume D to be a unit in R; note that in case of a field R this is equivalent to assuming that the characteristic of R does not divide D and so characteristic zero is always ok. Now we look for a monic polynomial

G = G{Y) = YN'D +

Yl

BiY^I0^

l
of degree N/D in Y with coefficients Bi in R such that GD is as close to being equal to F as possible. As (E15) below shows, if we interpret this as requiring degy(F — GD) < N—(N/D) then a unique G exists, and we call it the approximate -D-th root of F (relative to Y) and denote it by App£>(F) or App£>iy(-F). Recall that for any m,n in N with n > 1, the n-adic expansion of m consists of writing m = Y^i>o rninl where integers 0 < rrii < n are the digits of the expansion. Likewise for any / , g in R[Y] with g monic of positive Y-degree, the p-adic expansion of /

§«: NOTES

59

consists of writing / = ^2i>0 fig1 where fi € R[Y] with degy/j < deg y g are the digits of the expansion. By (E16) below these expansions exist and are unique. Moreover, if / is monic of Y-degree N > 0 and the Y-degree of g is N/D £ N+ where D £ N + is a unit in R, then fD = 1 and fi = 0 for all i > D; we try completing the -D-th power by putting Tf(g) = Tf:y(g) = g + (fo-i/D) and calling it the /-Tschirnhausen of g (relative to Y); see Krull's charming little book [Krl] for the reference to the 1683 work of Tschirnhausen who was a friend of Leibnitz. By (El 7) below, starting with any monic g of degree N/D and applying 77 to it N/D times will produce the approximate D-th root of / . EXERCISE (E15). Let F be a monic polynomial of degree N > 0 in Y over a ring R. Let D > 0 be an integer such that D divides TV and D is a unit in R. Show that there exists a unique monic polynomial G of degree N/D in Y over R such that degy(F - GD) < N - (N/D). Hint: With display as in (D7), the last condition gives the equations Ai — DBi + Pi(Bi,..., Bi_i) for 1 < i < N/D where the coefficient of YN~l in GD equals DBi + Pi(Bi,..., -Bj-i) with Pi a polynomial over Z; since D is a unit in R, these can be solved successively (in a unique manner). EXERCISE (E16). Given integers m > 0 and n > 1, show the unique existence of the n-adic expansion of m. Given univariate polynomials / , g over a ring R with g monic of positive degree, show the unique existence of the g-adic expansion / = Yli>o fi9% °f /• Show that if / is monic of degree N > 0 and the degree of g is N/D where D is a positive integer factor of N, then fo = 1 and / , = 0 for all i > D, and moreover: App£>(/) = g •*=> fn-i — 0. EXERCISE (E17). Let / , g be univariate monic polynomials of positive degrees N and N/D over a ring R where D is a positive integer which divides N, and is a unit in R. Let 17(3) = g, and let / = J2o
60

LECTURE L2: CURVES AND SURFACES

integers; all the rest is the work of man. §9: C O N C L U D I N G N O T E When there are several things which we do not know, we call them X, Y, Z and so on. This leads to one or more polynomial equations in these variables. One equation in two variables gives rise to a plane curve. One equation in three variables produces a surface in three space. Two equations in two variables give rise to the finite number of points in which the two curves intersect. Two equations in three variables produce one or more space curves in which the two surfaces intersect. Bezout makes a proper counting of the intersections.

Lecture L3: Tangents and Polars §1: SIMPLE G R O U P S After proving the simplicity of the alternating group An for n > 5, Galois enlarged the list of finite simple groups by showing that the projective special linear group PSL(2,p), which we shall define in a moment, is simple for any prime p > 5. Jordan extended this by proving the simplicity of PSL(n, p) for all n > 3; his proof may be found in his 1870 book [Jor] which was the first expansion of Galois' ideas on equation solving. Then Moore proved the simplicity of PSL(2, q) for any power q of p with q> p; indeed this was his main aim in the paper [Mol] which he presented at the Chicago World's Fair of 1893 which was the birthplace of the Galois field G F ( Q ) . Finally, in his influential 1901 book [Die], Dickson established the simplicity of PSL(n, q) for all n > 3 and q > p. Eventually we shall prove all these simplicity statements, [cf. L6§6(E46)]. To define PSL we start by introducing determinants and matrices. Given any ring R, for any integers m > 0 and n > 0, by an m x n matrix over R we mean a "rectangular array" A = (Aij) with Aij G R for 1 < i < m and 1 < j < n. By the (i, j)-th entry of A we mean the element A^. The 1 x n matrix whose (1, j)-th entry is A^ is called the i-th row of A, and the raxl matrix whose (i, l)-th entry is A^ is called the j - t h column of A. The set of all m x n matrices over R is denoted by MT(m xn,R). For any B = (By) G MT(m xn,R), we define the sum A + B = {{A + B)ij) G MT(m x n, R) by putting {A + B)^ = Atj + Bij. Clearly MT(m x n, R) becomes an additive abelian group whose 0 is the matrix with all entries reduced to the 0 of R. Next, for any integer r > 0 and any B = (Btj) G MT(n x r,R), we define the product AB = ((AB)„) G MT(m x r,R) by putting (AB)ij = ^2i 0 and any C G MT(r x s,R) we have (Ml)

(AB)C =

A{BC).

Now assume m = n. Then MT(n x n, R) becomes a skew-ring whose 1 is the matrix In = ((ln)ij) where ( l n ) y = the Kronecker 6ij which equals 1 or 0 according as i = j or i ^ j . The determinant of A is defined by putting

(M2)

det(A) = ^2 sgn(^) I I
A

™«)

\
where Sn is the permutation group (= the symmetric group) on ( 1 , 2 , . . . ,n), and sgn(
det(AB) = det(^)det(B). 61

62

LECTURE

L3: TANGENTS

AND

POLARS

Recalling that U(R) is the group of all units in R, it is easily seen that (M4)

f/(MT(n x n, R)) = {A e MT(n xn,R):

det(A) e U(R)}.

The group U(MT(n x n,R)) is called the n-dimensional general linear group over R and it is denoted by GL(n, R). The kernel of the determinant map GL(n,R) —> U(R), i.e., the homomorphism given by A \—> det(yl), is called the n-dimensional special linear group over R and is denoted by SL(n, i?). In other words SL(n,R) = {A e GL(n, .R) : det(A) = 1}. Being the kernel of a homomorphism, SL(n, R) < GL(n, R), i.e., SL(n, i?) is a normal subgroup of GL(n, R). Note that the determinant of a 0 x 0 matrix is 1; namely |5o| = 1 because |0 0 | = 1, and rii 0, we have an injective homomorphism fc* —> GL(n, k) which sends every a £ kx to the matrix whose (i,i)-th entry is a for all i and whose (i, j)-th entry is 0 for all i ^ j . The image of this homomorphism is called the n-dimensional homothety group over k and is denoted by HL(n,k). In other words, HL(n, k) is the group of all nonzero scalar matrices. As we shall eventually see, an n x n matrix induces a linear transformation of the n dimensional vector space kn over k. The transformation induced by a nonzero scalar matrix sends a geometric configuration to a configuration which is similar to it, like two similar triangles. For the use of the word homothety for such a transformation, see the last chapter starting on page 267 of the classic Pure Geometry book [Ask] by Askwith published in 1921 (first edition 1903) by the Cambridge University Press. We can easily see that (M5)

HL(n, jfe) < GL(n, jfc)

and we define PGL(n, k) = GL(n, fc)/HL(n, k) and PSL(n, k) = SL(n, k)/(SL(n, k)n HL(n, k)), and call these the n-dimensional projective general linear group over k and the n-dimensional projective special linear group over k respectively. In case k is the Galois field GF(q) where q is a power of a prime p, we may write GL(n, q), SL(n,q), PGL(n,g) and PSI^n,*?) in place of GL(n,k), SL(n, fc), PGL(n, k) and PSL(n, k) respectively. As said above, eventually we shall prove the simplicity of PSL(n, q) for n > 2 with (n,q) ^ (2,2), (2,3). Now the list of finite simple groups can be enlarged by considering finite orthogonal groups, i.e., the finite analogs of groups of distance preserving transformations of the ordinary euclidean space. The said orthogonal groups, as well as the related symplectic groups and unitary groups, are based on the geometry of quadrics; later on we shall give a detailed treatment of these groups. Collectively, the finite linear groups GL(n, q), SL(n,q), PGL(n,q) and PSL(n, q), together with the finite orthogonal, symplectic and unitary groups, are called finite classical groups. If we do not insist the field k to be finite then they are simply

§2: QUADRICS

63

called classical groups. This terminology goes back to Hermann Weyl's book [Wey]. Dickson [Die] called all of them linear groups. §2: QUADRICS To motivate the geometry of a quadric, let me start with the following: HOMEWORK PROBLEM (HI). Draw the two tangent lines to a circle D from a point A, and let n be the line joining the two points of contact. Call n the polar of A, and A the pole of n . Show that if the polar of A passes through a point B then the polar of B passes through A. If done pedantically, by analytic geometry, this may take five pages. The pure geometry solution given on page 14 of Askwith's book [Ask] cited above takes one page. Soon I shall show you a half-line proof. That proof will be based on homogeneous coordinates which are the mainstay of analytic projective geometry. Pure geometry may be regarded as synthetic projective geometry. Affine, i.e., ordinary, space is converted into projective space by adding points at infinity to it. In the above problem, the circle may be replaced by any conic such as an ellipse or parabola or hyperbola, or even by a quadric or a hyperquadric. A quadric is a surface in 3-space given by a quadratic equation, such as a sphere or an ellipsoid or a paraboloid or a hyperboloid. In case of a quadric, the polar of a point A is a plane which intersects the quadric in the points of contact of various tangent lines from A to the quadric. More generally a hyperquadric in AT-space is given by a quadratic equation in AT variables, i.e., it is a hypersurface of degree 2; similarly, a hyperplane is a hypersurface of degree 1. Now the polar of a point A is the hyperplane which intersects the hyperquadric in the points of contact of the tangent lines from A to the hyperquadric. This gives rise to the following: GENERALIZED HOMEWORK PROBLEM (H2). In case of a hyperquadric, show that if the polar hyperplane of a point A passes through a point B then the polar hyperplane of B passes through A. The half-line solution I spoke of actually works in the general hyperquadric case! So what are tangent lines, tangent cones, singular points, singular curves, and so on? Seeing that analytic geometry, like its sister discipline of projective geometry, has mostly been eliminated from college courses, let me first review these concepts from the viewpoint of calculus and then from the viewpoint of analytic geometry. In calculus we consider a plane curve C given by an explicit equation Y = $(X), and introduce the concept of the tangent T to C at a point P = (U, V) of C with V = $(t/) as the limit of chords (i.e., secant lines) of C through P. Then we

64

LECTURE

L3: TANGENTS

AND

show that the slope of T is the value of $'(X) = Y - V = $'({/) (X — U) as the equation of T. If C F(X, Y) = 0 then we use implicit differentiation to subscripts indicate partial derivatives. This yields

POLARS

^ at X = U. This gives us is given by an implicit equation get FxdX + FydY = 0 where ^ = =pL. If Py(£/, V) ^ 0

then the slope of T is "ffiffff giving Y - V = ~Fy(u,v) (X ~ U) a s t h e equation of T, and now by clearing the denominator and bringing things to one side we get FX(V, V)(X -U) + FY(U, V)(Y -V)

=0

as the equation for T. Likewise, if FX(U,V) ^ 0 then for the "antislope of T" (= the tan of the angle T makes with the Y-axis) we have %y\(u,v) = ~FYWV) giving X — U = ~F^m V\ (Y — V) as the equation of T, and again by clearing the denominator and bringing things to one side we get the same equation for T as displayed above. Thus the calculus method works if either Fx or Fy is nonzero at P . But if Fx and Fy are both zero at P then it breaks down and the books simply declare that P is then a singular point of C. Note that F(U, V) = 0 ^ FY(U, V) is exactly the condition under which, according to the Implicit Function Theorem, F(X,Y) = 0 can be solved near P giving Y = $(X) with V = $(£/). Similarly, if F(U,V) = 0 ^ FX{U,V) then we solve F{X,Y) = 0 to get X = * ( F ) with U = ^(V). The point P is singular exactly means that the Implicit Function Theorem does not work at P . §3: H Y P E R S U R F A C E S As I recently became aware, the modern calculus treatment of tangents to a surface S : F(X, Y,Z) = 0 in 3-space is even more troublesome. What is done is to introduce the gradient to S at a point P of S as the vector (Fx, Fy, Fz) evaluated at P . The gradient is also called the normal to S at P , and the plane perpendicular to it is called the tangent plane to S at P . More generally, given a hypersurface S : F(X\,..., XN) = 0 in TV-space, the gradient to S at a point P = (Ui,..., UN) of S is the vector of partials (Pxi, • • •, FxN) evaluated at P; thinking of P as a variable point, the gradient of S at P is the vector (Fu1,..., FuN) where Fut is the partial derivative of F(U\,..., UN) with respect to Ui. Again, the gradient is also called the normal to S at P , and the hyperplane Tp perpendicular to it is called the tangent hyperplane to S at P . Clearly Fux (Xi —Ui)-\ hFu N (X^ — UN) = 0 may be taken as the equation of Tp. Alternatively, Tp is defined to be the hyperplane spanned by the tangent lines to the various curves on S through P . Both these definitions break down if all the N partials are zero at P , i.e., if P is a singular point of S. The alternative definition has the further difficulty of having to define tangents to space or hyperspace curves, and of proving that sufficiently many of them can be drawn on the surface or hypersurface S. All these difficulties vanish if, instead of calculus, we follow the method of analytic geometry.

§3: HYPERSURFACES

65

To explain the method of analytic geometry, let us intersect the hypersurface S:F(X1,...,XN)=0 in iV-space by a line L through a point P =

(UU...,UN)

of S. Let the line L be given parametrically by the equations L : Xi = Ui + Rtt for 1 < i < N where t is a parameter and the components of the vector R = ( P i , . . . ,RN) are proportional to the "direction cosines" of the line; for the idea of direction cosines see the 1903 book [Bel] of R. J. T. Bell on Three Dimensional Analytic Geometry; in effect, they are the cosines of the angles a line makes with the various coordinate axes. To find the points of intersection, we substitute the equations of L in the equation of S. This gives F(Ui + Rrf, ...,UN + RNt) = h{t) = t"/i*(t) with h*(0) ^ 0 where fi = ordth(t). Now t = 0 is a root of h(t) of multiplicity fi, and so we visualize that /x points of S have coalesced at P . We call fj, the intersection multiplicity of S and L at P and we denote it by intp(5, L). To indicate the dependence of fi on P and R, we may write HP,R for fi. Expanding F around P we have

F{XU ... ,xN) = ^Gh...iN(x1

- u,r • • • (XN - uNy».

Let v = ordpF i.e., let v be the smallest value of i\ + • • • + i^ with G^...^ ^ 0. We call v the multiplicity of S at P and we denote it by multp.S. Geometrically v can be characterized by noting that, with P fixed, it is the minimum of fip,R taken over all R, i.e., taken over all lines through P . To indicate the dependence of v on P , we may write up for v. We call P a simple or double or triple or ... point of S according as vp = 1 or 2 or 3 or ... ; by a singular point of S we mean a point of S which is not simple. The line L is said to be tangent to 5 at P if its intersection multiplicity with S at P is greater than the multiplicity of S at P , i.e., if fip,R > up. Let

GV{XU...,XN)=

Yl

Gil...iN(X1-U1r...(XN-UN)i».

Now a cone with vertex P is a hypersurface passing through P such that, for every point P' ^ P on it, the entire line PP' is on it. Clearly G„ = 0 is a cone with vertex P , and a line L through P is on it iff L is tangent to S at P . We call Gv = 0 the tangent cone of 5 at P . If P is a simple point of S, i.e., if v = 1, then for

66

LECTURE L3: TANGENTS AND POLARS

1 < i < N we have F ^ = G0...010...0 with 1 in the i-th place, and hence the said cone is reduced to the hyperplane FVl(Xi

-Ui)

+ --- + FUN(XN

-UN)

=0

which we call the tangent hyperplane of S at P. Note that so far we did not assume F to be a polynomial. So at least when P is the origin ( 0 , . . . ,0), the above discussion applies to an analytic hypersurface, i.e., to the case when F is a (formal) power series. Now suppose that F(X\,..., Xjy) is a polynomial of degree d. Then d is also called the degree of the hypersurface S; if JV = 2 then S is a curve of degree d, and if N — 3 then S is a surface of degree d. To characterize d geometrically, we note that now h(t) is a polynomial of degree d, and hence it has d roots giving d points of intersection of S with L. This is so for most values of Ui and R4. For special values of Ui and Ri, the equation h(t) = 0 may have less than d distinct roots because some of them may be multiple roots. Thus d may be characterized as the maximum number of points in which a line meets 5; here and in what follows, the line is assumed not to lie on S. Moreover, if there are c distinct roots and their multiplicities are e\,..., ec, then for the sum e = e\ + • • • + ec we have e = d, and for the corresponding points P i , . . . , Pc in which S meets L we have intp^S, L) = e;. Thus the sum e of the intersection multiplicities of S and L at their points of intersection equals the degree of S. This is the first case of Bezout's Theorem, which is the oldest Theorem of algebraic geometry and was enunciated by Bezout in his 1770 book [Bez]. Here is an example which shows that we need to be careful in this. Namely, with TV > 1 and d > 1, let F(XU...,XN)

= XtlX2

+ X3 + • • • + XN - 1

a n d P = (1,1,0, . . . , 0 ) . Then h(t) = (1 + J R 1 i) d " 1 (l + Rat) + R3t + --- + RNt - 1 = Rf~1R2td

+ (terms of degree


Therefore if either Ri = 0 or R2 = 0 then the sum e is smaller than the degree d of S, because some points of intersection of S and L "have gone to infinity." To take care of this we must go over to projective spaces and homogeneous coordinates. For an elementary treatment of all these things, you may like to browse in my 1990 AMS book [A04]. §4: H O M O G E N E O U S COORDINATES Again let F(Xi,...,

XN) be a polynomial of degree d and consider the hyper-

67

§4: HOMOGENEOUS COORDINATES

surface 5 in JV-space given by F(X\,...,

Xjv) = 0. Expanding F we have

F(Xi, . . . ,Xpf) — 2_j^h—iN^l

• • -^N •

Collecting terms of like degree we have d

F(Xu...,XN)

=

J2Fj(Xi,...,XN)

where Fj(Xi,...,XN)

=

2_^

Fil...iNXll1 ..

.X$

is a homogeneous polynomial of degree j , i.e., every term in it is of degree j . For some j it may happen that Fj has no term in it, i.e., Fj may be reduced to zero. But Fd is not reduced to zero because F is of degree d. Now for very large values of the variables, the values of the terms of degree d are much bigger than the values of the terms of degree < d, i.e., Fd dominates the rest of F. Therefore, heuristically speaking, we may say that Fd = 0 gives the points at infinity of S. Algebraically, we "homogenize" F by multiplying terms of degree j by the (d—j)th power of a new variable XN+I to get a homogeneous polynomial f(Xi,..., X;v+i) of degree d. By "dehomogenizing" / , i.e., by substituting XJV+I = 1 in it, we get back F. Thus d

f(Xi,...

,XN+i)

= f^X^jFjjXi,.. j=o

.,XN)

and F(Xi,...

,XN) = f(Xi,...

,XN,1).

Alternatively, we substitute Xi = Xi/xM+i

for

1< i
in F and then we clear the denominators by multiplying by a;^ +1 to get f{xi,...,

XN+I)

= X%+1F(XI/XN+I,

• • •,

xN/xN+i)

d

= YlXN~+lF3iXl>--->X")3=0

By substituting xjv+\ = 0 in f(xi,..., XJV+I) we get Fd(x\,... ,XN), and so we may say that Fd = 0 is the intersection of / = 0 with the "hyperplane at infinity" given by xN+i = 0.

LECTURE L3: TANGENTS AND POLARS

68

Geometrically, let A^ = A f = kN = {P = (Ui,...,

UN) : Ui G k for 1 < i < TV}

be the affine TV-space over the field k. (If you do not want to be bothered by "fields" and such, you may simply take k be the set of all real numbers R, and then the affine TV-space is your familiar M.N). We obtain the projective TV-space pN _ pjV o v e r fc w n e n w e a U g m e n t A^ by the "hyperplane at infinity" in the following manner. First, instead of representing a point P of KN by a single Ntuple (Ui,..., UN), let us represent it by all the (TV + l)-tuples u = (ui,..., UJV+I) such that UN+I ^ 0 and Ui = Ui/u^+i f° r 1 < » < N; these are called homogeneous coordinates of P. Now the remaining (TV + l)-tuples, i.e., the (TV + l)-tuples u = (u\,..., UAT+I) with UJV+I = 0, represent points of the hyperplane at infinity, again with the understanding that different (./V+l)-tuples represent the same point iff they are proportional. In other words, let us say that two (N + l)-tuples ( u i , . . . , UJV + I) and (u[,..., U'JV+I) a r e equivalent iff they are proportional, i.e., iff (u[,..., u'N+1) = (cu\,..., CUJV+I) for some nonzero constant c. Now P ^ may be identified with the set of all equivalence classes of (iV + 1)-tuples, with the understanding that the zero (N + l)-tuple ( 0 , . . . , 0) is excluded from consideration; in other words Pw = P^ = ( ^ + 1 \ { ( 0 , . . . , 0 ) } ) / ~ where ~ is the said equivalence relation. A projective hypersurface S of degree d, i.e., a hypersurface in the projective space P " , is given by an equation f(xi,..., XJV+I) = 0 where f(xi,..., XN+I) is a nonzero homogeneous polynomial of degree d; this is unambiguous because for any projective point P we have f(u\,..., u^+i) = 0 for one homogeneous coordinate tuple ( u i , . . . , UN+I) of P iff f(u\,..., u'N+1) = 0 for every homogeneous coordinate tuple ( u i , . . . ,u'N+1) of P; when this is so we say that P G S; otherwise P $. S. In particular, a projective hyperplane is given by a homogeneous linear equation a\X\-\ hfliv+i^iv+i = 0 where the coefficients a\,... ,ajv+i are determined up to proportionality, and at least one of them is nonzero; if a, ^ 0 for some i < N then this corresponds to the hyperplane a\X\ H h CLNXN + ajv+i = 0 in AN; if a< = 0 for alH < N then we get the hyperplane at infinity given by XN+I — 0. The portion of S "at finite distance," i.e., S (~l AN, is the affine hypersurface of degree d given by the equation F(Xi,... ,XN) = 0 where F(Xi,... ,XN) = f(X\,... ,Xff,l); the remaining portion of S consists of the points of S lying on the hyperplane at infinity. Any hyperplane H in PN is clearly a copy of P w _ 1 , and its complement PN \ H is a copy of A^; in particular, this is so for the hyperplane Hi : xt = 0, i.e., P w \ Hi is a copy A ^ of Af; moreover, P ^ = U ^ j ^ P ^ \ Hi), i.e., the projective TV-space can be covered by TV + 1 copies of the affine TV-space; in other words

Pf = uf = + 1 (P JV \^) = uf = + 1 A^ Therefore the definitions of simple point, singular point, multiplicity, intersection

69

§4: HOMOGENEOUS COORDINATES

multiplicity, tangent cone, and tangent hyperplane can be extended to projective hypersurfaces. Let u = ( u i , . . . ,UN+I) be homogeneous coordinates of a point P of S. Let x = (xi,...,xN+i) and fx = (fXl,... ,fXN+1) and fu = (fUl,... ,fUN+1), and as usual let us put X • fx = XifXl H

h XN+lfxN

+1

and X • Ju

=

X\JUl

+ •• •+

XM+1JUN+I

and so on. By Euler's Theorem on Homogeneous Polynomials we have x-fx = df(x), and hence (because P G S) we get u • fu = 0. For a moment assume that u^+i ^ 0 and let Ut = Ui/uN+i for 1 < i < N. Then by the equation f(x\,... ,XM+I) = xfj+lF(xi/xN+i,... ,XN/XN+I), we see that fUi = uN~^lFui for 1 < i < N; since P is singular for S iff F\ji = 0 for 1 < i < N, in view of the equation u • fu = 0 we conclude that: P is singular for S iff fUi = 0 for 1 < i < N + 1. In case P is a simple point of S, we know that the tangent hyperplane Tp of S at P has the afHne equation Fj/i(^i — Ui) + • • • + FUN(XN — UN) = 0; since u • fu = 0 and fm — u%~+iFui for 1 < i < N, we conclude that Tp is given by the homogeneous equation x • fu = 0. By symmetry it follows that both these conclusions remain valid without assuming UN+I ^ 0. Thus, for any point P of S with homogeneous coordinates u = (m,..., UN+I), we have that: (11) P is a singular point of 5 iff fUi = 0 for 1 < i < N + 1, and (12) if P is a simple point of S then the tangent hyperplane Tp of S at P is given by the homogeneous linear equation x • fu = 0. For any point A of PN with homogeneous coordinates y = (yi,... ,yjv+i), let us define the polar 11^ of A (relative to S) to be the hypersurface of degree d — 1 defined by the homogeneous equation y • fx = 0. By (II) we see that: (13) if P is any singular point of S then P e 11^. By (12) we see that: (14) if P is any simple point of S then: P G ILA <£> A £ Tp. By (14) we see that: (15) if P is any simple point of S with P ^ A then: P G n ^ <^> the line AP is tangent to 5 at P . It follows that if S is a nonsingular (= having no singular point) hyperquadric then the present definition of polar coincides with the definition given when we assigned Homework Problems (HI) and (H2). To do the Homework Problems (HI) and (H2), assume that S is a nonsingular hyperquadric. Let A and B be any points in PN with homogeneous coordinates y = (j/i,..., j/jv+i) and z = (z\,..., ZN+I) respectively. Then the polars HA and n ^ are given by the equations y • fx = 0 and z • fx = 0 respectively. We can write f(Xi, . . . , XN+l) =

2J l
aiX

^+

H2 l<j<j'
bjj'XjXf

70

LECTURE L3: TANGENTS AND POLARS

with constants
yfz=

]T

2aiyizi+

b

3j'(yjzj' +Vj'zj) =z-

Y^

l
fv

l<j<j'
Therefore

B eUA^y-

fz = 0^ z-

fy=0^AeUB.

§5: S I N G U L A R I T I E S Reverting to affine equations, let S : F(X\,..., XN) = 0 be a hypersurface in the affine iV-space over a field k. Given a point P of S, by a translation of the coordinate system, we may assume it to be the origin ( 0 , . . . , 0). This amounts to making the /^-automorphism of the polynomial ring k[X\,..., Xn] given by Xi — i > Xi + Ui where ( f / i , . . . , UN) are the original coordinates of P. Let multpS = v. Then F(Xi,...,

XN) = F„(Xi,...,

XN) + (terms of degree > v)

where F„(Xi,..., XN) is a nonzero homogeneous polynomial of degree v. First let us think of S as an analytic hypersurface at P, i.e., let us assume F to be a power series. Let fc[[Xi,..., XN]] be the ring of all formal power series in X\,..., XN with coefficients in k; here formal means we are disregarding questions of convergence, and we are going to do only algebraic manipulations; for instance, we permit the power series ^2,ilXl whose radius of convergence is zero. We shall discuss the notion of radius of convergence later on. Later on we shall also show that power series rings over a field are UFDs. Using this fact we can write F = E0E[' ... E? where r\,... ,rt are positive integers, E\,..., Et are pairwise coprime irreducible power series with £ i ( 0 , . . . ,0) = • • • = Et(Q,.. • ,0) = 0, and EQ is a power series with E0(0,...,0) ^ 0. We call Si : Ex = 0 , . . . , St : Et = 0 the analytic branches of S (at P). Let multpSj = Ui for 1 < i < t. Then r\V\ + • • • + rtvt = v, and for 1 < i < t we have Ei{Xi,...,

XN) = EiVi {Xi,...,

XN) + (terms of degree > u^

where EiVi {X\,..., XN) is a nonzero homogeneous polynomial of degree Vi. For the "initial form" Fu of F we have the factorization FV{XU ...,XN)

= EQ(0, • • •, 0) Yl

EiVi{Xu...,

XN)ri

with £ 0 ( 0 , . . . , 0) ^ 0.

l
We say that S has a t-fold NC (= Normal Crossing) at P if all the analytic branches have a simple point at P and their tangent hyperplanes are linearly independent,

§5:

71

SINGULARITIES

i.e., if v\ = • • • = vt = 1 and upon letting Ea =

Yl

E

'ijXJ

with E

'ij

e

k

l<j
we have that the rank of the t x N matrix {E'^) is t. Note that the rank of an m x n matrix (a^) over k is the largest nonnegative integer r for which there exist sequences 1 < p\ < • • • < pr < m and 1 < q\ < • • • < qr < n such that the determinant of the r x r matrix (aPiqj) is nonzero. Also note that if S has a t-iold NC at P then obviously t < N. We may say that, in the sense of analytic geometry, S has a simple point at P iff it has a 1-fold NC at P. Now let us think of S as an algebraic hypersurface, i.e., let us assume F to be a polynomial of some degree d. Since the polynomial ring k[X\,..., X^] is a UFD, we can write

F = r 0 rf... r£where p\,..., pT are positive integers, T i , . . . , TT are pairwise coprime irreducible polynomials with T i ( 0 , . . . , 0) = • • • = r r ( 0 , . . . , 0) = 0, and To is a polynomial with r 0 ( 0 , . . . , 0) ^ 0. We call Ei : Ti = 0 , . . . , E T : TT = 0 the algebraic branches of S at P. Concerning these two factorizations, you may do the following Homework Problem, where we recall that a partition of a set I f is a collection of nonempty subsets of W whose union is W and which are pairwise disjoint, i.e., have pairwise empty intersection; we may express this by writing W = Llig/ W» where Wi are the subsets indexed by an indexing set / . HOMEWORK PROBLEM (H3). Show that the analytic factorization is a refinement of the algebraic factorization, i.e., there is a partition { 1 , . . . , i] = ]Ji
LECTURE L3: TANGENTS AND POLARS

72

a power series of even order with constant term 1 has a power series square root. In case of characteristic zero, this follows from Newton's Binomial Theorem for fractional exponents thus. By the said Theorem we have the power series identity

(Ti)

(i + xy = i + Y.

r(r

~ 1 ) --., ( r ~ i

+ 1)

x*

where r is any rational number [cf. §12(E16)]. For X we may substitute any power series in X of positive order. Therefore by taking r = 1/n we see that any power series A(X) whose constant term is 1 has an n-th root for every positive integer n. If the order of the power series B(X) is en for some integer e > 0 and the coefficient of Xen in B(X) is 1 then we can write it as B(X) = (Xe)nA(X) with A(0) = 1. Therefore B(X) has an n-th root. It follows that Y2 - X2 - X3 = [Y - X + (terms of degree > 1)] x [Y + X + (terms of degree > 1)] and Y2 - X* - X5 = [Y - X2 + (terms of degree > 2)] x [Y + X2 + (terms of degree > 2)]. Consequently, at a node we have a 2-fold NC, but at a tacnode we do not have an NC. At a cusp we of course do not have an NC. §6: HENSEL'S LEMMA A N D N E W T O N ' S T H E O R E M The fact that by L2§3(T1) we can find an n-th root of any A(X) € k[[X}] with A(0) = 1, and for the said n-th root a(X) € fc[[X]] we have Q(0) = 1, can be restated by saying that the polynomial Yn — A{X) is divisible by Y — a(X) in /c[[X]][y]. Moreover by carrying out the division we get the factorization Yn - A{X) = [Y - a(X)} x [Y"" 1 + (31(X)Yn~2 + ••• + /3„_i(X)] where 0i(X) e K[[X}] with A(0) = 1 for 1 < i < n - 1. Around 1900 this was generalized by Hensel into his celebrated lemma which has many incarnations. Let us now state and prove the following simplest version of it, where we note that two univariate polynomials over a field are coprime means they have no nonconstant common factor. BASIC HENSEL'S LEMMA (T2). Let F(X, Y) = Yn + Ai (X)Yn~1

+ --- +

An(X)

§6: HENSEL'S LEMMA AND NEWTON'S THEOREM

73

be a monic polynomial of degree n > 0 in Y with coefficients Ai{X) in fc[[X]] for 1 < i < n where k is any field. Assume that F(0,Y)

=

G(Y)H(Y)

where G(Y) and H(Y) are coprime monic polynomials of positive degrees r and s in Y with coefficients in k. Then there exist unique monic polynomials G(X,Y) and H(X, Y) of degrees r and s in Y with coefficients in A;[[X]] such that F(X,Y)

=

G(X,Y)H(X,Y)

and G(0, F ) = G(Y)

and

#(0, y) =

H(Y).

PROOF. Rewriting F, G, H as power series in X with coefficients polynomials in Y we have 'F(X, Y) = FQ(Y) + F1(Y)X < G(X,Y)

= G0(Y) + Gx(Y)X

H(X, Y) = H0(Y) + H1(Y)X

+ • • • + Fi(Y)Xi i

+ ••• + Gi{Y)X

i

+ ••• + Hi{Y)X

+ ... + ... + ...

where Fo(Y) = F(0, Y) = a monic polynomial of degree n over k GQ(Y) = G(Y) = a monic polynomial of degree r over k HQ(Y) = H(Y) = a monic polynomial of degree s over k Fi(Y) G k[Y] of degree < n for alii > 0 Gi(Y) e fc[F] of degree < r to be found for alH > 0 KHi(Y)

e /s[y] of degree < s to be found for alii > 0

and where by the coprimeness we can find P(Y) and Q(Y) in k[Y] such that (1)

P(Y)Go(Y)

+ Q(Y)H0(Y)

= l.

The requirement F = GH is equivalent to saying that for every / > 0 we want to satisfy the equation (2)

£

GiWHjiY)

= Ft{Y).

For I = 0 this is given. Now let / > 0 and suppose we have found d and Hj for all i < I and j < I such that the above equation is satisfied for every value of I smaller than the given value. By sending the known terms to the right we see that (2) is equivalent to requiring that (2')

Hl(Y)G0(Y)

+ Gi(Y)H0(Y)

= F[{Y)

74

LECTURE L3: TANGENTS AND POLARS

where F{(Y) = Ft(Y) -

Yl

Gi(Y)Hj(Y)

e k[Y] is of degree < n.

i-\-j=l with i < / and j
Upon letting ff/(y) = P(Y)F[(Y) (!')

and GJ(Y) = Q(Y)F{{Y),

by (1) we get

^ / ( ^ ) G o ( r ) + GKy)ffo(^) = F/(Y).

Dividing fl/(y) by flo(y) we find G'{{Y) and JJj(y) in k[Y] such that #/(Y) = G'l'(Y)H0(Y) + Hi(Y) and the degree of Ht{Y) is < than the degree of H0(Y) which is 5. Now upon letting Gt{Y) = G't(Y) + G'/(Y)G0(Y), by (1') we get (1")

Hl(Y)Go(Y)

+ G«(y)JJoOO =

Fl(Y).

In the above equation, the degrees of the first and the last terms are < n and so is the degree of the middle term GI{Y)HQ{Y), but in it the degree of HQ{Y) is s = n — r and hence the degree of the other factor Gi(Y) must be < r. Thus the degree conditions are satisfied and hence (1") yields (2'), which completes the induction on I. The uniqueness can also be proved by induction thus. Given I > 0, suppose G i ( y ) , . . . , G / _ i ( y ) and H!(Y),...,Hi-i(Y) are unique. Let Gf(Y) and flf(y) be any polynomials of degrees < r and < s respectively such that (2*)

H?(Y)G0(Y)

+ GUY)H0(Y)

= F/(Y).

Subtracting (2*) from (2') we get (2**)

Hr(Y)G0(Y)

=

-Gr(Y)H0(Y)

where G?(Y) = Gt(Y) - Gj(Y) G k\Y] is of degree < r and H?*(Y) = Ht(Y) H*(Y) £ k[Y] is of degree < s. Since Go(Y) and HQ{Y) are coprime and by the above equation GQ(Y) divides Gi*(Y)Ho(Y), it must divide G**(Y); but the degree of Gf*(Y) is smaller than the degree of Go(Y) and hence we must have Gi*(Y) = 0, i.e., G[(Y) = Gt(Y). Similarly Hfiy) = Hi(Y). This proves uniqueness. COMPLETING THE POWER. Now we shall use Hensel's Lemma to prove Newton's Theorem stated in L2(T1), i.e., in (Tl) of Lecture L2. An important ingredient of the proof will be Shreedharacharya's completing the square method in its incarnation of "completing the n-th power." By this we mean, given an n-th degree monic polynomial F(Y) = Yn + AxYn~l + • • • + An with coefficients A\,..., An in a ring R in which n is a unit (for instance R could be a field whose characteristic does not divide n), "by completing the n-th power"

§6: HENSEL'S LEMMA AND NEWTON'S

THEOREM

75

we can write

with A2, • •., An in R, and this gives us F(Y) = F(Y - (Ai/n))

= Yn + A2Yn~2 + ••• + ! „ .

The advantage of "killing" the coefficient of y n _ 1 is to detect multiple roots. So for a moment suppose that the coefficients A\,... ,An belong to an algebraically closed field K of characteristic 0, and let a\,... ,an be the roots of F in K, i.e., F(Y) =

(Y-ai)...(Y-an).

Assume that A\ = 0 but Aj ^ 0 for some j with 2 < j
ax)n = Yn - naiYn-1

+ ...

which would give us the contradiction A\ = —na.\ ^ 0. Therefore upon relabelling a\,..., an we can arrange matters so that for some positive integer r < n we have a\ = • • • = aT 7^ ar+j for 1 < j < s = n — r. Now upon letting G(Y) = (Y-ai)...(Y-ar)

and

H(Y) = (Y - ar+1)...

(Y -

ar+s)

we see that F(Y) = G(Y)H(Y) where G(Y) and H(Y) are coprime monic polynomials of positive degrees in Y with coefficients in K. Thus we have proved the following: LEMMA (T3). If F{Y) = Yn + £ i < i < « AiYn~i is a monic polynomial of degree n > 0 with coefficients A^ in an algebraically closed field K of characteristic 0, and if Ax = 0 but Aj ^ 0 for some j with 2 < j < n, then F ( F ) = G(Y)H(Y) where G(F) and i? (10 a r e coprime monic polynomials of positive degrees in Y with coefficients vaK. PROCESSES ON ROOTS. To put the completing the power method in proper perspective, again consider a monic polynomial

F(Y) = Yn+ J2 AiY"-* l
of degree n > 0 in Y with coefficients Ai in a field K, and let Q 1 ; . . . , an be the roots of F in an overfield L of K, i.e.,

F(X)= n (Y-ai). l
76

LECTURE L3: TANGENTS AND POLARS

To "increase the roots" by B G L, by putting = Yn+

F*(Y) = F(Y-B)

]T

AtY"-*

l
with A* E L we get f~(Y)=

J ] ( ^ - ( a i + B)). l
Likewise to "multiply the roots" by 0 ^ C G L, by putting F**(Y) = C m F ( y C - 1 ) = Y " +

A**Y n _ i

^ l
with A** G L we get F**(Y) =

J]

(Y-Ca,).

l
As the third "standard process" on roots, to "reciprocate the roots" we assume An 7^ 0 and then by putting F***(Y) = YnF(Y~1)

= 1 + AiY + • • • +

AnYn

i.e., by "reading F backwards" we get F***(Y) = An

n

(Y-a/a,)).

l
With this preparation let us now prove a slight variation of Newton's Theorem which we call: BASIC NEWTON'S THEOREM (T4). Let there be given any monic polynomial F(X, Y) = Yn + Ai (X)Yn~1

+ --- +

An(X)

of degree n > 0 in Y with coefficients A\(X),... ,An(X) in k((X)) where k is an algebraically closed field of characteristic 0. Then for some integer m > 0 we have a factorization F(Tm, Y)=

[ J (Y -

Vi{T))

with

Vi{T)

£ fc((T)).

l
Moreover, if At{X)

G k[[X\] for 1 < i < n then y,(T) G k[[T]\ for 1 < i < n.

For the reason given above we call our proof: SHREEDHARACHARYA'S PROOF OF NEWTON'S THEOREM. We make induction on n. For n = 1 we have nothing to show. So let n > 1 and assume true

§7-- fJVTJEGR-AL,

r}EFJ,&r*fD&WC7&

TT

for all values of n smaller than the given value. By completing the n-th power we have F*(X,Y)

= Yn+

= F(X,Y-(A1(X)/n))

A*{X)Yn~i

£ \
where A*(X) G k((X)) with Af (X) = 0. If yl*(X) = 0 for 2 < j < n then we have nothing to show. So assume that A*(X) ^ 0 for some j with 2 < j < n. Let u = min 1 < i <„ —

oi&xA*{X) :

i

.

Then u — v/w where v and w are integers with w > 0. Note that if Ai(X) £ fc[[X]] for 1 < i < n then clearly v > 0. Let l
where A**(Z) = A*(Zw)Z~iv

6 k[[Z]} for 1 < i < n with ^ i * ( ^ ) = °

A**(0) ^ 0 for some j with

and

2<j
Now first applying (T3) and then (T2) we get F**(Z,Y)

=

F'(Z,Y)F"{Z,Y)

where F'(Z, Y) and F"(Z, Y) are monic polynomials of degrees n' > 0 and n" > 0 in Y with coefficients infc[[£]]. By the induction hypothesis we can find integers m! > 0 and m" > 0 such that

F'{Tm',Y)=

J ] (Y-y'AT))

and F"(T m ",y)= JJ {Y-y'*(T))

l
l
with j/i(T) and y"(T) in &[[T]]. Upon letting rn = wm'm" positive integer and F*(Tm,Y)=

ft

(Y-y*(T))

with

we see that m is a

y*(T) e k((T))

l
and if w > 0 then y*(T) € fc[[T]] for 1 < i < n. It follows that F(Tm, Y)=

I ] (Y - yi(T))

with

y < (T)

G fc((T))

l
and if A^X)

€ fc[[X]] for 1 < i < n then yt(T) Gfc[[T]]for 1 < i < n. §7: I N T E G R A L D E P E N D E N C E

The last sentence of Theorem (T4) is a bit informal. It does not make completely clear whether the roots yi(T) asserted in the factorization display are automatically in k[\T]] when the coefficients Ai(X) are assumed to be in fc[[X]], or whether they

78

LECTURE L3: TANGENTS AND POLARS

can be so chosen. Of course the uniqueness of roots does show the two statements to be equivalent. To prove the apparently stronger statement, i.e., the implication At{X) G k[[X]] for all i => j/i(T) G k[[T]] for all i, let us introduce the ideas of integral dependence and integral closure. To go back in history, what is y/2 and why is it not rational? At any rate, the need to know y/2 arose since, by the Pythagoras Theorem, it is the length of the hypotenuse of an isosceles right angle triangle whose two equal sides have length one. Now l 2 = 1, 22 = 4 , . . . , and hence there is no integer whose square is two. Suppose there is a rational number t whose square is two, i.e., t satisfies the equation y2 — 2 = 0. We can write t = u/v where u and v are integers with v ^ 0; we may assume that this is a reduced form of t, i.e., u and v have no common prime factor; we may also assume that v > 0. Since (u/v)2 — 2 = 0 we get u2 = 2v2 which shows that every prime factor of v divides u. Therefore we must have v = 1. Thus t must be an integer which we have shown to be impossible. The belief that although y/2 is not rational, it is some sort of a number, makes it desirable to "complete" the field Q to get the field R; we can then accommodate y/2 — 1.4142..., n — 3.1415..., and so on. The above argument can be generalized to show that Z is a normal domain, i.e., any element t in its quotient field Q which is integral over it belongs to it. An element t in an overring 5 of a ring R is said to be integral over R if it satisfies a monic polynomial equation over R, i.e., if F(t) = 0 for some monic polynomial F(Y) = Yn + AiYn~l -\ h An of some degree n > 0 with coefficients Ai,...,An in R. We may say that t/R is integral to mean that t is integral over R. A subset T of S is integral over R means every t G T is integral over R; again we may indicate this by saying that T/R is integral. By the integral closure of R in S we mean the set of all elements in S which are integral over R. It can be shown that sums and products of integral elements are integral; moreover, any t € R is integral over R since it satisfies the equation Y — t = 0; consequently [cf. §10(E2)]: J the integral closure of a ring R in an overring S is a subring of S, 1 and R is a subring of the said integral closure. It can also be shown that [cf. §10(E2)] J if an overring 5 of a ring R is integral over R, and (^ an overring T of S is integral over S, then T is integral over R. A ring R is integrally closed in an overring S means R coincides with its integral closure in S. Thus a domain is normal means it is integrally closed in its quotient field. Since Z is a UFD, its normality will follow by showing that (J3)

every UFD is normal.

§7: INTEGRAL

DEPENDENCE

79

To put this in proper perspective, let us first show that J for any valuation v : K —> G U {00} of any field K, ) the valuation ring Rv = {x G K : v(x) > 0} of v is normal. To prove this let us extend v to a valuation w of K{Y) thus. For any

A = A{Y) = J2 AiYi G K[Y]

with

^ G /if

let w{A) = min

U(J4,)

taken over all i

and note that if A = 0 then u>(A) = oo and if A ^ 0 then IU(J4) £ G , = the value group of v. For verifying that w satisfies the valuation axioms, we observe that the valuation axioms imply that (J5)

v(l) = 0

and for any finite number of elements x\,..., (J6)

v(xi H

xm in K we have

1- xm) > min(u(a;i),..., v(xm))

and (J7)

if v(xi) < v(xi) for 2 < i < m then v(x\ + • • • + xm) = v{x\).

Now given any B = ^ BjYj G K[Y] with Bj G K we clearly have w(A + J3) > min(ty(i4), iu(B)). Moreover, if A ^ 0 ^ £ then, upon letting i' = min{i : w(A) = v(Ai)} and f = min{j : w(B) = v(Bj)}, and upon letting C = AB = ^ C i ^ 1 with C; G AT, for all I we have

i+j=l

and hence v(Ci) > w(A) 4- w(B), and we also have CV+j/ =Ai>Bj> +

]T i+j=i'+j'

and hence v(d'+ji) (J8)

A B

iJ

and either i < i' or j < j '

= w(A) + w(B). Thus we have

w(AB) = w(A) + w{B) for all A ^ 0 ^ £ in AT[y].

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LECTURE L3: TANGENTS AND POLARS

Therefore we may unambiguously put w(A/B) that

= w{A) — w(B), and then it follows

w : K(Y) -> G U {00} is a valuation of K(Y) (J9)

such that w(x) = v(x) for all x G K, and the value group of w coincides with the value group of v.

Given a domain R, let us say that R is overnormal to mean that if A(Y), B(Y), C(Y) are monic polynomials in V with coefficients (*)

in the quotient field K of R such that A(Y)B(Y)

= C(Y) e R[Y],

then A(Y) € R[Y] and B(Y) G R[Y}. Using (J8) let us show that J ^ or a n y v a m a t i ° n v : K ^> GU {00} of any field K, [the valuation ring Rv = {x € K : v(x) > 0} of v is overnormal. Namely, since v(l) = 0, we see that if A = A(Y) G K[Y] and B = B(Y) G K[Y] are monic in Y then w(A) < 0 and w(B) < 0, and if C = C(Y) G RV[Y] is monic in Y then w(C) = 0, and hence if also AB = C then by (J8) we get w(A) = 0 = w(B), and therefore A(Y) G RV[Y] and B{Y) G i? v [y]. Obviously (Jll)

every overnormal domain is normal.

In greater detail, let y in the quotient field K of a domain R satisfy C(y) = 0 with nonconstant monic C{Y) G R[Y]; then C(Y) = A(Y)B(Y) with monic A(Y) = Y -y G K[Y] and monic B(Y) = C(Y)/A(Y) G ^ [ F ] , and hence assuming R overnormal we get A(Y) G R[Y], i.e., y £ R. Now (J4) follows from (J10) and ( J l l ) . Given a ring R, let us say that R is a valuation ring to mean that R is a domain which is the valuation ring of some valuation of its quotient field. Given a domain R, let us say that R is an intval domain to mean that R is the intersection of a nonempty family of valuation rings of valuations of its quotient field. Now obviously if a domain R with quotient field K is the intersection of a nonempty (J12)

family of normal (resp: overnormal) domains with quotient field K, then R is normal (resp: overnormal).

By (J10) and the overnormal part of (J12) we see that (J13)

every intval domain is overnormal.

Likewise by (J4) and the normal part of (J12) we see that (J14)

every intval domain is normal.

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81

§8: U N I Q U E FACTORIZATION D O M A I N S Let R be a UFD with quotient field K. Recall that U(R) is the set of all units in R. Let W be a set of irreducible elements in R such that no two elements of W are associates of each other and every irreducible element in R is an associate of some element of W. Then every x £ Kx can be uniquely expressed as x = ux J | zVz{x) zew

with

ux e U{R)

and

vz{x) e Z

such that vz(x) = 0 for all except a finite number of z, with the understanding that in the product the terms which are reduced to 1, i.e., those for which vz(x) — 0 should be disregarded; also put vz(0) = oo. Then vz : K y-> ZLI {oo} is a valuation of K with value group Z, and clearly we have (J15)

R=

f]

RVz

zew

and hence (J16)

every UFD is an intval domain

and now (J3) follows from (J14). As above, we get a valuation wz : K(Y) \-> ZU{oo} when for every A = A(Y) = ^2 Aiyi

G K[Y]

with

Ai€K

we put wz(A) = min vz(Ai) taken over all i and so on. In particular, as in (J8), we have wz(AB)

= wz(A) + wz{B) for all A ^ 0 ^ B in R[Y].

This being so for all z e W, we get the GAUSS LEMMA which says that (J17)

c(AB) = c(A)c(B) for all A + 0 ^ B in R[Y]

where we define the content c(A) of 0 ^ A — A(Y) e R[Y] by putting c(A) = gcd(A 0 , Ai,...) with gcd as in L1(R4). Note that this defines c(A) up to associates in R, and (J17) means that for any values of c(A) and c(B), c(A)c(B) is a value of c(AB). As a consequence of (J17) it can be shown that (J18)

R is a UFD ^> R[Y] is a UFD.

For any positive integer r, by induction and (J18) we see that (J19)

R is a UFD => R[XU...,

Xr] is a UFD

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and hence in particular (J20)

if is a field =» K[XX,...,

Xr] is a UFD.

As a consequence of (Jl) and (J20) it can be shown that: for any field K and positive integers r,mi,... the if-homomorphism a : K[Xi,...,

,mr,

Xr] -> K[Ti,...,

Tr]

with Xi >-> JT™' is injective, and by identifying K(Xi,..., (J21)

Xr)

{ with its image under the monomorphism /? : K (Xu.

. . , * r ) -> K(TU ...,Tm)

induced by a,

we have that [K(T\,... ,TT) : K(X\,...

,Xr)\ = n\\.. .mr,

and

K[Tu...,Tr) = the integral closure of K[X\,...,

Xr] in K{T\,...,

Tr).

§9: R E M A R K S REMARK (Rl). [Factorization of Univariate Power Series]. Recall that an ED is a domain R with a euclidean function, i.e., a function : Rx —» N such that for all x,y in Rx we have: (Gl) y € xR =^ (a;) < <j>(y), and (G2) 2/ ^ xR => y = qx + r for some q £ R and r £ Rx with ^>(r) < (a;); moreover, if the r (and hence also the q) is unique then we call R a special ED. We have noted that the univariate polynomial ring over a field is a special ED. Now we observe that the univariate power series ring R = K[[X]} over a field K is even a simpler example of a special ED. To see this, it suffices to take (j>(A) = ordx^l for all A = A(X)£RX. It follows that: R is a PID and UFD with U(R) = the set of all power series of zero order, and XlR with i varying in N are exactly all the distinct nonzero ideals in R. Every nonzero nonunit element in R is an associate of a unique power X1 of X with integer i > 0. Every irreducible element in R is an associate of X. Every nonzero element h = h(X) in the quotient field K((X)) of -R has a unique factorization h = aXe where a G U{R) and e is an integer; moreover h e R-& e > 0. The only nonzero prime ideal in R is XR and it is maximal. For any nonzero / = f(X) G R of some order n > 0, clearly K n (fR) = {0} and hence we may identify K with a subring of the residue class ring L = R/(fR), and then upon letting x to be the image of X under the residue class map R —> L we see that l,x,..., xn~l is vector space basis of L over K and hence [L : K} = n; moreover L is a field <=> n = 1. For any 5 C R we put 0 0 0 ( 5 ) = 0 if S C {0} and otherwise we put GCD(S) = the unique power X1 of X which is a generator of the ideal in R generated by S, and for any finite or infinite sequence 3/1,2/2> • • • in i? we put GCD(yi,y2, • ••) =

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83

GCD({j/i, 2/2, • • •}); n ° t e that then the GCD is obviously a gcd. Likewise, for any finite sequence yi,.--,ya of elements in Rx with s > 0, by L C M ^ , . . . ,y„) we denote the unique 1cm of j/i, • • •, ys which is a power X% of X. To make the reference to the ring R explicit we may write GCD# and LCM^ instead of GCD and LCM respectively. Finally we note any 0 ^ ( e K((X)) has a unique reduced form C = £/Xl with £ € R and i £ N, and we call this the reduced form of £. It follows that (J22)

K is a field =» K[[X]\ is a UFD.

To justify the last sentence of (T4), as a consequence of (Jl) and (J22) it can be shown that: for any field K and positive integer m, the i^-homomorphism a : K[[X\] -> K[[T\] with X \-> Tm is injective and, by identifying K((X)) 1/3: K((X))

with its image under the monomorphism

-> if((T)) induced by a,

we have that \K((T)) : K{(X))] = m ,and K[[T\] = the integral closure of K[[X)} in A"((T)). REMARK (R2). [Kronecker's Theorem]. Later on we shall prove the converse of (J14) saying that [cf. L4§12(R7)] (J24)

every normal domain is intval.

By (J13) and (J24) we get KRONECKER'S THEOREM which says that (J25)

every normal domain is overnormal.

Note that by (J13) and (J16) we already get the special case of (J25) saying that (J26)

every UFD is overnormal.

§10: A D V I C E TO T H E R E A D E R As I have said before, an elementary treatment of many of the things said above, can be found in my 1990 book [A04] entitled Algebraic Geometry for Scientists and Engineers. Although this Engineering Book is not a logical part of the present book, I recommend it as collateral reading. §11: HENSEL A N D W E I E R S T R A S S We now turn to Hensel's Lemma for more variables, and at the same time we shall deal with the very versatile WPT = Weierstrass Preparation Theorem

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discovered by Weierstrass around 1870. As a companion to W P T we shall also prove the equally versatile WDT= Weierstrass Division Theorem. Before stating these we note that, following the treatment given in my Local Analytic Geometry book [A06], here our scheme of proof will be WDT => W P T =>• Hensel, all for any number of variables. For a comparison between Hensel's Lemma and WPT, and for a proof of bivariate W P T similar to the proof of Basic Hensel's Lemma (T2) given above, see pages 119-124 of my Engineering Book [A04]. To get ready for stating WDT and WPT, let Xu ...,Xm,7bem+l variables over a field k where m is any nonnegative integer, and consider the power series ring S = k[[Xu...,Xm,Y}}

= R{[Y}}

with

R=k[[Xi,...,Xm]].

As we have seen before, S = S(Xi,..., Xm, Y) G S is a unit in S iff 6(0,..., 0) ^ 0. Likewise e = e(X\,..., Xm) G R is a unit in R iff e ( 0 , . . . , 0) ^ 0. Equivalently h = h(Xi,.. .,Xm) e f i i s a nonunit in R iff h(0,... ,0) = 0, i.e., iff h e M{R) where M(R) is the ideal in R generated by X\,... ,Xm. Note that M(R) is the kernel of the epimorphism R —» k given by h H-> h(0,..., 0) and hence M(R) is a maximal ideal in R. Since M(R) is the ideal of all nonunits, it is the unique maximal ideal in R; see observation (1*) below. We define the Weierstrass degree of any / = f{X\,..., Xm, Y) G S relative to (Y, R) by putting wideg(/) = widegy ifl / = o r d y / ( 0 , . . . , 0, Y). If / * £ S is of the form

/ • = Yd + J2 fjYd~J

with

/ / G M(R)

for

1<3 < dG N

i<j
then we say that / * is distinguished or, in greater detail, we say that / * is a distinguished polynomial (of degree d) in Y over R. Note that then d = d e g y / * = wideg y>fi /* with the understanding that for any ring R and any power series A = ^2ieN AtYl £ R[[Y}} with Ai G R we put degy .A = max SuppyA where by convention the said max equals - c o iff the Supp is empty, i.e., iff A = 0, and similarly the said max equals co iff the Supp is an infinite set, i.e., iff A 0 R\Y). Recall that elements / , g in a ring R are said to be associates in R if for some elements / ' , g' in R we have / = gg' and g = / / ' . Now let us state WDT and W P T together with some supplements.

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SUPPLEMENTED WEIERSTRASS PREPARATION THEOREM (T5). Let X\, . • •, Xm, V b e m + 1 variables over a field k where m is any nonnegative integer, and let S = k[[Xi,. ..,Xm, Y}] = R[[Y}\ with R = k[[Xi,..., Xm}}. Then we have the following. WDT (T5.1). Given any g e S and any / e S with wideg(/) = d G N we can uniquely write g = qf + r where q £ S and r £ R[Y] with degyr < d. W P T (T5.2). Given any / e S with wideg(/) = d £ N we can uniquely write / = Sf* where 6 is a unit in S and / * is a distinguished polynomial of degree d in Y over R. SUPPLEMENT (T5.3). If g e R[Y] and / € S is distinguished then the equation g = qf + r in (T5.1) is the division identity in R[Y], i.e., q S R[Y}. Moreover, if g is also distinguished and r = 0 then 9 is distinguished. SUPPLEMENT (T5.4). If / , g in S are distinguished then: / and g are associates in S & f = g. SUPPLEMENT (T5.5). If / e S is distinguished and / = Fx...Fe with Fi,...,Fe in S then, for 1 < i < e, Fi is an associate in S of a distinguished F* £S , and we have / = Ff . . . Fe*. We shall deduce (T5) from a more general version of it. To start this off, we observe that the above rings S and R are complete Hausdorff quasilocal rings in the following sense. A quasilocal ring is a ring R with a unique maximal ideal, which we denote by M(R). It can easily be seen that J a ring is quasilocal iff all the nonunits in it form an ideal; lathis ideal is then the unique maximal ideal in that ring. As a consequence of (1*) and the valuation axioms L2(V1) to L2(V3) it can be seen that J the valuation ring Rv of any valuation v of a field K [is a quasilocal ring with M{RV) = {x £ K : v(x) > 0}. For ideals B, C in any ring R, the product BC is the ideal in R defined by BC = the set of all finite sums ^ bet with bi e B and c, € C. Similarly for the product B\... Bn of a finite number of ideals B i , . . . , Bn in R. If B\ = • • • = Bn then we write Bn for Bi.. .Bn. In particular B° = R. Note that products and sums of ideals are connected by the distributive law: (Bi -\ (- Bn)C = B\C -\ (- BnC. Let R be a quasilocal ring. R is Hausdorff means f l i ^ ^ ^ ) * = 0. For any h € R we define the .R-order of /i by putting ordfl/i = max{i e N : / i £ M ^ ) * }

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and we note that then R is Hausdorff means for any h G R we have: ord^h = 0 0 0 h = 0. A sequence x = (xi)i NE and j > NE we have xt - Xj G M(R)E. The sequence x is convergent means it converges to a limit £ G R, i.e., for every E eN there exists NE eN such that for all i > NE we have £ - Xj G M(R)E; we indicate this by some standard notation such as Xi —> £ as i —> 00 or limj_»0Oa;i = £• If i? is Hausdorff then a limit when it exists is unique. R is complete means every Cauchy sequence in it has a limit in it. If R is a complete Hausdorff quasilocal ring and Dj G M(R)u(ri with u(j) —> 00 as j —» 00 then a sequence like ]Cj'<,i> VjNote that if R is a quasilocal ring then the univariate power series ring S = R[[Y]] is a quasilocal ring with M(S) = M(R)S + YS, and if R is complete and Hausdorff then so is 5 . Clearly the rings S and R of (T5) are complete Hausdorff quasilocal rings. In particular R is Hausdorff because for any h G R we have ordn^ = ord(x1,...,J\:m)^The above power series rings are actually local rings in the following sense. If all ideals in a ring are finitely generated then we call it a noetherian ring. We shall eventually give a proof of the famous Hilbert Basis Theorem which says that a polynomial ring over a noetherian ring is noetherian, and then from this, by using WPT, we shall deduce that a power series ring is also noetherian. By a local ring we mean a noetherian quasilocal ring. Eventually we shall prove the Krull Intersection Theorem which says that every local ring is Hausdorff. For S = R[[Y]] with R quasilocal, the above concepts of wideg and distinguished apply except that now for any / = J2i>o ^% e & w ^ t n /i € i? we have wideg(/) = w i d e g y i j / = o r d y ^ ( / ) where i\) : S -> {R/M{R))[[Y]\ is the natural map given by i/>(/) = E,>o
/ is distinguished <=> deg(/) = wideg(/) < 00 and / is monic in Y.

Also note that for any / , / ' in S we have (4«)

deg(//') = deg(/) + deg(/')

and (5«)

wideg(//') = wideg(/) + wideg(/').

Finally note that 5 = Y^i>o < ^ *

G

& w i t ^ ^i G -R is a unit in S iff £0 is a unit in R.

Now let us state the said generalization of (T5).

