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M, where each E,- = E. If J — {j\, • • •, jc] C { 1 , . . . , nd}, we denote by cpj the restriction of (p to Ey-, © • • • © E7i.. Let / be a minimal subset of { 1 , . . . , nd} with the property that 1. Moreover, in any such isomorphism, both the integer n, and the isomorphism type of D are uniquely determined by A Proof: By (1.6), (1.7), (1.8) it suffices to show {==>)• If .4 is a ring and a e A, let Xa : A -> A be the endomorphism of right yt-modules given by A' into a Z-algebra A', either rk z (A). In practice, maximal Z-orders often appear in a slightly disguised form: let E be a finite algebraic extension field of Q, and let R C E be the subring of algebraic integers in E. Let A be a simple E-algebra, and, let A c A be an order over R. We can regard A as a Z-order by restricting scalars from 7? to Z. If A is a central simple E-algebra, then A <8>E E = Mn(E) and, for any /?-basis [€\, ...,€ni) for A, the reduced discriminant disc(ei,..., e,,*) is defined and takes values in R, so that the reduced discriminant disc(A) is also well defined as an element of R/(R*)2. Then we have: Proposition 7.6: Let E be a finite algebraic extension field of Q, and let R C E be the subring of algebraic integers in E. Let A be a central simple E-algebra, A] where A, is a Z-order with rk z (Ai) = rk z (A) such that |A|/ • Amax into a maximal order A m ax> • Amax is an imbedding into a maximal order A max, with finite co-index j . It follows from Fadeev's Theorem that Amax is nondegenerate, so that Disc(Amax) ^ 0. However, Disc(A) = y'2Disc(Amax), so that A is nondegenerate. This proves (iii) = > (i). Finally, it is clear that (ii) =>• (iii). For if c(A) is bounded, let A2 is a finite index imbedding with co-index k > 1, then ^ o (p : A -» A2 is a finite index imbedding with co-index jk > j = max{c(A)}, contradicting the definition of max{c(A)}. This completes the proof. • The study of maximal orders in a general semisimple algebra reduces to that of maximal orders in the simple factors; let A be a finite-dimensional semisimple algebra over Q with Wedderburn decomposition A = . Lemma 19.5: If C is a class of A-modules satisfying property T(2), then J e C is injective relative to C if and only if it is strongly injective relative to C. Proof: (=$•) Suppose that J is injective relative to C. Let A' be the homomorphism E of Id,,, where [ ] : Hom A (D, B) -> W'(A, B) is the quotient map. • B factors as Q, A. : Q —> B, and Q is projective. Consider the extensions ^ ( P ) = (0 -> Q -> Xo ->• A -> 0) and ^ ( P ) = (0 -> B -» £ 0 -> A -^ 0) where Xo - iim(^r, j) = ( g © Po)/Im(Vf x - / ) and £ 0 = lim( • A' is a Ahomomorphism, then there exists a morphism (p : P —> E lifting D is a Ahomomorphism, then there exists a morphism (p : E —>• P extending ip; if ^ is any other extension of E lifting Id^ P Q and j : Q —> N', with Q projective. Without loss, we may suppose that Q is finitely generated, and so Q € C. Since n > 1, we have an exact sequence in C thus 0 -> DB 4 />„_, ^ • / maps HomzlH^(M, TZ(N)) into HomZ{G](£(M), N). It suffices to show that Hom°[W] (M, ft(AO) -> HomJ [C] (f (Af), W); / t-> / is surjective. Thus suppose the Z[G]-homomorphism g : £(M) ->• TV factorizes as g = \js o y where y' : ^(M) -»• Q, \jr : Q ->• TV and where g is projective over Z[G]. Let i : M -*• TZ{Q) be the composite of j with the canonical inclusion M -> £(M);x h^ A: ® 1. Put / = TZ(x//)oi; then /factorizes through the Z[//] Z[G] is a Z[C]-homomorphism, then cp = «E for some neZ. (33.2) If xj/ : Z[G] —>• Z is a Z[G]-homomorphism, then \jr = ne for some n eZ. As in Chapter 4, we write a % 0 when the Z[G]-homomorphism a factors through a projective. We compute the endomorphism ring Endx>er(Z) of the trivial module Z in the derived category. Observe that Endz[G](Z) = Z; by contrast we have: Proposition 33.3: Let Z denote the trivial module over Z[G] where G is a finite group; then there is an isomorphism of rings EndDer(Z) = Z/|G| Proof: We may identify a e Endz[G](Z) with the integer a(l); under this identification we have a % 0 <=>• a = n\G\ for some n e Z Clearly a e EndZ[Cj(Z) factorizes through a projective module if and only if it factorizes through some Z[G] m . Suppose that a factorizes through Z[G]"\ thus er(Z) -» Z/|G|, must coincide, and we have: Proposition 33.7: If / e D,,(Z) then there exists a unique ring isomorphism KJ J. Proof: We first show (•<=). Assume there exists a surjective Z[G]-homomorphism J and that J is stably equivalent to I(G)*. Then 7 is a minimal representative of the stable module [I(G)*]. Likewise, I(G)* is also a minimal representative of [I(G)*]. It follows by the Swan-Jacobinski Theorem that J 8 Z[G] = I(G)* © Z[G] By semisimplicity of Q[G], it follows that J ®Q =Q\G\ I*(G)(g)Q. In particular, J ® Q contains no trivial summand, so that the kernel of Id : Q[G] -> 7 (g) Q is the trivial one-dimensional module over Q[G]. Hence we have a short exact sequence of Z[G]-modules 0 -*• Z - • Z[G] - ^ 7 ^ 0 Dualizing, we obtain another short exact sequence 0 -»• 7* -»• Z[G] - • Z -»• 0 As we observed in the proof of (35.1), it follows that J* = I(G), and hence 7 = I(G)*. This proves the implication (<=); the converse follows immediately from (35.3). D I(G). On dualizing, we have a commutative diagram * I*(G) -» 0 : follows from (37.7). V\ («), and this completes the proof. • We saw in Section 22 that elements of Dn(Z) can be represented by modules of the form N (B P where N e Q,,(Z) and P is finitely generated projective. Passing to the corresponding rational theory, we may identify ft,,(Z) ® Q with fin(Q). Also, by (16.1), PQ is free over Q[G]. Thus elements of Dn(Q) are represented by modules of the form NQ © PQ where A'Q e ^,,(Q) and PQ is free over Q[G]. Previously we saw, (29.1), that £2,,(Q) is represented by Q if n is even, and by IQ(G) if n is odd. It follows easily that: Proposition 40.2: A cohomological period of afinitegroup is necessarily even. We say that n > 0 is a. free cohomological period (or just/ree period) of G if Qn+k(Z) = £2*(Z) for all k e Z. Likewise, there are a number of ways of characterizing this possibility. Consider the following conditions on G: Fi(n): nn+k(Z) = nk{Z) for all integers it; Tiin): Q.n+k(Z) = Q.k(Z) for at least one nonzero integer k; J--${n): There exists an exact sequence in ,F(Z[G]) of the form 0 -> Z -> F,,_| ->• 0(1) = {±1} is the sign representation, a(x) = +\;a(y) = —1, then i = • 0(2) x 0(1) C 0(3) takes values in 50(3). If v : Spin(3) ->• S0(3) is the spin double covering, then Q(4n) = v~'(t(Z?2n))- That is, Q(4n) is the spin double cover of the dihedral group D2n. The algebraic Cayley complex of the presentation given above extends one step to the left to give a resolution of free period 4: (see [12], Chapter XII) C*(L) we mean a commutative diagram of Z[G]homomorphisms C.(tf) if • C*(L) be the postulated chain homotopy equivalence. Since • K\, ft : L\ -*• K, inducing the identity on TT\. Let S : K\ —> K, y : L —>• L\ be homotopy inverses for a, /), and put • M' is an isomorphism; (iii) cp+ : N ^- N' is surjective; and (iv) Q is projective. Then <po : E -*• Q is surjective and there is an isomorphism ljr : £ —*• E+(£') where ^ = Ker(<po). Proof: The surjectivity of N' ® K is also an isomorphism, so that TJT : £ -> £+(£') is an isomorphism over cp as claimed. • In the following discussion, M, M', N, N' e ^"(A). We denote by v(M) the smallest integer n > 1 for which there exists a surjective A-homomorphism Proposition 51.2: Suppose that £ e Stab1 (A/, N) and £' e Stab'(M', N'); if (p : M ->• M' is an isomorphism, then for each ix > v(N'), there exists a finitely generated stably free module S such that N -X F 4 M ->• 0) and £' = (0 -^ A?' -4 F ' 4A/' —>• 0). Since F is projective, £+(£') where S = Ker( v(N') there exist (i) a finitely generated stably free module S; (ii) an internal stabilization V off; and (iii) a internal quasi-stabilization Q of £' with the property that • E; equivalently, when there is a finite presentation 5 of G and a homotopy equivalence q> : C*{Q) -» E. If E = (0 -»• 7 -» F 2 -»• ^i -> ^b ->• Z ->• 0) is an algebraic 2-complex, then E is said to be minimal when J e £23(Z) is minimal. Say that G has the realization property when each algebraic 2-complex over Z[G] has a geometric realization; likewise say that G has the minimal realization property when each minimal algebraic 2-complex over Z[G] has a geometric realization. We arrive at: Theorem II: Let G be a finite group; then G has the realization property if and only if all minimal algebraic 2-complexes are realizable. Proof: If G has the realization property, then clearly G has the minimal realization property. It suffices to show the converse. Suppose that, for each N e ^ ( Z ) of minimal height and each £ e Stab3(Z, A0, the homotopy type of £ is geometrically realizable. We must Q -*• Q on to the trivial Q[G] module Q. Letting r\ : P —> Q be the Z[G] homomorphism given by r](x) = 0 I {X) induces an isomorphism It is tempting to restate (59.3) by saying that q> : C»(X) -> (X) is a weak homotopy equivalence. The difficulty is that C*(X) and (X) are slightly different sorts of things; (X) is both augmented (by Co(X) —>• Z) and co-augmented (by X2(X) ->• C2(X)) whereas C*(X) is merely augmented. In comparing the two, we should, strictly speaking, drop the co-augmentation, but as remarked in Section 47, this should cause no real confusion. Observe that there is a natural transformation of (augmented, co-augmented) chain complexes v : C*(K) -»• (X) described by the following diagram TZ2{K) E^.(£) be a weak homotopy equivalence. In particular, A is an epimorphism, then rk z (A) < |G|. Since A is stably equivalent to I*(G), then A is torsion free over Z, and a straightforward calculation of Z-ranks shows thatrk z (A) = rk z (F(G)) = |G| - 1. From the fact that A and I*(G) are stably equivalent, it follows from Wedderburn's Theorem for modules over Q[G], that A S+(E) be a weak homotopy equivalence. In particular, E has the property that Ker(^r o tp : n2(K) ->• J is isomorphic to Z[G]". Choose a Z[G] basis a,, a 2 , . . . , am for Ker(/), and form X(a) by attaching 3-cells £<3\ £< 3 ) ,..., E%ho K by means of ct\, a2,..., a m ; that is X(a) = /: UB1 £i 3 ) UO2 £ f • • • Uam £<3) Then it is easily checked that Hr(X(a); B) = 0 for all r > 3, and it is straightforward to see that (X) is homotopy equivalent to E. However, by hypothesis, X(a) is homotopy equivalent to a finite 2-complex, K say, which thereby provides the required geometrical realization of E. •
(x • a)j =
xn) and a = (a,-7-)i
(1.5) D" is a simple module over Mn(D); moreover, End/W/|(D)(D") = D. (1.6) Mn(D) has property S.
Orders in semisimple algebras
15
Let D\, D2 be division rings, and suppose that
ka(x) = ax then A. : A —> End^(^4); a M> Xa, is an isomorphism of rings. Now assume that .4 has property 5. Since A is finitely generated over itself, by the single element 1.4, then by hypothesis A = E*'0 for some simple ,4-module E. Thus End^(yl) = Mn(D) where D — End^(E). By Schur's Lemma, D is a division ring, and the result follows since A = End.4(.4). • However, if D is a division ring, then Dopp is also a division ring, and Mn{D)opp = Mn(Dopp). It follows that this result is left-right symmetric, and property S could as well be defined in terms of left modules. The characterization given in (1.9) is 'extrinsic', relying on properties of modules to give information about the algebra. There is a corresponding 'intrinsic'
16
Stable Modules and the D(2)-Problem
treatment, which we outline below. A ring .4 is said to be simple when it contains no non-trivial two-sided ideal. The basic example of a simple ring is given by: Proposition 1.10: For any division ring D, the ring Mn(D) of n x n matrices over D is simple. Proof: Considered as a D-bimodule M,,(D) is free with basis [E(i, j)]i
and we write X e M,,(D) in the form X = where Xrs e D. Let I be a nonzero two-sided ideal in M,,(D), and X e I be a nonzero element. In particular, suppose that X,y ^ 0; then £ ( / , ; ) = Xy 1 £(i > i)X£O',y)€l Hence for all r, x
£(r, s) = E(r, i)E(i, j)E(j, s) e I and I = Mn(D), since I contains the canonical basis.
•
The converse of (1.10) fails to be true without imposing some sort of finiteness condition. The most general of these, the so-called Artinian condition, need not concern us here, though we consider it briefly in Section 4. Instead, we impose a more straightforward finiteness condition by requiring rings to be algebras of finite dimension over a field. We then have: Proposition 1.11: Let A be an algebra of finite dimension over a field k; then the following conditions are equivalent: (i) A has property S; (ii) A is simple; (iii) A = Mn{D) for some division algebra D of finite dimension over k. Proof: We have shown already that (i) •£=> (iii) and that (iii) => (ii), so it suffices to show that (ii) =$> (i). Since A is finite dimensional over k, so is every right ideal of A. Let M be a nonzero right ideal in A of lowest possible dimension, and put aeA
Orders in semisimple algebras
17
Clearly J is a nonzero two-sided ideal; since A is simple it follows that J = A. Moreover, since A is finite dimensional over k there exists a finite subset [ a \ , . . . , a n ) s u c h that A — a\M + \- anM. Suppose that [a\,..., an] is a minimal subset with this property. Since M is simple it is straightforward to see that we have a direct sum A = a\M © • • • ®anM Thus A has property S by (1.2). This completes the proof.
a
2 Semisimple modules The classification of Section 1 extends in a straightforward fashion to modules over a direct product Mdl(D\)x • • • x A/
e2 = ( 0 , l , 0 , . . . , l ) ; . . . ;
em = (0, 0 , . . . , 1)
Clearly (2.1)
er.es = Srs.er
If M is an ,4-module, we obtain a collection M\,..., Mm of .4-modules thus Mr = M.er = {x.er : x e M) It is straightforward to check that M is the internal direct sum (2.2)
M
lf
Although the canonical injection ir : Ar —> A is not a ring homomorphism in the strict sense (the identity goes to a central idempotent rather than the identity), the formalism nevertheless applies to give a module Mr = i*{M) over the direct factor Ar. The projection nr : A —> Ar is a bona fide ring
18
Stable Modules and the D(2)-Problem
homomorphism, and we see easily that Mr = n*r{Mr)
(2.3) From the identity
Id = i[ O7T\ -\
Mm °X,n
we obtain: Theorem 2.4: Let A = A\ x • • • x Am be a direct product of rings; then any «4-module M can be expressed as a direct sum
It follows that (2.5) Let M, N be modules over A = A\ x • • • x Am; then M =.4 N -<=£• Ms =_4s Ns for each s Provided the context is clear, we shall ignore the distinctions between Mr, Mr and n*(Mr), and regard Mr as a module over both A and Ar. The problem of classifying modules over a product ^4, x • • • x Am is now reduced to the corresponding problem for each of the factors. We shall say that a nonzero module M over A\ x • • • x Am is supported on the rth factor when Ms = 0 for s ^ r. From (2.4) and (2.5), it follows that: (2.6) If A = A\ x • • • x Am is a direct product of rings, then any simple (nonzero) «4-module is supported on some unique factor. A right ,4-module M is said to be semisimple when it can be written in the form M = 0 / 6 / Ni for some collection (A^,),s/ of simple ^-modules N,-, and finitely semisimple when M = ®™_i N; for some finite collection (//,•))<,•<„ of simple .A-modules. Here we shall only be concerned with finitely semisimple modules. If ./V is any module over A, we denote by N^ the d-fo\d direct sum N(d)
= N 0 • • • ® JV
A module of this form is said to be isotypic, or, more precisely, isotypic of type (N,d). It is straightforward to see that End^(A^) is naturally represented as a matrix ring EndA(Nw)
= Md(EndA(M))
Orders in semisimple algebras
19
If M is an ,4-module, by an isotypic decomposition for M is a direct sum M — ®"'=\ Mj, where each M; is isotypic of type (Nj, d,), and where the isomorphism types Nj are pairwise distinct; that is
M = Nf ° © • • • 0 N^m) where N\,..., Nm are simple and Nj ^ Nj for i ^ j . By collecting together isomorphic summands, we see that: Proposition 2.7: An .4-module M is finitely semisimple if and only if it admits an isotypic decomposition. It is an easy consequence of Schur's Lemma that in such an isotypic decomposition, Hom^(A^/(/l), N^j)) = 0 whenever i ^ j . It follows that: Proposition 2.8: For a module M admitting the isotypic decomposition M = N\ © • • •ffiN\f^ there is an isomorphism of rings x ••• x
Mdm(Dm)
where D, is the division ring D, = We say that the ring A is right semisimple (resp. left semisimple) when it is semisimple as a right (resp. left) module over itself. Evidently A is right semisimple if and only if A"pp is left semisimple. When A is right semisimple, then A is automatically finitely semisimple as a module over itself, for, if / : A -> ©,e/M,- is an isomorphism of right ,4-modules where each M, is simple, then the index set / must be finite. In fact, if 7rr : ©, e /M, -> Mr and ir : Mr -> ©,S/M,- denote, respectively, the projection on to the rth summand and the inclusion of the rth summand, and, if J — {r e I : Jr r (/(U)) ¥" 0}) then J is finite by definition of®; it clearly suffices to show that J — I. Write A = /~'(©/<=./M,), B = /~'(©, e /\yM,). Then A, B are right ideals of A with A n B = (0); moreover, 1^ e A. If / \ J ^ 0, we may choose 6 e B such that ft 7^ 0. Since A is a right ideal, b = \j^.b e A. Hence A D B contains a nonzero element b; contradiction. Thus / = J, and for any ring A we have (2.9)
A is right semisimple <=>• A is right finitely semisimple.
A similar argument holds for left modules. Let A be a right semisimple ring; then, by (2.8), End^(.4) is isomorphic as a ring, to a direct product Mjx (D\) x • • • x Mdm (Dm), where each D,- is a division ring. However, for any ring ^4, there is a ring isomorphism End^(A) = A given by the evaluation map ev : End^(A) -» A, ev(a) = a ( l ^ ) . It follows that,
20
Stable Modules and the D(2)-Problem
when A is right semisimple, A is isomorphic to a direct product
A=
Mdl(D{)x---xMdm(Dm)
where D\,..., Dm are division rings. It follows from the results of Section 1 both that the converse is true and that the result is left-right symmetric; that is: Theorem 2.10: The following conditions on A are equivalent: (i) (ii) (iii) (iv) (v)
.A is right semisimple; A is left semisimple; every finitely generated left .4-module is semisimple; every finitely generated right ^4-module is semisimple; A is is isomorphic to a direct product
where D\,...,
Dm are division rings.
Let A be an algebra of finite dimension over a field k\ we say that A is semisimple when it satisfies any of the equivalent conditions of (2.10). (We note that the general notion of 'semisimple ring', namely that the Jacobson radical should be zero, allows, in the absence of any finiteness condition, for more complicated examples than the above. However, in our development, we shall neither meet nor need anything more general than finite products of matrix rings as above.) From (2.4), (2.5), and the results of Section 1, we obtain the following classification theorem: Theorem 2.11: Let M be a finitely generated module over Mdl(D\) x • • • x Mdl (Dm) where each D, is a division ring; then
where £,• is a simple module over Mdi{Dt), and n,•• : A -*• Mdi(Dj) is the projection on to the /th factor; with the given ordering of the factors on A, the sequence (/j,\,..., /xm) is uniquely determined by M. Furthermore, the simple modules over A are precisely those of the form JT*(I!,•). The elucidation of the structure of finite-dimensional semisimple algebras is due to Wedderburn [79]. By a Wedderburn decomposition of a finitedimensional ^-algebra A we mean an isomorphism of ^-algebras .4 = Mdl (D\) x • • • x Mdm(Dm) where each D, is a division algebra. It is not difficult to see that such a product structure is essentially unique; that is:
Orders in semisimple algebras
21
Theorem 2.12: Let A be a finite-dimensional /t-algebra possessing two product structures
i= \
i=l
then ix = m, and for some permutation a of [l,...,m], <5, = da(l ) and A,- = £>„((). Thus up to a permutation, we may speak of the Wedderburn decomposition of A. In the special case where the base field k is algebraically closed, every finite-dimensional division algebra over k is actually isomorphic to k, so that the Wedderburn decomposition then simplifies to: Theorem 2.13: Let k = k be an algebraically closed field and A a finitedimensional A>algebra; then there is an isomorphism of fc-algebras A = Fl/li Mdt(k) for some sequence of positive integers (d,-)i
3 Projective modules and semisimplicity As we have seen in Section 1, the theory of modules over a field, or more generally over a simple finite-dimensional algebra, is straightforward. Over a general ring A however, the situation is usually far more complicated. One still has the obvious analogue of vector spaces; these are the so-called 'free modules' of the form
More general, but almost as convenient, is the notion of projective module. An ,4-module M is said to be projective when it is a direct summand of a free module; that is, when there exists a module N such that M © N is a free over .4. Since a direct sum of free modules is free, we see that: (3.1) A direct sum of projective modules is projective. Suppose that each short exact sequence 0->Q-*M->P->0 splits. Let be a set which generates P over A, and let e : FA(A) —>• P A.€A
be the canonical epimorphism from the free module on A. Then 0 -> Ker(e) -* FA(A) -* P -> 0 is exact, so that P © Ker(e) = FA(A), and P is projective.