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ABSTRACT WEIERSTRASS PREPARATION THEOREM (T6). Let R be a complete Hausdorff quasilocal ring and consider the univariate power series ring S = R[[Y]]. Then we have the following. (T6.1). Given any g £ S and any / £ S with wideg(/) = d £ N we can uniquely write g = qf + r where q £ S and r G R\Y] with degyr < d. (T6.2). Given any / G S with wideg(/) = d £ N we can uniquely write / = Sf* with 5 a unit in S and /* a distinguished polynomial of degree d in Y over i?. (T6.3). If g £ i?[y] and f £ S is distinguished then the equation g = qf + r in (T6.1) is the division identity in R[Y], i.e., q £ R[Y). Moreover, if g is also distinguished and r = 0 then q is distinguished. (T6.4). If / , g in S are distinguished then: / and g are associates in S <=> / = g. (T6.5). If / G 5 is distinguished and / = F i . . . Fe with F j , . . . , F e in 5 then, for 1 < i < e, Fi is an associate in 5 of a distinguished F* £ S , and we have / = F 1 *...F*. Items (T6.1) to (T6.5) are identical with items (T5.1) to (T5.5) respectively, and hence (T5) follows from (T6). To prove (T6.1) we write

/ = E^ yi

with

i>0

^SJR

and upon letting

f = J2f
and

/= E /'yi

i>d

0
we see that / is a unit in S and hence upon letting G =g

and

F = -///

and

Q=

9

/

that

qf = (Yd - F)Q and

(•)

G = J2i>0 and

G yi

i

with

Gi£R

for all i

F = ]T\> 0 FiY1 with F* G M(R) for all i and therefore (T6.1) is equivalent to the following: LEMMA (T7). Let R be a complete Hausdorff quasilocal ring and consider the univariate power series ring S = R[[Y]]. Let G, F in S be as in (•), and let d be any nonnegative integer. Then degy[G — (Yd — F)Q] < d for a unique Q £ S.

LECTURE L3: TANGENTS AND POLARS

Moreover for this Q we have Q = £ u > 0 Q(") where Q G M(R)U[[Y}] are given by the recursive formulas Qe (••)

for all e > 0;

— Gd+e

^(u+1) _ y , „ n(u) r Ve — 2s0<j
Q

e

=

Ee>oQ^y

for all u > 0 and e > 0; for all u > 0.

PROOF OF (T7). To prove uniqueness it suffices to show that d e g y ( r d - F)Q Q

= 0.

If for some u > 0 we have Q G M{R)U[[Y}}, i.e., upon letting Q = £ \ > 0 QjY* with Qi G -R we have Qi G M(i?) u for all i, then for any e > 0, by the above degree condition, we get 0 = coeff of Yd+e in (Yd - F)Q = Qe + terms in and hence Qe G M(R)U+1. e > 0 we have Qe G M{R)U To prove existence we E„>o Q ( u ) w e h a v e d e Sv[G degv

G

M(R)U+1

Thus by induction on u we can conclude that for all for all u > 0 and hence Qe = 0. Therefore Q = 0. try to find Q{u) G M(.R)"[[y]] such that for Q = - ( F d - F)Q] < d, i.e., l u) F Y,Q{u)\Yd+\Y,Q - \ iu>0


lu>0

Solving degy[G - Q(°>yd] < d w e get Q<°> = £ e > 0 Q i ' y e w i t h Qe = G d+e for all e > 0. It remains to find Q G M(i?)"[[y]] for all u > 1 such that degy

£QM i«>i

£Q(U)

<

i>0

Since F G M(i?)[[y]], we can successively find Q{u) by "solving modulo M(R)U." Upon doing this by recursively letting Qi" + 1 ) = J2o<j 0 and e > 0, we see that the above degree inequality is satisfied by taking Q^ =

£e>o£eU)yeforallu^0PROOF OF (T6). As said above, (T6.1) is equivalent to (T7). The first assertion in (T6.3) follows from the uniqueness part of (T6.1) and the second assertion follows from (3*) to (5*). To prove (T6.4) let / , g in S be distinguished, and assume that they are associates in S. Upon relabelling / , g we may suppose that wideg(g) = e < d = wideg(/). Let ip : S —» (i?/M(i?))[[y]] be the natural map whose restriction to R is the residue class epimorphism R —> R/M(R). Since / and g are associates we have g = ff for some / ' G S. Applying tp to the equation g = ff we get the equation

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Ye = Ydip(f) in (R/M{R))[[Y]} and hence e = d. Now g - f = / ( / ' - 1) with degy(
in (T6).

Next we come to: ABSTRACT HENSEL'S LEMMA (T8). Let R = k[[Xi,.. .,Xm}} where m is any nonnegative integer and k is an algebraically closed field, or more generally let R be any complete Hausdorff quasilocal ring whose residue field R/M(R) is algebraically closed. Let : R —» R/M(R) be the residue class epimorphism and let ~4> : R[Y] -* (R){Y} be denned by putting ^ ( / ) = J2i>o (fi)yi for a11 / = E i > 0 fiY* G R[Y] with fi G R. Let F(Y) = Yn + Y^

AjYn-j

with

n G N+

and

A, G R.

l<j
Then we have the following. (T8.1) Let a i , . . . , ah be the distinct roots of (j)(F(Y)) in
HF(Y))=

n (Y-air. \
Then there exist unique monic polynomials F\{Y),..., Fh(Y) of degrees e i , . . . , e„ in y over i? such that F(Y) = Fi(Y).. -Fh(Y) and ~fi(Fi(Y)) = {Y - a t ) e i for l(R), then there exist unique monic polynomials G(Y) and ff(F) of degrees r and s in Y with coefficients in R such that F(K) = G(Y)H(Y) with 0(G(K)) = G(F) and 4(H(Y)) = F ( F ) . PROOF. By (T6.2) and (T6.3) we see that (T8.3)

(T8.2) is true when H(Y) = Ys.

Now it is easy to see that (T8.3) =$• (T8.1) => (T8.2).

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§12: DEFINITIONS A N D EXERCISES DEFINITION (Dl). [Minors and Cofactors]. Consider an m x n matrix

(An

A=

/ill

\A„

Aln\

... Aij ...

.. .

-**-ij , . .

•

Jly

-^mrt/

over a ring R. By the r x s submatrix of A located at the ( « i , . . . , ur; v\,..., vs)th spots, where r > 0 and s > 0 with 1 < u\ < • • • < ur < m and 1 < v\ < • • • < vs < n are integers, we mean the r x s matrix B = {Bpq) over R given by Bpq = AUpVq; if r = s then we may call B a square submatrix of A of order r. By the rank of A, denoted by rk(j4), we mean the largest order of its square submatrix whose determinant is a unit in R. By the row rank of A, denoted by rrk(j4), we mean max r such that A has an r x n submatrix of rank r. By the column rank of A, denoted by crk(A), we mean max s such that A has an m x s submatrix of rank s. If A is a square matrix, i.e., if m = n, then for 1 < i < n and 1 5: J < n, by the (i,j)-th square submatrix of A we mean the (n — l ) x ( n — 1) square submatrix of A obtained by deleting the i-th row and j-th. column, i.e., located at the ( 1 , . . . , i - l , i + l , . . .,n; 1 , . . . ,j-1, j + 1,.. .,n)-th spots, and (—l) t+J times its determinant is called the (i,j)-th cofactor of A. The term cofactor is motivated by (El) below. The word minor is used as a synonym for the determinant of a square submatrix. Thus, by a minor of A of order r we mean the determinant of a square submatrix of A of order r. Likewise, if m = n then the (i,j)-th cofactor of A is (_iy+J times the (i,j)-th minor of A. EXERCISE (El). Show that the determinant of an n x n matrix A = {Aij) over a ring R can be calculated by expanding in terms of any row or any column, i.e., show that for 1 < i < n and 1 < j < n we have det(A) 2^,\<j
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of the matrix of cofactors of A, i.e., if we denote the adjoint of A by A* = (A*^) and the (i,j)-th cofactor of A by Cy- then A*j = Cji. For elements ai,..., an in R, by d i a g „ ( a i , . . . , a „ ) or diag(ai,... ,a„) we denote the nxn diagonal matrix whose (i,j)-th entry is a, or 0 according as i = j or i ^ j . By the nxn identity matrix we mean the matrix d i a g n ( l , . . . , 1); we may sometimes denote this by l n . For the mxn matrix A = (A^) and any r € R, by r A we denote the mxn matrix whose (i,j)-th entry is rAij. This makes the set MT(m x n, R) of all m x n matrices over R into an .R-module. For a field k, it converts MT(m x n, A;) into an mn dimensional vector space over k. EXERCISE (E3). Show that units in the group o f n x n matrices over a ring R are those matrices whose determinants are units in R, i.e., prove claim (M4). Hint: for an n x n matrix A = (Aij) over R, show that AA* = A*A — det(vl)l„ where A* is the adjoint of A and l n is the nxn identity matrix. EXERCISE (E4). Show that for n x n matrices over a field k with positive integer n, the nonzero scalar matrices form a normal subgroup of the group of all matrices of nonzero determinant, i.e., prove claim (M5). EXERCISE (E5). Prove Euler's Theorem on Homogeneous Polynomials, i.e., show that for any homogeneous polynomial H = H(Z\,..., ZN) of degree d in a finite number of variables Z\,..., ZN over a field k we have ]CiA: Rn ^ Rm given by cf>A(y) = Ay for all y € Rn = MT(n x 1, R). Now A i-> cj>A gives a bijection of MT(m x n,R) onto the set of all -R-linear maps Rn —> Rm. Moreover, for any B £ MT(n x r, -R) we have 4>AAB- In c a s e R is a field k, we have rrk(^4) = dimfeim((/>/i) = crk(^4) = n — dimfcker(0^). In particular if -R is a

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92

field k and m = n then: A is a linear fc-automorphism of Rn «=> det(A) ^ 0. DEFINITION (D4). [Linear and Polynomial Automorphisms]. Consider the rings P = k[Xi,.. .,Xn] C k[[Xi,... ,Xn}] = S where Xi,...,Xn are a finite number of variables over a field k. Let A = (Ay) be the m x n matrix depicted in (Dl) where we take R = k and m = n. Let N = n2 and let ( P such that c ^ X , ) = YLi<j 0}. Hence by (E8) below, cr^ uniquely extends to an automorphism T,I : S —> 5. We call T^ a fc-linear automorphism of S. If .A can be chosen so that for a given nonzero F*(Z\,..., ZJV) £ T we have F * ( a i , . . . , ajv) ^ 0 then we say that TA is a generic &:-linear automorphism of S. Given any 0 ^ F £ S, upon letting ordgF = v, we have that v is a nonnegative integer and we can uniquely write F = F(XU ...,Xn)=

FV(XU ...,Xn)

+ (terms of degree > v)

where Fv = FV(X\,..., Xn) s P i s a nonzero homogeneous polynomial of degree v. We call Fv the initial form of F relative to S or relative to (Xi,..., Xn), and denote it by infos-F or info^,...^,!)-^ w e a l s o P u t infosO = 0. Geometrically speaking Fv = 0 is the tangent cone of the analytic hypersurface F = 0 at ( 0 , . . . , 0). Now suppose n > 0. If for F"(Zi,..., ZN) = Fv(Zln,..., Znn) we have F"(au..., aN) ^ 0 then clearly TA(F)(XI,

..., Xn) = aA(Fv)(Xi,...,

Xn) + (terms of degree > v)

where OA(FV)(XI, . . . , Xn) £ P is a nonzero homogeneous polynomial of degree u with <JA(FV)(0, . . . , 0,1) ^ 0, and hence OV&STA(F) = v with inhsTA(F) = GA(FV) and upon writing ( X i , . . . , Xm, Y) for ( X i , . . . , X„) we get wideg(r J 4(F)) = v. If fc is infinite then by taking G = F*F'F" and applying (E6) we will have found a generic fc-linear automorphism TA '• S —> S such that wideg(r/i(F)) = v. Without assuming k to be infinite, we claim that we can always find a polynomial k-automorphism r : S —> 5 such that wideg(r(F)) < oo, where an automorphism r : S —> S1 is called a polynomial /c-automorphism if it is a /c-automorphism with T(P) = P; note that then we must have T(P+) = P+. TO see this, for any E = ( e i , . . . , e n ) € N™ with e n = 1 let S. Writing F = F(X1,...,Xn)= with FiGk

^FiX*

and X* = X[l ... Xln", we put Supp(F) = {i € N n : F4 ^ 0} and call

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AND EXERCISES

93

this the support of F. Note that Supp(F) is nonempty because we assumed F ^ 0; also note that if F e P then Supp(F) is finite. Let us lexicographically order N n by saying that for i ^ j in N™, upon letting t to be the smallest subscript with it ^ jt we have: i < j •& it < jt- In this ordering any nonempty set has a smallest element and any nonempty finite set has a largest element. So let r be the smallest element of Supp(F), and in case of F £ P let s be the largest element of Supp(F). Upon letting deg and degx n be the {X\,..., X n )-degree and the X„-degree respectively, by (E9) below we see that if

{

^n—vrn—v

&n—u -> 2-/0
for 1 < u < n — 1

then deg(aE(Xn)

= degXn(aE(Xr))

for all r^r*e and hence wideg(Ts(F)) of F G P, if e

n—u >

degXn(aE(Xr'))

Supp {F)

= degXn(aE{Xr))

max

<

= e\r\ + ••• + enrn < oo, and in case

s ' e S u p p ( F ) 2-^0
e

™—v\sn-v ~

s

n—v)

for 1 < u < n — 1 then deg(aE(X°))

= degxJaE(X°))

for all s^s'

e Supp (F)

>

degXn(aE(X°'))

and hence deg(aE(F)) = degXn(crE{F)) = degXn(aE(Xs)) = eisi + • • • + ensn. Thus, in addition to finding E so that wideg(rE(F)) < oo, in case of F £ P we have also found E so that deg(cr£(F)) = degXn(aE(F)), i.e., aE(F) is regular in Xn in the following sense. Given any 0 ^ F £ P, upon letting deg(F) = fx, we have that fj, is a nonnegative integer and we can uniquely write F = F(XU ...,Xn)

= Fli(X1,...,Xn)

+ (terms of degree < /x)

where FM = F^(Xi,... ,Xn) £ P i s a nonzero homogeneous polynomial of degree (j,. We call Fu the degree form of F relative to P or relative to (Xi,..., Xn), and denote it by defopF or defo(x1,...,xn)-P1i w e a l s o P u t defopO = 0. F is regular in Xn means deg(F) = deg x (F), i.e., equivalently Fu(0,... ,0,1) ^ 0. Geometrically speaking F^ = 0 is the infinite portion of the hypersurface F = 0, and F is regular in Xn means the X n -axis, i.e., the line X\ = • • • = X n _ i = 0 does not meet this portion. If k is infinite then by taking Fu for F" in the above argument, we can arrange that VA{F) is regular in Xn and has the same degree as F . All this can be applied to any finite family of polynomials or power series by letting F be the product of that family.

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EXERCISE (E8). In (D4) show that any k-linear automorphism a : P -> P with c(-P+) = P+ can be uniquely extended to an automorphism r : S —> S. EXERCISE (E9). In (D4) show that (*) => (**), and show that (') => (")• EXERCISE (E10). Let Yi,...,Yn be a finite number of indeterminates over a ring R, and let I be the ideal in S = R[[Y\,..., Yn]] generated by Y\,..., Yn. Show that then S is /-complete and /-Hausdorff according to (D5) below. Also show that if R is quasilocal then S is quasilocal with M(S) = M{R)S + I. Finally show that if R is complete Hausdorff quasilocal then so is S. EXERCISE ( E l l ) . Give details for the proof of Abstract Hensel's Lemma (T8). DEFINITION (D5). [Hausdorff Modules and Bilinear Maps]. Let us now generalize the notions of Hausdorff and complete from the case of quasilocal rings to more general situations. So let / be an ideal in a ring R and let V be an ZJ-module. V is /-Hausdorff, or Hausdorff relative to / , means (li^PV = 0, where for any ideal J in R, by JV we denote the submodule of V given by JV = {J2jivi '• Ji G J and Vi G V (finite sums)}. For any h G V we define the (/?,/)-order of h by putting ord( fl /)/i = max{j G N : h G PV} and we note that then V is /-Hausdorff means for any h £V we have: oid^nj-jh = oo «=> h = 0. A sequence x = (xi)i NE and j > NE we have xt - Xj G IEV. The sequence x is /-convergent, or convergent relative to / , means it converges to an /-limit £ G V, i.e., for every E G N there exists NE G N such that for all i > NE we have £ — Xi G IEV; we indicate this by some standard notation such as Xj —> £ as i —> oo or limj^ooXj = £. If V is /-Hausdorff then a limit when it exists is unique. V is /-complete means every /-Cauchy sequence in it has a limit in it. If V is /-complete and /-Hausdorff and yj G /"(•?') V with u(j) —> oo as j —-> oo then a sequence like Yli'<ji' Vj- ^ R is quasilocal with I = M(R) and V = i? then, according to the previous definitions of the above notions, the adjective "relative to /" may be dropped. Let U, V, W be modules over R. A map / : V x W —> U is /Z-bilinear means for every » e 7 the map W —> L/ given bywi-» /(u, w) is .R-linear and for every w £W the map V —* U given by t> t-> f(v,w) is .R-linear. In a natural manner we can regard U/(IU),V/(IV),W/{IW) as modules over the ring R/(IR), and then in a natural manner / induces a (/?///?)-bilinear map (V/(IV)) x (W/(IW)) -> U/(IU). The following three Exercises deal with the above set-up. In particular, in (E14) we remove the hypothesis of R/M(R) being algebraically closed from (T8.2). EXERCISE (E12). [Completeness Lemma]. Let / be an ideal in a ring R,

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let U, V, W be any /^-modules, and let n be any positive integer. (E12.1) Show that if V is /-Hausdorff then Vn is /-Hausdorff. (E12.2) Show that if V is /-complete then Vn is /-complete. (E12.3) Show that if (xt) is an /-Cauchy sequence in V such that some subsequence (xr(j)) has an /-limit x in V, then x is an /-limit of the sequence (a;,). (E12.4) Show that for any it-linear map : W -> V we have is surjective then we have {InW) = InV. (E12.5) Show that for any ^-bilinear map 9 : V x W - • £/ we have 0 ( / n K x P W ) C /"?/. (E12.6) Show that if W is /-complete and there exists a surjective .R-linear map W —> V then V is /-complete. (E12.6*) Show that (E12.6) is not true with Hausdorff replacing complete. (E12.7) Show that if V is a finite /^-module and R is /-complete then V is /-complete. HINT. (E12.1) to (E12.5) are straightforward. For (E12.6*) take W = R = k[X] where X is an indeterminate over a field k, let / = XR, and let

V = R/(X - 1)R. (E12.7) follows from (E12.2) and (E12.6). To prove (E12.6), let (xi) be any Cauchy sequence in V. Then we can find positive integers r(l) < r(2) < . . . such that for all j , m, n in N+ with m > r(j) and n > r(j) we have xm - xn G P'V. Pick any y\ G 0 _ 1 ( x i ) . For all j > 1, by (E12.4) there exists 2/j G (rj). Then by (E12.4), £ is an /-limit of (xr(j)) in V, and hence by (E12.3), £ is an /-limit of (xi). EXERCISE (E13). [Hensel's Sublemma]. Let / be an ideal in a ring R such that R is /-complete. Let U, V, W be finite /^-modules such that U is /-Hausdorff. Let : R -> R = R/I, fa : U - • !7 = E//(J^), 0y : V -» V = V/(/V), <j>w :W^>W — W/(IW) be the canonical epimorphisms. Let 9 : V x W —> [/ be an /^-bilinear map and let 9 : V x W —> U be the /i-bilinear map induced by #. Let there be given F GU with G G 7 and # G W such that (i) fa(F) =j?(G,/7) and (ii)J/_= ^(G, W)_+ 0(7,H\_ where as usual 0(5", IF) = {0(G, W) : W G F } and 9(V,ll) = {9{V,H) : V G F } . Show that then there exist G G (<M - 1 ({G}) and H G ( ? w ) - 1 ( W ) s u c h that F = 9(G,H). HINT. Let G0 = H0= 0. By induction on I > 1 we find (i*) G/ G (0 V ) _1 ({G"}) and Hi G ( ^ ^ ( { t f } ) such that: (ii*) G1-G1-1 G J'- X V and /// - / / ; _ i G J ' " ^ with: (hi*) F - 9(Gi,Hi) G IlU. By (i) we can do it for 1 = 1. So let Z > 1 and assume we have done it for I — 1. Then F — 0(G;_i,//;_i) G Il~1U and hence F — 0(Gi-i,Hi-i) = J2£jXj with £j G / i - 1 and Xj G Z7 where the sum is finite. By (ii) we find yj G V and Zj G W such that Xj — 9(G,Zj) — 9(yj,H) G IU. Now it suffices to take G; = Gi~\ + Y^^jVj a n d Hi = H1-1 + £ £ j z j - I n v i e w °f (ii*)) by

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(E12.7), (Gi) and (Hi) have /-limits G and H in V and W respectively. In view of (i*), by (E12.4) we see that G £ ( ^ v ) - 1 ({ : R —> R — R/M(R) be the residue class epimorphism and let (f>: R[Y] —> R[Y] be the obvious extension of cf>. Let F(Y) = Yn+

J2

AjYn-j

with

n £ N+

and

A, £ R.

l<j
Assume that v : V -» V = V/{IV), ~<$>w : W —> W = Wy(iW) be the epimorphisms induced by <£. Let ^ : V x W —> U be the i?-bilinear map given by multiplication, i.e., given by (V, W) H-> VW. Let 0 :V xW -* U be the i?-bilinear map induced by 6. In view of (E12.1), by (E13) we find G(Y), H(Y) in R[Y] of degree < r and < s respectively such that $(G{Y)) = G{Y) and ~$(H{Y)) = H{Y) with F{Y) = G(Y)H(Y). It follows that the degree of G(Y) is r and therefore for the coefficient G' of Yr in it we have {H') = 1. Therefore G',ff' are units in R with G'/T = 1. Now it suffices it to take G{Y) = G{Y)/G' and H{Y) = H(Y)/H'. EXERCISE (E15). Let D be a derivation of a field K. Let y, z be nonzero elements of K such that zr = y where r is a rational number. Show that then D(y) = r z r - 1 D ( « ) . OBSERVATION (01). I am not saying that given y and r this defines z uniquely. The matter is made precise by saying that for some way of writing r = m/n with m € Z and n € N + , and for some x £ K with y = xm, we have z = xn, and then interpreting the equation D{y) = rzr~1D{z) as the equation zD(y) — ryD(z). EXERCISE (E16). Prove Binomial Theorem (Tl) for fractional exponents. In other words, for any rational number r = m/n with m £ Z and n £ N+, upon letting

y(X) = i + £

r(r 1)

- -.,(r-*

+ 1)

X* e MM

%12: DEFINITIONS AND EXERCISES

97

where X is an indeterminate over a field k of characteristic zero, show that y{X) is the unique element in A;[[X]] with y(0) = 1 such that (1 + X)r = y(X)

HINT. By taking F(X,Y) there exists a unique

(1 + X)m =

or equivalents =Yn-{l

+ X)m

y{X)n.

in Basic Hensel (T2) we see that

y[X) = 1 + 5 ] Vix* e Ml*]]

with

Vi €

k

i>0

such that (l + X)m = y(X)n i.e.,

{i + xy = i + YlViXiek[[x}}i>0

For any i > 0, in view of (E15) and (01), differentiating both sides i times with respect to X and then putting X = 0 we get r(r - 1 ) . . . (r - i + 1) = i!j/j. EXERCISE (E17) Prove Binomial Theorem for integral exponents. In other words, show that for any n £ N and any elements x, y in any ring R we have xlyn-1 0
where (™) is the number of ways of choosing i things out of n things, i.e., the size of the set of all i size subsets of a set of size n. Also show that /n\ \iy

n! i\(n — i)\

n(n — 1 ) . . . (n — i + 1) i\

HINT. In addition to the usual high-school proof, give a second proof using (E16). OBSERVATION (02). Newton first proved (E17), then he generalized it to (E16), and finally he generalized it to (T4). EXERCISE (E18) Show that a univariate polynomial ring over a UFD is a UFD, i.e., complete the proof of (J18). Also complete the proofs of claims (J21) and (J23) about power series rings.

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OBSERVATION (03). From this lecture, it only remains to do Homework (H3) and to complete the proofs of claims (Jl), (J2), and (J24). §13: NOTES NOTE (Nl). [Preparation for W P T ] . To take care of the finiteness of wideg in WPT, consider the power series ring S = fc[[-X"i,... ,Xn]] in a finite number of variables X\,..., Xn over a field k. Given 0 ^ F = F(X\,..., Xn) € S, we want to find an automorphism r : S —> S such that after taking (Xi,...,Xm,Y) for (Xi,... ,Xn) we would have wideg(r(F)) < oo. If k is infinite then by (D4) and (E8) we can achieve this by means of a generic fc-linear automorphism r = TA of S; geometrically speaking what we do is to rotate coordinates so that a generic line not on the tangent cone of F = 0 becomes the X n -axis, i.e., becomes the line Xi = • • • = X n _ ! = 0. Without assuming k to be infinite, by (D4) and (E9) we can always achieve it by means of a polynomial fc-automorphism T = TE of S; what we do is that instead of varying the coefficients of the equations of the automorphism over the field, we suitably vary their exponents over N+ which is an infinite set. NOTE (N2). [Abhyankar's Proof of Newton's Theorem]. What we called Shreedharacharya's Proof of Newton's Theorem, is also known as Abhyankar's Proof of Newton's Theorem; see pages 575 and 659 of volume II of Jacobson's Basic Algebra Book [Jac]. §14: CONCLUDING N O T E Analytic geometry was initiated by Descartes in 1637. Descartes' work was pushed further, first in France (1760-1790) by Bezout, then in England (1840-1890) by Cayley, Sylvester, and Salmon, then in Germany (1860-1910) by Plucker, Klein, and Noether, and simultaneously in Italy (1860-1920) by Cremona, Castelnuovo, Enriques, and Severi, and finally in the United States (1930 onwards) by Zariski, Weil, Chevalley, and others. Somewhere along the line, in this long march, analytic geometry changed its name to algebraic geometry. By stressing synthetic arguments and augmenting ordinary space by inserting points at infinity, analytic geometry is metamorphosed into projective geometry. As said above, analytic geometry and projective geometry have mostly been eliminated from the college curriculum in the last forty years or so. But I was lucky. In my time I had four years of analytic geometry. Also I had two years of projective geometry. In the one year course in Bombay College, my text for projective geometry was the classic Pure Geometry book [Ask] of Askwith. This was followed up by the one year projective geometry course I had at Harvard with Zariski where our text was the equally famous book [VYo] of Veblen and Young. Out of all these illustrious names, whom shall we elect as the father of algebraic geometry? By reading the relevant material in Klein's

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famous Entwicklung der Mathematik book [Kle] of 1910 which is reconfirmed in Karen Parshall's recent book [PaR] on Sylvester's Correspondence, I am convinced that George Salmon of Dublin, Ireland, is the true father of algebraic geometry and his 1852 book [Sal] on Higher Plane Curves is the cradle of that subject. In any case, on page 17 of Askwith's book, you will find Salmon's generalization of the pole-polar property of a circle.

Lecture L4: Varieties and Models §1: RESULTANTS A N D D I S C R I M I N A N T S One way of proving Bezout's Theorem, and of doing Homework L3(H3), is by using resultants and discriminants. Assuming n, m to be nonnegative integers, the y-Resultant of two polynomials f(Y)=a0Yn

+ a1Yn-1+---

m

g(Y)=b0Y

m 1

+ b1Y -

+ an

+ --- + bm

is the determinant R-esy(/,ff) = det(Resmaty (/,#)) of the n + m by n + m matrix / ao a\ • 0 ao a\

Resmaty(/,#) =

0 0 • b0 h • 0 b0 h

an 0 • • an 0

ao oi • • bm 0 •

b0 h

\ 0 0

brr,

0

0 0

bm J

where the first m rows consist of the coefficients of / and the last n rows consist of the coefficients of g. More precisely, the first row starts with the coefficients of / , these are shifted one step to the right to get the second row, shifted two steps to the right to get the third row, and so on for the first m rows, then the (m + 1)st row starts with the coefficients of g, these are shifted one step to the right to get the (m -f 2)-nd row, and so on for the next n rows. The matrix is completed by stuffing zeroes elsewhere. The determinant Resy(/, g) is sometimes called the Sylvester resultant of / and g because it was introduced by Sylvester in his 1840 paper [Syl] where he enunciated the following: BASIC FACT (Tl). If the coefficients aj, bj belong to a domain R then we have: Resy(/, g) = 0 ^ n + m ^ 0 and either a 0 = 0 = bo or / and g have a common root in some overfield of R.

100

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101

In case n > 0, the F-Discriminant of / is defined to be the F-Resultant of / and fY, i.e., DisCy(/)=ReSy(/,/y). where we view fy to be the polynomial fY(Y)

= na0Yn-1

+ (n - l ^ F " " 2 + • • • + a„_i

i.e., we let the discriminant to be the determinant of the appropriate 2n — 1 by 2n — 1 matrix without considering whether nao equals zero or not. From the Basic Fact (Tl) we deduce the following: COROLLARY (T2). If n > 1 and the coefficients a* belong to a domain R then: Discy (/) = 0 •*=> either ao = 0 or / has a multiple root in some overfield of R. OBSERVATION (01). [Resultant and Projection]. If XU...,XN are indeterminates over a field k with N £ N + and R is either the polynomial ring k[X\,... ,XN] or the power series ring k[[Xi,... ,XN}}, then Hesy(f,g) equals a polynomial or power series $ = &(Xi,..., XN). If ao and bo are in kx with nm ^ 0 and k is algebraically closed then, in the polynomial case, by the Basic Fact it follows that the hypersurface $ = 0 in the N-space of (X\,..., XN) is the projection of the intersection of the hypersurfaces / = 0 and g = 0 in the (TV + l)-space of (Xi,..., XN, Y). Moreover, without assuming k to be algebraically closed but assuming that ao and &o are nonzero constants, in the polynomial case as well as the power series case, by the Basic Fact it follows that: $ is identically zero (i.e., $ is the zero element of R) •*=> / and g have a nonconstant common factor in R[Y]. OBSERVATION (02). [Discriminant and Projection]. Again ifXx,...,XN are indeterminates over a field k with N e N+ and R is either the polynomial ring k[X\,..., XN] or the power series ring fc[[Xi,..., XN]], then Discy (/) equals a polynomial or power series A = A ( X i , . . . , XN)- NOW if ao is in fcx with n > 1 and k is algebraically closed then, in the polynomial case, for all values (Ui,... ,UN) of (Xi,..., XN) in k, the equation / = 0 has n roots which may or may not be distinct, and by the Corollary it follows that these roots are not distinct iff A ( f / i , . . . , UN) = 0. In other words, when we project the hypersurface / = 0 in (N + l)-space onto the TV-space, above most points there lie n points, and A = 0 is the locus of those points above which there lie less than n points. Moreover, without assuming k to be algebraically closed but assuming that ao is a nonzero constant, in the polynomial case as well as the power series case, by the Corollary it follows that: A is identically zero -S4- / has a nonconstant multiple factor in .ft[y]. OBSERVATION (03). [Isobaric Property]. View the coefficients a^bj as indeterminates over Z. Give weight i to a*, and j to bj. Then 0 ^ Resy(/,g) €

LECTURE L4: VARIETIES AND MODELS

102

Z[ao) • • •, o-n, bo, •.., bm] is isobaric of weight mn, i.e., for the weight of any monomial r aQ°... ajj" &o° ...b&r occurring in R e s y ( / , g) we have ( E o < r < n v ) + (Eo< s<m S39> ~~ mn. In particular, the principal diagonal affb^ has weight mn, and it does not cancel out because there is no other term of 6TO-degree n in the resultant; the principal diagonal of an N x N matrix (Ay-) is the term A11A22 • • -ANN; [cf. §12(R6)(14) and its proof following §12(R6)(15)]. The resultant being isobaric of weight mn is the fundamental fact behind various cases of Bezout's Theorem. The following two Observations, where we use the set-up of Observation (01), illustrate this for plane curves and general hypersurfaces respectively. OBSERVATION (04). [Plane Bezout]. Let N = 1 with X = Xx, and assume that ao, bo are nonzero elements in k, and / , g are polynomials of total (X, Y)degrees n and m respectively. By the isobaric property we see that then always d e g x $ < mn and "in general" d e g x $ = mn. Hence the n-degree plane curve / = 0 meets the m-degree plane curve g = 0 in mn points "counted properly." The possibility of d e g x $ < ran is explained by saying that some intersections have "gone to infinity." OBSERVATION (05). [Hyperspatial Bezout]. Let TV be general and assume that cio,bo are nonzero elements in k, and / , g are polynomials of total {X\,... ,X;v,y)-degrees n and m respectively. By the isobaric property we see that then always deg( Xl xN)$ < rnn and "in general" deg(x 1 ,...,x JV )^ = mnHence, in the (N + l)-dimensional space, the n-degree hypersurface / = 0 and the m-degree hypersurface g = 0 meet along a "secundum" (= a subvariety of dimension two less than dimension of the ambient space) which projects onto the (mn)-degree hypersurface $ = 0 in A^-dimensional space. Again the possibility of deg(x 1 ,...,x JV )^ < mn s a y s t n a t s o m e intersections have "gone to infinity." EXAMPLE (XI). [Resultant and Discriminant in terms of Roots]. If the coefficients a*, bj belong to a domain R and ao ^ 0 =fi bo then, upon writing

f(Y) = ao J ] (y-ai)

an

d

g(X) = b0 ]\

l
with ot\,...,an

l<j
and / 3 i , . . . , Pm in an overfield of R, we have

Resy(/,fl)=<« n n («<-&) l<j
l<j<m

\
= (-irn&o n Mi) l<j
(Y - 0j)

§1: RESULTANTS AND DISCRIMINANTS

103

and

Discy(/) = (-l)"/2aS

I]

(^-«i)2-

l
EXAMPLE (X2). [Quadratic Resultant]. Considering the quadratic polynomials / ( F ) = aY2 + bY + c

and

ff(F)

= a ' F 2 4- b'Y + c'

and calculating the 4 x 4 determinant (a 0 a' \0

b c 0\ a bc b' d 0 a'6'c7

we get Resy (/, g) = (a2c'2 + a'2c2) + (b2a'c' + b'2ac) - {abb'c' + a'b'bc) - 2aca'c'. EXAMPLE (X3). [Quadratic Discriminant]. Considering the quadratic f(Y) = aY2 + bY + c and calculating the 3 x 3 determinant a b c^ 2a b 0 ,0 2a bj we get Discy(/) = — a(b2 — Aac). EXAMPLE (X4). [Cubic Discriminant]. Considering the cubic f(Y) = a0Y3 + axY2 + a2Y + a 3 and calculating an appropriate 5 x 5 determinant we get Discy(/) = —ao(a\a\ — Aa^a\ — ka\az — 27a^a\ + 18ao«ia2«3)EXAMPLE (X5). [Special Quartic Discriminant]. Considering the quartic f(Y)=Y4+pY2

+ qY + r

and calculating an appropriate 7 x 7 determinant we get Discy(/) = 16p4r - 4 p V - 128pV + lUpq2r - 27q4 + 256r 3 .

104

LECTURE L4: VARIETIES AND MODELS EXAMPLE (X6). [General Quartic Discriminant]. Considering the quartic f{Y) = aQY4 + aiY3 + a2Y2 + a3Y + o 4

and calculating an appropriate 7 x 7 determinant we get Discy(/) = a0(256a0ia4i - 192a^aia 3 a| - 128aga|a| + 144aga2a§a4 - 27ala\ <

+lAAaoa\a,2a\ — 6000^0304 — 800001030304 + 1800010203 + 16aoa|o4 —4000^03 - 27a\a\ + 1801020304 — 4afa 3 - 40^0^04 +

ajalal).

OBSERVATION (06). [Future Plans]. Further details of the concrete explicit approach to equation solving expounded in this section, including proofs of the various claims, will be given by and by. First let us push forward the abstract approach via ideals and varieties. §2: VARIETIES The concepts of curves, surfaces, and hypersurfaces give rise to the idea of a variety. A variety consists of the common solutions of a finite number of polynomial equations in a finite number of variables. To study these in any detail we need to build up some facts about ideals in a ring. For instance, why we do not get anything extra by allowing an infinite number of equations is explained by proving that every ideal in a multivariate (finitely many variables) polynomial ring over a field is finitely generated. This is called the Hilbert Basis Theorem. It is one of the numerous versatile theorems of commutative algebra obtained by Hilbert [Hil] in the period 1890-1920. As said before, a ring in which every ideal is finitely generated is called a noetherian ring. This is in honor of Emmy Noether who, in 1925-1935, stressed the importance of that property. In a moment we shall prove the Hilbert Basis Theorem, and at the same time we shall prove that the multivariate power series ring over a field is also noetherian. A variety can be decomposed into its (finitely many) "irreducible components." The ideal theoretic counterpart of this will be the "primary decomposition" of ideals in a noetherian ring which we shall take up next. This will also set up an inclusion reversing correspondence between irreducible varieties and prime ideals. Modding out the polynomial ring by the prime ideal corresponding to an irreducible variety will produce the "affine coordinate ring" of the variety; its quotient field is the "function field" of the variety. Global properties of a variety, such as its dimension and degree, are reflected in the affine coordinate ring and the function field. Local properties of a variety "near a point," such as whether the point is simple or singular, are reflected in the properties of the "local ring" of the point on the variety. The construction of the local ring generalizes the formation of the quotient field of a domain. By taking the local ring of an irreducible subvariety of an irreducible variety we can detect whether most points of the subvariety are simple

§3: NOETHERIAN RINGS

105

points of the original variety; in technical terms, this will be so iff the said local ring is "regular." After talking about "localization" we shall resume the discussion of varieties, first in affine space, and then in projective space. The idea of localization will also enable us to reinterpret varieties as "models" which are suitable collections of local rings. In turn, the language of models will facilitate the discussion of various operations on varieties, such as the blowing-up of subvarieties of a variety for resolving its singularities. §3: N O E T H E R I A N R I N G S Let us now prove the: HILBERT BASIS THEOREM (T3). For any noetherian ring R and any positive integer m, the multivariate polynomial ring R[Xi,... ,Xm] is noetherian. PROOF. Clearly R[XU..., Xm] = R[X1:..., Xm-i][Xm} and hence by induction the general case follows from the case of m = 1. So assume m = 1 and let X = X\. Given any ideal J in S = R[X] we want to show that it is finitely generated. Let d denote the X-degree. For every i G N let J, = {r G R : d(s — rXl) < i for some s G J } . For all r, r' in R and s, s' in J we have: (i') d{s - rX{) < i and d(s' - r'X*) < i =» d((s + s1) - (r + r ' ) X ' ) < i with s + s' G J; (ii') d(s - rXl) < i => d((r's) - (r'r)Jf*) < i with r's G J; and (iii') d(s - rX*) < i =^ d(sX - rXi+1)
(')

a finite number of generators (fij)i<j<e(i) °f J% < together with elements (sij)i<j<e(») m J such that d(sij — rijX1) < i for 1 < j < e(i).

Let Joo = UjeN«/j- Then JQQ is an ideal in R. Again since R is noetherian, we can find a finite number of generators (rj)i<j<e of J ^ together with l(j) G N and (sooj) G J such that d{sooj - TjX1^) < l(j) for 1 < j < e. Take / G N such that I > l(j) for 1 < j < e and let Sj = SoojX1"1^. Then for 1 < j < e we have Sj G J and d(sj - rjX1) < I. Now J for every i > I in N we may take e(i) = e j^and for 1 < j < e we may take r^ = Tj and s^ = SjXl~l. For every i G N, by (') we see that J given any s G J with d(s) = i there exists Oj G R such that \ d ( s - Ei<j< e (i) airijxi)


and hence d s

( ~ Ei<j<e(«)

a

^ ) < *'

106

LECTURE

L4: VARIETIES

AND

MODELS

In view of (1') and ("), by induction we see that (2')

{

given any s G J with d(s) > I there exists aJC G R such that d s

c l - E i < j < e AjSij) < I where Aj = Ei< c
(

In view of (1'), by induction we also see that ,o/\

J given any s G J with d(s) < I there exists bij € R such that [S

=

£^0
£->l<j<e(i)

Vijsij-

By first applying (2') and then (3'), we see that the elements (sy)i|* i for some s G J } . For all r, r' in R and s, s' in J we have: (i*) d(s - rX*) > i and d(s' - r'X*) > i => d((s + s') - (r + r')**) > i with s + s' G J; (ii*) d(s - rX*) > i => d({r's) - (r'r)X{) > i with r's G J; and (iii*) d(s - rX*) > i => d(sX - r X i + 1 ) > i + 1 with sX G J. Therefore Jj is an ideal in R with Jj C Ji+i- Since i? is noetherian, we can find a finite number of generators {fij)\<j<e{i) °f ^ (*)

together with elements (sy)i<j< e (j)

m

^

such that d(sy — r^ X*) > i for 1 < j < e(i). Let JQO = UjeN^i- Then J ^ is an ideal in R. Again since R is noetherian, we can find a finite number of generators (r.j)i<j<e of Joo together with l(j) G N and (sooj) £ J such that d(sooj - TjX1^) > l{j) for 1 < j < e. Take I G N such that I > l(j) for 1 < j < e and let Sj = SoojX1-1^. Then for 1 < j < e we have Sj G J and d(sj - rjX1) > I. Now for every i > I in N we may take e(i) = e and for 1 < j < e we may take r^ = rj and s^- = SjX1

§4: ADVICE TO THE READER

107

For every i £ N, by (*) we see that

{

given any s £ J with d(s) = i there exists a,j £ R such that <*(* - £ i < j < e ( 0 a3rUxi)

>{and

hence d

( s _ Ei<,'< e «) aJsij) > *•

In view of (1*) and (**), by induction we see that J given any s £ J with d(s) > I there exists a,jC £ R such that \

s

= Y,l<j<eAiSU

w h e r e

A

J = T,l
In view of (1*), by induction we also see that J given any s £ J with d(s) < I there exists 6y € i? such that ["V s

—

2^0 I-

By first applying (3*) and then (2*), we see that the elements (sij)i<j< e (i) constitute a finite number of generators of J. §4: A D V I C E TO T H E R E A D E R The next long section on Ideals and Modules is rather heavy abstract dry stuff. You may prefer to skip it in a first reading and return to it as necessary. At any rate, to soften the blow here is a brief summary. The totality of common solutions of a bunch of polynomial equations / ; = fi{Xi,...,Xff) = 0, with I varying over an indexing set L, in a finite number of variables X\,... ,XN over a field k, is unchanged by replacing the polynomial family (/j)iet by another family which generates the same ideal I in the polynomial ring R = k[Xi,..., X^}. By Hilbert's Basis Theorem (T3) we can choose the other family to be finite. Thus there is no loss of generality to let the variety V of common solutions be denned by a finite number of equations. The variety V may be decomposed into its irreducible components; for instance the variety in the {X,Y) plane given by the equation (X2 + Y2 - 1)(Y2 ~ X2) = 0 has the circle X2 + Y2 = 1 and the pair of lines Y = ±X as its irreducible components. Algebraically this corresponds to making a primary decomposition of / in R which generalizes the fact that an integer is a product of powers of prime numbers. The said decomposition amounts to writing / = Q\ D • • • fl Qn with "primary" ideals Qi,- • -,Qn- The "radical" of Qi is a prime ideal Pi. The pair prime-primary ideal generalizes the pair prime number and its power. The primary decomposition has certain uniqueness properties. The operations of sums, products, and quotients of integers are generalized to sums, intersections (as well as products), and colons (or quotients) of ideals. Most of this further generalizes from ideals to modules. At once dealing with modules has the extra advantage that in various arguments the roles of the first and the second elements in a product rs are separately stressed as r belonging to the ring and s belonging to the module.

LECTURE H: VARIETIES AND MODELS

108

Two circles meet at a point of tangency with "intersection multiplicity" two, a tangent meets a cubic curve at a point of inflexion with intersection multiplicity three, and so on. To algebracize intersection multiplicity we introduce the concept of the length of a module, which itself generalizes the idea of the dimension of a vector space. To formalize the idea of the dimension of a variety we introduce the ideas of the height and the depth of an ideal as well as the idea of the dimension of a ring. §5: IDEALS A N D MODULES To continue the general discussion of ideals and modules started in LI §4 and LI §5, let V be a module over a ring R. The sums and intersections of any (indexed) family of submodules (UI)IEL of V are again clearly submodules of V, where the latter is set-theoretic while the former is defined to be the union U(f//1 -\ h Uin) taken over all finite sequences l\,..., ln in L. Note that the members of an "indexed family" need not be distinct, i.e., we may have Ui — Ui* for I ^ /* in the "indexing set" L. For any B c R and any submodule C of V we put BC = < Y J 6jCj : bi G B and c* G C (finite sums) > and we note that this is a submodule of C. For any elements a, ai, 0 2 , . . . in R and any submodule C of V we put aC = {a}C

and

(ai, 0,2,...)C = {0,1,0,2, ••-}C

and we note that these are again submodules of C. For any submodule U of V we put radyC/ = {r e R: reV C U for some e G N+} and we call this the radical of U in V, and we note that it is an ideal in R. By taking V = R, all the above applies to ideals in R. In particular, sums and intersections of any families of ideals in R are again ideals in R, and the radical radfl/ = {r £ R : r e G I for some e G N + } of any ideal J in R is an ideal in R. The product of any finite number of ideals in R was already introduced in the paragraph following item (2*) of L3§11, and in case of two ideals B and C it agrees with the above definition. For any B C R or a G R, and any C C V, we put (C:B)v

= {vGV:

B{RV) C C}

and

(C : a)v = {v €V : a(Rv) C C}

and we call these the colons (or quotients) of C by B and C by a in V respectively, and we note that if C is a submodule of V then these are submodules of V which

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AND

MODULES

109

contain C. For any B c V or a £ V, and any C C V, we put ( C : B ) f t = { r € / J : r( J RS) C C}

and

(C : a) a = {r 6 R : r(ifa) C C}

and we call these the colons (or quotients) of C by B and C by a in i? respectively, and we note that if C is a submodule of V then these are ideals in R. Note that for V = R the two meanings of colon coincide. An element x in R is a zerodivisor in R means xy = 0 for some ?/ € Rx = i?\{0}. Now nonP means the negation of a property P, and so nonzerodivisor means an element which is not a zerodivisor. Note that a domain is a nonnull ring having no nonzero zerodivisor. An element x in R is a nilpotent means xe = 0 for some positive integer e. Just as in the ring of integers, a prime number generates a prime ideal, a positive power of a prime number generates an ideal which is primary as well as irreducible in the following sense. An ideal / in the ring R is said to be primary if I ^ R and for any elements r, s in R with rs £ I and s $ I we have re £ I for some positive integer e. Just as the primeness (resp: maximalness) of I is equivalent to the residue class ring R/I being a domain (resp: field), so the primaryness of / is clearly equivalent to R/I being a nonnull ring in which every zerodivisor is nilpotent. An ideal / in the ring R is said to be irreducible if I ^ R and / cannot be expressed in the form I = J\ C\ • • • fl Je where e is a positive integer and J\,..., J e ideals in R with I r £ P. (iii*) Q is primary and P is prime with radfl<5 = P. By a cyclical proof (i*) =>• (ii*) => (iii*) => (i*) we see that these three conditions are equivalent. When they are satisfied, we say that Q is primary for P, or that Q is P-primary. OBSERVATION (08). [Colons and Intersections of Primary Ideals]. As easy consequences of (07) we see that for ideals P, Q in a ring R we have the following:

110

LECTURE H: VARIETIES AND MODELS

(I*) Q is P-primary and B <£_ Q for ideal B in R =$• (Q : B)R is P-primary. (2*) Q = Qi n • • • n Qh where Q, is P-primary for 1 < i < h G N + => Q is P-primary. (3*) P is a maximal ideal in R with P c rad^Q ^ R =*> Q is P-primary. (4*) P is a maximal ideal in R and Pe Q is P-primary. OBSERVATION (09). [Primaries a n d Powers of Primes]. In connection with the above claim (4*), let us construct some relevant examples. EXAMPLE (X7). Here is an example of (•') a maximal ideal P and a P-primary Q which is not a power of P . Let e > 1 and m > 1 be integers, and let R = k[[Xi,..., Xm]] where X\,..., Xm are indeterminates over a field k. Let P = M(R) = {X\,..., Xm)R, let W be the set of all monomials of degree e in X\,..., Xm, i.e., set of all expressions X\* ... X%? with nonnegative integers ii,... ,im for which i\-\ \-im = e. Let W be a subset of W which contains X{,..., X^ but is different from W, and let Q be the ideal in R generated by W. EXAMPLE (X8). Also here is an example of (•") a nonmaximal prime P with P which is not P-primary but r a d ^ P = P . In the notation of the Example (X7), let e', e" be positive integers with e' + e" = e, let (f>: R —> R = R/(Xf — X\ Xf )R be the residue class map, and let P be the ideal in R generated by (Xi) and (f>(X2). OBSERVATION (010). [Factor M o d u l e s a n d I s o m o r p h i s m Theorems]. For a module V over ring R, and a submodule U of V, the additive abelian factor group V/U is made into an .R-module in a unique manner so that the residue class map (of additive abelian groups) V —> V/U becomes an P-epimorphism; the resulting P-module V/U is called the factor module; note that U is the kernel of the residue class epimorphism V —> V/U. In Ll§4 we have already dealt with the special case of this when V = R and U = an ideal / in R, and then we made R/I into a ring; now R/I also becomes an P-module. For any P-epimorphisms <\> : V —> V and V* where V, V',V* are P-modules, if ker(4>*) C ker(0) then clearly there exists a unique P-epimorphism <j>' : V* —* V such that (j> = 4>'(j>*; conversely if there exists an P-epimorphism 4>' : V* —> V such that ^ = 4>'(p* then clearly we must have ker(*) C ker(^). (1*') For any P-epimorphism (j>: V —> V (of P-modules) and any P-submodule T of V, let ker(<£) = U with 0(T) = T', and let a : T -> V be the map induced by (f>, i.e., given by a(x) = <^>(:r) for all x £ T; then clearly a is an P-epimorphism with

§5: IDEALS AND MODULES

111

0 _ 1 ( T ' ) = U + T and ker(a) = UC\T, and hence induces a unique .ft-isomorphism a' : T/(U f l T j ^ r such that a = a1 a* where a* :T -> T/(U n T) is the residue class epimorphism; also clearly <j)(U + T) = T" and for the map P : U + T —• T' induced by 0 we have that /3 is an i?-epimorphism with ker(/3) = U and hence 0 induces a unique ^-isomorphism ft' : (U + T)/U —* T" such that P = /?'/?* where /?* : C + T —> ([/ + T)/{7 is the residue class epimorphism; thus 4> induces the ^-isomorphism (a')" 1 / 3 ' : {U + T)/U -+ T/(UnT). Note that in all this, by taking 0 to be the residue class epimorphism V —> V/U we can let U be any preassigned .R-submodule of V. (2**) For any ^-epimorphism

V (of i?-modules) with ker(<£) = [/, clearly W H-+ 0(W) gives a bijection of the set of all i?-submodules of V containing U onto the set of all .R-submodules of V ; this bijection is inclusion preserving, i.e., for any submodules W C W* of V we have <^(W) C (W*). Moreover, for any Rsubmodule W of V with U cW, upon letting 0(W) = W, we have 0 _ 1 ( W ) = W, and ! induces an i?-epimorphism W —> W' with kernel [/, and it also induces ^-isomorphisms W/U -> W and K/W -> V'/W. Assertions (1**) and (2**) are sometimes called the First and the Second Isomorphism Theorems. We may use them tacitly, i.e., without explicit mention. OBSERVATION (Oil). [Abelian Groups and Isomorphism Theorems]. Any additive abelian group G may be viewed as a Z-module by declaring ng = 9 + 9 + • • • + 9 taken n times; this for all n £ N and g e G; the definition is completed by putting {—n)g = —{ng) for all n € N + and g € G. This makes all considerations of i?-modules over a ring R, for instance the above two theorems of (O10), applicable to additive abelian groups. In particular an ideal 7 in a ring R and a subring H of R are both additive subgroups of R and hence their sum I + H is defined as an additive subgroup of R; actually I + H is a subring of R and I is an ideal in I + H; likewise I fl H is an ideal in H. The said two theorems may be supplemented by the following Third, Fourth, and Fifth Isomorphism Theorems which may also be used tacitly. (3**) Let 4> '• R —* R' be a ring epimorphism with ker() = I. Then for any subring H of R and the additive group epimorphism a : H —* H' = we have that H' is a subring of R' with cj)~l(H') = I + H and ker(a) = In H, and hence induces a unique ring isomorphism a' : H/(I D H) —> H' such that a = a'a* where a* : H —» H/{I fl H) is the residue class epimorphism; also clearly 4>{I + H) = H' and for the map /? : I + H —> H' induced by <j> we have that j3 is a ring epimorphism with ker(/3) = I and hence induces a unique ring isomorphism P' :{I+H)/I^> H' such that P = P'p* where /?* : I + H -> {I+H)/I is the residue class epimorphism; thus induces the ring isomorphism (a')~1P> : (I + H)/I —>

LECTURE H: VARIETIES AND MODELS

112

H/(I fl H). Also J — i > { J) gives an inclusion preserving bijection of the set of all ideals in R containing / onto the set of all ideals in R'. Moreover, for any ideal J in R with I c J, upon letting <j>{J) = J', we have ~l{J') = J, and induces an additive group epimorphism J —> J' with kernel / , and it also induces an additive group isomorphism J/I —> J' and a ring isomorphism R/J —• R'/J'. (4**) Let : R —> R' be a ring epimorphism. Then for any ideal J' in R' upon letting J = 0 _ 1 ( J ' ) we have that J is primary (resp: prime, maximal) iff J' is primary (resp: prime, maximal), and for any ideal J in R' upon letting J = ~l{J ) we have that rad« J = J iff rad^/J' = J , and for any family of ideals ( J[)I^L in R' upon letting J; = 0 _ 1 (J ( ') we have that J = X)jg£ ^ ^ J' = SigL ^/> anc ^ w e n a v e that J = n ( e L J ( iff J ' = n J e L J/. (5**) Let <j> : V —> V be an i?-epimorphism of .R-modules where i? is a ring. Then for any i?-submodule W of V upon letting W = (j>~1(W) we have that W is primary iff W' is primary, and for any family of ii-submodules (W()ieL of V upon letting Wi = (j)-l{Wl) we have that W = £ / e L Wj iff W = £ ; e z , W/, and we have

that w = nl£Lwt iff w = nl€Lw;. OBSERVATION (012). [Annihilators a n d Colons]. For a module V over a ring R, colons in R can be expressed in terms of annihilators thus. For any B dV or a € V, we put a n n f l 5 = {r &R: r(RB) = 0}

and

ann R o = { r 6 f l : r(Ra) = 0}

and we call these the annihilators of B and a in R respectively, and we note that they are clearly ideals in R. Obviously ann#.B = (0 : B)R

and

ann^a = (0 : CL)R

and more generally for any submodule C of V we have ann f l ((C + RB)/C)

= (C : B)R

and

ann f l ((C + Ra)/C) = [C : a)R.

If / is any ideal in R with I C ann^V then the .R-module structure of V induces a unique (i2/7)-module structure on V such that, upon letting (j> : R —> R/I be the residue class epimorphism, for all r £ R and v G V we have
§5.- IDEALS

AND

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113

OBSERVATION (013). [Primary Submodules]. To generalize the material of (07) and (08) to modules, let P be an ideal in a ring R and let Q be an Rsubmodule of an .R-module V. Again by a cyclical proof we see that the following three conditions are mutually equivalent, and when they are satisfied, we say that Q is primary for P, or that Q is P-primary; [cf. §13(E11)]. (i") Q is primary with radv<5 = P. ( i f ) P C ra.dvQ ^ R and: r G P and s<=V\Q with rs G Q => r G P. (in") Q is primary, annj^V/Q) is primary (as ideal), and P is prime with radyQ = PAs easy consequences of the above equivalence we have the following [cf. §13(E11)]: (1") Q is P-primary and B (Q : B)R is P-primary. (2") Q = Qi n • • • n Qh where Q, is P-primary for 1 < i < h G N+ =*> Q is P-primary. (3") P is a maximal ideal in R with P C iadvQ ^ R => Q is P-primary. (4'*) P is a maximal ideal in R and PeV C Q ^ V" (for instance Q = P e 7 ) with e 6 N+ => Q is P-primary. OBSERVATION (014). [Noetherian Modules with ACC a n d MXC]. Generalizing the noetherian condition on a ring R, an jR-module V is said to be noetherian if every submodule U of it is finitely generated, i.e., U = RU for some finite subset U of V; let us call this the NNC = noetherian condition on V. In the proofs of the Hilbert Basis Theorems (T3) and (T4), we implicitly used the fact that the NNC is equivalent to the ACC = ascending chain condition which says that for any ascending chain Ui C U2 C . . . of submodules of V we have Ue = Ue+\ = Ue+2 = ... for some e G N + . Yet another equivalent condition is the MXC = maximal condition which says that every nonempty family (UI)I^L of submodules of V has a maximal element, i.e., there exists I' G L for which there is no I" G L with Ui> C Uv and Ui> ^ Ui". To prove the equivalence of the three conditions we go in the following cyclical manner: (5*) NNC =* ACC =*• MXC => NNC. To see that NNC => ACC, given any ascending chain U\ C U2 C . . . of submodules of V let U = UJ6N + Ui\ then U is a submodule of V and hence by NNC we can find a finite set of generators u i , . . . ,u<j of U; now Uj G C/e(j) for some e(J) G N + , and it suffices to take e G N + with e > e(j) for 1 < j < d. To see that ACC => MXC, let (Ui)i£L be any nonempty family of submodules of V; take any 1(1) G L; if U^i) is not maximal then for some 1(2) G L we have Ui(i) C Z7/(2) with U^i) ^ ^(2)! if £/((2) is not maximal then for some Z(3) G L we have t/;(2) C £/;(3) with £/((2) 7^ ^(3)i

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and so on; by ACC this must stop and we get V G L with Uv maximal. To see that MXC => NNC, let U be any submodule of V; the family (= the set )U of all finitely generated submodules of U is nonempty since it contains the zero module; by MXC this family contains a maximal member U'; for any x GU\U', the module U" = U' + Rx is clearly in the family and for it we have U' C U"\ therefore by the maximality of U' we must have U' = U", i.e., x G U'; thus U = U' and hence U is finitely generated. OBSERVATION (015). [Irreducible and Primary Submodules]. Let us show that for a module V over a ring R we have the following: (6*) V satisfies MXC =*> every submodule U of V can be expressed as an intersection U = r\\ 1, let e > 0 be an integer, let P = M(R) = (X\,... ,Xm)R, let W be the set of all monomials of degree e in X\,..., Xm, let W\,..., Wn be a finite number of nonempty subsets of W such that W\C\---C\ Wn = 0. By (4*) we know

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115

that P e + 1 is P-primary and it can easily be seen that Pe+1 ^ ( P e + 1 + WiR) for 1 < i < n but P e + 1 = n 1 < i < n ( P e + 1 + WiP) and hence P e + 1 is not irreducible. OBSERVATION (016). [Spectrum and Irredundant Decomposition]. For a ring R we put spec(P) = the set of all prime ideals in R and we call this the spectrum of R, and we put mspec(P) = the set of all maximal ideals in R and we call this the maximal spectrum of R. For any J c R we put vspec fi J = {P G spec(P) : J C P } and we call this the spectral variety of J in R, and we put mvspec fi J = mspec(P) n vspec fl J and we call this the maximal spectral variety of J in R, and we put nvspec^ J = the set of all minimal members of vspec K J i.e., the set of all P in vspec# J such that for all P ' ^ P in vspec^ J we have P' <£_ P , and we call this the minimal spectral variety of J in R. We put svt(P) = the set of all spectral varieties in R i.e., the set of all subsets of spec(P) of the form vspec^J with J varying over the set of all subsets of R. For any W C spec(P) we put ispec^W =

r\P€wP

and we call this the spectral ideal of W in R; note that by convention: ispec^.W = R & W = 0. We put rd(P) = the set of all radical ideals in R i.e., the set of all ideals J in R such that r a d ^ J = J. Later on we shall show that, when R is a multivariate polynomial ring over a field, ispec^ gives a bijection svt(P) —> rd(P) whose inverse is given by vspec#. First we want to set up the machinery for proving partial uniqueness of the primary decomposition hinted at in (015). For any R-module V we put &SSRV

= {P G spec(P) : ann/ja is P-primary for some a £ V}

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116

and we call this the associator of V in R; note that we automatically have a ^ 0 because the annihilator of 0 is R. We also put nassftV = the set of all minimal members of ass^V i.e., the set of all P in ass^1!^ such that for all P' ^ P in ass^V we have P' <£ P, and we call this the minimal associator of V in R. Note that for any P-submodule U of V we clearly have assR(V/U)

C vspec R (ann/{(V/t/)).

For any P-submodule U of V, by an associated (resp: associated minimal, associated embedded) prime of U in V we mean a member of assfl(V/l7) (resp: a member of nassR(V/U), a member of &ssR(V/U) \ nassR(V/U)). These terms are motivated by an assertion about an irredundant primary decomposition of an P-submodule U of V, i.e., an expression of the form:

' U = ^i
= {Pi,.. .,Pn}

and nassR(V/U)

= nvspec fl (ann fl (V A /t/)).

Clearly: (8.1*) In (•) we have vspecR(imnR(V/U))

= Ui
Therefore, assertion (8*) is equivalent to the following assertion: (8.2') In (•) we have {P € spec(i?) : (U : a)R is P-primary for some a &V} = { P i , . . . , P„}. As a supplement to (8.2*) we shall also prove: (8.3*) If R is noetherian then in (•) we have {P € spec(P) :(U:a)R

= P for some a €V}

= {Pu...,

Pn).

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Before proving (8.2*) and (8.3*), let us prove the following claim: (8.4*) In (•), for any i £ { l , . . . , n } and a £ f~lje{i n}\{i}Qj with a g Qi we have that (U : CL)R is Pj-primary. Moreover, if (U : a)# 7^ Pj then for some x £ R with xa $? Qj we have (£/ : a)# ^ (£/ : XO)R. Finally, if R is noetherian then for some z £ R with za ^ Qi we have ([/ : ZCL)R = Pi. Namely, in view of (ii'") we see that a n n ^ V / Q j ) C (U : a)# C Pi and hence Pi C radfl(C/ : a)R^ R. Moreover: r £ R\Pi

and s £ i? with rs £ (U : a)R

=> r(sa) <=U cQi ' =3- sa € Qi [by (ii'*) because r £ Pi] =$• sa £ U [because U = Di s G (*/ :a)R and hence by (ii*) we see that (U : O)R is Pj-primary. Moreover, if (U : a),R ^ Pj then for some x,y in R\(U : O)R we have yx £ (U : a)n; now: yx £ (U : CI)R =*> y(a;a) £ U => y £ (U : XCL)R => ([/ : a)# ^ (£/ : xa)^; also: x g (U : CL)R and a £ <~}j£{i,...,n}\{i}Qj => xa $ Qi. If (U : X\O)R ^ Pi with xi = x then replacing a by x\a we find X2 £ R such that a^aria 0 Qj and (U : XICL)R ^ (U : a^xia)^. And so on. Now clearly (U : CI)R C (U : XIO)R C (U : X2X\O)R C It follows that if R satisfies ACC then this must stop and we find z = xexe-\... x\ £ R such that za £ Qi and (U : ZO)R = Pj. To prove (8.2 s ) and (8.3*), given i £ { 1 , . . . , n}, by the irredundancy we can find a as in (8.4*) and then by that claim we see that (U : CL)R is Pj-primary, and if R is noetherian then after replacing a by za with suitable z £ P we have (JJ : O)R = Pi. Conversely let P £ spec(P) be such that (U : O,)R is P-primary for some a £ V. Now P = rad fi (l7 : O)R = r\i
LECTURE

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and tass coincide. (9*) If R is noetherian then in (•) we have tass#(V/[/) =

assrt(V/U).