22
Stable Modules and the D(2)-Problem
Conversely, suppose that P is projective. Let N be an .A-module such that, for some set X, P © N = FX(A). Making the identification P © N = FX(A), for each x e X choose p(x) e P, and t(x) e N such that
p{x) + t(x) = x Given an exact sequence of ^-modules £ — (0 -> Q -*• M A- P -» 0), for each x e X choose an element s(x) 6 M such that n(s(x)) = p(x). Then the correspondence x i->- s(x) defines a homomorphism of ^-modules 5 : Fx(A) —> M, and, by restriction to P, a homomorphism s : P —> M; s splits £ on the right, so that: (3.2) An A- module P is projective if and only if each short exact sequence of the form O - ^ g - ^ M - ^ - P - ^ O splits. This has the following consequence: Proposition 3.3: Let A be a finite-dimensional algebra over a field k; if every finitely generated module over A is projective, then A is semisimple. Proof: Let M be a finitely generated ^-module; we must show that M is a direct sum of simples. Since M is automatically a finite-dimensional vector space over k we may proceed by induction on dim^M). If dim,t(M) = 1, then M is simple, and there is nothing to prove. Suppose proved for all modules L such that dim^(L) < m, and let dim^(M) = m. If M is simple, there is nothing to prove. If M is not simple, then it has a proper nonzero submodule M\. However, by the projectivity hypothesis and (3.2), the exact sequence 0 -+ M, ->• M ->• M/M, - • 0 splits, so that M = M\ (&M/M\. However, dim^(Mi) < manddim^(M/Mi) < m, so that, by induction, both Mi and M/M\ are direct sums of simple modules. Thus M is also a direct sum of simples, and this completes the proof. • The converse to this is also true, as we now proceed to show. Consider first the case where A — Mn{D) where D is a division ring. Let R(;) C Mn(D) be the right ideal consisting of matrices concentrated in the ith-row thus irE(i,
r);
Xir e
We note that there are isomorphisms R(l) = R(2) = • • • = R(«) over Mn(D), and, as we have already seen, any finitely generated Mn(D)-module M is isomorphic to a direct sum of copies of R(l) M =
Orders in semisimple algebras
23
Since M,,(D) = R(l) © R(2) 0 • • • 8 R(«) we see that R(l) is projective, and so we obtain: Proposition 3.4: If D is a division ring, then any finitely generated module over M,,(D) is projective. More generally, by (2.6), if £ ) , , . . . , Dm are division rings, then any simple right (resp. left) module over a finite product Mdl(D\) x • • • x Mdm(Dm) is isomorphic to a simple right (resp. left) ideal supported on a single factor, and from the argument already given, is necessarily projective. Since, by (2.11), any finitely generated module over M^ (D\) x • • • x Md,,,(Dm) is a direct sum of simple modules, then we have: Proposition 3.5: Let A be a finite-dimensional semisimple algebra over a field k; then every finitely generated module over A is projective. From (3.3) and (3.5), together with the results of Section 2, we obtain one final charactization of semisimplicity: Theorem 3.6: Let A be a finite-dimensional algebra over a field k. The following conditions on A are equivalent: (i) every finitely generated right .4-module is projective; (ii) every finitely generated left ^4-module is projective; (iii) A is isomorphic to a product A = fl/'Li Mdj (£>,-) where each D, is a finitedimensional division algebra over k, and each dj is a positive integer.
4 The Jacobson radical Let A be a finite-dimensional algebra over k; a two-sided ideal a in A is said to be nilpotent when a" = 0 for some n > 1. If a, b are nilpotent two-sided ideals in A, then so is a + b. In fact
r=0
In fact, if a" = 0 and bM = 0, then (a + b)N+M
= 0. In particular:
Proposition 4.1: Let A be a finite-dimensional algebra over a field k; then A has a maximal two-sided nilpotent ideal, the radical rad(.4).
24
Stable Modules and the D(2)-Problem We shall need the following also:
Lemma 4.2: Let A be a finite dimensional algebra over a field k such that rad(.4) = 0; then any simple right ideal Q C A contains a nonzero element e such that e2 = e. Proof: If a e Q we have a homomorphism of .4-modules a : Q -> (2 given by a(x) = ax. Since Im(a) is a right ideal of A and Q is simple then either (I) Im(a) = {0} or (II) a : Q —> (2 is an isomorphism for at least one a e Q. Put
then r is a two-sided ideal which contains Q and hence is nonzero. If (I) above holds, then T2 = 0, and thus rad(.4) ^ 0, contradiction. Hence there exists a € Q such that a : Q —> Q is an isomorphism. In particular, there exists e e Q such that ae = a. Evidently e •£ 0. Also ae 2 = a, and so e2 — e e Ker(a) C Q. However Ker(a) is a right ideal in A, and by choice of a is contained properly D in Q. Since Q is simple, Ker(a) = {0}, and e2 = e as claimed. As a corollary, we get: Theorem 4.3: If A is a finite-dimensional algebra over a field k, then A is semisimple <=^ rad(A) = 0 Proof: Suppose A is a finite-dimensional semisimple algebra over k. Let A = A\ ® • • • © Am be the Wedderburn decomposition of A into simple two-sided ideals A\,..., Am. Since each Ai is simple, it is straightforward to see that any nonzero two-sided ideal a has the form a = Ait ® • • • ® A;n for some unique sequence i\ < • • • < in. However, the simple summands A\,..., A,,, have the property that i
if i = j
from which it follows that
in particular, no nonzero two-sided ideal of A is nilpotent, and rad(A) = 0.
Orders in semisimple algebras
25
To prove the converse, we first prove, by induction on dim^P), that any right ideal P is a direct sum of simple right ideals p = P[ e • • • e pn If dim^(f) = 1 there is nothing to prove. Thus suppose that dimk(P) = N and that the result is established for all right ideals of dimension < N. If P is simple, there is nothing to prove. If P is not simple, choose a nonzero right ideal P\ c P of lowest possible dimension. Then P\ is simple, so that, by (4.2), there exists e e P\ such that e2 = e. Then P decomposes as a direct sum of right ideals
P =eP®(\
-e)P
Now e € P\ C P and e2 = e; from the simplicity of P\ it now follows that eP = P\. However, (1 - e)P is aright ideal withdim^((l - e)P) < N, so that, inductively, (1 — e)P decomposes as a direct sum of simple right ideals
giving the required decomposition into simples P =
P\@---@Pn
The theorem as stated now follows on, taking P = A.
D
The finiteness condition we have imposed, namely that of being a finitedimensional algebra over a field, is not the most general that can be imagined. For example, a finite product of semisimple algebras over different fields is still semisimple. The most general finiteness condition in our context is the so-called Artinian or descending chain condition. A ring A satisfies the right Artinian condition when each descending sequence • • • c P,,+\ C P,, C • • • C P\ C PQ of right ideals has the property that for some N, Pr = PN for N < r. We shall not need, and do not prove, the appropriate generalization of (4.3), which is: Theorem 4.4: Let A be a ring; then A is semisimple (that is, semisimple as right module over itself) if and only if A is right Artinian and rad(A) = 0. There is a corresponding, and equivalent, statement for left Artinian rings. It is perhaps worth pointing out that the Artinian condition by itself is not left-right symmetric; nevertheless, in conformity with (2.10), the combined condition 'Artinian + rad = 0' is left-right symmetric.
26
Stable Modules and the D(2)-Problem 5 Nondegenerate algebras
Let R be a commutative integral domain of characteristic zero with field of fractions k; by an /?-order, we mean an /?-algebra whose underlying /^-module is finitely generated and free. In the special case where R = k, a fc-order is simply anfc-algebraof finite dimension. When A is an 7?-order, and x e A, we denote by x the homomorphism of right modules x : A —> A given by
x(y) = xy There is a symmetric bilinear form f3A : A x A —»• A given by t, y) = We say that A is nondegenerate as an /^-algebra when fiA is nondegenerate as a symmetric bilinear form; that is, when the correlation 01 : A -> A*;
p*A(x)(y) = 0A(x, y)
is injective, where A* denotes the /?-dual of A. Proposition 5.1: Let k be a field of characteristic zero, and let .4 be a finitedimensional algebra over k; if A is simple, then A is nondegenerate. Proof: Put L
C, y)
= 0 for all y e A}
It is straightforward to see that A1 is a two-sided ideal in A. Since A is simple, then either AL = A or AL = {0}. However, #4(1,1) = dim^C^l) ^ 0 so that l^t £ X x , and A1 ^ .A. Thus AL = {0}, and «4 is nondegenerate. D Nondegeneracy is preserved under a direct product: Proposition 5.2: Let R be a commutative integral domain of characteristic zero, and let A|, A2 be /?-algebras whose underlying /?-modules are finitely generated and free; then Ai x A2 is nondegenerate -<==>• Ai, A2 are each nondegenerate. Proof: It is straightforward to see that there is an isomorphism of symmetric bilinear forms (A, x A 2> J8A 1 XA 2 ) = ( A I , J 8 A 1 ) - L ( A 2 , ^ A 2 )
•
Orders in semisimple algebras
27
In consequence, we obtain: Theorem 5.3: Let k be a field of characteristic zero, and let A be a finitedimensional algebra over k; then A is nondegenerate <=^ A is semisimple Proof: If .4 is semisimple, then A is isomorphic to a finite direct product A = A\ x •••xA
m
where each At is simple, and hence, by (5.1), nondegenerate. Thus A is nondegenerate by (5.2). To establish the converse, it suffices to show that rad(.4) = 0 whenever x A — 0. Thus suppose that a is nonzero two-sided ideal of A such that a"+1 = 0 for some n > 1. Put m = maxfr : ar ^ 0}, and let {e"\ . . . , e™ } be a basis for am. For each k in the range 0 < k < m — 1, let { e , , . . . , ekdk} be a linearly independent set in a* which projects to a basis in a^/a*"*"', where by convention we take a0 = A- Then
is a basis for A. Moreover, if x e a, it is straightforward to check that, with respect to this basis, x is represented by a matrix in triangular block form whose diagonal is identically zero. Consequently Tr(x) = 0 for all x € a. However, if x e a, then xy e a for all y e A, so that pA(x, y) = Tr(x)>)) = Tr((xy)) = 0 for all x e a and so a C A1. In particular, rad(.4) c A1. However, if A is nondegenerate, then rad(.4) = A1 = 0, and hence A is semisimple. This completes the proof. • We consider the effect of 'extension of scalars'. Let R be a commutative integral domain of characteristic zero. Proposition 5.4: Let R c S be a ring extension; if A is an /?-order, then As — A (gifl 51 is an 5-order, and A is nondegenerate «=>• As is nondegenerate Corollary 5.5: Let R be a commutative integral domain with field of fractions k, and let A be an /?-order; then A is nondegenerate <=> A^ is semisimple
28
Stable Modules and the D(2)-Problem 6 The discriminant
If ( e i , . . . , €„} is an R basis for A, the discriminant Disc(ei,..., e,,) is defined by Disc(e, , where /},-_,• = jSA(e,-, e,-) — Tr(e,e/). If [E\, ...,£"„} is also an R basis for A, and, if A = (otij)\
then Disc(ei, ...,€„), (6.1)
Disc(£i , . . . , £ „ ) are related by
D i s c ( e , , . . . , €„) = det(A) 2 Disc(£, , . . . , £ „ )
It follows easily that: Proposition 6.2: The following conditions on an fl-order A are equivalent: (i) A is nondegenerate; (ii) D i s c ( 6 ] , . . . , €„) ^ 0 for some R-basis {€\,..., en] of A; (iii) Disc(ei, ...,€„) ^ 0 for every R -basis [€\,..., €„} of A. This gives an absolute invariant of /J-orders; if A is an /?-order and { e i , . . . , €„} is an /?-basis for A, we define the discriminant, Disc(A), to be the image of D i s c ( e i , . . . , €„) in the multiplicative monoid R/(R*)2 obtained as the quotient of R by the subgroup of squares of units. As an example, we compute fiA in the case where A = Mn(R)\ let (£(/, j))[
so that X e Mn(R) is represented in the form
We wish to compute the trace Tr(X) for X e Mn(R); observe that the matrix X has a trace in the usual sense
Orders in semisimple algebras
29
In fact, we have: Proposition 6.3: If X e Mn{R) then Tr(X) = Proof:
With respect to the basis (£(/, j))\<jj
rs
and a straightforward comparison shows that _{xrk H(X)kl,rs
ifs =
( 0
if 5 ^
In particular, Ai(X)^;,*/ = X ^ , so that
as desired. One can now compute the discriminant Disc(£(7, j)\
il
follows easily from (6.3) that
fi(E(i, j), E(k, /)) = n SjME(il)) so that
I
n if / = / and /' = k 0
otherwise
If we put B(X, Y) = p(X, YT), we see that , j), E(k, /)) =
f n if 0', j) = (k,l) 0
otherwise
30
Stable Modules and the D(2)-Problem
so that B is represented by a diagonal n2 x n2 matrix with the constant entry '«' on the diagonal. Thus
det[(B(E(i,j),E(kJ))ijM]=n"2 It follows that det[()B(£(i, j), E(k, l))ijikl] = an"2 where a is the sign of the 'transpose' permutation (i, j) (->• (j, i) on { 1 , . . . , n] x { 1 , . . . , n); this permutation is obviously the composition of "(n~X) transpositions, so that a = (— 1)" T , which is the required result. • It follows already from (5.1) that Mn(R) is nondegenerate when R is a commutative integral domain of characteristic zero. The above computation gives an alternative proof of this result. Let A be a simple algebra whose centre is the field E; we may make the identification A ® E E = M n(E). If X e A, then Tr(X) = «tr(X
=
det((bjj)\
We take disc(.A) to be the class in E/(E*) 2 of disc(ei ,...,€„) for any E-basis {
disc (E(i, j)]
7 Z-orders 2
In the case where R = Z, (Z*) is the trivial group, and we identify Z/(Z*) 2 with Z; that is: Proposition 7.1: If A is a Z-order, then the discriminant Disc(A) is a welldefined integer.
Orders in semisimple algebras
31
We also have the following: Proposition 7.2: Let A be a Z-order; then the following conditions on A are equivalent: (i) (ii) (iii) (iv)
Disc(A) ? 0; A is nondegenerate; AQ is nondegenerate; AE is nondegenerate for any extension field E of Q.
Let A, Q be Z-orders in the same Q-algebra A, and suppose that A C S ; then, since A and £2 both span A over Q, rk z (A) = rkz(£2) and the index j = |£2/A | is finite. It follows from (6.1) that: (7.3)
Disc(A) = j2D\sc(£2)
from which we easily deduce (7.4)
disc(A) = /disc(fi)
We say that an integer j > 1 is a co-index for the Z-order A when there exists a pair (A',
32
Stable Modules and the D(2)-Pwblem
and let A C A be an order over R. If disc(A) is represented by a unit in R*, then A is maximal considered as a Z-order. We denote by c(A) the set of co-indices for A. We say that A has the bounded co-index property when c( A) is a finite subset of Z+. It follows easily from (7.3) that: Proposition 7.7: A nondegenerate Z-order has the bounded co-index property. The converse is also true as we now see: Proposition 7.8: Let A be a Z-order; if A is degenerate, then c(A) is infinite. Proof: Wefirstshow that there exists an injective ring homomorphism
Orders in semisimple algebras
33
(iii) there exists a finite index imbedding
(iv) AQ is semisimple. Proof: (i) and (ii) are equivalent, by (7.3) and (7.8). (i) and (iv) are also equivalent, by (5.3). We show that (iii) ==> (i). Suppose therefore that
A\®---®Am
where A\,... ,Am are the simple two-sided ideals of A. Let A c A be a maximal order; then A is a direct sum A = A, ©•••© A,,, where A, c A; is a maximal order.
8 Examples If R is a commutative ring and a, b e R*, we denote by (^r) the quaternion algebra over R, with /?-basis {1, /, j , k], and multiplication given by iz = a-l;
j 2 = b-l;
ij =
-ji=k
When R — K is a field, ( ^ ) is a simple K-algebra, and is a division algebra if and only if the equation
ax\ + bx\ — abx\ = 0
34
Stable Modules and the D(2)-Problem
has no nonzero solution x = (x\, xi, x$) e K3. We denote by H the standard quaternion algebra
over R. Following Swan [59], we exhibit two distinct maximal orders in a quaternionic division algebra Q-algebra ("'^T1), where A" is a a totally real number field of degree 4 over Q. Our examples will be quaternionic conjugates of Swan's. Let A be an algebra over a commutative ring /?; we denote the group of units of A by U(A). In the case where A = H, U(H) is the product Spin(3) x R + , where Spin(3), the unit sphere in H, is the simply connected covering of the group SO(3) of rotations in R3. If R = Q and A is a Z-order in A we denote by Uo(A) the intersection of U(A) with the the commutator subgroup of U0(A) = U(A) fl [\J(A), U(A)] The ring homomorphism A -> A <8>Q R; X I->- X
x H d
then R), V(A <E>Q R)] = Spin(3) x • • • x Spin(3)
which is compact. Since Uo(A) imbeds as a discrete subgroup in a compact group, it is finite; to summarize:
Orders in semisimple algebras
35
Proposition 8.2: Let K be a totally real algebraic extension of finite degree d over Q, let A be a quaternion algebra of the form A — ( ~ K ~ ' ) » and let A C A be a Z-order; then Uo(A) is finite. Note that, if A is a Z-order in the quaternion algebra A = ( ~ K ~ ' X then, since K is totally real, U(A) actually imbeds as a group of units in U(H). In particular, the finite group Uo(A) imbeds as a subgroup of £/(H). Now the isomorphism types of finite subgroups of t/(H) are known ([82], p. 88); with three exceptions, a finite subgroup of H* is either cyclic or generalized quaternion. The exceptions are: (i) T*, the binary tetrahedral group of order 24; (ii) O*, the binary octahedral group of order 48; and (iii) /*, the binary icosahedral group of order 120. We proceed to our examples: put £ — e x p ( ^ ) and put K = Q(f + £); then K is a totally real number field of degree 4 over Q whose ring of integers R takes the form R = Z[x)/(x4 - Ax2 + 2) Observe that the minimal polynomial x4 — 4x2 + 2 over Q factorizes as x4 - Ax2 + 2 = (x - z)(x + r)(x - f)(x + f) where r = ^/2 + *J2 and f = ^2 — ^/2. Moreover, the Galois group Gal(A7Q) is cyclic of order 4 and a generator acts on the roots of x4 — Ax2 + 2 — (x — r)(x + T)(X — f )(x + f) by means of r i-»- —f;
f i->
T
Since K is totally real, the quaternion algebra A = (~y~') is a division algebra of dimension 16 over Q. Next define a, fi e A by a =
f+ir ; 2
Then it is easy to compute
We put A = spanR{l, a, p, af)}.
p H
=
(r-r)(l-y) 2
36
Stable Modules and the D(2)-Problem
Proposition 8.3: A is a subring of A. Proof: Since A is an /?-submodule of A, it suffices to show that A is closed with respect to multiplication. The product afi is in A by definition, and the products a2, 01, fia all belong to A as we see from the following identities a2 = - 1 + ( T 3 - 3 T ) Q ;
f}2 = (4 - r 2 ) + (4r - r3)/S 0a = (3 - T 2 ) + (4r - r 3 )a + (r 3 - 3r)j8 - a? Straightforward substitions now reveal that a2ft, a/32 and afia all belong to A. For £ e A we denote by pt= : A ->• A the mapping P* (*) = *£ To show that A is a subring of A, it suffices to show that p%(A) C A for any f € A. Since the products a 2 , /6a, a/3a all belong to A, we see easily that P(,(A) c A. Likewise, a/3, ft2, aft2 all belong to A, so pp(A) C A. Clearly
/MA) C A since pap = pp o pa. Upon writing an arbitrary element £ e A in the form £ = co • 1 + c\ • a + c-i • P + CT, • a/6, with c, e R, we see that PS = c 0 • Id + c\ • pa + c2 • pp + c3 • pap Hence p^(A) c A, and A is closed under multiplication.
•
a, /3 are units in A, since a 16 = /3 16 = 1; taking Q to be the subgroup of U(A) generated by a, /6, we obtain: Proposition 8.4: Q is contained in Uo(A) and is isomorphic to the generalized quaternion group 2(32). Proof: Note that the commutator quotient group of [/(A) is a torsion free abelian group. It follows that every element offiniteorder in U( A) is contained in Uo( A). The verification that Q = 2(32) is straightforward. • Now put © = span R {l,£, r],^rj), where f = ^ p and r\ = ^ p ; then %n = ( '~'~ ; + ' 7 ) . Observe that i e 0 , since
likewise j , ij e &, and a somewhat easier computation than (8.3) shows: Proposition 8.5: © is a subring of A.
Orders in semisimple algebras
37
In fact, straightforward computation shows that the reduced discriminant equals — 1 for A and 0 , so that: Proposition 8.6: A and 0 are both maximal orders in A. However: Proposition 8.7: A ^ 0 . Proof: Observe that the elements £, r\ are units in 0 , since £ 8 = if — 1. Consequently %r\ e Uo(0), since we have
A straightforward calculation shows (§>?)3 = — 1; hence (£?7)2 has order 3, and the finite group Uo(0) has order divisible by 3. However, since Uo(A) contains a subgroup isomorphic to (9(32) it follows that Uo( A) has order divisible by 32. Suppose that 0 = A, then Uo(A) = Uo(0) has order divisible by 96. This eliminates the possibility that U0(A) is one of the exceptional groups I*,O*,T*.
Since Uo(A) is non-abelian, it must therefore be a generalized quaternion group, and so must have an element of order 48. In particular, A must contain a copy of the field Q[x]/(c48(x)), where Q ( X ) denotes the dth cyclotomic polynomial, that is, the product cd(x) = [~[(x — £), where f ranges over the primitive dth-roots of unity. In this case, c48(x) = x 16 — x 8 + 1, and so dimQ(Q[x]/(c48(x)) = 16 = dimQ(A
Chapter 2 Representation theory of finite groups
We review briefly the representation theory of a finite group G over a field k of characteristic zero; expressed differently, the module theory of the group algebra k[G]. The case k — Q gives a first approximation to the module theory ofZ[G].