This follows from (8.2*) and (8.3'). OBSERVATION (018). [Tacit Use of PIC and P M C ] . In the proof of (8.4*) in (016) we cited (ii*) and (ii") when we should have really said by the equivalence of conditions (i*) to (in') asserted in (07) and by the equivalence of conditions (i /# ) to (iii") asserted in (013). These are conditions for an ideal to be P-primary and for a module to be P-primary. We may call these PIC (= Primary Ideal Conditions) and PMC (= Primary Module Conditions) respectively and we may use them tacitly. OBSERVATION (019). [Multiplicative Sets and Isolated Components]. By a multiplicative set in a ring R we mean a subset S of R with 1 € S such that: x, y in S =>• xy € S. For a submodule U of a module V over R we put [U : S]v = Uses(U : s)v and we call this the isolated 5-component of U in V; note that [U : S]y is a submodule of V with U C [U : S]V- To distinguish this from (U : B)v for B c R we observe that (U : B)v = DbeB(U : b)v For a prime ideal P in R we put [U:P}v

=

[U:R\P]v

and we call this the isolated P-component of U in V; there should be no confusion between these notations because always l € S and 1 ^ P . More generally, for any prime ideals P i , . . . , P„ in R with n £ N+ we put [U:(Pi,...,Pn)]v

=

[U:ni
and we call this the isolated ( P i , . . . , P n )-component of U in V. In (016), we used the parenthetical colon operation (:) to prove the uniqueness of the primes Pj occurring in (•), and now we shall use the bracketed colon operation [:] to prove a partial uniqueness of the primaries Qi. The proof applies to a primary decomposition which need not be irredundant, i.e., when for a submodule U of V we have: U = ni
(CONVENTION: ni
where n G N and Qi is a primary submodule of V for 1 < i < n and rad/jQi = Pi for 1 < i < n.

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119

Now the uniqueness assertion is the following: (10*) In (••) let 1 < i(l) < i(2) < ••• < i(h) < n be any sequence of integers with h £ N+ such that: j £ { 1 , . . . , n} with Pj C P,(/) for some I £ { 1 , . . . , h} =>• j £ {i(l),...,i(h)}. Then we have [[/ : (P»(i),... ,Pi(h))]v = ni SnPm ^ 0. Then U = Q'nQ". We want to show that [U : S]v = Q'• Now

{

v£[U:S}v

=>• =*•

xv £ U for some x e S C r\i
To prove the other implication, for 1 < / < n — h we can first take yi £ SflPj^ then we can find e(l) £ N+ such that y f ( ' V c Qj(i). Now y = Y\l
=•

and £ S

yv £ Q' n Q" = U

=> v £ [U : S]y. OBSERVATION (020). [Zerodivisors and Quasiprimary Decomposition]. To generalize the contents of (016) to (019), PIC and PMC suggest the following weakenings of primaries in a ring R and in an i?-module V. Given ideals P, Q in R we say that Q is P-quasiprimary if P is prime with Q ^ R and: r £ R and s £ R\Q with rs £ Q =>• r £ P. More generally, given an ideal P in R and a submodule Q of V we say that Q is P-quasiprimary if P is prime with Q ^ V and: r £ R and s e F \ Q with rs e Q =>• r € P . By fixing s £ R \ Q in the above implication about ideals we see that r £ Q^>rs£Q=>r£P and hence: Q is a P-quasiprimary ideal in R =>• Q C P . Similarly, by fixing s e V \ Q in the above implication about modules we see that r £ annft(VyQ) ^>rs£Q=$>r£P and hence: Q is a P-quasiprimary submodule of V =*> &nnji(V/Q) C P . Moreover in the ideal case rad^Q = R iff Q = R, and in the module case T&djiQ = R iff Q = V. Therefore in (07) we see that (ii*) =^ Q C .P, and in (013) we see that (ii /# ) =£> annfl(Vy<2) c P . This remark was implicitly

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120

used in the cyclical proofs asserted in (07) and (013). By a zerodivisor of V we mean r £ R such that rv = 0 for some 0 ^ v € V, and we put ZR(V) = the set of all zerodivisors of V. We also put SR(V) =

R\ZR(V)

and we note that, by the nonP convention, a nonzerodivisor of V means an element of SR(V); we also note that SR(V) is a multiplicative set in R. For a submodule U of V, members of ZR(V/U) or SR(V/U) may respectively be called zerodivisors or nonzerodivisors mod U. In particular, for an ideal I in R, members of ZR(R/I) or SR(R/I) may respectively be called zerodivisors or nonzerodivisors mod I. [The following six terms: loass, lmass, lpass, zoass, zmass, and zpass, are used only in this Observation (020), with the single exception of lmass to be used in the proof of (024)(20*). So the readers NEED NOT MEMORIZE them]. For any ideal I in R and multiplicative set S in R with J fl S = 0, we put loass^(7, S) — the set of all ideals J in R with I
\oassR(I,SR(R/I))

and we call this the zerodivisor overideal associator of J in R, and we put zmass/{(7) = lmassfi(7,

SR(R/I))

and we call this the zerodivisor maximal associator of I in R, and we put zpass i j(J) = lpass

R(I,SR(R/I))

and we call this the zerodivisor prime associator of I in R. Note that an ideal J in R consists only of zerodivisors mod I iff JDSR(R/I) = 0; in this case we may say that J is a zerodivisor ideal mod I. Similarly, a member of zoass#(/) (resp: zmassfl(J),

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zpassfl(/)) may be called a zerodivisor overideal (resp: zerodivisor maximal ideal, zerodivisor prime ideal) mod 7. In (12*) below we shall show that lmassfl(7, S) C lpass fl (7,5) and hence in particular zmass^(7) C zpass fl (7). In (13*) below we shall give a quasiprimary decomposition for any nonunit ideal 7 in R, and (14*) below we shall generalize it to a quasiprimary decomposition for any submodule U of V with U ^ V. To achieve the said generalization, we note that for any submodule U of V with JJ ^ V we clearly have &imR(V/U) D SR{V/U) = 0 and we put zoassv(C/) = loassR(<mnR(V/U),

SR(V/U))

and we call this the zerodivisor overideal associator of U in V, and we put zmassv(CZ) = lmassR(annR(V/U),

SR(V/U))

and we call this the zerodivisor maximal associator of U in V, and we put zpassv(U)

=

lpassR{annR{V/U),SR(V/U))

and we call this the zerodivisor prime associator of U in V. Note that an ideal J in R consists only of zerodivisors mod U iff J n SR(V/U) — 0; in this case we may say that J is a zerodivisor ideal mod U. Similarly, a member of zoassy(£7) (resp: zmassy(CZ), zpassy(t/)) may be called a zerodivisor overideal (resp: zerodivisor maximal ideal, zerodivisor prime ideal) mod U. By a principal component of a nonunit ideal 7 in R we mean an ideal in R of the form [7 : P]R for some P £ zmassft(7); in (13*) we shall show that [7 : P]R is P-quasiprimary. Likewise, by a principal component of a submodule U of V with U 7^ V we mean a submodule of V of the form [U : P)v for some P G zmass\/(C/); in (14*) we shall show that \U : P]v is P-quasiprimary. Note that since 1 belongs to every multiplicative set S in 7?, for any ideal 7 in R we clearly have 7 c [I: S]R and hence in particular for every prime ideal P in R we have 7 C [7 : P]R. Similarly, for any submodule U of V we have U C [U : S]v and U C [U : P]v(12*) For any ideal 7 in R and multiplicative set S in R with 7 n S = 0 we have 0 ^ lmassii(7,5) C lpass R (7, S) C vspec fi (7) C spec(i?) and for any ideal 7i in R with 7 C h and 7i n S1 = 0 we have lmassfi(7i,5) C

lmassR(I,S).

In particular, for any nonunit ideal 7 in R we have 0 y£ zmassfi(7) c zpass fi (7) c vspec fl (7) c spec(i?) and for any ideal 77 in R we have 77 n SR(R/I)

= 0 => i7 c P for some P e zmass fi (7).

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LECTURE L4: VARIETIES AND MODELS

More generally, for any submodule U of V with U / V we have 0 ^ zmassv(U)

C zpass v (C) c vspecR(annfi(V/C/')) c spec(i?)

and for any ideal H in R we have if n SR(V/U)

= 0 =4- if C P for some P € zmass v (t/).

(13*) Given any nonunit ideal I in i?, let W = zmassR(I) with W = vspecR(I), and for every P G spec(P) let I[p] = [I : P]R. Then for every P € spec(i?) \ W we have 7[PJ = R, for every P G W the ideal I[p] is P-quasiprimary, and we have the quasiprimary decompositions I = P\p^wI[P] = <^PewI[p] = ^Pespec(R)I[P] where the first equation may be paraphrased by saying that I is the intersection of its principal components. Moreover, for every P G nvspec fl (7) the ideal 7[pj is P-primary. (14*) Given any submodule U of V with U =£ V, let W = zmassy([/) with W = {P G spec(P) : [/[p] ^ V} where for every P G spec(P) we are putting U[p] = [U : P]v- Then for every P G W the submodule £/[pj is P-quasiprimary, and we have the quasiprimary decompositions U = r\p&wU[p] = ^PeW'U[p] = ^Pespec(R)U[p] where the first equation may be paraphrased by saying that U is the intersection of its principal components. Also W C vspec fl (annij(V/[/)), and if V is a finitely generated jR-module then W = vspec i j(annii(V/[/)). Moreover, for every P G nvspecij(ann/{(V/C/)) the submodule U[p] is P-primary. PROOF OF (12*). To prove the first display, let us start by noting that lmassft(7, S) is nonempty because of Zorn's Lemma. To show that any P in lmassp_(/, 5) is prime, given any Xi G R \ P for 1 < i < 2, by the maximality of P we can find yi G (xiR + P) C\S, and then t/j = nxi +pi with ri G R and pi G P, and now 2/12/2 G S c R\P and 2/12/2 -r\riX\X2 = rixip2 + ^2^2Pi +PiP2 G P , and hence x\Xi £ P. The rest of the first display is straightforward and so is the second display. The third display follows by taking SR(R/I) for S in the first display, and the fourth display follows from the second display by also taking H +1 for I\. The fifth display follows by taking &niiR(V/U) for I and SR(V/U) for S in the first display, and the sixth display follows from the second display by also taking H + I for/i. PROOF OF (13'). Follows by taking (U,V) = (I,R) in (14*). PROOF OF (14*). Let us put I = a.nnR(V/U). Since U C U[P] for every P in spec(P), we have U C r\pew>U[pj = nPespec(R)U[p] C Dp^\yU[p]. Hence to prove the "quasiprimary decompositions" it suffices to show that u G C\pewU[P] =*> u £U.

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Now given any u £ Hp&wU[p], upon letting H = (U : U)R, we see that: P £W =>u£ U[P] ' => for some r G R \ P we have ru £ U and hence r £ H => H

£P.

Consequently by the last display in (12*) we conclude that H fl hence u £ U. Let there be given any prime ideal P in R. Now r £ R\P

SR(V/U)

^ 0 and

and s € V with rs £ U[P]

=>• a;rs G [/ for some x £ =4> (a;r)s G [/ with xr £

R\P R\P

^S£U[P].

This shows that U[p] is P-quasiprimary for every P € W'. In view of PMC, it only remains to establish the following three implications. (a) I £ P => U[P] = V. (b) U[p] - V and V = Rv\-\ l-Rvh with h in N + and vx,..., vh in V => J • P c radyC/jpj. To prove (a) note that:

'/£P ^ • s V c t / for some s £

R\P

< =>• V C (f/ : s)v for some s £

R\P

=>Vc[U:P]v ^U[P]=V. To prove (b) note that: U\p\ =V = Rvi + • • • + Rvh for some v\,...,

Vh in V with h G N +

=> XjUj G U with Xj G R \ P for 1 < i < h < =>• a;V C t/ where x = x\...Xh => x £ I with x £

£

R\P

R\P

=>I<£P. To prove (c) assume that P G nvspeCp(J). Given any y £ P let S be the smallest multiplicative set in R containing y and R\P. If J were disjoint from S (i.e., if / n S = 0) then, by the first display in (12 0 ), there exists P' £ spec(i?) with I c P' and P'nS = 0. But P' D S = 0 implies that P ' D ( # \ P ) = 0, i.e., P' C P which in view of the minimality of P implies that P' = P and hence y £ P' which is a contradiction. Therefore / is not disjoint from S, i.e., j / n z £ I for some

124

n G N and z£ R\P. y G rady[/[ P ].

LECTURE H: VARIETIES AND MODELS

It follows that ynzV

c U and hence ynV C U[P]. Therefore

OBSERVATION (021). [Length of a Module]. The dimension of a vector space V over a field R can be characterized as the maximum length of strictly increasing finite chains of subspaces VQ C V\ C • • • C Vn. Generalizing this we define the length IR{V) of a module V over a ring R to be the maximum length n G N of a finite sequence (a)

VoCViC-cV„

of submodules of V with

0 = Vo^Vi^---^Vn-!^Vn

= V.

We call such a sequence a normal series of V of length n. When there does not exist a bound for the lengths of such sequences then we put £R(V) = oo. Note that iR(V) = 0 o V = 0. By a simple module we mean a module of length 1, i.e., a nonzero module which has no nonzero submodule different from the whole module. By a composition series of V we mean a normal series (a) such that the factor module Vi/Vi-i is simple for 1 < i < n. Two normal series (a) and

(<*')

Vo'cv/c-.-cK'

of V are equivalent means n = n' and there is a permutation a of { 1 , . . . , n} such that the factor modules V^/V^-i and Vi'/Vi'_1 are isomorphic for 1 < i < n. The normal series (a') is a refinement of the normal series (a) means n' > n and there is a subsequence 0 < j i < • • • < j n - \ < n' such that VJ. = V% for 0 < i < n. To slightly generalize the above concepts, let W be any submodule of V. By a normal series between W and V of length n G N we mean a sequence (a) of submodules of V with

W = V0jtV1^---^Vn.1^Vn

= V.

This is called a composition series between W and V if the factor module Vi/Vi-i is simple for 1 < i < n. Two normal series (a) and (a') between W and V are equivalent means n = n' and there is a permutation cr of { 1 , . . . , n} such that the factor modules Va{%)IVa(i)-\ and Vi'/Vi'_1 are isomorphic for 1 < i < n. The normal series (a1) between W and V is a refinement of the normal series (a) between W and V means n' > n and there is a subsequence 0 < j \ < • • • < jn-i < n' such that Vj. =VitorO
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if (a) is a normal series of V of length n, then (a) can be refined to a composition series of V of length at most n'. Namely, if (a) is not a composition series then Vi/Vi-i is not simple for some i £ { 1 , . . . , n} and we can insert a submodule between Vi-i and Vi which is different from both. Now we must have n + 1 < n', and so on. (15*) If the module V has a composition series then any two composition series of V are equivalent (and hence in particular have the same length which equals £R(V)), and every normal series of V can be refined into a composition series of V. PROOF. By induction on the nonnegative integer n let us show that if V has a composition series (a) of length n then for the length n' of any normal series (a1) of V we have n' < n. This being obvious for n < 1, let n > 1 and assume true for all smaller values of n. If n' < 1 then we have nothing to show. So assume n' > 1. If V^,_j = V n _i then VQ C V/ C • • • C VrT(/_1 is a normal series of V n _i of length n' — 1 and Vb C Vi C • • • C Vn-\ is a composition series of Vn-\ of length n — 1, and hence by the induction hypothesis n' — 1 < n — 1 and therefore n' < n. If K ' - i ^ K - i with Vr^,_1 C Vn-i then V0' C V{ C • • • C V^,_x C V n _i is a normal series of V„_i of length n' and Vb C V\ C • • • C V„_i is a composition series of Vn-i of length n— 1, and hence again by the induction hypothesis n' < n— 1 and therefore n' < n. Finally suppose that VT^,_1 ^ V^_i. Then, since there is no submodule between Vn-i and V other than these two, we must have Vn-i + V^,_1 — V, and hence by the First Isomorphism Theorem (1**), the factor modules V/Vn-i and Vn'-i/(Vn-i n V ^ j ) are isomorphic, and therefore Vr^,_1/(Vrn_i n VrT(,_1) is simple. Since Vb C Vi c • • • C Vn-\ is a composition series of V n _i of length n — \ and V^-i n Vrf-i C V n _i with Vn-i n V^/_! ^ V^-i, by the induction hypothesis it follows that the length of any normal series of Vn-i H V^,_1 is at most n — 2, and hence V n -i fl V^/_1 has a composition series of length at most n — 2; since V^/_ 1 /(K l _i Pi V^/_1) is simple, it follows that V^,_1 has a composition series of length at most n — 1. Since VQ C F / C • • • C \rll,_1 is a normal series of V^,_1 of length n' — 1, by the induction hypothesis we get n ' — 1 < n — 1 and hence n' < n. This completes the induction. It follows that every composition series of V has length n, and any normal series of V can be refined into a composition series of V. It only remains to show that, assuming (a) and (a') to be composition series of V of lengths n and n' with n = n', there is a permutation a of { 1 , . . . ,n} such that the factor modules VCT(j)/Vra(j)_i and V{/V(_l are isomorphic for 1 < i < n. Again we make induction on n. For n < 1 we have nothing to show. So let n > 1 and assume true for all values of n smaller than the given one. If V^_x = Vn-\ then deleting the last terms from (a) and (a') we get two composition series of V n _i of length n — 1, which are equivalent by the induction hypothesis, and hence (a) and (a') are equivalent. Now assume that V^_t ^ Vn-\. Then V^_1 + Vn-\ = V and hence, upon letting V*. = V^l_1 fl V„_i, by the First Isomorphism Theorem (1**), the two modules Vn-\/V*. and V/V^_x are isomorphic, and so are V^-i/V*, and

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V/Vn-i.

L4: VARIETIES

AND

MODELS

Therefore by taking a fixed composition series

v0* c v; c • • • c v:,

(P)

of V*. we get the two equivalent composition series

(7)

V0* c V,* c • • • c v:. c K-i c V

and

(7')

V0* c V{ c • • • c V£ c K_! c v

of V. By deleting the last Vn-i which are equivalent equivalent; clearly we must Therefore (a) and (a') are

terms in (a) and (7) we get two composition series of by the induction hypothesis, and hence (a) and (7) are have n* = n - 2. Similarly (a') and (7') are equivalent. equivalent.

(16*) If W is a submodule of V such that there is a composition series between W and V then any two composition series between W and V are equivalent (and hence in particular have the same length which equals £R(V/W)), and every normal series between W and V can be refined into a composition series between W and V. If W is a submodule of V with residue class epimorphism <j>: V —» V/W, then, for any normal series (a) between W and V, the series ^(Vo) C 4>{Vi) C • • • C 0(V„) is a normal series of VyW and for 1 < £ < n the factor modules (j){Vi)/(p(Vi-\) and Vi/Vi_i are isomorphic; in particular: (a) is a composition series between W and V & 4>{VQ) C (Vi) C • • • C <j>(Vn) is a composition series of V/W. PROOF. The first sentence follows by taking V/W for V in (15*). The second sentence follows by the Second Isomorphism Theorem (2**). (17*) For any submodule W of V we have £R(V) = £R(W) + £R(V/W), where by convention 00 = 00 + 00 = 00 + integer. In greater detail, if (a) is a composition series between W and V, and Wo CW\C---C Wm is a composition series of W, then Wo C W\ C • • • C Wm C Vi C • • • C Vn is a composition series of V. PROOF. Follows from (16*). (18*) For any submodules U and W of the module V we have £R(C/) + lR(W) = £R(U + W) + £R(U n W^), again with the understanding that 00 + 00 = 00 + integer. PROOF. Taking (U + W, U) for (V, W) in (17*) we get £R(U + W)= £R(U) + £R((U +

W)/U).

The factor modules ([/ + W)/U and W/{U D W) are isomorphic by the First Isomorphism Theorem (!*'), and hence by taking (W, U D W) for (V, W) in (17*) we

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127

get

IR(W) = eR(u nw) + eR((u + w)/u). We are done by adding the two equations, and taking care of infinities. OBSERVATION (022). [Heights and Depths of Ideals, and Dimensions of Rings]. Replacing chains of submodules by chains of prime ideals we get the definitions of heights and depths of ideals and the definition of the dimension of a ring. By the dimension dim(P) of a ring R we mean the maximum length n £ N of a finite sequence (6)

Po C Px C • • • C P„

of prime ideals in R with P0 + P\ + • • • ± Pn-l ± PnWe call such a sequence a prime sequence in R of length n. When there does not exist a bound for such sequences then we put dim(P) = oo provided R is not the null ring; if R is the null ring then we put dim(P) = —oo. By the height htRP (resp: depth d p t ^ P ) of a prime ideal P in R we mean the maximum length of a prime chain (<5) in R with P = Pn (resp: P = PQ), again with the understanding that when there does not exist a bound for such sequences then we put htflP = oo (resp: dptRP = oo). We define the height htft J and the depth dpt# J of a nonunit ideal J in R by putting htRJ = min{ht R P : P £ vspec f l J} £ N U {oo} and dptRJ = max{dpt f i P : P £ vspec f l J} € N U {oo} and we note that for J £ spec(P) these reduce to the previous definitions. Also note that for a nonunit ideal J in R, dpti? J is the maximum length of a prime chain (6) in R with J C Po. Finally note that for an ideal J in R, vspecRJ is empty iff J = R, and let us complete the definitions of height and depth by putting htRR = —oo = dptRR and let us observe that now d p t ^ J = dim(i?/J). EXAMPLE (X10). Taking R to be the n-variable polynomial ring k[Xi,..., Xn] over a field k and Pt to be the ideal in R generated by Xi,...,Xi, and putting X\ = • • • = Xi = 0 we get an epimorphism R —> k[Xi+i,..., Xn] with kernel Pj for 0 < i < n which shows that (5) is a prime sequence in R. It follows that dim(P) > n

128

LECTURE H: VARIETIES AND MODELS

with htflPi > i and dptflPj > n - i for 0 < i < n. Later on we shall show that these three inequalities are actually equalities, [cf. L5§5(Q10)(T47)]. OBSERVATION (023). [Modular Law]. The following Modular Law gives a useful identity concerning a module V over a ring R: (19*) For any submodules T, U, W of V with U C T we have

TD{U + W) = U + (Tr\W).

To see this note that the RHS is obviously contained in the LHS. Conversely, if v is any element in the LHS then, because v G U + W, we get v = u 4- w with u £U and w &W. Now w = v - u € T because v eT and u eU CT. Thus i n e T n i y and hence w £ U + (T n W). OBSERVATION (024). [Generalities on Prime Ideals]. In the following assertions (20*) to (23*) we give some useful properties of prime ideals. (20*) In any ring R, the set of all nilpotent elements, i.e., the radical of the zero ideal, equals the intersection of all the prime ideals, with the understanding that if there are no prime ideals (i.e., if the ring is the null ring) then the said intersection is the unit ideal. [Note the analogy with L2(R5) about the existence of maximal ideals]. PROOF. Since any prime ideal contains 0, it must contain every nilpotent element. Conversely, given any nonnilpotent element v in R we want to find a prime ideal P not containing it. We can do this by letting 1 = 0 and S = {vn : n G N} in (12*), and taking P G lmass f l (/,S). (21*) Let P be a prime ideal in a ring R, and let J\,..., Jn be ideals in R with n G N+ such that Ji n • • • f~l Jn C P. Then for some i € { 1 , . . . , n) we have Ji C radft Ji C P. Moreover, if J\ n • • • O J„ = P then for some i G { 1 , . . . ,n} we have Ji = P. PROOF. If Ji <]L P for 1 < i < n then by taking Xi G Ji\P we get x\ ... xn G Jifl • • -fl J„ C P which contradicts the primeness of P. Therefore for some i G { 1 , . . . , n} we must have Ji C P and by taking radicals we get rad# Ji C P. The rest is obvious. (22*) Let J be an ideal in a ring R, and let P i , . . . , Pn be prime ideals in R with n G N + such that J C Pi U • • • U P„. Then for some i G { 1 , . . . , n} we have J c PiPROOF. By discarding some of the primes P i , . . . , P „ we may assume that

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129

Pj ^ Ui». We claim that: (24*)

DCC o MNC.

Namely, by reversing inclusions and changing maximal to minimal in the proof of ACC =^ MXC given in (014) we get DCC => MNC. Conversely, if V does not satisfy DCC then we can find an infinite sequence U\ D U2 D • • • of submodules of V with Ui 7^ U2 ^ •. • and clearly the family (C/();eN+ has no minimal element. We say that V is an artinian it-module to mean that V satisfies the DCC and hence the MNC. We say that R is an artinian ring to mean that R is an artinian .R-module. In assertions (25*) to (33*) below, we shall prove some general properties of artinian and noetherian modules. (25*) Let W be any submodule of V. Then V is an artinian module <=» W and V/W are artinian modules. Similarly V is a noetherian module <=> W and V/W are noetherian modules. PROOF. If DCC holds in V then it obviously holds in W and by the Second

130

LECTURE L4: VARIETIES AND MODELS

Isomorphism Theorem (2**) it also holds in V/W. Conversely, assume DCC holds in W and V/W, and let U\ D U2 D ... be any decreasing sequence of submodules of V. Then Ui n W D U2 n W D •.. and {Ux + W)/W D (U2 + W)/W D ... are decreasing sequences of submodules of W and V/W respectively, and hence for some e e N+ we have Ue n W = Ue+i n W = Ue+2 r\W = ... and Ue + W = Ue+\ + W = Ue+2 + W = ... . For every integer d > e we get V d = Ud n (J7d + W) <

because C/d c C/d + W

= UdD (Ud+1 + W)

because Ud + W = Ud+i + W

= Ud+i + (Ud n W)

by Modular Law (19*)

= Ud+i + (Ud+1 n W)

because f/d n W = C/d+1 n W

= Ud+\

because Ud+i C\W C Ud+i.

Therefore Ue = Ue+i = Ue+2 = ••• • For the noetherian case, in the undisplayed material change DCC to ACC and D to C, and in the displayed material change (d,d+l) to (d+l,d). (26") Assuming V = V\ + • • • + Vs, where V\,...,VS are a finite number of submodules of V, we have the following. If the modules V\,..., Vs are artinian then so is V. Similarly, if the modules V i , . . . , Va are noetherian then so is V. PROOF. Since, Vi + • • • + Va = (Vi -f • • • + V s _i) + Vs, by induction the general case follows from the s = 2 case. This case follows from (25*) by noting that, in view of the First Isomorphism Theorem (1*'), the modules (Vi + V"2)/Vi and V2/(Vi n V2) are isomorphic. (27*) If R is an artinian ring and V is a finitely generated i?-module then V is an artinian module. Similarly, if R is a noetherian ring and V is a finitely generated .R-module then V is a noetherian module. PROOF. In view of (26') it is sufficient to show that, assuming V = Rx with x G V, if DCC (resp: ACC) holds for ideals in R then DCC (resp: ACC) holds for submodules of V. So assume DCC (resp: ACC) for ideals in R, and given any descending (resp: ascending) sequence ({7j)ieN+ of submodules of V = xR let Ji = {a £ R : ax £ Ui}. Then (Jt)ieN + is a descending (resp: ascending) sequence of ideals in R, and for all i e N+ we have Ui = {ax : a€ J , } . By DCC (resp: ACC) for ideals we can find e G N+ such that J e = Je+i = Je+2 = • • • - I n both the cases it follows that Ue = Ue+i = Ue+2 = ... . (28*) (R(V) < 00 «4- the module V is artinian as well as noetherian. PROOF. If ^ H ( V ) = n < 00 then every descending or ascending sequence of

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131

submodules of V has at most n distinct members, and hence V is artinian as well as noetherian. Conversely suppose that V is artinian as well as noetherian. Let UQ — V. If UQ = 0 then £R(V) = 0. If UQ ^ 0 then, because V is noetherian, we can take a maximal member U\ in the family of submodules of Uo different from U0. If U\ = 0 then £R(V) = 1. If U\ ^ 0 then, again because V is noetherian, we can take a maximal member Ui in the family of submodules of U\ different from U\. And so on. Since V is artinian, this must stop. Thus we get n G N together with a decreasing sequence Uo D UI D • • • D Un of submodules of V with V = Uo ^ U\ 7^ • • • 7^ Un-\ ^ Un = 0 such that Ui/Ui+i is simple for 0 < i < n. It follows that IR(V) = n < 00. (290) If R is a field then d i m ^ y = IRV and we have that: V is an artinian module <^> d i m ^ ^ < 00 <=$• V is a noetherian module. PROOF. Obvious. (30*) If there are maximal ideals Pi,...,Pn in R with n € N + such that P i . . . Pn = 0 then: R is an artinian ring o R is a noetherian ring. PROOF. Let Vi = P i . . . P n _ t . Then 0 = V0 C Vi C • • • C V„ = R is an increasing sequence of ideals in R. For 1 < i < n we have P n _j + i c ann/j(Vi/Vi_i) and hence the i?-module Vj/Vj_i may be regarded as a vector space over the field R/P„-i+i. Therefore our assertion follows from (29*). (31*) If R is any ring and J is an ideal in R with IRJ — 1 then a n n # J is a prime ideal in i?. PROOF. Now J 7^ 0 and hence ann# J ^ .R. Given any a,b in R\ ann« J, let A = aR and £? = bR. Then A J and B J are nonzero ideals contained in J. Since IRJ = 1, we get AJ = J — BJ. Consequently ABJ = J and hence a& belongs to R \ aniiR J. (32#) If R is a domain then: R is an artinian ring •& R is a, field. If R is an artinian ring then every prime ideal in R is maximal. PROOF. Assume R to be an artinian domain, and let 0 ^ x e R. Then xR D x2R D ... is a decreasing sequence of ideals in R, and hence for some e G N + and y G R we must have xe = y:r e+1 which gives xe(l — yx) = 0. Since xe =£ 0 we get ya; = 1. Thus R is a field. The rest is now obvious. (33*) R is an artinian ring iff it is either the null ring or a zero-dimensional noetherian ring.

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PROOF. The case of a null ring being obvious let us suppose that R is nonnull. First assume R to be a zero-dimensional noetherian ring. Then by (23#) we can find a positive integer n and maximal ideals P i , . . . , Pn in R such that P i . . . Pn c 0 and hence P i . . . Pn = 0. By (30*) we see that R is artinian. Conversely assume R to be artinian. Then we can take a minimal element 7 in the set of all ideals in R which can be expressed as nonempty finite products of prime ideals. We shall prove that 7 = 0 and, in view of (30*) and (32*), this will show that R is a zero-dimensional noetherian ring. Suppose if possible that 7 ^ 0 . Let A = ann#I. Then A is a nonunit ideal in R, and hence we can take a minimal element B in the nonempty set of all ideals in R which contain A but are different from A. Let P = (A : B)R and let : R —> R/A be the residue class epimorphism. Now <j){R) is an artinian ring and ^ , ( R ) 0 ( B ) = 1. Therefore by (31*) we see that ann^,(fl)0(B) is a prime ideal in (a) where 4>: 72 —• 7?/A and ^> : R —> # / B are the residue class epimorphisms (for => take a, b as in the above paragraph; for <$= write (1 — b) + b = 1). Clearly A, B are comaximal <=> no maximal ideal in R contains both A and B; hence if Ai,..., An, B\,..., P m are ideals in R with n,m in N+ such that Ai, Bj are comaximal for each i,j then A i . . . A„, B i . . . Bm are comaximal and so are Ai D • • • n An, B\ n • • • n B m . The above equality of the product and intersection can be generalized by showing that for any finite number of pairwise comaximal ideals A\,..., An we have J T 1 < i < n A j = rii 1 and assume for all smaller values of n. Then by the induction hypothesis

ni
and

rii
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133

and by assumption A\ + A2 = R and hence 'r\i
={Ax+A2){C\i = Ax (ni
cA1{jlmAi)+A2(u^2A) =

lll<j
and obviously rii rad^A and rad^i? are comaximal «• no maximal ideals in R contains both A and B •£> there are elements o, b in R such that : R-^ R/A and V : R -> # / # are the residue class epimorphisms. If A i , . . . , An, B\,..., Bm are ideals in R with n , m in N + such that Ai and Bj are comaximal for each i,j then A\.. .An and £?i... B m are comaximal and so are A\C\ • •• C\An and B\ n • • • D B m . (35*) If A i , . . . , 7 a n arc a finite number of pairwise comaximal ideals in a ring -R then rii
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LECTURE H: VARIETIES AND MODELS

§6: P R I M A R Y D E C O M P O S I T I O N The fact that every positive integer can be written as a product of powers of prime numbers can be reformulated by saying that every nonzero ideal in the ring of integers can be expressed as a finite product, and hence finite intersection, of ideals generated by powers of prime numbers. In (015) to (020) we have generalized this to obtain primary and quasiprimary decompositions of ideals in a ring R. Let us restate these generalizations by first recalling some relevant definitions. The radical of an ideal 7 in R is denned by r a d ^ I = {r G R : re G 7 for some e G N + } . By spec(J?) and mspec(P) we denote the set of all prime and maximal ideals in R respectively. We put vspec/j/ = { P e spec(P) : 7 c P } , and nvspec#7 = set of all minimal members of vspecfl/. 7 is primary means I ^ R and: r £ R and s G R\ I with rs G 7 => re G I for some e G N+. If 7 is primary then radij/ is prime and we say that 7 is (radfl7)-primary. We say that 7 is P-quasiprimary if P G spec(P) with 7 ^ R and: r £ R and s G R\I with rs G J =$• r G P; then 7 is P-primary iff 7 is P-quasiprimary and P C rad#7; see (07). For any a G R we put (7 : a) f l = {r G 7? : ra G 7}. We put a,ssR(R/I) = {P G spec(7?) : (7 : a)/j is P-primary for some a G P } and tassn(R/I) = {P G spec(P) : (7 : O)R = P for some a G P } ; we also put nass#(P/7) = set of all minimal members of &SSR(R/I); members of assij(P/7) and nass/i(P/7) are called associated primes and associated minimal primes of 7 in R respectively. By a multiplicative set in R we mean a subset S of R with 1 G S such that: x,y in S1 =>• xy G S1. For any multiplicative set 5 in P we put [7 : S]R — U a e s ( 7 : O)R = the isolated S1component of 7 in P . For any P G spec(P) we put [I : P]R = [I : R\ P]R = the isolated P-component of 7 in R. More generally for any P i , . . . , P„ in spec(P) with n G N + we put [7 : ( P i , . . . ,P„)]ij = [7 : r i i < i < n ( P \ P)]fl = the isolated ( P i , . . . , P„)-component of I in R. An irredundant primary decomposition of 7 is an expression of the form 7 = ni
(CONVENTION: ni
where n G N and Qi is a primary ideal in R for 1 < i < n

(t)

such that n j e { i n}\{i}Qj *t- Qi f° r a n i G { 1 , . . . , n } , and upon letting radijQi = Pi we have Pj ^ Pj for alH ^ j in { 1 , . . . , n } .

A (not necessarily irredundant) primary decomposition of 7 is an expression of the form I = ni
(tt)

(CONVENTION: f)i
where n G N and Qi is a primary ideal in R for 1 < i < n and radftQj = Pj for 1 < % < n.

Given (ft), first by (2*) of (08) we arrange the fourth line of (f) to hold, and

§6: PRIMARY

DECOMPOSITION

135

then by discarding some of the Qi we arrange the third line of (f) to hold. Thus from a primary decomposition we extract an irredundant primary decomposition. Therefore by (6*) and (7#) of (015) we get the first sentence of the following Decomposition Theorem (T5) concerning the existence of (|); by (12*) and (13*) of (020) and (35*) of (026) we get the second sentence of (T5) and also Decomposition Theorem (T8). For primes occurring in (f), by (8.1*) to (8.3*) of (016) we get the following Uniqueness Theorem (T6), and for the primaries occurring in (f), by (10*) and (11*) of (019) we get the following Partial Uniqueness Theorem (T7). PRIMARY DECOMPOSITION THEOREM FOR IDEALS (T5). If R is noetherian then every ideal I in R has an irredundant primary decomposition. Without assuming R to be noetherian, if for an ideal I in R we have that (f f f)

vspecRl is a finite subset of mspec(P)

then I has an irredundant primary decomposition (f) for which Qin---nQn

=

Qi...Qn-

PRIME UNIQUENESS THEOREM FOR IDEALS (T6). In decomposition (f), nvspecft/ = na.ssp(R/I) and &SSR(R/I) = { P i , . . . , P n } . If R is noetherian then in decomposition (f), a.ssR(R/I) = tassR(R/I). If a nonunit ideal I in R satisfies (f f f) then in its decomposition (f), nvspecRI = nass#(P/7) = assR(R/I). PRIMARY UNIQUENESS THEOREM FOR IDEALS (T7). In (ft) (and hence in (f)) let 1 < i(l) < i(2) < • • • < i(h) < n be any sequence of integers with h G N+ such that: j G { l , . . . , n } with Pj C P ^ ) for some I G {l,...,h} =$• j G {i(l),...,i(h)}; then we have [I : ( P i ( i ) , . . . ,Pi(h))]R = ^i S 0 Pm = 0; then we have [/ : S]R = C\i
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§6.1: P R I M A R Y D E C O M P O S I T I O N FOR MODULES Turning to a submodule U of an P-module V, the radical of U in V is defined by radyt/ = {r £ R: reV C U for some e 6 N + } . C/ is primary means U ^ V and: r € P and s £ V \U with rs € £/ => r £ r&dyU. If U is primary then radyt/ G spec(P) and we say that (7 is (rady[/)-primary. We say that U is P-quasiprimary if P G spec(P) with U ^V and: r € P and s£V\U with rs G U => r G P ; then U is P-primary iff U is P-quasiprimary and P C radyC/; see (013). For any a G P we put ([/ : a)v = {v G V : at> G U}. For any a G V we put ([/ : a)R = {r G R : ra G C/}. We put &ssR(V/U) = {P G spec(P) : ([/ : O)R is P-primary for some a £ V} and tassfi(V^/C/) = { ? £ spec(P) : (U : O,)R = P for some a G V}; we also put n a s s ^ V / t / ) = set of all minimal members of &SSR(V/U); members of &ssn(V/U) and nassfl(V/{7) are called associated primes and associated minimal primes of U in V respectively. Note that ann^(V r /f/) = {r £ R : rV C U}. For any multiplicative set S in R we put [U : S]v = Uaes{U : a)v = the isolated 5-component of U in V. For any P G spec(P) we put [U : P]y = [U : P \ P]v = the isolated P-component of U in V. More generally for any P i , . . . , P„ in spec(P) with n £ N+ we put [U : ( P i , . . . , P„)]v = [U : C\i
(to

(CONVENTION: ni
where n £ N and Qi is a primary submodule of V for 1 < i < n such that r\j£{x

n}\{i}Qj

€• Qi f° r aH « £ {1, • • • , n } , and

upon letting x&dyQi — Pi we have Pj ^ P, for alH ^ j in { 1 , . . . , n}. A (not necessarily irredundant) primary decomposition of U is an expression of the form U = D^KnQi

(ttO

(CONVENTION: n ^ o Qi = V)

where n £ N and Qi is a primary submodule of V for 1 < i < n and radyQj = Pj for 1 < i < n.

Given (ttO> first by (2'*) of (013) we arrange the fourth line of ( f ) to hold, and then by discarding some of the Qi we arrange the third line of ( f ) to hold. Thus from a primary decomposition we extract an irredundant primary decomposition. Therefore by (6*) and (7*) of (015) we get the first sentence of the following Decomposition Theorem (T5') concerning the existence of {]'); by (12*) and (14*) of (020) we get its second sentence and also the Decomposition Theorem (T8'). For primes occurring in ( f ) , by (8.1*) to (8.3*) of (016) we get the following Uniqueness Theorem (T6'), and for the primaries occurring in (f), by (10*) and (11*) of (019) we get the following Partial Uniqueness Theorem (T7')-

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LOCALIZATION

137

PRIMARY DECOMPOSITION THEOREM FOR MODULES (T5')- If V is noetherian (i.e., every submodule of V is finitely generated) then every submodule U of V has an irredundant primary decomposition. Without assuming V to be noetherian, if for a submodule U of V we have that (f f f)

vspecR(annR(V/U))

is a finite subset of mspec(i?)

then U has an irredundant primary decomposition ( f ) . PRIME UNIQUENESS THEOREM FOR MODULES (T6 7 ). In decomposition (f), nvspecR{annR{V/U)) = nassR(V/U) and assR{V/U) = { P i , . . . , P n } . If R is noetherian then in decomposition ( f ) , assR(V/U) — tassR(V/U). If a submodule U of V with U ^ V satisfies (f f f ) then in its decomposition ( f ) , nvspecR(annR(V/U)) = nassR(V/U) = assR(V/U). PRIMARY UNIQUENESS THEOREM FOR MODULES (T7'). In (ft') (and hence in (f)) let 1 < i(l) < i(2) < • • • < i(h) < n be any sequence of integers with fceN+ such that: j £ { 1 , . . . ,n} with Pj C P^i) for some I G { 1 , . . . , h} => j £ {i(l),...,i(h)}; then we have [U : ( P i ( 1 ) , . . . ,Pi(h))]v = C)i 5 n Pm = 0; then we have \U : 5]v = rii
LECTURE H: VARIETIES AND MODELS

138

recall that a multiplicative set in R is a subset S of R with 1 £ S such that: s £ S and s' £ S => ss' G 5. Still more generally we may let any multiplicative set 5 in R play the role of denominators and thereby obtain the localization Rs of R at S thus. We define i?g to be the set of all equivalence classes of pairs (u,v) £ Rx S under the equivalence relation: (u, v) ~ (u\ v') 4=> v"{uv'-u'v) = 0 for some v" £ S. The equivalence class containing (u,v) is denoted by u/v or ^, and we add and multiply the equivalence classes by the rules ^ + &• = »i«a+"a«i and ^ x ^ = H i a . Also we "send" any •*

V\

V2

V1V2

^1

V2

VlV-2

J

u £ R to u/\ G i?s- This makes i?s into a ring and u i - m / i gives the "canonical" ring homomorphism (j): R -> AsClearly 0(5) C ?7(fls)

and

ker(<£) = [0 : S]R

where we recall that for any ideal / in R we have defined

{

[/ : S]R

= {r £ R : rs G / for some s £ S} = the isolated 5-component of / in R.

The localization of R at SR(R) is called the total quotient ring of R and denoted by QR(i£). Clearly [0 : SR(R)]R = 0 and hence the canonical map of R into QR(i?) is injective. By "identifying" every u £ R with u / 1 G QR(ii), we may and we shall regard QR(.R) to be an overring of R. Note that then SR(R) C U(QR(R)) and every element of QR(i?) is of the form u/v with u £ R and v G SR(R); also note that QR(i?) is its own total quotient ring. Conversely, if T is an overring of R with SR(R) C U{T) then clearly there is a unique i?-injection of QR(i?) into T whose image is the set of all u/v with u £ R and v £ SR(R); we may call this image the total quotient ring of R in T and again denote it by QR(i?). Note that if R is a domain then QF(f?) = QR(.R), and in this case the said image may be called the quotient field of R in T. In particular we may talk of the quotient field of a domain in an overfield. Let us revert to the general multiplicative set S in R and note that: ker() = 0 <=> S C SR(R). If S C SR(R) then clearly there is a unique i?-injection of Rs into QR(i?); we call its image the localization of R at S in QR(i?) and continue to denote it by Rs; let us call this the GOOD case; thus in the good case Rs is an overring of R. In the general case the passage from R to Rs can be achieved in two steps. First we take the residue class map a : R —» R/[0 : S]R. Now a(S) is a multiplicative set in a(R) with a(S) C 5 Q ( fl )(a(i?)) and we take a(R) c Q = a{R)a(s) a s m the good case, and we consider the map 7 = /3a : R —> Q where 0 : a(R) —» Q is the natural injection. Now we may identify Rs with Q via the unique isomorphism ip : Rs —> Q such that ip4> = 7.

§7: LOCALIZATION

139

The isomorphism ip is generalized in the following Characterization Theorem (T9) for the localization Rs- In Theorems (T10) to (T12) we give some basic properties of localization. CHARACTERIZATION THEOREM FOR LOCALIZATION (T9). Let 7 : R —> Q be a ring homomorphism. Then: there is an isomorphism ip : Rs —> Q with ipcfr = 7 <4> ker(7) = [0 : S]R with j(S) C U(Q) and every element in Q can be written as 7(u)/7(v) with u £ R and v £ S. Moreover, if 7(5) C U(Q) then there is a unique homomorphism ip : Rs —• Q with %l>4> = 7. PROOF. Straightforward. TRANSITIVITY OF LOCALIZATION (T10). Let 7 : R -» Rs> and <j>' : Rs ^ (Rs)t(S') D e the canonical maps where 5 ' is a multiplicative set in R with 5 C S". Then there is a unique isomorphism ip : (Rs)ct>(S') ~* Rs1 with V>' is an isomorphism. PROOF. Follows from (T9). PERMUTABILITY OF LOCALIZATION AND SURJECTION ( T i l ) . Let S : R —> R' be a ring epimorphism with ker(<$) D S = 0, and consider the map 7 = 'S : -R —+ ^ ( ^ i f s ) where $' : 5(R) —> S(R)s(s) 1S the canonical homomorphism. Then there exists a unique epimorphism ip : .Rs —> S(R)s^s) with ^i<^ = 7. PROOF. Follows from (T9). IDEAL CORRESPONDENCE FOR LOCALIZATION (T12). For any ideal I in R let 7s = (s) : (t,s) £ I x S}. Moreover:

is^Rs^ ms = ®. (b) For any ideal I in R we have I[s] = ~1(Is)(c) An ideal I in i? is a contracted ideal (i.e., for some ideal J in Rs we have I = 0 - i (j)) i f f / e C . (d) Every ideal J in Rs is an extended ideal (i.e., for some ideal I in R we have J = 0(J)fl s ). (e) I *—> Is gives a bijection of C onto E, and its inverse is given by J 1—> 0 _ 1 ( J ) . The sets C and £" are closed with respect to the ideal theoretic operations of radicals, intersections, and quotients, i.e., for instance: I £ C => TSARI £ C; recall that quotients = parenthetical colons, where for ideals I,H in R we have

LECTURE L4: VARIETIES AND MODELS

140

(I : H)R = {r £ R : rh £ / for all h £ H}. Moreover, the said bijection commutes with these operations, i.e., for instance: I G C =$• r a d # s i s = (radR-T)s. (f) If P and Q are ideals in R such that P is prime and Q is P-primary with Q n S = 0 then: P D S = 0, P and Q belong to C and both contain ker(^), Ps is prime, and Qg is (Pg)-primary. (g) A primary ideal, and hence in particular a prime ideal, in R is a contracted ideal iff it belongs to C iff it is disjoint from S. Moreover, P i-> Ps gives a bijection of C n s p e c ( P ) onto spec(P,s). Likewise, given any P € Cnspec(ii), Q H-> QS gives a bijection of all P-primaries in R onto the set of all (Ps)-primaries in Rs, and this bijection preserves intersections and quotients. (h) Given an irredundant primary decomposition / = Di S fl P m = 0. Then Is = <^i<j) = the intersection of all primary ideals in R which are disjoint from S, and also ker(^) — the intersection of those primary ideals which occur in an irredundant primary decomposition of 0 in R and which are disjoint from S. PROOF OF (a). By the definition of Rs, or by (T9), every element of Rs can be written as (j)(r)/<j>(s) with (r,s) G fix S. Therefore, since Is = (j>(I)Rs, every element of Is can be written as a finite sum Ylict>{ti)(i>{ri)l{si)- Now this sum equals (t)/(j)(s) :(t,s) £ I x S}. It follows that

1G Is » 1 = 4>{t)I'4>(s) for some (t,s) £ I x S < •£> (t - s) — 0 for some (t,s) £ I x S <=> s'(t - s) = 0 for some (t, s,s') £ I x S x S

<^/nS^0. V

Consequently: Is ^ Rs & I n 5 = 0.

because ker() = [0 : S]R

§7:

141

LOCALIZATION

PROOF OF (b). For any r G R we have

•& 4>(r) G i s <=> (r) = (t)/(s) for some (t,s) £ I x S

by (a)

O (rs - t) = 0 for some (t, s) e I x S <=> s'(rs - t) = 0 for some (£, s,s')e . ^ G PROOF

I xS xS

by (T9)

HI-

OF (c). If I G C then, upon letting J = 7 S , by (b) we see that I is

contracted. Conversely suppose I = (j)~l{J) for an ideal J in Rs- Then Is C J and hence 0 _ 1 ( / s ) C <j>~l{J) = 7. Therefore by (b) we get /[$] c / and hence J[s] = P PROOF OF (d). Given an ideal J in Rs let 7" = _1(J). By (T9), any u G J can be expressed as u = (f)(r) / (f>(s) with (r, s) £ Rx S, and multiplying this by 0(s) we get (^(r) G J. Consequently r £ / and hence u G i s - Thus J C Is and obviously i s C J. Therefore J = Is. PROOF OF (e). In view of (c) and (d), this follows from the following Lemma (T13.3). PROOF OF (f). Let P and Q be ideals in R such that P is prime and Q is P-primary with Q n S = 0. Then s G P n S = * > s e G Q n S f o r some e G N + which contradicts the assumption that Q D 5 = 0. Consequently P n S = 0, and hence P and Q belong to C, and therefore by (c) they are contracted ideals and hence they contain ker((f>). Now by (a) and (e) we see that Ps C r&dRsQs ^ Rs and hence, in view of PIC = (07), for proving that Ps is prime and Qs is (Ps)—primary, it suffices to show that: x G Rs and x' G Rs\ Qs w r t n xx' G Qs =$• x G P s - Given a;, a;' as postulated, by (a) we can write x = {s),x' = (s'),xx' = <j>(r")/4>(s") with r, r', r" in R and s, s', s" in S such that r' $ Q and r" G Q- Equating two values of xx' we obtain (j>(rr')/(j>(ss') = (r)/<j>(s) G P s . PROOF OF (g). In view of (c), (e) and (f), this follows from the following Lemmas (T13.1) and (T13.2) together with the additional observation that, say by (08), if Q, Q are P-primaries then so are QnQ and (Q : Q)R, except that: if Q C Q then (Q :Q)R = R and Qs C Qs with (Qs : QS)RS = Rs = RePROOF OF (h). Let 7 = ^i<j
142

LECTURE L4: VARIETIES AND MODELS

the second equation is an irredundant primary decomposition in Rs and we have 1 = Is- Also clearly: I = 7 <^> S n Pi = 0 for 1 < i < n. By (T7) we have I = I[s]. It only remains to show that Is = J- By the following Lemma (T13.5) we have Is C J. Conversely, by (a), any x £ J can be expressed as x = {t)/<j>(s) with (t, s) e 7 x S. We can take s' = Ilie{i n}\{*(i) i(h)} si w i t h s'i e s ' n Q/ s't G J and hence by (a) we get x = <j>(ts')/(j>(ss') G Is-

and

then

PROOF OF (i). Follows from (d). PROOF OF (j). Assuming R to be noetherian, by (T9) we know that ker(^) = Os and, in view of (T5), by taking I — 0 in (h) we see that ker(0) = the intersection of those primary ideals which occur in an irredundant primary decomposition of 0 in R and which are disjoint from S, and hence by (g) we see that ker(^) = the intersection of all primary ideals in R which are disjoint from S. LEMMA ON CONTRACTED AND EXTENDED IDEALS (T13). Let 0 : P -> T be a ring homomorphism. For any ideal I in R let Ie — <j>(I)T, and for any ideal J in T let J c = 0 _ 1 ( J ) ; note that if 9 is the natural injection of R into an overring T then Ie = IT and Jc = J n R. Let C be the set of all contracted ideals in R (relative to 9), i.e., ideals which can be expressed as J c with ideal J in T. Let E be the set of all extended ideals in T (relative to 9), i.e., ideals which can be expressed as Ie with ideal I in R. Then we have the following. (T13.1) For any ideals I C 7 in R we have Ie cT. For any ideals J C J in T we have Jc C J°. For any ideal / in R we have / c (Ie)c and Ie = ((/ e ) c ) e . For any ideal J in T we have ( J c ) e C J and J c = ((J c ) e ) c . (T13.2) If P, Q are ideals in T such that P is prime and Q is P-primary, then Pc is prime and Qc is (P c )-primary. If J = C\i Ie gives a bijection of C onto E, and its inverse is given by J >—> J c . The set C is closed under the ideal theoretic operations of radicals, intersections, and quotients. Moreover, if the set E is also closed under one of these three operations then the said bijection commutes with it. Finally the set E is always closed under the ideal theoretic operations of sums and finite products. PROOF. The proofs of (T13.1) and (T13.2) being straightforward, we shall prove (T13.3). Clearly I € C => I = {Ie)c- Also clearly J € E => J = ( J c ) e . Therefore / — i » Ie gives a bijection C —> E whose inverse is given by J — i > Jc. For any ideals / and J in R and T respectively, we clearly have (T13.4)

(rad f l /) e C rad T (7 e )

and

(rad T J) c = rad f l (J c ).

From the second part of the above display it follows that C is closed under the op-

143

§7: LOCALIZATION

eration of radicals, and if E is also closed under it then the said bijection commutes with it. For any families of ideals (Ii)ieL and (Ji)ieL in R and T respectively, we clearly have

(T13.5)

( r w , - ) ' c nj€LJf

(n,eLJj)c = nleLjf

and

and (T13.6)

(ZieLlir

= TieLIf

and

£ i e L J? C ( £ ; € L J ; ) C

and if the indexing set L is finite then we also have (T13.7)

O W 0

e

= r W f

and

UieL ^ ^ (UieL Ji)°

From the second part of (T13.5) it follows that C is closed under the operation of intersections, and if E is also closed under it then the said bijection commutes with it. From the first parts of (T13.6) and (T13.7) it follows that E is closed under the operation of sums and finte products respectively. To turn to quotients let I, I be any ideals in R and let J, J be any ideals in T. Then clearly (J : 7)% C (Ie : T)T

(T13.8)

and

C (Jc :

(J : lfT

T)R.

We claim that J£E^-(J:J)CT

(T13.9)

= (JC:

T)R.

Namely, assuming J e E, we have (Jc : T)R1

= (Jc : T)R{Ty

= ((J c : T)R{T)f

C (Jc)e

(where the first equation is because of the assumption J € E and the second equation is because of the first part of (T13.7)) and hence we get

(Jc:jyRC(J:J)T and therefore we have ((J c : T)R)C

C ((J :

J)T)C

and hence, because obviously ( J c : J )« C ((J c : J )eR)c, we get ( J c : J C ) f l C ((J : J)T)C and therefore, because of the second part of (T13.8), we conclude that ( J : J)CT = (Jc :

T)R.

144

LECTURE L4: VARIETIES AND MODELS

This proves (T13.9). By taking J = Ie and J = f in (T13.9) we see that (T13.10)

( / : 7

/eCand7GC^{

2f

G C a n d

_

By (T13.10) we conclude that C is closed under quotients, and if E is also closed under it then the said bijection commutes with it. §7.1: LOCALIZATION AT A P R I M E IDEAL Given any prime ideal P in R we get a particularly interesting case of localization by taking S = R\P. We put Rp =

RR\P

and call this the localization of R at P. Let $ : R^

RP

be the canonical map and note that now $(R \P)C

U(RP)

and

ker($) = [0 : P]R

where, for any ideal I in R, by definition [I:P]R

= [I:R\P]R = {r € R : rs € I for some s £ R\P} = the isolated P-component of I in R.

Note that if ZR(R) C P, where ZR(R) is the set of all zerodivisors in R, then we are in the good case, and Rp may be regarded as a subring of QR(-R). The following Theorems (T14) to (T17) follow by taking S = R\P in the above Theorems (T9) to (T12) respectively. CHARACTERIZATION THEOREM FOR PRIME LOCALIZATION (T14). Let 7 : R —> Q be a ring homomorphism. Then: there is an isomorphism ip : RP -> Q with ip$ = 7 <S> ker(7) = [0 : P}R with >y(R \ P) C U(Q) and every element in Q can be written as 7(u)/7(i>) with u G R and v £ R\P. Moreover, if j(R \ P) C C/(Q) then there is a unique homomorphism ip : Rp —> Q with •0$ = 7. TRANSITIVITY OF PRIME LOCALIZATION (T15). Let 7 : R -> i? P / and $ ' : JRP —> (Rp)^(p') be the canonical maps where P ' is a prime ideal in R with P ' C P . Then there is a unique isomorphism tp : (Rp)<s>{p>) -* Rp> with ^>$'$ = 7.

§7.1: LOCALIZATION AT A PRIME IDEAL

145

PERMUTABILITY OF PRIME LOCALIZATION AND SURJECTION (T16). Let 5 : R —• R' be a ring epimorphism with ker(5) c P , and consider the map 7 = $'<5 : Pi —> 5(P)<5(P) where $ ' : <5(P) —> (5(P)<s(p) is the canonical homomorphism. Then there exists a unique epimorphism tji : Rp —> £(P),5(p) with ?/>$ = 7. IDEAL CORRESPONDENCE FOR PRIME LOCALIZATION (T17). For any ideal 7 in R let Ip = 3>(I)Rp, and note that the two meanings of Rp do coincide. For any ideal 7 in P let 7jp] = [I : P]p, and let C be the set of all ideals 7 in R for which 7[pj = 7. Finally let E be the set of all ideals in Rp. Then we have the following. (a) For any ideal 7 in R we have IP = ($(*)/$(s) : (t,s) € I x (R\ P)}. Moreover: IP ^ RP <=> 7 C P . (b) For any ideal 7 in R we have 7[pj = $ _ 1 ( 7 p ) . (c) An ideal 7 in R is a contracted ideal (i.e., for some ideal J in Rp we have 7 = $ - i ( J ) ) iff J e C . (d) Every ideal J in P p is an extended ideal (i.e., for some ideal 7 in R we have J = $(/)PP). (e) 7 (—• Zp gives a bijection of C onto 75, and its inverse is given by J H-» $ _ 1 ( J ) . The sets C and £ are closed with respect to the ideal theoretic operations of radicals, intersections, and quotients, i.e., for instance: I € C => radp,/ S C; recall that quotients = parenthetical colons, where for ideals I, H in R we have (I : H)R = {r 6 R : r/i € 7 for all h G 77}. Moreover, the said bijection commutes with these operations, i.e., for instance: 7 G C =>• radp p 7p = (rad#7)p. (f) If P and Q are ideals in R such that P is prime and Q is P-primary with Q C P then: P C P , P and <5 belong to C and both contain ker($), P F is prime, and QP is (Pp)-primary. (g) A primary ideal, and hence in particular a prime ideal, in R is a contracted ideal iff it belongs to C iff it is contained in P . Moreover, P 1—> Pp gives a bijection of Cflspec(P) onto spec(Pp). Likewise, given any P e Cflspec(P), Q y-> QP gives a bijection of all P-primaries in R onto the set of all (Pp)-primaries in P p , and this bijection preserves intersections and quotients. (h) Given an irredundant primary decomposition 7 = Di P m C P . Then 7p = r\i<j
146

LECTURE H: VARIETIES AND MODELS

Some obvious consequences of the above Theorem (T17) are worth repeating as: LOCAL RING CONSTRUCTION THEOREM (T18). RP is a quasilocal ring, i.e., a ring with a unique maximal ideal M(Rp). We have M(RP)

= (P)RP

with

^(MiRp))

= P.

The mapping P i-> $(P)RP gives a bijection of {P G spec(i?) : P C P} onto spec(i?p) whose inverse is given by $ _ 1 , and hence in particular, referring to (022) for the definitions of dimension and height, we have dim(i?p) = h t ^ P . If R is noetherian then Rp is a local ring, i.e., a noetherian quasilocal ring. If ZR(0) C P (resp: if R is a domain) then regarding RP to be a subring of QR(i?) (resp: of QF(i?)), for any ideals I and J in R and Rp we have Q(I)RP

= IRP

and

$-*( J) = J n R.

§8: A F F I N E VARIETIES Postponing the discussion of varieties in projective space to a later opportunity, we shall now discuss varieties in affine space. Let k c

K

CA

be fields and let iV be a nonnegative integer. Consider the affine iV-space A^ = KN = {a = (ai,...,

aN) : cci € K for 1 < i < N}

over K, and the N-variable polynomial ring P>N,k = k[Xi,.

over k. For any J c

BJV,A

..,XN]

let

V K (J) = { Q £ A f : f(a) = 0 for all / G J } where for / = f(X\,..., Xfj) G BN,\ and a = (a\,..., ajv) G A ^ we are writing / ( a ) = f(oti,..., ajv). We call this the affine variety defined by J. For polynomials / , g,... in BN<\ we may write YK(f, g,...) in place of V K ({/, g,...}) and we may informally call this the affine variety / = g = • • • = 0. By a variety in A ^ defined over k we mean a subset of A ^ which can be expressed in the form VK(J) for some J C R>N,k- We put avtfc(A^) = the set of all varieties in A ^ defined over k. Members of avtfc(A^) may be called varieties in Aj^. For any U C A ^ we put MU) = {/ G BN:k : f(a) = 0 for all

aeU}

§8: AFFINE VARIETIES

147

and we note that this is an ideal in Bjv,fc; we call it the ideal of U in B^tk- For points a,/3,... in A ^ we may write lk(a,P,...) in place of lk{{a,(3,...}). Given U,U' in avtfe(A^), U' is a subvariety of U means U' C U, and U' is a proper subvariety of U means U' C U with U' ^ U. Given U in avtfc(A^), [/ is reducible means ?/ is the union of two proper subvarieties; U is irreducible means it is nonempty and nonreducible. We put iavtfc(A^) = the set of all irreducible members of avtfc(A^). Referring to (022) for the definition of the dimension dim of a ring, for any U in avtjfc(A^) we define the dimension of U by putting dim(U) =

dim(BN,k/lk(U)).

Given U £ avtfc(A^), for any / e BN / ( a ) and, letting a vary only over U, this gives a polynomial map U —> /t. Since two such maps on {/ coincide iff the difference of the corresponding members of B;v,fc belongs to the ideal Ifc(i7), the ring of all these K-valued polynomial maps on U may be identified with the residue class ring Bjvjfc/Ijt(C/). Indeed this is the basic motivation behind the concept of residue class rings. We denote -Bjv,fc/2fc(f/) by k[U]* and call it the affine coordinate ring of U. In case lk{U) is not the unit ideal in i?jv,/t, we may and we do identify k with a subfield of k[U]* and we note that then, according to (027), k[U]* becomes an affine ring over k, and hence in particular it is a noetherian ring. In case U € avtfc(A^) is such that !&([/) is a prime ideal in i?;v,fc, we denote the quotient field of k[U}* by k{U)*, and we call k(U)* the function field of U (over k). We denote the localization of BN,W at lk(U) by Rk(U)*, and call it the local ring of U (over A;); note that by (T18) this is indeed a local ring. We denote the residue field Rk(U)*/M(Rk(U)*) of Rk(U)* by k{U)\ and we call this field fc(t/)» the alternative function field of U (over fc); again we may and we do identify k with a subfield of k(U)K Let $[/ : BNtk ~> k[U)* and *c/ : RkiP)* -> k{U)* be the residue class epimorphisms. Now clearly there is a unique homomorphism ifru '• Rk{U)* —» k(U)* such that for all / G BN,k and g € BNtk\h(U) we have ipu(f/g) = $u(f)/$u(g)Obviously im(ipu) = k(U)* and kev(ipu) = M(Rk(U)*) and hence there is a unique isomorphism (f>u : k{U)^ —> k(U)* for which we have 4>u^u = i>u- We call ipu and <j>u the natural epimorphism and the natural isomorphism respectively. Note that A^ C A^, and the ideal Ifc(a) of any point a in A ^ is the maximal ideal in B^,k generated by Xi - a\,..., XN — OLN\ also VK(Ifc(a)) = {a}, and writing a in place of {a} we get k = k[a\* = k(a)* C Rk(ct)* = the local ring of a. The above notions are particularly relevant when k or K is algebraically closed, or more precisely, when K contains an algebraic closure of k. To take care of fields which may not be algebraically closed, we recall that analogously, for any subset J of any ring A, in (016) we have defined the spectral variety, the maximal spectral

LECTURE L4: VARIETIES AND MODELS

148

variety, and the minimal spectral variety of J in A by putting vspec A J = { P e spec(i4) : J C P) and mvspec^ J = mspec(A) D vspecAJ and nvspec^ J = the set of all minimal members of vspec^ J where spec(A) and mspec(J4) are the sets of all prime and maximal ideals in A respectively. Moreover, for any U C spec(A) we have defined the spectral ideal of U in A by putting ispec^f/ = C\pei/P and we have put rd(.A) = the set of all radical ideals in A i.e., ideals which are their own radicals. We have also put svt(j4) = the set of all spectral varieties in A i.e., the set of all subsets of spec(yl) of the form vspecAJ with J varying over the set of all subsets of A; members of svt(A) may be called varieties in spec(A). Now we put msvt(j4) = the set of all maximal spectral varieties in A i.e., the set of all subsets of mspec(yl) of the form mvspec^*/ with J varying over the set of all subsets of A; members of msvt(A) may be called varieties in mspec(j4). Given U, U' in msvt(A) or svt(A), U' is a subvariety of U means U' C U, and U' is a proper subvariety of U means U' C U with V ^ U. Given U in msvt(A) or svt(A), U is reducible means U is the union of two proper subvarieties; U is irreducible means it is nonempty and nonreducible. We put imsvt(A) = the set of all irreducible members of msvt(A) and isvt(A) = the set of all irreducible members of svt(A). For any U in msvt(A) or svt(A) we define the dimension of U by putting dim((7) = dim(A/ispec A [/). If A is an affine ring over k then for any U in msvt(A) or svt(A) we denote A/ispecA(U) by k[U]* and call it the affine coordinate ring of U. In case ispec^C/

§S: AFFINE

VARIETIES

149

is not the unit ideal in A, we may identify k with a subfleld of k[U]* and we note that then k[U]* becomes an affine ring over k, and hence it is a noetherian ring. In case ispec^C/ is a prime ideal in A, we denote the quotient field of k[U]* by k(U)*, and we call k(U)* the function field of U (over k). We denote the localization of A at ispec^J/ by Rk(U)* and call it the local ring of U (over k); we note that by (T18) this is indeed a local ring. We denote the residue field Rk(U)*/M(Rk(U)*) by &(£/)* and we call it the alternative function field of U (over k); we may and we do identify k with a subfleld of k{U)K Let $ y : A -> k[U\* and *c/ : Rk{U)* -> fc(C/)" be the residue class epimorphisms. Clearly there is a unique homomorphism V>u : Rk{U)* —> &(£/•)* such that for all / S A and 5 € A \ ispec^t/ we have < M / / 3 ) = $ u ( / ) / $ u ( 0 ) . Obviously im(V>c/) = fc(E/)* and ker(V^) = M(Rk(U)*) and hence it follows that there is a unique isomorphism [/ : &([/)* —> k(U)* for which we have (pu^u = i>u- We call I/JU and <j>u the natural epimorphism and the natural isomorphism respectively. As GEOMETRIC MOTIVATION for the definitions, consider the variety / = 0 in A ^ where / is a nonconstant polynomial in Bx,k- For N = 1 this is a point-set (= a finite set of points) on the line, for N = 2 it is a plane curve, for N = 3 it is a surface, for N = 4 it is a solid, and in general it is a hypersurface, i.e., something defined by a single equation. For instance, by taking / = Xf + • • • + Xfj — 1 we get a point-pair, a circle, a sphere, a solid ball, and so on. Intuitively, a point-set is zero-dimensional, a curve is one-dimensional, a surface is two-dimensional, a solid is three-dimensional, ..., a hypersurface is (N — l)-dimensional. The hypersurface is irreducible if the polynomial / is irreducible. This gives rise to the INTUITIVE DEFINITION of an r-dimensional irreducible algebraic variety U in the affine AT-space A ^ as a geometric object which can be parametrized by an irreducible hypersurface in (r + l)-space. Here by a geometric object we mean something defined by a bunch of polynomial equations. Let P be the ideal generated by these polynomials in Bjv.fc- The irreducibility of U suggests that we require P to be a prime ideal. The dimensionality r of U suggests that we require it to equal trdegfcfc(t/)* where trdegfc denotes transcendence degree over k and k{U)* is the quotient field QF(k[U}*) of k[U}* = BNik/P. Referring to (022) for the definitions of the depth dpt and the height ht of an ideal, eventually we shall prove the following [cf. L5§5(Q10)(T47)]: DIMENSION THEOREM (T19). For any prime ideals P C P' in B = BN: B —• B/P be the residue class epimorphism, we have trdeg f c QF(5/F) = dim(B/P)

= dptBP

= N-

htBP

and trdeg fc QF(B/P) = htHB)4>{P') + dptHB)
150

LECTURE L4: VARIETIES AND MODELS

Eventually we shall also prove the following [cf. L5§5(Q32)(T144)]: PRIMITIVE ELEMENT THEOREM (T20). If k is algebraically closed then for any prime ideal P in B = B^,k: upon letting : B —> B/P be the residue class epimorphism, and after changing the variables by a linear transformation, it can be arranged that (Xi),... ,<j>(Xr) is a transcendence basis of QF(B/P) over k and either N = r or <j>(Xr+\) is a primitive element of QF(B/P) over QF((t>(Br(Brtk))((Xr+i)). [In the latter case we can find an irreducible polynomial F(Xi,... ,Xr+i) in P n 2?r+i,fc and then the irreducible variety U in A^ defined by P is "parametrized" by the irreducible hypersurface F = 0 in A£ +1 ; in the former case U is "parametrized" by A£]. Now (T20) provides the rationale behind calling the transcendence degree as dimension. In turn (T19) connects it to the ring-theoretic definition of dimension which we have chosen for U in avt or msvt or svt. At any rate, as suggested by the above GEOMETRIC MOTIVATION, in avt or msvt or svt, we call U a point-set or a curve or a surface or a solid according as dim(f/) = 0 or 1 or 2 or 3 respectively. Another rationale for choosing the ring-theoretic definition of dimension can be found in the following Theorems (T21) and (T22) according to which: a curve has points on it, a surface has curves on it, a solid has surfaces on it, and so on. Most parts of the following Inclusion Relations Theorem (T21) for the operators V, I, vspec, and ispec are straightforward. We shall soon complete the proof by establishing the remaining parts [see the cf. between (T21) and (T22)]. INCLUSION RELATIONS THEOREM (T21). Recall that A is any ring. Let '(R,S,V,V',V",I)

stand for

^ (5 iV;fc ,A^,V K ,avt fc (A^),iavt fc (A^),I fc ) or (A, mspec(j4), mvspec A , msvt (A), imsvt(^l), ispec^) or (A, spec(A), vspec A , svt(A), isvt(A), ispec^) and let I' be the set of all ideals J in R such that J = I(U) for some U C S. Then we have the following. (T21.1) For any subsets J and J' of R we have: J C J' =4> V(J') C V(J), and we have: rad f i (Ji?) = radfl(J'H) =5> V(J) = V(J'). (T21.2) For any subsets U and U' of S we have: U C U' => I(U') C I(U). (T21.3) For any family of ideals (Ji)ieL in R we have V(J2leL J{) = n/ e z,V(Ji), and if the family is finite then we have V(r\i£L Ji) = V(ILeL Ji) = ^ieLV(Ji). (T21.4) For any family of subsets (Ut)ieL of S we have I(UleLUi) = DleLI(Ui). (T21.5) For any J c R we have J C I(V(J)) and: J = I(V(J)) •& J 6 / ' => J = radR(JR).

§S; AFFINE

VARIETIES

151

(T21.6) For any U C S we have U C V(I{U)) and: U = V(I(U)) &UeV. (T21.7) V" = {UeV : I(U) G spec(ii)}. (T21.8) Assuming R is noetherian (this is certainly so in case R = BN,H), every U € V can be expressed as a finite union U = Ui ispecBiV U gives a bijection svt(S;v,fc) —* rd(j9jv,fc) whose inverse is given by J t—> vspec Bjvfc J. Now this assertion is the eighth amongst the following eight equivalent versions (T22.1) to (T22.8) of the famous Hilbert Nullstellensatz which we shall soon prove [cf. (T49) and (T50) of L5§5(Q11), and the material between (T47) and (C15) of L5§5(Q10]. HILBERT NULLSTELLENSATZ (T22). (T22.1) A field which is an affine ring over k is algebraic over k. (T22.2) For any maximal ideal J in BAT,*:, the field B^^/J is algebraic over k, i.e., over the image of k under the residue class map B^,k —> B^^/J. (T22.3) For any maximal ideal J in B^,k there is a unique set of ./V generators (fi(Xi, • • •, Xi) G Bi,k)i
X?i+gi(X1,...,Xi)

=

where n, is a positive integer and gi{X\,..., fe&Xj9i{Xi,.

-.,Xi)<

Xi) G Bj,fe with n-j

for

1 < j < i.

(T22.4) If k is algebraically closed then the mapping a H-» Ij.(a) gives a bijection A-^ —> mspec^A^ifc). (T22.5) If K contains an algebraic closure of k then for any ideal J in Bjv.fc we have that Ifc(VK(J)) = ia,dBNkJ. (T22.6) If K contains an algebraic closure of k then the mapping U <-^> fk(U) gives inclusion reversing bijections avtfc(A^) -> rd(Bjv,fc)

and

iavt fc (A^) -> spec(BN>k)

whose inverses are given by J — i » VK(J). (T22.7) The mapping U t—»ispecBjv kU gives inclusion reversing bijections msvt(i?jv,fc) —> rd(Sjv.fc)

and

imsvt(BN,k)

—> spec(BNtk)

whose inverses are given by J — i > mvspec Bw J. (T22.8) The mapping U H-> ispec Bjv kU gives inclusion reversing bijections svt(BN,k)

-> rd(BN>k)

and

isvt(Bjv,fc) -» spec(BN>k)

152

LECTURE H: VARIETIES AND MODELS

whose inverses are given by J — i > vspec Bjv J. §8.1: SPECTRAL A F F I N E SPACE Henceforth we shall consider points and varieties in A% rather than in A^. As we have already noted, the ideal Ifc(oj) of any point a in A% is the maximal ideal in Bjv.fe generated by X\ - a\,..., XN — a AT. In other words, a is the intersection of the TV hyperplanes Xi - a i = 0 , . . . , XN - a^ = 0. Also clearly a ^ a' in A^ => Ifc(a) ^ lk{a'). So via Ifc we may ENLARGE A ^ into spec(BAr,fc) which we may denote by (Aj^)CT and call it the spectral affine TV-space over k. The mapping a t-> Ifc(a) gives an injection A^ —• (&%)". The image of Aj^ under this injection may be denoted by (AN)P
where ss stands for isomorphism; here it is set-theoretic isomorphism, i.e., bijection. Now (T22.3) says that a point of (Aj^)*"7 is the intersection of TV hypersurfaces; moreover, these hypersurfaces are hyperplanes, i.e., they are of degree one, iff it is a point of (A%y. §8.2: MODELIC SPEC A N D MODELIC A F F I N E SPACE To describe another way of enlarging A^, for any domain A we put %J(A) = the set of all localizations Ap with P varying over spec(A) and we call this the modelic spec of A. By (T18) we see that Ap is a quasilocal ring for which M(AP) = PAP and M(AP) n A = P with dim(AP) = h t ^ P , and if A is noetherian then Ap is a local ring R. Thus P i-> Ap gives an inclusion reversing bijection of spec(^l) onto %3(A), and the reverse bijection is given by R *-* M(R)C\A. We denote VB{BNik) by (A%)s and call it the modelic affine TV -space over k. Now P \—» (T3;v,fc)p gives an inclusion reversing bijection (A^Y —> (AN)6 whose inverse is given by R i-> M(R) f~l BNtk. The images of (Af)"" and (A%y under this map may be denoted by (A%)pS and {ANYS and called the rational modelic affine TV-space over k and the minimal modelic affine TV-space over k respectively; note that because of the inclusion reversing property, (A^)MlS is the set of all minimal members of (A^)s. Now the isomorphisms (A?)" « (A?)"

and

( A ? ) " ' « (A?)"*

and

give rise to the enlargements A ? « (A»y* c (A?)"* C (A%)s.

(A»y

« (A?)'

§S.3: SIMPLE POINTS AND REGULAR LOCAL RINGS

153

Given any U £ (A^)CT, by (T22) we see that k(U)* = k &U £ (A^)pa whereas k(U)*/k is algebraic <^> U £ (Af )^CT; consequently, points in (A^)p(7 may be called rational and points in (A^)'" 7 may be called algebraic. Similarly, for any R £ (A%)s, after identifying k with a subfield of R/M(R), we have that R/M(R) = k & R £ (A%)p6 whereas (R/M(R))/k is algebraic &• R £ (A^)""5; consequently, points p6 in (A%) may be called rational and points in (Aj^)Ml5 may be called algebraic. By (T22) we also see that if K contains an algebraic closure of k then "points" of (A^)CT are the prime ideals of "irreducible varieties in A^ defined over k," and the corresponding "points" of (Af?)6 are their local rings. §8.3: SIMPLE P O I N T S A N D R E G U L A R LOCAL RINGS The embedding dimension of a local ring R is defined by putting emdim(-R) = the smallest number of elements which generate M{R). We shall soon prove the [cf. L5§5(Q7)(T29)]: DIM-EMDIM THEOREM (T23). For any local ring R we have emdim(i?) > dim(i?) and hence in particular (since R noetherian => M(P) is finitely generated) emdim(i?) and dim(i?) are nonnegative integers.

A local ring R is said to be regular if emdim(i?) = dim(.R). Concerning regular local rings we shall soon prove the following theorem, where we recall that a domain is normal means it is integrally closed in its quotient field (see L3§7), and for any element x in any quasilocal ring R we have put OIARX = max{i G N : x £ M(RY} where the max is taken to be oo if the set of i is unbounded (see L3§11). [cf. L5§5(Q4)(T7), L5§5(Q14)(T64), L5§5(Q15)(T71), L5§5(Q17)(T81.7)]. ORD VALUATION THEOREM (T24). Let R be any local ring. Then: (T24.1) For all 0 ^ x £ R we have ord R :r £ N, i.e., n i e N M ( i ? ) i = 0. (T24.2) If R is regular then for all nonzero elements x and y in R we have ordR(xy) = ordure + ord^y; consequently R is a normal domain and we get a valuation ord# : QF(R) —> Z by putting ordR(x/y) = ordfii - ordRy. [We shall continue to use this extended meaning of ord#]. (T24.3) For any x £ M(R) \ ZR(R) we have that: (i) R/{xR) is a local ring whose dimension is one less than the dimension of R, (ii) if R/(xR) is regular then ordfla; = 1, and (iii) if ord/ja: = 1 and R is regular then R/(xR) is regular.

154

LECTURE H: VARIETIES AND MODELS

[Note that if R is regular, or more generally if R is a domain, then the condition x e M(R) \ ZR(R) is equivalent to the condition 0 ^ x e M(R)]. (T24.4) If R is regular then its localization Rp at any prime ideal P in it is regular. Theorems (T19) and (T22.3) tell us that, the localization of B^,k at any U € (A^)MCT is an iV-dimensional regular local domain, i.e., equivalently, every R S (AjV)M<5 -1S a n jV-dimensional regular local domain. Geometrically this says that every point of (Aj^)^ is simple, and so is every point of (A^)M<S. By (T24.4) it follows that the localization of Bjv.fe at any U € (A^)CT is a regular local domain, i.e., equivalently, every R £ (A^)s is a regular local domain. EXAMPLE (Xll). Consider a hypersurface S : F = 0 in A ^ with F = F(Xi,... ,XN) £ Bjv,fc\fc and let a = ( a i , . . . , a ^ ) be a point of S, i.e., o n , . . . ,ajv are elements in k with F{a\,..., OJ AT) — 0. Expanding F around a we get FiX,,. ..,XN)

= J2 Gn-iN (*i - "I)* 1 •••{XN-

aN)iN

with Gi1...iN € /c. Let i' be the smallest value of i\ -\ HTV for which Gix,.,iN ^ 0. In L3§3 we have put ord Q F = u, called ^ the multiplicity mult Q S' of S at a, called a a i/-fold point of S, and said that a is a simple or singular point of S according as v = 1 or v > 1. In terms of (formal) partial derivatives we have dF v > 1 (i.e., a is a singular point of 5) <=> — -

= 0 for 1 < i < N.

In the present notation ovdR(ayF = v, and by (T24) the local ring is regular iff v — 1, i.e., iff a is a simple point of S.

R(a)*/(FR(a)*)

§9: MODELS As described in L3§4, we can construct the projective A^-space P ^ over a field k by patching up N +1 copies of the afflne A'-space A ^ over k. Similarly, by patching up several modelic specs we can construct a model. More precisely, we introduce the following definitions. Let A be a subring of a field K. Referring to L2§3 for the definition of a valuation v of K and the definition of the valuation ring Rv of v, recall that, according to L3§7 and L3§11, Rv is a quasilocal domain with quotient field K, and by a valuation ring we mean the valuation ring of some valuation of its quotient field. By a valuation of K/A (K over ^4) we mean a valuation v of K such that A C Rv. By a valuation ring of K we mean a valuation ring with quotient field K, and by a valuation ring of K/A we mean a valuation ring of K which contains A. We put JK(A') = the set of all valuation rings of K

§9: MODELS

155

and %\(K/A) = the set of all valuation rings of K/A and we call these the Riemann-Zariski space of K and the Riemann-Zariski space of K/A respectively. We also put 9\'(K) — the set of all quasilocal domains with quotient field K and OK'(K/A) = the set of all members of D\'(K) which contain A and we call these the quasitotal Riemann-Zariski space of K and the quasitotal Riemann-Zariski space of K/A respectively. Finally we put 9\"(K) = the set of all quasilocal domains which are subrings of K and m"(K/A)

= the set of all members of
and we call these the total Riemann-Zariski space of K and the total RiemannZariski space of K/A respectively. A quasilocal ring R is dominated by a quasilocal ring S means R is a subring of S and M(R) c M(S) or equivalently M(R) = RC\ M(S); alternatively we may say that S dominates R; we may indicate this by writing R <S

or

S > R.

This converts Vt(K),iR(K/A),W(K),W(K/A),W'(K),W'(K/A) into posets (= partially ordered sets). A valuation v dominates a quasilocal ring R means Rv dominates R. We shall soon prove the following Theorems (T25) to (T28) about valuations; [cf. §12(R7)]. VALUATION MAXIMALITY THEOREM (T25). V\(K) (resp: V\(K/A)) is the set of all maximal members of W(K) (resp: W(K/A)), i.e., R in W(K) (resp:
156

LECTURE L4: VARIETIES AND MODELS

VALUATION CHARACTERIZATION THEOREM (T28). A domain R is a valuation ring iff for all x, y in R we have either x £ yR or y £ xR. Equivalently, a domain R with quotient field K is a valuation ring of K iff for every x £ Kx we have either x £ R or 1/x £ R. To proceed with the definition of models, by a premodel of K (resp: K/A) we mean a nonempty subset E of W(K) (resp: ^ ' ( i f / A ) ) . The premodel E is irredundant means any member of 91{K) (resp: Di(K/A)) dominates at most one member of E. Note that a quasilocal domain S can dominate at most one member R of an irredundant premodel E, and if R exists then we call it the center of S on E. To see the uniqueness of R, let S dominate another member R' of E; identifying K with a subfield of the quotient field L of S, by (T26) we can find a valuation ring W of L dominating 5 and then by (T27) W n K is a valuation ring of K which dominates R as well as R' and hence R = R'. By a semimodel (resp: model) of K/A we mean an irredundant premodel E of K/A which can be expressed as a union E = UI E). A set E' of quasilocal rings dominates a set E of quasilocal rings (or E is dominated by E') means every member of E' dominates E\ we may indicate this by writing E < E' (or E' > E). A set E' of quasilocal rings properly dominates a set E of quasilocal rings (or E is properly dominated by E') means E' dominates E and every member of E is dominated by some member of E'. A semimodel or model of K/A is complete means it is dominated by 9t(K/A). Note that if K is the quotient field of A then V(A) is a complete model of K/A. Also note that if E is a semimodel (resp: complete semimodel) of K/A then E dominates (resp: properly dominates) 5J(A). A model E (of K/A) is said to be normal (resp: noetherian) if every R £ E is normal (resp: noetherian). A model E is said to be nonsingular if every R £ E is a regular local ring. Note that by (T24), every nonsingular model is a normal noetherian model. By the dimension dim(S) of a model E we mean max{dim(i?) : R £ E}; note that then

dim(E) <ENU{oo}.

§£U: MODELIC PRO J AND MODELIC PROJECTIVE SPACE

157

§9.1: MODELIC PROJ A N D MODELIC P R O J E C T I V E SPACE As we have seen in §8.1, for the iV-variable polynomial ring B;v,fc over a field k, the modelic spec ^J(Bj^tk) is bijective with spec(B/ftk) (i.e., there is a bijection between the two), and hence if k is algebraically closed then its portion consisting of AT-dimensional members is bijective with the affine space Aj^. We proceed to show that similarly a certain portion of a certain model is bijective with the projective space Pj£\ So for any family (xi)i^A of elements in K, with Xj ^ 0 for some j G A, we put W(A; (xt)leA)

=

(J

V(A[(xi/xj)i^])

jgA with xj#0

and we call this the modelic proj of (xi)ieA over A. Here A[(xi/xj)ieA] denotes the smallest subring of K which contains A and which contains XI/XJ for all / G A. If A is a finite set, say A = { 1 , . . . , n}, then we may write %D(A; x\,..., xn) instead of W(A; (XI)IZA) and call it the modelic proj of (xi,... ,xn) over A. Clearly for all i,i' in A with Xi ^ 0 ^ xv we have QF(A[(xi/xi)ieA\) = Q,F(A[(xi/xi')iGA\) and, upon letting K' be this common quotient field, W(A; (xi)i£A) is obviously a premodel of K'/A. Moreover, for any i ' e A with Xi< ^ 0, upon letting yi = xi/x^, we see that (j//);eA is a family of elements in K' such that yj ^ 0 for some j G A and W(A; {xi)iGA) = W(A; {yi)ieh)- According to Theorem (T29), which is stated below and which we shall soon prove, W(A; {XI)I&A) is actually a semimodel of K'/A and if A is finite then it is in fact a complete model of K'/A; [cf. §12(R8)]. By a projective model of K/A we mean a premodel E of K/A such that E — W{A; xi,..., xn) for some finite number of elements x\,..., xn in an overfield of K with Xj ^ 0 for some j G { 1 , . . . , n } ; by what we have just said, E is then indeed a complete model of K/A and we have E = W(A; y\,..., yn) for a finite number of elements yi,. • • ,yn in K at least one of which is nonzero. MODELIC PROJ THEOREM (T29). Given any family (xi)ieA of elements in K, with Xj ^ 0 for some j G A, let E = W(A; (xi)ieA) and let K' = Q,F(A[(x[/xi)[e\]) where i G A with Zj ^ 0; as noted above the QF is independent of the choice of i. Also let yi = xi/x^ for any fixed i' G A with x? ^ 0. Then we have the following. (T29.1) E is a premodel of K'/A and E = W{A; (xi/x)i5A) for all 0 ^ x G K. In particular {yi)i&A is a family of elements in K' such that yj ± 0 for some j G A, and we have E — W(A; {yi)i£A). (T29.2) Given any R G E and any subring S of K such that 5 is a quasilocal ring dominating R, there exists j G A with Xj ^ 0 such that xi/xj G S for all / G A. Moreover, for any such j we have R = BQ where B = A[(XI/XJ)I£A] and Q = BnM{S).

LECTURE L4: VARIETIES AND MODELS

158

(T29.3) E is a semimodel of K'/A, and if A is finite then E is in fact a complete model of K'/A. To justify the terms modelic proj and projective model, we proceed to show that there is a natural injection of the projective space P ^ into a projective model. Referring to L3§4 for details, recall that

Pf = (fc W + 1 \{(0,...,0)})/~ where for u = (u\,... ,UJV+I) a n d u ' = (u[,... ,u'N+1) mkN+1\{(0,.. .,0)} we have: u ~ v! o ( u ' j , . . . , Mjv+i) = (CMI, • • •, CUJV+I) for some 0 ^ c € k. Thus Pj^ is the set of all equivalence classes under this equivalence relation. Let Hi be the hyperplane in Pj^ consisting of all equivalence classes /3 £ Pj^ of those u e kN+1 for which Ui = 0, and let A j ^ be the copy of Aj^ consisting of those v = (vi,..., VN+I) € kN+1 for which Vi = 1. Now f3 H-> (ui/m,..., WJV+I/MJ) gives a bijection 0* : P ^ \Hi -* Aj^, and identifying Pj^ \ i?j with Aj^ via this bijection we have

i f = u£L|1(if \H i ) = u£t1A&Now Bjv+i,fc =fc[-X^i,• • •, -X^AT+I] and we take its quotient field k(X\,..., .Xyy+i) to be .ftf; also we take k to be A. For 1 < z < N + 1, upon letting Yu = Xi/Xi (with 5^j = 1), we get a copy -Bjv,fc,i = fe[Yi,,..., Y}v+i,i] of 5jv,fc whose modelic spec V3(BN (A^)" 5 be the natural injection whose image is (Aj^)'*5. For all i , j in {1,...,JV + 1} we clearly have QF(.Bjv,fc,i) = QF(BNtk,j) a n d we let K' stand for this common subfield of K. Then 2U(/c; Xi,..., XAT+I) is a projective model of K'/k. We denote 2B(fc; X\,..., Xj\r+i) by ( P ^ ) 5 and call it the modelic projective JV-space over k. By the definition of 2B we get

(Pf)*=

U

(A&)'.

l
We put

(Pf)^=

|J l
(A^)' 5

and ( P ^ =

|J

(A»tirS

l
and we respectively call these the rational and minimal modelic projective iV-spaces over k. By §8.2 and §8.3 it follows that (P^)* is a nonsingular iV-dimensional complete model, and for any R e (P^) 5 , after identifying A; with a subfield of R/M(R), we have that: R € (P^) p<5 <£> R/M{R) = k, and we have that: R € (^Nys ^ dim(il) = N & R is a minimal member of (PN)6 & R/M{R) is algebraic over k. We claim that there exists a unique injection 9 : P ^ —> (P^) 15 , such that for 1 < i < N + 1 we have 6(a) = 0j(a) for all a £ A ^ ; note that then the image of 9

%9.2: MODELIC BLOWUP

must be (F^)pS.

159

After proving this claim we will have

p^ «(v[*y6 c (p%rs c (P£V extending the enlargements of §8.2 from the affine case to the projective case. The uniqueness in the claim follows from the existence. To prove the existence, let (3 be the equivalence class of u — (u\,..., UJV+I) € kN+1 \ { ( 0 , . . . , 0 ) } . For any i £ { 1 , . . .,N+l} withuj ^ 0, let R = 0i(v) where v = (UI/UJ, . . . ,wjv+i/ui) € A ^ , and define 9{(3) = R. To show that 9 is well-defined, for any i' G { 1 , . . . , TV + 1} with Ui> ^ 0, let R' = 9i'(v') where v' = (u\/ui', . . . , ujv+i/ u i') G ^fcV- What we must prove is that R = R'. Since R and R' both belong to the irredundant premodel (P^)' 5 , it suffices to find a local domain 5 which dominates both R and R'; take 5 to be the localization of BN+\^ at the maximal ideal generated by X\ — u i , . . . ,XM+I - WJV+I- This defines 9 as a map from Pj^ to (P^)' 5 . To see that it is injective, let (3* be the equivalence class of u* — {u\,... ,u*N+l) € kN+1 \ { ( 0 , . . . , 0 ) } , such that 9{(3) = 9(f3*). We have to show that (3 = (3*. This is obvious if u* ^ 0, because 6i : Aj^ —> (Aj^)"5 is injective. Moreover, ut ^ 0 with 0(0) = 9i{v) = R =$• Xj/Xi e R for all j but Xj/Xi e M(fl) for exactly those j for which itj = 0; therefore, since 8(/3*) = 9(/3) = R, we must have u* ^ 0. §9.2: M O D E L I C B L O W U P The following Theorem (T30), which we shall soon prove, says that the modelic blowup, which we shall now introduce, is nothing but the coordinate free incarnation of the modelic proj; [cf. §12(R9)]. By and by we shall see that it is also the main key to the desingularization of curves and surfaces in particular and varieties in general. For any nonzero A-submodule P of K we put

W(A,P) = | J WiAlPx-1}) O^xEP

and we call this the modelic blowup of A at P. Here ^ P r r - 1 ] denotes the smallest subring of K which contains A and which contains y/x for all y £ P. MODELIC BLOWUP THEOREM (T30). Recalling that A is a subring of a field K, and P is a nonzero j4-submodule of K, we have the following. (T30.1) For any 0 ^ x 6 P we have (A[Pa;- 1 j)P = (A[Px~l])x and hence RP — Rx for every R e 5J(A[Pa; _1 ]). In particular, if P is a nonzero ideal in A then for every R e V(A[Px~1}) we have that PR is a nonzero principal ideal in R. (T30.2) Given any family (x{)i£A of generators of P, for all 0 ^ x € P we have AlPx'1} = A[(xi/x)leA], and we have W(A\ (xi)leA) C W(A,P). (T30.3) For any family (xi)ieA of generators of P we have W(A,P)

=

W(A;(xl)l&A).

160

LECTURE L4: VARIETIES AND MODELS

(T30.4) For all x ^ 0 ^ y in P we have QF(A[Pa;- 1 ]) = Q F ^ P y " 1 ] ) and letting K' denote this common QF we have that W(A, P) is a semimodel of K'/A, and if P is a finitely generated ^-module then 30(A, P) is a projective model of K'/A. In particular, if P is a finitely generated ideal in A then 2B(A, P) is a projective model of QF(A)/A. UP = Ax for some 0 ^ x E K then 2U(A,P) = 3D(A;x) = 33(A). (T30.5) If P is an ideal in A then {R £ 33(A) :PR = R} = {Re W(A, P):PR

= R}.

(T30.6) If A is quasilocal and P is an ideal in A then: P is a principal ideal in A <=> 2»(A, P ) = 33(A) « 4 e 2U(A, P ) .

§9.3: B L O W U P OF SINGULARITIES We shall now make some EXAMPLES to indicate how modelic blowup helps out in desingularizing plane curves and surfaces in 3-space. EXAMPLE (XI2). [Nodal a n d Cuspidal Cubics]. Consider the nodal cubic / = 0 with / = Y2 - X2 - X3 and the cuspidal cubic g = 0 with g = Y2 - X3 discussed in L3§5. For A take the bivariate polynomial ring k[X, Y] over a field k. For P take the ideal of the origin in the (X, y)-plane, i.e., the maximal ideal in A generated by (X,Y). Now W(A,P) = 33(A') U 33(A") where A' and A" are the bivariate polynomial rings k[X', Y'\ and k[X", Y"\ where (X' = X, Y' = Y/X) and (X" = X/Y,Y" = Y) respectively. Substituting the first set of equations in / we get / = X'2(Y'2-l-X') and hence, in the (X',y')-plane, the "total transform" of the nodal cubic consists of the "exceptional line" X' = 0 together with the "proper transform" Y'2 — 1 — X' = 0 which is a parabola meeting the exceptional line in two distinct points. Likewise, substituting the first set of equations in g we get g = X'2(Y'2 - X') and hence, in the (X', Y')-P l a n e . the "total transform" of the cuspidal cubic consists of the "exceptional line" X' = 0 together with the "proper transform" Y'2 — X' = 0 which is a parabola meeting the exceptional line at a single point where it is tangent. At any rate, in both the single blowup has resolved the singularity. You may reconfirm this by taking a similar look in the (X",y")-plane. We shall talk more about exceptional lines and proper transforms later. In the meantime you may like to glance at the relevant pictures on pages 2, 35, 131-134, and 154-155 of my Engineering Book [A04]. EXAMPLE (X13). [Higher Cusps]. Consider the plane curve ga = 0 with gs = Y2 — X2s+l where s > 1 is an integer; at the origin this has a double point which is a higher cusp, say a cusp of height s (pictured on pages 155-156 of [A04]). With notation as in the above Example (X12), substituting the first set of equations in gs we get g3 = X'2(Y'2 - x /2 <»- 1 > +1 ) and hence, in the (X', y')-pfcuie, the total

§m- EXAMPLES AND EXERCISES

161

transform of our curve consists of the exceptional line X' = 0 together with the proper transform Y'2 — ^ / 2 ( s _ 1 ) + 1 = 0 having a cusp of height s — 1. Repeating the process we get a cusp of height s — 2, a cusp of height s — 3, ..., a cusp of height 1 (like Y2 — X3 = 0), and finally a simple point. In Max Noether's picturesque terminology, we say that the plane curve gs = 0, in addition to having a double point at the origin, has one double point in its first neighborhood, one double in the second neighborhood, ..., and one double point in the (s — l)-th neighborhood, thus making a total of s — 1 double points infinitely near to the origin. So (including the origin) we have a totality of s double points. By and by we shall give precision to the idea of infinitely near points. In the meantime you may enjoy reading about them, from a somewhat heuristic viewpoint, in Lecture 19 on Infinitely Near Singularities (pages 145-158) of the Engineering Book [A04]. EXAMPLE (X14). [Circular Cones and Fermat Cones]. Consider the horizontal circular cone X2 — Y2 — Z2 = 0 (description on pages 197-198 of [A04]), or more generally the Fermat Cone / = 0 with / = Xn±Yn±Zn, where n > 1 is an integer which is nondivisible by the characteristic (discussed on page 202 of [A04]). For A take the trivariate polynomial ring k[X, Y, Z] over a field k. For P take the ideal of the origin in the (X, Y, Z)-space, i.e., the maximal ideal in A generated by (X, Y, Z). Now W(A, P) = *0(A') U 9J(J4") U <0(A'") where A', A", and A'" are the trivariate polynomial rings k[X',Y',Z'], k[X",Y",Z"}, and k[X'",Y'",Z'"] where {X' = X,Y' = Y/X,Z' = Z/X), (X" = X/Y,Y" = Y,Z" = Z/Y), and (X'" = X/Z, Y'" = Y/Z, Z'" = Z) respectively. Substituting the first set of equations in / we get / = X'n(l ± Y'n ± Z'n) and hence, in the (X',y',Z')-space, the total transform of the Fermat Cone consists of the exceptional plane X' = 0 together with the proper transform 1 ± Y'n ± Z'n = 0 which is a nonsingular "cylinder." §10: EXAMPLES A N D EXERCISES EXERCISE (El). [Algebraic Closure]. In L2(R6), while establishing the existence of an algebraic closure K of any given field K, we found an overfield K of K such that: (b) K is algebraic over K and contains a splitting field of every nonconstant monic polynomial F over K. The implication (b) => (bb), where (bb) says that K is algebraically closed, follows from the fact that an algebraic extension of an algebraic extension is an algebraic extension, which was proved in L1(R7)(C3). To have another transparent proof of the implication (b) =3- (bb), show that: given any algebraic field extension K/K and given any nonconstant monic G in i^[y], there exists a nonconstant monic F in K\Y] such that G divides F in /f[V]. EXERCISE (E2). [Integral Closure]. Recall that: an element t in an overring S of a ring R is said to be integral over R if it satisfies a monic polynomial equation

LECTURE U: VARIETIES AND MODELS

162

over R, i.e., if F(t) = 0 for some monic polynomial F(Y)=Yn

(»)

+ A1Yn-1

+ --- + An

of some degree n > 0 with coefficients A\,..., An in R; this is indicated by saying that t/R is integral; a subset T of S is integral over R means every t G T is integral over i?; this is indicated by saying that T/R is integral; finally, by the integral closure of R in S we mean the set of all elements in S which are integral over R. In L3§7 we asserted without proof that (Jl)

the integral closure of a ring R in an overring S is a subring of S, and R is a subring of the said integral closure

and (J2)

if an overring S of a ring R is integral over R, and an overring T of S is integral over S, then T is integral over R.

Note that assertions (Jl) and (J2) are the integral extension analogues of the algebraic extension assertions L1(R7)(C5) and Ll(R7)(C3) out of which the last was cited in (El) above. Also note that to prove (Jl) it suffices to show that iiti,...,tm (JO)

are any finite number of elements in a ring S

^ which are integral over a subring R of S then the ring R[ti,...,

t m ] is a finitely generated .R-module and is integral over R.

As an exercise, prove (JO) and (J2) in the following five STEPS (E2.1) to (E2.5), where S is an overring of a ring R. STEP (E2.1). For t G S let V be an fl[t]-submodule of S with ann f l [ t ] 7 = 0 such that V is finitely generated as an i?-module, and let J be an ideal in R such that tV C JV. Then F(t) = 0 for some F(Y) with Ai £ J f for 1 < i < n. HINT. Let Xu...,Xn be ^-generators of V with n e N+. Then for 1 < i < n we have tXi = Y^\
§i0: EXAMPLES AND EXERCISES

163

STEP (E2.2). For any t e S the following four conditions are equivalent. (1) t/R is integral. (2) The ring R[t] is finitely generated as an i?-module. (3) R[t] C T for some subring T of S such that T is finitely generated as an .R-module. (4) There exists an i?[t]-submodule V of S with a,nnR[t]V = 0 such that V is finitely generated as an .R-module HINT. (1) =» (2): from F(t) = 0, by induction we get tn+i € R+Rt+- • • + Rtn~1 for all i G N, and hence ( l , t , . . . , t n _ 1 ) are fl-generators of R[t\. (2) => (3): take T = R[t]. (3) =*> (4): take V = T and note that then 1 e V and hence for any r in ann fi(t] V we have r = r l = 0. (4) => (1): take J = Rin STEP (E2.1). STEP (E2.3). If S is a finitely generated ii-module, and an overring T of S is a finitely generated S-module, then T is a finitely generated il-module. HINT. If (2Jj)i<j (1) of (E2.2) it suffices to show that if elements * i , . . . , t m of S are integral over R then R[ti,..., tm] is a finitely generated .R-module. Make induction on m. For m — l this follows from (1) =$> (2) of (E2.2). Also m - 1 =>• m follows from the m = 1 case and (E2.3). STEP (E2.5). (J2). HINT. Assume S/R and r / 5 are integral. Then for any t € T we have F(f) = 0 for some F{Y) as in (|t) above with Ai,..., A„ in 5. Now by (E2.2) and (E2.4) we see that R[A\,..., An] is a finitely generated .R-module and R[t, A\,..., An] is a finitely generated R\A\,..., A n ]-module. Therefore R[t, A\,..., An) is a finitely generated .R-module by (E2.3), and hence t/R is integral by (E2.2). EXERCISE (E3). [Conditions of Integral Dependence]. (i) Note that in (E2.1) as well as (E2.2)(4), if either V ^ 0 and S is a domain, or if 1 e V (for instance if V is an overring of R), then the condition ann fl [ t ]V = 0 is automatically satisfied. (ii) Show that if R is noetherian then (2) of (E2.2) is equivalent to saying that: (2') R[t] is contained in a finitely generated i?-submodule of S. (iii) By an example show that if R is nonnoetherian then (2') 7^ (2).

164

LECTURE H: VARIETIES AND MODELS

EXAMPLE (X15). [Nonarchimedean Valuations]. Let v be a valuation of a field K such that the value group Gv of v is nonarchimedian, let a, (3 be elements in Gv such that a > n/3 > 0 for all n £ N+, let x, y be elements in the valuation ring R = Rv of v with v(x) = a and v(y) = /?; see L2§7(D2), L2§3, L3§7. Let t = l/y and V = Ru with u = 1/x. Then t/R is nonintegral but R[t] is contained in the finitely generated R-module V. EXERCISE (E4). [Cramer's Rule for Linear Equations]. Establish the Rule cited in (E2.1) above, i.e., given a finite number of simultaneous linear equations ^

AijXj = Yi

for

1< i < n

l<j
where the entries of the nxn matrix A = (Aij) as well as the elements Xj, Yj belong to a ring R, show that (E4.1) where the nxn

det(A)Xj=det(B^')) matrix B^

for

l<j
= {B\l ) is given by BU)=Uu

il

iil^j

\H=j

\Yi

i.e., B^ is obtained by replacing the j - t h column of A by the column vector (Y\,... ,Yn). Hence in particular, if det(^4) is a nonzerodivisor in R then, in the total quotient ring QR(i?), we have (E4.2)

Xj = det(B^)/det(A)

for

1 < j < n.

HINT. Let A^ be the matrix obtained by multiplying the j - t h column of A by Xj. Then B^ is obtained by adding to the j-th column of A^ linear combinations of the remaining columns. Hence by L3(E1) we get det(A ( j ) ) = det(A)Xj

and

det(B^)

= det(^>).

EXERCISE (E5). [Solutions of Homogeneous Linear Equations]. Consider the system of homogeneous linear equations Y^

aijXj = 0

for

1< i < M

l<j
where a = (a^) is an M x N matrix over a field k with M, N in N+. Let S be the solution space of the system, i.e., the set of all X = (X\,... ,XJV) S Aj^ satisfying the system; clearly S is a fc-linear subspace of A ^ . Let n be the rank of a, and

§10: EXAMPLES AND EXERCISES

165

note that then 0 < n < min(M, N). Show that if M = N then: 5 = { ( 0 , . . . , 0)} <^ n — N, i.e., <*=> det(a) ^ 0. More generally, without any assumption, show that dimfcS = N — n by finding a fc-linear injection cj) : A ^ ~ " —> Aj^ with im() = 5. Moreover, do this by establishing the explicit formulas which are described below. By relabelling ay we may assume that det(A) ^ 0 where A = (Aij) is the n x n matrix with A^ = a^ for 1 < i < n and 1 < j < n, i.e., A is the left upper corner submatrix of a. Let Y

i = -

Y,

aixXx

f r

°

1 i M

<< -

n<X
For 1 < j < n let B^ be the nxn matrix obtained by replacing the j - t h column of A by the column vector (Yi,.. .,Yn). Now for every Z = (Z\,..., ZN-U) 6 A j ^ _ n we put <j>(Z) = x = (xi..., XJV) € Aj^ where

Xj

_ fdet(B^)/det(A)

iil<j
~~ [Zj-n

iin<j
HINT. By the above displayed definition of Xj, and by looking at the expansion of d e t ( I ? ^ ) by its j-th column, we see that cp is an injective fc-linear map. So it only remains to show that its image is S. Now the given homogeneous system is clearly equivalent to the nonhomogeneous system y ^ AijXj =Yi

for

1< i < n

l<j
together with the extra equations ^21<j
Yi-

^2

aydet(B^)/det(A) = 0.

l<j
So given any such i, let C be the (n + l)x(n + l) matrix obtained by bordering A by the last column ( Y i , . . . , Yn,Y) and the last row (an,..., ain, Yi), and for n
^ l<j
and this completes the proof.

a«det(B W ) )

166

LECTURE H: VARIETIES AND MODELS

EXERCISE (E6). [Conditions for a Common Factor]. Let n and m be positive integers, and let / and g be polynomials of degrees < n and < m in Y with coefficients in a field k, respectively. Show that e

f + 9^ — 0 f° r some e and h in k[Y]

with degye < m — 1 and degyh < n — 1 < such that at least one of e and h is nonzero <*=> either degy/ < n and degyg < m or / and g have a common root in some overfield of k. HINT. k[Y] is a UFD. Moreover, if f,g have a nonconstant common irreducible factor then in its splitting field we get a common root for them. EXERCISE (E7). [Sylvester Resultant]. For polynomials / = f{Y) = a0Yn + a i F " - 1 + • • • + an g = g(Y) = bQYm + hY™'1

+--- + bm

with coefficients in a domain R, where n, m are nonnegative integers, establish the Basic Fact (Tl), i.e., show that: Resy(/, g) = 0 <=> n+m ^ 0 and either oo = 0 = &o or / and g have a common root in some overfield of R. HINT. The case when mn = 0 being straightforward, assume mn ^ 0. Let A = (Aij) be the (m + n) x (m + n) matrix over k = QF(i?) whose transpose is Resmaty(/, g). In view of (E6) it suffices to show that det(j4) = 0 iff ef + gh = 0 for some e and h in k\Y] with degye < m — 1 and degyh < n — 1 such that at least one of e and h is nonzero. Now e = e (Y) = XxYm~x

+ X2Ym~2

n

h = h(Y) = Xm+\Y ~

+ • • • + Xm

+ Xm+2Y

+ • • • + Xm+n

where elements X i , . . . , Xm+n, at least one of which is nonzero, are to be found in k, such that ef + gh = 0. Upon letting

ef + gh=

Yl

Ai"" +n-i

l
we see that for 1 < i < m + n we have l<j<m+n

Hence by (E5), det(A) = 0 iff there exists ( X x , . . . , Xm+n) € A ^ + n \ {(0,...,0)} for which D\ = ••• = Dm+n = 0. This completes the proof.

§m- EXAMPLES AND EXERCISES

167

EXERCISE (E8). [Elementary Properties of Determinants]. Prove the following properties of det(A) where A = (Ay) is an n x n matrix over a ring R which were used in (E4) to (E7) above. [(E8.1) to (E8.3) are vacuous for n = 0]. (E8.1) If a row (or a column) of A is multiplied by an element of R then det(j4) gets multiplied by that element. (E8.2) If to a row (or column) of A we add an i?-linear combination of the remaining rows (or columns) of A then det(A) is unchanged. (E8.3) If the rows (or columns) of A are permuted according to a permutation a then det(^4) gets multiplied by sgn(cr). In particular if we interchange two rows (or columns) then det(A) gets multiplied by —1. (E8.4) If A is replaced by its transpose then det(A) is unchanged. EXERCISE (E9). [Blowup of a Point in the Plane and Three Space]. In Examples (X12) and (X13) of §9.3 check what happens in QJ(A"), and in Example (X14) of §9.3 check what happens in 5J(A") and V{A'"). EXAMPLE (X16). [Blowup of a Line in Three Space]. In the Examples cited above we blew up a point and studied the effect on singularities of curves and surfaces. Now let us blowup the line L:X

= Y = 0

in the (X, Y, Z)-space over a field k, and study what effect it has on a surface singularity at the origin Q : X = Y = Z = 0. So consider the surface S:f

=

Y*-X2Zn~2=0

of degree n > 5 with double line L through the triple point Q, and the nonconical cubic surface T:g = Y2-

X2Z = 0

with double line L through the double point Q, discussed on pages 204-205 of my Engineering Book [A04]. Let P be the ideal generated by (X, Y) in A = k[X, Y, Z\. Then W{A,P)=V3(A')UV3{A") where A' and A" are the trivariate polynomial rings k\X', Y', Z] and k[X", Y", Z] where (X',Y') = (X,Y/X) and (X",Y") = (X/Y,Y) respectively. Substituting the first set of equations in / we get / = X'2(X'Y'3

-

Zn~2)

and hence, in the (X', Y', Z)-space, the total transform of S consists of the exceptional plane X' = 0 together with the proper transform X'Y'3 — Zn~1 = 0 which

168

LECTURE L4: VARIETIES AND MODELS

has a quadruple point at the origin. Likewise, substituting the first set of equations in g we get g = X'2(Y'2

- Z)

and hence, in the (X', Y', Z)-space, the total transform of T consists of the exceptional plane X' = 0 together with the proper transform Y'2 — Z = 0 which is a nonsingular parabolic cylinder. The reason why the singularity of T was resolved but the singularity of S was worsened (a triple point becoming a quadruple point) is that for T we were in the equimultiple case because the multiplicity of Q = 2 = the multiplicity of L, whereas for S we were in the nonequimultiple case because the multiplicity of Q = 3 > 2 = the multiplicity of L. EXAMPLE (X17). [Tight Associators and Monomial Ideals]. In §5(017) we showed that for a submodule U of a module V over a noetherian ring R, the associator assji(V/U) coincides with the tight associator tassR(V/U), i.e., if P is a prime ideal in R such that (U : a)# is P-primary for some a G V then (U : b)R = P for some b £ V. Here is an example that in case R is nonnoetherian, this need not be so even when V is R and U is an ideal Q in R. So let R be the polynomial ring k[Xi,X2,X3,...] in an infinite number of indeterminates (Xi)i^n+ over a field k, and let P and Q be the ideals in R generated by Xi, X2, X3,..., and X?, X$,X$... respectively, i.e., by (Xi)ieN+ and (Xf)ieN+ respectively. In a moment we shall show that P belongs to assR(R/Q) but not to tassn(R/Q). Actually we shall do this in a somewhat more general set-up. We start by proving some claims about monomial ideals. So let (Xj), e / be any family of indeterminates over the field k and consider [cf. L6§6(D10)] the polynomial ring Rj = k[(Xi)i£i]. Let W be the set of all maps w : I —+ N whose support supp(w) = {i G I: w(i) ^ 0} is finite, and for any such w write Xw = Yli€l X™ '. Note that then the set of all monomials M = {Xw : w € W} is a (free) vector space basis of Ri over k, and for any / G Rj, after writing / = Ylwew awXw with aw G k we have that its support supp(/) = {Xw : w € W with aw ^ 0} is finite. For any u and t in W we write u > t to mean that u(i) > t(i) for all i £ I, and we observe that this is so iff Xu = XfXw = Xt+W for some w G W, where we define t + w £ W by putting (t + w)(i) = t(i) + w(i) for all i £ I. For any T C W, let L(T) =

{Xt:teT}

be the corresponding set of monomials, let f = {u G W : u > t for some * G T} be the corresponding "saturated" subset of W, and let N(T) = {/ G P 7 : supp(/) C L(T)}.

§10: EXAMPLES AND EXERCISES

169

Note that then N(T) is the fc-vector-subspace of Rj generated by L(T), and L(T) is a fc-vector-space-basis of N(T). Let us consider a monomial ideal in Rj, i.e., an ideal generated by L(T) for some T C W; let us denote this ideal by J(T), i.e., J(T) =

L(T)Rj.

We claim that for any T C W we have (1)

J(T)=N(f)

and

J(T)nM

= L(f).

To prove this, note that obviously N(T) c J ( T ) . Conversely, any / £ J(T) can be written as / = X^ter' ^ ' / t where T' is a finite subset of T and ft £ Ri- Now we can find a finite subset W of W such that for all i £ T" we have ft = 23™ew" atwXw with atli) £ /c. Now T* = {t + w : (t, w) £ T" x W'} is a finite subset of T and we have / = J2uer* buXu where bu = T,{(t,w)eT<xW:t+w=u}atw G ^ for all u £ T*. Therefore supp(/) c T. This proves the first part of (1), and the second part follows from it. Consider the ideal in Rj generated by {Xi)i£P where I' is any subset of J, i.e., the ideal J(c(I')) where c(I') =

{c(j):j£l'}

and where c(j) is the characteristic function of j £ 7, i.e., c(j) e W is defined by putting

c(j)(i) = ll

\0

[ii=j

iii^j.

For any T C\W consider the condition: (D)

for all j in some infinite subset / ' of I we have t(j) ^ 1 for all t eT.

We claim that then: (2)

(tl) =» (J(T) : J(c(I)))Rl

= J(T).

To prove this, assuming (fl), given any / e f l / \ J(T) we want to find g £ J{c(I)) with fg £ J{T). Now supp(/) is a nonempty finite set and hence we may label its elements as t i , . . . , tn where n £ N + and ti,...,tn are pairwise distinct members of W. Since / 0 J(T), by (1) we know that some U does not belong to T and hence by relabelling we may assume that t\ & T. Since I' is an infinite set, we can find j £ I' such that ti(j) = 0; then (ti +c(j))(j) = 1; since tx £ f, by (fl) it follows that *i + c 0 ) ^ T. Let g = Xj. Then ti + c(j'),..., tn + c(j) are exactly all the distinct elements of supp(/#), and hence by (1) we see that fg & J(T). Also obviously g e J(c(l)). For any T c W consider the conditions: (»')

for every t e T we have t(i) ^ 0 for some i s /

170

LECTURE L4: VARIETIES AND MODELS

and (((")

for every j £ I there exists dj £ T such that: dj(i) =/= 0 iff j = i.

By (1) we see that: (3)

(«) + (f) => J(T) C J(c(/)) ^ J(T).

Now J(c(7)) is the kernel of the fc-epimorphism Ri —> k with Xi H-» 0 for all i £ 7, and hence J(c(I)) is a maximal ideal in /?/ and therefore by §5(08)(3*) we see that:

(4)

m + (n =• lJ{T)

is J c 7

( ( ))-p rimar y with

\vspec f l / J(T) = mvspec*, J(T) = {J(c(/))}. In view of Theorems (T5) and (T6) of §6, by (2) to (4) we conclude that: (5)

(tt) + (if) + (II") => ass fl/ (Ri/J(T)) = {J(c(I))} but tass*,(Ri/J(T)) = 0. As a special case of (5), assuming I to be infinite, upon letting P to be the ideal in Rj generated by (Xj)j e j and Q to be ideal in Ri generated by ( X " ' ) ^ / where n* € N + with n, > 1 for infinitely many i, we get assn,(Ri/Q) = {P} but tassfl 7 (i?//Q) = 0. In particular we have the beginning illustration when I = N+ with P and Q generated by (X,)i € N + and (Xf)i&^+ respectively. EXAMPLE (X18). [Isolated Components of a Module]. As we noted in §5(O20)(14*), given any submodule U of a module V over a ring R with U ^= V, if V is finitely generated then: P £ vspec(annfi(T^/J7)) =4- [[/ : P]v ^ V. We shall now show by an example that this is not true without finite generation. Let V be the bivariate polynomial ring k[X, Y] over a field k. Then U = k[X,XY] is a subring of V, and R = k[X] is a subring of U. So we may regard V as a module over R, and U a submodule. Note that by putting primes on X and Y, the pair U C V may be identified with the pair A C A' of Examples (X12) and (X13) of §9.3. For any / = Y^aijXW £ V with atj £ fc let supp(/) = {(i,j) £ N x N : atj ^ 0}. Clearly U = {/ £ V : i > j for all (t, j ) £ supp(/)}. Any 0 7^ g £ i? can be written as a finite sum g = J2 biX% where bi £ k with 6e ^ 0 for some e £ N; obviously (e,e + 1) belongs to supp(#y e + 1 ) and hence gYe+1 £ U with Ye+l £ V; therefore ann/^V/lO = 0. Thus for the zero ideal P in R we have P £ vspec(ann^(F/t/)). For any / £ V we can find a nonnegative integer d which exceeds the degree of / and then by the above criterion we get Xdf £ U with XdeR\P. Therefore [U:P]v = V. EXAMPLE (X19). [Annihilator of a Primary Module]. As we noted in §5(013), given any submodule U of a module V over a ring R: U is primary => a,niiii(V/U) is primary. Here is an example showing that the converse is not true. Let R be the bivariate polynomial ring k[X, Y] over a field k, and let U and V be the ideals in R generated by (X2,XY) and (X,Y) respectively. Then clearly

%11: PROBLEMS

171

annfi(V7{7) is the ideal in R generated by X which is prime and hence primary. But Y eR\iadR(a,nnR(V/U)) = R\ (XR) and X £ V\ U with YX e U. Therefore U is not primary. §11: PROBLEMS To initiate a new column, occasionally I shall list Problems (PI), (P2), . . . . A Problem is an Exercise which has not been completely worked out. By solving one of these, sometimes the student may get a mild satisfaction, sometimes a Ph.D. thesis, and sometimes fame. If there is some imprecision in the statement of a problem, a part of the exercise is to make it precise. PROBLEM (PI). [Explicit Equations of Integral Dependence]. Redo items ( J l ) , (J2) mentioned in (E2) together with the analogous items (C3), (C5) of L1(R7) cited there, by finding explicit formulas. In other words, let Fi(Y) = Yni + AnY"*-1 + ••• + Aini with Aij in a ring R be the equations of integral dependence satisfied by a finite number of elements U in an overring S of R for 1 < i <m. Find explicit formulas for B\,..., Bni and C\,..., Cn" so that t\-\ \-tm and t\... tm satisfy the equations G{Y) = Yn' + BtY71'-1 + ... + Bn, and

H(Y) = Yn" + dr""- 1 +... + c»« of integral dependence over R. Also let s € S satisfy F*(Y) = Ym+ UY™-1 + --- + tm and find formulas for Di,...,

Dn* so that s satisfies the equation

E(Y) = Yn' + D j F " * - 1 + . . . + £>„. of integral dependence over R. [SATISFACTION]. HINT. Think of the Aij as indeterminates over Z. Let G{Y), H(Y), and E(Y) be the minimal polynomials of t\ H h t m , t\.. .tm, and s over Q((^4ij)i, belong to Z[(Ay)i
172

LECTURE

L4: VARIETIES

AND

MODELS

PROBLEM (P3). [Resolution]. Let K/A be as in the above (P2). Show that there exists a nonsingular projective model of K/A. [FAME]. §12: R E M A R K S REMARK (Rl). [Laplace Development]. To describe the Laplace development of a determinant cited in L3§12(E2), which generalizes the development according to a row or column described in L3§12(E1), let A = (Aij) be an n x n matrix over a ring R, say as depicted in L3§12(D1) with m = n. Let r = ( r i , . . . , rt) be a sequence of positive integers with n + • • • + r^ = n. Recall that Sn is the set of all bijections a of {l,...,n}. Let Sn(r) be the set of all a € Sn such that
det(A)= Yl sgnO"-1) II d 6 * ^ ) TeSn(r)

l
and for any r in Sn(r) we have det(A)=

^
sgn^r-1)

J]

det(4;)).

l
These may respectively be called the Laplace development of det(A) according to the row partition (r\,...,rb) relative to the permutation a, and the Laplace development of det(A) according to the column partition ( r i , . . . , 77,) relative to the permutation (T,S); when a or T is the identity permutation we may drop the reference to it. REMARK (R2). [Block Matrices]. Let A = (Aij) be an m x n matrix over a ring R, say as depicted in L3§12(D1). Let (qi,..., qa) and ( r i , . . . , r&) be positive integers with 51 + • • • + qa = m and n + • •• + rb = n. The definition of a matrix over a ring introduced in L3§1 can be generalized in an obvious manner to define a matrix over (having entries in) any set. In particular we get an a x b matrix A = (Auv) where Auv = ((Auv)ij) is the qu x rv matrix over R given by putting (Auv)ij

= Aqi-i

\-qu~i+i,ri-\

\-rv-i+j

i.e., the (i, j)-th entry of Auv equals the (q\-\ \-qu-i+hi"i-\ \-rv-i+j)-ih entry of A. We call A a block matrix over R, or in greater detail a (qi,..., qa) x ( n , . . . , n)

173

§12: REMARKS

block matrix over R, and in an obvious sense we have the equation A = A which, for making clear that the LHS is an ordinary matrix while the RHS is a block matrix, may be written as -block

A.