9 Group representations Let G be a group and A be a commutative ring; by a (G, A)-representation, or Arepresentation if the group G is understood, we mean a pair (V, p), where V is an A-module and p : G -> GLA(V) is a group homomorphism. If (V, p), (W, a) are (G, A)-representations, then by a (G, A)-morphism * : (V, p) -*• (W, a) we mean an A-linear map H> : V —*• W such that *V(p(g)(v)) = cr(g)(4'(ij)) for all g e G, v € V. If (V,, p,), e / are (G, A)-representations, we define the direct sum representation 0 , e / ( V , , p^ thus
16/
where 0 i e / p,- : G -> GL A (®, 6 / V,-) is the homomorphism (©; 6 /P/)(g)0)j = When V is free module of rank n over A, GLA(V) is isomorphic to GL,,(A), the group of invertible n x n matrices over A, and a (G, A) representation can be interpreted in explicit coordinate form as a homomorphism p : G —>• GL,,(A). Recall the group algebra construction; if G is a group and A is a commutative ring, the group algebra A[G] consists of all A-valued functions with finite support defined on G; it is naturally an A-module under pointwise addition and scalar multiplication. 38
Representation theory of finite groups
39
If g e G, we denote by g the element of A[G] defined thus fl
ifjc = *
The set {g}gsc is a basis for A[G] over A, leading to the canonical representation of elements in A[G] as finite sums of the form
geG
in particular, A[G] is a free module over A. Moreover, A[G] acquires the structure of an A-algebra in which the multiplication is given by
fteG
Here the multiplicative identity is 1, so that the inclusion A -> A[G]isA. i->- A.I. If (V, p) is an A-representation of G, we associate with (V, p) a right A[G]module V(p) whose underlying A-module is V, and on which A[G] acts by means of
Conversely, if V is a finite-dimensional right A[G]-module, we associate with V a finite-dimensional A-representation py : G —> G L A ( V ) by means of Pv(g)(y) = v • g"' This correspondence between A-representations of G and right modules over A[G] is clearly 1-1. Any group ring A[G] admits a canonical involution r : A[G] ->• A[G] given by
geG
When (V, p) is an A-representation, an A-submodule W c V is said to be (G, p)-invariant (or just G-invariant, if p is understood) when, for all geG, p(g)(W) C W. Since \v — P(g)p(g~l), this is equivalent to requiring that P(g)(W) = W. A G-invariant subspace V of V defines the subrepresentation (V, p) of (V, p) by requiring that p : G -> GL^(V) be the homomorphism
P(g)(v) — p(g)(v).
40
Stable Modules and the D(2)-Problem
Being a module over itself, A[G] gives rise, via this correspondence, to a representation of particular importance, the regular representation p reg : G -> GLA(A[G]); that is Preg(g)(v) = g.V
Any representation p : G -+ GLA(V) extends to a homomorphism of Aalgebras, denoted by the same symbol, p : A[G] -> EndA(V); thus
We make no notational distinction between appearances of p as a group representation and as an algebra representation. Suppose that A = k is a field; a £-representation (V, p) of G; (V, p) is said to be irreducible over k when the only G-invariant fc-linear subspaces of V are {0} and V itself, and completely reducible over k when there is an isomorphism in {G,k)
where each (V,-, p,) is irreducible over k. We see easily that: (9.1) The correspondence between ^-representations of G and right modules over k[G] preserves direct sums, and irreducible (G, k) representatations correspond to to simple &[G]-modules. 10 Maschke's Theorem A classical result of Maschke reduces the study of (G, ^-representations to the Wedderburn theory of semisimple algebras developed in Chapter I (although historically, the development went the other way). Theorem 10.1: (Maschke): Let G be a finite group and let k be a field whose characteristic does not divide the order of G; then every k[G]-module is projective. Proof: Let 0 - > - A A - B - ^ - C — > 0 b e a n exact sequence of /c[G]-modules. Forgetting the G-actions, this is an exact sequence offc-modules,and so splits. In particular, there exists a ^-linear map s : C -> B such that p o s = 1C- The map t : C —> B defined by t(x) = YlgeG s(x8)g~l is clearly fc-linear, and the computation
_
\yS{{Xh)g)g^hAh [geG J ^
[geG
' \h = t(x)h J
Representation theory of finite groups
41
for x € C, h e G, shows that t is also &[G]-linear. Moreover, for all x e C
geG
so that p o t(x) = ^,g€c X8g~l = |G| • x. However, |G|.l is invertible in k, and the map a : C -» B given by
is ^c[G]-linear, and /? o a = \c- Thus C is projective since each exact sequence of &[G]-modules ending in C splits, and this completes the proof. D Applying Theorem (3.5) to (10.1) above, we see that: Corollary 10.2: Let G be afinitegroup and let k be a field whose characteristic does not divide \G\; then k[G] is semisimple. Corollary (10.2) is false if the characteristic of k is allowed to divide the order of G. As an example, we may take F2[C2] where F 2 is the field with two elements. In this case F 2[C 2] = {0, 1,7, 1 + r } and J = { 0 , 1 + T] is a (two-sided) ideal which does not possess a complementary ideal. Since k[G] is finite dimensional over k, (10.2) has the following consequences. Corollary 10.3: Let G be afinitegroup and let k be a field whose characteristic does not divide the order of G; then every finitely generated fc[G]-module is semisimple. Corollary 10.4: When G is a finite group, and k is a field of characteristic coprime to the order of G, there is an isomorphism of ^-algebras
k[G] =
Mttl(Dl)x---xMnm(Dm)
for some positive integer m(= m{G, k)) and some sequence (£),-, n,-)i
42
Stable Modules and the D(2)-Pmblem
Proposition 10.5: Let G be a finite group, and let k be a field of characteristic coprime to \G\; then all (G, ^-representations are completely reducible. Indeed, this was the original formulation of Maschke's result [37]. Suppose that the characteristic of k is coprime to \G\. From the uniqueness of the Wedderburn decomposition
each simple right module over k[G] can be identified with a simple right ideal N, = D"' in Mni (D\). Hence each simple right ideal N of k[G] is isomorphic to Ni, for some unique/, 1 < i < m. The trivial one-dimensional representation of G occurs with multiplicity = 1 in k[G], allowing us to write the Wedderburn decomposition in the form k[G]=kxMn2(D2)x--xMnm(Dm) Here we are taking D\ = kandn\ — 1 for the factor of the trivial representation. So far our results have only assumed that A: is a field of characteristic coprime to the order of G. From now on, without further mention, the field k is assumed to be of characteristic zero. When k = k is algebraically closed, the only division algebra of finite dimension over k is k itself, so that the Wedderburn decomposition takes the form k[G] = Mdl(k) x Md2(k)• • • x Mdm{k) where d\ = 1. Here the number m of simple factors is equal to the number of conjugacy classes of G, and the integers d, are called the degrees of the absolutely simple representations of G. Clearly
11 Division algebras over Q If G is a finite group and K is a field of characteristic zero, the theorems of Maschke and Wedderburn imply the existence of a product decomposition of K[G] in the form
where D\,..., Dm are finite-dimensional division algebras over K; as above, n\ = 1 and D\ = K. Eventually, we wish to drop the condition that K be a field and study modules over Z[G]; there are three cases of increasing interest, and difficulty, which may
Representation theory of finite groups
43
be seen as successive approximations to Z[G]. When k = C, the only finitedimensional division algebra over C is C itself, and the complex Wedderburn decomposition of a finite group G then assumes the form C[G] = C x Md2(C) • • • x
Mdm(C)
where the number m of simple factors equals the number of conjugacy classes ofG. Slightly more complicated is the case where K = R; then the real Wedderburn decomposition takes the form R[G] = R x Mni(D2)
x •• •x
Mllm{Dm)
where for / > 2, D,- is either R, C or H. As we shall see, in Chapter 3, when discussing the Eichler condition, a particular difficulty arises in the case where H is a direct factor of R[G], that is when D,- = H and «,• = 1 for some i, A rather closer approximation to Z[G] is Q[G]; in this case, there are infinitely many isomorphically distinct finite-dimensional division algebras over Q. The degree of complication, though infinite, is not completely arbitrary however, and there is at least one a priori restriction which is imposed on the type of the division algebra Z), occurring in a Wedderburn decomposition. In fact, for any totally real field K, the canonical involution where
a =
on K[G] is positive in the sense that ax{a) is a positive element of K whenever a ^ 0. A theorem of Albert ([1]) (see also [51]) now implies an isomorphism of involuted K-algebras (K[G], r) = (Mni(£>,), f,) x • • • x (M,,m(Dm), fm) where each D,- is afinite-dimensionaldivision algebra over K, and r,- is a positive involution on Z),. In particular, the division algebras £>, which occur in a rational Wedderburn decomposition Q[G] = Q x Mn2(D2) x • • • x Mllm{Dm) admit positive involutions. We proceed to sketch Albert's classification of these positive division algebras. First recall the construction of cyclic algebras; fix the following notation: 1Z: a commutative ring, n: an integer > 2, s: a ring automorphism of TZ satisfying s" — Id, a: a nonzero element of 71 such that s(a) = a.
44
Stable Modules and the D(2)-Problem
The cyclic algebra Cn(1Z, s, a) is the two-sided free 7£-module of rank n, with basis ([Xr])o
(X e U)
C,,(TZ, s, a) is an algebra over the fixed point ring E = [x e 71: s(x) = x] with multiplication determined by
i
[Xr+[] a[X°]
0
We note that when 1Z. — K is a field and s has order n (rather than merely satisfying s" = Id) then the fixed point field SK = E is actually the centre of C«(K, s, a); indeed, in some definitions it is required that s has order n precisely to guarantee this outcome. However, we find it more useful to work with the weaker condition s" = Id. This construction is natural with respect to direct products; that is, we have: Proposition 11.1:
Albert's classification of simple Q-algebras which admit a positive involution is as follows: let K be a subfield of R, and let A be a finite-dimensional semisimple K-algebra; by an algebra involution x on A, we mean an isomorphism of A with its opposite algebra such that r 2 = 1A; T is said to be positive when trK(x r(x)) > 0 for all nonzero x e A, where 'tr K ' denotes 'reduced trace'. If A is expressed as a sum of simple ideals A — Ai © A2 © • • • © A,,, then by uniqueness of the Wedderburn decomposition, an algebra involution T induces an involution r* on the index set {1, 2 , . . . , « } by the condition that r(A ; ) = A r , (l) Ifr*(;) ^ /, it is immediate that, for any x e A,-,xr(x) = 0. Hence the positivity condition forces r* = Id; that is: Proposition 11.2: Let r be a positive involution on finite-dimensional semisimple A"-algebra .4; then there is an isomorphism of involuted /("-algebras (A, r) = (A,, n ) © (A2,
T2 )
© • • • © (An, r n )
in which each (A,, T,) is a simple positively involuted algebra.
Representation theory of finite groups
45
If D is a finite-dimensional division algebra over K, an involution a on D extends to an involution a on M,,(D) thus HiXij)) = (cr(Xji)) with transposed indices as indicated. By the Skolem-Noether Theorem, each involution on Mn{D) has this form. Moreover, a is positive if and only if a is positive. An involution r of a simple algebra A is said to be of the first kind when it restricts to the identity on the centre Z of A; otherwise, t is said to be of the second kind. A quaternion algebra ( ^ ) admits two essentially distinct involutions of the first kind, namely conjugation, c, and reversion, r, defined thus
c(x0 +x\i + x2j + xik) = XQ - x\i - x2j - x^k r(xo + x\i + X2J + x-$k) = xo+x\i - x2j + x^k Albert showed that a positively involuted division algebra (D, r), of finite dimension over Q, falls into one of four classes; here E and K are algebraic number fields: I D = E is totally real and r = 1 E ; II D = ( ^ ) , where E is totally real, a is totally positive, b is totally negative, and r is reversion; III D = ( ^ ) , where E is a totally real, a and b are both totally negative, and T is conjugation; IV D — C,,,(K, s, a), where s is an automorphism of K, of order in, whose fixed point field E is an imaginary quadratic extension, E = Eo(^/b), of a totally real field Eo, and a e E; moreover, if L is a maximal totally real subfield of K, there exists a totally positive element d € L such that
12 Examples of Wedderburn decompositions In what follows, we give the rational Wedderburn decompositions for some familiar finite groups, in particular, cyclic, dihedral and quaternionic groups.
Cyclic groups We write the cyclic group C,, in the form C, = {x\x"
= 1)
46
Stable Modules and the D(2)-Problem
denoting by Q[x] the polynomial ring in 1-variable x, we may identify Q[C,,] with the quotient Q[x]/(x" - 1). For each positive integer d we put Q.(d) = {£ e C : frf = 1 and ord(f) = d) and put
It is well known that each Q ( X ) is an irreducible polynomial over Q, so that
Q(d) = Q[x]/(cd(x)) is a field. Moreover, the factorization of x" - 1 into Q-irreducible factors is given by
x
" - ' = n c"M d\n
so that the decomposition of Q[C,,] = Q[x]/(x" — 1) into simple factors, that is the rational Wedderburn decomposition of Cn, is given by (12.1) d\n
If t,d is a primitive
The dihedral group of order 6 The smallest non-abelian group is D(,, the dihedral group of order 6. We write D6 — {\,x, x2, y, xy, x2y] with the relations x 3 = 1, y2 = 1 and yx = x2y. There are three conjugacy classes {1};
{x,x 2 };
[y,xy,x2y]
and consequently three isomorphism classes of C-representations, (V,-, p,), (1 < / < 3), with dim(V,) = d,. Then d2 + d2 + d\ = 6, so that, up to order, we may suppose that d\ = d2 = 1 and d3 = 2. Let pi be the trivial representation, then we may take
and
Pi(x)
= (°
l
V
) = (° M
Representation theory of finite groups
47
In particular, every simple representation of De is defined over Z, and hence over Q, and the rational Wedderburn decomposition is (12.2)
Q[D 6 ] = Q x Q x M2(Q)
It follows that for any field k of characteristic 0 k[D6] =k xk x M2(k)
The general dihedral group For each n > 3, the dihedral group D2n is defined by the presentation
If 72. is a commutative ring then for any n > 2, the group ring TZ[D2n] can be described as a cyclic algebra in terms of the group ring of C,, thus
where^: lZ[Cn] —*• TZ[Cn] is the involution given on group elements by
We specialize to the case where H = Q; as we have seen
d\n
Under the isomorphism Q[C n] = Q[x]/(x" — 1) £ Y\d\n Q(d)tne canonical involution^: Q[C n] —*• Q[C,,] induces an involution/^ : Q(d) —*• Q(d), which is the identity for d = 1, 2, and complex conjugation otherwise; for any n > 2, we obtain (12.3) d\n
Moreover, it is straightforward to see that [QxQ d = l , C2(Q(d), Yd, 1) = [M(Q())
d>3
Thus, for any n > 2, we obtain the following complete decomposition formula ^(QxQxn^^QW) J = { [ Q x Q x Q x Q x rid|n,d>3 A/2(Q(Mrf))
nodd « even
48
Stable Modules and the D(2)-Pmblem The quaternion group of order 8
2(8), the quaternion group of order 8, is given by the following presentation g ( 8 ) = (X, Y : X4 = 1; X2 = Y2 = (XY)2;
XY = YX3)
The elements are represented in the normal forms Xr, Xs Y where r,s = 0, 1,2, 3, and there are five conjugacy classes
{1};
{X2}; [X,X\
{Y,X2Y};
{XY, X3Y)
Clearly d2 + d2 + d2 + d\ + dj — 8, so that, up to order, we may suppose d\ = d2 = di, = d\ = 1 and d$ = 2. The four inequivalent one-dimensional representations are x\, Ti, T3, T4, where T\ is the trivial representation and
x2{X) = -\;
x2{Y)=l\
r3(X)=l; r 4(X) = - l ; In particular, each r,- is defined over Q. Over any field F of characteristic zero, (2(8) admits an irreducible representation a in the unit group (~'jr~1)* by means of
o(X) = i;
cr(Y) = j
and we get the decomposition (12.4)
F[G] = F x F x F x F x ( ~
' ~
In the cases F = Q or R, (~'p~') is a division algebra. When F = C, (~'p*~') = M2(C), giving the complex Wedderburn decomposition C[Q(8)] = C x C x C x C x M2(C)
The general quaternion group For any integer n > 3, the generalized quaternion group Q(4n) is defined by the presentation Q(4n) = (a,b\
a2" = b2,
aba = b)
Denoting the canonical generators of D2n by £, r\, there is a non-trivial central extension 1 -+ C2 ->• g(4n) ^ D2n -^ 1
Representation theory of finite groups
49
defined by the correspondence \lrn(a) = £; irn(b) = r\. This will enable us give an expression for Q[(2(4«)] directly in terms of Q[£>2«]. We denote by Q(n) the quotient
Q(n) = Q[x]/(x" + ]) and by an the involution on Q{n), namely an(xr) =
-x"~r
Theorem 12.5: For each n > 2, there is an algebra isomorphism
Q[Q(4n)] = Q[D2n] 0 C2(Q(n>, an,-\) Proof: Taking the presentation Q(4n) = {a,b \ a2n = b2,
aba = b)
we let z denote the central element z = a'" = b2, and denote by 0 the translation map 6 : Q(4n) -»• Q(4n); 6(y) = zv Then 02 = Id, so that we have a decomposition
2(4/0 = J+ e y_ as a direct sum of two-sided ideals, where J+, /_ denote respectively the +1 and —1 eigenspaces of 9. We may take bases for J+, J- as follows J+ = spanQ I arbs (
J:0
1|
It is easy to check that \[rn induces an algebra isomorphism xj/n : 7 + —> Q[Z?2,i]In fact
i/fn is automatically multiplicative, and we have an algebra decomposition
Q[Q(4n)] S Q[D2n] 0 JTo complete the proof, it suffices to show that
50
Stable Modules and the D(2)-Pwblem
Put
5 = spanQ Ia r (
J : 0 < r < n - \\
We view S as a ring in which ^ , the central idempotent of Q[ Q(4n)] generating J- as an ideal, acts as the identity. It is straightforward to see that, as rings, S = Q(n>. J- is a free module of rank 2 over S, generated by {X°, X 1 }, where X° = ( - ^ ) and X1 = b(^). However, conjugation by X' on S corresponds to the action of an on Q{n), and (X 1) 2 = - X ° . Thus
J-=C2(Q{n),
D
The Wedderburn decomposition of Q[/?2n] has already been given. We can analyse the structure of the summand C2(Q(n>, crn, — 1) further as follows; from the identities
d\2n
and
d\n
it follows by uniqueness of factorization that
d\2n,d\fi
Hence
(12.6)
Q(n)^
f[
Q(d)
d\2n,d\A
Under the isomorphism Q[Cn ] = Q[X]/(JC" — 1) = \\d\n Q(d) the canonical involution^: Q[Cn] -^ Q[C,,] induces an involution yd : Q(d) ->• Q(J) which is the identity for J = 1,2, and complex conjugation otherwise; we obtain: Corollary 12.7: For each n > 2, there is an algebra isomorphism Q[G(4n)] = Q[D2n] x
[] d\2n,d\fi
C2(Q(rf), yrf, - l )
Representation theory of finite groups
51
Writing
^
(2nr\
=
C0S
{-T)+ls
for r coprime to d, we have: Proposition 12.8: (rd - t;d)2 =
-4sin(2f^)2.
C2(Q(d), y, —1) is a free module of rank 2 over Q(£/). In the case d = 2, it is straightforward to check that C2(Q(2), y, — 1) is isomorphic to the field QA/—T. By contrast, when d > 3, C2(Q(d), yrf, — 1) is a free module of rank 4 over Q ( ^ ) . As a basis over QOd) we may take 1 = X°; i = (£/ — £/); y = X1 ; k = (frf - frf)X'. It is easily checked that ij = -ji
=k
and that i2 = -4s(df;
j 2 = -1
where 5(c?) = sini}^). We have proved (12.9) Finally, we get the complete decomposition formula for Q[Q(4n)]. Q[G(4n)] Q
x Q x nrf|B,d>3 A/2(Q(^» x Q(i) x n ^ ^ ^ ^ i S f 1 ) » o d d
Q x Q x Q x Q x FL| M > 3 A/2(Q(/*
OQ
R= Hx
where v(d) is the degree of Q(/zj) over Q.
x H
Chapter 3 Stable modules and cancellation theorems
Modules M, N over a ring A, are said to be stably equivalent when M © A'" = N © A". In the case when A is a nondegenerate Z-order, the fundamental cancellation theorem of Swan and Jacobinski gives conditions under which one may argue in the opposite direction, from stable equivalence to isomorphism. As was pointed out by Dyer and Sieradski [16], stable modules have the natural structure of a directed tree, by drawing an arrow N\ —>• N2 between modules Wi, N2 whenever N2 = N, © A. We also give an example of Swan to show that, even in the simplest case of projective modules, cancellation may fail if the hypotheses of the SwanJacobinski Theorem are not satisfied. 13 Schanuel's Lemma We begin with a basic result from module theory, which is usually known as 'Schanuel's Lemma' [58]. Proposition 13.1: LetO ->• D -4 P - i A -> OandO -» D' -4 P' -U A ->• 0 be short exact sequences of A-modules in which P and P' are projective; then D © P' = D' © P Proof: Form the fibre product Q = px p> = {(X, y)ePx /.&•
P': f{x) = g(y))
There is a short exact sequence 0-^-D'^-Q^-P-^-O where n(x, y) — x. Since P is projective, the sequence splits, so that Q = D' © P. Likewise, the short exact sequence 0-*D-*Q^*P'->0 with n'(x, y) — y also splits, since P' is projective, and Q = D © P'. Now Q = D' © P = D © P' as claimed. D 52
Stable modules and cancellation theorems
53
14 The structure of stable modules Until further notice, A will denote a Z-order. By a A-lattice, we mean a Amodule whose underlying abelian group is finitely generated and torsion free. We denote by J-(A) the class of all A-lattices. It can be regarded as a full subcategory of the category of A-modules; that is, if M, N are A-lattices Hom^ (A) (M, N) = HomA(M, N) We begin by considering the various cancellation properties possessed by Alattices. Here we are not so much concerned to investigate the detailed structure of individual modules as to draw general combinatorial conclusions from the ability to cancel in certain ways. We put Stab(.F(A)) = where the stability relation ' ~ ' is defined by M ~ N <=> M © A" = N © A'" for some m, n Say that a A-lattice M has the cancellation property when, for any N e such that rk z (M) < rkz(/V) /V © A"1 = M © A" = > N = M © A""'" It is a straightforward observation that: Proposition 14.1: If M e ^"(A) has the cancellation property, then M @ Ah has the cancellation property for each b > 0. We shall say that the module M e J-(A) has the weak cancellation property when M © A has the cancellation property. It follows trivially from (14.1) that: Proposition 14.2: If M e !F(A) has the cancellation property, then M has the weak cancellation property. The converse to (14.2) is definitely false. Specific counterexamples will be discussed in Section 17 and Section 18. We shall say that a A-lattice M is minimal, when rk z (M) < rk z(A0 for all N e [A/]. Proposition 14.3: Let M e .F(A); then the following conditions are equivalent: (i) (ii) (iii) (iv)
every non-minimal module N e [M] has the cancellation property; every module /V e [M] has the weak cancellation property; every minimal module Mo € [M] has the weak cancellation property; at least one minimal module Mo e [M] has the weak cancellation property.