Note that the "blocks" Auv are submatrices of A. We define sums and products of block matrices by an obvious generalization of the definitions for the sums and products of matrices over a ring given in L3§1. For instance, if B =uock B where B = (B^) is an m x n matrix over R and B = (Buv) is an (qi,...,qa) x ( l , • • • ,r&) block matrix over R then A + B = ((A + B)uv) is the (qi,... ,qa) x ( r i , . . . ,rb) block matrix over R defined by putting {A + B)uv = Auv + Buv, and we clearly have A + B =biock A + B. Likewise, if B —block B where B = {B^) is an n x o matrix over R and B = (Buv) is an ( r i , . . . , n,) x ( s i , . . . , sc) block matrix over R with si + • • • + sc = o then AB = ((AB)UV) is the {q\,..., qa) x ( s i , . . . , s&) block matrix over R defined by putting (AB)UV = Yli<w
where A' = (A'^), A* = (A*j), A** = (A*j), are m' xn,m*x over R with m' + m* = m, n* + n** = n, and

Ajj

—<

n*, m* x n** matrices

A'i;j

if 1 < i < m' and 1 < j < n

A-i-m'j A*lm,j_n,

if m'
and 1 < j < n* and n* < j
or equations of the type A'

I A"

where A' = (A'^), A" = ( 4 £ ) , A* = (A*^), A** = {A*j), are m' x n', m' x n", m* x n*, m* x n** matrices over R with m' + m* — m, n' + n" = n, n* + n** = n,

174

LECTURE LJ,: VARIETIES AND MODELS

and A1

if 1 < i < m! and 1 < j < n'

A"

if 1 < i < m' and n' < j
A*i—m',j

if ml < i < m and 1 < j < n*

A*. •m'

J—n*

if m! < i <m and n* < j < n.

For the sake of clarity, in cases as above, we may write A =qbiock (—) in place of

A={-). REMARK (R3). [Products of Determinants]. Referring to L3§12(E2), here is the proof of the product formula for determinants using the Laplace development. Given nxn matrices A,B over a ring R, let C = (Cy) be the (2n) x (2n) matrix over R which is given in terms of an (n,n) x (n,n) block matrix as C = (-i B) where 1 and 0 are the nxn identity and zero matrices. For n < j < 2n, we add to the j-th column of C the linear combination Xalock form as D = (^\ A 0 B ) and hence by (E8.2) we get det(C) = det(D). By applying the Laplace row development according to the row partition (n, n) we get det(C) = det(A)det(.B) and det(D) = det(AB). Therefore det(AB) = det(A)det(B). REMARK (R4). [Solutions of Nonhomogeneous Linear Equations]. To generalize (E5), let y]

ciijXj =bi

for

1< i < M

l<j
be a system of nonhomogeneous linear equations where a = ( a y ) is an M x N matrix over a field k with M, N in N, and a = (a%j) is the M x (N + 1) matrix over k obtained by augmenting a by the last column (&i,... , 6 M ) , i-e., for 1 < i < M we have a^ = a^ or bi according as 1 < j < N or j = N + 1. Let S be the solution space of the system, i.e., the set of all X = (Xi,..., XN) € A^ satisfying the system. Note that if S is the solution space of the corresponding homogeneous system

^2

a

ijxi = 0

for

1< « < M

l<j<JV

and if S is nonempty then it is clearly an additive coset of S in Aj^, i.e., for any X in S we have S = {X + X : X £ S}. Let n and n be the ranks of a and a respectively. Then clearly n
175

%12: REMARKS

1 < i < n and 1 < j < n, i.e., A is the left upper corner submatrix of a. Let Yi = bi-

Y^

a

^A

for

l
n<\
For 1 < j < n let B be the nxn matrix obtained by replacing the j-th column of A by the column vector (Y\,..., Yn). Now for every Z = (Z\,..., Zjv-n) £ A^~™ we put (Z) = x = (xi... ,X~N) G A^ where f det(B{i))/det{A)

if 1 < j < n

[Zj-n

iin<j
We CLAIM that S is nonempty iff n = n, and when that is so, <j> is an injective map whose image is S. PROOF OF THE CLAIM. By the above displayed definition of Xj, and by looking at the expansion of det(B ) by its j-th column, we see that 4> is an injective map. So it only remains to show that S is nonempty iff n = n, and when that is so, S is the image of
=Yi

for

1< i < n

l<j
together with the extra equations Y1KJ
Y^

aijdet(Bb))/det{A)=Ofoin
l<j
So given any such i, let C be the (n + 1) x (n + 1) matrix obtained by bordering A by the last column (Yi,..., Yn, Yi) and the last row (an,...,a;„, Yi), and for n < I < N + 1 let C be the (n + 1) x (n + 1) matrix obtained by bordering A by the last column ( a n , . . . , a n i , a u ) and the last row (an,...,ain,au). Expanding by the last column we see that det(C) = det(C ) - J2n
^ \<j
and this completes the proof.

a^de^B•t/k

176

LECTURE L4: VARIETIES AND MODELS

REMARK (R5). [The How and W h y of Cramer and Sylvester]. To solve the simultaneous linear equations

£

AijXj =Yi

for

1< i
l<j
we subtract An times the first equation from An times the i-th equation for 2 < i < n. This "eliminates" Xi and produces n — 1 equations in the n — 1 variables X2, •.., Xn. Iterating this procedure n times, the last step produces a value of Xn. Substituting this in the first n — 1 equations gives us n — 1 equations in the n — 1 variables X\,... ,Xn-i. Iterating this entire procedure produces a value of Xn-i, and so on down the line until we get values of all the variables X\,...,Xn. By condensing the whole hullabaloo in one giant step we end up with Cramer's Rule. This "explains" (E4). Similar "explanations" apply to the systems of linear equations dealt with in (E5) and (R4). Turning to the higher degree simultaneous equations f(Y) = a0Yn + a1Yn-1 + --- + an g(Y)=b0Ym

+ b1Ym-1 +

--.+bm

dealt with in (E7), since there is only one variable Y, how shall we eliminate it by itself? Well, we do it by letting the degree play the role of a variable. To wit, assuming n > m, we multiply the first equation by 60 and the second by aoYn~m, and then subtract the second from the first. This replaces the equations of degrees n, m by equations of degrees n — 1, m thereby decreasing the sum n + m. Iterating this procedure several times we make the sum of the degrees zero, i.e., we eliminate Y and get hold of an expression involving only the coefficients ao, • • •, an> bo, • . . , bm, which, if everything is nice and dandy, ought to produce something resembling the Sylvester Resultant. In (E7) we proved the Basic Fact (Tl) about the Sylvester Resultant. Here is an easier proof of that part of (Tl) which says that if n,m are positive integers and / , g have a common root in some overfield of R then Resy(/, g) = 0. Namely, call that root Y. Multiplying / by Ym~l ,Ym~'2,... ,Y,l and g by yn-i Yn~2,... ,Y, 1 we get m + n homogeneous linear equations in the m + n "variables" Ym+n~1,Ym+n'2,... ,Y, 1. This "solution vector" is a nonzero vector because it ends with 1. Therefore by (E5) the determinant of these equations, which is nothing but Resy(/,g), is zero. Q.E.D. In greater detail, the matrix T = (7V,)i|^™+^ of the said equations is nothing but Resmaty(/, g), and the equations are

V-

T. Ym+n-i

i<^+„ ^

=

{Ym-if{Y)=* \Y"-ig(Y)=0

ifl<»<m ifl
Note that T is the transpose of the matrix A of the HINT to (E7). Letting Y revert to its role of an indeterminate, and without assuming / , g to have a common root,

§12: REMARKS

177

but continuing to assume n,m to be positive integers and disregarding the = 0 (twice) of the above display, the said display says that by adding to the last column of T certain il[Y]-linear combinations of the remaining columns we get the column vector ( y ™ - 7 ( y ) , . . . , Yf(Y), f(Y), Yn~lg{Y),..., Yg{Y), g(Y)). Expanding the determinant of this new matrix by its last column we get an identity of the form f Resy(/,
+ v(Y)g(Y)

with u(Y), v(Y) in R[Y]

[of F-degrees < m - 1 and < n — 1 respectively. By letting Y be a common root of / , g, this gives yet another proof of the above mentioned part of (Tl). Having proved the Basic Fact (Tl), let us briefly comment on the remaining items of §1. Corollary (T2) follows by noting that a nonconstant univariate polynomial over a field has a multiple root iff the polynomial and its derivative have a common root. Observations (01) and (02) follow by noting that the operations of forming the resultant or the discriminant commute with the operation of giving values to the auxiliary variables X\,..., X^. The Isobaric Property (03) and the Product Formula (XI) will be proved in the next Remark (R6), where we shall also prove a product formula for the resultant matrix. Examples (X2) to (X6) are straightforward computations. Bezout Observations (04) and (05) will be dealt with later on. This takes care of the Future Plan Observation (06). REMARK (R6). [Product Formula for the Resultant Matrix]. To formulate the last identity (5.1) of the above Remark (R5) more precisely, and to prove the Product Formula (XI) of §1 together with its generalizations, for nonnegative integers n,m, in addition to the polynomials f(Y)=a0Yn

+ a1Yn-1 + --- + an

g(Y)=b0Ym

+ b1Ym-1 +

---+bm

with coefficients o o , . . . , an, bo,..., bm in a ring R, which is assumed to be nonnull, let us consider polynomials F(Y) = A0Yn + AxYn-1 m

G(Y) = B0Y

whose coefficients Ao,---,An,Bo,...,Bm

+ --- + An 1

+ BtY™-

+ ... + Bm

are indeterminates over R. Let

R = R[AQ, ... ,An,B0,

. •. ,Bm].

Since Resmaty-(/, g) and Resy(/, g) depend not only on / , g but also on n, m, strictly speaking we should write Resmaty (/, g) and Res y l ' m ) (/,g) for them and call them the Y-Resultant Matrix and the Y-Resultant of / , g relative to n, m respectively, with the understanding that the reference to n, m may be dropped when it

178

LECTURE

H:

VARIETIES

AND

MODELS

is clear from the context. As a more precise version of (5.1) we have the identity: ' R e s ^ ' m ) ( F , G ) = U(Y)F(Y) (6.1)

{

U{Y) = E i < , < m ^

m

+ V(Y)G{Y)

with

J

Y.m+i
- ^ and V{Y) =

in R[Y] of y-degrees < m — 1 and < n — 1 respectively, where Ctj S ~R is the (i, j)-th cofactor of Resmat^' m ) (F, G)

In the rest of this Remark, as an abbreviation, we denote Resmaty the (m + n) x (TO + n) matrix T{n>m\f,g)

(/, g) by

(T^m\f,g)ij).

=

and we note that then T(n'm) (/, g)ij € fl and

T^ n ' m ) (F, G)« € P .

We also put P(">m> (/, 5 ) = R e s ^ m ) (/,
= p(".™)(F, G)

and we note that then P{n'm\f,g)€R

and

Q(A0,...

,An,B0,...

,Bm) e R.

Finally, we let „ a

J a,i if i e { 0 , . . . , n} '~\0

if

ieZ\{0,...,n}

and, for any nonnegative integers u,v, we let S^n'u'v^(f) u x u matrix obtained by putting

= (S^n,u'v^(f)ij)

be the

and we call S^n'u'v^(f) the (n, u, u)-spread of / . The resultant matrix can be expressed in terms of the spreads by the block matrix identity:

(

c(n,m,m+n)r

f\\

g(m,n,m+n)

f„\ ) •

As direct consequences of the spread notation we have the following two relations. Relation between Polynomial Addition and Matrix Addition: (1)

if n = m then

S{n'u'v){f

+ g) = S{n'u'v)(f)

+

S{m'u'v)(g).

§12: REMARKS

179

Relation between Polynomial Multiplication and Matrix Multiplication: (2)

m+n u v

if integer q > n + u then

S^

' ' \fg)

=

S^'^ifiS^^ig).

Before using (0) to (2) for obtaining a product rule for T, let us observe some easy properties of P . Since a 0 x 0 determinant is 1, we see that: (3)

if n = 0 = m then

P ( n ' m ) (/, g) = 1.

The determinant of a triangular matrix, i.e., a matrix having zeroes above or under the principal diagonal, equals the product of the diagonal entries. Also the zeroth power of anything (including zero) equals 1. Hence: if n = 0 then

P ( n ' m ) (/, g) = a%

ifm = 0then

P{n'm)(f,g)

(4) whereas (5)

= b%.

By respectively expanding according to the first or the last row and then calculating with triangular determinants, i.e., determinants of triangular matrices, we get: p(n,m)(f

( 6 )

Q) =

\Eo
if m = 1

lEo^j-ir^ar* if n=i.

By (E8.1) we see that: (7)

for all c, d in R we have

p("' m > (c/, dg) = cmdnP{n'm)

(f,g).

Expanding by a row and invoking (E8.2) we see that the determinant of an N x N matrix (with positive integer N) is zero if either a row consists only of zeroes or if two rows are identical, and hence: (8)

if either mn^0

= f-goim^0

= fom^0

= g then p( n - m )(/, g) = 0.

If mn = 0 then by (3) to (5), and if mn ^ 0 then by (E8.3), we get the Permuting Rule for Resultant: P(m'n)(gJ)

(9)

= ( - i r n P ( " ' m ) (/,
Also we clearly have the Universality Property of Resultant: (10)

P{n'm)U,g)

= Q(a0, ...,an,b0,...,

bm).

LECTURE L4: VARIETIES AND MODELS

180

Next we record the following five properties (11) to (15) of the polynomial Q. As notation, coeS(U, V) denotes the coefficient of a monomial U in a polynomial V = V(Xi,... ,Xr,Yi,... ,YS) in indeterminates Xi,...,Xr,Yi,...,Ya over the ring R. Recall that V is homogeneous of degree t in (X%,... ,Xr) means that for every monomial U = X^ .. .XpYJ1 .. .Yj° with coeS(U,V) ^ 0 we have ii + ••• + ir — t. More generally, V is isobaric of weight r in (Xi,.. .,Xr) when we give weight Uj to Xi for 1 < i < r means that for every monomial U = XI1 ...XJrYJ1 ...Yj- with coeff^.V) ^ 0 we have «i»i + ••• + urir = T; clearly this is so iff as a polynomial in an extra indeterminate Z we have V(Z^XU .. .,Z^Xr,Yu. ..,Ya) = ZTV(XU.. .,Xr,Ylt.. .,YS). Diagonal Property of Resultant: (11)

coeffWB£,Q) = l

and

coe«(A™BZ,Q) = ( - l ) m n .

First Homogeneity Property of Resultant: (12)

Q is homogeneous of degree m in ( A o , . . . , A n ).

Second Homogeneity Property of Resultant: (13)

Q is homogeneous of degree n in (Bo,...,

Bm).

Weight Property of Resultant:

{

Q is isobaric of weight mn in ( A o , . . . , A„, Bo,.. •, Bm) when Ai and Bj are given weights i and j for 0 < i < n and 0 < j < m respectively.

Special Property of Resultant: if coeS(U, Q) ^ 0 for a monomial nK,

(,15)

\U = A«... A^Bl0 ...Bk<jL {Ao"B^, A-5 0 "} < then in < m and j m < n and in + j m <max(m, n) and if also m^n

then in + j

m

<max(m, n).

To prove these, by looking at the resultant matrix depicted at the beginning of §1 we easily get (11) to (13), whereas (15) follows from (12) to (14). So it only remains

%12:

181

REMARKS

to establish (14). To do this, first we note that . . . , ZnAn,

(Q(Z°AQ,

ZmBm)

Z°B0,...,

\= det (Tn>m{Z°AQYn

+ ••• + ZnAn,

Z°B0Ym

ZmBm))

+ ••• +

and now for 1 < i < m we multiply the i-th row of this matrix by Zl and for I < j < n we multiply its (m + j)-th row by Zi and then upon denoting the resulting matrix by T we get det(f) = ZSQ(Z°A0,...,

ZnAn,

ZmBm)

Z°B0,...,

where 5 = $ Z 1 < i < m * + Yli<j however, T is the same as the matrix we obtain when for 1 < I < m + n we multiply the Z-th column of Tn'm(F, G) by Zl and hence det(f) = ZCQ(A0, ...,An,B0,...,

Bm)

where e = ^Zi ^ v summing the three arithmetic series we conclude that £

_ g

=

(m+n)Tm+n+l) _ [mCm+1)

+

nfcH^j

=

^

^

h e n c e

b y

^

a b o y e

t w Q

displayed equations for det(T) we get (14). To start towards the product rule, for a polynomial h(Y) = c0Ym+n + c 1 r + n - 1 + • • • + cm+n with Co, ci,...,

cm+n in R, we record the following two sum rules.

First Sum Rule for Resultant: P{n'm+n)(f,h

(16)

P{n'm+n\f,h).

+ fg) =

Second Sum Rule for Resultant: (17)

p(m+n,m)(h

+

jgg^

=

p(m+n,m)^

gy

To prove these, by (0) we have T(m+n,m)(h

*•

„\ - I ^

W\

Vl>y) — I g ( m , m + n , 2 m + n ) / „ \ J

and by (1) and (2) we have {m+n,m)(hn T 1

, f„

\ -rJyi9)—\

^ - (S^m+n^m+n\h)

+ g(m+n,m+n,2m+n)i\

mm

S^

' +n\f)S^m+^m+"){g)\ J

and hence by adding certain linear combinations of the last m+n rows of the matrix T^m+n'm\h,g) to its first m rows we obtain the matrix T^m+n'm\h + fg,g) and

182

LECTURE

L4: VARIETIES

AND

MODELS

from this we conclude that these two matrices have the same determinant which establishes (17); now by (9) and (17) we also get (16). Recalling that u, v are any nonnegative integers, let also w be any nonnegative integer, and consider the following two shift rules. First Shift Rule for Resultant: P(n+W'm){f,g)

(18)

= {-l)wmh%

P{n'm){f,g).

Second Shift Rule for Resultant: (19)

p(n,m+W)ifig)

= ag >p(n."O(/ t 0).

/

,

\

/

ci(ni"'i1Tl+7l + 'w)/<,\

\

The proof of (19) follows by noting that T^m+W\f,g) = { — ^ f ^ 4 b ) where 0UtV denotes the uxv matrix with all entries zero; moreover, by (9) and (19) we get (18). Recall that lu is the u x u identity matrix, and note that for 1 £ R[Y] we have degyl = 0 < w. About these special matrices we have the obvious Relations: ®u,v + J = J + 0u,v for every uxv

matrix J ,

Ou,vJ — Ou,w for every v x w matrix J, . , (20)

{

J0U v = 0W „ for every w x u matrix J , \UJ = J for every uxv

matrix J ,

Jlu = J for every v x u matrix J, S(w,u,u+v+w)^ = (0uw^ i u ) o U i V ) as block matrix. The ENLARGED RESULTANT MATRIX is the (u + m + n + v) x (u + m + n + v) matrix defined as a block matrix by putting 0U ,m+n o„ 0m+n,uT^m\f,g)0m+n,v Vv,u

Vv,m-{-n

*-v

and we note that clearly (say by Laplace development) (20)

det (r^n'm^

(f,gj)=

P ( n ' m ) (/, g)

and (21)

det(r
=ctf.

%12:

183

REMARKS

By analogy with the equation

(

q(n,m+w,m+n+w)

I f\

o(m+w,n,m+n+w)

I „\

we introduce the SKEWSHIFTED RESULTANT MATRIX by putting

(

q(n,m,m+n+w)

( f\ ^

7

V'

a(m,n+w,m+n+w)

l„\

and we note that these two (m + n + w) x (m + n + w) matrices are clearly related to the (m + n) x (m + n) matrix T{-n'm\f,g) by the formulas c(n,w,m+n+w)

( f\

\

( 0 w ,lT.-4lp)) and fOQ/\ [

'

rp[n,w,m}( f

\ _ ( *•

U,9)

~

'

\0w,n\

l / i 9) \ Vm+n,w

\

S(^,m+w){g)J-

As an immediate consequence of (22') we have the Skewshift Rule for Resultant: (19')

det ( T ^ m ] ( / , 2 ) ) =

bZP{n'm\f,9).

Finally we introduce the ENLARGED SKEWSHIFTED RESULTANT MATRIX which is the (u + m + n + w + v) x (u + m + n + w + v) matrix defined as a block matrix by putting ,v

0m+n+w,uT^^(f,g)0m+n+w,v Vv,u

Uv,m+n+w

*-v

and we note that by (19') we have (20')

det (T^u'n'w'm'v\f,

g)) = 6™ p( n - m )(/, g)

and (21')

det M u '°'"' m '"] (1, g)) = bnm.

Now upon letting n',n*,m',m* n = n' + n*

be any nonnegative integers such that and

m = m' + m*

184

LECTURE

L4: VARIETIES

AND

MODELS

and upon considering any polynomials f'(Y)=a'0Yn'

+a'1Yn'-1

+

---+a'n,

g'(Y) = b'0Ym' + fciY™'-1 + • • • +

C

and = a*0Yn' + alY"*'1 + ••• + < .

f*(Y)

g*(Y) = b*0Ym' + b{Ym''x with coefficients a'0,...,a'n,,b'0,...

+ ... + b*m.

,b'm,,a^,...,a*n,,bl,..

/ = /'/*

and

.,b*m. in R such that

g = g'g*

we can state the following two product rules. Product Rule for Resultant matrix: (23)

T{m'n''n'fi\f\l)T{n'm\f,g)

T^'n''m'n'){f',g)T{n''m+n'){f\g).

=

Skew Product Rule for Resultant matrix: (23')

y[0,O,m',m*,n]/^ q*)T^n'm\f

n) = y ( m * > " > ' n ' ' 0 ) ( f q'\ J<[n,m',m"] / t

*\

To prove (23), we have the following equations where the first and the last equations are deduced by items (0) and (Oi) whereas the middle two equations are deduced by block multiplication followed by items (2) and (2o).

(

lm

0m,n

\

/q{n,m,m+n)(

0n;m 5("*."'-")(D

*

f\\

m+n)y\)

c(n,7n,7n+n) / f\ 0(771+71*,n',m+n) ( f*sj\ o{m-j-n' ,n* ,m+n) o(n',m,ro+n')/f/\ *-> U / c(m ) n',m+n')/' / , , \ *-> U/J Un* ,771+71'

/„\ n \ , N u m,n* \ / o f ^ m - t - r ^ m - r - n W i - * ^ n , * I I , ^ ' u n ' , n * I I g(m+Ti',n*,m+n)/'„} J-n* /

^

RHS of (23).

To prove (23'), we have the following equations where the first and the last equations are deduced by items (0') and (0[) whereas the middle two equations are deduced

185

%12: REMARKS

by block multiplication followed by items (2) and (2o).

f

£

'

(1) 0 m - i n \

s qfam^m+n),

*\\

o(n,m" ,m+n) I f\ c(m* +n,m',m+n) I f a*\ g(m,n,m+n)/g\ J-TTl*

fl vm',m*

Vm* ^m''-\-n c(«,"X* ,m+n)( f\ c(n,TB',m'+n)/f\ I I ^' " "'w; I I , o U) l \ u q(m',m'+n,m+n)(„*\ n,m'+n)fqi\l \

= RHSof(23').

Taking determinants of all the matrices in (23), by (19), (20) and (21) we get (24)

afP^m\f,g)

= a*0n'>("'•"*)(/''

,g)p(n">m\f\g)

and taking determinants of all the matrices in (23'), by (19'), (20') and (21') we also get (24')

b^P^m\f,g)

= b^'P^m'\f,g')P^m'\f,g*).

By the "principal of universal identities" let us show that the factors a™ and b^i can be cancelled out from the above two equations. So consider the polynomials F'(Y) = A'0Yn' + A'^Y71'-1 + ••• + < , G'{Y) = B'0Ym' + B j y " 1 ' - 1 + ... + B'm, and F*(Y) = A*QYn" + A^Y71'-1 + • • • + A*n. G*(Y) = B*QYm' + BIY™'-1

+ --- + B*m.

whose coefficients A' = (A'0,..., A'n,), B' = {B'Q,..., B'm,), A* = (A%, ...,A*n.) and B* = (BQ , . . . , B^,) are indeterminates over R which are assumed to be "independent" of the previous indeterminates A = (Ao,..., An) and B = ( S o , . . . , Bm). Then F'(Y)F*{Y)=

Y^

Fi{A',A\B)Yn-i

0
G'(Y)G*(Y)= J2 Gj(A,B',B*)Ym-j 0<j<m

LECTURE L4: VARIETIES AND MODELS

186

where Fi and Gj are polynomials over R in the exhibited indeterminates. Let Q{A',A*,B)

P(n''m){F',G)P{n"'
=

Q'(A, B', B*) = p("- m ')( j p, G')P{n'm'\F,

G*)

and Q(A',A*,B)

=

Q(F0(A',A*,B),...,Fn(A',A*,B),B0,...,Bm)

Q'(A, B', B*) = Q(A0, ...,An,

G0(A, B', B*),...,

Gm(A, B', B'))

where Q, Q ,Q, Q' are polynomials over R in the exhibited indeterminates. Now clearly Q(A' ,A*,B) = P< n ' m ) {F'F*, G) Q'(A,B',B*)

= P ( "' m ) (F,G'G*)

and AQ71 and B^ are nonzerodivisors in R[A', A*,B] and R[A, B', B*} respectively, and hence by (24) and (24') we get (25)

Q(A',A*,B)=Q(A',A*,B)

and (25')

Q'{A,B',B*)

=

Q'{A,B',B*).

Upon letting a = (a0,... ,an), b = (b0,... ,bm), a' = (a'0,... ,a'n,),b' = (b'0,... a* = ( o 5 , . . . , a ; . ) , b* = (bl,...,b*m.), we clearly have

,b'm,),

P ( " ' m ) (/, 9) = Q(a', a*,b) = Q'(a,b',b*) p(n\m)

( / / ) g)pin>

,m)

p(n,m')

( / ) g,)p{n,m*)

( /

. ^ g)

=

( / ? ff,} =

g ( f l / > fl* ( ft) g'^

^

^

and therefore upon substituting small letters for the corresponding big indeterminates in the polynomial identities (25) and (25') we get (26)

P{n'm)(f,g)

=

P{n''m)(f',9)Pin*'m)(r,9)

P ( " ' m ) (/,
P{n'm,)(f,9')P{n'mn(f,9*)-

and (26')

By repeated applications of (26) and (26') we get the

%12:

REMARKS

187

Product Rule for Resultant: if u, v, n(i),m(j)

are in N

with n = E i ^ u ^ W and m = E i < j < « m ( i ) (27)

{ and / ( i ) (y), s ( i ) (y) are in R[Y] with / = Ui
f(i) ^ d 5 = n K i < . 9U)

then P("- m )(/, f f ) =

Ili&KuYlimvPW'^Hf®,^)-

By (3), (4), (5), (6) and (27) we get the following three special product rules. First Special Product Rule for Resultant: (28)

if f(Y) = a0 Ui
R

Second Special Product Rule for Resultant: (29)

'if g(Y) = 60 UiKjKmV ~ Pi) w i t h & i n R mn 6&Ill<,-<m/(^)t n e n p(«.m)(/, fl ) = ( - l )

Third Special Product Rule for Resultant:

(30)

'if f(Y) = a0 Ui
- «i) and g(Y) = b0 Yl^^Y

= a^UiKiKnUi^Km^i

- Pi)

- Pi)-

By (8) and (27) we get the Tiny Resultant theorem: if q, n', ml are in N with q > 0 and n = q + n' and m = q + m' (31)

and / = hf and g = hg' where h, / ' ,
By (8) and (31) we get the Small Resultant theorem: if R is a domain with m + n ^ 0 and / and g have a (32)

nonconstant (i.e., of positive y-degree) common factor in R[Y] thenP("' m >(/,0) = O.

188

LECTURE L4: VARIETIES AND MODELS

By (32) we get the Little Resultant theorem: if R is a domain with m + n ^ 0 and f(-y) = 0 = 3(7) (33)

for some 7 in an overfield of R thenP("> m )(/,#) = 0.

In (E7) of §10 we proved the Basic Fact (Tl) stated in §1. Now, in view of (3) to (5), by (28) or (29) we get another proof of (Tl) which we restate as (35). The version (34) is clearly equivalent to (35). First Version of Resultant theorem: (34)

if R is a field then: p(n,m) = o<=>m + n ^ O a n d either a0 = 0 = b0 or / and g have a nonconstant common factor in R[Y}.

Second Version of Resultant theorem: if R is an algebraically closed field then: (35)

p(n,m) = 0 < ^ m + n ^ 0 a n d either aQ = 0 = b0 or / and g have a common root in R.

REMARK (R7). [Intval Domains, Divisibility Rings, Maximality Rings, and Equivalent Valuations]. From L2§3, L3§7, and L3§8, recall various terms related to valuations, and note that out of assertions (Jl) to (J26) about valuations made in L3§§7-9, we still have to prove that: (J24)

every normal domain is intval

where a domain is intval means it is an intersection of a nonempty family of valuation rings of its quotient field. We shall now prove this as well as the Valuation Theorems (T25) to (T28) stated in §9. First let us introduce some more concepts. By a divisibility ring we mean a domain in which, out of any two nonzero elements, at least one divides the other. In other words, a domain R with quotient field K is a divisibility ring means it satisfies one and hence all of the following five equivalent conditions: (1) (2) (3) (4) (5)

For any For any For any For any The set

nonzero element x in K we have either x € R or 1/x G R. nonzero elements x,y m K we have either x/y £ R or y/x € R. nonzero elements x, y in R we have either x/y £ R or y/x € R. principal ideals I,J'mR we have either I c J or J C I. of all principal ideals in R is linearly ordered by inclusion.

%12:

REMARKS

189

To see the equivalence of (1), (2), (3), in (1) change x to z and then write z = x/y. The equivalence of (3), (4), (5) is obvious. Given any divisibility ring R with quotient field K, we introduce an ordered abelian group T(R) together with a valuation JR : K —> T(.R)u{oo} in the following manner. We call these the divisibility group of R and the divisibility valuation of K induced by R respectively. It will turn out that the value group of 7# is T(R), and the valuation ring of 7^ is R. As a group T(R) equals the factor group U(K)/U(R) written additively, where we recall that U(R) denotes the group of all units in R. The restriction of j R to U(K) = Kx is the residue class epimorphism and we stipulate that JR(0) = 00. For any x,y in Kx we define: 1R{X) < JR(V) «• y/x e R. Since R is a divisibility ring, it follows that the relation < on T(R) is well-defined (i.e., is independent of the representatives in Kx) and converts it into an ordered abelian group. Using the assumption that R is a divisibility ring, we also see that JR is a valuation of K with value group T(R) and valuation ring R. By a divisibility ring of a field K we mean a divisibility ring with quotient field K. We may restate the above construction by saying that: if R is a divisibility ring of a field K then (7.1)

the divisibility group T(R) is an ordered abelian group and the divisibility valuation 7^ is a valuation of K with value group T(R) and valuation ring R.

Note that for any valuation v and any nonzero x, y in Rv we have either v(x) < v(y) or v(y) < v(x) and hence either y/x G Rv or x/y e Rv, and therefore Rv is a divisibility ring. This may be taken as a motivation for the above definition of a divisibility ring. Also note that for any valuation v of a field K, the restriction of v to Kx gives an epimorphism Kx —> Gv with kernel U(RV) and for any x, y in Kx we have: v(x) < v(y) <£=> y/x € Rv. This motivates the above definitions of the divisibility group and the divisibility valuation of a divisibility ring. Thus we have shown that: (7.2)

I a domain is a valuation ring { J^iff it is a divisibility ring.

Valuations v and w of a field K are equivalent means there is an order (preserving) isomorphism 8 of the value group Gv onto the value group Gw such that for all x S Kx we have 6(v(x)) = w(x). By what we have said above, it follows that: J every valuation is equivalent to the I divisibility valuation of its valuation ring

190

LECTURE L4: VARIETIES AND MODELS

and (7-4)

I valuations of a field are equivalent < [ iff their valuation rings coincide.

We claim that for any valuation v and for any ideals / and J in Rv we have: (*) either I c J or J C I. To see this, first note that: (**) x G J and y G Rv with v(x) < v(y) => y £ I (because y = x{y/x) with y/x G Rv). Now suppose J £ I and take y £ J\I. Then for every x £ I, by (**) we get u(x) > v(y) and hence by applying (**) to J we get x £ J. This proves (*). Now in view of the equivalence of (1) to (5), by (7.2) and (*) we see that: (7.5)

I a domain is a valuation ring { I iff the set of all ideals in it is linearly ordered.

By a maximality ring of a field K we mean a subring R of K such that R is a quasilocal domain which is not dominated by any quasilocal domain S which is a subring of K with S ^ R. We claim that: J a subring R of a field K is a valuation ring of K 1 iff it is a maximality ring of K. Namely, if R = Rv for some valuation v of K and S is a quasilocal domain dominating R such that S is a subring of K with S ^ R then we can find x £ S\R; clearly x G X \ -R =>• w(x) < 0 => (l/x) G M(i? t) ) =» (l/x) G M(S) (because S dominates R) => x ^ S* which is a contradiction. Conversely assume that R is a quasilocal domain which is not dominated by any quasilocal domain S which is a subring of K with S ^ R. Given any u £ K \ R let (j> be the unique i?-epimorphism of the univariate polynomial ring R[X] onto R[u] which sends X to u. Let J be the ideal in R[X] generated by ker() and M{R). If J ^ -R[^] then by Zorn's Lemma we can find a maximal ideal N in -R[X] with J C N, and this would give us a maximal ideal (./V) in R[u] whose intersection with R equals M(R); localizing R[u] at (j>(N) gives us a quasilocal domain R[u]^N) which dominates R and is a subring of K different from R; this is a contradiction. Therefore J = R[X], and hence we can find a positive integer n, elements a0,-.. ,an in R, and elements po ... ,pn in M(R) such that do + a\u + • • • + anun = 0 and (a 0 + aiX H 1- anXn) + (po+PiX -\ \-pnXn) = 1. The last equation implies that a0 = 1 -p0 £ R\M(R) and a, = —pi £ M{R) for 1 < i < n. Consequently a0 is a unit in R, and upon letting bi = ai/ao we get bi £ M(R) for 1 < i < n with

(!/«)"+ £ fc(l/«)n-< = 0. Ki
By minimizing n we may assume that 1/u does not satisfy any equation of this type of degree smaller than n. Suppose if possible that 1/u g- R. Then in a similar

§i2: REMARKS

191

manner we can find a positive integer m together with elements c; e M(R) for 1 < i < m such that

+ J3 c^-^O l
and again we can assume m to be minimal. Without loss of generality we may assume that m>n. Multiplying the above displayed equation for 1/u by cmun and then subtracting it from the above displayed equation for u we get d0um+

Yl

dium-i

=0

l
with d0 € R\M(R) unit do we obtain

and dt € M(R) for 1 < i < mum +

J2

1. Dividing throughout by the

em™"1 = 0

l
with ej £ M(R) for 1 < i < m — 1. Since u ^ 0, by the above equation we must have m > 1. Now dividing by u we get the equation u" 1 - 1 +

^

e^u" 1 - 1 -' = 0

l
which contradicts the minimality of m. This shows that R is a divisibility ring with quotient field K and hence we are done by (7.2). Given any chain (under domination) of quasilocal domains which are subrings of a field K, their union is clearly a quasilocal domain (whose maximal ideal is the union of the maximal ideals of the members of the chain), which is a subring of K and which dominates every member of the chain. Thus the said union is an upper bound of the chain. Therefore, in view of (7.6), by Zorn's Lemma (see L2§5) we conclude that: J if R is a subring of a field K such that R is quasilocal I then R is dominated by some valuation ring of K. We claim that J if P is a prime ideal in a subring A of a field K then \A is contained in a valuation ring T of K with P = M(T) n A. This follows by taking R = Ap in (7.7). Next we claim that: J if y is a nonunit in a subring A of a field K then [A is contained in a valuation ring T of K with y £

M(T).

This follows from (7.8) by noting that by Zorn's Lemma there exists a prime ideal (in fact a maximal ideal) P in R with y e P.

LECTURE L4: VARIETIES AND MODELS

192

If w is a valuation of a field L and K is a subfield of L then we get a valuation v of K by putting v(x) — w(x) for all x G K, and clearly Rv = Rw !~\ K and iJ w dominates R„. Thus we have shown that: .

J if S is a valuation ring of a field L and K is a subfield of L y then 5 n K is a valuation ring of K dominated by S.

If R is a valuation ring of a field if and L is an overfield of K then by (7.7) R is dominated by some valuation ring S of L and by (7.6) and (7.10) we must have R — SC\K. Thus we have shown that: if R is a valuation ring of a field K and L is an overfield of K (7.H)

then R is dominated by some valuation ring S of L and for any such pair we have R = 5 PI K.

This completes the proof of Theorems (T25) to (T28) of §9. If a subring R of a field K is integrally closed in K then for any x G K\Rwe must have x $ R[l/x] (because otherwise we can write x = qo + q\(\/x) -\ h qn(l/x)n n with n in N and qo,...,qn in R, and multiplying by x this would give an equation xn+1 — qoXn — qix11"1 qn = 0 for x over R), and therefore by taking A = R[l/x] and y = 1/x in (7.9) we can find a valuation ring T of K with R[l/x] C T such that 1/x G M(T) and hence such that x £T. Since every valuation ring is normal, it follows that: J if a subring R of a field K is integrally closed in K then I i? is the intersection of all valuation rings of K which contain it. This completes the proof of (J24). REMARK (R8). [Modelic Proj]. Let A b e a subring of a field K. Given any family (X;); 6 A of elements in K, with Xj ^ 0 for some j G A, let E = W(A; (XI)I&A) and let K' = QF(j4[(i|/ij)igA]) where t £ A with xt ^ 0, and note that K' is clearly independent of the choice of i. Also let yi = xi/xi> for any fixed i ' s A with Xi> ^ 0. Theorem (T29) of §9.1 says that then we have the following. (T29.1) E is a premodel of K'/A and E = W{A; (xt/x)leA) for all 0 / x G K. In particular (yi)ieA is a family of elements in K' such that yj ^ 0 for some j G A, and we have E = 221(^4; (yi)ie\)(T29.2) Given any R G E and any subring S oi K such that S1 is a quasilocal ring dominating R, there exists j G A with Zj 7^ 0 such that £;/:rj G S for all / G A. Moreover, for any such j we have R = BQ where B = A[(XI/XJ)I&A] and Q = BnM(5). (T29.3) E1 is a semimodel of K'/A, and if A is finite then E is in fact a complete model of K'/A.

%1S: REMARKS

^

193

PROOF OF (T29.1). It suffices to note that if xt ^ 0 then for all/ G A we have = a;,/^ and hence A[(^)ieA} = A[(xi/xi)leA].

PROOF OF (T29.2). Since R G W(A; {xi)ieA), there exists f G A with xr ^ 0 such that R G 33(B') where B ' = A[(xi/xj')i&A], and then we have xijxy G P for all I G A, and P = B'Q, where Q' = B' n M(R). Since 5 dominates P , we get a;//a;j' G 5 for all / G A, and Q' = B' n M(S). Therefore the first assertion follows by taking j = j ' . To prove the second assertion, given any j G A such that Xj =fi 0 and if/xj G S for all Z G A, let B = A[(xi/xj)i^A] and Q = B n M(5). Then 1 XJ/XJ' and xy /XJ are both in S and hence they are units in 5. Since Xj/xy G P , 5 dominates P , and Xj/xy is a unit in 5, we get that Xj/xy is a unit in R and hence XJ>/XJ G P ; consequently XI/XJ = {xi/xy){xy/XJ) G R for all / G A and hence B C R; since 5 dominates R and <2 = B(~\M(S), we get that <3 = Bf)M(R) and hence B Q C R. Again since XJ>/XJ G BQ, S dominates BQ, and xy/xj is a unit in S, we get that xy /XJ is a unit in -BQ and hence Xj/xy G -BQ; consequently xi/xy = (XI/XJ)(XJ/XJ>) G B Q for alU G A and hence B' C B Q ; since 5 dominates BQ and Q' = B ' n M(5), we get that Q' = B' n M(B Q ) and hence B Q , C B Q . Since -R = B Q , , we conclude that R = BQ. PROOF OF (T29.3). The first assertion follows from (T29.1) and (T29.2). To prove the second assertion assume that A is a finite set. In view of the first assertion it suffices to show that if V is any valuation ring of K' containing A then V dominates %0(A; (xi)ie\). Since A is a finite set, there exists j G A with Xj jL 0 such that XI/XJ G V for all / G A. Let B = A[(xi/xj)iex\ and Q = BC\M{V). Then V dominates BQ and BQ G W(A; (xi)ieA). REMARK (R9). [Modelic Blowup]. Let A be a subring of a field K, and let P be a nonzero A-submodule of K. Theorem (T30) of §9.2 says that then we have the following. (T30.1) For any 0 ^ x G P we have (A[Pa;- 1 ])P = {A[Px~l))x and hence RP = Rx for every R G V{A[Px~x}). In particular, if P is a nonzero ideal in A then for every R G 2J(A[.Pa:-1]) we have that PR is a nonzero principal ideal in R. (T30.2) Given any family {xi)ieA of generators of P, for all 0 ^ x G P we have A[Px~l] = A[{xi/x)l€A], and we have W(A; (ZJ)J€A) C W(A, P). (T30.3) For any family (:E/); 6 A of generators of P we have W(A,P)

=

W(A;(xl)l€A).

(T30.4) For all x ^ 0 + y in P we have Q F ^ P a ; - 1 ] ) = QF(A[Py- x ]) and letting K' denote this common QF we have that W(A, P) is a semimodel of K'/A, and if P is a finitely generated ^4-moduIe then %B(A, P) is a projective model of K'/A. In

194

LECTURE H: VARIETIES AND MODELS

particular, if P is a finitely generated ideal in A then W(A, P) is a projective model of QF(A)/A. UP = Ax for some 0 ^ x G K then W(A, P) = W(A; x) =
= R}.

(T30.6) If ^4 is quasilocal and P is an ideal in A then: P is a principal ideal in A «• 2J(A, P) = 9J(A) o ^ G 2»(A, P ) .

PROOF OF (T30.1). Obvious. PROOF OF (T30.2). The first assertion is obvious. The second follows from it. PROOF OF (T30.3). In view of (T30.2) it suffices to show that W(A,P) C W(A; (xi)ieA). So let any R G W(A, P) be given. Then there exists 0 / i e P such that R = BQ where B = AlPx'1] = A[{xi/x)i€K} C R and Q = B n Af (P). Since 0 ^ i £ P , there exists a nonempty finite subset A' of A such that x = J2i£\> rixi with ri G A. Then 1 = J2I&A> ri{xi/x) a n d n G R and (zj/a:) G P for all I G A', and hence there exists j G A' such that Xj/x $. M(R). Consequently Xj ^ 0, and Xj/x and X/XJ are units in R. In particular XI/XJ = (XI/X)(X/XJ) G R for alH G A and hence B' C R where B ' = A[(a;j/a;j)jeA]. Upon letting Q' = B ' n M ( P ) we get that B'Q, G W(A; (xi)ie\) and P dominates BQ,. Since z/z.,- G P Q , , R dominates BQ,, and X/XJ is a unit in R, we get that X/XJ is a unit in B'Q, and hence Xj/x G P Q , . Consequently xi/x = (XI/XJ)(XJ/X) G P Q , for all I £ A and hence P C B'Q,; since P dominates B'Q, and Q = P fl M(R), we get Q = B D M(B'Q,) and hence S Q C B'QI, i.e., RCB'Q,. Therefore R = B'Q,, and hence P G 2ZJ(A; {xi)t€A). Thus 22J(AP)c21J(^;(^);eA). PROOF OF (T30.4). Follows from (T30.3) and (T29). PROOF OF (T30.5). Assume P is an ideal in A. First let P G <8(A) such that PR = P; then P ^ M(R) and hence there exists 0 ^ x G P such that x g M(R); now A[Pa: _1 ] c P and hence, upon letting Q = A[Px~l\ n M ( P ) , we get P = (AIPX-^Q G 23J(AP)- Conversely let R G 22J(^,P) such that PR = P; now W(A,P) dominates 9J(A) and hence there exists R' G 5J(A) such that P dominates P'; since R dominates R' and PR = R, it follows that PR' = R'\ therefore R' G W(A, P) by what we have already proved; since P dominates P as well as R', and by (T30.4) W(A, P) is an irredundant premodel of the quotient field of A, we must have R = R'\ consequently R G %3(A). PROOF OF (T30.6). Assume A is quasilocal and P is an ideal in A. Now: P is a principal ideal in A => P = xA for some 0 ^ x € A => 2U(A, P) = 2U(A; a;) = 2J(v4).

§i3: DEFINITIONS AND EXERCISES

195

Also clearly: W(A,P) = A £ W(A,P). Finally: A £ W(A,P) =• A £ WiAlPx-1}) for some 0 ^ x £ P => AlPx'1} C A ^ AWx'1} = A => PA = P(A[P:r _ 1 ]) = x{A[Px~l\) = xA =$> P is a, principal ideal in A. §13: DEFINITIONS A N D EXERCISES EXERCISE (E10). [Hilbert Basis Theorem]. Show that an alternative proof of the polynomial case of the Hilbert Basis Theorem, i.e., Theorem (T3) of §3, can be obtained by slightly modifying the proof of the power series case, i.e., the proof of Theorem (T4) of §3. HINT. In the proof of (T4), everywhere change [[]] to [], and in the second summation of display (2*) change < oo to < deg(s). EXERCISE ( E l l ) . [Primary Modules]. Let P be an ideal in a ring R and let Q be an Pi-submodule of an Pi-module V. Recall that Q is primary if Q ^ V and for any r £ R and s £ V with rs £ Q and s $. Q we have r £ radyQ where radyQ = {r £ R : reV C Q for some e £ N+}. In §5(013) (which obviously generalizes §5(07) and §5(08) from ideals to modules) we said that the following three conditions are equivalent and when they are satisfied Q is called P-primary. (i") Q is primary with radyQ = P. (ii") P C radyQ ^ R and: r£R and s£V\Q with rs £ Q => r £ P. (iii/—) Q is primary, &niiR(V/Q) is primary (as ideal), and P is prime with radyQ = PWe also stated the following four consequences of the said equivalence: (1") Q is P-primary and B <£. Q for submodule B of V =3- (Q : B)R is P-primary. (2") Q = Qi D • • • n Qh where Qt is P-primary for 1 < i < h £ N + => Q is P-primary. (3") P is a maximal ideal in R with P C radyQ ^ R =>• Q is P-primary. (4") P is a maximal ideal in R and PeV cQ^V (for instance Q = PeV) with e £ N+ => Q is P-primary. The exercise is to give the details of the proof. HINT FOR THE EQUIVALENCE. The definition of primary can be paraphrased in terms of zerodivisors by saying that

(11.1')

Q is primary <=*> Q ^ V and

ZR(V/Q)

C radyQ

196

LECTURE L4: VARIETIES AND MODELS

and similarly for an ideal J in R (11.2')

J is primary <=> J ^ R and ZR(R/J)

C rad^J.

Likewise (11.3')

(ii") » P c radyQ ^ i? and ^ ( V / Q ) C P.

Also clearly (11.4')

Q ± V & r&dvQ + R

and (11.5')

radvQ = T&dR&imR(V/Q)

and (11-6')

ZR{R/onnR(V/Q))

C

ZR(V/Q).

Putting annj^V/Q) = J, by (11.1') to (11.6') we see that (i") => (ii") =}• Q is primary and a.nnR(V/Q) is primary, and obviously (iii") =>• (i"). To complete the equivalence proof, we need to show that (ii") =*• P is prime with radyQ = P. For any r in vadyQ, by (11.5') we have re £ ann^(V r /Q) for some positive integer e; letting e to be the smallest such, and assuming radvQ ^ R, by (11.4') we get r 6 " 1 £ ann jR (V/Q); it follows that r e V C <2 and re~lV <£ Q; since re~lV £ Q, we can find t £V with r e - 1 £ ^ Q, and since r e V C Q, we get r e £ S Q; thus: 11.7'

f for some e € N + and £ £ P we have r e rad^Q ^ ii =>• I e , \r t £ Q with re~H £ Q.

Assuming (ii"), in the above set-up, by taking s = re~1t we have s £ V \ Q with rs £ Q and hence we get r £ P. Thus (11.8')

(ii") =• P = radvQ.

Again assuming (ii"), given any r\ £ R\P and r-i £ R with r £ P where r = r?,r\, in view of (11.8'), by taking s\ = {r2T\)e~1t in (11.7') we get si £ V \ Q with r2TiSi £ Q; since r\ £ R\P and si £ V \ Q, by (ii") we get S2 £ V \ Q where S2 = T\SI; since r2 £ R and s2 £V\Q with T2S2 £ Q, by (ii") we get t2 £ P . Thus (ii") => P is prime. HINT FOR THE FOUR CONSEQUENCES. To prove (1"), assume that Q is P-primary and B <£_ Q for submodule B of V. We want to prove that (Q : B)R is P-primary. By the definition of (Q : B)R we get (Q : B)RB C Q; in view of (ii") and because Q is P-primary, the last inclusion and the noninclusion B • p e V C Q for some e £ N+ => p e P C Q (because B c V) => pe £ (Q : B)R; thus:

§i3: DEFINITIONS AND EXERCISES

197

radv<2 C radfl(<3 : B)R; since Q is P-primary, we also have P C r&dyQ, and hence: (b) P C rad fi (Q : B)R. By (a) and (b) we get: (c) radR(Q : B)R = P. Now: rGR and s G R\(Q : B)R with rs G (Q : £ ) # =>• r G i? and s G R such that for some v G B we have sv £ Q but rsu e Q ^ r g rad„<3 =>• r G P by (c); thus: (d) r G R and s G R\(Q : B)R with rs G (Q : B)R => r G P. in view of (ii"), by (c) and (d) we see that (Q : B)R is P-primary. To prove (2"), assume that Q = Q\ n • • • f~l Qh where Qi is P-primary for 1 < i < h G N+. Then radyQ — (radyQi) n • • • (~1 (radyQh) and hence rad^Q = P . Now: rGR and s G ^ \ Q with rs G Q => r G R and for some i G { 1 , . . . , h} we have s G y \ Q i with rs G Qi =>• r £ radyQ» = P by (ii"). Therefore again by (ii") we conclude that Q is P-primary. To prove (3"), assume P is a maximal ideal in R with P C iadvQ ^ R- Let r G R\ P and s G V be such that rs G Q. In view of (ii"), it suffices to show that then s G Q. Since P is maximal, we must have P + Rr = R and hence we can write 1 = c + dr with c G P and d G R. Since c G P C rad^Q, we can find m G N + with c m V C Q. Raising the equation 1 = c + rfr to the m-th power we get 1 — lm = cm + d'r with d' G R and multiplying throughout by s we obtain s = cms + d'rs G Q. To prove (4"), in view of (3"), it suffices to note that P C radvPeV. DEFINITION (Dl). [Support of a Module]. The support of a module V over a ring R is defined by putting s u p p f i ^ ={PG

spec(P)

:[0:P}v^V}.

EXERCISE (E12). [Support a n d Annihilator]. Show that for the support of a module V over a ring R we have the following. (E12.1) If V = RB with B c V then s u p p l y = \Jb&B vspec fi (ann fl &). (E12.2) If V is finitely generated then s u p p l y = vspec i j(ann/jK). HINT FOR (E12.1). Assuming V = RB, for any P e spec(P) we have P G supply &[0:P}v^V <=» there exists b G B such that b g [0 : P]y < <=> there exists b G B such that b g (0 : s)v for all s G R \ P «=> there exists b G B such that bs ^ 0 for all s G R \ P O there exists b G B such that &xmRb C P .^

p

£ Ubes vspec^ann^fc).

198

LECTURE L4: VARIETIES AND MODELS

HINT FOR (E12.2). Assuming V = RB with finite set B = { 6 1 , . . . , &„} where n G N+, for any P G spec(P) we have P e supply < «=> P G vspec H (ann fl 6j) for some i

[by (E12.1)]

<^> ann/{6j C P for some i and obviously P G vspecfl(ann/{V) <=$> arniflV' C P and hence it suffices to show that ann#6j c P for some z <=> ann/jV C P. The =>• part of the above is obvious because for all i we have ann^V C ann#&i. Moreover: the negation of the LHS => for all i we have ann#6j • for each i there is S, G P \ P with Sj&j = 0 = * > S I . . . S „ G P \ P with si... snV = 0 =>• the negation of the RHS. EXERCISE (E13). [Support and Quasiprimary Decomposition]. Let U be a submodule of a module V over a ring R. Show that: (E13.1) If U ^ V and V is finitely generated then SuppR(V/U)

= {P£ spec(P) : [U : P]v * V}

and suppR(V/U)

= {P G spec(P) : [f : P]v is P-quasiprimary}

(E13.2) If P is a minimal element of s\xppR{V/U)

then [U : P]v is P-primary.

HINT. Use the above Exercise (E12) and Theorem (T8') of §6.1. DEFINITION (D2). [Homogenization and Dehomogenization]. Given a polynomial / ( Y i , . . . , Yjv) in indeterminates Y i , . . . , Yjv over a field fc, taking a nonnegative integer d > deg(f), we can "homogenize" / to get a homogenous polynomial F — F{X\,..., XJV+I ) of degree d in indeterminates X\,..., XN+I over k by the substitution Yj = Xi/X^+i, i.e., by putting F(Xi,...

,XN+I)

= XN+1/(XI/XN+I,-

••

,XN/XN+I)-

§J3: DEFINITIONS AND EXERCISES

Conversely, given any homogeneous F = F{X\,... "dehomogenize" it by putting f(Yu.

..,YN)

= F(Y1, ...,YN,1)

199

,Xjv+i) of degree d we can

= XJid+1F(XiXN+i,...

,YNXN+1,XN+1).

EXERCISE (E14). [Embedding Projective Space Into Projective Model]. In §9.1 we gave a natural injection 6 from the projective iV-space Pj^ over a field k into the modelic projective TV-space (P^)*5 = W(k;X\,... ,XN+\). There we used the irredundancy of 2B. Without using the said irredundancy, give a more elementary construction of the said injection on the following lines by using homogenization-dehomogenization. Given u = ( u i , . . . , ujv+i) G A ^ + 1 \ { ( 0 , . . . , 0)} we want to find its image in W(k;X\,...

,XN+I)

= Ui
with BN,k,i = k\Xii' ••••> YN+i,i]

where

Yu =

X\/Xi.

For i, i' in { 1 , . . . , N + 1} with m ^= 0 ^ Ui> consider the local rings Rk(y)* and Rk(y')* where v = ( u i / w i , . . . , UN+i/ui) G Aj^ and v' = (u 1 /u i /,...,u;v+i/ u «') G A ^ , . If we show that these local rings coincide then we can send the equivalence class of u in P ^ to that common local ring. So the exercise is to show that Rk{v)* =

Rk(vTHINT. The case of i = i' being trivial, suppose i ^ i'. Relabelling suitably we can arrange that i = 1 and i' = 2. Disregarding the coordinate Yu which identically equals 1, we have -Bjv.fc = BN,k,i — k[Yi,...,

Yff]

where (YUY2, ...,YN)

= (X2/X1,X3/X1,

...,

XN+1/X!)

and V = ( u 2 / u i , U 3 / W l , - - - . UN+l/ui)

G Aj^.

Likewise, disregarding the coordinate Yu> which identically equals 1, we have •"JV.fc

=

B{q,k,i'

=

k\Yi, . . . , XJVJ

where (^l'> Y2> •••' YN)

=

{Xl/X<2, X3/X2,

...,

XN+1/X2)

200

LECTURE L4: VARIETIES AND MODELS

and v' = (ui/u2,u3/u2,...

,uN+i/u2)

eAy.

Now let us note that the local ring Rk(u)* is the set of all quotients of polynomials F(Xi,... ,XN+I)/G(XI, ... ,XN+I) with G(u\,... ,UJV+I) / 0, the local ring Rk(v)* is the set of all quotients of polynomials f(Y%,..., Y^)/g(Yi,..., YN) with g(v\,..., vpf) ^ 0, and the local ring Rk(v')* is the set of all quotients of polynomials f'(Y{,..., Y[<)/g'{Y{,..., y/r) with g'(v[,..., v'N) ^ 0. For f/g in Rk(v)*, homogenizing, i.e., multiplying / and g by Xf for large enough positive integer d we get f/g = F(X\,... ,XM+I)/G(X\, . . . , X J V + I ) where F,G are homogeneous of degree d with G(ui,... ,UN+I) ^ 0, and hence F/G £ Rk(u)* • Dehomogenizing, i.e., dividing F and G by X2d we get F/G = f'{Y{,... ,Y^)/g'{Y{,... ,Y^) with 5'(ui,...,«Ar) + 0, and hence f/g' G Rk(v')*. Thus Rk(v)* C Rk(v')*, and by symmetry Rk(v')* C Rk(v)*. Therefore Rk{v)* = Rk{v')*. §14: N O T E S NOTE (Nl). [Princeton Book]. My Princeton Book [AOl] provides a rapid introduction to parts of this and some of the following Lectures. In particular, Observation §5(O20)(13*) and Theorem §6(T8) stem from Chapter I Section 2 of that book, and Observation §5(O20)(14*) and Theorem §6.1(T8') are their modulifications. In [AOl] these items were motivated by the desire of giving a unified treatment of ramification of local rings which are always noetherian and valuation rings which are not (unless the value group is a subgroup of Z). NOTE (N2). [Resolution Book]. The language of models was introduced at the end of my Princeton Book [AOl] and was further developed in my Resolution Book [A05]. In particular the material of §8 and §9 is mostly taken from [A05]. More material from [A05] will be included in later Lectures. As a motivation for the idea of an irredundant premodel introduced in §9, the proof of the existence of a natural injection of a projective space into a projective model, which is given in §9.1 and which uses the irredundancy, should be compared with the longer but more elementary proof of the same which is given in §13(E14) and which uses the technique of homogenization-dehomogenization. NOTE (N3). [Kyoto P a p e r ] . The treatment of resultants given in §12(R6) is taken from my Kyoto Paper [A03]. The same is true of the material on Approximate Roots given in (D7) and (E15) to (E17) of L2§7. Later Lectures will contain further material from that paper. A significant feature of the treatment of resultants given in §12(R6) is the establishment of product rules for the resultant matrix and the ensuing deduction of the product formula for the resultant itself. This makes the whole theory applicable to polynomials over a ring with zerodivisors. The methods

§15: CONCLUDING NOTE

201

used are longer but more elementary since they depend only on the grade-school operations of addition and multiplication and do not involve root-extractions. NOTE (N4). [Valuations and L'Hospital's Rule]. The L'Hospital Rule from calculus says that if x(t),y(t) are infinitely differentiable functions of a real variable t which are not identically zero then the limit of x(t)/y(t) as t tends to zero always exists although it may be infinite. Motivated by this, what we have called a divisibility ring in §12(R7) could have been called a L'Hospital ring, and then assertion (7.2) of §12(R7) would have said that a L'Hospital ring is nothing but a valuation ring. The situation for two or more variable functions is quite different, as shown by the quotient X/Y which takes all possible values as X and V both tend to zero. This fundamental indeterminacy is mitigated by the introduction of valuations which may be viewed as an algebraization of the idea of limits. As we shall see in later lectures, the said indeterminacies can be "removed" by means of the modelic blowups introduced in §9.2 NOTE (N5). [What Remains To Be Done]. Referring to §1(07) and the last para of §12(R5), the only items from §1 which remain to be dealt with are Bezout Observations (04) and (05). The only other assertions from the present Lecture which remain to be proved are Theorems (T19) to (T24) of §8; Theorems (T25) to (T30) of §9 were proved in (R7) to (R9) of §12. Finally, referring to L3§12(03), we proved (Jl) and (J2) in §10(E2) and we proved (J24) in §12(R7), and so the only item from Lecture L3 which remains to be dealt with is Homework (H3). §15: CONCLUDING N O T E It is good to keep a balance between algebra and geometry. Let logical formality and narrative discourse follow each other.

Lecture L5: Projective Varieties §1: D I R E C T SUMS OF MODULES Before studying projective varieties which are defined in terms of homogeneous polynomials, we prepare some algebraic background. Given any polynomial in a finite number of variables X\,..., Xm over a ring R, by collecting terms of like degree in it we can uniquely write it as a finite sum of homogeneous polynomials. This is formalized by saying that, as an .R-module, the polynomial ring S = R[Xi, • •., Xn] is a direct sum

S = £

St

0
where Si is the set of all homogeneous polynomials of degree i, including the zero polynomial. Moreover, for all fi G Si and fj G Sj we have fofj G Si+j. This is expressed by saying that S is a graded ring over R with gradation ( S i ) ^ . Quite generally, a module V over a ring R is a direct sum of a family (Vj);g/ of submodules if it is the sum

and the sum is direct, i.e., for every finite subset I' of the indexing set / we have: ]T\ e I , Vi = 0 with Vi G Vi for all i G / ' =» vt = 0 for all i € V. Note that V = Y^isi ^ means every v G V can be written as v = ^2ieIi Vi for some finite subset I' of I and some u< G Vi for all i G I'. The fact that the sum is direct is expressed by writing something like V — V^ Vi

(direct sum)

iei

or something like

In case / is a finite set, say I = { 1 , 2 , . . . , n}, we may write something like V = Vi 9 V 2 ® •••®Vn. These are sometimes called internal direct sums to distinguish them from the external direct sums which we shall now introduce and which we shall again denote by ®. To stress the fact of being internal, the above displays may be written as V = y~] Vi

(internal direct sum)

202

§/: DIRECT SUMS OF MODULES

203

or V =0

Vi

(internal)

iei

or V = Vi © V2 © • • • © Vn

(internal).