54
Stable Modules and the D(2)-Problem
Proof: We first show that (i) ==> (ii). Thus suppose (i) holds; if N e [M] is non-minimal, then, by hypothesis, it has the cancellation property and hence the weak cancellation property by (14.2); if N is minimal, then N ® A, being nonminimal, has the cancellation property. Either way, N has the weak cancellation property, which is the desired conclusion. Moreover, it is clear that (ii) =>• (iii) ==>• (iv). Thus we must show that (iv) =>• (iii) = > (ii) =>• (i). (iv) =>• (iii) Suppose that Mo, N e [M] are both minimal, and that Mo ® A has the cancellation property. We must show that N ® A has the cancellation property. Now rkz(A') = rk z (M 0 ), and since N and Mo are stably equivalent it follows that
M0®Aa = N®Aa for some a > 0. If a — 0, 1, there is nothing to prove; if a > 2, then Mo 8 A 8 A 0 " 1 = N 8 A 8 A 0 " 1 so that, since Mo 8 A has the cancellation property M0
e
A
=
N
®A
and so N ® A has the cancellation property as required. (iii) = > (ii) Suppose that Mo e [M] is minimal and that N e [M] is nonminimal, then rkz(Mo) < rkz(A0, and, since A' and MQ are stably equivalent N 8 A" = Mo 8 Ab for some a, b with a < b. If a = 0, there is nothing to prove. So suppose that 1 < a, then 1 < a
for some b, c with 0 < b. By hypothesis, Mo 8 A has the cancellation property, so that Mo 8 Ab has the cancellation property by (14.1). Hence N = Mo 8 Ah, and so Af also has the cancellation property. This completes the proof. a
55
Stable modules and cancellation theorems
If M e F(A), we say that the stable module [M] has the weak cancellation property when any, and hence all, of the four conditions of (14.3) are satisfied. From the proof of (14.3) we may draw the following conclusions: Proposition 14.4: Suppose that the stable module [M] has the weak cancellation property for M e F(A), and let Mo e [M] be minimal: (i) if N e [M] is minimal, then N © A = Mo © A; (ii) if N e [M] is not minimal, then N = Mo © A" for some n > 1. When M e .F(A), the stable module [M] has a natural combinatorial structure which we represent as a directed graph in which the vertices are the modules N e [M], and where we draw an arrow A' —> N © A. It is clear that there are no loops in the corresponding undirected graph, so that [M] is a tree. In the cases which interest us, the number of prongs is finite and the structure of the graph is more specific. Represent the natural numbers N as a tree with one end in which there is no branching away from the main stem. By a fork, we shall mean a tree T for which there exists a surjective map of graphs A.T : T -> N with the property that for all r > 1, the segments [r, r + 1] in N are covered precisely once. The name 'fork' conveys the essence of the definition, namely, that there should be no branching above level 1. It is easy to check that, for any fork T there is a unique surjective 'level function' T —>• N, so that Xj is intrinsic
(14.5)
• T
0
N
Proposition 14.6: Let M e F(A); then the stable module [M] has the weak cancellation property if and only if [M] is a fork.
56
Stable Modules and the D(2)-Problem
Proof: The implication (==>) follows from (14.4), and the converse is trivial. • In the special case when [M] = N we say that [M] is straight
(14.7)
[M] = N
The stable module [M] is said to have the strong cancellation property when every N e [M] has the cancellation property. A straightforward chase of definitions now shows: Proposition 14.8: Let M e ^(A); then [M] has the strong cancellation property if and only if [M] is straight. In Chapter 9, we shall encounter a number of important cases where stable modules do indeed have the strong cancellation property, and whose underlying tree structures are thereby straight.
15 The Swan-Jacobinski Theorem A is now assumed to be a nondegenerate Z-order. It is convenient to regard A as imbedded in AR = A
Let V; be a simple right ideal in A,-. Then any simple module over AR is isomorphic to V,- for some unique index i. Putting £),• = EndAR(V,), we see that Dj is a finite-dimensional division algebra over R, so that, in particular, D, is isomorphic to one of R, C or H = (~'^~1). Now suppose that M is a A-lattice and put MR = M ® R, so that MR has an isotypic decomposition
M R ^ v, (/l) © • • •
Stable modules and cancellation theorems
57
where, to allow for the case that some multiplicities might be zero, we adhere to the convention that V(0) = 0. Note that the ring EndAR(A^R) is also a semisimple algebra, and decomposes as a direct sum of two-sided ideals, thus EndAR(MR) = £ , © • • • © £ „ where £>,• = Mf. (£>,) is the ring of f; x f; matrices over D,; here, of course, the convention takes the form that M0(D) = 0. We say that M is an Eichler lattice when no simple factor Bj is isomorphic to H. The following special case of the Swan-Jacobinski Theorem will suffice for our purposes: (compare [14], vol. 2 (51.28), p. 324). Theorem 15.1: Suppose that M is an Eichler lattice over A with the property that M = Mo © A for some module MQ. Then M has the cancellation property. From the point of view of cancellation properties, it is convenient to recast the definition. If M is a A-lattice, we shall say that M is pre-Eichler when M © A is Eichler. The Swan-Jacobinski Theorem then implies the following, which is the conclusion from this discussion that we use most frequently: Theorem 15.2: Let M be a A-lattice; if M is a pre-Eichler lattice, then M has the weak cancellation property. The question of whether a A-lattice M satisfies the Eichler condition depends only upon the isomorphism type of M
where each V, is simple over AR and V, ^ V) for i ^ j . Moreover, the isomorphism types of V\,..., Vn, and their multiplicities e\,..., en, are uniquely determined up to order. We say that the order A satisfies the Eichler condition when, considered as a A-module, A is an Eichler lattice. Clearly one now has: Proposition 15.3: Let A be a semisimple order and let A R = A, ©•••©A« be the Wedderburn decomposition of AR into a direct sum of simple-two-sided ideals. Then A satisfies the Eichler condition if and only if no simple two-sided ideal A, is isomorphic to H. From the above discussion, we see immediately that: Proposition 15.4: If A satisfies the Eichler condition, then every M e satisfies the Eichler condition.
58
Stable Modules and the D(2)-Pmblem
One may express the Eichler condition for A more explicitly; if the Wedderburn decomposition of AR takes the form A R = Mdl (R) x • • • x M ^ R ) x Mei (C) x • • • x Meb(C) x Mu (H) x • • • x M/r.(H) then A satisfies the Eichler condition when either c = 0 or each f, > 2. Under the hypothesis that A satisfies the Eichler condition, it follows that each M e F(A) has the weak cancellation property. Though satisfyingly general, this statement is, however, of limited utility, since in some cases, which we shall encounter, although A itself may fail the Eichler condition, nevertheless the lattices of most interest to us may still possess the weak cancellation property. To see how this may arise, we need to analyse the situation further. Let
A R = v,(ai) e • • • e v™ be the decomposition of AR into isotypic modules; we say that the simple module V, is quaternionic when D, = EndAR(V,) = H; V, is then said to be good when a-, > 2; otherwise, when a, = 1, Vj is said to be bad. When M is a A-lattice we write MR = M
where we adhere to the convention that V(0) = 0. In this case, EndAR((A7ffiA)R ) decomposes as a direct sum of two-sided ideals, thus EndAR((Af 0 A)R ) 2S A\ ® • • • ® A where Ai is the ring Ai = Ma.+hj(Di) of (a, + b,) x (a,- + /?,) matrices over D,i. Then M © A satisfies the Eichler condition precisely when a, +b, ^ \ for each simple quaternionic module V,. Since a, > 1 and fo, > 0, the possibility that a, + bj = 0 does not occur, and M ® A satisfies the Eichler condition precisely when a, + b; > 2 for each simple quaternionic module V,-. This is automatically satisfied when V, is good; when V,- is bad, we have a,- = 1 and for such modules we require b, > 1, that is: Theorem 15.5: Let A be a semisimple order and let M e T{A); then M has the weak cancellation property provided either (i) A satisfies the Eichler condition or (ii) each bad simple quaternionic AR-module has multiplicity > 1 in MR. We say that A has the cancellation property property for free modules when M © A = A(k) © A =$• M = Aik)
Stable modules and cancellation theorems
59
Since any finitely generated free module belongs to the stable module [A] containing A itself, it follows easily from (15.3) and (15.5) that: Proposition 15.6: The following conditions are equivalent for any semisimple order A: (i) (ii) (iii) (iv)
A has the cancellation property for free modules; the module A has the cancellation property; the stable module [A] has the strong cancellation property; each finitely generated stably free module over A is actually free.
From this we see that: Proposition 15.7: If A has the Eichler property, then A has the cancellation property for free modules. 16 Finitenessof AT0(Z[G]) In connection with the stability relation, there is a classical invariant of rings obtained from equivalence classes of projective A-modules; thus let P(A) denote the collection of finitely generated projective modules over A. Then P(A) is closed with respect to direct sum. Consequently, the set P(A)/ ~ of stable classes in P(A) forms a commutative monoid with addition derived from ©
in which the class of any free module represents zero. Since for any projective P there is a projective Q such that P © Q is free, this monoid is actually a group, the reduced projective class group of A, and denoted classically by C( A), but nowadays, in view of its status within algebraic K-theory, more usually by A'o(A). For any finite group G, K0(Z[G]) is finite. This was first proved by Swan in [57]. Here we content ourselves with pointing out the main landmarks in the proof. The main result needed is Swan's fundamental observation that projective modules over Z[C] are locally free. We shall subsequently need to use a special case of this, which we note now: Theorem 16.1: Let G be a finite group, and let P be a finitely generated projective module over Z[G]; then for some S > 1 P ® Q = Q[G]S For a proof see [57], and also [14], Vol. I, Section 32.
60
Stable Modules and the D(2)-Problem
Swan also gives (see [59]) the following representation theorem for projective modules over a semisimple order: Theorem 16.2: Let A be an order in a semisimple Q-algebra; then each finitely generated projective module P over A can be represented in the form P = J © A" where J is a projective ideal in A. Nowadays, (16.2) is proved using Swan's general local freeness theorem in conjunction with the Swan-Jacobinski Theorem. In particular, it applies in the case A = Z[G] where G is finite. An immediate consequence is that any class in KQ(Z[G]) can be represented by a module J satisfying rkz(V) = \G\. However, by the Jordan-Zassenhaus Theorem ([14] Vol. I, Section 24) there are only finitely many isomorphism classes of Z[G]-lattices of a given finite rank. As an immediate consequence we have: Corollary 16.3: If G is a finite group, then the reduced projective class group F 0 (Z[G]) is finite. This has the following obvious but useful consequence: Corollary 16.4: If P is a finitely generated projective module over Z[G], then for some n > 1, P ( n ) = P ® • • • © P is stably free over Z[G]. n
Although we make no use of the fact, we note that in the case where A is a maximal order every A-lattice is A-projective ([14] vol I (26.12)).
17 Non-cancellation and an example of Swan Let A be a ring. We say that two right (resp. left) ideals /, J of A are coisomorphic when A / / and A / 7 are isomorphic as right (resp. left) A-modules. Elementary considerations show that isomorphism does not imply coisomorphism; for example, when A = Z, any two nonzero ideals are isomorphic, whilst co-isomorphic ideals are necessarily identical. In general, co-isomorphism does not imply isomorphism either. Nevertheless a weaker statement is true. Say that two ideals /, J are stably isomorphic when / © A = J © A as A-modules. Then Schanuel's Lemma gives immediately (17.1)
co-isomorphic ideals are stably isomorphic.
We describe in detail an example, due to Eichler-Swan, of co-isomorphic left ideals which are not isomorphic. First let A be the maximal order constructed
Stable modules and cancellation theorems
61
in Chapter 1, Section 8. To recall the details briefly, put K = Q(£ +1,), where £ = exp(^-). Then K is a totally real number field of degree 4 over Q whose ring of integers R takes the form R = Z[x]/(x4 - 4x2 + 2) Let A be the quaternion division algebra /-l.-l * = ( Then A = span^fl, a, (1, aft], where -f+iT :
/> =
We have seen already that A is a maximal order in A. Observe that 17 factorizes in R, thus IV = P\P2P3P4 where pi = 1 +
2T;
/?2 = 1 — 2f; /?3 = 1 - 2r; p 4 = 1 + 2f
Likewise, —1 + 4y factorizes in A as — 1 + 4y = «ia2«3«4 where a{ = \
; :: —
; a22 = 1
T
; «33 = 1 HH T
;; T
; «4
1H H T
In fact each a,- e A as we have a, = ( l - 2 r + r 3 ) + ( l - r 2 ) J 6 a2 = ( l + 4 r - r 3 ) - / 6 a3 = ( 1 + 2 T - T
3
) + (T2-1)/3
04 = (1 - 4 r + r 3 ) + ^ Proposition 17.2: pt e A (a,) for each /. Proof: Ifr) — rjo + r)\j, where rjj e K, we write ij = rjo — r/ij. We may express this in terms of ft rather than j thus: write f — fo + ?i$, where f, € A"; then | = fo + (4T - T3)$, - f,^
62
Stable Modules and the D(2)-Problem
In particular, if £0> fi € R, then both § and f e A. We define M, e /? for / = 1,2, 3,4 by u, = 3 + 4r-r 2 - r 3 ; -•1
U2 =
2
-r3;
-4r-r 2 + r3;
«3=3 UA
+ 2 T + :I
-•1 - 2 r + 7:
=
2
+ T3.
A straightforward computation reveals that, for each i /?, = (uidi)aj
and this completes the proof.
D
For any rational prime q, we denote by F a the field with q elements. Proposition 17.3: /?/(/?,) = F17 for r = 1, 2, 3,4. Proposition 17.4: A/(/?,-) = M 2 (F, 7) for i = 1, 2, 3, 4. It follows that any two simple modules over A/(/?,) are isomorphic. One may be more precise; if V, is a simple module over A/(/?( ), then dimR/(Pi.)(V;) = 2, and: Proposition 17.5: Let £/,• be a nonzero module over A/(/?,); then dimRnPi)(Ui)
< 4 «=> (/,- =A/(Pi) V;
Since /?,• e A(a,), then A/A(a,) is a module over A/(/?,-), and hence over R/(Pi). The sequence of A-submodules A(p,0 C A (a,-) C A gives an exact sequence of A-modules 0 -» A{ai)/A(Pi)
-> A/A(p,-) -> A/A(a,) -» 0
However, the inclusion A(a,) c A is proper, so that A/A(a,) ^ 0. Likewise, the inclusion A(a,-)/A(/?,-) is also proper, so that A(a,)/A(/7,) is also nonzero, and, consequently dimR/(p,)(A/A(a,-)) < dimR/(Pi)(A/A(pj)) Thus from (17.5), we see that:
=4
Stable modules and cancellation theorems Proposition 17.6: A/A(a,)
=A/( PI-)
63
Vj.
We define fii =
A(fa-p,p<x-i)
From the identity pi = (r 2 — l){a(fa — /}) — (/3a — f)}, we see that /?i e fi|. Proposition 17.7: £2| and A(ai) are co-isomorphic left ideals in A; in fact, we have A / Q , = A / A ( a i ) = Vi Proof: The sequence of A-submodules(/?i) c S2i C A gives an exact sequence of A-modules
However, the inclusion Q\ c A is proper; for example, a g Si{, so that A / Q | ^ 0. Likewise, since P g (p\), the inclusion (p\) C S2i is also proper, so that Q\/(p\) is also nonzero, and, consequently dim R / ( / ) | ) (A/^i) < dimfi /(pl)(A/(p1)) = 4 Hence from (17.5) we see that A / Q | = V\. The result now follows since, by (17.6), A/A(fl,)= V,. D FromSchanuel'sLemma, weseethatQi ©A S A(ai)©A.Moreover,since A(a\) = A, we also obtain (17.8)
Q, © A = A ® A
However: Proposition 17.9: £2\ ^ A as a left A-module. Proof: Put Q.2 = A(f, /3), and observe that the mapping
is an isomorphism of A-modules. It suffices, therefore, to show that Q2 is not isomorphic to A. Putting £ = {a e A : £22v C ^2} it is clear that E is an order in A. If ^2 = A, then there must exist exist x e £2 such that ^2 = Ax. If a € £ , then, in particular xa €
AJC
64
Stable Modules and the D(2)-Problem
Now x is necessarily nonzero, so JC"1 exists in A, since A is a division algebra. Thus a e x~'Ax, and £ c JC-'AJC. However, it is evident that x~] Ax C £. Thus ~L(Q.) — x~l Ax = A. If 0 c A is the maximal order of Section 8, it is easy to check that 0 c £, and, since 0 is a maximal order, it follows that 5 = ©. However, 0 ^= A by (8.7), so the result follows from the argument above. • Observe that one may produce examples of co-isomorphic right ideals by applying a suitable anti-involution. A convenient choice is the anti-involution 6 : A ->• A given by 6(x0 + x\i + x2j + x3k) = xo—xii+
x2j - x3k
Since 8(j) = j , we see that 6{a\) = a\ and 6(0) = ft. Putting A = 0(A), we see easily that A is a maximal order in A, and that: Proposition 17.10: The right ideals (ai)A and (r, /6)A in A satisfy: (i) (fli)A © A = (r, 0)A@ A and (ii)
18 Non-cancellation over group rings Swan's example to show non-cancellation over a semisimple order can be elaborated to show non-cancellation holds over some integral group rings. The first case discovered was Z[<2(32)]. First recall the structure of Q[Q(4n)]; for each n > 2, there is, by (8.5), an algebra isomorphism Q[G(4n)] = Q[D2)!] 0 C2(Q(«>, crH, - 1 ) When n is even
where /u,,/ = & + t,d, and fa — e~r, and where s(d) — sin{^j). In the case n — 8, there is a single factor, and we have
A: where, continuing with the notation of Section \1, K = Q(/ii6)- However,
Stable modules and cancellation theorems 2
65
= 2 + V2 = r 2 is a square in K. Thus Q[G(32)] = Q[D, 6 ] 0
Under this product structure, Z[Q(32)] imbeds in Z[D\(,] x A, where A c (~y~') is the maximal order of Section 17. Moreover, if F c A is the image of Z[g(32)] under projection, then T = Z[Q(32)]/(y2 + 1) Since the reduced discriminants of the Z-algebras Z[Q(32)] and Z[D\(,] are both powers of 2, and since the reduced discriminant of A is — 1, it follows that the index of F in A is also a power of 2. Let J c Z[Q(32)] be the left ideal J = Z[Q(32)](y + 4), and put T = Z[Q(32)]/J. In Z[(2(32)] we have -255 = / - 256 = (y2 + \6)(y + 4)(y - 4) so that J has finite index in Z[g(32)]. The prime factorization of 255 is 255 = 3 - 5 - 1 7 so that T decomposes as T = M © N, where M consists of 17-torsion, and N is a direct sum of 3 and 5-torsion. Since y2 is in the centre of Q(32), y2 s 16 = (-4) 2 on Z[Q(32)]/J. Since 16 = — l(mod 17) and 16 = l(mod q), when q — 3, 5, we see that M and A'are respectively the — 1 and +1 -eigenspaces of y2 in T. It follows that M is a module Z[Q(32)](y2-\). overF = Z[£(32)](.y 2+1) and N is a module over Z[£>,6] = 2 Since T is a quotient of Z[Q(32)](y - 1), it follows also that M is a quotient of F. Moreover, since the index of F in A is a power of 2, and hence coprime to 17, we may identify M with A/A(y + 4), and so regard M as a module over A. Likewise TVis a quotient of Z[D]e]; we denote the respective projections by itM : A ->• M and nN : Z[D\(,] -*• N. We saw in Section 17 that there is a factorization in A -1+4;
-a\a1al!aA
where
. a\ — 1
0' ~ ])
, ; a2 = 1
0' ~ D
, , 0-1) ; a3 = I -\
, , (J ~ 1) ; a4 = 1 H
r r f r Since - 1 + Aj — j(j + 4), we see easily that M = A/A(j + 4) = A/A(a,) © A/A(a 2 ) © A/A(a 3 ) © A/A(a 4 )
66
Stable Modules and the D(2)-Problem
Let n,: : A —*• A/A(a,) be the projection, so that TTM = it\ x ^2 x ^3 x ^4However, we have seen that A/ A{a\) = A / f i | . Let n[ : A -> A/^2, and put 7r^ = JT[ x 7T2 x 7T3 x 7T4; then 7r^ is also a projection TT'M : A —> M Finally, we denote by 7r : Z[Q(32)] -*• T = M © N the canonical projection and by TT' : Z[Q(32)] ->• M © N the restriction of n'M x nN : A x Z[£>i6] ->• M®N to Z[Q(32)] (identified with its image in A x Z[D|6] under the canonical imbedding). Again since the index of V in A is coprime to the exponent of M, n' : Z[g(32)] -> M © N is surjective. Put 7 = Ker(jr')- Then by Schanuel's Lemma Ker(Tr') © Z[Q(32)] S Ker(Tr) © Z[g(32)] However the mapping Z[g(32)] -*• Ker(7r); x i->- x(y + 4) gives an isomorphism Ker(;r) = Z[G(32)]. Thus
* © Z[fi(32)] = Z[fi(32)] © Z[fi(32)] where * = Ker(7r')- However, * is not isomorphic to Z[Q(32)] since by extending scalars to A we get * ®z[g(32)] A = £2i and Qt is not isomorphic to A. The above example of non-cancellation is the original one given by Swan [59]. As before, one obtains a corresponding statement for right ideals by applying an anti-involution, for example the canonical anti-involution, on Z[Q(32)]. It follows from the Swan-Jacobinski Theorem that the presence of a quaternionic order is an essential feature of non-cancellation. Perhaps the most basic cancellation property is the cancellation property for free modules. Subsequently, Swan ([63]) gave a systematic treatment of non-cancellation in integral group rings, in which he shows there are precisely seven exceptional binary polyhedral groups which do possess the cancellation property for free modules, namely the binary tetrahedral, octahedral, and icosahedral groups T*, O*, /*, and four quaternion groups Qs, <2i2, 2i6, 620- We say that the remaining binary polyhedral groups are typical. Swan shows that a finite group G which has a typical binary polyhedral group as a quotient fails to possess the cancellation property for free modules. The converse is not true; for example, Q% x Ci fails to possess the cancellation property for free modules, and this is the smallest such example. We shall consider non-cancellation phenomena again in Chapter 9.
Chapter 4 Relative homological algebra
In [83], Yoneda showed how to formulate the classification of n-fold module extensions in cohomological terms. Here we present a version of Yoneda's Extension Theory which is appropriate for our problem. We begin by introducing the notion of a tame class. This, approximately, is a class of modules closed with respect to short exact sequences, which contains all finitely generated projective modules, and relative to which projective modules are injective. From the outset, we formulate matters within the 'derived module category' of a tame class C; that is, the quotient category obtained by setting projective modules equal to zero. The derived category approach permits the explicit construction of a sequence of 'derived functors' D,, : Per(C) -> £>er(C) for n > 0: M and D,,(M) are connected by an exact sequence 0 -)• Dn (M) -> />„_, -*
• Po -* M - • 0
where each Pr is projective. Cohomology is then introduced as the 'nth derived functor' of Horn by means of H"(M, N) = HomVer(Dn(M),
N)
We show that this formulation is equivalent, for a tame class, to the the traditional Eilenberg-Maclane definition via the homology of the chain complex obtained by applying Horn to a projective resolution. The above then becomes, in effect, a 'corepresentation formula'. The relative injectivity of projectives shows that each D,, is a self-equivalence of categories, and has an inverse functor D_,, which gives a corresponding 67
68
Stable Modules and the D(2)-Problem
'representation formula' U\M,
N) = H o n W M , D_B(A0)
Yoneda's Theorem can also be expressed, relative to such a projective n-stem, as follows P = (0 - • D n (M) ->?„_!-»•
• Po -> M -* 0)
an arbitrary extension A = (0 -* J -> An_! ->
> Ao ->• M -* 0)
with Ar e C is classified by EndDer (/) = The objects of the derived category can be identified with 'hyper-stable modules' ; that is, with equivalence classes of modules M e C under the relation M ~ ~ N <=>• M ® P\ = N © P2 where P\, P2 are finitely generated projectives. For later applications, this relation is too coarse for us, the appropriate stability notion being that previously considered in Chapter 3, namely M ~ ,/V M © Fx = N 0 F2 where F\, F2 are finitely generated free, or, what is equivalent, stably free. It is thus necessary to refine the construction M M> D,,(M). TO any module M e C, we associate a well-defined stable module Qn(M) (n > 0) by requiring that M is connected to some J e £2,,(M) by an exact sequence F = (0 - • J -> Fn_i ->
>• F o -^ M -> 0)
where each Fr is finitely generated free. When G is a finite group, the stable modules Qn(Z) over Z[G] will subsequently assume a primary significance.
19 The derived category of a tame class A will denote an associative ring with unity. We say that a A-homomorphism / : M\ -> M2 factors through a projective module, written ' / % 0', when / can be written as a composite / = /J o a, for some projective module P, and some A-homomorphisms a : M\ ->• P and /} : P ->• M2. The relation ' « ' is additive; that is: Proposition 19.1: Let f,g:M^-Nbe g « 0, then / + g % 0.
A-homomorphisms; if / % 0 and
Relative homological algebra
69
Proof: Let / = a o /} be a factorization through the projective P and g = y o S be a factorization through the projective Q; then
is a factorization of / + g through the projective P ® Q.
•
Proposition 19.2: Let / : M -* N be a A-homomorphism; if / % 0, then -/«0. We extend «»to a binary relation on HOITIA(M, A') by means of f za g <^=> f -
g
%0
With the definition so extended, % is an equivalence relation. Moreover, it is compatible with composition; that is: Proposition 19.3: Let / , / ' : Mo -> M\, g, g' : M\ -»• M2 be A-homomorphisms; if / « / ' and g % g', then g o / % g' o / ' . For any class C of A modules, there is a well-defined category, the derived module category Der(C) of C, whose objects are (right) A-modules, and in which, for any two objects M, N, the set of morphisms Homi>er(C)(M, N) is given by Hom 0er (M, N) = HomA (M, N)/ % It should cause no confusion to call 'Der(C) simply the 'derived category' of C (see note at end of Section 22). The additivity property of % shows that: Proposition 19.4: For any class C, and any M,N e C, HomDerCoC^* N) has the natural structure of an abelian group. Suppose that C is a class of A-modules. A module J € C is said to be injective relative to C when any short exact sequence of the form
with M\, M2 € C splits. The class C is said to be tame provided it is closed under isomorphism, that is, if M = M' and M e C, then M' e C, and it also satisfies the following properties T(0) — T(4): T(0): Each M e C is finitely generated over A. T(l): C contains all finitely generated projective A-modules.
70
Stable Modules and the D(2)-Problem
T(2): If 0 —^ K —> M —>• <2 —> 0 is an exact sequence of A-modules and Q e C, then M e C <=> A: e C T(3): If M € C, then M is injective relative to C 4> M is projective. T(4): If M € C, then there exists an exact sequence of A-modules ()->• M -* F - » L ^ 0 where L e C and F is finitely generated free over A. A module J e C is said to be strongly injective relative to C when, given any short exact sequence of the form 0->K-!*L-!+M-+Q with K, L,M eC, and any A-homomorphism
be an exact sequence in C, and let q> : D —*• J be a homomorphism. We must produce a homomorphism
: ^ D )
of L ® J; then the sequence
is exact, where L' = {L © 7 ) / A , 7 : J ->• L' is the homomorphism y(-«) = [('(x), 0], and 7r : L' -» M is the homomorphism ^ I j . t f ] = p(y)
Relative homological algebra
71
put
M
-¥
^ y4
M
-•
L'
4
and the fact that po j — Idy, it follows immediately that q> = <J> o i as required, proving ( = > ) . (•<^) If J is strongly injective relative to C and
is an exact sequence in C then the mapping p : L -> J extending Idy : J -*• J splits £ on the left, showing that J is injective relative to C. This completes the proof. • In the derived category of a tame class C, it is not necessary to postulate that the projective modules through which morphisms are factorized need to be finitely generated. Moreover, since projective modules are direct summands of free modules which are also projective, the condition ' / % 0' is entirely equivalent to the requirement that / : M\ -> M2 factors through a free module. In summary, we have: Proposition 19.6: Let M be a finitely generated A -module, and l e t / : M ->• N be a A-homomorphism; then the following conditions are equivalent: (i) (ii) (iii) (iv)
/ / / /
factors factors factors factors
through through through through
a projective module; a finitely generated projective module; a free module; a finitely generated free module.
We extend these considerations to exact sequences; denote by Extg the collection of exact sequences of A-modules and homomorphisms E = (0 -> E+ -> Eo -» £_ -> 0) in which the modules E+, Eo and £_ are all in C; Ext^. can be regarded as a category by taking morphisms to be commutative diagrams of A-homomorphisms thus
72
Stable Modules and the D(2)-Problem
The projection £_ -» (A
£L
w_ £'
defines a functor <w_ : Ext^. C. When A, B e C, ExtUA, B) will denote the full subcategory of Ext£ whose objects E satisfy E- = A and E+ = B. Let Projc denote the full subcategory oof Ext
M ^ 0)
in which D, P, and M are objects in C, and where P is projective. Any M e C admits such a projective cover. The following proposition is fundamental: Proposition 19.7: Let
\
KA X
M'
be a morphism in Ext£, in which P is projective; then
Proof: Consider the special case where /i_ = 0; then \mQi) c Im(7) and /i + admits the factorization D -V P -> A" through the projective module P, where A. = j~xh. In general, suppose that /i_ = a o /3 M where 2 is projective. By the universal property for Q, since /? is surjective there exists a homomorphism a : 2 -> X such that a = r\ o a. Put ^ = a o / J o e i P - ^ X s o that the following diagram commutes
4, 0
Relative homological algebra
73
Consequently, we have a commutative diagram
ih+
o
A:
ih-h
-4
10
XA
so that by the above special case, h+ :
M'^ K factors through a projective. D
We also have the dual statement: Proposition 19.8: Let K -+ 0
4- M ->• 0 /
be a morphism in Ext^, in which P is projective; then /z_f- ^ 0
?> / z _ ^ 0
Proof: Consider the special case where h+ = 0; then, since /j|im(y) = 0. there exists a unique homomorphism A. : M' ->• P such that/z = A. o 77. Then/i_ 0/7 = e o / j = e o X o rj so that /i_ = e o A. since rj is an epimorphism. However, e o A. is a factorization through the projective P, which is the desired conclusion. In general, we may only assume that h+ factorizes as h+ = \x o A., thus
where Q is projective. Since Q is strongly injective relative to C, there exists a homomorphism v : X -» Q such that A. = v o /, thus
Put h — jo/Mov:
X->- P; then o / — j o fxo v o i = j o 1x0X
= J ° h+
74
Stable Modules and the D(2)-Problem
Moreover, rj o j — 0 implies that r\ o h — 0, so that the following diagram commutes
'O^KM
M'^A
X\
4.0
ill- h
D4
/>4
M ^
0
The conclusion h- » 0 follows by the above special case.
•
Taking (19.7) and (19.8) together, we obtain: Corollary 19.9: Let C be a tame class, and let
4, /J+
4- h
\, h-
be a morphism in Proj,!.; then /i + »0
20 Derived functors When C is a tame class, its derived category admits a sequence of functors, the so-called 'derived functors' D,, : £>er(C) -+ Der(C) which we now describe. In the simplest case, if 0 —> D —> P —> M —y 0 is a projective cover and f : M —*• M' is a A-homomorphism, then by the universal property for projective modules, for any short exact sequence
there exists a A-homomorphism / : P ->• X making the following diagram commute
-
M'
Relative homological algebra
75
We thus get a commutative diagram 0->
D -s M D^ M^ 0 iw+(f) if if
0
M' ^ 0
K
where a> + (/) is the restriction to D. The existence of projective covers shows that &>_ : Projg -»• C is surjective on objects. Moreover, if / : M -> M' is a morphism in C, and 7-\ Q are any objects in Proj^ such that a>_CP) = M and (O-(Q) = M', then there exists a morphism / : V -> Q in Proj1 such that « _ ( / ) = / ; that is: Proposition 20.1: &>_
C is an epifunctor.
We construct a functor D| : I?er(C) -*• Ver(C) in the following way: for each M e C w e make a definite choice of a projective cover VM = (0 -» D,(M) -> P M - • M - • 0) For each morphism / : M lifting / ; that is
N in C there exists a morphism / : VM ->
0 / =
if 0
Suppose that / : VM -* VN is another morphism lifting / ; on considering the difference / - / , it follows from (19.7) that &>+(/) - co+(f) % 0, and that the class of [«+(/)] e Der(C) is uniquely determined by that of / . Thus the correspondence M ->• D](M); f h-» [&;+(/)] determines a functor Di : I>er(C) -> Der(C). Changing the projective cover used to define D| does not change the isomorphism type of Di(M) within 2?er(C); it does, however, change the isomorphism by which the identification is made. Thus suppose that VM = (0
Df
P -• M
QM = (0
Dp
Q ->• M ->• 0 )
and
are both projective covers of M, then there is a morphism Id :
QM
76
Stable Modules and the D(2)-Problem
lifting Idw; that is (Q
_> Df -»• P -> M -* o\ 4- a P g J, Id I Id
Id =
-* Dp -* Q -* M -> It is easy to check that, in the particular case where Q = V, one has a-p-p % Id. It follows that: Proposition 20.2: a-pQ is an isomorphism in the derived category. Proof: a-pQ o agp = a-p-p % Id and likewise ctgp o a-pQ « Id.
•
We also have a functor o>+ : Proj^. -—>• C given by D D'
0 -»• D ' -»• P ' - • M ' -»• 0
If Af' is a A-module then, by property T(4), there exists an exact sequence in C of the form 0 -+ M' -» P'
0
with P' projective. Suppose that M, M' are A-modules and that / : M —> M' is a A-homomorphism. Let
be an exact sequence in C with P' projective and let
be any short exact sequence in C begining in M. By the strong relative injectivity of the projective module P', there exists a A-homomorphism / : X -> P' making the following diagram commute P' jof
f
X
M
Relative homological algebra
11
We get a commutative diagram O^M^>X4
4/
Q ->
0
1/
0 -> A" -4 P' \
Q' -> 0
We see that: Proposition 20.3: a>+ : Proj' (C) -> C is an epifunctor. We construct a functor D_i : £>er(C) —>• "Der(C) in the following way: for each M e C choose an exact sequence in C VM = (0 - • M -> P -»• C -*• 0) with P projective, and for each A-homomorphism f : M -*• N choose a specific morphism / : VM -*• VN such that co+(f) — / ; then define
and D_,(/) = [«_(/)] It is straightforward to see that on the category Ver(C) we have D| o D_| % D_i o D | % Id that is: Proposition 20.4: Let C be a tame class; then up to a natural equivalence, the functors Di,D_i : X>er(C) ->• Vtr(C) are mutually inverse additive equivalences of categories. For any integer n > 2, we obtain functors D,,, D_,, : Ver(C) -> Ver(C) as follows Dn = D, o D i o
oDi
and D_,, = D _ , oD_i o
oD_i
78
Stable Modules and the D(2)-Problem
Obviously D,, and D_,, are also additive self-equivalences of the derived category £>er(C). It follows that for all M,N eC ), D,(A0) = HomVer(M, N) = In particular, we obtain the following adjointness formula (20.5)
Homper(D,,(M), N) = HomVer(M, D_n
The rcth-cohomology group TC"(M, N) of M with coefficients in TV is defined thus (20.6)
H"(M, N) = HomDer(D,,(M), N)
From (20.5) and (20.6) it is clear that (20.7)
W"(M,A0
Though not essential to our arguments, it is true that, for n > 1, cohomology defined in the above way for modules in a tame class coincides with the standard Eilenberg-Maclane definition in terms of projective resolutions. The details are given in Section 27 below. With this interpretation, (20.6) above is a Corepresentation Formula, and (20.7) is the corresponding Representation Formula.
21 The long exact sequence in cohomology Let C be a tame class of A-modules, and fix a short exact sequence £ within C
It is straightforward to show that, for any A-module M, the sequence 0 -» HomA (M, A) 4 HomA (M, B) -5- HomA(M, C) is exact. Passage to the derived category yields a more restricted statement: Proposition 21.1: For any A-module M, the sequence Hom Per (M, A) 4 H o n W M , B) 4 Hom 0er (M, C) is exact.
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Proof: Clearly pj* = 0. Let p e HomA(M, B) and suppose that p*(P) «* 0; it suffices to show that there exists a e HomA(JW, A) such that P «s /*(a). That is, we have a commutative diagram as follows, where P is projective B
M
Since p is surjective, there exists X : P —> B such that pok — x; thus po p = TOO, and hence p o ^ — p okoaAx. follows that we get a mapping P - X o a : M -> Ker(/?) = Im(i) Put a = i~l(P — Xoa) so that a : M ->• A. Then /3 - i*(a) = A.ocr. However A. o a is a factorization through the projective P, and, as required, p ss /*(a) for some a e Hom A(M, A). D It is not in general true that/, : Honvper(M, A) —>• HomDer(A/, 6)isinjective. This will become clear (see (21.5) below) as we proceed to extend the above exact sequence to the right. Within C, we fix a module M and a projective cover V = (0 —> D —> P —*• M —*• 0); any A-homomorphism y : M —> C admits a lifting to a morphism of exact sequences V —> £ of the following form
and, as we have already seen, the correspondence y i->- ai + (y) gives a welldefined additive homomorphism 3 : Homx)er(M, C) —> Homx)er(D, A); with this notation: Proposition 21.2: The sequence H o n W M , B) 4 HomDer(jW, C) 4- Hom Per (D, A) is exact.
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Stable Modules and the D(2)-Problem
Proof: Let b e HomA (M, B); since n o j = 0 then b o n o j = 0 so that p*{b) fits into the following commutative diagram 0 - > D -»• p -> M -> 0 4- 0 [b o n 4. p •(*)
0->
p
B
A
0
c
Thus 0 : D -*• A represents 3 o p*(b); that is 3 o p*(b) «B 0, and so 3 o /?* = 0. Conversely, suppose that y e HomA(M, C) is such that 3(y) % 0; we must show that there exists b e Hom A (M, B) such that /?*(b) % y. Consider first the special case where we have a commutative diagram of the following form
I
A
>J^ U
I
Q
-J, D
I
,.
X V
that is, where we may actually take the zero map 0 : D —> A to represent d(y). Since j8|im(y) = 0, there exists a unique homomorphism b : M —> B such that f) = b o n. However yon
= pofiso
that y on — p ob on. Since n is an
epimorphism, the desired conclusion p*(b) = y now follows. In general, however, we can only assume that we have a commutative diagram
1«
irP
lY
O ^ A 4 S 4 C ^ O where a factors through a projective module Q; that is a = fx, o X where X : D —*• Q and /x : Q —>• A for some projective module Q € C. However, Q is injective relative to C, so there exists a homomorphism v : P -> g making the following diagram commute
D
Q
Relative homological algebra Then /xovoj
— fxoX = a. Put
J3 = ^ - I O I I I O D : / ) - >
81 B SO that
P o j = p o j — i o ix o v o j = i o a — i o n, o A.
= /oa — ioa
However, the diagram below commutes
;o o->
ip iy S 4 c-^ o
A4
We are thus in the special case treated above, and the stated conclusion follows. D Proposition 21.3: The sequence HomVer(M, C) X Hom Per (D, A) 4- Homp er(D, B) is exact. Proof: If y '• M ->• C then y lifts to a commutative diagram
la
Clearly /*(a) = /3 o y is a factorization of /*(a) through the projective P, so that i»(a) « 0. By definition, however, a represents 9(y) so that (*(3(y)) «* 0, and we have shown that /» o 3 = 0 in Homi>er(D, B). Conversely, suppose that /*(a) % 0 where a : D —*• A, and let i o a = x o a be a factorization of i o a through a projective Q Q
i
D
oa
B
82
Stable Modules and the D(2)-Problem
Since Q is injective relative to C, o factorizes thus j D
Q Put ft = z o X : P —> B; then fi o j = r o A. o j — TOO = i o a. In particular, since poi = 0 , then pofioj = poioa = 0; that is, (/7o>8)|imO") s 0 a n d P°fi induces a unique homomorphism y : M —> C making the following commute
o-»
D4
/> A
A/-» o
Thus a % d(y) for some y e HomA (M, C), as claimed.
D
We obtain the basic exact sequence in cohomology: Theorem 21.4: Let £ = (0 -> A -V B -4- C - • 0) be an exact sequence in C, where C is a tame class of A-modules; then, for any M e C, the sequence below is exact H°(M, A) 4- W°(Af, B) -^ W°(Af, C) 4- H\M,
A) 4 W'(A/, B)
The boundary map 3 : H°(M, C) ->• 1-0 {M, A) depends on a specific choice of projective cover V = (0 - • D -^ P ->• A/ -^ 0). If Q = (0 ->• D' -^ 2 ^^ Af -> 0) is another choice of projective cover for A/, it is straightforward to see that dQ = {aVQ\
o dV
where (apg)* : HomperC^, A) ->• HomperC^'. ^) is the isomorphism in the derived category which was discussed in Section 20. It is clear from the definition that H"(M, A) = H°(Dn(M), A) = HomDer(D,,(M), A); moreover, it is also evident D n+ i = D| o D n . It follows immediately that: Theorem 21.5: Let £ = (0 -> A -4 B -*• C -*• 0) be an exact sequence in the tame class C; then, for any M e C, the sequence below is exact for all n e Z. > H"(M, A) X H"(M, B) -^ H"(M, C) 4- Hn+l(M, A) 4 H"+\M, B) %• • • •
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22 Stable modules and the derived category For a tame class C, the derived module category 2?er(C) can be regarded as the quotient category obtained from C by setting all projective modules equal to zero. It is useful to have some criterion for characterizing projective modules in this context. Proposition 22.1: If C is a tame class then the following conditions on a module L e C are equivalent: (i) (ii) (iii) (iv) (v) (vi)
L is projective; Id/. % 0; H o n W M , L) = 0 for all M eC; Hom Per (L, M) = 0 for all M eC; there exists an integer n such that H"(M, L) = 0 for all M e C; there exists an integer n such that H"(L, M) — 0 for all M e C.
Proof: (i) =>• (ii) is obvious. (ii) => (i) Suppose that Id^ % 0, then Idi. factorizes. Thus Id/. = a o £ where a : P -+ L and ft : L -+ P where P is projective. In particular, a is surjective and P is injective. Put A" = Ker(a); then there exists a short exact sequence 0 ^ K ->• P A L - • 0 which is split on the right by /3; that is, a o fi = Id;.. Thus P = K ® L, and L, being a direct summand of a projective module, is itself projective. (ii) => (iii) Let / € HomVet(M, L); then ldLof Idt o / % 0 so that / « 0.
= f. Thus, if ]dL % 0, then
(iii)=)> (ii) If Hom;Der(M, L) = 0 for all L e C, then, in particular, Homx>er(L, L) = 0. Thus Idi % 0. (ii) => (iv) This follows by the same proof that (ii) =>• (iii) on replacing \dL o f by f oldL. (iv) =^ (ii) This follows by the same proof as (iii) => (ii). The equivalence of (iii) with (v) is clear, since H"(M, L) = Hompe^D,^^), L). Likewise, (iv) and (vi) are equivalent since W'(L, M) = Hom-per(i,,
• Recall that, in Chapter 3, we introduced a stability relation ' ~ ' on A-modules by means of M ~ N
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Stable Modules and the D(2)-Problem
When C is a tame class we put Stab(C) = C/ ~ In addition to the stability relation ~ , there is an analogous, but coarser, relation which arises naturally in connection with the derived category. We define the hyper-stability relation ' ~ ~ ' on C by writing M,
~~
M2
«=> Mi © P\ =
M2
©
P2
for some finitely generated projective modules Pi, P2. Denote by Hyp(C) the set of equivalence classes of C under ~ ~ Hyp(C) = C/ The objects in Hyp(C) will be called hyper-stable modules.* If M e C, denote by {M) € Hyp(C) the class of M under ~ ~ . Hyper-stable modules are precisely the same as isomorphism classes in X>er(C): Proposition 22.2: Let C be a tame class, and let M, N e C; then M =Ver N
Proof: (•*=) Let P e C be projective. For any M e C, let iM : M -* M ® P denote the inclusion IM(X) = (x, 0), and let TZM : M © P —>• M denote the projection TTM(X, y) = x. Then nM oi M = Id^ whilst iM onM - ldM «» 0, so that IM and nM are mutually inverse isomorphisms in Der(C). (=>•) Suppose that / : M ->• A^ and g : N -+ M are A-module homomorphisms such that
Let
F(m,q)-
f(m) +
* We apologize for this neologism but it seems difficult to avoid something like it. If, as we wish to, we follow the the usual convention within Algebraic K-Theory, then 'stabilize' is already reserved to mean 'addition of a free summand'. See the note on terminology at the end of this section.