Recall that W = U\ x • • • x Un is the cartesian product (see L2§1) of a finite number of sets Ui,...,Un and it consists of all n-tuples u = (ui,...,un) with Ui G Ui. Assuming U\,... ,Un to be i?-modules, we convert W into an i?-module by componentwise addition and scalar multiplication, i.e., by denning u + u' = (ui+Ui,... ,un + u'n) for all it' = (u[,... ,u'n), and ru = {rui,... ,run) for all r £ R. With this module structure we call W the (external) direct sum of U\,..., Un and indicate this by writing W = U\ © U2 © • • • © Un

(external)

where we shall usually omit the adjective external. If U\ = • • • = Un — U then, as in the case of cartesian products, we put W = Un and call it the (module theoretic) direct sum of n copies of U. More generally, the cartesian product of a family (Ui)i£i of sets Ui, where / is any indexing set (which need not be finite), is denned by putting IT Ui = the set of all maps <j>: I —» M Ui such that cf>(i) G Ui for all i & I. iei

i€l

In case the Ui are i?-modules, this is converted into an f?-module again by defining componentwise addition and scalar multiplication, i.e., for any tj> € Yiiei Ui w e P u ^ ((i) for all i £ I; this module is called the (external) direct product of the family, where the adjective external is usually omitted. The set of all <j) G Elie/ ^» whose support supp(0) = {i £ I : (i) ^ 0} is finite is called the (external) direct sum of the family and is denoted by ^H Ui

(external)

where the adjective external will usually be omitted. Note that this is a submodule of the direct product Y\isI Ui and is hence an .R-module. For any j G / we get an injective map (called the natural injection) \ij : Uj —> Ilie/ U*tys a v m g that for every x G Uj we have fij(x)(i) = x or 0 according as i = j or i ^ j , and we get a surjective map (called the natural projection) Vj : Y\i&1 Ui —> Uj by saying that for every y € YlieI Ui we have Vj(y) = y{j)- Clearly Vi\ii is the identity map of Ui. All this works for sets Ui. In the module case these maps are Rhomomorphisms, and they give rise to an obvious .R-monomorphism (again called the natural injection) JJ,J : Uj —• ®i^iUi and an obvious i?-epimorphism (again

204

LECTURE L5: PROJECTIVE

VARIETIES

called the natural projection) Vj : © i e / A —> Uj. Note that upon letting Ui to be the image of /x, we have ^ A i€I

(external) = ^

A

(internal).

iel

If the set Ui = a set U for alH e 7 then the cartesian product YlieI Ui is denoted by U1 and is called the set-theoretic 7-th power of U, and if U is an 7?-module then the module U1 is called the module theoretic J-th power of U. Finally, if the module Ui = a module U for alM € 7 then the module © i £ / A is denoted by U^ and is called the module theoretic restricted 7-th power of U. If {ai)iei is a family of 7?-homomorphisms on : Ui —• A where (Di)iGi is a family of .R-modules, then we get an 7?-homomorphism Y\ieI Ui —> YlieI A or ©,e/A —> © i e / A which sends any 0 in Y[iei A or © i e / A to i/> in flie/ A or © i e / A given by ?/)(i) = cti((j>(i)) for all i £ 7; we call this the direct product or the direct sum of the family (aj)jg/ and denote it by I l i e / a ' o r ®i£iai respectively. Note that the kernels of these maps are f ] i e / k e r ( a , ) and ©j e /ker(aj), and their images are Y[ieI im(ai) and ©i € /im(ai), respectively. Also note that for the definition of the ma P Yliel ai : Yliel A —* Hie/ A , it suffices that the a, : Ui —> Di be set-theoretic maps; in case Ui — U for all i G 7, the diagonal map [/ —» C/7 sends any a; e £/ to <j> £ U1 with (i) = x for all i € I, and multiplying it from the left by I L e / ai w e get a map U —> flie/ A which we call the diagonal product of the family {ai)i^i\ in case of modules, the diagonal map is an 7?-monomorphism and the diagonal product is an 7?-homomorphism. Note that if the indexing set 7 is finite then the direct product coincides with the direct sum. In particular, if 7 is finite then the diagonal product U —> Y\ieI Di = ©i € /7)j may be called the diagonal direct sum of the family (aj)j e /. If Ti is a submodule of A for all i e l , then clearly Yli€l Ti and ©ie/Tj are submodules of l i t e / A and © i S / A respectively, and upon letting /3j : Ut —> Uj/T, be the residue class epimorphism we see that Yliei Pi '• l l i e / A —* I l i e / ( A / 7 i ) and ©ig//?, : ©jg/A —> ®iei(Ui/Ti) are 7Z-epimorphisms with kernels ] l i e / ^ a n d ©ie/Ti respectively. For finite cartesian products Ui x •• • x Un, and for the corresponding external direct sums, we used the tuple notation. In case of indexed families (C/j)jG/, for cartesian products Y\ieI Ui and the corresponding external direct sums, we used the functional notation. The two are related by saying that the n-tuple u = ( u i , . . . , un) corresponds to the function (= map) u : { 1 , . . . ,n} —> Ujg/A with u(i) written as Ui. The concepts of the above several paragraphs carry over to the tuple notation. For instance the natural projection U\(B- • -®Un —> Uj picks out the j - t h component, and the natural injection Uj —» U\ ® • • • © Un puts zeros around it. Given maps oti : Ui —» A , their product a\ x • • • x an : U\ x • • • x Un —* D\ x • • • x Dn as

well as their direct sum a\ ® • • • ® an : U\ © • • • © Un —> £>i © • • • © Dn is given by ( u i , . . . ,u n ) H-> (ai(ui),... ,an(un)). Moreover, in case of U\ = • • • = Un = U,

%1: DIRECT SUMS OF MODULES

205

the diagonal maps U —> U\ x • • • x Un and U —> C/i © • • • © f/n are given by u t-> (M, . . . , u), and the diagonal product U —> £>i x • • • x D„ and the diagonal direct sum {/ —> £>i © • • • © Dn are given by u i-» ( a i ( u ) , . . . , a„(u)). We use the notation o?i © • • • © a n for the last map, i.e., we write a\ © • • • © an : U —> -Di © • • • © Dn

(diagonal direct sum)

where, as said above, a i © • • • © an(u) = (cti(u),... ,an(u)); we may drop the parenthetical phrase "diagonal direct sum" when clear from the context; note that now the maps on : U —> Di are from a common set U, and hence the notation a.\ © • • • © an : U —> D± © • • • ® Dn will not be confused with the similar notation a i © • • • © Q„ :U\®---®Un—>D\®---®Dn where the maps a» : U, —> Di are from differently labelled sets Ui. For the above diagonal direct sum we clearly have (1)

ker(ai © • • • © an) = ker(ai) n • • • D ker(a n ).

Most of the discussion of the above several paragraphs applies with internal direct sums replacing external direct sums. For instance the inclusion map Vj —> ®ieiVi is the natural injection, and the natural projection ©jg/Vi —> Vj is the obvious epimorphism whose kernel is ®i^i\{j}Vi. It is usually clear from the context whether we are dealing with internal or external direct sums. For instance when the constituents Vi are submodules of a module V then it must be internal, and when the modules Ui are not given to be submodules of some module then it must be external. For the lengths of finite (internal or external) direct sums of .ft-modules U\,..., Un, by induction applied to L4§5(021)(18 # ) we see that (2)

IR(UI

© • • • © [ / „ ) = W i ) + • • • + tR(Un)

with the usual convention about sums involving oo. All the above matter applies to additive abelian groups by viewing them as modules over Z. EXAMPLE (XI). [Annihilators and Direct Sums]. In L4§10(X18) we gave an example of a submodule T of a non-finitely-generated module U over a ring R together with a prime P in R such that a,miR(U/T) c P but [T : P]u = U, and in L4§10(X19) we gave an example of a nonprimary submodule W of a module V over a ring R such that aniiR(V/W) is a primary ideal in R. Now using (external) direct sums we give other such examples. First let x b e a nonzero element in a domain R such that r\ien{^lR) — 0; for instance R = the polynomial ring in one or more indeterminates X, Y,... over a field k, and x = X. Let U = ffiieN^ and T = ©ieN^i with Ui = R and I* = x'R. Let P be the zero ideal in R. Then P is a prime ideal in R. Given any O ^ r e f i we have r g xnR for some n £ N; we can take £ U with (f>(i) = 1 or 0 according as i = n or i ^ n, and then r € U with (r)(n) £ Tn and hence rcj) $. T\ consequently

206

LECTURE L5: PROJECTIVE

VARIETIES

r G" annn(C//T). Thus annfl([//T) = P. Given any ip G U we can find m G N such that i/)(i) = 0 for all i > m; now a:mi/> G T with x m G R\ P and hence V G [T : Pjy. Thus [T:P]u = U. Next let P be any domain having a nonzero prime ideal P ; in other words, let R be any domain which is not a field. Let OB and I s be the images of 0 G R and 1 G R in E = R/P. Let V = R © E and let W be the zero submodule of V, i.e., W = 0 © 0j5- Then the zero element in V is ( 0 , 0 B ) and for any r G R we have: r ( l , 0 £ ) = ( 0 , 0 B ) =>• r = 0; therefore ann#(V7W) is the zero ideal in R which is prime and hence primary. But we can take 0 ^ t £ P and then (0, 1B) G V \ W with i(0, 1 E ) = ( 0 , 0 E ) G W but no power of t belongs to a,rmR(V/W). Therefore W is not a primary submodule of V. §2: G R A D E D R I N G S A N D H O M O G E N E O U S IDEALS We are now ready to formalize the concept of a graded ring. Recall (see LI §4) that an additive abelian monoid is a nonempty set I together with an element 0 and a binary operation + which to every pair of elements i, j in I associates a third element i + j such that for all i, j , k in I we have: i + 0 = i, i + j = j + i, and (i + j) + k = i + (j + k). For instance, Z and N are additive abelian monoids, and for any positive integer d, so are Z d and Nd. By a graded ring S over a ring R we mean an overring S of R together with a family (5j)j g / of i?-submodules of S, where I is an additive abelian monoid (see Ll§4), such that S = Y^ Si

(internal direct sum of .R-modules)

with So = R and such that for all fi G Si and /_,- G 5,- with i, j in I we have fifj G S»+j. The family (5i)j g / is called the gradation (or grading) of S, and I is called the type of S. Elements of Si are called homogeneous elements of degree i. Collectively, elements of Ujg/Si are called homogeneous elements. The direct sum assumption tells us that every s G S has a unique expression

s = J2si with Si £ Si; the element SJ is called the homogeneous component of s of degree i, and collectively the elements st are called the homogeneous components of s. Note that Si — 0 for almost all (i.e., all except a finite number of) i. Also note that we have Cauchy Multiplication, i.e., if tj are the homogeneous components of t G S,

§2: GRADED RINGS AND HOMOGENEOUS IDEALS

207

and (st)k are the homogeneous components of st, then

(st)k = Y,

Sit

i-

i+j=k

An ideal J in S is said to be a homogeneous ideal if it satisfies one and hence all of the following three mutually equivalent conditions: (i) J = H,iei(J l~l Si) (as additive groups). (ii) The homogeneous components of every element of J belong to J. (iii) J is generated (as an ideal in S) by homogeneous elements. The unique decomposition s = Y2iei si yields (i) <=> (ii). Obviously (ii) =>• (iii). To prove (iii) => (ii), let (xj)jgi, be a family of generators of J with xi G S^i)- Any 1/6 J can be expressed as y = Y^ieL zixi where zi G S with zi = 0 for almost all L Writing y = Y,iei Vi where y, G Si with yt = 0 for almost all i, and zj = J2j&i zh where zij G Sj and z^ = 0 for almost all j , we get

and hence y, =

V,

z x

lj l

e

^

f ° ra u

i G /.

j+d(0=i

Thus we have shown that (1)

conditions (i), (ii), (iii) are mutually equivalent.

As in the case of an element, we call Si the homogeneous component of S of degree i, and more generally, for any homogeneous ideal J in S, we call J n Si the homogeneous component of J of degree i. We use i-th homogeneous component as a synonym for homogeneous component of degree i. Also we may say component instead of homogeneous component. By a graded subring of S we mean a subring T of S such that T is a graded ring of type I over T (1 R with gradation (T n Si)ie.rGiven any other graded ring S' of type / over a ring R' with gradation (5,-)^/, by a graded ring homomorphism : S —> S' we mean a ring homomorphism such that {Si) C S{ for all i G J. Clearly: if

S' is any graded ring homomorphism (2)

then im() is a graded subring of S' and ker(0) is a homogeneous ideal in S.

208

LECTURE L5: PROJECTIVE

VARIETIES

Conversely, given any homogeneous ideal J in S, upon letting tp : S —> S/J be the canonical epimorphism and ( 5 / J ) , = ip(Si), we can convert S/J into a graded ring of type / over ip(R) with grading (S/J)iei, and then clearly ip : S1 -> S / J becomes a graded ring epimorphism. Thus: if J is any homogeneous ideal in any graded ring S (3)

then the canonical epimorphism ip : S —> S/ J is a graded ring epimorphism. In view of (1) we see that:

if (Ji)i£L is any family of homogeneous ideals in S, (4)

then the ideals J2ieL ^ anc ^ ^leLJi and if the family is finite

m

S are homogeneous,

then the ideal YlleL Ji in S is also homogeneous. In view of (1) and (2) we see that: if (j>: S —> S' is any graded ring epimorphism (5)

then J — i > (/)(J) gives a bijection of the set of all homogeneous ideals in S containing ker(<j>) onto the set of all homogeneous ideals in S".

COMMENT (CI). [Alternative Definition of Graded Rings]. The assumption So = R implies that in the definition of a graded ring S it is not necessary to mention the subring R. At any rate, alternatively, a graded ring S of type / may be defined to be a ring S together with a family of additive subgroups ( S ^ g / , where the type I is an additive abelian monoid, such that we have the internal direct sum of additive abelian groups S = ^2ieI Si with 1 G S0 (where 1 is the identity element of S) and such that for all fi 6 Si and fj € Sj with i,j in / we have fifj £ Si+j. Note that in the alternative definition So is a subring of S, and hence the alternative definition is equivalent to saying that, in the previous sense, S is a graded ring of type / over So- Also note that if the monoid / is cancellative (i.e., i + j = i + k in / => j = k) then the condition that 1 € So is automatically satisfied. To see this let l j be the homogeneous component of 1 of degree j , and for any y € S let yi be the homogeneous component of y of degree i. Then j/j = j/,1 = Yljei V^J and hence by comparing homogeneous components of degree i we get j/j = yjlo because by cancellativeness i + j = i =>• j = 0. By summing over i £ I we get y = ylo- Therefore 1 = 1 0 € SQ.

§3: IDEAL THEORY IN GRADED RINGS

209

COMMENT (C2). [Cancellative Monoids and Ordered Monoids]. In a moment we shall show that if / is cancellative then for any homogeneous ideals J and H in S, the ideal ( J : H)s is also homogeneous. To introduce another useful property of monoids, recall that (see L2§3) an ordered abelian group is an additive abelian group G with a linear order < such that: i < i' and j < j ' in G => i+j M Ma+jiH \-jb < i[-\ \-i'a+j'i^ l-J^I note that j < f means j < j ' and j ^ j ' . Analogously, the (additive) abelian monoid J is said to be ordered if there is a linear order < on it such that: i\ = i[,..., ia = i'a,ji < j[,. • • ,jb < Jl in J with a e N and b € N+ ^ h + • • • + ia + j x + • • • + j b < i[ + • • • + i'a + j[ +' • • + fbIn a moment we shall show that if / is ordered then the radical of any homogeneous ideal in S is again homogeneous. Note that if / is ordered then it is obviously cancellative. COMMENT (C3). [Integrally or Nonnegatively or Naturally Graded Rings]. If / c Z, i.e., if / is a submonoid of Z, then we say that S is a subintegrally graded ring. If J C Z and Si = 0 for all i < 0 in / , then we say that S is a nonnegatively graded ring. If / = Z, then we say that S is an integrally graded ring. Finally, if 7" = N, then we say that S is a naturally graded ring. Note that in all these four cases, I is an ordered abelian monoid. §3 IDEAL THEORY I N G R A D E D R I N G S Again let I be an additive abelian monoid, and let S be a graded ring of type / over a ring R with gradation (5j)i e /. Given any y e S let j/i be the i-th homogeneous component of y, and let the support of y be defined by supp(y) = {i £ I: yi ^ 0}. Note that then V=

^2

Vi

with

0^yi£

Si.

iGsupp(y)

We claim that: assuming I to be cancellative, (1)

for any homogeneous ideals J and H in S, the ideal ( J : H)s is homogeneous.

Namely, for any generating set L of H we clearly have ( J : H)s — OxeL(J '• x)s and hence, in view of (iii) and (4) §2, it suffices to show that, for any j G I and x € Sj, the ideal (J : x)s is homogeneous. Given any y £ S we get ^ i6supp(y)

xyi = xy G J

with

xyi G Sj+i

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LECTURE L5: PROJECTIVE

VARIETIES

and, by cancellativeness, for all i ^ i' in supp(y) we have j + i^ j + i', and therefore the homogenity of J yields xyt £ J for all i G supp(y); consequently, j/ 4 G ( J : i ) s for all i € supp(y). Thus ( J : a;)s is homogeneous. Next we claim that: assuming I to be ordered, (2)

for any homogeneous ideal J in S, the ideal rads J is homogeneous.

Namely, by induction on the size of supp(?/) we shall show that y £ rads J =$• y% £ radsJ for all i £ supp(2/). If the size is zero then y = 0 and we have nothing to show. So let the size be positive and assume true for all smaller values of the size. Let d be the smallest element of supp(y). Since y £ rads J, we have yn £ J for some n £ N + , and raising both sides of the displayed expansion of y to the n-th power we see that Vn = Vd + YlUi

with

Vd e 5"d

and

u

i

£

S

i

i£A

where A is a finite subset of {i £ I : i > nd}. Consequently by the homogenity of J we get y% £ J and hence yd £ r a d s J and therefore y - yd £ rads J. Clearly supp(j/ - yd) = supp(y) \ {d} and (y — yd)i = y% for alii > d in J, and hence we are done by the induction hypothesis. To prepare for the next two claims, given any z £ S we can write z=

2_]

z

i

v^ith

0 ^ Zi £ S{.

iGsupp(z)

The usual criterion for a prime ideal may be sharpened by claiming that: assuming / to be ordered, (3)

for any nonunit homogeneous ideal J in S, we have that: J is prime •& for all homogeneous elements r, s in S with s & J and rs £ J we have r £ J.

To prove this, assume that for all homogeneous elements r, s in S\J we have rs £ J. It suffices to show that then for all nonzero elements y, z in S \ J we have yz £ J. Let d and e be the smallest elements of supp(y) and supp(,z) such that yd 0 J and ze 0 J respectively. Then ydze £ Sd+e \ J and yz = x + ydze + YlieB vi w ^ n x £ J and t>j £ Si for alH G B where B is a finite subset of {i £ I: i > d + e}. Therefore yz 0 J. Likewise, the usual criterion for a primary ideal may be sharpened by claiming

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211

that: assuming / to be ordered, (4)

1

for any nonunit homogeneous ideal J in S, we have that: J is primary <£> for all homogeneous elements r, s in S with s £ J and rs £ J we have r € r a d g j .

To prove this, assume that for all homogeneous elements r, s in S with s £• J and rs £ J we have r £ rads J. It suffices to show that then for all nonzero elements y, z in S with z g" J and yz £ J we have ?/ £ rads J. Let e be the smallest element of supp(z) such that ze $ J- Upon replacing z by z—^2{iEsuPp(z):i<e} z*> without loss of generality we may assume that for the smallest element e of supp(,z) we have ze g J. Let the size of supp(y) be c, and label the elements of supp(j/) as d(l) < • • • < d(c). By induction on b with 0 < b < c we shall show that y^Ci) £ radg J for 1 < i < b. For 6 = 0 we have nothing to show. So let b be positive and assume true for all values of b smaller than the given one. Now X^i
5Z Vd(i

£ J

,l
for some m € N + . Let x

=[y-

Z ) y^) l
Then xz € J and xz = y™{b)ze+ YlWi tec

with

V7(b)ze £ Smd{b)+e

and

Wi € Si

where C is a finite subset of {i £ I : i > md(b) + e}. Therefore y™tb\Ze £ J and hence y
generated by all homogeneous elements in H C S, for any prime (resp: primary) ideal J in S, the ideal (J)* is also prime (resp: primary).

To see this, let J be any prime (resp: primary) ideal in S, and let there be given any homogeneous elements in S with rs £ (J)* and s g" (J)*. Now clearly r,s are elements in S with rs £ J and s £ J. Therefore, since J is prime (resp:primary), we get r £ J (resp: rn £ J for some n £ N + ) . Since r is homogeneous, we must

212

LECTURE L5: PROJECTIVE

VARIETIES

have r G (J)* (resp: rn G (J)*). By (3) (resp: (4)) It follows that (J)* is prime (resp: primary). Recall that a primary decomposition of an ideal J in S is an expression of the form J = Qi n • • • n Qn where Qi,...,Qn are primary ideals in S with n G N, and this is an irredundant primary decomposition means the prime ideals r a d ^ Q i , . . . , r a d s Q n are pairwise distinct and for 1 < i < n we have

J ^ Qi n • • • n Qi-i n Q<+1 n • • • n Q„. If the primary ideals <3i, • •., Qn are all homogeneous then we call this a homogeneous primary decomposition or homogeneous irredundant primary decomposition respectively. We claim that: assuming I to be ordered, if a homogeneous ideal J in S has a primary decomposition in S then J has a homogeneous irredundant primary decomposition in S and all the associated primes of J in S are homogeneous. To see this, letting J = Qi D • • • D Qn t>e a primary decomposition in S, since an intersection of homogeneous ideals is homogeneous, by (5) we get a homogeneous primary decomposition J=(Qi)*n---n(Qn)* in S. By L4§5(08)(2*), in the last decomposition we can arrange that the prime ideals rads(Qi)*,.. •, ra.ds(Qn)* are pairwise distinct, and then by discarding some of the {Qi)* we can arrange that for 1 < i < n we have

J ± (Qi)* n • • • n (Qi_i)* n {Qi+ly n • • • n (Qny. Now, in view of L4§6(T6) by (2) we see that all the associated primes of J in S are homogeneous. COMMENT (C4). [Positive Portion and Irrelevant Ideals]. Assuming / to be ordered, we define the positive portion Q(S) of S by putting

n(S) = Yl

Si

{iel:i>0}

and we note that this is an i?-submodule of S, and if S is nonnegatively graded then it is actually a homogeneous ideal in S. Assuming I to be ordered, an ideal J in 5 is said to be irrelevant or relevant according as £l(S) C r a d g J or Q(S) <£. r a d g j .

§3: IDEAL THEORY IN GRADED RINGS

213

COMMENT (C5). [Relevant Portion and Isolated System of Prime Ideals] . Assuming S to be nonnegatively graded and noetherian, given any ideal J in S, let J = Q\ n • • • n Qn be an irredundant primary decomposition of J in 5 with ia.dsQi = Pi- By a suitable labelling we may assume that Q\,..., Qm are relevant and Qm+i, • • •, Qn are irrelevant. It follows that P i , . . . , Pm is an "isolated system of prime ideals" of J in S, i.e., for 1 < i < m and m + l<j
Qn to be homoge-

COMMENT (C6). [Homogeneous Rings and Semihomogeneous Rings]. Examples of nonnegatively graded noetherian rings are provided by homogeneous rings over noetherian rings. By a semihomogeneous ring S over a ring R we mean a naturally graded ring S over R such that for its grading (5j), e / with I = N we have S — R[Si] as a ring; this is clearly equivalent to saying that there is an il-epimorphism _R[(X;)/ £ L] —> S of graded rings where R[(Xi)ieL] is the polynomial ring in a family of variables over R which is naturally graded by putting R[{Xi)i£L)i = the set of all homogeneous polynomials of degree i (including the zero polynomial). By a homogeneous ring S over a ring R we mean a semihomogeneous ring S over R such that Si is a finitely generated .R-module; this is clearly equivalent to saying that there is an iZ-epimorphism R[Xi,..., XN} —> S of graded rings where R[X\,..., XN] is a finite variable polynomial ring over R which is naturally graded by putting R[Xi,..., X^]i = the set of all homogeneous polynomials of degree i (including the zero polynomial). Needless to say that a homogeneous ring over a noetherian ring is noetherian. Also needless to say that a homogeneous (resp: semihomogeneous) domain is a homogeneous (resp: semihomogeneous) ring which is a domain. For a moment suppose that S is a noetherian semihomogeneous ring over R, let J be a homogeneous ideal in S, and let the notation be as in Comment (C5) above. By the noetherianness we can find do G N + such that for all d > do in N + we have the inclusion Cl(S)d C Qi of homogeneous ideals for m + 1 < i < n, and hence we have the equality S^ = QiCiS^ for m + 1 < i < n, and therefore we have the equality J n Sd = (J)" n Sd- Instead of saying that there is d0 G N+ such that for all d > do in N+ we have J n Sd = {J)" ("1 Sd, for brevity we may say that J n Sd = (J) 8 n Sd for ALL LARGE ENOUGH d G N+. Now let if be a homogeneous ideal in S such that HC\Sd= (JynSd for all large enough d G N + . Then we can take e G N + such that H n Sd = (J)" n Sd = J n Sd for all d > e in N+. Given any i G { 1 , . . . , m}

214

LECTURE

L5: PROJECTIVE

VARIETIES

we have Q(S) <£ Pi and hence there exists t(i) G Si \ Pi\ for any y e H D Si with I G N, we have yi(i) e G HnSe+l = ( J ) ' nS e +/ = J n S e + ; with i(z) e 0 P and hence y G Q,; this being so for 1 < i < m we get y G (J)"; therefore by the homogenity of H we get H c (J) 8 . Applying the same result to (H)* we get (ff)* C (J) 8 . By symmetry we obtain (J) 8 C (i?)11. Therefore (.H")8 = (J)". Thus we have shown that:

assuming S to be a noetherian semihomogeneous ring over R, for any homogeneous ideal J in S we have J C) Sd = (J)* n Sd for all large enough d G N + , (7)

and moreover (J)" is the largest such homogeneous ideal, i.e., if H is any homogeneous ideal in S such that HDSd

= (J)* n Sd for all large enough d G N+

t h e n t f c (J)" and (H)« = (J)«.

Again suppose that S is a noetherian semihomogeneous ring over R, let J be a homogeneous ideal in S, and let the notation be as in Comment (C5) above. Now (J : £l(S)d)s is an increasing sequence of ideals in S and hence by the noetherianness of S we can find e G N+ such that (J : fl(S)d)s = ( J : tt{S)e)s for all d > e in N+. Moreover, by the said noetherianness, for all large enough d G N+ we have Q(S)d C Qi for m + 1 < i < n and hence (Qj : fi(5)d)5 = S for m + 1 < i < n. For i G { 1 , . . . ,m} we have fi(S') ^ P; and hence there exists t(i) G Si \ Pi and therefore for all d G N+ we get Qi C (Qi : fi(S')d)5 C (Qi : £(i) d )s C Qi and hence (Qi : Cl(S)d)s = Qi- Since taking colon clearly commutes with finite intersections, for all large enough d G N + we have ( J : n{S)d)s = (Qi D • • • D Q„ : fl(S)d)s =

(Qi: n(S)d)s n • • • n (Qn : ft(S)d)s = Qi n • • • n Q m = (j)«. Thus we have shown that:

assuming 5 to be a noetherian semihomogeneous ring over R, (8)

for any homogeneous ideal J in 5 we have (J)" = ( J : fi(S)d)s for all large enough d G N+.

In the following two claims (9) and (10), for any F in the polynomial ring R[Xi,..., XN] in a finite number of indeterminates Xi,..., Xjv, we shall write F = Y^, Fi where P; is a homogeneous polynomial of degree i (or the zero polynomial).

§3: IDEAL THEORY IN GRADED RINGS

215

We claim that: assuming S to be a semihomogeneous ring over R, and x\,...,

XN to be a finite number of elements in Si,

for any homogeneous ideal J in S we have that:

(9)

F £R[Xi,...,XN]

withF(xi,...,xN)

=> F(tx\,...,txN)

£ J for all t £ R.

e J

Namely, F(XI,...,XN)

= YjFi(xii---ixN)

with Fi(xi,...,xN)

£ S,

and

(')

F(txi,...,txN)

=

'EFiixi^.^XN)?

and so the homogenity of J tells us that: F(x\,.., J for all t £ -R.

,XN) £ J =)• F{tx\,...,

tejv)

£

As a partial converse to (9) we claim that: assuming 5 to be a homogeneous ring over an infinite field R, and xi,..., (10)

XN to be a finite number of -R-generators of Si,

if J is an ideal in S such that: F £ R[Xi,...,XN]

mthF(xi,...,xN)

€ J

^> F ( t e i , . . . , txN) £ J for all t £ R, then J is homogeneous. Namely, in view of (') it suffices to show that, for any F £ R[Xi,..., XN], the (finite-dimensional) -R-vector-subspace U of S generated by F(txi,..., tx^teR coincides with the i?-vector-subspace V of S generated by Fj(:ri,..., XN)i^n- Clearly V is of some finite dimension v and by (') we have U C V. If £7 ^ V then we can take a basis zi,..., zv of V such that z\,..., zM is a basis of [7 for some /J, < v, and writing any v G V as v = a\Z\ + • • • + avzv with ai,...,av in R, we would get a linear map V —> R given by u H-> a„ which is zero on [/ but not on V. Thus it suffices to show that if <j> : V —> R is a linear map which is zero on U, it must also be zero on V. So given any linear map (/> : V —> R which is zero on U, let Ci = <j>(Fi(x\,... ,XN)). Suppose if possible that c, ^ 0 for some i and let 6 be maximum with eg ^ 0. Then 5 g N and

G(y) = ^ C l r i is a univariate (nonzero) polynomial of degree 5. Since \R\ > 6, we can find t £ R with G(t) ^ 0. But by (') we have G(t) = <j>(F(txi,..., fcrjv)) and by the zeroness

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of (f) on U we get 4>(F{tx\,..., tx^)) = 0 which is a contradiction. Therefore c* = 0 for all i and hence d. Then, as shown in L4§6,

{

&ssR(R/I) = tassR(R/I)

= { P i , . . . , Pe}

and nvspec fl / = nassR(R/I)

= {Pi,...,

Pd}

where for the definitions of the terms ass, tass, nass, and nvspec we refer to (016)

(Ql) NILPOTENTS AND ZERODIVISORS IN NOETHERIAN RINGS

217

and (017) of L4§5. Let (f> : R —» R/I be the canonical epimorphism, and let NR(R/I) be the set of all x £ R such that (j>{x) is nilpotent. Then clearly (2)

NR(R/I)

= rad fl 7 =

Pln---nPd.

As a companion to this result, in a moment we shall show that (3)

ZR{R/I)

= Pi U • • • U Pe

where ZR(R/I) is the set of all zerodivisors mod / as denned in L4§5(O20). Actually, we shall prove the following more general result: ZERODIVISOR THEOREM (Tl). For any submodule U of any module V over a noetherian ring R, we have ZR{V/U)

= UPetasSR(V7t/)P

with

(1')

tassR{V/U)

= 0 <# U = V.

Moreover, assuming the module V to be finitely generated, there is an irredundant primary decomposition U = Qifl- • -r\Qe where Q i , . • • ,Qe are primary submodules whose radicals P\,... ,Pe are distinct prime ideals in R, which are labelled so that Pi,..., Pd are minimal and Pd+i,... ,Pe are embedded; with such labelling we have UssR(V/U)

(2')

= t&ssR(V/U) =

{Pu...,Pe}

i and [ n v s p e c ^ a n n ^ / t / ) ) = nassR(V/U)

= {PU...,

Pd}.

PROOF. It suffices to prove the first assertion; the second assertion starting with "Moreover" follows from L4§6.1. Clearly: U = V => ZR(V/U) = 0 = t a s s f l ( ^ / ^ ) . So assume that U ^ V. Then the set of all ideals in R of the form (U : O)R with a € V \ U is a nonempty set of nonunit ideals. Since R is noetherian, the said set contains a maximal member P. Now P = (U : O)R for some a € V \ U. Given any r e R and s £ R\P with rs £ P, we have sa £ V \U and P C (U : SCL)R, and hence by the maximality of P we must have P = (U : SO,)R; now by the equation r(sa) £ U we get r £ P. Thus P is prime and hence P belongs to tassR(V/U) which shows that the latter is nonempty. Clearly for every P in tassR(V/U) we have P = (U : CL)R for some a £ V \U, and hence P C ZR(V/U). Conversely, given any s £ ZR(V/U) we have sb £ U for some b £ V \ U, and by what we have just proved we have tass#((f/ + Rb)/U) ^ 0, and hence we can find r £ R together with P £ spec(-R) such that P = (U : C)R where c = rb; now c £ V \U and hence P £ ta,ssR(V/U); also sc £ U and hence s£ P.

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COROLLARY (T2). In the situation of (Tl), assuming V to be finitely generated, for any ideal J in R we have that; J
& {U : J)v

= U «- J <£ Pi for 1 < i < e.

PROOF. In view of L4§5(024)(22'), this follows from (Tl). COMMENT (C7). [Maximality and Conditions for Prime Ideals]. Prom the above proof of (Tl) as well as from several other instances such as L2§5(R5) and L4§5(O20)(12*), a common message is that a maximal element in a set of ideals satisfying certain conditions tends to be prime. In checking the primeness of a nonunit ideal P in a ring R, we can use one of the following three conditions, each of which is easily seen to be equivalent to the primeness of P: (i) x e R and y € R\ P with xy £ P =$> x £ P; (ii) / and J are ideals in R with IJ c P and J <£. P =» / C P; (iii) / and J are ideals in R which contain P but are different from P => IJ is not contained in P. Out of this (i) was stated as (A2) in Ll§10. As an exercise, you may shorten the proof of L4§5(O20)(12*) by using (ii) or (iii) instead of (i). Note that L2§5(R5) corresponds to the S = {1} case of L4§5(O20)(12'). COMMENT (C8). [Embedded Prime and Primary Components]. As said above, in L4§6 we have shown that every ideal / in a noetherian ring R has an irredundant (finite) primary decomposition I — Q\ n • • • D Qe where Qi,...,Qe are primary ideals whose radicals P\,... ,Pe are distinct prime ideals which depend only on / and not on the particular decomposition. Moreover, after suitable labelling we may assume that Pi,... ,Pd are the minimal primes of / , and Pd+i, • • • ,Pe are its embedded primes, i.e., for 1 < i < e we have Pj C Pi for some j j^ i •£> i > d, and then, as shown in L4§6, the primaries Q\,..., Qj are uniquely determined by I. As an application of the Krull Intersection Theorem, in Theorem (T13) of (Q4) we shall show that the primaries Qd+i, • • •, Qe are NOT determined by / . Actually, from the Krull Intersection Theorem we shall first deduce Theorem ( T i l ) on the intersection of all primaries belonging to the same prime, and then deduce (T13) from ( T i l ) . Moreover, from ( T i l ) we shall also deduce Theorem (T12) which says that the members of a,ssR(R/I) can be any preassigned finite set of primes which are not minimal primes of 0, and hence in a domain they can be the members of any finite set of nonzero primes. This shows the existence of ideals with embedded primes in any noetherian ring of dimension at least two. QUEST (Q2) Faithful Modules and Noetherian Conditions Let D be a module over a ring R. We say that D is a faithful module to mean that ann^D = 0. As noted in L4§5(012), D may be regarded as a module over the residue class ring R/(0 : D)R, and then it is clearly faithful. In a moment we shall

(Q3) JACOBSON RADICAL, ZARISKI RING, AND NAKAYAMA LEMMA

219

show that if D is a noetherian module over R then the ring R/(0 : D)R must be noetherian. First note that by the Isomorphism Theorems:

{

if C is any submodule of D then: D is noetherian & C and D/C are noetherian.

In view of (1), by induction on n we see that: if D is the (internal or external) direct sum D\® • • • ® Dn of a finite number of modules D\,...,

(2)

Dn then:

D is noetherian <=> Z?i,..., Dn are noetherian. Obviously: if vi,...,

vn are any finite number of generators of D

then upon letting a, : R —> Z), = Rvt (3)

\ be the i?-epimorphism given by 11—» Vi we have that: the diagonal direct sum a\ © • • • © an : i? —> £>i © • • • © .Dn is an i?-homomorphism with kernel (0 : D)R.

If £> is noetherian then it is generated by a finite number of elements vi,...,vn, and hence by (2) and (3) (and because submodules and homomorphic images of noetherian modules are noetherian) we see that: (4)

if the module D is noetherian then so is the module R/(0 : D)R.

If J is an ideal in R then clearly: the i?-module R/J is noetherian <=> the ring R/J is noetherian; consequently by (4) we conclude that: (5)

if the module D is noetherian then the ring R/{0 : D)R is noetherian.

QUEST (Q3) Jacobson Radical, Zariski Ring, and Nakayama Lemma The Jacobson radical of a ring R is defined by putting

jrad(ii)

p|

P

P€mspec(R)

and we observe that clearly: (1)

jrad(ii) = R & mspec(fi) = 0 spec(fi) = 0 O R = 0.

By a Zariski ring we mean a pair (R, J) where R is a noetherian ring and J is an ideal in R with J C jrad(i?). This generalizes the idea of a local ring, since for any local ring R, the pair (R,M(R)) is clearly a Zariski Ring. After proving the Krull

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Intersection Theorem in (Q4), in (Q5) we shall use it to give a Characterization Theorem for Zariski rings. As a preliminary to the said characterization, for any ideal J in any ring R we put 1- J = {1-x

:x G J}

and we note that (2)

if J c jrad(i?) then every element of 1 — J is a unit in R

and J if V is a finitely generated ideal in R with JV = V \then (1 - J) n (mnRV)

^ 0

and more generally J if V is a finitely generated module over R with JV = V {then (1 - J ) n (arniflV) ^ 0. To prove Claim (2), if x e R is such that 1 — x in a nonunit in R then we can find a maximal ideal P in R with 1 — x £ P, and now we must have x $ jrad(i?) because otherwise we would get x G P which would yield the contradiction 1 £ P. Claim (3) follows by taking t = 1 in the HINT to L4§10(E2.1), and noting that for the nxn matrix A of the said HINT we have det(A) e ann# V and det(A) = 1 — x with x S J. Claim (4) also follows from the same HINT, in view of the following: COMMENT (C9). [Cramer's Rule for Modules]. Cramer's Rule given in Exercise (E4) of L4§10 remains verbatim valid by letting the elements Xj, Yi of that Exercise belong to an i?-module. In view of (2), by taking V/U for V in (4) we get the very useful: NAKAYAMA LEMMA (T3). If U is a submodule of a finitely generated module V over a ring R such that V = U + JV for an ideal J in R with J C jrad(ii), then V = U. QUEST (Q4) Krull Intersection Theorem and Artin-Rees Lemma In Theorem (T24.1) of L4§8.3 we asserted that if R is local then n n g N M ( i ? ) n = 0. This is the famous Intersection Theorem of Krull which we now proceed to prove. Heuristically it says that, in a good situation, a function (such as the one defined by a power series) which is not identically zero has a definite order. Actually we shall give a modification of Krull's original proof by means of the so called:

(Q4) KRULL

INTERSECTION

THEOREM

AND ARTIN-REES

LEMMA

221

ARTIN-REES LEMMA (T4). Given any ideals V, W, J in a noetherian ring R, there exists a € N such that for all c > b > a in N we have

jcWnV = Jc-b(Jbwnv). PROOF. Take a finite set of generators x\,..., XN of J. Let S = R[Xi,..., XN] be the N variable polynomial ring regarded as naturally graded with grading (S'i)igN where Si is the set of all homogeneous polynomials of degree i (including the zero polynomial). For every i S N we have an i?-epimorphism 4>% : Si —> J1 given by f(Xi,..., XN) H-> f(xi,..., xN). Since J W l~l V C J\ upon letting Hi = ( J r ' f J W n y ) we get Ht C St with i(Hi) = J W n V. By Hilbert Basis Theorem (see L4§3(T3)), S is noetherian and hence we can find a e N and a finite number of elements (yi,j)i<j 6 > a in N and any z £ Hc we can write z=

5J

ti,j,iyi,j

with

t j j ^ e 5/

0
for some e > c in N. Equating like degree terms we get Z

=

/ j 0
^i,i,c—iVi,3

and hence

0c(*)e I J3 ^ ( ^ n v ) \0<2
crV^nl^). J

C «/ c " (, (J ( Wn1< / ). Since the other inclusion is obvious, we

get Jcw nv = jc'b{Jbw n V).

ARTIN-REES COROLLARY (T5). Given any ideals W and J in a noetherian ring R, upon letting V = p | JnW we have jmy

PROOF. In (T4) take m =

= v

c-b.

for

all

m

e

N

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KRULL INTERSECTION LEMMA (T6). Given any ideals W and J in a noetherian ring R, upon letting V = p|

JnW

we have that: V = o « . (i - J) n Zfl(W) = 0. PROOF. By (Q3)(3) and (T5) we get V ^ 0 =• (1 - J ) n Z«(V) / 0 and obviously ZR{V) C Z f l (W). Conversely suppose there exists x G J and O ^ j e W with (1 — x)y = 0. Then xy = y and hence by induction on n we see that xny = y for all n e N. Therefore y € n „ e N J " W = V and hence V / 0. KRULL INTERSECTION THEOREM (T7). For any local ring R we have

p | M{R)n = 0. nGN

PROOF. In view of (Q3)(2), this follows from (T6) by taking W = R and J = M(R). CLOSEDNESS COROLLARY (T8). For any ideals I and J in a noetherian ring R, in the notation of (1) and (3) of (Ql) we have that: p | (/ + Jn) = I o Pi + J ± R for 1 < i < e.

PROOF. Taking W = R in (T6) we see that: (*)

p | Jn = 0 «• (1 - J) n Za(fl) = 0, n£N

Taking i?/7 in (*) we see that: (**)

p ) (I + Jn) = I & (l - J) n ^ ( i J / J ) = 0.

Now in view of (**) our assertion follows from (1) and (3) of (Ql). CLOSURE COROLLARY (T9). Let J be any ideal in any noetherian ring R. For any ideal I in R let us put I = nnen(I + Jn)- Then we have the following. (T9.1) Let I = Qi fl • • • fi Qe be any primary decomposition (which need not be irredundant and which exists by L4§6) where Qi,...,Qe are a finite number of primary ideals whose radicals P\,... ,Pe in R are prime ideals labelled so that Pi + J ^ R for 1 < i < d and Pi + J = R for d + 1 < i < e. Then upon letting

(Q4) KRULL INTERSECTION

THEOREM AND ARTIN-REES LEMMA

223

A = Qi n • • • n Qd and B = Qd+i n • • • (~l Qe we have I = AnB with 1 = A = A and £? = R. (T9.2) L e t / i , . . . ,Ih be any finite number of ideals in R such that I = JiD- --n/^. Then7 = 7in---nlft. PROOF. To see that (T9.2) follows from (T9.1), for 1 < j < h we can take a primary decomposition Ij = Qj\ n • • • n Qje(j) where Qji,..., Qje(j) a r e a finite number of primary submodules of V whose radicals are prime ideals Pji,..., Pje(j) in R labelled so that Pjt + J ± R for 1 < i < d{j) and Pji +J = Rioi d(j) + 1 < i < e(j). Now applying (T9.1) to I as well as Ij we get

1

= n

n Qj* and

I

n

J=

l
^

l<»
and hence / = Ji n • • • Pi IhTo prove (T9.1), first note that clearly I = AnB. for 1 < i < d, and hence

3

Q

By (T8) we have Q{ = Qi

= n ( ( n Qi)+Jn)cn(n(^+jn)) n€N \ \ l < i < d

/

/

n€N \ l < i < d

=^ /

But obviously A C A, and hence A = A. By L4§5(026) we see that B + Jn = R for all n £ N, and hence B~ = R. For all n € N we have 4 + J n = AR + Jn = A(B + Jn) + Jn = AB + JnCl

+ Jn

and therefore by intersecting as n varies over N we get A C I. But clearly I C A, and hence I = A. DOMAINIZED KRULL INTERSECTION THEOREM (T10). For any nonunit ideal J in any noetherian domain R we have p | Jn = 0. Moreover, if J ^ 0 then for all a =£ b in N we have Ja ^ Jb. PROOF. In view of (Q3)(2), the first assertion follows from (T6) by taking W — R. To prove the second assertion, assume that J ^ 0. Now if Ja = Jb for some a T^ b in N, say with a < b, then by induction on m we would get J a + m ( f e - a ) = J a for all raeN, and by the first assertion this would tell us that Ja = 0, which would be a contradiction.

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INTERSECTION OF PRIMARIES THEOREM ( T i l ) . For any prime ideal P in a noetherian ring R, the intersection of all P-primary ideals in R equals [0 : P]R. COMMENT (CIO). [Power of Krull Intersection]. Note that, upon letting 0 = Q*n- • -n<3* be an irredundant primary decomposition of 0 in R labelled so that for the radicals Pf,..., P* of Q\,..., Q* we have P* c P for 1 < i < S and P* £ P for S + 1 < i < e, by L4§5(019) we have [0 : P}R = P* where P* = Q\ f~l • • • n Q*s. Also note that, for the canonical homomorphism $ : R —-> Rp = the localization of R at P , by L4§7.1 we have [0 : P]R = ker(3>). Finally note that a weaker version of ( T i l ) was proved in L4§7.1(T17)(j) where, upon letting P = the intersection of all primary ideals in R which are contained in P , it was shown that P = P*. On the other hand, ( T i l ) says that, letting P = the intersection of all P-primary ideals in R, we have P = P*. The said result is weaker than ( T i l ) because obviously P C P . This shows the power of the Krull Intersection Theorem. PROOF OF ( T i l ) . Let H be the set of all P-primary ideals in R, let H' be the set of all M(Pp)-primary ideals in Rp, and let $ : R —> Rp be the canonical homomorphism. Then by L4§7.1(T16)(g) we see that Q' »-» $ - 1 (<2') gives a bijection H' -> H, and <&-1{C\Q>£H'Q') = nQeHQ. By L4§7.1(T16)(i) we know that RP is a local ring and hence by (T7) we get n „ e j v M ( i i p ) n = 0. By L4§5(08), every positive power of M(Rp) belongs to H', and by noetherianness every member of H' contains a positive power of M(Rp). Therefore DQ'^H'Q' = 0. Consequently ker($) = PlQeijQ, and as noted above [0 : P]R = ker($). EXISTENCE OF EMBEDDED PRIMES THEOREM (T12). Let P i , . . . , P e with e > 0 be any finite number of distinct prime ideals in a noetherian ring R which are all of positive height, i.e., none of which is a minimal prime of 0 in R. Then we have the following. (T12.1) There exists an ideal I in R whose associated primes are exactly the given ideals P i , . . . , P e . (T12.2) Assume that e > 1 with P e <£. Pi for 1 < i < e - 1, and let J = Qi n- • -flQe-i be an irredundant primary decomposition in R such that the radicals of Q\,..., Qe-i are P 1 ; . . . , P e - i respectively. Then there exists a (P e )-primary ideal Qe in R with J (£. Qe. Moreover, for any such Qe, upon letting / = JnQe, we have that the primary decomposition / = Q\ n • • • fl Qe is irredundant. [In (T12.2) the positive height assumption for P e is not needed], [cf. L5§6(E3)]. PROOF. (T12.1) is obvious for e = 1 because then we can take I = P\. Therefore (T12.1) follows from (T12.2) by induction on e. To prove (T12.2), first note that because P i , . . . , P e _ i are of positive height, by ( T i l ) there exists a (Pe)-primary ideal Qe in R with J (f_ Qe. It only remains to show that <3i n • • • n Qi-i n Qi+1 n • • • n Qe <£ Qi for 1 < i < e - 1. By relabelling we

(Q5) NAGATA 'S PRINCIPLE OF IDEALIZATION

225

may suppose i = 1. By the irredundancy of J = Q\ n • • • D Qe-i there exists r G (Q 2 n • • • n Qe-i) \ Q\- Since Pe
rse ( Q 2 n - - - n Q e ) \ g i . NONUNIQUENESS OF EMBEDDED PRIMARIES THEOREM (T13). In any noetherian ring R we have the following. (T13.1) Let I = Qi n • • • n Qe be any irredundant primary decomposition of an ideal I in R with e > 0 such that, upon letting P i , . . . , P e be the radicals of Qi,... ,Qe respectively, P e is an embedded prime of I. Then there exists an infinite sequence Qe = Qeo D Qei D Qe2 D ... of (P e )-primary ideals in R with Qeo 7^ Qei ¥" Qe2 ^ • • • such that I = Q\ C\ • • • C\ Qe-i O Qen is an irredundant primary decomposition of I for all n G N. (T13.2) Let Q and Q be any primary ideals in R whose radicals P and P are such that P C P with P ^ P. Then there exists a P-primary ideal Q' with Q' C Q such that <2' ^ Q and Q D Q' = Q ("1Q. PROOF. (T13.1) obviously follows from (T13.2). To prove (T13.2), in view of the Isomorphism Theorems of L4§5(011), by passing to R/(QC\Q) we may suppose that QnQ = 0. In view of this equation, by ( T i l ) the intersection of all P-primary ideals is 0. Since P C P with P ^ P , we also have Q / 0. Therefore Q <£ Q" for some P-primary ideal Q", and we are done by taking Q' = Q n Q". QUEST (Q5) Nagata's Principle of Idealization We shall now modulize most of the material of the previous Quest (Q3) by means of the above mentioned principle which converts the study of submodules of a module D over a ring R into the study of those ideals of a ring R which are contained in a certain ideal D of R. To do this we consider the (external) .R-module-theoretic direct sum ~R= R®D which we convert into a ring by declaring that for all (r, v) and (r', v') we have (r, v)(r',v')

= (rr',rv'

+r'v).

We let R = a(R)

with natural injection

a : R —> R

and we note that R is a subring of R isomorphic to R, and a is an injective ring homomorphism. We also let D = (3(D)

with natural injection

/3 : D —> R

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§5: MORE ABOUT

IDEALS

AND

MODULES

and we note that D is an ideal in R, and C H-> /3(C) gives a bijection of the set of all i?-submodules of D onto the set of all ideals in R contained in D; moreover, if B C D is any set of i?-module generators of C, then the two subrings R[P(B)} and R[/3(C)} of ~R coincide. It follows that (1)

if D is a finitely generated module then R is an affine ring over R

and hence I if D is a noetherian module and R is a noetherian ring { [then R is a noetherian ring. We are now ready to prove: (2)

MODULIZED ARTIN-REES LEMMA (T14). Given any submodules V and W of a noetherian module D over a ring R and any ideal J in R, there exists a G N such that for all c > b > a in N we have JCWDV

Jc-b(JbWDV).

=

PROOF. By (Q2), via the canonical epimorphism (j>: R —> R = R/(0 : D)R we may regard D as a module over the noetherian ring R and then J = { J) is an ideal in R with J^fiW n V ) = J^PW n V) for all i,j in N. Therefore, replacing R by i? we may assume that R is noetherian. Now in the above notation let V = fi(V), W = P(W), and J = a(J)R. Then V,W,J are ideals in the noetherian ring R, and for all i, j in N we have J^^Wf)V) = ^{PWC\V)). Thus we are done by (T4). MODULIZED ARTIN-REES COROLLARY (T15). Given any submodule W of a noetherian module D over a ring R and any ideal J in R, upon letting U = p | JnW we have JmU = U

PROOF. In (T14) take m =

for all

m € N.

c-b.

MODULIZED KRULL INTERSECTION LEMMA (T16). Given any submodule W of a noetherian module D over a ring R and any ideal J in -R, upon letting V = p) J n W neN

(Q5) NAGATA'S PRINCIPLE OF IDEALIZATION

227

we have that: V = 0 <s> (1 - J ) n ZR(W)

= 0.

PROOF. By (Q3)(4) and (T15) we get V ^ 0 =>- (1 - J) n Z fl (V) 7^ 0 and obviously ZR(V) C Z A ( W ) . Conversely suppose there exists x € J and 0 ^ y £W with (1 — x)y = 0. Then xy = y and hence by induction on n we see that xny = y for all n G N. Therefore y e n „ e N J " W = V and hence V ^ 0. MODULIZED KRULL INTERSECTION THEOREM (T17). Let J be an ideal in a noetherian ring R with J C jrad(i?). Then for any finitely generated R-module D we have

P| JnD = 0. n€N

In particular we have

f] Jn = 0. nGN

PROOF. In view of (Q3)(2), the first assertion follows by taking W = D in (T16). The second assertion follows by taking D = R in the first assertion. MODULIZED CLOSEDNESS COROLLARY (T18). Given any submodule U of a finitely generated module V of a noetherian ring R, and any ideal J in R, in the notation of (1') and (2') of (Ql) we have that: f](U

+ JnV) =U<^Pi

+

J^R{ovl
nGN

PROOF. Taking W = D in (T16) we see that for any finitely generated module D over R we have: (')

f)jnD

= 0**(l-J)n

ZR{D) = 0.

Taking D = V/U in (') we see that: (")

f | (U + JnV) = U<*(1-J)n

ZR(V/U)

= 0.

Now in view of (") our assertion follows from (1') and (2') of (Ql). MODULIZED CLOSURE COROLLARY (T19). Let J be any ideal in any noetherian ring R. For any submodule U of a finitely generated module V over R let us put U = r\ne^(U + JnV). Then we have the following.

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(T19.1) Let U = Qi D • • • n Qe be any primary decomposition (which need not be irredundant and which exists by L4§6.1) where Qi,...,Qe are a finite number of primary submodules whose radicals P i , . . . , Pe in V are prime ideals in R labelled so that Pi + J ^ R for 1 < i < d and P, + J = R for d+ 1 < i < e. Then upon letting A = Qi n • • • n Qd and B = Qd+1 n • • • n Qe we have U = A n B with U = A = A and B = V. (T19.2) Let U\,...,Uh be any finite number of submodules of V such that U = Uin---r\Uh. Then U = Ui n • • • C\Uh. PROOF. To see that (T19.2) follows from (T19.1), for 1 < j < h we can take a primary decomposition Uj = Qji n • • • n Qje(j) where Qj\,..., Qje(j) are a finite number of primary submodules of V whose radicals are prime ideals Pji,..., P,e(j) in R labelled so that P^ + J ^ R for 1 < i < d(j) and P* + J = R for d(j) + 1 < i < e(j'). Now applying (T19.1) to U as well as Uj we get

fl Q^ and ^ = H ^

^=0

l<j
l
and hence TJ = U\ D • • • D t//j. To prove (T19.1), first note that clearly U = AnB. for 1 < i < d, and hence

n€N \ \ l < i < d

/

/

n£N \ l < i < d

By (T18) we have Qt = Q,

/

But obviously A C A, and hence A = A. Since P is noetherian, we can find c(i) G N with Pf{i)V c Qi for d+l
n„€N(f/ + Jnv) = u.

(Q6) COHEN'S AND EAKIN'S NOETHERIAN

THEOREMS

(v) For any finitely generated R-module V we have DnetiJnV (vi) For any ideal / in R we have nne^(I + Jn) = I.

229

= 0.

PROOF. By (Q3)(2) we get (i) =» (ii). By (Q3)(4) we get (ii) => (hi). By (T18) we get (iii) => (iv) by noting that: if I is any ideal in R with I + J = R then upon letting V = R/I we have JV = (I + JR)/I = (I + J)/I = V and hence (iii) tells us that V = 0, i.e., I = R. By taking U = 0 we get (iv) => (v). By taking V = R/I we get (v) => (vi). Finally to show that (vi) => (i), by taking J e mspec(i?) in (vi) we get I + Jn ^ R for some n e N and hence Jn C I and therefore J c / . QUEST (Q6) Cohen's and Eakin's Noetherian Theorems After proving Lemma (T21), as an application of the Maximality Principle stated in (C7), in (T22) we shall give a proof of Cohen's Theorem which says that if all the prime ideals in a ring are finitely generated then so are all the ideals. In L4§5(025) (27*) we showed that if an overring of a ring is a finitely generated module over the ring then the noetherianness of the ring implies the noetherianness of the overring. Eakin's Theorem says that conversely the noetherianness of the overring implies the noetherianness of the ring. As an application of assertion (Q2)(5) about faithful modules, in (T23) we shall give Formanek's proof of Eakin's Theorem. LEMMA (T21). Let J be an ideal in a ring R such that, for some a G R, the ideals J + aR and (J : CL)R are finitely generated. Then J is finitely generated. PROOF. We can take a finite number of elements x\,..., xn in J and r\,..., r„ in R such that the elements Xi + ar\,... ,xn + arn generate J + aR. We can also take a finite number of generators yi, • • • ,ym of (J : a)R. We claim that elements xi,...,xn, ayi,..., aym generate J. They are certainly in J. Conversely, given any z G J, we clearly have z € J + aR and hence we can write z = b\(xi + ari) -\ h bn(xn+arn) with&i,... ,bn in JR. Nowa(6inH h6„r„) = z-b\X\ bnxn e J and hence b\r\ + • • • + bnrn £ (J : O)R and therefore 6irH

1- bnrn = ayi H

Thus, upon letting di = ac\,... z = b\xx H h bnxn + diyi -\

h cmym

with

c i , . . . , c m in R.

,dm = acm, we have found di,... h dmym.

,dm in R with

COHEN'S NOETHERIAN THEOREM (T22). A ring R is noetherian iff every prime ideal in it is finitely generated. PROOF. The only if part being obvious, assume that every prime ideal in R is finitely generated, and let W be the set of all ideals in R which are not finitely generated. Suppose if possible that W is nonempty. Now the union of any linearly

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ordered (by inclusion) set of members of W clearly belongs to W and hence, by Zorn's lemma, W has a maximal member J. Clearly J is a nonunit ideal which is not prime. Therefore we can find elements a,b in R\ J with ab G J. Since a £ R\J, the ideal J + aR properly contains J (i.e., contains J but is different from J); since b G R\J with ab € J the ideal ( J : O)R also properly contains J. Therefore by the maximality of J, the ideals J + aR and (J : CL)R are finitely generated, and hence by (T21) so is J, which is a contradiction. THEOREM (T23). For any ring R we have the following. FORMANEK (T23.1) Assume that there exists a faithful i?-module V such that V is finitely generated and the MXC (= Maximal Condition) holds for the set T all submodules of the form IV with / varying over the set of all ideals in R (i.e., every nonempty subset of T has a maximal member). Then R is noetherian. EAKIN (T23.2) Assume that there exists a noetherian overring S of R such that S is a finitely generated ii-module. Then R is noetherian. PROOF. Clearly (T23.2) follows by taking V = S in (T23.1). In view of (Q2)(5), to prove (T23.1), it suffices to show that V is a noetherian module. Assume the contrary, and let fl be the set of all submodules of V of the form IV with ideal I in R such that the .R-module V/(IV) is nonnoetherian. Then fi is nonempty and hence has a maximal member JV with ideal J in R. Let W = V/(JV) and T = R/(0 • W)R. Then W is a faithful finitely generated nonnoetherian T-module such that for every nonzero ideal / in T the T-module W/(IW) is noetherian. Let x i , . . . , xn be a finite set of T-generators of W, and let A be the set of all submodules W* of W for which the set {a € Tx : ax\ G W*,.. .,axn e W*} is empty. Then 0 G A and the union of any nonempty linearly ordered (by inclusion) subset of A clearly belongs to A. Therefore by Zorn's lemma, A has a maximal member W. If the T-module W/W were noetherian then T would be a noetherian ring by (Q2)(5) and hence W would be a noetherian T-module by L4§5(025)(27*). Therefore W/W is a nonnoetherian T-module and hence W has a submodule U with W C U and W 7^ U such that the T-module U/W is not finitely generated. By the maximality of W there exists a G T x such that aW C U. Now the T-module W/(aW) is noetherian and hence its T-submodule U/(aU) is finitely generated. Therefore the T-module U is finitely generated, and hence so is U/W. This is a contradiction. QUEST (Q7) Principal Ideal Theorems In L4§8 we indicated (although not yet proved) a correspondence between varieties in affine iV-space over a field and ideals in the iV variable polynomial ring over that field. More generally there is such a correspondence between subvarieties of a variety and ideals in the affine coordinate ring of that variety. Geometrically speaking, a hypersurface, i.e., a subvariety defined by one equation, has codimension

(Ql) PRINCIPAL IDEAL THEOREMS

231

one, i.e., dimension one less than the dimension of the ambient space or the ambient variety. Similarly the codimension of a subvariety defined by a certain number of equations is less equal that number, and in good situations equality prevails. The various principal ideal theorems proved by Krull strengthen this experience by showing that there is a parallel situation for ideals in a noetherian ring. Namely, the height of an ideal generated by a finite sequence of elements is less equal the number of elements in that sequence, and if each of those elements is a nonzerodivisor modulo the previous elements then the height of every minimal prime of the ideal equals that number. We start off by proving the case of one element. PRINCIPAL IDEAL THEOREM (T24). Let x be an element in a noetherian ring R, and let P be a prime in nvspec^a;. Then h t # P < 1 with equality in case x is a nonzerodivisor in R. COMMENT (CI 1). [Minimal Primes]. We are writing vspec^a; and nvspecflX instead of vspec#{x} and nvspec/{{:r} respectively. A prime ideal P in a ring R is said to be minimal over an ideal I in R if it belongs to nvspec#7; similarly, P is minimal over x means it belongs to nvspecRX. Clearly a prime ideal in R is of height zero iff it is minimal over 0; let us denote the set of all these by nspec(P) and call it the minimal spectrum of (R). By (Ql) we see that: (1) if R is noetherian then every element of such an ideal is a zerodivisor in R. By (Ql) it also follows that: (2) if R is noetherian then every P in vspecfli contains some P in nvspec^/. To prove (2), since / C P, in the notation of (Ql)(l) and (Ql)(2) we have P i . . . Pd C Pi n • • • n Pd = radfli" C radtfP = P , and hence by (Q1)(C7) we get Pi C P for some i £ {1 , d}. PROOF OF (T24). Let $ : R -> S = RP be the canonical homomorphism and let y = $(a;). Then 5 is a local with dim(S') = ht/jP and y £ M(S) with M(S) £ vspecsy. In view of item (1) of the above Comment, it only remains to show that dim(S) < 1. In other words, given any prime ideal J in S different from M(S), we want to show that hts J = 0, i.e., upon letting T be the local ring Sj, we want to show that dim(T) = 0. Let a : S —> T be the canonical homomorphism, and for every n £ N + let J„ = a-l{M{T)n); by L4§5(08) and L4§7.1(T17) we see that Jn is a J-primary ideal in S. In S we have the descending chain of ideals yS + Ji D yS + Ji D yS + J3 D Since M(S) is minimal over y, the local ring S/yS is artinian and hence for some n € N+ we must have yS + Jn = yS + Jn+iNow any z G J„ can be written as z = ys + t with s € S and t € Jn+i\ this gives ys € Jn', since M(S) is minimal over y, we must have y ^ J; since Jn is J-primary, we get s G Jn. Thus J„ = yjn + Jn+i- Consequently, since y e M(S), by (T3) we get J„ = J „ + 1 . Therefore by J4§7.1(T17) we conclude that M(T)n = M(T)n+1. n Consequently, again by (T3) we get M(T) = 0. Therefore dim(T) = 0.

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COROLLARY (T25). Let H and I be ideals in a noetherian ring R, let x G R, and let P G nvspecR(H + xR) with / c P be such that for every P' G nvspec^H we have I <jL P'. Then P G nvspec f l (i/ + / ) . PROOF. Apply (T24) to the images of x and P in

R/H.