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85
Put K — Ker(F), and let j : K -> M\ be the inclusion; then: (I) K e C; (II) F is surjective; (III) F : M\ -*• N is an isomorphism in X>er(C). By (III), for any module L e C, the induced map F* : Homx>er(L, Mj) -> Homper(L, N) is an isomorphism of abelian groups. In particular, F*: W°(L, M,) -» W°(L, A') is an isomorphism. Moreover, since H°(Di(L), - ) = H\L,-), it is also true that the induced map F* : HX{L,M\) ->• W'(L, AO is an isomorphism. From the long exact coefficient sequence n°(L; Mi) -^ W°(L; AT) 4- W'(L; AT) 4- W'(L; M,) -5- W'(L; A') we see that H](L; K) = 0 for all modules L e C. Thus £ is projective by (22.1). In particular, K is also injective relative to C, so that the exact sequence O - H - . / f ^ A f i ^ A ' - ^ O splits, and M{=N®K. Hence M © g = N 0 K and A/ ~ ~ A' as required. D Thus, when C is tame, isomorphism classes in Der(C) are parametrized by hyper-stable modules so that the derived functors Dn give rise to correspondences Dn : C -». Hyp(C) If C is a tame class, then Stab(C) is naturally an abelian monoid under '©'. A case of particular interest is the class C — P(A) of finitely generated projective modules. Then Stab(P(A)) is simply the reduced projective class group A'o(A) of A. In general, for any tame class C, Hyp(C) is also an abelian monoid under ©, and the correspondence [M] i->- (M) gives a monoid homomorphism /x : Stab(C) —> Hyp(C) given by ju.([A/]) = (M), whose kernel is easily seen to be K0(A). Thus, fx fails to be injective precisely when K0(A) ^ 0. A projective cover 0 - » £ 2 - » S - » A - » 0 i s said to be stably free when 5 is stably free. ifO^-Q-^S^-A^-OandO-^Q'^-S'^A-^O
are
stably free covers, Schanuel's Lemma gives an isomorphism £2 © S' = ft' © 5; hence ft ~ ft'. By restricting to stably free covers, the constructions D,, are modified to produce correspondences ftn : C -+ Stab(C) We proceed as follows; to each module M e C, we associate a sequence (ftr(A/))r>o of stable modules defined by the condition that ft,,(M) is the stable
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Stable Modules and the D(2)-Pmblem
class [D] of any module D e C for which there exists an exact sequence of the form 0 -> D -* Sn-\ ->
> So -* M -> 0
where each Sr is a finitely generated stably free module over A. Likewise, to each module M e C, we associate a sequence (Q_r(/W))r>o of stable modules defined by the condition that Q-,,(M) is the stable class [D] of any module D e C for which there exists an exact sequence of the form 0 -> M -> So ->
> 5,,_i -> £> -> 0
where each Sr is a finitely generated stably free module over A. In effect, we produce liftings of D,, through fi Stab(C)
C
Hyp(C)
The following is clear: Proposition 22.3: For any module M e C the following relations hold:
(i) nm(&n (ii) nn(Q-n(M))
= [M\.
The correspondences (£2,,),,<=z are no longer functors on the derived category; in particular, we have not defined the action of morphisms under Q,,.
A note on terminology In the world of Algebraic K-Theory, 'stabilization' has a reserved meaning, namely 'addition of a free summand'. We have accordingly chosen 'hyperstable' to connote 'addition by a projective summand'. In Carlson's book [11], what we have called the 'derived module category' is called the 'stable module category'. In that case, there is no confusion since projectives are then necessarily free; 'hyper-stable module category' seemed too much. We have used 'derived' because it objectifies the 'derived functor' construction. Our 'derived module category' should not be confused with 'derived categories' of chain complexes [23], [22], although the relation is close. In any case, as anyone
Relative homological algebra
87
familiar with derived categories will know, there are already so many variations (positive, negative, bounded • • •) that one more will not hurt.
23 Module extensions and Ext1 Let C denote a tame class of A-modules; associated with Extg are a number of natural constructions. Pullback Given a A-homomorphism / : A\ —>• A 2 there is a 'pullback functor' / * : E\tlc(A2, B) -> Ext^A,, B) denned as follows; if E = (0 -> B -+ Eo 4 A2 -> 0) e Extlc(A2, B), we put /*(E) = (0 -> B -> F o -4 /I, -» 0), where F o is the fibre product nJ and e : FQ —• A| is the projection e(x, y) = _y. If g : A2 —> A3 is a Ahomomorphism, it is straightforward to see that (g o /)*(E) = / * o g*(E). There is a natural transformation fif : f* -*• Id defined by 0^
M where /K,O : ^b ~^ £^0 is the projection /J,Q(X, y) = X.
Pushout Let A, B\, B2 be A-modules; if / : B\ —> B2 is a A-homomorphism, the 'pushout' functor /* : Ext^A, B}) -> Ext^(A, B2) is defined thus. Let E = (0 - • B{ - • £ 0 4 A - • 0) e and put
ME) = (0 ^ B2 - i Fo 4 A -»> 0) where fb is the colimit Fo = Hm(/, 1) = (S 2 © £ 0 )/Im(/ x -1)
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Stable Modules and the D(2)-Problem
and j : B2 —*• FQ is the injection j(x) = [x, 0]. It is straightforward to see that, if g : B2 -> B3 , then
(g o fUE) = g* o /,(E) There is a natural transformation vy : Id —>• /„ obtained as follows E
; v/ = /»(E)
/ 0 -> Bi -» £ 0 -* A ->
; /
1 vo ; w
\ 0 -* B2 - • Fo - • A - •
where y0 : £0 -> Fo is the inclusion vo(x) = [0, x].
Direct product Let A,, A2, Bi, B2 be A-modules, and let E(r) e Extc(A r , Br ) for r = 1, 2 E(r) = (0 - • Br - • £(r) 0 - • A r -> 0) The direct product E(l) x E(2) is the extension (0 -+ Bj x S 2 -^ £ ( l ) 0 x E(2)o -^ A! x A 2 ^ 0) The sequence E(l) x E(2) is exact, and we get a functorial pairing x : Ext^(A,, B,) x Extc(A2, B2) -> Ext^(A, ® A2, Bx 0 B2) If E, F e Ext^(A, B), a morphism ^0 : E -> F is said to be a congruence when it induces the identity at both ends thus E
By the Five Lemma, congruence is an equivalence relation on Ext^.(A, B). We denote by Ext^(A, B) the collection of equivalence classes in Ext^(A, B) under the relation of congruence. Elementary considerations show that Extlc(A, B) is equivalent to a small category, so that Ext^.(A, B) is actually a set. There is a natural group structure on Ext^(A, B) obtained as follows; direct product gives a functorial pairing x : Ext Ext^(A, 8 A2, B, 0 B2) Let A, Bi, B2 be A-modules; there is a functorial pairing, external sum 0 : Ext£(A, Bx) x Ext^,(A, B2) ->• Ext^A, B, © B2)
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89
given by £, © £ 2 = A*(£, x £ 2 )
where A : A —> A x A is the diagonal. The addition map + : B x B —*• B can also be regarded as a A-homomorphism a : B © B —> B;
a(b\, fo2) = b\ + £2
Combining external sum with pushout, we obtain the so-called 'Baer sum'; let £r e E\t[c(A ,B)forr=
1, 2, and define the Baer sum £, + £ 2 by
£1 + £2 = <x*(£\ © ^2) (= «*A*(£, This gives a functorial pairing + : Extc(A, B) x Extc(A, B) -> Ext^A, B) It is straightforward to see that congruence in Ext^ is compatible with Baer sum, and that: Proposition 23.1: Ext],{A, B) is an abelian group with respect to Baer sum. Proof: The identity is given by the trivial extension
If £ = (0 -» B -V X -4- A -> 0), then the congruence class —£ is represented by (0 -+ S -V X ^> A ->• 0). The required congruence ** : £ + ( - £ ) ->• T is induced from the morphism 0-> B® B ^> X x x -+ A -^ 0 1 + 0^
5^
1^
1 Id
fi©A^A->0
where ty(x\, x2) = (x\ + x2, p(x)).
D
Projective covers have some notable invariance properties under isomorphisms, inclusions and projections. For the purposes of the discussion, fix a projective cover M,D)
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Stable Modules and the D(2)-Problem
Let a : D —*• D' be a morphism in C; then we have a natural map
We claim: Proposition 23.2: With the above notation, a»(P) is a projective cover if and only if [a] is an isomorphism in Der(C). Proof: (-£=). Suppose that a : D -> D' gives an isomorphism in 'Der(C). We must show that the colimit L — \'™(a, i) is projective. For any X e C, there is a commutative ladder of coefficient sequences
n°(x,
M)
-> n\x,
4- id
D)
-> n\x, p) -> n\x,
4- «*
4- («o)*
M)
->
H2(X, D)
4- w
r
4- «* 2
H°(X, M) -> W(X, D') -> •H'(A , L) ->• H\X,
M) -> H (X, D')
in which Id and a* are isomorphisms; then (a 0)* : H\X, P) -*• H](X, L) is also an isomorphism. However, P is projective so that we have H[(X, L) = H'(X, P) = 0 for all X e C, and L is projective by (22.1). This completes the proof of (•<=). (=>•) To show the converse, note that, if P = (0 ->• D -» P ->• A -> 0) is a projective cover and /n : P ->• P is a lifting of the zero map 0 : A -*• A, then /i+ : D —>• D factors through the projective P. Consequently, given a lifting ofIdA p
4- = p
P-
/o ->• D ->
4- on) P _»
0 4-Id A ^ 0
then a) — Id factors through P; that is, [co] = [Id] in Endi>er(£)). Now suppose that a : D —»• D' is a A-homomorphism such that L = '""(a, (') is projective, then there exists a morphism a*(P) -> P lifting Id
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91
By the argument above, both a o fi — Ido< and fi o a — Ido factor through projectives, so that [at] is an isomorphism in Der(C), with inverse [/3]. This D proves (=>•) and completes the proof. The proof of the corresponding statement for pullbacks is dual to the above; we leave the details to the reader: Proposition 23.3: If fi : M' —> M is a homomorphism in C, then /S*(P) is a projective cover if and only if [/5] is an isomorphism in Der(C). Let Q € C also be projective, let io '• D —> D © Q be the inclusion i{d) = (d, 0), and let nM '• M © Q ->• M be the projection n(m,q) = m; then we have (23.4) (23.5)
(ioUP) e Proj^(M, D ® Q) 7r^(P) e Proj'(M © Q, D)
Now suppose that P' = (0 -> D © 2 -> P ->• M -»• 0) is an extension in P j c in which 2 is also projective, then (23.6)
(TTDUV) € Proj^M, D)
Finally, if P" = (0 -> D - • P -^ M © 2 ->• 0) is an extension in Proj^ in which 2 is also projective; then (23.7)
iV(P") e Proj^(M, D)
The proofs of (23.4)-(23.5) are completely straightforward. Those of the dual statements (23.6), (23.7) are only slightly less obvious, but do require specific appeal to the properties T(2) and T(3) of the tame class C. We leave them to the reader. For any module M e C, D|(M) represents all modules D' e C which are isomorphic in Der(C) to some particular module D e C which occurs in a projective cover P = (0 - • D -+ P ->• M ->• 0). By (22.2), we can identify Di(Af) with the class (D) under the hyper-stability relation ' ~ ~ ' . These identifications are completely compatible with the construction of projective covers, since by (22.2), (23.2), (23.4), (23.6) it follows easily that: Proposition 23.8: Let M, D be modules in C which occur in a projective cover P = ( 0 ^ D - » . P - > M - * 0), and let D' e C. Then the following statements are equivalent: (i) D' ~ ~ D; (ii) £>' e D,(M); (iii) there exists a projective cover of the form P' = (0 -> D' -» P' —> M -> 0).
92
Stable Modules and the D(2)-Problem The dual statements likewise follow easily from (22.2), (23.3), (23.5), (23.7):
Proposition 23.9: Let M, D be modules in C which occur in a projective cover P = (0 -> D ->• P -> M ->• 0), and let M' e C; then the following statements are equivalent: M; (i) M' (ii) M' e D_,(D); (iii) there exists a projective cover of the form P' = (0 -» D -»•/"-»• Af' ->• 0). Fix A, fi e C, and make a specific choice of projective cover
Then D is a representative of D| (A), and there is a mapping extp : HomA(£>, 5) -> Ext
p E It follows from (19.7) that, if i/r is any other lifting of Id^, the difference
for any such lifting
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93
Proposition 23.11: The following diagram commutes Extlc(A,B)
extp
/
\
cP
Hom A (D, B)
W(A,B)
The proof of the following is straightforward: Proposition 23.12: Let E , F e Ext^A, B); if E « F then cP (E) = cP (F). It follows that c P induces a mapping, denoted by the same symbol c P : Ext^A, B) -» HomVer(D, B) Composing extp : Hom A (D, B) -> Ext^(A, B) with the quotient map
Ext^A, fl) -+ Ext^A, 5)/ « = Ext^A, 5) we obtain a function HoiriA(D, fi) —> Ext,l(A, 6), also denoted by 'extp'. The following is the main technical result of Yoneda Theory: Theorem 23.13: The mapping extp : HomA(D, B) ->• Ext^A, B) factors through Homper(O, B)\ that is, there is a mapping e P : HomOer(jD, B) -> Ext^A, #) such that extp = ep o [ ]. Proof: We have to show that extp : Hom A (D, B) -> E\tlc(A, B) factors through Homx>er(O, B). Suppose that
4.1= where A,o[^, p] — [^-(g), p]. Q, being projective, is injective relative to C, by property T(3), so that i/^CP) splits. Let s : A —> XQ be a right splitting
94
Stable Modules and the D(2)-Problem
of i/r*(P); then A.o o s : A -> Eo is a right splitting of
Then there is a congruence if; :
E as can be seen by the following B
E
Here i™ = i'm(
24 General module extensions and Ext" As above, C is a tame class of A-modules; for n > 2, Ext£ will denote the category whose objects are exact sequences in C of the form E = ( 0 - > E+ - • £ „ _ , - * • • • - » Eo -+ £ _ - + 0 )
and whose morphisms are commutative diagrams of A-homomorphisms E h =
4- hF_ -+ 0
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95
As in the case of Ext', there are functors «_, &>+ : Ext£ —>• C given by
4- "«-i
4K
c -
t
and 0-> K a>+ I
—>
x4- ftj. J, 4- «„_! K-\
0-» K
J, «n
x «_
i = 4- hj
c -
' - »
If A, B are A-modules we denote by Ext£(/t, S) the full subcategory of Extji whose objects E satisfy £_ = A and E+ = B. The constructions introduced for Ext1 all have analogues for Ext" (n > 2).
Pullback \ff:A\ —¥• A2isa A-homomorphism, we obtain a functor/* : Ext£(/42> B) Ext£(/t|, B); letE e Ext2(y42, 5)
E = (o _,
B
-» £„_, ^ ' • • • 4 £ 0 4. ^ 2 -*
and put /*(E) = (o -> fi -*. FB _, ^ ' • • • -^ F0 4 A, where Fo is the fibre product Fo = £o x A\ = {{x, y): T](x) = f(y)}, 5, : F\ -> F o isthemap5|(x) = (9|(x), 0);e : F o -> /li is the projection e(x.y) = y, Fr = Er for r > 1; F r = £ r for /• > 1, and, finally, Sr = 9r for r > 2. It is straightforward to check that, if g : A2 —»• A3, then
Observe also that there is a natural transformation /^y : / * —>• Id obtained as ^ F n _ , -> Id 4, /U«-l £„_, - • . . . where /tir = Id for r > 1, and /XQ : Fo -> £0 is the projection /io(-*, y) — x.
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Stable Modules and the D(2)-Problem Pushout
If A, B\, Bi are A-modules and / : B\ —> B 2 isa A-homomorphism, we obtain a functor /„ : Ext£(A, B,) -+ Ext£(A, B2) as follows; let E e Ext£(A, B,)
E = ( 0 _ B, - i £„_, *>' ... 4 £ 0 A A and put /.(E) = (o -+ B2 -4 F,,_,
^ . . 4 F
O
4 /
where F r = Er and <5r = dr for r < n - 2; Fn_i is the colimit Fn-\ = ] ™(/, 0 = (B2 8 £,,_,)/Im(/ x - i ) 7 : B2 ->• ^n-i is the injection y'(x) = [x, 0], and 5n_) : Fn_i ->• Fn_2 is the map 8n-\[x, y] — 9,,_i(_y). It is straightforward to see that, if g : Bi -> B 3, then
(f ° /)* =8*°f* In this case there is a natural transformation vy : Id —> /* obtained as follows E
where vr = Id for r < n — 2, and v n_| : £ 0 ->• F o is the inclusion vn_i(x) = [O.JC].
Direct product Let A1, A 2 ,fi,, S 2 be A-modules, and let E(r) e Ext£(A,., Br) for r = 1,2 E(r) = (0 - • Br -> £(r) B _, -»
>• £(r) 0 - • A r ^ 0)
The direct product E(l) x E(2) is the sequence (0-» B , x f i 2 ^ F.(l),,_,x£(2) n _, - •
• £ ( l ) o x £ ( 2 ) o ^ A,xA 2 ->0)
which is easily shown to be exact. We obtain a functorial pairing x : Extg(Ai, Bi) x Ext£(A2, B2) ->• Ext^(A, © A2, B, © B2)
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If E, F e Ext£(A, B), a morphism
_>
B I
- • £„_, TJ
I
^ Id 4- ^n
In the case n — 1 we observed that elementary congruence is an equivalence relation. This fails to be true however when n > 2. We write E ~> F when there exists an elementary congruence
Yoneda product If A, B, C are A-modules we define a pairing o : Ext£'(5, C) x Extg(A, B) -» Extc'+"(A, C) as follows. Let E, = (o - • C -4 Ym-\ V • • • 4- Zo -^ B -> 0) 6 Ext?(i4, B)
E2 = ^0 -> fi -> X,,_i ^ ' • • • i x o l A ^ O J e Ext^(B, C) and let E, o E 2 = (o - • C -> Z«_i *"4."' • • • 4 Z o -» /I - • 0) e Ext™+"(A, C) be the extension denned by (Z r , 6r) = (Xr, 3r2) for r < n - 1, Z r = yr_,, for n — 1 < r < m + n — 1, 8,, = r\ o i, and Sr = 3r'_n f o r n < r < m + / i - l . The Yoneda product is compatible with congruence, and descends to give a pairing of set valued functors o : Ext£(fl, C) x Ext£G4, B) -» Ext^'+"(A, C)
98
Stable Modules and the D(2)-Problem
We shall eventually impose a group structure on any Ext£(A, B), with respect to which these pairings are bi-additive. This point will not be pursued here, but will emerge later, as a consequence of Yoneda's classification of extensions. 25 Classification of general module extensions If M is a A-module, by a projective fresp. free) n-stem of M we mean an exact sequence of A-homomorphisms of the form P = (0 -> K -* P n _, ->•
> Po -> M -* 0)
where P r is projective (resp. free) for 0 < r < n — 1. A projective 1 -stem is the same as a projective cover. We denote by Proj£ the full subcategory of Ext£ whose objects are projective n-stems, and we denote by Proj£(M, D) the corresponding subset of Ext£(M, D). Projective n-stems exist; in fact we have: Proposition 25.1: LetCbeatameclassof A-modules. Then any module A eC has a free n-stem for each n > 1. Proof: This is true for n = 1, since any A 6 C is finitely generated, and there exists an epimorphism r\ : Fn -*• A where F(0) is a finitely generated free Amodule. Putting K — Ker(>;), the exact sequence P(l) = (0 - • K -> 7^(0) ->• A -» 0) is a free 1-stem, and, since C contains all finitely generated projectives, it contains F(0), so that K e C by property T(2). By induction, there is a free (n — l)-stem P(n - 1) = (0 -> D - • F(n - 1) ->
• F(l) -^ A" -» 0)
over /iT, and the Yoneda product ¥(n — 1) o P(l) is a free rt-stem over A.
D
In dealing with projective n-stems, there is a simple technique which facilitates arguments by induction, namely 'cutting and splicing'. Given a projective n-stem P = (0 -> N -> P,,_, -> • • • -^ Po - • W -> 0) by cutting at an intermediate point we form two shorter projective stems P, = (0 -). N - • /»„_, - •
• Pm -* /C - • 0)
K -»• P m _ , - •
>• P o - • A^ -»• 0 )
P 2 = (0 ^
where A1 = Im(Pm ->• P m_i) = Ker(Pm_i ->• Pm-i). The reverse process, 'splicing', re-combines the two shorter sequences by means of the Yoneda product P = P, o P 2
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Let A, A', B be A-modules, let P be a projective n-stem over A, and let E e Ext£(/4', B); if cp : A -> A' is a A-homomorphism, then a morphism q> : P -> E is said to be a ///h'«g o/V when
p
10 = E
41 $0
4- (f+ -J, (Pn — I
O -» B ->• £•„_, ->
•!• ^
> Eo -* A' -> 0 i
The following is easily proved by induction, using cutting and splicing, starting from the corresponding result for projective covers (19.7): Proposition 25.2: Let P = (0 -> D -»• P,,_: -> • • • ->• P o -» £ -» 0) be a projective n-stem over /4, and let E e Ext^(A', B); if
• ^o ->•
A^0)
so that /) e C is a representative of D,,(A). Pushout defines a mapping extp : HomA(£>, B) -> Ext2(/1, B) by means of ext P (/) = /»(P) There is also a function c P : ExtJi(A, B) -»• HomDerC^. 5) = H"(A, B) obtained as follows: for each E e Ext£(A, B), we may, by (25.2) above, choose a morphism
/ 0 ->
D ->
PB_, - •
4-
\ 0 ->• B - »
£n_i ->
•• • -•
Po - •
A
l
4- I d
•• • -^ £
0
-•
->• ^ ->• '
100
Stable Modules and the D(2)-Problem
If i/r is any other lifting of Id^, the difference
cP(E) = [
E\t"c(A,B)
extp
/
\
cP
Hom A (D, B)
* H"{A,B)
The proof of the following is entirely straightforward: Proposition 25.5: Let E, F e Ext£(A, B); if E s» F then cP(E) = c P (F). It follows that cp induces a mapping, denoted by the same symbol c P : Ext£(A, B) ->• Homper(£>, fi) Composing extP : Hom A (D, B) -> Ext^/l, B) with the quotient map , B) ->• E\fc(A, B) = Extnc(A, B)/ « we obtain a function Hom A (D, B) -*• E\\.nc{A, B), also denoted by 'ext P '. We have the following extension of (23.13): Theorem 25.6: For any n > 1, extP : Hom A (D, B) -> Exl"c(A, B) factors through HomDer(D, B)\ that is, there is a mapping e P : HomperCA B) -> Ext£(A, i8) such that extp = e P o [ ]. Proof: The statement for n — 1 is true by (23.13). The case of n-fold extensions reduces to (23.13) using an explicit form of dimension shifting. Let A, B e C and let P =
Dn -» />„_, ^ '
». Po - • A -». o)
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be a specific choice of projective «-stem over A. We split this at the (n — 1) stem to get a projective (n — 1 )-stem over A P = (0 -* £>„_, -> P,,_2 ->
• Po -> A -> 0)
where £),,_i = Im(3,,_i) = Ker(3n_2), and a projective cover P(l) = (0 -> Da -> />„_, -* £>n-i - • 0) which are reassembled into the original by using the Yoneda product P = P(l)oP If
By the Main Theorem for n — 1, if
Thus the construction ip >-*• extp(
factors through Hom-per(D, B).
e P : H o n W A B) -> Ext^(A, B) such that extp = ep o [ ]. This completes the proof of (25.6).
D
Using (25.6) we now prove: Theorem 25.7: Let A,B, Dbe modules in C with D e Dn(A), and let P e Ext£(j4, D) be a projective n-stem; then c P : ExtJi(A, B) -+Hn(A, B) is bijective with cjT1 = ep. - The map e P : HomDer(£>, B) -> E*fc(A, B) is surjective. If E e , B), then, by (25.2), we choose a morphism ip : P -> E lifting Id^ D-+ E
B
- > Po -»•
A -•
• - > £ o ->• ^ - > •
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Stable Modules and the D(2)-Problem
Then there is an elementary congruence the following
E, as can be seen by
Po^
E
A-*
0\
I IdB 4, Vn-1 4- fn-2
4- ^0 4- Id/i
B -4 £„_, -> Ea-2 -*•
£0 -> A -> 0 /
\o
Here 1™ = lim(
D'
-»• p,;_,
0)
be specific choices of projective n-stems over A, A'. Then / lifts to a commutative diagram p
0
Pn-\^
4-/ =
4-/o 4 - /
I fn-l
p'
- • P^ - > A ' - >
Note that the following diagram commutes Ext"(A', B) - A Hom Per (D', fi)
/* I
W+(/)*
i
Ext"(A, B) - A HomDer(£>, B) and that:
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Theorem 25.8: (Yoneda [83]) If for each A e C, PA is a projective n-stem over A, then c = (cpA)Aec defines a natural equivalence
Since the functor H"(—, —) is group valued, we obtain a natural group structure on Ext£(A, B), by requiring that cp be a group isomorphism; it is now clear also that: Proposition 25.9: The pairing o : Ext£(fi, C) x Ext£(A, B) -* Ext£'+"(A, C) given by the Yoneda product is bi-additive with respect to the natural group structures. An extension E e Ext£(A, B) is said to be almost free (resp. almost projective) when Ej is free (resp. projective) for j ' ^ n — 1. When n = 1, every object in Extg(A, B) is almost free. If we repeat the proof of (25.7) taking a free n-stem P, then the extension E = (
• ^o -> A -+ 0
where FQ, ..., Fn_2 are free and P is projective; that is: Proposition 25.11: IfE e Proj£(A, B) is a projective n-stem, then there exists an almost free extension E also belonging to Proj£(A, B) such that E % E . Proof: When n = 1, the statement is empty and we can take E = E and \}r — Id. When n = 2, write E = (0 -> B -4 Ei A Eo 4 A -+ 0) Since EQ is projective, we can choose a complementary module D such that Eo 0 D is free. Put
where /; \
/ an \
= (P, 0)
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Stable Modules and the D(2)-Problem
Then E is almost free, and the projections £,• © D —> £7 induces an elementary congruence E ->• E. Suppose proved for n — 1, and let n > 3. Decompose E into Yoneda factors, thus E = E+ o E_ where E_ e Proj 2(A, C), and E+ e Proj"~ 2 (C, B). Let E_ e Proj 2(A, C) be an almost free extension congruent to E_, and put F = E + oE_ e Proj"(A, B). We may now decompose F thus F = F+ o F_ where F_ € Free'(A, D) is a free extension, and F + e Proj"~'(£>, B). By induction, there exists an almost free extension L e Proj"~'(D, B), which is congruent to F+. Putting E = L o F_ e Proj"(A, B) gives the result. • 26 Classification of projective n-stems Fix a specific projective «-stem P = (0 -> D -> />„_, -^
> P o -> M ->• 0)
the set of endomorphisms of D in the derived category 2?er(C) is then naturally a ring. We put A(D) - Aut Per (D) that is, A(D) is the unit group of the ring End-per(D). Then Yoneda's Theorem (25.7), shows that the correspondence a H> CC*(P) defines a bijection ep : Endper(^) ~* Ext"(Af, D). The main result proved in this section ((26.5) below) is that ep restricts to give a bijection e P : A(D) -> Proj^(M, D) allowing us to parametrize projective n-stems by the elements of A(D). Note that (23.8), (23.9) have analogues for general projective «-stems: Proposition 26.1: Let M, D be modules in C which occur in a projective nstem P = (0 -* D -* />„_, -> • • • Po -> M ->• 0) and let D' e C; then the following statements are equivalent: (i) D' ~ ~ D; (ii) £»' e D n (M); (iii) there exists a projective n-stem of the form
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105
Again there is a dual statement: Proposition 26.2: Let M, D be modules in C which occur in a projective nstem P = (0 -> D -» />„_! -» • • • Po -» M -> 0) and let A/' e C; then the following statements are equivalent: (i) A/' M; (ii) Af 6 D_B(D); (iii) there exists a projective n-stem of the form P' = (0 -> D -> P,;_, -> • • •
PQ
->• A/' -> 0)
The proofs of (26.1), (26.2) follow easily by induction from those of (23.8), (23.9) by decomposing the projective n-stem into the Yoneda product of a projective cover and a projective (n — l)-stem. Morphisms a : D —*• D', fi : M' —> M in C give functors or, : Ext"(A/, D) -+ Ext"(A/, D');
fi* : Ext"(M, D) -^ Ext"(A/', D)
It is not generally true that that Proj"(—, —) is stable under all these functors. However, generalizing (23.2) and (23.3), Proj"(—, —) is stable under functors a*, fi* where a, fi are isomorphisms in the derived category. When n > 2, the statement for Proj"(—, - ) corresponding to (23.2) reduces immediately to the case where n — 1, and hence to (23.2), on decomposing an n-fold extension P as a Yoneda product P = P+ o P_ where P + is a 1-fold extension and P_ is an (n - 1 )-fold extension. We leave the details to the reader, and note the result. (26.3) Let P e Proj"(M, D) and let a : D -> D' be a A-homomorphism; then a*(P) € Proj"(M, D') if and only if [a] is an isomorphism in Per(C). The above result is fundamental in the detailed classification of projective n-stems both now and later. Though we shall not need to use it, the corresponding statement for pullbacks is also true, and is easily reduced to (23.3). We leave the details to the reader. (26.4) Let P e Proj"(A/, D) and let /S : M' ->• M be a A-homomorphism; then p*(P) e Proj"(A/', D) if and only if [fi] is an isomorphism in Per(C). We can express (26.3) in terms of the following (left) module structure of Ext£(M, D) over the ring EndOer(£») , D) -> Ext£(A/, D) [a] * [£] = [«.(£)]
106
Stable Modules and the D(2)-Problem
ep then takes the form ep([a]) = [a] * [P] We will show that Proj£(M, D) forms a single orbit under the group action A(D) x Ext£(M, D) -* Ext£(Af, £>) obtained by restriction. Theorem 26.5: Let P € Proj£(A/, D) be a projective n-stem; then ep restricts to give a bijection e P : A(D) -> Proj£(M, £>) Proof: We first treat the case n — 1. If Q also belongs to Proj^M, D), then there is a morphism u> : P -> Q which lifts Id w . Let a = co+ : D ->• D be the A-homomorphism so obtained. Then by (23.2), [a] is an isomorphism in "Der(C). Furthermore, Q is congruent to a*(P), so that Proj^(M, D) consists of a single orbit under the action of A(D) on Ext1 (M, D). Thus ep gives a surjection e P : A(D) -> Proj^M, D) As in the proof of (25.7), ep is injective on Endper(D), and so gives a bijection e P : A(D) -> Proj^(M, D) as claimed. This completes the proof for n = 1. As in the proof of Yoneda's Theorem, the proof for n > 1 is reduced to the case n = 1 by dimension shifting. In effect, the proof for the action of A(Dn) on Proj£(M, Dn) is the • same as that for the action of A{Dn) on Proj^(D,,_i, Dn). Let M, D e T(Z[G]) and suppose that P = (0 - •
D ->• P B _ , - •
>. P o - • W - •
0)
is a projective n-stem. Since the functor Dn is an equivalence of categories on Z>er(C) Hom Per (D, D) = HomVer(M, M) so that we have both an isomorphism of rings Endi>er(D) = Endper and of groups A(D) = A(M)
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An isomorphism Endper^) = Endx>er(M) niay be calculated explicitly in the following way. If a : D -> D is a Z[G]-homomorphism we may extend a to a morphism in Ext"(M, D)thus
p
1,5 = P
r~*
la
Pn-\ " •
» />„-»•
4- a«-i
4- ao
lo->
>
0
4-a
Po^
0
We write = [a] Proposition 26.6: /cp is an isomorphism of rings KP : End-per(D) = EndVeT(M) Proof: Note that KP is automatically bijective, since D,, is an equivalence of categories. For Z[G]-homomorphisms a, /3 : D -+ D, we obtain commutative diagrams thus p
0^
4- a = P
0 -•
D^
/>„_, -
4, a
4. a,,_i
D -*• />„_,
4-ao 4-a
-
and
4- P = P
4 - ^ 4 - P«-\ \ O - ^ D ^ />„_, -
From the commutative diagrams P
D^
Pn_, ->
D ^
Pn-\
•/>()-•
M-»
O\
M -^
(
4-ex' P and
-»• ••• ->•
PQ
->•
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Stable Modules and the D(2)-Problem
we see that KP{a + P) — KP(a) + Kp(/3) and KP(a o ft) = KP(a) o KP(P). Finally, note that the commutative diagram
0- » D->
p 4, Id =
/><>-> M ^
4, Id
P
0-
4, Id />„_, - •
4. Id 4, Id •
Po^> M ->
ov
shows that /cp(Id) — Id, which is the final requirement for KP to be a ring isomorphism. D Recall also, that if Q = (0 -> D ->• g,,_i -»• • • • -> g 0 -> A/ -> 0) is another element of Proj"(A/, D) then there is a unique automorphism A-PQ of M in the derived category making the following diagram commute 0
0
4-
4, Id 4- K-\ 0 - ^ D - *• PP,,_, > B _, -->•
>• Po -+ M ->
0
Clearly A.QP = A.pQ and KQ is related to «:p by means of
(26.7)
KP(a) =
27 The standard cohomology theory of modules We recall briefly the basics of the Eilenberg-Maclane cohomology theory of modules [12], [34]. Let M be a A-module; a resolution of M is an exact sequence of A-homomorphisms A
=
•••
A
A xt
n+1
in which M-4 = M. We abbreviate this to A = (X^1 —> M). By a morphism of resolutions, cp : A ^ B, we mean a collection (<pr) of A-homomorphisms completing a commutative diagram
A I
vA n
A
_± "*•
4-
A
vA n-1
~
0
4-
We say that a resolution A — (XA ->• M) is projective (resp. /ree) when each Xp4 is a projective (resp. free) A-module. The following is standard:
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109
Proposition 27.1: Every A-module has a free (and hence a projective) resolution. Projective resolutions will be denoted by A — (Pf ->• M) or (P* -*• M) if A is clear from context, and free resolutions will be denoted A — (FA —> M) or (F, -> M). Suppose that A = (XA - • MA) and B = (Xf -* M s ) are resolutions, and q>, \jr : A -> i3 are morphisms such that ^ w = ^rM\
(ii) <pr-i,r
= 3f+1 Wr + //.-19,"4 for all r > 1.
If / : M-4 —> M B is a A-homomorphism, a morphism
110
Stable Modules and the D(2)-Problem
If P is projective, then as a projective resolution of P we may take V = (
>0-»0->0-+
• Z5 4- P - > 0)
where e = Id/.. It follows that: Proposition 27.4: If P is projective, then for any module C, H"(P, C) = 0 when n > 1. Furthermore, when P is projective, the functor B (-»• Hom A (B, P) is ejcacf, from which we see: Proposition 27.5: If P is projective, then for any module N, H"(N, P) = 0 when n > 1. We now relate cohomology in this definition to that of our earlier definition in terms of the derived category. Thus suppose that C is a tame class, and / : M\ -»• M2 is a morphism in C which factors through the projective module P, then the induced map on cohomology / * : H"{Mi, N) —> H"(M\, N) factors through H"(P, N) — 0. Hence, for n > 1, H" considered as a functor of the first (contravariant) variable factors through the derived category; that is, if f,g:M^-Nare morphisms in A^odA such that / « g, then we have equality of induced maps / * = g* : H*(N, B) —> H*(M, B) for any fixed module B. This extends to the following: Theorem 27.6: Let C be a tame class; for any A-modules M, N, there is a (canonical) surjective homomorphism v : H"(M, N) -> H"(M, N) Proof: Since C is tame and M e C, we may suppose, without loss of generality, that M has a projective resolution V = (P* ^ AO in which each Pr is finitely generated
If / : A -> B is a A-homomorphism we denote by fN HomA(A, TV) the induced map /%) Put
= ao /
: Hom A (B, N)
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111
Then we have H°(M,N)
= Zl;
H"(M,N)
= Z"/B"
(n>\)
Put D,, = Im(9,, : Pn -»• Pn-\, and factorize 9,, : Pn -*• P,,_i as 9,, = i,, o nn where in : Dn C Pn_i is the inclusion, and nn : P,, -*• Dn is the canonical surjection. From 9,, o 9,,+i = 0 it is straightforward to see that lm(jr^) C Z". This inclusion is actually equality; to observe the inclusion in the opposite direction, note that z 6 Z" is a map z : Pn —*• N satisfying z ° 3n+i = 0, and so induces a homomorphism z : Pn/Im(9,,+i) -> A^,andsoalso| : P,,/Ker(dn) -> A^sincelm(3,,+|) = Ker(9,,).Via^n, we may identify P,,/Ker(9,,)withlm(9,,) = Dn, under which identification n*(z) = z, so that Z" C Im(^*), and so, as claimed, we have
In particular, we have a surjective homomorphism
Put P(Dn, N) = {Xe Hom A (D,,, N):
k^
0}
so that HomPer(D,,, A0 = Hom A (A,, N)/V(Dn,
N).
We claim that
(jtnN)~\B")cV(Dn,N) For, if/8 € « ) - ' ( £ " ) , then /3 o jrB = 9,f («) for some a e Hom(Pn_|, A'); that is P o nn = a o dn = a o in o nn
But nn is surjective, so that /S = a o in
Hence fi factors through the />„_,, and {iz%)-\B") C P ( A i , A') as claimed. Recalling that Hom(Dn, N)/V{DIU N) = Hom0er(D,,(M), N)
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Stable Modules and the D(2)-Problem
it follows that there is a short exact sequence
0 -* V(Dn, N)/(n»)-\Bn)
-* Hom(Dn,
-> H o n W D ^ M ) , N) -*• 0 But Hom(Dn, A') = (n^)~l(Z")
and n^ induces an isomorphism
(n?)~\zn)/(it?)~\ffl)
= Zn/B" = H"{M,N)
so that the exact sequence becomes 0 -»> P(D n , AO/(;r,f )~'(e") -» Hn(M, N) -» HomBer(D,,(M), AT) - * 0 This completes the proof.
D
When C is a tame class and n > 1, the functor M K>- H"{M, —) is corepresentable on the derived category, with co-representing object D,,(M); that is: Theorem 27.7: Let C be a tame class; if n > 1 then v : tf"(M, AO -» HonvDer(D,,(M), fi) = W"(M, W) is an isomorphism for any M,N e C. Proof: As before, let V = (P* -» M) be a resolution of Af byfinitelygenerated projectives. Put Dn = Im(3n : Pn ->• Pn-i), and factorize 3,, : P,, -> P,,_i as 3n = <„ o nn. The surjectivity of v follows from the fact, observed above, that (n*yl(Bn) C V(Dn, N). To show v is an isomorphism, it now suffices to show that (nn)*(V(Dn, N)) c B". Thus suppose that / : Dn —> N factors through aprojective, say f = j ocp, where
£>„_, - • 0
<2 is projective, so that, by T(3) and (19.5), Q is strongly injective relative to C, and there exists a homomorphism
so that ( T « ) ' ( / ) = / o 7rn = ./ o
Relative homological algebra
113
and, consequently
(*»)*(/) = 0«)*(y o o) so that (nn)*{V{Dn, N)) C B" as desired.
D
We note that the conclusion of (27.7) fails for n — 0; in general, H°(M, N) is a proper quotient of H°(M, N). In particular, as we shall see in Chapter 6, when A = Z[G] is the integral group ring of a finite group G and Z is the trivial Z[G]-module structure on the group of integers, then H°(Z, Z) = Z/|G|isfinite whilst H°(Z, Z) = Z is infinite. In connection with the translation functors D,, we have the phenomenon of dimension shifting which holds independently of any proof that v is an isomorphism. Proposition 27.8: For any modules M, N e C, we have H"(Dm(M), N) = H"+m(M, N)
Chapter 5 The derived category of a finite group
We specialize the theory developed in Chapter 4 to the case where A = Z[G], the integral group ring of a finite group G. We begin by showing that the class .F(Z[G]) of Z[G]-lattices is tame. It follows that for coefficients N e T{Z[G]), the group cohomology H"(G, N) = H"(Z, N) is both representable, H"(G, N) = Hom Per (Z, Q-n(N)), and co-representable, H"(G, N) = Homx>er(£2n(Z), N), in the derived category, by means of the stable module construction fin(—). The stable modules (£2n(Z)),,ez then become our primary object of study; in particular, we analyse the tree structure introduced in Chapter 3. Since Z[G] is an order in the semisimple algebra Q[G], the cancellation theory developed in Chapter 3 also becomes available, and we deduce fron the Swan-Jacobinski Theorem that the stable modules ^2«+i(Z) always have the structure of a fork. Though less crucial for our study, the structure of the even modules £22n(Z) is also transparent. Finally, in Sections 31 and 32 we consider the relations which hold between the derived category of the finite group G and those of its subgroups.
28 Lattices over a finite group Throughout, G will denote a finite group, A will denote a commutative ring, and A[G] will denote the group algebra of G over A. We denote by .F(A[G]) the category of A[G]-lattices; that is, right A[G]-modules which are finitely generated and projective as modules over A. A fortiori, they are finitely generated over A[G]. More generally, if A is an algebra over A, which is finitely generated as an A-module, we denote by J~(A) the category of right A-modules which are finitely generated and projective as modules over A. When the algebra structure of A over A is understood we shall refer to elements of J-(A) simply as A-lattices. 114
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By an involution x on a ring A, we mean an isomorphism r : A —> Aopi> of A with its opposite ring such that x o r = Id. If A is any ring, and M is a right A-module, we denote by M" the abelian group HomA(A/, A) with a left A structure given by
When A admits an involution x, we can convert this to a right action '•' by means of
If G is a finite group, the group ring A[G] admits a natural involution k \-+ X given by
We begin by showing that J-(A[G]) is a tame class. In this we are aided by the fact that the natural involution on A[G] allows us to introduce a duality into J"(A[G]); let M € .F(A[G]), and convert the left module AT into a right module by means of the canonical involution; we get the following explicit model for AT; elements of AT are sequences a = (ag)geG, with each ag e HomA{M, A), satisfying the condition ax(mg~x)
<Xgx(m) =
Moreover, the right G-action on M" is given by (a •h)x=
ahx
Within .F(A[G]) however, there is another possible model for the dual module; we denote by M* the right A[G]-module whose underlying A-module is HomA(M, A) and whose G-action is given by
The two constructions are naturally equivalent. This special case of 'Shapiro's Lemma' is effected explicitly by means of the isomorphism
L:M~-*M*;
L((ag)geG) = a,
Since it is a matter of taste which model to employ, we use M (->• M* for simplicity. There is a natural transformation v : M -> M** given by v{x)(a) = a{x). Since M is free over A, v is easily seen to be an isomorphism over A; moreover,
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v is easily shown to be equivariant with respect to the actions of G. It follows that: Proposition 28.1: For all M e J"(A[G]), v : M -> M** is an natural isomorphism of A[G]-modules. Consider first the case where a module M e !F(\[G]) is free, rather than merely projective, over A. In matrix terms, M determines, and is determined by, a representation p : G ->• GL m(A), where m = rk A (M). The dual module M* then corresponds to the representation p* : G -*• GLm(A) denned by p*(g) = p(g~') r where fjj is the transpose of the matrix /u,. (28.1) is simply a reflection of the fact that p** = p. Although M = M**, it is not true in general that M* = M. This fact does not appear to be as well known as perhaps it should be; the point is perhaps best illustrated by reference to the most important examples with which we shall have to deal: Example 1: The regular representation The regular representation is simply the matrix description of the free module of rank 1; that is, the group ring A[G] considered as a module over itself. If G is a finite group with \G\ = N, (left) translation in the group ring gives rise to the (left) regular representation k : G -> GL^(A); that is: A.(g)(x) = gx for g e G and x € A[G]. Each k(g) is a permutation matrix and so satisfies the orthogonality condition A.(g)~' = Hg)T, and so X = k*. Expressed in terms of modules, we have that A[G] =A[G] A[G]*. In particular, we may legitimately identify A[G] with A[G]* by confusing the canonical basis [g}g€G with its dual basis {g*)g<=cExample 2: The trivial representation The coefficient ring A can be considered as a module over A[G] in which each group element g e G acts as the identity. In matrix terms, the trivial representation is given by the trivial homomorphism x : G ->• GLi(A). Since we have z(g) = r*(g) = Id for all g e G, we see also that A =A[G] ^*The fact that the two most obvious examples are self-dual is perhaps deceptive. More typical are: Example 3: The augmentation ideal and its dual The regular representation and the trivial representation are related by the augmentation homomorphism, which is the ring homomorphism e : A[G] -*• A defined by ^{YlgeGasS) = £ » e G a g . The kernel of e is called the aMgmerafafion ideal of G and is denoted by I A (G). Clearly, I A (G) e T{k[G]). In general, the augmentation ideal and its dual IA(G)* are not isomorphic. For us, the principal case of interest is when A = Z; then we write the integral
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augmentation ideal simply as I(G). To anticipate matters slightly, it can be shown that the following statements (I)-(IV) are equivalent: (I) (II) (III) (IV)
KG) =z t c] KG)*; I(G) ~ I(G)*; G has cohomological period 2; G is cyclic.