COROLLARY (T26). Let H and P be ideals in a noetherian ring P such that P is prime with h t ^ P > e € N+ and for every P ' G nvspec^if we have P <£ P'. Then there exists a sequence of prime ideals PQ C P I C • • • C Pe = P in R with PQ ^ P\ ^ • • • ^ Pe such that for every P ' G nvspec fl i? we have Pi (jLP1. PROOF. We make induction on e. The assertion being obvious for e = 1, let e > 1 assume for e — 1. Since ht/jP > e, there exists a sequence of prime ideals P<5 C Px' C • • • C Pe' = P in P. with P^ P[ ^ • • • ^ P'e. By L4§5(024)(22 # ) there exits x G P\P e '_2 s u c ^ ^ a t for every P ' G nvspec f l # we have x $ P'. By (Cll)(2) there exits P* G nvspec fl (Pg_ 2 +a:P) with P* C P . Applying (T24) to the images of x and P* in R/P^2 w e see that P'e_2 ^ P* ^ P. It follows that ht f l P* > e - 1 and for every P' G nvspecRH we have P* <£ P'. Therefore by the induction hypothesis there exists a sequence of prime ideals PQ C P I C • • • C P e - i = P* in R with Po ¥=• Pi 7^ • • • 7^ -Pe-i such that for every P ' G nvspecRH we have Pi 0 and assuming true for n — 1 we shall show it for n, and then we shall be done by induction. Let H be the ideal in R generated by x\,..., xn-\. If P G nvspec^fl" then by the induction hypothesis h t # P < n — 1. So assume the contrary. Then clearly for every P' G nvspec fi i? we have P <£_ P'. Given any e G N + with h t # P > e we want to show that e < n. By (T26) there exists a sequence of prime ideals P 0 C Pi C • • • C P e = P in R with P 0 7^ Pi ^ • • • ^ Pe such that for every P' G nvspecRH we have Pi <£. P'. Taking I = P\ and x = xn in (T25) we get P G nvspecR(H + Pi). Consequently by applying the induction hypothesis to the image of P in R/P\ we get e — 1 < n - 1, and hence e < n. This completes the proof of the height inequalities h t ^ P < n and htRI < n. To prove the height equality htRP = n under the nonzerodivisor assumption, we

(Q7) PRINCIPAL IDEAL THEOREMS

233

again make induction on n. The n = 0 case being obvious, let n > 0 and assume for n - 1. Given P G nvspec fi J, upon letting H be the ideal in R generated by x\,.. .,xn-i, by (Cll)(2) there exists P' G nvspec K # with P' C P. Applying (T24) to the images of xn and P in R/H we see that P' ^ P . By the induction hypothesis h t ^ P ' = n — 1 and by the already proved height inequality h t # P < n. Therefore h t # P = n. As a consequence of (T27) we have the following theorem which includes a partial converse of (T27). HEIGHT THEOREM (T28). Let R be a noetherian ring. Then for any nonunit ideal / in R we have ht/{7 G N; recall that the height of the unit ideal is defined to be —oo. In particular, for any prime ideal P in R, upon letting h t ^ P = n, there exists a sequence of prime ideals PQ C P I C • • • C P„ = P in R with P 0 ^ P x ^ • • • ^ P n , and for any such sequence of prime ideals we have htflPj = j for 0 < j < n. Moreover, given any such sequence of prime ideals, there exist elements xi,...,xn in R such that upon letting Ij be the ideal generated by x\,..., Xj, for 0 < j < n we have Pj G nvspec R 7j and ht^Pj = j for all Pj S nvspec/,-. PROOF. Everything except the last sentence immediately follows from (T26). The last sentence being obvious for n = 0, let n > 0 and assume it for n — 1. Then there exist elements xi,...,xn-\ in R such that upon letting Ij be the ideal generated by x\,..., Xj, for 0 < j dim(P). Moreover, dim(fl) can be characterized as the smallest number of elements in R which generate an M(P)-primary ideal. As another consequence of (T27) and (T28) we now prove the: DIMENSION LEMMA (T30). Let S be the polynomial ring R[Xu...,Xm] or the power series ring R[[Xi,..., Xm]} where X\,..., Xm are a finite number of indeterminates over a noetherian ring R. Then we have the following. (T30.1) S is noetherian, and for any ideal / in R, upon letting Ij be the ideal

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in S generated by Xi,...,Xj and J we have Ij D R = I for 0 < j < m, and hence for any ideal H ^ I in R we have HS ^ IS. Moreover, if I ^ R then Jo C h C • • • c J m C S with Jo ^ h £ ••• ^ Im ^ S and htsIj = j + htRI for 0 < j < m. Furthermore, if J G spec(P) then Ij G spec(S) for 0 < j < m. Finally, assuming I ^ R but without assuming J G spec(P), if (Li)i: P —> P = P / J be the canonical epimorphism. Let 5 = R[X] or P[[^]] respectively. Then: J is prime <=> P is a domain •<=> S is a domain. We get an epimorphism ip : S —> S by letting i / ) ( ^ o e X e ) = J ^ 0 ( a e ) X e for all Y^ aexB G S with ae G P . Clearly ker(^j) = Jo and hence: J is prime iff Jo is prime. We get an epimorphism 6 : S —• P by letting # ( ^ 6 e X e ) = bo for all ^ beXe G 5 with 6e G P . Clearly ker(0) = X S = V(JL) and hence: J is prime iff Ji is prime. In view of (Cll)(2) it now follows that: (•) if (J<j)i j + n for 0 < j < 1. By (T28) we have J G nvspec f l (a;i,... , x „ ) P for

(Q7) PRINCIPAL IDEAL THEOREMS

235

some xi,..., xn in R, and then by (•) we get IQ € nvspec f l (a;i,..., xn)S and I\ £ nvspec R (a;i,..., xn, X)S, and hence by (T27) we get htslj < j + n for 0 < j < 1. Therefore htslj = j + n for 0 < j < 1. This completes the proof of (T30.1). In view of the argument of the first sentence of this paragraph, to prove (T30.2) it suffices to show that dim(S') < 1 + dim(i?). We shall do this first in the polynomial case and then in the power series case. For a moment suppose we are in the polynomial case. Since the intersection of any prime in S with R is a prime in R, continuing the assumption that / is prime with htft/ = n, and letting J be any prime in S with J D R = I and J ^ IS, in view of (T30.1) it suffices to show that htg J = I + n. In the above notation let J = ip(J). Then J is a nonzero prime ideal in S with J D R = 0. Let QF(iZ) = K. Then the localization of S at the multiplicative set R is the univariate polynomial ring K[X] over the field K, and hence J = iP(F(X))K[X] with F(X) e J of degree E > 0 in X such that the coefficient of XE in F belongs to R \ I. Upon letting B = F(X)S we get J € nvspec s (/S' + B). By (T28) we have J e nvspec fl A where A = (xi,..., xn)R for some n,..., xn in R. It follows that (see HINT in (C13) below): (••) J £ nvspecs(AS + B). By (••) and (T27) we get h t s J < 1 + n. By (T30.1) we have htglS = n and hence h t ^ J > 1+n. Therefore h t s J = 1 + n. Henceforth suppose we are in the power series case. To complete the proof of (T30.2) it suffices to show that given any prime ideal J in S there exists a prime ideal I in R with J c IS + XS; to do this, by L2§2(ii) we see that the ideal {/(0) : f(X) e J} in R is not the unit ideal, and hence by Zorn's Lemma it is contained in a maximal ideal / in R. To prove (T30.3), also assume that R is a local ring. Then in view of L2§2(ii) and L3§11(1*) we see that 5 is a local ring with M(S) = M(R)S+XS, and now clearly J C IS = XS. Upon letting emdim(-R) = d, it only remains to show that emdim(S) = l + d. We can write M(R) — ( z i , . . . , Zd)R with z i , . . . , zd in R. Then M(S) = ( z i , . . . , zd, X)S and by (T30.1) we get M{S) ^ ( z i , . . . , Zd)S. For 1 < i < d we also have M(R) ^ {z\,..., -Zj-i, Zi+i,..., Zd)R and hence by (T30.1) we get M(S) ^ ( z i , . . . , Zi-i, zi+i,. ..,zd, X)S. Therefore by (T3) we conclude that emdim(5) = 1 + d [cf. (C14) below], COMMENT (C12). [Multivariate Ideal Extensions]. In the situation of (T30), for any A C R let A[XX,..., Xm] or A[[Xi,..., Xm}] respectively denote the set of all f{Xi,..., Xm) £ S such that the coefficient of any monomial Xf1 . . . X%"in / belongs to A. Then, as indicated in the proof of (T30.1), for any ideal I in R we have IS = I[X\,..., Xm] or I[[Xi,...,Xm}] respectively. COMMENT (C13). [Univariate Ideal Extensions]. Let S be the univariate polynomial ring R[X] over a noetherian ring R, let J be a prime ideal in S, and let / = J n R. Then J is a prime ideal in R, and as indicated in the proof of the polynomial case of(T30.2), we have the following. (1) If J = IS then hts J = htRI.

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(2) UJ^IS then h t s J = 1 + ht f l 7 and there exists F{X) € J of degree E > 0 such that the coefficient of XE in F belongs to R \ I. HINT. Here is the detailed proof of claim (••) which says that: given respective prime ideals I and J in a noetherian ring R and a noetherian overring S with J C\ R = I, and given respective ideals A and B in R and S with I G nvspecflvl and J G n v s p e c ^ J S + B), we have J £ nvspec s (^45 + B). We can prove this by a double application of a criterion which follows from L4§6(T5) to L4§6(T7) and which says that, for ideals A and I in a noetherian ring R with I prime we have: I G nvspec flJ 4 •*=> I = iadR[A : I]R. In our case AS + B C J and hence rads[i45 + B]s C J. Conversely, given any z G J, because of J G nvspec s (/5 + 5 ) , we can find n G N + and 6 G -B such that zn = b + s\Ci H h s e c e with si,...,se in 5 and c i , . . . , c e in I. Because of I G nvspecRA, we can find m G N + and t £ R\I such that ic™ = r^ai with n £ R and aj G ^4 and for 1 < z < e. It follows that for all large enough positive integers /^,v we have t^z" £ AS + B. Clearly t^ £ S\J and hence 2 £ rads[^45 + B]sCOMMENT (C14). [Module Generation For Local Rings]. In the last sentence in the proof of (T30) when we said by (T3) we meant by the following consequence of (T3). Let V be a finitely generated module over a local ring R, and let x\,..., xm and y\,..., yn be finite sequences of generators of V, which are both assumed to be irredundant, i.e., no member of either sequence can be omitted for it to remain a system of generators of V. Then m = n. This follows by noting that V/M(R)V is a finite dimensional vector space over the field R/M(R), and for any submodule U of V, by (T3), we have: V = U + M{R)V =» U = V. Consequently, upon letting a : V —* V/M(R)V to be the canonical epimorphism and v to be the dimension of the vector space a(V), we see that: xi,...,

xm are irredundant generators of V

<^> a(xi),...,

a(xm)

are irredundant generators of a(V)

•*=> a(xi),...,

a(xm)

is a basis of a(V)

•<=> m = v . V

As an illustration that this is not true without the localness hypothesis, take V = R = Z with (xi,...,xm) = (1) and (yi,...,yn) = (2,3). QUEST (Q8) Relative Independence and Analytic Independence In continuation of (T26) and (T27), in Theorem (T34) below we shall show that an ideal I generated by a finite number of elements noetherian ring R has height equal to that number n iff the elements are independent in a sense to be defined is a moment. The said theorem will be preceded by Lemmas (T31)

(Q8) RELATIVE INDEPENDENCE AND ANALYTIC INDEPENDENCE

237

to (T33). Out of these, Lemma (T33) will relate the independence of the elements xi,... ,xn to the modelic blowup of the ring R at the ideal I as described in L4§9.2 which is the coordinate free incarnation of the modelic proj of the ring R at the ideal / as described in L4§9.1; unlike in these references, here the ring R will not be assumed to be a domain. In Theorem (T35) we shall show how this concept of independence is related to the notion of analytic independence of power series which too we shall introduce in a moment. We shall consider the polynomial ring R[Y\,... ,Yn] in a finite number of variables over an arbitrary ring R which need not be noetherian. For any A C R we let A[Yi,...,

Yn]i = the set of all homogeneous members of R\Y\,...,

Yn]

of degree i all of whose coefficients belong to A (including the zero polynomial in case 0 € A) and we put A[Y1,...,Yn]o0

=

\jA\Y1,...,Yn]i.

Let J be an ideal in R. For elements xi,..., / =

xn in R let

(xi,...,xn)R.

The elements x\,...,xn are said to be independent over (R,J), relative to (R, J ) , if J C J and f F(Yi,...

,Yn) e R[YU .. ^Yrjoo with F(xlt...

or independent

,xn) = 0

\^F(Y1,...,Yn)£j[Y1,...,Yn}o0. The elements x\,... ,xn are said to be independent over R, or independent relative to R, if I ^ R and they are independent over (R, rad#J). More precisely, we may say that the sequence xi,...,xn is independent over (R, J) or R and so on. In case R is the power series ring X [ [ X i , . . . , Xm]] in a finite number of variables over a field K, elements x\,... ,xn in M{R) are said to be analytically independent over K if there is no nonzero / ( Y i , . . . , Yn) e K[[Yi,..., Yn}} with f(x\,..., x„) = 0. In all the above situations, dependent means not independent. LEMMA (T31). For any elements x\,..., xn in R we have the following. (T31.1) For the .R-epimorphism a : R[Y\,..., Y„] —> R which sends Yi to Xi for 1 < i < n, we have (H)

ker(a) = (Yt - xu...

,Yn - xn)R\Yi

,...,Yn]

and condition (f) is equivalent to the condition:

(f)

ker(a) nR[Y1,...,Yn]00

c J[YU...,

Yn\

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(T31.2) Assuming I C J, condition (f) is equivalent to the condition: \F{YU...,

Yn) G R[YU ..., Yn]i with i G N and F{xu. • •, xn) G Ii+l

\^F(Y1,...,Yn)GJ[Y1,...,Y„]i. PROOF OF (T31.1). Clearly (f) o (]'). To prove (tt), by induction on n we shall show that any / ( Y i , . . . , Yn) G R[Yi,..., Yn] can be written as / ( Y i , . . . , Yn) = r + Ili 0 and assume for n — 1. By division algorithm in Yn over i?[Yi,..., Y n _i] we can write / ( Y i , . . . , Y n ) = h ( Y i , . . . , y„_i) + (Y„ - a: n )5 n (Yi,..., Y„) with h ( Y i , . . . , Y n _i) G Ji[Yi,..., Y„_i] and g n ( Y i , . . . , Yn) G i?[Yi,..., Yn]. By the induction hypothesis we can write h{Y1,..., Y n _i) = r + X)i
F(YU. .. ,yn) = Fen,... ,y„) - G ^ , ... ,y„) we get F(x\,...

,xn)

= 0; hence F ( Y i , . . . , Y n ) G J [ Y i , . . . , Y„]oo and therefore

F(Y!,...,Y„)G jryi,...,^]*.

LEMMA (T32). Elements R iff the elements Z, R\Z\.

noetherian ring R are independent over independent over the univariate polynomial ring

X j , . . . , %n in. &

PROOF. Upon letting H = ZR[Z] + IR[Z], by (T30.1) we know that it I ^ R then H ^ R[Z], and clearly R n ra,dR[Z] H =

V&ARI

and hence if the elements Z, independent over R[Z] then the elements x\,..., xn are independent over R. Conversely assuming that the elements xi,...,xn are independent over R, let F(Y0,..., Y„) G R[Z] [Y 0 ,..., Yn]j with i e N be such that F(Z, x i , . . . , £„) = 0. Expanding F we have F(Y0,...,Y„)= £ 0
£

Fln...jn(Z)Yfrt...Yt -+jn=i-l

(Q8) RELATIVE INDEPENDENCE AND ANALYTIC INDEPENDENCE

with Fijlm„jn(Z) sides we get

€ R[Z) and substituting (Z,xi,...,xn)

£ 0
£

239

for (Yo,... ,Yn) on both

fiil..jB(z)z'a^..4n=o

h+-+jn=i-l

and comparing coefficients of Zl we obtain £ Fijl...jn(0)x{1...xi»€P+1-1

for

0
and hence by (T31.2) we get Fiji• • -in(°) € rad#.T

for all

l,ji,...,jn

and therefore Fih...jn e rad fi [ 2 ]H

for all

2,j

BLOWUP LEMMA (T33). For elements x i , . . . , x „ in a nonnull noetherian ring R with n > 1, assume xi is a nonzerodivisor in R. Let L be the subring of AT = QR(i?) given by L = R[x2/xi,... ,xn/x\}, let S be the polynomial ring R[T2,... ,T„], let 0 : S —> L be the i?-epimorphism which sends Tj to Zj/xi for 2 < i < n, let B = ker(/3), let C = (xiT 2 - x 2 , . . . , x i T n - Xn)^, let R' be the subring of K given by i?' = R[xn/x\\. Then we have the following. (T33.1) C C B and xfB c C for some m € N+. (T33.2) If rad/j/ = J then condition (t) is equivalent to the condition: (t")

Be

PS for all P e vspec^/.

Hence the elements independent over R iff I ^ R and (t") holds. (T33.3) If the elements independent over R then the elements xi, • •., x„_i are independent over R'. (T33.4) If htRI = n then the elements independent over i?. (T33.5) If the elements xi,...,xn are independent over R then h t ^ / = n. PROOF OF (T33.1). Obviously C C B. Now L C R C K where £ is the localization of R at the multiplicative set (xi)jeN- Extend /? to an Repimorphism a : R[T2,... ,Tn] —> R with a(Ti) = x;/xi for 2 < i < n. Given any z e B, clearly z €. ker(a) and hence by (T31.1)((t) we can write « = E 2 < i < „ ( T i - xix^1)gi{T2,...,Tn) with ffi(T2)...,t„) 6 i?[T 2 ,...,T„] for 2 < i < n. Therefore x\z G C for some e e N + . It follows that x^B C C for some m € N + . PROOF OF (T33.2). For a moment assume that rad#J = J and (t); then given any P e vspec^Z and 0 ^ f(T2, •.. ,Tn) € B of degree i, letting F(Yi,..., Yn) =

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Yjf(Y2/Y1,...,Yn/Y1) G R\Y1,...,Yn]i, by (T31.1)(f) we get F(Yu...,Yn) G J [ Y i , . . . ,Yn]i- hence F(YU... ,Yn) € P[YU ... ,y„] i ; therefore f(T2,... ,Tn) G P S . Conversely assume rad#7 = J and ( f ) ; then by (Ql)(2) and (Cll)(2) we get J = ripGvspecR/-P and hence, by (C12), all the coefficients of any / ( T 2 , . . . ,T„) G B belong to J; given any J , (yi,...,Y r „) G # [ ¥ ! , . . . , y„]oo with F{xi,...,xn) = 0, letting f(T2,..., T„) = F ( l , T 2 ) . . . , T„) we clearly have / ( T 2 , . . . , Tn) G B and hence F ( y , . . . ,Yn) G J f y , . . . ,y„]oo. PROOF OF (T33.3). Assume the elements independent over B. Let B ' = ker(/3') where /3' is the B'-epimorphism of S' = R'[T2,.. • ,T n _i] onto L which sends Tj to Xj/xi for 2 < i < n — 1. Let 0 : 5 —» 5 ' be the B-epimorphism which sends ( T 2 , . . . , T „ ) to ( T 2 , . . . , T n _ i , x „ / x i ) . Then clearly 0(B) = B'. Let J' = ( x i , . . . , x „ _ i ) B ' . Then clearly V = IR! ^ B'. Given any P' G vspec fi ,P, upon letting P = P' n B, we clearly have P G vspec fi 7 and hence by (T33.2) we get B C PS and therefore B' C P ' S ' . Consequently, again by (T33.2) we see that the elements x i , . . . , xn-i are independent over R'. PROOF OF (T33.4). Assume htRI = n. Given any P G vspec fi 7, in view of (T33.2), it suffices to show that B C PS. By (T30.1) we have PS G spec(S) and clearly C C IS C PS. Consequently by (Cll)(2) we can find Q G nvspec^C with Q C PS. By (T27) we get h t s Q < n - 1 and by (T30.1) we get hts(IS) = htRI > n. Therefore IS <£ Q. Consequently x\ 0 Q because otherwise IS C C C Q. Therefore by (T33.1) we get B C Q and hence B C PS. PROOF OF (T33.5). Assuming the elements x\,...,xn to be independent over R, by induction on n we shall show that h.tRI = n. For n = 1 this follows from (T24). So let n > 1 and assume for n — 1. By (T27) we have htRI < n. Therefore, given any P G vspec fl 7, it suffices to show that h t ^ P > n. Let V = (xi,... , x n _ i ) B ' . Then clearly I' = IR' ^ R'. In view of the induction hypothesis, by (T33.3) we get htfl-J' = n - 1. Let P' = PR'. We shall show that then P' G vspec fl ,J' with htRP > htfl'P' and this will complete the proof. Let D — R[Tn] C S and let 6 : D —» R' be the B-epimorphism which sends Tn to xn/x\. Then clearly I' C 6{PD) = P'. Also S(y) = f3{y) for all y G D, and hence ker(J) = Bf\D. By (T30.1) we have PD = (PS)nD G spec(£>) with htD{PD) = h t f l P , and by (T33.2) we have B c PS. Therefore P£> G vspec D ker(J) with P ' = 8{PD) G vspecR,I', and it suffices to show that h t p (PD) > htR>P'. Let /i = htR>P'. Then we can find a sequence of prime ideals ker(<5) C Pi C P 2 C • • • C Ph = PD in D with P1 ^ P 2 ^ •.. ^ p h . Since xi is a nonzerodivisor in R, it follows that a nonzerodivisor in D; since xiT„ - x„ clearly belongs to ker(J), by (Cll)(2) and (T24) there exists a prime ideal PQ in D with P 0 C Pi and PQ ^ P\. Therefore h t s ( P B ) > h.

(Q9) GOING UP AND GOING DOWN THEOREMS

241

RELATIVE INDEPENDENCE THEOREM (T34). Elements xu ... ,xn in a noetherian ring R are independent over R iff ht^Z = n. PROOF. The assertion being vacuous if R is the null ring, we may assume R to be nonnull. Let n* = 1 + n and let I* be the ideal in the univariate polynomial ring R* = R[Z] generated by the sequence (x\,..., a;*.) = (Z,xi,..., x„). Then n* > 1, the element x^ is a nonzerodivisor in the nonnull noetherian ring R*, by (T30.1) we know that ht f l .7* = n* o ht f i J = n, and by (T32) we know that the elements x\,. •., a;*. are independent over R* •& the elements x\,...,xn

are independent over R.

Thus we are reduced to (T33.4) and (T33.5). ANALYTIC INDEPENDENCE THEOREM (T35). In case R is the power series ring i^[[Xi,..., Xm]] in a finite number of variables over a field K, and the elements x\,... ,xn belong to M{R), we have the following. (T35.1) If the elements x\,...,xn are independent over (R,M(R)) then they are analytically independent over K. (T35.2) If m = n and the ideal I is M(i£)-primary then the elements x\,..., xn are analytically independent over K. PROOF OF (T35.1). Assume xi,...,xn are analytically dependent over K. Then there exists 0 ^ f{Y1,..., Yn) e K[[Yi,..., Y„]] with f(xi ,...,xn) = 0. Now let i be the order of / . Then i € N and f(Yu..., Yn) £ M{Rf \ M(R)i+1. Consequently f(Yu...,Yn) = F(Y1,...,Yn)£R[Y1,...,Yn]i\M(R)\Y1,...,Yn]i. Therefore dependent over (R,M(R)). PROOF OF (T35.2). Follows from (T34) and (T35.1). QUEST (Q9) Going U p and Going Down Theorems In (T30.4) we showed that a finite variable polynomial ring over a field has dimension equal to the number of variables. We want to generalize this by showing that the dimension of an affine domain over a field equals the transcendence degree of its quotient field. To do this we need the going up and going down theorems for prime ideals in integral extensions of rings; the domain case of these theorems is due to Krull, and then they were generalized by Cohen-Seidenberg to rings with zerodivisors. These are applied via Emmy Noether's Normalization Theorem which we shall take up in the next Quest and which says that every affine ring is an integral extension of a polynomial ring. Note that an integral extension of a ring is an overring which is integral over the ring. Also note that a prime ideal Q in an overring S of a ring R lies above a

242

L5: PROJECTIVE VARIETIES §5: MORE ABOUT IDEALS AND MODULES

prime ideal P in R means P = Q n R; we may also express this by saying that P lies below Q in R. We extend the notion of a normal domain to a normal ring by saying that the ring R is normal if it is integrally closed in its total quotient ring QR(R). Clearly the quotient field of a domain is the same thing as its total quotient ring, and hence a domain is a normal domain iff it is a normal ring. Note that if a multiplicative set T in a ring R is such that TnZs(S) = 0 where Zs(S) is the set of all zerodivisors in an overring S of R, then the localization RT may be regarded as a subring of QR(S'). In particular this is so for the localization RP at a prime ideal P in R with (R\P)r\ ZS(S) = 0. See L4§7. LOCALIZATION OF NORMALITY LEMMA (T36). For a normal ring R, the localization RT at a multiplicative set T in R with T n ZR(R) = 0 (hence in particular the localization Rp at a prime ideal P in R with ZR(R) C P) is a normal ring. PROOF. Given any y in QR(i?) which is integral over RT, we can write

yn + J2

x

iyn~l = °

with

n € N

+

l
and Xi £ RT- We can find t £ T with txt £ R for 1 < i < n. Multiplying the above equation by tn we get

(ty)n + J2 (*<^)(*y)n-i = 0 l
with tlXi £ R as an equation of integral dependence for ty £ R over R. Consequently by the normality of R we get ty £ R and hence y £ RTPRESERVATION OF INTEGRAL DEPENDENCE LEMMA (T37). For any integral extension S of a ring R we have the following. (T37.1) If tp : S —> S" is any ring homomorphism then tp{S) is integral over tp(R). Equivalently, if J and / are ideals in S and R with I = J C\ R then after identifying R/I with a subring of S/J we have that S/J is integral over R/I. (T37.2) If ip : S —> S T is the natural homomorphism where T is any multiplicative set in R, then SV is integral over ip(R)^(T)- [Referring to L4§7, upon letting : R —» /?T be the natural homomorphism, there is a unique isomorphism 8: RT -> ip{R)^(T) such that 0(0(:r)) = V(z) for all x £ R, and if T n ^ ( S 1 ) = 0 then we may regard i?r and ST as subrings of QR(5) with RT C 5T] • PROOF. Let

yn+

J2 XiV11^ = ° KKtt

with

n e N

+

(Q9) GOING UP AND GOING DOWN THEOREMS

243

and Xi G R be an equation of integral dependence for y G S over R. In case of (T37.1) we get

l
with ip(xi) G ip(R) as an equation of integral dependence for ijj(y) over ip{R). In case of (T37.2), for any t G T we get

my)/m}n+

£ [^(xl)/^(ti)][v(2/)/V'W]"-i = o l
with ip(xi)/ip(tl) over ip(R)^T)-

£ ip(R)i/,(T) as an equation of integral dependence for ip(y)/ip(t)

PRESERVATION OF FIELDS AND MAXIMAL IDEALS LEMMA (T38). For any integral extension S of a ring R we have the following. (T38.1) If S is a domain then: R is a field o 5 is a field. (T38.2) If P is a prime ideal in .ft lying below a prime ideal Q in S then: P is maximal •*=> Q maximal. PROOF. (T38.2) follows by taking (R/P,S/Q) for (S,R) in (T38.1). To prove (T38.1) first suppose that S is a domain and R is a field. Then for any 0 ^ y £ S we have that y/R is integral and hence y/R is algebraic and so 1/y G -ff[j/] C 5". Therefore S is a field. Conversely suppose that S is a field and let 0 ^ £ G .ft. Then x~x G 5 and hence x_1/R is integral, i.e., (z-1)"+ £

z i (a;- 1 ) n - < = 0

with

n G N+

l
and Zj G R, and multiplying by x n _ 1 we get x~x

= -

£

Z i ^ " 1 G JJ.

l
Therefore Pi is a field. LYING BELOW LEMMA (T39). Given any overring S of a ring R and any prime ideal P in R, we have the following. (T39.1) P lies below some prime ideal in 5 iff (PS) nR = P. (T39.2) If P does not lie below any prime ideal in S, then there exists a prime ideal Q in 5 together with elements y G S, u G P\Q, and v G R\P, such that Q n R C P and v -uy £Q. PROOF. The only if part of (T39.1) is obvious. To prove its if part let W be the set of all ideals Q in S with Q D R = P and assume that (PS) flfl = P , and

244

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VARIETIES §5: MORE ABOUT IDEALS AND MODULES

to prove (T39.2) let W be the set of all ideals Q in S with QnRc P. Then W is nonempty (in the first case it contains PS and in the second case it contains 0). In both the cases the union of any linearly ordered (by inclusion) subfamily of W clearly belongs to W. Hence by Zorn's Lemma W contains a maximal member Q. Clearly 1 ^ Q. If Q were not prime then there would exist elements r, t in S\Q with rt G Q; now the ideals Q + rS and Q + tS properly contain Q (i.e., contain Q but are different from it) and hence by maximality they respectively contain elements r',t' belonging to R\P; for suitable a, b in S we have r' —ar G Q and t' — bt G Q and this implies r't' — abrt G Q which gives the contradiction r't' G (QC\R)\P. Thus Q must be prime. This proves (T39.1). To prove (T39.2), assuming Q n R ^ P, we need to show the existence of w G S, u G P \ Q, and v G R\P, with v — uy G Q. Since Q (~) R C P with QflR^ P, there exists u e P\Q. Now Q + US' properly contains Q, and hence by maximality we have (Q + US') C\R(£ P, and therefore we can find y G s and v € R\P with v — uy £ Q. LYING ABOVE THEOREM (T40). Given any integral extension 5 of a ring P and any prime ideal P in R, there exists a prime ideal Q in 5 lying above P. PROOF. Suppose if possible that there is no prime in S lying above P. Then by (T39.2) there exits a prime Q in S together with elements y G S, u G P\Q, and v G R\P, such that Q (1 Re P and v — uy £ Q. Since i / € S and 5/i? is integral, we have

yU+

H

a;iyn_i = 0

with

n G N+

l
and Xi G il. Multiplying by un we get l
and hence, because w — uy G Q, we get vn + ^

( u ^ K " ' G Q.

l
Since the above LHS is in R, it must be in P; since u G P , we get vn € P and hence u e F which is a contradiction. GOING UP THEOREM (T41). Given any integral extension S of a ring R, given any prime ideals P* C P in R, and given any prime ideal Q* in 5 lying above P*, there exists a prime ideal Q in 5 lying above P such that Q* C.Q. PROOF. By (T37.1) S/Q* may be regarded as an integral extension of R/P*, and P/P* may be regarded as a prime ideal in R/P*, and so our assertion follows from (T40).

(Q9) GOING UP AND GOING DOWN

THEOREMS

245

PROPER CONTAINMENT LEMMA (T42). Given any integral extension S of a ring R we have the following. (T42.1) Assuming S to be a domain, for any nonzero ideal J in S we have

JflR^O. (T42.2) If Q is a prime ideal in S lying above a prime ideal P in R and J is any ideal in S properly containing Q (i.e., containing Q but different from it), then J fl R properly contains P. PROOF. (T42.2) follows by applying (T42.1) to S/Q and invoking (T37.1). To prove (T42.1), assume S to be a domain, fix 0 ^ y £ J, and let yn + ^2

Zjj/™-* = 0

with

n £ N+

and

x{ £ R

l
be a smallest degree equation of integral dependence for y over R. Then 0 ^ xn £ JnR. RADICAL DESCRIPTION LEMMA (T43). Given any integral extension S of a domain R such that S is a domain, and given any ideal H in R, we have r a d s (ifS) = {y£S:yn + J2i
z m + J2

tiz7n i

~ =°

with

m e N

+ a n d f*€

F

l
and then by taking z = yd and n = dm we will get yn + ^Zi
"iU m _ i = 0

with

m G N+

and

xiyn~'1 = 0 with

u* G ii

l
be an equation of integral dependence for v over R we get -2m + ]Ci
+ ^2

r w9

J ~J = °

witil

?£^+

and

Tj £ H.

i
Let the products (z l u^)o m can be written as a linear combination of 1, z,..., zm~l with coefficients in H, and every power uP with j > q can be written as a linear

246

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VARIETIES §5: MORE ABOUT IDEALS AND MODULES

combination of 1, w,..., w9^1 with coefficients in H. Therefore every product ziw:> with i + j>m + q — 1 can be written as a linear combination of f \ , . . . , fmq with coefficients in H. Hence for p = m + q — 1 we have (z + w)pfi =

NJ

hijfj

with

hij G H

and

1 < i < mq.

l<j<mij

Therefore, letting Sy be the Kronecker symbol and considering the determinant D of the (mq) x (mq) matrix ((z + w)pSij — hy), by Cramer's Rule we get Dfi = 0 for 1 < i < mq. Since one of the fi equals 1, the determinant D is zero and this gives us an equation of degree p for z + w of the desired type. GOING DOWN THEOREM (T44). Given any integral extension S of a normal domain R such that S is a domain, given any prime ideals P* C P in R, and given any prime ideal Q in S lying above P , there exists a prime ideal Q* in S lying above P* such that Q* C Q. PROOF. Let T be the multiplicative set in S consisting of elements y which can be expressed as y = ab with a € R\P* and b G S\Q. Suppose if possible that T n (P*S) ^ 0, i.e., for any such elements y, a, b we have y = ab G P*S. Then by (T43) we have A(y) = 0 for some univariate polynomial A(Y) over R of the form AkYk

A(Y) = ] T

where

n G N+

and

^ fc G P*

0
with An = 1. Let B(y) =

^

B

*yi

where

m e N

and

+

B e

i QF(#)

0<s<m

with Bm = 1 be the minimal polynomial of y over QF(P). Then C(Y) = A(Y)/B(Y)

= Y^

C YJ

J

where

1e

N

and

C

J

e

QF(#)

0<J<9

with Cq = 1. By Kronecker's Theorem (see L3§8(J25) and L4§14(N5)) B(Y) and C(Y) belong to R\Y]. Clearly there is a unique minimal i with Bi 0 P* and a unique minimal j with Cj $. P*. Now Ai+j = BiCj + terms in P*, and hence Ai+j £ P*, and so i + j = n, i.e., i = m and j = q. Therefore Bi G P* for 0 < I < m - 1. Clearly

B'(Y) = J2

B yl

'i

with

B

'i = Bi/am~l

0
is the minimal polynomial of b over QF(R), and hence again by Kronecker's Theorem (see L3§8(J25) and L4§14(N5)) we get B[ G R for 0 < / < m-1. Since Bt = am-lB[

(Q10) NORMALIZATION

with a£ R\P*

THEOREM AND REGULAR POLYNOMIALS

247

and Bi £ P*, we must have B{ € P*. Consequently

bm = -

J2

B b e P S c

'i '

*

Q

0
which is a contradiction. Thus we have shown that T n (P*S) = 0. Hence by L4§5(O20)(12#) there exists a prime ideal Q* in S with TnQ* = 0 and P*S C Q*. Now T n Q* = 0 =» Q* n (S \ Q) = 0 => Q* C Q. Also

T n Q* = 0 => (Q* n R) n (i? \ p*) = 0 =*• Q* n R c P* where the first implication follows by noting that if there existed

a' e (Q*nR)n(R\p*) then by letting 6' = l e S \ Q w e would get y' = a'b' G T n Q*. Since

P* c (P*S) n R c Q* n R, we must have Q* D R = P* • DIMENSION COROLLARY (T45). For any integral extension 5 of a ring R we have the following. (T45.1) Given any finite sequence Po C Pi C • • • C P n of prime ideals in R with Po 7^ Pi 7^ • • • ^ Pn, there exists a sequence Qo C Q\ C • • • C Qn of prime ideals in S with Q0 ^ Qi ^ • • • ^ Qn such that Pt- = Qt,D R for 0 < i < n. (T45.2) Given any finite sequence Q0 c Q\ C • • • C Qn of prime ideals in 5 with Qo y£ Qi T^ • • • 7^ Qn; upon letting P, = Qj n R for 0 < i < n, we have that Po C Pi C ••• C P n is a sequence of prime ideals in R with P 0 ^ Pj ^ • • • ^ P„. (T45.3) dim(S) = dim(P). (T45.4) For any prime ideal Q in S lying above any prime ideal P in R we have dptsQ = d p t K P and h t s Q < h t f l P . (T45.5) If 5 is a domain and R is a normal domain, then for any prime ideal Q in S lying above any prime ideal P in R we have htsQ = h t ^ P . PROOF. The first assertion follows from (T40) and (T41). The second assertion follows from (T42). The third and the fourth assertions follow from the first two assertions. The fifth assertion follows from (T42) and (T44). QUEST (Q10) Normalization Theorem and Regular Polynomials The above theorem, proved by Emmy Noether in 1926, says that any affine ring A over a field k has a normalization basis over k, by which we mean a finite number

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of elements Y\,... ,Yr in A which are algebraically independent over k and which are such that the ring A is integral over the subring k[Yi,... ,Yr\. By definition the affine ring A is a homomorphic image (B) of a finite variable polynomial ring B = k[Xi,..., X/v] under a fc-epimorphism <j>: B —* A. The basic idea behind the Normalization Theorem is the fact that, by applying a k-automorphism 8 to B, any given nonzero member of B can be made regular in X^. Recall that a nonzero polynomial F of degree d in X\,..., X^ over k is regular in X^ means F = cXdN + ^

FiXfT*

where

F

i

e

M*i> • • •. Xff-i]

l
with deg(i?j) < i and O ^ c e f e . This was proved in L3§12(D4) where it was noted that if k is infinite then 8 can be chosen to be A;-linear, i.e.,

8(Xi) = Yi

MijXj

for

1< i < N

l<j
where M = (My) is an N x N matrix over k with det(M) ^ 0; note that if N = 0 then 0 ^ F = c € k with d = 0. In Theorem (T47) we shall show that the length r of any normalization basis of A over k equals the dimension of A, and is hence independent of the particular normalization basis. Actually, in view of (T45), this follows from (T30.4). However we want to make (Q9) and (Q10) independent of (Ql) to (Q8). NOETHER NORMALIZATION THEOREM (T46). Every affine ring A over a field k has a normalization basis Y i , . . . , Yr over k. SUPPLEMENT. Given any fc-epimorphism : B —» A where B is a finite variable polynomial ring k[X\,... ,XN], and given any nonunit ideal I in B with ker(4>) C / , we can choose the normalization basis Y i , . . . , YT of A over k, for some integer 0 < r < N, together with a fc-automorphism r of B and elements Z\,..., ZM in B which are algebraically independent over k so that: B is integral over the ring C = k[Zu. ..,ZN] with ker() f l C = (Zr+1,..., ZN)C and (<£(Zi), • • •, 4>{Zr)) = ( Y i , . . . , Y r ), and for some integer 0 < s < r we have I DC = {Zs+i,..., ZN)C with (ZU...,ZS) = ( r ( X i ) , . . . , r ( X s ) ) and {I) n E = ( Y s + 1 , . . . , YT)E where E = k[Yu...,Yr]. ADDENDUM. If k is infinite then r can be chosen to be fc-linear. PROOF. We shall first consider the Special Case when is the identity map of B = A. Note that in this case r = N and we need to find a fc-automorphism r of B (which is fc-linear if fc is infinite) together with elements Z\,..., Z^ in B which are algebraically independent over fc such that B is integral over C = k[Z\,..., Z^\ and for the given nonunit ideal / in B we have I D C = (Za+i,..., ZN)C with (Zi,..., Zs) = ( T ( X I ) , . . . , T(XS)) for some nonnegative integer s < N. To satisfy the "need" we make induction on N. This being trivial for N = 0, let TV > 0 and

(Q10) NORMALIZATION

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assume for N — 1. If I = 0 then we have nothing to show. So also assume 1^0 and fix any 0 =£ Z^ S / . Since I ^ B, by L3§12(D4) there is a fc-automorphism 6 of B (which is fc-linear if fc is infinite) such that 9{ZN) = cXdN+

Yl

F X

i N~i

where

Ft € B' = k[Xu...,

XN^}

l
with d € N+ and 0 ^ c G fc. Let / ' = 0(7) n B'. Then V is a nonunit ideal in B' and hence by the induction hypothesis we can find a fc-automorphism r ' of B' together with elements Z[,...,Z'N_1 in B' which are algebraically independent over k such that B' is integral over C = k[Z[,... ,Z'N_1] and for the nonunit ideal I' in B' we have V D C = (Z's+1,..., Z'^_X)C' with (Z[,..., Z's) = (T'(X\), ... ,T'(XS)) for some nonnegative integer s < N — 1. Let Zi = 6~1{Z'i) for 1 < i < N — 1. By the above displayed equation we see that 6~1(X^) is integral over 6~1(B')[ZN], and hence the elements Z\,..., ZJV in -B are algebraically independent over A; and the ring B is integral over the subring C = k[Z\,..., Z/v]. Since I' flC" = (Z's+1,..., Z'N_1)C, by the above displayed equation we also see that IDC = (Zs+i,..., ZN)C. Obviously there is a unique A;-automorphism r of B such that T(XN) = e-\XN), and T(Xi) = fl-VU^i)) for 1 < i < N - 1. Clearly ( Z i , . . . , Zs) = (T(XI), ..., T(XS)), and if k is infinite then r is fc-linear. Having completed the proof of the Special Case, let us turn to the General Case. Applying the Special Case to (£,ker(<^)) we can find a fc-automorphism a of B (which is fc-linear if fc is infinite) together with elements T i , . . . , TN in B which are algebraically independent over k such that the ring B is integral over the subring D = k[Tu ...,TN] and we have ker(cp) DD = ( T r + 1 , . . . , TN)D with (T1,...,Tr) = (a(Xi),..., a(Xr)) for some nonnegative integer r < N. Let B' = k[Ti,..., Tr] and I' = IDB'. Applying the special case to (£?', / ' ) we find a fc-automorphism r ' of B' (which is fc-linear if fc is infinite) together with elements Zi,...,Zr in B' which are algebraically independent over fc such that B' is integral over C = k[Z\,...,ZT\ and we have V n C = (Zs+1:..., Zr)C with (Zu ...,Z.) = ( r ' ( T i ) , . . . , T'(TS)) for some nonnegative integer s < r. Let Zi = Tt for r + 1 < i < N. Then the elements Z\,...,Z^ are algebraically independent over fc, the ring B is integral over the subring C = k[Z\,..., ZN], and we have ker(^) DC = (Zr+i,..., ZN)C with I n C = (Z8+1,..., ZN)C. Let Yi = (Zi) for 1 < i < r. Then Y1:... ,Yr is a normalization basis of A over fc, and we have cj>(I) C\E = {Ys+i,..., Yr)E where E = k[Yi,..., YT\. Obviously there is a unique fc-automorphism r of B such that r{Xi) = T'(a(Xi)) or a{Xi) according a s l < i < r o r r + l < i < W . Clearly (Z\,..., Zs) = (T(X\), ... ,T(XS)), and if fc is infinite then r is fc-linear. As a consequence of (T45) and (T46) we shall now prove Theorem (T47) which includes L4§8(T19). For convenience we introduce some: TERMINOLOGY. By a saturated chain of prime ideals in a ring R we mean a

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finite sequence Po C Pi C • • • C P„ of prime ideals in R with P0 ^ Pi ^ • • • j= Pn such that there is no prime ideal P* in R with Pj C P* C Pj+i and Pt ^ P* ^ Pi+1 for some i G { 0 , . . . , n — 1}; we call n the length of the chain; if P C Q are prime ideals in R with P = P 0 and Q = P„ then we say that Po C Pi C • • • C P„ is a saturated prime ideal chain between P and Q; if there is no prime P in P with P C Po and P ^ Po and there is no prime Q in R with Pn C Q and P„ ^ Q then we say that Po C Pi C • • • C P n is an absolutely saturated prime ideal chain in R. By an infinite chain of prime ideals in a ring R we mean a sequence of prime ideals (Pi)j€N in R such that for all i G N we have Pi C P i + i with P» ^ Pi+V, if P C Q are prime ideals in R such that P C P* C Q for all z G N then we say that this is an infinite prime ideal chain between P and Q. For a subdomain R of a domain R', by trdeg^P' we denote the transcendence degree of QF(P') over QF(P). More generally, for a prime ideal P' in a ring R' lying above a prime ideal P in a subring R, after identifying R/P with a subdomain of R'/P', by t r d e g R / p P ' / P ' we denote the transcendence degree of Q F ( P ' / P ' ) over Q F ( P / P ) ; in case R is a subfield k of R', we write trdeg^P'/P' in place of trdegk/oR'/P'\ in case P' and P are the respective maximal ideals in quasilocal rings R' and R, we may write restrdeg^P' in place of trdegR/PR'/P' and call this the residual transcendence degree of R' over R. EXTENDED DIMENSION THEOREM (T47). Given any affine ring A over a field k, we have the following. (T47.1) For any nonunit ideal J in A, upon letting J\,..., Jm be all the distinct minimal primes of J in R we have dim(A/J) = dptAJ = maxi<j< m trdeg fc (A/J,) G N. Hence, for any prime ideal P in A we have dim(A/P) = d p t ^ P = trdeg fc (A/P) G N. In particular, if A is a domain then dim(A) = trdegfeA G N. (T47.2) Every pair of prime ideals P c Q in A has the property which says that: (*) there is no infinite chain of prime ideals between P and Q, and any two saturated chains of prime ideals between P and Q have the same length. Moreover: (**) this common length equals dim(A/P) — dim(A/Q). (T47.3) If A is a domain then the length of every absolutely saturated chain of prime ideals in R equals the dimension of R. (T47.4) If A is a domain then for every nonunit ideal H in A we have htA.H" + d p t A # = dim(A). (T47.5) If A is a domain then, for any prime ideal P in A, the length of every saturated prime ideal chain in A between 0 and P equals h t A P , and the length of

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every saturated prime ideal chain in A between P and any member of mvspecAP equals dpt^P(T47.6) If A is a domain and A' is an afflne domain over A then for any prime ideal P' in A' lying above any prime ideal P in A, upon considering the localizations R' = A'P, and R = Ap, we have dim(/?') + restrdeg^-R' = dim(P) +

tidegRR'.

(T47.7) In the situation of (T46) we have dim(A) = r and d p t ^ ( J ) = s. PROOF. Clearly dim(A/J) = dpt A J = maxi<,< m dim(j4/Ji) and hence to prove (T47.1) it suffices to show that if A is a domain then dim(A) = N where AT = trdeg fc A We shall do this by induction on N. By (T46) we can arrange that A is an integral extension of B — k[X\,... ,XN] and then by (T45.3) we have dim (A) = dim(B). If N = 0 then obviously dim(B) = 0 and hence dim(A) = 0. So let N > 0 and assume that our assertion is true for all smaller values of N. By L4§5(O22)(X10) we have dim(B) > N. Suppose if possible that dim(B) > N, and let Q0 C Q1 C • • • C QN+I D e a sequence of prime ideals in B for which we have QQ ^ Qx =£ • • • ^ Qjy+i- Then B/Qx is an afflne domain over k with trdegfc(-B/<51) < N and hence by the induction hypothesis d\m{B/Qx) < N which is a contradiction because obviously dim(P>/Q1) = dptBQx > N. This completes the proof of (T47.1). Continuing to assume A to be a domain, let 0 = PQ C P I C • • • C Pn be any absolutely saturated prime ideal chain in A. Since dim(j4) = N, we must have n < N. By induction on N we shall show that n = N, which will prove (T47.3). This being obvious for N = 0, let N > 0 and assume for all smaller values of N. Since dim(^4) > 0, the domain A cannot be a field, and hence we must have n > 0. For 0 < i < n let Qi = Pj n B. Then 0 = Qo C Qi C • • • c Qn is a prime ideal sequence in B, and by (T45.2) we have <5o ¥" Qi ¥" • •' ¥• Qn', we CLAIM that this is an absolutely saturated chain in B. First note that P n is clearly a maximal ideal in A, and hence Qn is a maximal ideal in B by (T45.4). Also clearly ht^Pi = 1 and B is a normal domain because it is a UFD; consequently htfiQi = 1 by (T45.5). By the Special Case of (T46) we can find a sequence of integers A'' > A r (l) > • • • > N(n) = 0 and, for 1 < i < n, elements X i v l , . . . , X^N in B which are algebraically independent over k such that B is integral over the subring Bt = k[Xiti,..., X^N] with Qt n Bt = (Xii/(Qi n Bi), it follows that B/Q\ is an (N — l)-dimensional affine domain over k, and hence by the induction hypothesis, every absolutely saturated prime ideal chain in it has length N — 1. Given any i e { 1 , . . . , n - 1}, let a, : A —> A/Pi and 0i : B —• B/Qi be the residue class epimorphisms, and let d = k[Xiti,..., X^^]. Then a^ and fli induce

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isomorphisms d —• a%{Ci) and d —» 0i(d), and the domains A/Pi and B/Qi are integral over the polynomial rings on{Ci) and /3j(Cj) respectively. Moreover, under the induced isomorphisms, the prime ideal Qi+\ DC, in d is sent to the prime ideals (Pi+i/Pi) C\ ai(d) and (Qi+i/Qi)C\Pi(d) respectively. Consequently, by applying (T45.5) twice we get htA/Pi(Pi+1/Pi) = h t C i ( Q i + i DCO = h t B / Q . ( Q i + i / Q 0 - Since the chain P 0 C Pi C • • • C Pn is saturated, we must have htA/Pi(Pi+i/Pi) = 1. Therefore ht B /Q ( (Q i + i/Qi) = 1- This being so for 1 < i < n - 1, our CLAIM is proved. Consequently by going mod Qi we see that Qi/Qi C ••• C Qn/Qi is an absolutely saturated prime ideal chain in B / Q i of length n — 1, and hence n - 1 = TV - 1. Therefore n = N. This completes the proof of (T47.3). Still continuing to assume A to be a domain, given any prime ideal P in A, we can clearly find saturated prime ideal chains (')

P0C---CP^

and

PoC-.-CPd

in A such that Po = 0 and Ph = P with htAP = h and such that PQ = P and Pd € mspec(A) with dptAP = cJ- This gives the absolutely saturated prime ideal chain (")

Po C • • • C Ph C Pi C • • • C Pd

in A of length h + d, and hence by (T47.3) we get h + d = N. In particular, given any nonunit ideal H in A, for every P in vspec^-ff we have htAP + d p t A P = N, and hence \AAH + dptAH = N. This proves (T47.4). Again assuming A to be a domain, given any prime ideal P in A, if (') are any saturated prime ideal chains in A such that P 0 = 0 with Ph = P and such that Po = P with Pd € mspec(A), then clearly (") is an absolutely saturated prime ideal chain in A of length h + d, and hence by (T47.3) we get h + d = N. But obviously h t A P > h and d p t ^ P > d, and by (T47.4) we have h t A P + d p t A P = n. Therefore we must have h t A P = h and d p t A P = d. This proves (T47.5). In (T47.2), the nonexistence of an infinite chain between P and Q follows because by (T47.1) we have dim(A/P) < oo. Moreover, given any saturated prime ideal chain P = Po C Pi C Pn = Q in A between P and Q, we get a saturated prime ideal chain 0 = PQ/P C P i / P C Pn/P = Q/P in A/P between 0 and Q/P, and hence by (T47.5) we obtain n = htA/P(Q/P) and by (T47.4) we have htA/P(Q/P) = d i m ( ^ / P ) - dim(yi/Q). This proves (T47.2). Once again assume that A is a domain and let P, R, A',P', R' be as in (T47.6). Then A' is also an affine domain over k, and hence by (T47.1) we get dim(A') - dim(A) = txdegRR' and restrdeg fl il' = dim(A'/P') - dim(A/P).

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By (T47.3) we also have dim(R') - dim(R) = [dim(A') - dixn(A'/P')} - [dim(A) - dim(A/P)]. Adding the three equations we obtain dim(R') + restrdegfl.R' = dim(fl) + trdeg^-R' which proves (T47.6). In the situation of (T46), A is integral over the r-variable polynomial ring E = k[Yi,..., Yr] and A/cp(I) is integral over an isomorphic image of the s-variable polynomial ring D = k[Yi,...,Ys], and hence by (T45.3) and (T47.1) we get dim(A) = dim(E) = r and dpt(A/0(J)) = dim(A/(J)) = dim(D) = s. This proves (T47.7). We shall close this Quest by deducing some parts of the Hilbert Nullstellensatz, i.e., L4§8(T22), as consequences of part (T47.1) of the above Dimension Theorem. The rest of the Nullstellensatz will be dealt with in the next Quest. PROOF OF (T22.1) TO (T22.6) OF L4§8. The first part (T22.1) says that a field which is an arfine ring over a field k is algebraic over fc; this follows from (T47.1) because the dimension of a field is 0, and algebraic means whose transcendence degree is 0. The second part (T22.2) says that for any maximal ideal J in the finite variable polynomial ring Bjv,fc = k[X\,..., Xpi], the field B^,klJ is algebraic over fc; this is simply (T22.1) in a different guise. The third part (T22.3) says that any such maximal ideal J has a unique set of N generators (fi{Xi,..., Xi) S Bitk)\
=

where n* is a positive integer and gi(X\,..., de

Sx; 9i{Xi,...,

Xi) < rij

X?i+gi(X1,...,Xi) Xi) e B ^ with for

1 < j < i.

As a preamble to the proof, let : B^tk —> Bj^^j 3 be the residue class epimorphism, let fc0 be the subfield 4>(k) of the field 0(Bjv,fc), and for 1 < i < N let ki = 4>(Bitk) and Xi = {Xi) together with 1% = {g{Xi, • • • ,Xi) € B iifc : degx.g(Xx,...

,Xt) < rij for 1 < j < i}.

By (T22.2) we know that fcjv is a finite algebraic field extension of fco (i.e., fcjv is an overfield of the field fco with [fc# : fc0] < oo), and hence for 1 < i < TV we see that ki = ki-i(xi) is a finite algebraic field extension of the field fcj_i. In proving our claim by induction on N, let us include the extra assertion which says that for 1 < i < N we have that: the polynomial fi(xi,...,Xi-i,Y) is the minimal

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polynomial of Xi over ki-i (and hence the field degree [ki : &i_i] equals n,), the polynomial fi(X\,..., Xi) is irreducible in Bitk, and g(Xi,...,

Xi) i-> g(xi, ...,Xi)

gives a bijection

7* -> fcj.

With this added task the AT = 0 case is obvious and for N > 0 the implication N — 1 => N can be proved thus. Let B = B^,k and A = B^-i^ with / = J n A Then

1 onto the field fc/v-i and the kernel of the induced epimorphism A —> /CAT-I is J. Consequently J is a maximal ideal in A. Therefore by the induction hypothesis / has a unique set of N — 1 generators (fi(Xi,..., Xi) £ .Bj,fc)i
=

X^+9i(X1,...,Xi)

where rij is a positive integer and gi(X\,... de

g x , 9i{Xi,...,

,Xi) £ Bi^ with

Xi) < rij

for

1 < j < i.

Moreover, for 1 < i < N — 1 we have that: the polynomial / , ( x \ , . . . , Xi-\, Y) is the minimal polynomial of x^ over fcj_i (and hence the field degree [ki : /CJ_I] equals rij), the polynomial fi(X\,..., Xi) is irreducible in Bj,fc, and g{Xi,...,

Xi) i-> g{x±,... ,Xi)

gives a bijection

7, —> fcj.

Upon letting n^ = [fc/v : fc^-i] we see that njv is a positive integer and we can find /JV = JN(XI, ... ,Xpf) £ B such that / w ( x i , . . . ,xn-i,Y) is the minimal polynomial of xn over kN-i and JN(XI,.

where gN(X\,...,

.. ,XN)

= X^N

+ gN(Xi,...

,XN)

X^) £ B with deg Xj gN(Xi,...,XN)

< rij

for

1 < j < N.

Clearly /N(XI, ... ,XN) is irreducible in B and belongs to J. Given any h £ B, by the division algorithm we can uniquely write h = qfjy + r with q, r in B such that degxNr < njv. From this it follows that g(Xi,..., XN) *-* g(%i, • • •, XN) gives a bijection 7 ^ —> k^. Moreover h £ J =$• r £ I =$• r £ (fi,..., JN-I)A =i>fce (/1, • • • i IN)B, and hence ( / 1 , . . . , /W)S =
= X*

+ Gi(Xu.

where Ni is a positive integer and Gi(X\,..., de g j f , Gi(Xlt...,

Xi) < Nj

..,Xi)

Xi) £ Bitk with for

1 < j < i.

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Now (IB) n A = I and hence there is a unique epimorphism 6 : B —> C = k^-i [Y] such that 0{XN) = Y and 9(z) = <j>(z) for all z € A. Let f(Y) = fN(x1,...,xn.1,Y)

and

F(Y) = FN(Xl,...

,xn-i,Y).

Then f(Y)C = 0(J) = F(Y)C and hence f(Y) = F(Y). Therefore NN = nN, and given any h e B, by the division algorithm we can uniquely write h = QF^ + R with Q,Rm B such that degx N i? < njv. From all this we see that ( F i , . . . , FJV-I)^4 = / . Hence by the uniqueness part of the induction hypothesis F, = /, with Ni = rij for 1 < i < N - 1. It is clear that also FN = fN. Thus (T22.3) has been proved. The fourth part (T22.4) says that if k is algebraically closed then the mapping a H-» Ifc(a) gives a bijection A^ —> mspec(Bjv,fc), where we recall that for every a = ( a i , . . .,aN) € A f = kN we have put I fc (a) = {/ e Bjv,fc : / ( a ) = 0}. Note that for f = f(Xi,...,XN)

G BNtk

and

a = (ai,...,aw)eAf = K"

where k C /t are any fields we write /(a) = /(ai,...,aiv). Clearly (T22.4) follows from (T22.3) or even from (T22.2). To see the latter, assuming k algebraically closed, let there be given any J in mspec^^fc). By (T22.2) the residue class epimorphism <j> : BN^ —* BN,IC/J maps k onto the entire image. Hence there exists a £ A^ such that <j>(Xi) = 4>(ai) for 1 < i < N. Now clearly we havelfc(a) = J = (Xi - Q i , . . . ,XN - aN)BNtkTo prove the rest of (T22) and some relevant portions of L4§8(T21), henceforth let k C K be fields, and let R = BNtk

= k[Xu.

..,XN]

and

S = k% = KN

where N is a nonnegative integer. For any U C S let I{U) = lk(U) = {f eR:

f(a) = 0 for all a € U}

and for any J c R let V(J) = VK(J) = {a£S:

f(a) = 0 for all / e J } .

Let I' = {I(U)

:UCS}

and rd(i?) = set of all ideals in R which are their own radicals. let V = avt fc (A^) = {V(J)

:JdR}

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and V" — set of all irreducible members of V' where U in V is irreducible means U is nonempty and cannot be expressed as union of two members of V which are subsets of U different from U. Now clearly

{

(1)

for any subsets J and J' of R we have: J C J' => V(J') C V(J) and we have: radR(JR)

= mdR(J'R)

=» V(J) = V(J')

and (2)

for any subsets U and U' of S we have: U C U' => /([/') C J(f/). We claim that for any family of ideals v(EieLJi)

(3)

=

(JI)IZL

in R we have

nieLv(jl),

and if the family is finite then we have

{v(nleL Ji) = v(UleL Ji) = yJieLVW and (4)

{

for any family of subsets {U{)ieL of S we have I(VI€LUI)

= n ;eL 7(C/()-

Concerning the first equation of (3), by the first implication of (1) we see that LHS C RHS; conversely: a € RHS => a € V(UieL-/() => a e LHS where the first implication is obvious and the second follows from the second implication of (1). Concerning the last line of (3), by the first implication of (1) we get the inclusions UieLV(Ji) C V(nj £ £,Jj) C VXPI/gL ^ ) i moreover, for any a € S we have that: a ^ UjgLV'(Jj) => for each I 6 L there exists /; G J; with ft(a) ^ 0 =^ / ( a ) 7^ 0 with / = X\leLfi G Y\l£LJi, and hence ^(flieL Jl) c U; e /,y(J;). Concerning equation (4), by (2) we get LHS C RHS; conversely: / G RHS => / ( a ) = 0 for every a G Ul€LUi =>fe LHS. We also claim that J for any J c f i w e have J C \ a n d : J = I(V(J))

I(V(J))

& J G V => J = r a d R J

and J for any U C S we have 17 C V{I{U)) [and: [/ = V{I(U))

**UeV.

The first line of (5) follows by first applying the first line of (1) and after that applying (2). The first line of (6) follows by first applying (2) and after that applying

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the first line of (1). Concerning the second line of (5), obviously J = I{V{J)) =>• J £ I1; moreover 'J el' =>J = I(U) for some

UcS

=*. V(J) = V(I(U)) by taking V of both sides ' =>UC V(J) by first line of (6) => I(V(J))

C J by (2) because I{U) = J

=> J = J(V(J)) by the first line of (5); also: / G radfi J with J = I(U) for some U c S =4- / n e J for some n G N + < => / " ( a ) = 0 for all a G £/ => / ( a ) = 0 for all a G U

and hence J = r a d ^ J . Concerning the second line of (6), obviously U = V(I(U)) => U G V; moreover 'UeV => U = V(J) f o r s o m e ^ C i? =» /([/) = J(F( J)) by taking J of both sides U j C /(£/) by first line of (5) =*• V{I(U)) cUby (1) because V{ J) = U => U = V(I(U)) by the first line of (6). Next we claim that (7)

V" = {U G V : I(U) G spec(fl)}.

To prove (7) let U G V. Assuming U to be irreducible, we have U ^ 0 and hence I(U) ^ R; moreover, for 1 < i < 2, given any fc G R\I(U), upon letting Ui = {a G U; fi{a) = 0}, we have Ui G V with Ui C U and f7j ^ [/; since U is irreducible, we must have t/i U J72 ^ U, and hence /1/2 ^ 7(£/); thus I(U) is prime. Conversely assuming I(U) to be prime, we have I(U) ^ R and hence U =/= 0; moreover, given any C/i,C/2 in V" with J7iUt/ 2 = *7, by (4) we have I{Ui)(M{U2) = I{U) and clearly I{Ui)I{U2) C I(Ui) n 7(£/2) and hence I(Ui)I(U2) C 7(t/); now assuming f/j ^ J7, by (2) and the second line of (6) we get 1(17) C 7(17!) with I(U) ^ I{U\), and hence the primeness of I(U) tells us that 7(L/2) C I(U) and therefore by the first line of (1) we get U2 = U; thus U is irreducible.

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Finally we claim that every U £ V can be expressed as a finite union /OA (8)

I <

U

= u^
and this decomposition is unique up to order if it is irredundant, i.e., if no Ui can be deleted from it, (where the irredundancy can be achieved by deleting some of the Ui, and after it is achieved, the remaining Ui are called the IRREDUCIBLE COMPONENTS of U). Before proving (8) we note that the noetherianness of R tells us that it has no strictly increasing infinite sequence of ideals, i.e., there does not exist any infinite sequence ideals J 0 C J\ C • • • C Jn C . . . in R with J0 ^ J\ ^ • •• ^ Jn^ ..., and hence by (2) and the second line of (6) we see that V has no strictly decreasing infinite sequence, i.e., there does not exist any infinite sequence ' W0 DWXD---D

WnD...

in V

^with W0 + Wl ± • • • ± Wn +

....

To prove the existence part of (8), suppose if possible that some U £ V has no irreducible decomposition, i.e., cannot be expressed as a finite union of members of V". Then U itself cannot be irreducible and we can write U = U\ U U^ with U\,U% in V such that U\ ^ U ^ U^\ note that the empty variety has the empty irreducible decomposition (with h = 0), and hence U is nonempty, and therefore U\,U2 are also nonempty. Clearly either U\ or U% has no irreducible decomposition (because if both did then putting them together would produce an irreducible decomposition of U). Say U\ does not have an irreducible decomposition. Let WQ = U and W\ = U\. In this manner we get an infinite sequence WQ D W\ D • • • D Wn D ... of members of V with Wo 7^ W\ 7^ • • • 7^ Wn T^ . . . such that none of the Wn has an irreducible decomposition. This contradicts (9) and so proves the existence part. To prove the uniqueness part of (8), in addition to assuming the decomposition displayed in (8) to be irredundant, let U = Ui<j
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BN+i,k' - k'[Xx,.. .,XN+1] and W = JT + (1 - XN+1f)T'. Then clearly there is no a' = K , . . . ,a'N+l) G A ^ + 1 such that f'(a') = 0 for all / ' G H'. Suppose if possible that H' ^ T". Then if' is contained in some maximal ideal H* in T", and by (T22.4) we have # * = (Xi - a[,.. .,XN+i - a'N+1)T'. But this gives the contradiction that f'(a') = 0 for all / ' G H'. Therefore we must have H' = T". Hence, upon letting T = BN+hk = k[Xu..., XN+i] and H = JT + (1 XN+1f)T, by the following Comment (C15) we get H = T. Consequently we can express 1 as a finite sum 1 = (1 - XN+if)g

+ Y^

with

fi9i

fit

and

J

9,9i'mT.

l
Substituting l/X^+\ for X^+i in the above identity and then multiplying both sides by X^+1 for a large enough n e N+ we get

J2 fiGi

X%+1 = (XN+i-f)G+

with

fi£j

and G,Gi in T.

l
Now substituting / for Xw+i in the above identity we obtain

/" = H fihi

with

/» € J

and h e R

i

\
and hence / " € J. The sixth part (T22.6) of (T22) says that if K contains an algebraic closure of k then the mapping U t-> I{U) gives inclusion reversing bijections V —> rd(i?) and V" —> spec(-B) whose inverses are given by J i—• V^(J). In view of (1), (2), (5), (6) and (7), this follows from (T22.5). COMMENT (C15). [Ideals Under Field Extensions]. In the above proof of (T22.5) we used the fact that: if k c k' are any fields and T = k[X\,..., XM] and T" = k'[Xi,..., XN] are finite variable polynomial rings, then for any ideal H in T we have (HT') C\T = H. To see this, we can take a vector space basis (zj)ie.j of k' over k with Zj = 1 for some j £ I. By expressing the coefficients of any / € T" in terms of this basis we can uniquely write / as a finite sum f = ^2zifi

with

fiET

where J is a finite subset of I with j £ J. Note that then /

T <=* / , = 0 for all i € J \ {j}.

e

Assuming / € ifT" we can write it as a finite sum / = Y,

F

i9i

with

Fi G T1

and

ffi

G H.

260

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VARIETIES §5: MORE ABOUT IDEALS AND MODULES

By taking J large enough, for all I £ L we have Fi = ^2 ZiFu

with

Fu G T.

ieJ

Now the uniqueness says that for all i £ J we have

fi = ^2FugieH. leL

Hence if / e (HT') n T then / = fiGH.lt

follows that ( F T ' ) n l = ff.

COMMENT (C16). [Supplement to Hilbert Nullstellensatz]. In the context of L4§8(T22.3), let cj> : B^,k -* B^^/J be the residue class epimorphism, let k0 be the subfield cj>(k) of the field {Xi) together with li = {9{Xi, ...,Xi)£

Bi,k : d e g x . 5 ( X i , ...,Xi)<

rij for 1 < j < i}.

Then, for 1 < i < N, as shown above: kt = /c,_j(a;j) is a finite algebraic field extension of the field fc,_i (i.e., kt is an overfield of the field fcj_i for which we have [hi : fcj_i] < oo and ki = ki-i(xi)) with [fcj : fcj_i] = m, the polynomial fi{x\,..., Xi-i, Y) is the minimal polynomial of Xi over fcj_i, the polynomial fi(Xi,..., Xj) is irreducible in Bitk, and g(X\,..., Xi) H-> 5(0:1,..., ij) gives a bijection 7* —> fcj. COMMENT (C17). [Decomposition of Ideals and Varieties]. In (1) to (8) above we have proved the first out of the three cases of the Inclusion Relations Theorem, i.e., L4§8(T21). The other two cases will be dealt with in the next Quest. In view of the portions of the Inclusion Relations Theorem and the Hilbert Nullstellensatz proved above, the decomposition of affine algebraic varieties into their irreducible components can be found thus. Let k C /t be fields such that K contains an algebraic closure of k. For any ideal J in the finite variable polynomial ring B = k[X\,... ,XN] take a primary decomposition J = ni<j< e Qi where Qi is primary for the prime ideal Pi in B. Then V K (P,) = VK(Qi) = an irreducible variety in A^ for 1 < i < e and V re (J) = Ui<j< e V K (Pj) is a decomposition of V K (J) into irreducible subvarieties. Moreover, if the primary decomposition of J is irredundant and is so labelled that Pi is a minimal or embedded prime of J according as 1 < i < d or d + 1 < i < e, then VK(J) = Ui
(Qll)

NILRADICAL,

JACOBSON

SPECTRUM,

AND JACOBSON

RING

261

QUEST ( Q l l ) Nilradical, Jacobson Spectrum, and Jacobson Ring At the end of this Quest we are going to give Two Supplements to the Dimension Theorem (T47) which slightly strengthen some parts of it. First we shall complete the proofs of the Inclusion Relations Theorem and the Hilbert Nullstellensatz, i.e., L4§8(T21) and L4§8(T22). To put the matter in proper perspective, we start by defining Nilradical, Jacobson Spectrum, and Jacobson Ring. Recall that in (Q9) and (Q10) we avoided using (Ql) to (Q8). In the current Quest we shall avoid using (Q2) to (Q8), except that we may use the incidental comment (Cll) of (Q7) but we shall not use Comment (C8) of (Ql). Also note that (Q9) to (Qll) are independent of §1 to §4 of this Lecture L5. After making some definitions, we shall prove the Nilradical Theorem (T48). Then we shall prove the Spectral Relations Theorem (T49) which includes the remaining two cases of L4§8(T21). After that we shall prove the Spectral Nullstellensatz (T50) which generalizes the remaining two parts (T22.7) and (T22.8) of the Hilbert Nullstellensatz of L4§8(T22), and adds yet another incarnation (T50.3) of it. Still after that we shall prove the Minimal Primes Theorem (T51) which says that (Cll)(2) remains valid without the noetherian hypothesis. Finally we shall prove the Supplementary Dimension Theorems (T52) and (T53). Recall that in (Q3) we defined the Jacobson radical of a ring R by putting

jrad(tf) =

f|

P.

P(Emspec(.R)

Now we define the nilradical of R by putting

nrad(tf) =

f]

P.

FGspec(R)

We also define the Jacobson Spectrum of R by putting jspec(-R) = < P £ spec(.R) : P = {

f]

Q\

Q£mvspec R P

J

and we note that then mspec(i?) c jspec(ii) C spec(R). The ring R is said to be a Jacobson Ring if jspec(i?) = spec(R). In analogy with the definitions of vspec^ J, mvspec#J, svt(-R), msvt(R), and imsvt(iZ) made in L4§8, for any J c f i w e put jvspec R 7 = {P € jspec(i?) : J C P) and call this the Jacobson spectral variety of J in R, and we put jsvt(ii) = the set of Jacobson spectral varieties in R

isvt(R),

262

L5: PROJECTIVE

VARIETIES §5: MORE ABOUT IDEALS AND MODULES

i.e., subsets of jspec(i?) of the form jvspec/{J for some J C R (we may call these varieties in jspec(-R)), and we put ijsvt(i?) = the set of all irreducible members of jsvt(i?) where U is irreducible means U is nonempty and U is not the union two proper subvarieties, i.e., subsets which belong to jsvt(i?) and are different from U. By an IRREDUCIBLE COMPONENT of U e jsvt(i?) we mean an irreducible subvariety U' of U such that U has no irreducible subvariety U" ^ U' with U' C U"; similar definitions hold for IRREDUCIBLE COMPONENTS of varieties in msvt(R) or svt(f?). Recall that rd(i?) = the set of all radical ideals in R and for any U C spec(iZ) we defined the spectral ideal of U by putting ispecfi£/ = npgt/P. NILRADICAL THEOREM (T48). For any ring R we have nrad(i?) = rad^O. More generally, for any ideal J in any ring R we have rad/jj = ispec R (vspec R J), i.e., radflJ is the intersection of all the prime ideals in R which contain J. PROOF. The first assertion is L4§5(O24)(20") which itself was a repetition of the first line of L4§5(O20)(12*). The second assertion follows by applying the first assertion to the residue class ring R/J. SPECTRAL RELATIONS THEOREM (T49). Given any ring R, let ' ( S , ^ ' , ! / " , / ) stand for (mspec(-R), mvspec R , msvt(i?), imsvt(i?), ispec R ) or (jspec(i?),jvspec fi ,jsvt(R), ijsvt(i?), ispec^) or (spec(-R), vspec^, svt(-R), isvt(ii), ispec fl ). In all the cases let I' stand for the set of all ideals J in R such that J = I{U) for some U C S. Then we have the following. (T49.1) For any subsets J and J' of R we have: J C J' =*> V(J') C V(J), and we have: radR(JR) = ia,dR(J'R) => V(J) = V(J'). (T49.2) For any subsets U and U' of S we have U C U' => I(U') C I(U). (T49.3) For any family of ideals (J;)/eL in R we have V(%2leL Ji) = r\i€LV(Ji), and if the family is finite then we have V(r\i£L Ji) = ^(ILeL «^) = ^ieLV(Ji). (T49.4) For any family of subsets {Ui)i€L of S we have I{Ui€LUi) = nieLI{Ui). (T49.5) For any J C R we have J C I(V(J)) and: J = I(V(J)) J £ / ' =*• J = radfl J. (T49.6) For any U C S we have U C V(I{U)) and: U = V(I(U)) &• U G V.

(Qll) NILRADICAL, JACOBSON SPECTRUM, AND JACOBSON RING

263

(T49.7) V" = {U £ V : I(U) G spec(P)}. (T49.8) Assuming R is noetherian, every U £ V can be expressed as a finite union U = L)i
W

and

I(V(T))

= T.

Moreover: (ii) if U £ V" is such that I(U) £ S then upon letting P = I(U) we have that P is the unique member of S with U = V(P). (T49.12) Without assuming R to be noetherian, in the last two cases, i.e., when S is either jspec(i?) or spec(i?), for any U £ V" we have that I{U) £ S and upon letting P = I(U) we have that P is the unique member of S with U = V(P). PROOF. The proofs of (T49.1) and (T49.2) are clear. Concerning the first equation of (49.3), by the first implication of (49.1) we see that LHS C RHS; conversely a £ RHS =» a £ V(UieLJi)

=>a£ LHS

where the first implication is obvious and the second follows from the second implication of (49.1). Concerning the last line of (49.3), by the first implication of (49.1) we get the inclusions UJ6LV(7,)

C V(nieL Ji) C V(Y[

Jt).

Moreover, for any a £ S we have that < =>• for each / £ L there exists // e J; with fi £ a ^ f £ a with / = n , 6 L fi G ILeL Ji and hence VillJOcUizLViJt).

264

L5: PROJECTIVE

VARIETIES §5: MORE ABOUT IDEALS AND MODULES

Concerning equation (49.4), by (49.2) we get LHS C RHS; conversely / G RHS =*• / G a for every a G UteLUi =>• / G LHS. The proofs of (T49.5) and (T49.6) are verbatim the same as the proofs of (5) and (6) given in the previous Quest (Q10), except that the references to (1), (2), (5), (6) should be changed to references to (49.1), (49.2), (49.5), (49.6) respectively, and the second braced display starting with "also" should be replaced by saying that, also: / G radfl J with J = I(U) for some U C S =$• fn £ J for some n G N+ < =» / " G a for all a G U => f G a for all a G U =>/€ J and hence J = rad# J. To prove (49.7) let U G V. Assuming U to be irreducible, we have U ^ 0 and hence /({/) 7^ R; moreover, for 1 < i < 2, given any fo e R\ I(U), upon letting [/< = {a G C/;/i G a } , we have Ui G V with Ui C U and C/j ^ [/; since 17 is irreducible, we must have U1UU2 ^ U, and hence /1/2 ^ -^(C^); thus /([/) is prime. Conversely assuming I(U) to be prime, we have I{U) ^ R and hence [/ ^ 0; moreover, given any Ui,U2 in V' with [/1 U t/2 = £/, by (49.4) we have I(Ui) n /(f/ 2 ) = /(I/) and clearly I(Ux)I(U2) C /(C/i) n /(C/2) and hence I(U\)I{U2) C / ( t / ) ; now assuming f/i 7^ C/, by (49.2) and the implication portion of (49.6) we get I(U) C I{U{) with I(U) ^ /(f/i), and hence the primeness of I(U) tells us that I{Ui) C /({/) and therefore by the first implication of (49.1) we get [/2 = U\ thus U is irreducible. The proofs of (T49.8) and (T49.9) are verbatim the same as the proofs of (8) and (9) given in the previous Quest (Q10), except that the references to (2), (6), (8), (9) should be changed to references to (49.2), (49.6), (49.8), (49.9) respectively. To prove (T49.10), by Zorn's Lemma, given any U G V" we have U C U* for some maximal U* G V", and clearly any such U* is an irreducible component of S. Given any P G U G V, replacing (U, S) by (V(P), U) in the previous sentence, we see that P G V(P) C U* for some irreducible component U* of U, and hence every U in V is the union of its irreducible components, [cf. §6(E13)]. The proof of part (i) of (T49.ll) is straightforward. To prove part (ii), let U G V" be such that upon letting P = I(U) we have P G S; then by (T49.6) we get U = V(P); conversely, if T G S is such that U — V{T) then by the last equation in part (i) we get I{U) = T and hence T = P. To prove (T49.12), given any U G V", upon letting P = I(U), by (T49.7) we see that P is a prime ideal in R. It follows that if S is speci? then P G S. On the other hand, if S is jspec(i?) then every Q G S is an intersection of members of

(Qll) NILRADICAL, JACOBSON SPECTRUM, AND JACOBSON RING

265

mspec(P), and hence by the equation P = I(U) = DQ£UQ with U C S we see that P is an intersection of members of mspec(-R); thus again P € S. The rest follows from part (ii) of (T49.ll). SPECTRAL NULLSTELLENSATZ (T50). Given any ring R, let '(S.V.V'.VVJstandfor (mspec(-R), mvspec^, msvt(P), imsvt(P), ispec fl ) or (jspec(P),jvspec fi ,jsvt(P),ijsvt(P), ispec fl ) or (spec(P), vspec^,svt(P),isvt(fl), ispec^). Then we have the following. (T50.1) If R is Jacobson or we are in the case of S = spec(.R), then for any ideal J in R we have I{V(J)) — rad#J. (T50.2) If R is Jacobson or we are in the case of S = spec(P), then the mapping U (-»I(U) gives inclusion reversing bijections V —> rd(i?) and V" —> spec(P) whose inverses are given by J H-> V(J). (T50.3) If R is an afflne ring over a field, then Pi is a Jacobson ring, i.e., every prime ideal in it is an intersection of maximal ideals. PROOF. The first part (T50.1) follows from (T48) and, in view of (T49.1), (T49.2), (T49.5), (T49.6) and (T49.7), the second part (T50.2) follows from the first part (T50.1). [cf. §6(E14)]. To prove the third part (T50.3), given any afflne ring R over a field k and any prime ideal H in R, we want to show that H = C\p£mvspecRHP' We can take a fc-epimorphism : B —> R where B = k[Xi,... ,XN] is a polynomial ring over k in a finite number of variables, and we can take an algebraic closure K of k. Let G = 4>~1{H) and J = DpemvspeCBaP. Then G C J with G £ spec(B) and J G rd(P). What we want to show is that G = J. Suppose if possible that G ^ J. Then by the sixth part (T22.6) of the Hilbert Nullstellensatz L4§8(T22), which we proved at the end of the last Quest, there exist a = (a\,..., Q;v) € nN such that a € VK(G)\VK(J), i.e., upon letting Q be the maximal ideal in C = n[X\,..., Xpf] generated by X\ — a\,..., X^ — &N we have G C Q with J <£ Q. Now C is integral over B and hence upon letting P = Q n B, by (T38.2) we see that P is a maximal ideal in B. Clearly G C P but J (Z! P which is a contradiction. MINIMAL PRIMES THEOREM (T51). Given any ring R, every prime ideal in R contains some minimal prime ideal in R, and we have rad^O = np e n s p e c (ft)P. More generally, given any ideal J in any ring R, every member of vspec^J contains some member of nvspec^J, and we have rad^J — C\p&mspecRjPPROOF. The second assertion follows by applying the first assertion to

R/J.

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L5: PROJECTIVE

VARIETIES

§5: MORE ABOUT

IDEALS

AND

MODULES

To prove the first assertion, by (T49.10) we see that for any H £ spec(P) there exists an irreducible component UH of spec(P) with H G UH and, upon letting PH = ispec^C/ff, by (T49.12) we see that PH G nspec(P)

with

PH C H.

It follows that nrad(#) = n p e n s p e c ( i l ) P and hence by (T48) we get radfiO = n PenS pec(fl)P.

FIRST SUPPLEMENTARY DIMENSION THEOREM (T52). Given any affine ring A over a field k, let (Ji)i
= dim(A).

Moreover, for any prime ideal P in A, the length of every saturated prime ideal chain P 0 C • • • C Ph in A with ht^Po = 0 and Ph = P equals h t ^ P , and the length of every saturated prime ideal chain Po C • • • C Pd in A with Po = P and dpt^Pd = 0 equals d p t ^ P . PROOF. Let dim(A) = n. For 1 < i < m, upon letting ^ : A —> A/Jt be the residue class epimorphism, by (T47.1) we get dim(i(J4)) = n. Let P be any prime ideal in A. We can take saturated prime ideal chains Po C • • • C Ph and Po C • • • C Pd in A with ht^Po = 0, Ph = P, htAP = h, P 0 = P , dpt^P^ = 0, and d p t ^ P = d; clearly we must have Po = Ji for some i and applying (T47.3) to <j>i(A) we get h + d — n, i.e.: (•) h t ^ P + d p t ^ P = n. In particular, given any nonunit ideal H in A we have h t ^ P + d p t ^ P = n for every P in vspecAH and hence htAH + dptAH = dim(A). Given any saturated prime ideal chain Po C • • • C Ph in A with ht^Po = 0 and Ph = P, we can take a saturated prime ideal chain Po C • • • C Pd in A with Po = P , dpt^Pd = 0, and d p t ^ P = d; clearly we must have Po = Ji for some i and applying (T47.3) and (T47.5) to (f>i(A) we get h + d = n with h t ^ ^ j ^ P ) = h and d p t ^ ^ c ^ P ) = d; obviously d p t ^ . ^ ^ P ) = d p t ^ P , and hence by (•) we get htAP = h. Given any saturated prime ideal chain Po C • • • C Pd in A with Po = P and dpt/iPd = 0, we can take a saturated prime ideal chain Po C • • • C Ph in A with ht^Po = 0, Ph = P, and h t ^ P = h; clearly we must have P 0 = Ji for some i and applying (T47.3) and (T47.5) to
(Qll) NILRADICAL, JACOBSON SPECTRUM, AND JACOBSON RING

and dpt^^^)i(P) = d; obviously dpt^(A)i(P) d p t A P = d.

267

= d p t A P , and hence by (•) we get

SECOND SUPPLEMENTARY DIMENSION THEOREM (T53). Let A be any affine ring over a field k. Then dim(A) is the maximum number of elements of A which are algebraically independent over k. Moreover if A* is any subring of A such that A* is also an afRne ring over k, then dim(A*) < dim(/l). PROOF. The second assertion follows from the first. To prove the first assertion, let dim(P) = n, let (Ji)i<j< m be all the distinct minimal primes of 0 in A, and let 4>i : A —> A/Ji be the residue class epimorphism. By (T47.1) we know that t r d e g ^ ^ ^ A ) < n for 1 < i < m with equality for some j = i. We can take elements Z\,..., Zn in A such that the elements j(Zi),..., (pj(Zn) are algebraically independent over i(Yr) in i(k), and hence r : B —* R where B = k[X\,... ,X^} is a finite variable polynomial ring over k, and ker(<^>) is the ideal Ik{U) of some U G avtfc(A^) for some overfield K of k containing an algebraic closure of k; we may then regard R as the coordinate ring of U.

268

L5: PROJECTIVE VARIETIES §5: MORE ABOUT IDEALS AND MODULES

QUEST (Q12) Catenarian Rings and Dimension Formula A ring A is said to be catenarian if every pair of prime ideals P c Q in it satisfies condition (*) of (T47.2) which says that: there is no infinite chain of prime ideals between P and Q, and any two saturated chains of prime ideals between P and Q have the same length. Thus the first part of (T47.2) says that an afHne ring over a field is catenarian. A ring A is said to be universally catenarian if it is noetherian and the polynomial ring over it in any finite number of variables is catenarian. Given an overdomain A' of a domain A, we say that the dimension formula (resp: the dimension inequality) holds for A relative to A' if for every prime ideal P' in A', considering the localizations R' = A'P, and R = Ap with P = P' n A, we have dim(i?') + restrdeg f i # = dim(i?) + trdeg^i?' (resp: dim(fZ') + restrdeg^i?' < dim(i?) + trdeg R i?'). We say that the dimension formula (resp: the dimension inequality) holds in a domain A if it holds for A relative to every afRne domain over A. Thus (T47.6) says that the dimension formula holds in any affine domain over a field. As we shall now see, the above concepts enable us to view parts of the Dimension Lemma (T30) and the Dimension Theorem (T47) from a somewhat more general perspective. Let us start off by giving two sharper versions of (C13). In Lemma (T54) we consider the behavior of prime ideals in a simple ring extension, i.e., in a ring extension A[x] obtained by adjoining a single element a; to a ring A, and in Lemma (T55) we shall generalize it to a ring extension obtained by adjoining a finite number of elements. SIMPLE RING EXTENSION LEMMA (T54). Let A and A' be noetherian domains such that A' = A[x) for some x £ A'. Let P' be any prime ideal in A' lying above any prime ideal P in A, and consider the localizations R' = A'P, and R = AP. Then restrdeg R .R' < 1

and

trdeg^i?' < 1

and dim(R') + TestrdegRR' < dim(R) +

trdegRR'.

Moreover, if trdeg^-R' = 1 then dim(ii') + restrdeg fl i?' = dim(it) + trdeg H .R'.

(Q12) CATENARIAN

RINGS AND DIMENSION FORMULA

269

PROOF. The first display is obvious, and the third follows from (C13). So assume x is algebraic over A. Let B = A[X] be the univariate polynomial ring over A, let (/>: B —> A' be the unique A-epimorphism which sends X to x, let H = ker(<^>), and let Q = <j>~l(P'). Now H is a nonzero prime ideal in B lying above 0 in A, and hence by (C13) we have hte-ff = 1; also clearly hts-H" + htA>P' < h t s Q ; consequently 1 + dim(-R') <

dim(BQ).

Obviously restrdeg^Bg = restrdeg^-R' and Q is a prime ideal in B lying above P in .A, and hence by (C13) we get dim(B Q ) + restrdeg^i?' = 1 + dim(il). The above two displays imply dim(-R') + restrdeg fi i?' < dim(i?), and this completes the proof. MULTIPLE RING EXTENSION LEMMA (T55). Let A and A' be noetherian domains such that A1 = A[x\,..., xm] for some m G N and xi,...,xm in A'. Let P' be any prime ideal in A' lying above any prime ideal P in A, and consider the localizations R' = A'P, and R = Ap. Then restrdegfi.R' < m

and

trdeg fi i?' < m

and dim(i?') + restrdeg R i?' < dim(i?) + trdeg fi i?'. Moreover, if either trdeg fl i?' = m or the polynomial ring B = A[X\,..., variables is catenarian, then

Xm] in m

dim(i?') + restrdeg fi i?' = dim(R) + trdeg^i?'. PROOF. The first display is obvious. The second display and the first case of the third display are trivial for m = 0 and for m > 0 their implication m — 1 => m follows from (T54), and so they are done by induction on m. So now assume that B is catenarian. Let : B —> A1 be the vinique yl-epimorphism which sends Xi to xt for 1 < i < m. Let H = ker(^), and let Q = 4>^1(P'). Now H and Q are prime ideals in B lying above 0 and P in A respectively, and hence by the first case of the third display, taking (A',P',A,P) = (B,H,A,0) and (A',P',A,P) = (B,Q,A,P), we respectively get htsH + trdeg^j/l' = m and hteQ + restrdeg^i?' = m + dim(i?).

270

LS: PROJECTIVE

VARIETIES §5: MORE ABOUT IDEALS AND MODULES

Since B is catenarian, by Lemma (T56.1) below we have hts<5 - n t ^ i ? = dim(it'), and hence by the above two displays we get dim(P') + restrdegfi.R' = dim(i?) + trdeg f i P' which completes the proof. CATENARIAN CONDITION LEMMA (T56). For any ring A we have the following. (T56.1) If A is a noetherian domain then: A is catenarian o <

for every pair of prime ideals P c Q in A we have htAQ = htAP + ht ( j 4 / p) (Q/P)

<^ for every pair of prime ideals P C Q in A with

^{A/P){Q/P)

= 1

we have ht^Q = 1 + h t ^ P . (T56.2) If A is catenarian then every homomorphic image of A is catenarian and so is the localization of A with respect to any multiplicative set in A. (T56.3) If A is universally catenarian then every homomorphic image of A is universally catenarian and so is the localization of A with respect to any multiplicative set in A. (T56.4) If A is universally catenarian then every affine ring over A is universally catenarian. PROOF. In (T56.1), to see that the first condition implies the second, we can take saturated prime ideal chains Po C • • • C Ph and Po C • • • C Pd in A with Po = 0, Ph = P, htAP = h, P 0 = P , Pd = Q, and ht{A/P)(Q/P) = d; clearly Po C • • • C Ph C Pi C • • • C Pd is a saturated chain of length h + d in A between 0 and Q and hence by the catenarianness of A we get htAQ = h + d. The second condition obviously implies the third. To see that the third condition implies the first, let P C Q be any prime ideals in A together with a saturated prime ideal chain Po C • • • C P„ in A between P and Q; then by the third condition we have ht^P, = 14- ht^Pj_i for 1 < i < n and hence ht^Q — htAP = n. The ideal correspondence between a ring and its homomorphic image given in the Third and the Fourth Isomorphism Theorems of L4§5(011), and the ideal correspondence between a ring and its localization given in L4§7(T12), yield (T56.2). A ring epimorphism <j>: A —» C uniquely induces an epimorphism of polynomial rings V •' A\Xi,..., Xm] —> C[X\,..., Xm] with ip(Xi) = Xt for 1 < i < m and ip(z) = {z) for all z £ A, and hence the first part of (T56.2) implies the first part of (T56.3). Similarly, for any multiplicative set S in any ring A, the canonical homomorphisms ( J 4 [ X I , . . . , xm])s uniquely induce an isomorphism ip : {A[Xi,..., Xm])s —> As[Xi,..., Xm] with tp(6(Xi)) =

(Q12) CATENARIAN

RINGS AND DIMENSION

FORMULA

271

Xi for 1 < i < m and ijj(6(z)) = : B —> C" = B/P be the residue class epimorphism, and let Q' = 4>{Q). Now Q' is a prime ideal in the noetherian domain C" and it suffices to show that the length of every saturated prime ideal chain in C between 0 and Q' equals htcQ''• For showing this it is enough to prove that C" is catenarian, and hence in view of (T56.1) it is enough to prove that, for any pair of prime ideals I' C J' in C", upon letting R' = C'j, and S' = C'j,, and upon letting V = D'L, where ft : C" -> D' = C'/V is the residue class epimorphism and V = '( we have dim(S") = dim(iJ') + dim(T'). Let H — P n A. Then H is a prime ideal in A, the domain C is an affine domain over the subdomain C = '(C) and T = DL where L = L ' f l C . Then the domain D is a homomorphic image of A, and the domain D' is an affine domain over D, and hence again

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by the dimension formula assumption we have dim(T') + restrdeg T T' = dim(T) 4- trdeg T T'. Now C is catenarian by (T56.2), and hence by (T56.1) we get dim(S) = dim(i?) + dim(T). Also clearly restrdeg T T' = restrdeg s S" and trdeg T T" = restrdeg fi i?'. By the above six displays we get dim(S") = dim(fi') + dim(T'), and this completes the proof. QUEST (Q13) Associated Graded Rings and Leading Ideals Let / be an ideal in a ring R. Recall that for any h e R we have put ord(yji/)/i = max{n e N : h € / " } , and the ring R is /-Hausdorff means: o r d ^ j j / i = oo <=> h = 0; see L3§12(D5). Also recall that in case R is quasilocal with I = M(R), o r d ^ j ) and /-Hausdorff are abbreviated to ord^ and Hausdorff respectively; see the first display after (2*) in L3§11. We define the associated graded ring of (R, I) to be the naturally graded ring grad(.R, I) whose underlying additive abelian group is obtained by putting grad(.R, I) = (${In/In+1)

(external direct sum of Z-modules).

neN

This is made into a ring by requiring that for all fm e Im and / „ £ In with m, n in N we have grad^i7)(/m) • grad^7)(/„) = g r a d ^ ( / m / „ ) where the central dot indicates multiplication and where grad^i7):/"^grad(fl,/) is the composition of A„ : / " -> (In/In+1)

and

/i n : ( / " / / " + 1 ) -> grad(fl, / )

where An is the natural surjection and /in is the natural injection. We also put grad(i?,J) n = g r a d ^ i 7 ) ( n

(Q13) ASSOCIATED GRADED RINGS AND LEADING IDEALS

273

and we observe that then ker (gradfoj)) =

In+1

and im (grad" R 7 ) J = grad(i?,7) n and moreover grad(ii,I) = Y J grad(i?,/)„

(internal direct sum).

We may call grad/ l f l / ) and grad(i?,/)„ the n-th graded map and the n-th graded component of (R, I) respectively, and for any h £ In we may call grad^fl t-. (h) the n-th graded image of h relative to (R, I). Note that the ring grad(il, I) is nonnull iff / ^ R. For any h G i? we put w iu\ (grad(R,i)W iford(RJ)/i = n e N v lefo ( / v ) (/i) = I ' 10 it oid^nj-fh = oo and we call this the leading form of h relative to (R,I). For any ideal J in R we put I the homogeneous ideal in grad(-R, I) ledi( / i /)(J) = < [generated by (Mo{RJ)(h))h£j i.e., equivalently f the homogeneous ideal in grad(i?, I) ledi( R / )(J) = < [ generated by U „ € N S r a a w ) (J

and we call this the leading ideal of J relative to (R,I). generators x = (XI)I£L of I and letting yi =

n

7

")

Taking any family of

gi&d\RJ){xi)

we see that grad(it, I) is a semihomogeneous (homogeneous if L is finite) ring over the ring grad(i?, I)0 with grad(JJ, J) = grad( J R,/) 0 [grad(ii,/) 1 ] = gia,d(R, I)0[(yi)ieL] and moreover considering the polynomial ring k\(Xi)iei\ with k = R/I as a semihomogeneous (homogeneous if L is finite) ring over the ring k (see (C6) of §3), we get a unique graded ring epimorphism ©?«,/)

:

M(*I)«6L]

- grad(ii, / )

with

k = R/I

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such that QfRj){z)

= Ho{z) for all z e k

and

9fRI)(Xi)

= yi for all I e L.

The homomorphism 0? f l 7N is said to be induced by (R, I) and x. Given any ring homomorphism : R —* R' and any ideal V in R! with (f)(1) C / ' , we get a unique graded ring homomorphism grad(R,/)(

giad(R',I')

such that, for every n £ N, the homomorphism gradfojjfa) : grad(fl,/)„ -> grad(i?',/')n induced by grad(flj)() are said to be induced by 4>. Note that upon letting J = kei((f>) and

H = ledi( fl) /)(J)

for the homogeneous ideal l e d i ^ / ^ J ) in grad(.R, /) we have that [cf. §6(E15)] (1)

\ed\{RJ)(J)

c ker(grad(Hi/)(4>))

and if is surjective with (f)(1) = / ' then grad(fli/)(
(2)

with ledi{RJ)(J)

= ker(grad (i j 7) (^))

and if (p is surjective with (f)(1) = I' and J is a principal ideal in R generated by an element h whose leading form le£o(R j)(h) is a nonzerodivisor in grad(.R, / ) (3)

then grad(/j)/)() is surjective and ker(grad( f l n ((/>)) is the principal ideal in gra.d(R,I) generated by

leio(Rj)(h).

In case R is a quasilocal ring with / = M(R), in the above three paragraphs we write R in place of (R,I), except that we write grad(i?) and grad(i?)„ in place of grad(ii,/) and grad(i?,/)„ respectively. A valuation function on a ring R is a map v : R —> Z U {oo} such that for all x, y, z in R we have: (i) v(xy) =v(x) +v(y),

(Q13) ASSOCIATED GRADED RINGS AND LEADING IDEALS

275

(ii) v(x + y) > mm(v(x),v(y)), and (iii) v(z) = oo & z = 0, with the usual understanding about oo. Thus in the notation of L3§11, a valuation function is like a valuation with R and Z playing the roles of the field K and the ordered abelian group G respectively. If v is a valuation function on a nonnull ring R then clearly R is a domain and we get a unique valuation v : QF(7?) —> Z U {00} such that for all x, y in Rx we have v(x/y) = v(x) — v(y); we say that the valuation v is induced by v; usually we denote v also by v. Considering the weaker property: (i*) v(xy) > v(x) + v(y), we see that ord(^j) satisfies (i*) and (ii); it satisfies (iii) iff R is 7-Hausdorff. We shall investigate this matter further in (T60) below and then again in the next Quest. Note that in case R is a local ring with 7 = M(R), (iii) is the Krull Intersection Theorem. LEMMA (T59). Let A = £ „ e N An be a naturally graded nonnull ring. Then A is a domain iff the product of any nonzero homogeneous elements in A is nonzero, i.e., iff: 0 ^ fm G Am and 0 ^ gn G An with m, n in N =$• fmgn ^ 0. PROOF. Given 0 ^ / = £ m e N / m e A and 0 ^ g = £ n e N S n G 4 with fm G j4 m and g„ G A„, let m and n be minimal with fm ^ 0 ^ gn- Then and / # = /mffn -I- Em+rKqeN ^9 w i t n ^ 9 e A> a n d hence

THEOREM (T60). Let 7 be an ideal in a ring ii. Then we have the following. (T60.1) 7 ^ R and o r d ^ / ) is a valuation function on R iff R is 7-Hausdorff and grad(7?, 7) is a domain. (T60.2) Upon letting R* = R/J* and 7* = 7/J* with J* = n n € N 7 n , we have that R* is 7*-Hausdorff, and if grad(7i, 7) is a domain then R* is a domain and ord( f l . ] / .) is a valuation function on R*. (T60.3) Let : R —> R' be a ring epimorphism, with J = ker(<j>), such that the leading ideal grad( f l / )(J) is a (homogeneous) prime ideal in grad(7Z,7). Then n„GN(7 + 7") is a prime ideal in R. PROOF. (T60.1) follows from (T59). To prove (T60.2) first note that 7?* is obviously 7*-Hausdorff, and then let * : R —* R* be the canonical epimorphism and assume that grad(i?, 7) is a domain. Now clearly ker(0*) = J* and 7* = (j>*{I) with grad( Ri/ ) (*) (J*) = 0, and hence grad(7i*, 7*) is a domain, because grad(iu)(*)(ker(*)) is the kernel of the induced epimorphism of grad(7?, 7) onto grad(7?*,7*). Thus we are reduced to (T60.1). To prove (T60.3), upon letting 7' = 4>(I), we see that grad(7?',7') is a domain, because grad(#j)(J) is the kernel of the induced epimorphism of grad(i?, 7) onto grad(i?',7'). Therefore by (T60.2) we see that R'/ n n eN 7 / n is a domain, and therefore nn€N7'™ is a prime ideal in R'. Clearly Dn&n(J + In) = ^^(rWN-f'"),

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L5: PROJECTIVE VARIETIES §5: MORE ABOUT IDEALS AND MODULES

and hence C\ne^{J + In) is a prime ideal in R. THEOREM (T61). Let R be a local ring with dim(R) = d and emdim(i?) = e, let x = (xi,... ,xm) be a finite number of generators of M(R), let X = (X\,... ,Xm) be indeterminates over the field k = R/M(R), and recall that 9fj : k[X] —> grad(i?) is a graded ring epimorphism where k[X] = ^2n€nH^}n is the naturally graded polynomial ring with k[X]n = the set of all homogeneous polynomials of degree n (including the zero polynomial). Then we have the following. (T61.1) k e r ( 9 | ) n k{X}x = 0 &• m = e. (T61.2) k e r ( 9 | ) = 0 <S> m = d. (T61.3) If m = e then: R is regular «• k e r ( 9 | ) = 0. (T61.4) R is regular O there is a permutation a of { 1 , . . . ,m) together with elements a^ in k such that the ideal ker(9fj) in k[X] is generated by the m — e elements Xa^ - J2i<j<e aijXo{j) for e + 1 < i < m. PROOF. Let J = M(R), let An : In -> (r/In+1) be the natural surjection, and n n+1 let fin : (I /I ) —> grad(ii,/) be the natural injection. Also let yi = /ii(Ai(a:;)) for 1 < I < m. To prove (T61.1), note that any z G k[X]\ can be expressed uniquely as z = a\X\ -\ ha m X m with a-^,..., am in k, and for 1 < I < m we can write a/ = Ao(6/) with bi e R. Observe that z = 0 «=> all the a; are zero <=> all the bi belong to / . Now QR(Z) = H\(Xi(b\Xi + • • • + bmxm)). Since Hi is injective, we see that: GxR(z) = 0 o Ai(6in + • • • + bmxm) = 0. By the Nakayama Lemma (T3) we know that: m > e <^> Xi(b\Xi -\ 1-bmxm) = 0 for some elements b\,..., bm in R at least one of which does not belong to / . This completes the proof of (T61.1). Now clearly m = d o h t ^ / = m, and hence (T61.2) follows from the Relative Independence theorem (T34). By definition R is regular iff e = d, and hence (T61.2) implies (T61.3). So it only remains to prove (T61.4). First assume that R is regular. Then e = d, and by the Nakayama Lemma (T3) we can find a permutation a of { 1 , . . . , m) together with elements bij in R such that x a(i) = Xa<j< e bijx k[X'} be the unique fc-epimorphism of graded rings which sends Xa^ to Xa(i) or J2i<j<e aijXc,(j) according as 1 < i < e or e + 1 < i < m. Then clearly 0% = 9 | A, and hence k e r ( 9 | ) = ker(A) = H. Conversely assume that there exists a permutation a of { 1 , . . . , m} together with elements a»j in k such that the ideal ker(9^) in k[X] is generated by the m - e

(QU) COMPLETELY NORMAL DOMAINS

277

elements Xa{i) -

^2

cnjXa{j)

for

e + 1 < i < m.

l<j<e

It follows that x' = (a;CT(i),... ,xa(e)) are generators of / modulo I2 and hence by Nakayama they are generators of / . Now with k[X'],H, A as above we have ©ft — ©fiA w i t n ker(A) = H. Consequently ker(0^') = 0 and hence R is regular by (T61.3). COROLLARY (T62). If it is a ci-dimensional regular local ring then grad(-R) is isomorphic t o a d variable polynomial ring over the field R/M(R), and hence in particular grad(il) is a noetherian normal domain. PROOF. Follows from (T61.3). COMMENT (C19). [Graded Rings of Polynomial Rings]. Let X = {Xt)ieL be a family of variables over a field k, consider the polynomial ring R = k[X], and let / be the maximal ideal in R generated by X. Then we may "identify" k with R/I as well as with grad(i?, J) 0 . Now upon letting Y = (Yi)ieL

with

y , = g r a d ^ i 7 ) ( X , ) = lefo (fl>/) (X,)

we see that grad(i?, / ) = the polynomial ring k[Y] and

©ffl,/) = HX] ^ k[Y] is the /c-isomorphism which sends Xi to Y; for all / s L. QUEST (Q14) Completely Normal Domains Having proved L4§8.3(T24.1) as the Krull Intersection Theorem (T7), in (T64) we shall prove L4§8.3(T24.2) which says that the order function of a regular local ring is a valuation and hence the said ring is a normal domain. For dealing with the normality part of this we introduce the concept of complete normality and prove a lemma about it in (T63). A domain R is said to be completely normal if: x e QF(.R) with dxn £ R for some d e Rx and all n € N => x G R. COMPLETE NORMALITY LEMMA (T63). For any given ring R we have the following.

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(T63.1) If R is a completely normal domain then R is a normal domain. If R is a noetherian normal domain then R is a completely normal domain. (T63.2) If I is an ideal in R such that for all z G R we have n „ e N ( z i ? + / n ) = zR, and grad(i?, I) is a completely normal domain, then R is a completely normal domain. (T63.3) For any ideal I in R we have that: I ^ R iff grad(i?, / ) is nonnull. (T63.4) If R is noetherian and I is any ideal in R then grad(.R, I) is noetherian. (T63.5) If R is noetherian and has an ideal I with 7 c jrad(i?) such that grad(.R, I) is a normal domain, then R is a completely normal domain. PROOF. To prove (T63.1), first assume that R is a completely normal domain, let x € QF(i?) be integral over R, and let

xm+ Yl aixm-i = 0 l
with m G N + and en G R be an equation of integral dependence for x. Write x = y/z

with

y GR

and

z £ Rx.

Let d=2m. Then clearly (ieflx

with

dxn € R

for

0 < n < TO.

By induction on n we shall show that for all m < n G N we have dxn G R. For n = mwe have already said this. So let n > m and assume for all smaller values of n. Multiplying the equation of integral dependence by dxn~m we get the equation

dxn = -

J2 o-idx71'1 \
whose RHS belongs to R by the induction hypothesis and hence so does the LHS. Therefore by complete normality we get x & R. Thus R is normal. Now assume that R is a normal noetherian domain, and let x G QF(i?) be such that dxn G R

for some

d £ Rx and all

n G N.

Then R[x] is contained in the finitely (by a single element) generated ^-module Rd_1, and hence (because of the noetherianness of R) by L4§5(025)(27*) we conclude that R[x] is a finitely generated i?-module. Consequently by L4§10(E2.2) we

(Q14) COMPLETELY

NORMAL

DOMAINS

279

see that x is integral over R, and hence (by the normality of R) we get x € R. Thus R is a completely normal domain. To prove (T63.2), let I be an ideal in R such that for every element z in R we have DneN(zR + In) — zR, and grad(f?, I) is a completely normal domain. By taking z = 0 we see that f]ne^In = 0, and hence by (T60.2) we see that R is a domain and o r d ^ j ) is a valuation function on R. To show that R is completely normal, let there be given any x G QF(R) such that dxn G R for some d G R* and all n G N. Write x = y/z

with

yeR

and

z e

Rx.

We want to show that x £ R, i.e., ?/ G zi?. By hypothesis nmen{zR

+ Im) = zR

and hence we are reduced to showing that y£zR

+ Im

for all

m G N.

We do this by induction. Since it is trivial for m = 0, let m > 0 and assume for m - 1 . Then y = az + 6

with

a G i?

and

6e/m_1.

Since cten G .R for all n G N, we get d(x - a)n G R for all n G N. Consequently, since x = y/z = a + (b/z), we get dbn G znR

for all

n G N.

Thus for all n G N we can write dbn = cn2™

with

c„ G i?.

Since o r d ^ j ) is a valuation function on R, taking leading forms relative to (R, I) preserves products, and hence for all n G N we have lefo(fl,/)(d) • (lefo (Ri/) (b))" = lefo (fli/) (c„) • (Mo{RJ){z))n Since grad(i?, I) is completely normal, this implies that lefo(fl i /)(6)/lefo (fli/) (z) G gr&d(R,I) and hence lefb(fl,/)(6) =lefo(fl,/)(c) • lefo (fli /)(z)

with

c G .R

.

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and therefore (because b e 7 m _ 1 ) by the definition of multiplication in grad(i?, J) we get

b-cz£lm. Thus y-{a

+ c)ze

Im

and hence y£zR

+ Im.

Part (T63.3) is obvious. Part (T63.4) follows from L4§5(027)(36*). In view of (T20) and (T60), by (T63.1) to (T63.4) we get (T63.5). ORD VALUATION LEMMA (T64). Let R be a regular local ring. Then for all nonzero elements x and y in R we have ovdR(xy) — ordRX + ordRy. Consequently R is a normal domain and we get a valuation oidR : QF(R) —> Z of QF(R) by putting ordR(x/y) = ordfl:c - ordRy. [We shall continue to use this extended meaning of ord#]. PROOF. Follows from (T62) and (T63). QUEST (Q15) Regular Sequences and Cohen-Macaulay Rings Let R be a ring. Let x = (a;i,..., xn) be a sequence of elements in R of length n S N, and for 0 < m < n let Im be the ideal in R generated by ( x i , . . . , xm). Let V be a module over R. Recall that SR(V) = R \ ZR(V) where ZR(V) is the set of all zerodivisors of V, i.e., elements r £ R with rv = 0 for some 0 ^ v G V. We say that £ is a F-regular sequence, if InV ^ V and for 1 < m < n we have xm £ SR(V/(Im-iV)). Moreover if J is an ideal in R with In c J then we say that £ is a ^-regular sequence in J. Furthermore if for every z € J ("1 ^ ( ^ / ( J n V ) ) we have J n y + zV = V then we say that £ is a maximal ^-regular sequence in J. By taking V — R we get the notions of i?-regular and maximal i?-regular sequences; note that now the condition xm £ SR(R/(Im-iR)) is equivalent to the condition m(xm) G S^m{R){(j)m{R)) where m : R —> R/Im-\ is the residue class epimorphism. We define the (R, J)-regularity of V by putting veg/R n V — the maximum length of a ^-regular sequence in J and we note that: this is in N U {—oo,oo}; it is —oo means V = 0; it is 0 means V ^ O and for every z G J n SR(V) we have zV = V; and it is oo means V =£ 0 and there is no bound on the said length. If R is local with J = M(R) then we call this

(Q15) REGULAR SEQUENCES AND COHEN-MACAULAY RINGS

281

the ^-regularity of V and denote it by reg#V, i.e., reg fl V = the maximum length of a V-regular sequence in M(R) and in this case we write reg(i?) in place of regfl.R, i.e., reg(i?) = the maximum length of an ii-regular sequence in M(R) and we call it the regularity of R; by (T27) this is a nonnegative integer < dim(i?). By parts (T66.4) and (T66.5) of Lemma (T66) below it follows that if R is noetherian, V finitely generated, and J is an ideal in R with JV ^ V, then there exist maximal ^-regular sequences in J and they all have the same length which equals reg(#,j)V. In particular if R is local then there exist maximal R-regular sequences in M(R) and the integer reg(-R) equals the length of every such sequence. The Generalized Principal Ideal Theorem (T27) inspires the following definitions of generating numbers, complete intersections, and CM (= Cohen-Macaulay) modules and rings. We define the generating number of V in R by putting gnb fl V = the smallest number of generators of V and we note that this is oo iff V is not finitely generated, and it is 0 iff V = 0. In particular this defines the generating number gnb#J of any ideal / in R by putting gnb fi 7 = the smallest number of generators of / . An ideal / in R is said to be an ideal-theoretic complete intersection in R if / ^ R and h t # / = gnbRI. A finitely generated -R-module is free means it is isomorphic to Rd for some d £ N , i.e., equivalently, it has a basis yi,. • • ,yd- In case R is local and V is finitely generated, V is said to be a CM module if reg^V = dim(i?/(0 : V)R). A local ring R is said to be a CM ring if it is a CM module over itself, i.e., if reg(i?) = dim(i?). Note that by (T27), for any local ring R we have the inequality reg(iZ) < dim(i?), and so the CM property says that equality holds; this may be compared with the fact that for any local ring R we have emdim(i?) > dim(i?) and R is regular means equality holds. A noetherian ring R is said to be a CM ring if, for every maximal ideal P in R, the localization Rp is a CM ring. An ideal / in a noetherian ring R is said to be unmixed if for every prime P in &SSR(R/I) we have h t ^ P = ht/j/, i.e., equivalently, if / has no embedded primes and all its minimal primes have the same height. The unmixedness theorem is said to hold in a noetherian ring R if every ideal in R, which can be generated by as many elements as its height, is unmixed; note that by (T27) this is equivalent to saying that if I is any ideal in R, which can be generated by e elements where the nonnegative integer e equals htjil, then / has no embedded primes. Amongst the various results about the above concepts which we shall now prove, in (T71) we shall prove the third part L4§8(T24.3) of the Ord Valuation Theorem.

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AND

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Before that, in (T67) and (T68) we shall relate regularity with the independence of (Q8) and the graded rings of (Q13). We start off with Lemmas (T65) and (T66). Out of this, the very versatile Lemma (T65) is of general interest. LOCALIZATION LEMMA (T65). For any ring R we have the following. (T65.1) If R is a domain then for any ideal I in R we have

n

(IRP)=i

PGmspec(H)

and in particular we have

f)

RP = R.

Pemspec(fl)

In both the above equations the intersection is taken in QF(i?). (T65.2) Without assuming R to be a domain, for any ideal / in R we have {r GR: cj>p{r) G (j>P{I)RP for all P G mspec(JR)} = / where Rp is the canonical homomorphism, i.e., equivalently we have

n

v-p]*=i

PGmspec(-R)

where the intersection is taken in R. PROOF. In the second equation of (T65.1), clearly RHS C LHS; to prove the other inclusion, given any x G LHS, upon letting

J = {z e Rx : xz e R] u {0} we see that J is an ideal in R with J $£ P for all P G mspec(R), and hence J = R by L2§5(R5), and therefore 1 € J, i.e., x G R. This proves the second equation of (T65.1), and hence the first equation of (T65.1) is subsumed under the first equation of (T65.2). In view of L4§7.1(T17)(b), the two equations of (T65.2) are equivalent to each other. In the second equation of (T65.2), clearly RHS C LHS; to prove the other inclusion, given any x G LHS, upon letting J = {z € Rx : xz G / } U {0} we see that J is an ideal in R with J <jL P for all P G mspec(i?), and hence J = R by L2§5(R5), and therefore 1 G J, i.e., x G I. REGULAR SEQUENCE LEMMA (T66). Let V be a module over a ring R, let x = (x\,... ,xn) be a ^-regular sequence in R of length n G N, let Im be the ideal generated by x\,... ,xm for 0 < m < n, and let I = In. Then we have the following.

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(T66.1) If j / i , . . . , yn are any elements in V with x\y\ + • • • + xnyn = 0 then yi e IV for 1 < i < n. (T66.2) If du...,d n are any elements in N+ then the sequence x-^1,..., x^n is V-regular. (T66.3) If n > 0 and xn € Sfi(Vy(J m V)) for 0 < m < n then the sequence (xn,xi,...,xn-i) is V-regular. (T66.4) If R is noetherian and J is an ideal in R with J V ^ V such that I C J, then there exists an integer n' > n together with elements xn+\, • • • ,xn' in J such that (x\,..., xni) is a maximal V-regular sequence in J. (T66.5) If R is noetherian, V is finitely generated, J is an ideal in R with JV =£ V such that I C J, and there exists a maximal F-regular sequence in J of length n, then £ is a maximal V-regular sequence in J. [If R is noetherian and J is an ideal in R with J V ^ V, then by taking n = 0 in (T66.4) we see the existence of maximal V-regular sequences in J, and moreover if V is finitely generated then by (T66.5) it follows that all maximal V-regular sequences in J have the same length and this common integer equals r e g ^ j j V ] . (T66.6) If R is noetherian, V is finitely generated, and J is an ideal in R with J V ^ V such that I C J, then reg (fl ,j) V = n + reg ( i J i J ) (V/(/V)). (T66.7) If i? is noetherian, V is finitely generated, and J is an ideal in R with J V 7^ V such that I C J, then upon letting n' = reg/ fl nV and v = gnb f l J we have v > n' >n and J has generators £ i , . . . , £„ such that (£i > • • • i fn') is a maximal V-regular sequence in J. (T66.8) Without assuming the sequence x to be V-regular, but assuming R to be noetherian and V to be finitely generated, if positive integer m < n is such that J m _i C ZR(V) and J m $! ZR(V), then for some elements r\,..., r m _ i in .R we have r i z i + • • • + r m _i:r m _i + xm e SR{V). (T66.9) Without assigning a V-regular sequence x in R, if R is noetherian, J is an ideal in R generated by v generators with i / £ N , and V is finitely generated with J V ^ V and reg( fi j)V = /x, then i/ > \x and J has generators f i , . . . ,&, such that ( £ i , . . . ,£M) is a V-regular sequence. PROOF. To prove (T66.1) we make induction on n. For n = 0 there is nothing to show. So let n > 0 and assume for n— 1. Let yi,..., yn in V with £i?/iH \-xnyn = 0. Since z n is a nonzerodivisor for V/(a;i,..., z„_i)V, we get yn = £ i < i < „ _ i xtZi with Zi e V. This gives X)i 0, and we may also assume d\ = d € N + with di = • • • = dn = 1 because then the sequence (x2,. -. ,xn) will be (V/(:rfV))-regular and we can repeat the argument. Now we

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make induction on d. For d = 1 we have nothing to show. So let d > 1 and assume that the sequence (x(~1,x2,... ,xn) is V-regular. Clearly xf is V-regular. So let 1 < i < n and assume that (xf,X2, • • •, aJi-i) is V-regular. Let there be given any elements s,ti,...,U-i in V such that XiS = Xit\+x2t2-\ \-Xi-\U-\. Then by the induction hypothesis we have s = xf _ 1 ui 4- x2u2 H h Xi-iUi-i with u\,..., u*_i in V. Therefore X*~l{x\t\

-XiUi)

+ X2{t2 -XiU2)

-\

\-Xi-i(ti-i

-XiUi-i)

= 0

and hence by (T66.1) we get Xiti

-XiUi

G

(xf~1,X2,...,Xi-i)V.

It follows that XiU\ G (a;i,..., Xi-\)V and hence u\ G (xi,..., Xi-i)V and therefore s G (xf, x2,..., a:j_i)V. This completes the induction on i and proves (T66.2). To prove (T66.3) suppose n > 0 and xn G 5 f l ( y / ( / m y ) ) for 0 < m < n. We make induction on n. The case of n = 1 being obvious, let n > 1 and assume for n — 1. Applying the induction hypothesis to the V/(/i ^-regular sequence (x2,..., xn) we see that the sequence (x\,xn, x2,..., xn-i) is ^-regular. Suppose if possible that x\ € Z]i(V/(xnV)). However 'x! G

ZR(V/(xnV))

=>• x\y = xnz for some y,z in V with y ^ xnV => z = x\w for some w &V (because xn G Si{(Vy(:riV))) => xi(y - xnw) = 0 =>• y — xnw = 0 (because xi G S/{(V)) => contradiction to y g xnV. Therefore x\ G SR.(V/(xnV)), and hence (xn,xi,... ,xn-i) is V-regular. To prove (T66.4) and (T66.5) assume that R is noetherian and let J be an ideal in R with JV 7^ V such that I c J. If £ n +i, xn+2,... are any elements in J such that the sequence (x\,... ,xn+i) is ^-regular for i = 1,2,... then, upon letting In+i = (^l, • • •, xn+i)R we have that In C In+i C / n +2 C . . . are ideals in R with In 7^ -^n+i 7^ ^n+2 7^ • • • and hence by the noetherianness of R this must stop at some integer n' > n. This proves (T66.4). To prove (T66.5) also assume that V is finitely generated and there exists a maximal V-regular sequence x' = (x[,... ,x'n) in J of length n. By induction on n we shall show that then £ is a maximal V-regular sequence in J. This being obvious for n = 0, for a moment suppose that n = 1. Then J C ZR(V/(X'1V)) and hence, by (Q1)(T1), J C UP where the union is taken over the finite nonempty subset tassRiV/ix^V)) of spec(P) and therefore, by L4§5(024)(22'), J c P for some P G tassR(V/(x'1V)). Consequently, there exists t G V \ (x[V) such that for every j G J we have jt = x[j* for some j * G V. In particular xi* = x[u for some « e 7 ; since t g x[V and zi G SR(V), we must have u ^ i i 7 . Multiplying the

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285

equation x\t = x[u by j we get x\ jt = x[ju, and multiplying the equation jt = x[j* by x\ we get xijt = x'xx\j*\ equating the two RHSs we obtain x[ju = x[xij* and hence x[(ju — x\j*) = 0; since x\ € SR(V), this yields ju = x\j* and therefore j £ ZR(V/(XIV)). Thus J C ZR(V/(XIV)), and hence x = (x\) is a maximal V-regular sequence in J. Now let n > 1 and assume for n — 1. For 0 < m < n, upon letting I^ = (x'1,..., x'JR, we have J (jL ZR(V/(ImV)) and J <jL ZR{V/(I'mV)). Hence by (Q1)(T1) and L4§5(024)(22') we can find < £ J such that for 0 < m < n we have < £ Z f l (V/(I m V)) and x*n # ZR(V/(I'mV)). It follows that (xu... ,xn^,x*n) are and (x^,... ,x'n_i,Xn) V-regular sequences in J. By (T66.3) we see that (x„,xi,... ,xn-i) and ( a ; * , ^ , . . . , a ^ - i ) a r e ^-regular sequences in J; the second is maximal by applying the n = 1 case to the (y/(/^_ 1 V))-regular sequences x'n and x*n. Consequently (x\,..., xn-i) and (x[,..., x'n_1) are {V/{x*ny))-regular sequences in J; the maximality of the second follows by the maximality of the Irregular sequence (a;* ,x[,... ,x'n_x) and hence the maximality of the first follows by the induction hypothesis. This in turn yields the maximality of the V-regular sequence {x\,..., xn-i, x*n). From this, again by the n = 1 case, as applied to the (y/(/ n _iy))-regular sequences xn and x*n, we get the maximality of the V-regular sequence x. This completes the proof of (T66.5). In view of the bracketed remark following (T66.5), the proofs of (T66.6) and the inequality n' > n of (T66.7) follow from (T66.4). The rest of (T66.7) follows from (T66.9) which itself will be deduced from (T66.8). To prove (T66.8), without assuming the sequence x to be V-regular, but assuming R to be noetherian, V to be finitely generated, and m to be a positive integer with m < n such that Im-\ C ZR{V) and Im <jt ZR(V), we want to show that for some elements r\,..., rm-\ in R we have r\X\ + • • • + rm-ixm-i + xm £ SR(V). Now for some s\,..., sm in R we have sixi + • • • + smxm € SR(V). By (Q1)(T1) we know that ZR(V) = Pi U • • • U Pc where P\,...,PC are prime ideals in R with c £ N+ such that for all i =^ j we have Pi <£_ Pj (namely, these are the maximal members of ass^(V)). Since Im-l C ZR(V) and sixi -\

1- smxm

£ SR(V)

we must have xm & Pi for some i. Therefore by suitably labelling Pi,..., Pc we can arrange that xm g" Pt for 1 < i < b and xm £ Pj for b + 1 < j < c where b is an integer with 1 < b < c. By L4§5(024)(22'), for 1 < i < b we can take t{ £ Pi such that U 0 Pj for b+ 1 < j < c. Let t = tx... tb. Then t £ Pt for 1 < i < b and t £ Pj for b + 1 < j < c. Now if 1 < i < b then t e Pi with xmg

Pi^> tsixi -\

1- tem_!:rm_i + xm g Pi

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whereas it b + 1 < i < c then 'sixi + ••• + smxm => s i i i H => tsia;i H _ => tsixi H

g ZR(V) = Pi U • • • U P c with xm G P,

h s m _ia; m _i 0 P* h is m _ia; m _i £ Pj h is m _i:r m _i + xm g Pi.

Thus isi^i H h t s m _ i x m _ i + x m ^ P* for 1 < i < c, and hence it suffices to take 77 = tsi for 1 < Z < m — 1. To prove (T66.9), without assigning a ^-regular sequence x in P, assume that R is noetherian, J is an ideal in R generated by v generators 771,..., j]v with v G N, and V is finitely generated with JV ^ V and reg( fl ^V = /x. We want to show that then v > fi and J has generators £ 1 , . . . , £„ such that ( £ 1 , . . . , £M) is a V-regular sequence. We make induction oni/. If v = 0 then /j = 0 and we have nothing to show. So let ^ > 0 and assume for v — 1. Let e be the largest nonnegative integer < 1/ such that (TJI, . . . ,r]e)R c ZR(V). If e = U then ju = 0 and again we have nothing to show. So assume that e < v. Then by (T66.8) we can find elements r\,..., re in R such that for £i = 7-1771 + • • • + rer]e + 7?e+i we have £1 e SR(V). Clearly J is generated by £1,772,.-., rje, r}e+2, • • •, f]v Let