Here (I) = • (II) is obvious; when the definition of cohomological period is known (see Chapter 7), (II) = » (III) should be clear from the Representation and Co-Representation Formulae, (20.7) and (20.6), once it is appreciated that
> En -> 0
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in which all objects are in ,F(A[G]); then everything we shall need is covered by: Proposition 28.4: If 0 —> EQ -> E\ -> •••-»• F.,, -> 0 is an exact sequence in ^(AtG]), then the dual sequence 0 ->• F* - » • • • - » F* ->• FQ ~^ 0 ' s a ' s o an exact sequence in .F(A[G]). Theorem 28.5: For any commutative ring A the class T{\[G]) of A-lattices over A[G] is tame. Proof: Evidently, F(A[G]) is closed under isomorphism, and property T(0) is true directly from the definition. It remains to show that T(l)-T(4) hold also: T(l): If P is a finitely generated projective module over A[G], then, a fortiori, P is finitely generated projective over A. Hence P e .F(A[G]). T(2): Let O - ^ K - ^ M - ^ g - ^ O b e a short exact sequence of A[G]-modules and Q e .^(AtG]). Then Q is projective over A, so that the sequence splits over A, and we have an isomorphism of A-modules M =A K 0 Q If M is a direct summand of a free A-module so also is K. Conversely, since Q is a direct summand of a free A-module, Am = Q @ Q' say, then M ® Q' =A K 0 A"1 Hence if K is a direct summand of a free A-module, so also is M. Thus J^-"(A[G]) satisfies T(2). T(3): If £ = (0 ->• P ->• M ^> 2 -» 0) is an exact sequence in .F(A[G]) with P projective, then the dual £ * = (0 -> Q* -> M* ->• P* -> 0) is also exact, and P* is projective. If s : P* ->• M* splits £* on the right, then identifying M** = M, P** = P, the dual map s* : M -> P splits £" on the left. Hence P is injective relative to JT(A[G]). Conversely, suppose that P is injective relative to .F(A[G]). Since P* is finitely generated, there is an exact sequence of A[G]-modules 0 -+ K -> F" -> P* -> 0 in which F" is free. By property T(2), A: e ^"(A[G]), so that this is an exact sequence in J"(A[G]). Making the identification P** = P, the dual exact sequence 0 ->• P ->• F" ->• /f* -> 0
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is also in .F(A[G]), and P is injective relative to .F(A[G]). Thus the dual sequence splits, P @ K* = F", so that P is projective. T(4): Let M e .F(A[G]); then M* e J"(A[G]). Since M* is finitely generated over A[G] there exists an exact sequence in J-(A[G]) of the form
where F" is the free module of rank n over A[G]. By T(2), we see that K € ^"(A[G]). Dualizing gives an exact sequence 0 - j . M** - > F" ->• A"* ->• 0
in ^(AfG]) in which the middle term is free. However, M** = M, so that we have an exact sequence 0 -> M -> F" ->• AT* ->• 0 in ^(AfG]) in which the middle term is free. This completes the proof.
•
Since J-(A[G]) is a tame class, then as in Section 20, for each module M e ^-"(A[G]) we construct a sequence of stable modules (£2n(M))nsz'- for n > 1, we require Qn(M) to be the stable class [D] of any module D e J^(A[G]) for which there exists an exact sequence of the form 0 ->• D ->• Fn_i ->
> F o -> M -> 0
where each Fr is a finitely generated free module over A[G]. Likewise, we require Q.-n{M) to be the stable class [D] of any module D e T{Z[G~\) for which there exists an exact sequence of the form 0 - > M - > F o -+
• • • - > F n _ , ->• D -»• 0
where each Fr is a finitely generated free module over A[G]. Duality is introduced into Stab(A[G]) by writing [MY = [M*] Observe that for any M e F(Z,[G~\), the tree structures on [M] and [M*] are isomorphic, even though, in general, the stable modules themselves are not. Proposition 28.6: For any module M e ^"(AfG]) we have the following relations: (i) Qm(Qn(M)) = (ii) S2n(M*) = St-n
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Stable Modules and the D(2)-Problem
The case of primary interest is M = Z, the trivial module Z over Z[G]. Note that for any finite group G it follows from the augmentation exact sequence 0 -» I(G) -> Z[G] 4 z ^ 0 that we have (28.7)
£2,(Z)
Moreover, the trivial module Z is self-dual, so that: (28.8)
n_,(Z)
29 The stable modules n n (Z) When F is a field whose characteristic does not divide \G\, the F-isomorphism type of the stable module fi,,(Z) is easily calculated. Since F[G] is semisimple, the augmentation exact sequence 0 -> I F (G) -* F[G] 4 F ^ 0 splits so that we have a direct sum F[G] considering, in sequence, the extensions
=F[C]
F©
IF(G).
In particular, by
0 ->• I F (C) -> F[G] -* F -* 0 and 0 -> F -» F[G] -* I F (G) -> 0 we see that the F-isomorphism type of the stable modules fi,,(Z) is determined as follows
I
[F]
if n is even
[IF(G)] if n is odd
Proposition 29.2: For each n > 0, Q 2n+i(Z) is a fork. Proof: Wedderburn's Theorem combined with (29.1) shows that [ I R ( G ) ] is the unique minimal representative of Q2«+i(Z) <8> R. If Mo is a minimal representative of fi2n+i(Z), it follows that Mo ® R = I R (G) © R[G]'" for some m > 0. Moreover, Wedderburn's Theorem, together with the isomorphism R[G] =R[G] R © IR(G), shows that every non-trivial simple module over R[G], a fortiori, every quaternionic simple module, occurs with multiplicity > 1 in IR(G). Hence every quaternionic simple module over R[G] occurs with multiplicity > m + 1 > 1 in MQ
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Typically, £22«+i(Z) can be pictured, thus
(29.3)
The above argument breaks down in the case of Qin{T). Nevertheless, the situation is still amenable to analysis, and, although the detailed structure of the even derived modules is not a crucial consideration in what follows, we sketch the results for completeness. In fact, there is only one extra complication; in addition to a fork, there is another possibility, which we call a crow's foot
(29.4)
This indicates that there is just one module at the minimal level, a finite number > 1 at level 1, and again just one module at every level > 2.
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Stable Modules and the D(2)-Pwblem
If Mo is a minimal representative of £22n(Z), then, by (29.1), there are essentially two possibilities for Mo ® R: either (*) M o ® R = R or (**) Mo
> A, -> Ao - • M -> 0
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with Ai e .F(Z[G]) and Proj"(M, A') is the subclass of Ext" (M, N) consisting of extensions in which each A,- is projective. Let M, N e J-(Z[G]). It follows from (24.1) that, for each n > 1, the following conditions are equivalent: (i) Proj"(M, AO ^ 0; (ii) N e D«(M); (iii) M g D_n(yV). Likewise Free"(M, TV) (resp. Stab"(Af, N)) will denote the subclass of Ext"(A/, A') where each Ai is free (resp. stably free). We have inclusions Free"(M, AO C Stab"(M, N) C Proj"(M, A7) C Ext"(M, A7) Slightly more delicate than the equivalence of the three conditions above is: Theorem 30.1: Let M,N conditions are equivalent:
e T(Z[G]); then for each n > 1, the following
(i) Stab"(M, N) # 0; (ii) N e n n (A/); (iii) M € J2_ B W. Moreover, these conditions are equivalent to: (iv) Free"(M, AO ^ 0 provided that e/r/ien (a) n > 2 or (b) « = 1 and G has the free cancellation property. Proof: We consider first the case where n > 2; then the equivalence of (ii) and (iii) is obvious, as are the implications (iv) =>• (i) and (iv) = > (ii). Thus it suffices to show that (ii) => (i), and (i) =>• (iv). (ii) = ^ (i): If F is a finitely generated free module over Z[G], and F, c F is a free Z[G] submodule, then an easy argument using the injectivity of F\ relative to ^"(Z[G]) or, what is the same, by double dualization using the universal property of free modules, shows that F/F\ is stably free over Z[G] if and only if F/F\ is torsion free over Z. Suppose that N e Sln(M)\ by definition, there exists a module D which occurs in an exact sequence 0 ->• D -> Z[G]Y - • F,,_2 ->
>• Fo -»• M -> 0
with y > 0, each Fr finitely generated free over Z[G], such that N © Z[Gf = D@ Z[G]* for some a, b > 0. Now D © Z[G]fc occurs in the exact sequence 0 -> D ©
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Stable Modules and the
D(2)-Problem
so that N © Z[G]a occurs in the exact sequence 0 - * N © Z[G]" - » Z[G]Y+b
-> F,,_ 2 ->
• F
We split this last sequence as a pair thus: (I) 0-+ N® Z[G]a -> Z [ G F + f c -> AT -> 0 (II) 0 - » AT -> F,,_ 2 ->
> F o - * M -> 0
Dividing through (I) by Z[G]'" gives an extension: (III) 0 - * AT -> ZtGl^+VZEG]" -> AT -> 0 in which N, AT e J"(Z[G]). Thus, Z [ G F + 7 Z [ G ] a is torsion free over Z, so = S is stably free over Z[G]. that, by the remark above, Z[G]y+b/Z[G]a Splicing (II) and (III) back together gives an element of Stab" (A/, N) 0 -> N -> 5 - • F n _ 2 ->•
> F o -» Af -> 0
as required. This completes the proof that (ii) = > (i). (i) = > (iv) Suppose that (0 -> A^ ->• 5,,_i ->• • • • ->• So ->• M -> 0) is an element of Stab"(M, TV) where each 5,- is finitely generated stably free. If n = 2 (this is only place where we need the hypothesis that n > 2), then there exists a > 0 so that 5,- ® Z[G]a is free for i = 1,2. Adding a summand of Z[G] fl to the middle terms gives an extension 0 - • N -+ Si © Z[G]° -> 5 0 0 Z[G]" - » M -s- 0 which defines an element of Free 2 (M, /V). If « > 2, we may first add a suitable free summand to Si and So to make So free; the result follows by induction after an easy splicing argument. This completes the proof that (ii) = > (i), and the proof of the Theorem when n > 2. In the case where n = 1, the equivalence of (ii) and (iii) is again obvious, and the proof that (ii) = > (i) given above goes through with a minor change of indexing. To show (i) = » (ii), suppose that O-*- N -> S -> M ^
0
is an extension in which S is a finitely generated stably free module. Then for some a > 0, S © Z[G] fl is free, and from the exact sequence 0 - * N © Z [ G f -+ S © Z [ G f -> A/ ->• 0 we see that A 7 ©Z[G]" e ^ i ( M ) . However, this easily implies that A7 e showing that (i) = > (ii). This completes the proof that (i), (ii) and (iii) are all
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equivalent in the case n = 1. Finally, if G has the free cancellation property, then (i) is equivalent to (iv). • We denote by Prof (Af, AO (resp. Stab" (M, AO, resp. Free" (/W, AO)theimage of Proj"(/W, N) (resp. Stab"(jW, N), resp. Free"(M, AO) under the canonical map Ext"(M, N) -» Ext"(M, AO Evidently Free"(M, AO C Stab"(M, A'); in fact, the argument of (30.1) also shows that for n > 2, Stab"(M, N) and Free"(M, AO represent the same congruence classes; that is: Proposition 30.2: For all n > 2, Free"(M, AO = Stab"(M, A').
31 Subgroup relations in the derived category If G is a finite group and H is a subgroup, there is an obvious 'restriction of scalars' functor
which, as the name implies, simply restricts scalars from Z[C] to Z[//]. There is also an 'extension of scalars' functor £GH : T{l{H\)
- • ^(Z[G])
given by £GH(M) = M ® Z[WJ Z[G]
Here the tensor factor Z[G] is simultaneously a left Z[//]-module and a right Z[G]-module; in particular, we have the following identity m ® h — mh
(m®gug2)
H^ m®g\g2
When G and H are unambiguous, we abbreviate to £ = £fi and 7?. = 72.^.
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For any right module Z[//]-module M, the Z[H]-dual Hom Z[//] (M, Z[//]) admits a natural left Z[//]-module structure, given by (h •
However, in the special case where M = Z[G], Homz[«](M, Z[//]) is naturally a (Z[//] - Z[G]) bimodule, with natural right Z[G]-module structure given by (
=
Evidently Z[G] also has a natural (Z[//] — Z[G]) bimodule structure, given by translation on either side. In fact, we have: Lemma 31.1: There is an isomorphism of (Z[H] — Z[G]) bimodules Z[G]4.Hom Z [ H ,(Z[G],Z[//]) Proof: Let p — [p],..., tient set H\G; that is
pn] be a complete set of representatives for the quo-
G = U"=1 Hpi where //p, D Hpj = 0 if i # ; Put A,- = pj" 1 . Then A. = {A.], . . . , ! „ } is a complete set of representatives for G/H. As a left Z[//]-module, Z[G] is free on {p\,..., p n }. Moreover, A.,,}, where as a left Z[tf]-module, Hom Z[//] (Z[G], Z[W]) is free on {A, ki : Z[G] ->• Z[H] is the right Z[//]-homomorphism given by
It follows that there is an isomorphism of left Z[//]-modules v : Z[G] - • HomZ[W](Z[G], given by
and straightforward computation shows that v is equivariant with respect to the right G action. D The functors £, H have a number of easily verified properties; they are additive, exact, and take free modules to free modules. It follows that they also take projective modules to projective modules. In addition, they have the unusual property that they are simultaneously mutual left and right adjoints without being mutually inverse; that is:
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Proposition 31.2: There are natural isomorphisms Hom Z[C] (£(M), N) = HomZ[W](M, TZ(N)) Homzm(TZ(N),
M) = HomZ[G](/V, £(M))
Proof: The first of these is entirely straightforward. In the second, it is useful to make use of a different model for £. Denote by £'{M) = Hom Z|W |(Z[G], M) where the right Z[G] structure on £'{M) is given by (a • g){x) = a{xg-x) There is a preliminary natural equivalence v, : £'(M) -> M ®zm
HomZ[Hi(Z[G],
Z[H])
By (31.1) above, there is now an equivalence v2 : M ® Z[H] Hom Z[W](Z[G], Z[H]) - • yW ® Z[H] Z[G] = f (Af) which is natural in M and with respect to the right Z[G] action. Then v = v2 o V, : £'(A/) -> £(M) is the desired natural equivalence. The adjunction isomorphism if : HomZ[G] (N, £\M)) -> HomZ |H] (^(^V), M) is given by [f(a)](n) = a(n)(l)
D
We need to distinguish between the derived categories and associated constructions corresponding to the different groups G, H. For an arbitrary group f we denote by: (i) X>er(r) the derived category of .F(Z[r]); (ii) Dj"(M) the nth derived object of the module M e .F(Z[r]); (iii) Q^(M) the nth stable module of M e Since the functors £, TZ are exact and preserve projectives they are definable at the level of the derived category; that is we have TlGH : Der(G) -» Per(//) and £°, :
VQT(H)
Moreover, they each commute with D n ,
-> I?er(G)
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Stable Modules and the D(2)-Pwblem
and £(DnH(M)) = DnG(£ (M)) Because they preserve free modules, they also commute with
and
£(^{M))
= Q?n(£{M))
Proposition 31.3: There are natural isomorphisms HomVeKG)(S(M),
N) ^
UomVer{H)(M,
HomVer(H)(n(N),
M) = HomVer{G)(N, £(M))
Proof: If T is a finite group, and M, N e J"(Z[r]), then Homx,er(r)(M, N) = Hom z i n (M, N)/Hom°Z[r](M, N) where Hom^rjCM, A') denotes the subset of Homz[n(M, N) consisting of homomorphisms which factor through a projective. It suffices to show that the adjunction isomorphisms induce bijections [ |
[
]
) , N)
and Hom°Z[H](Tl(N), M) <-^ HomZ[G](N, E{M)) Suppose f : M -¥• 1Z(N) admits a factorization f — cpoi, where i : M -> P, cp : P —> 1Z(N) and P is projective. Then the adjoint morphism / : £{M) —> N factorizes as / =
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129
projective TZ(Q). It is straightforward to see that g = / , and the correspondence is surjective as claimed. Thus the adjunction map / i-> / induces a bijection Hom°Z[H](M, H{N)) <—• Hom°Z[C](S(M),
N)
as claimed. The bijectivity of HomZ[H](TZ(N), M) <—> HomZ[C](/V, £(M)) follows by a similar argument. • Combined with (25.7) and with the representation and co-representation formulae (20.6), (20.7), adjointness gives rise to the following Eckmann-Shapiro relations in cohomology (31.4)
ExfZ[H](TZ(M), N) = ExtJ[C](A/, £(N))
and (31.5)
ExtZiC](£(A), B) = Ext'Z[H](A, H{B))
When M is the trivial Z[G]-module Z, TZ(M) is the trivial Z[//]-module Z, and the first of these specializes to the classical 'Shapiro's Lemma' (31.6)
H"(H, N) = U"(G,£(N))
32 Restriction and transfer Adjointness gives rise to some natural transformations. Thus fix a Z[G]-module A; since UomZiH](Tl(A), K(A)) = HomZ|G] (A, £H(A)) then, corresponding to I d ^ ) we obtain a Z[G]-homomorphism S : A -* £U{A) Let d = [G : H] be the index of // in G, and let { p i , . . . , p
S(a)= > apf'
®pi
/=]
In addition, HomzrH](7?.(A), TZ(A)) = Hom Z[C |(f TZ(A), A), so that, again corresponding to Idft^), we also obtain a Z[G]-homomorphism 6 : £TZ(A) ->• A
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Stable Modules and the D(2)-Problem
€ is the canonical 'contraction map' given on primitive tensors by e(a
€ o S(a) = da for all a € A
The composite S o e : £TZ(A) —»• £7£(A) is computed as follows; the general element in £ 1Z(A) can be represented canonically as
where z,- e A. Then e(z) = 5Zf=1 z,, and d
S€(z) = When A is a (right) lattice over Z[G], we can give a 'coordinatized' description of£7£(A)asaforrnofwreaf/ipro£/McrasfoHows: let{pi,..., pj] be a complete set of coset representatives for H\G. Let E Ej by means of
Then £TZ(A) can be described as A(d) = A d
with typical element written as a = {a\, a-i,..., aj) and G-action given by
The mapping 4>(a,, a2,...,
ad) =
^djpr
is the desired isomorphism of right Z[G]-modules f 7?.(A) admits another G-action, conjugacy, as follows: if g e G, we define the conjugacy map cs : £ 1Z(A) —> £1Z(A) by 8
Id
1
\
c I ^ a z / o f
/
d
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The elements of £1Z(A)) which are fixed by each cg are those of the form 8(a) for some a € A. Moreover, cg induces an automorphism cl on any Homper(C)(M, £1Z(A)) and hence on any H"(H, 1Z(A)), via the EckmannShapiro relation, H"(H, ll(A)) = U"(G, £TZ(A)). An element z e H"(//, H(A)) is said to be stable when cl{z) = z for all g e G. Observe that, for a stable element z, i*t(z) = dz where d is the index [G : H). For any Z[G]-lattice TV, there is both a canonical 'Shapiro isomorphism'
^ : H"(G, £1l(N)) = H"(//, TZ(N)) and a transformation 5* : H"(G, A') -» H"(G, £Tl{N)) induced from the homomorphism of coefficients 8 : N —> £TZ(N). If / denotes the inclusion H C G, it is usual to write /* = s o <5, : H"(G, N) -> H"(//, 7e(/V)) ('* is the restriction homomorphism. Likewise, we have a transformation e» : H"(G, £TZ(N)) -+ H"(G, A^) induced from the homomorphism of coefficients e : £7l(N) -*• N. It is usual to write ( = e , o r ' : H"(H, 1l(N)) -> H"(G, A^) t is the transfer homomorphism. The following is clear: Proposition 32.2: Let*' : H c G be the inclusion of a subgroup H in the finite group G; then, for any Z[G]-lattice N, the composite ( o i * : H"(G, AO ->• H"(G, N) is multiplication by rf = [G : / / ] . From this follows the well-known fact that, for n > 0, H"(G, AO consists entirely of torsion of order dividing \G\. In fact, taking H = {1) to be the trivial subgroup, we see that e, o 5* : H"(G, A') -»• H"(G, A') factors through 0 = H"(H, 1l(N)) and has the effect of multiplication by the index, |G|, of {1} inG. Corollary 32.3: If G is a finite group and n > 0, then for each Z[G]-lattice N, H"(G, N) is a module over Z/|G|.
Chapter 6 ^-invariants
Having established the general structure of £2,,(Z) in Chapter 5, we now turn to the problem of classifying extensions 0 -» J -* An_, -*
• Ao -* Z -* 0
where J e £2n(Z) and /4r e .F(Z[G]). In Chapter 4, this problem was reduced, in principle, to that of computing the group Ext"(Z, J). Here we compute Ext"(Z, J) = H"(Z, O,(Z)) = Z/|G| where Z/|G| arises as the endomorphism ring, in the derived category, of the trivial module Z. In Section 34 we show how the method of ^-invariants, due originally to Maclane and Whitehead [35], allows a classification of projective «-stems by the group of units (Z/|G|)*. In Sections 35-37 we give our version of Swan's explicit description of onedimensional extensions over Z using the ^-invariant method. This is extended to general extensions in Section 38. Finally, in Section 39 we give Tate's generalization of the action of Endper(Z) on extensions. We introduce the Tate cohomology groups Ylk(G, A) for all integer indices k, positive and negative, and show how H*(G, A) becomes a graded module over the graded ring H*(G, Z).
33 Endomorphisms in the derived category Let G be a finite group, and let Z denote the trivial module over Z[G]. Note that there are Z[G]-homomorphisms £ : Z -*• Z[G] and e : Z[G] ->• Z given by ) \g€G
()
)
\geG
/
The following two propositions are easily verified. 132
geG
k-invariants
133
(33.1) If
z We may write
so that
134
Stable Modules and the D(2)-Problem
Conversely, if a = m\G\, then a factors through Z[G] since a = me o E; that is: a factors through some Z[G] m <=>• a = n\G\ for some « and the result now follows.
•
Corollary 33.4: A(Z) = (Z/|G|)* Let M, D e .F(Z[G]) be such that Proj"(M, D) ^ 0 and let P = (0 -* D -> />„_, ->
• P o -> Af -> 0)
be a projective rc-stem over M terminating in D; recall that in Section 26 we showed that P gives rise to a ring isomorphism Kp : Endper(D) ^> EndperCA^). Moreover, if Q = (0 - • D - • g n _, - •
• go - • M - • 0)
is another element of Proj"(M, D), the ring isomorphisms KP, KQ are related by (33.5)
where APQ = AQP is the unique automorphism A.PQ of M in the derived category making the following diagram commute 0 -> D -»• 2,,-i -^ 4, Id
>• 2o -»• A^ -». 0
\, kn-\
\.
0 ->• D -> />„_, ->•
XQ
-I A.PQ
>• Po -> ^ ->• 0
In the case where End-Der(M) is commutative, conjugation is trivial, and so (33.6)
Kp = KQ
that is, we have a ring isomorphism K = Kp : Endx)er(£)) —»• Endper(M) which is independent of the particular choice of projective n-stem P used to construct it. Though, we have not stressed the point, K may perhaps still depend on n; that is, the length of the projective stem. However, in the case of most interest to us, namely M = Z, the independence of K is absolute. Then the ring Endper(Z) = Z/1G | is not only commutative, but, as is easily seen, its only ring automorphism is the identity; thus any two ring isomorphisms \fr,
:
k-invariants
135
In particular, (33.7) permits the unambiguous identification Endx>er(Z) =
34 ^-invariants and the action of Autx>er(/) In Section 26 we showed how to classify projective «-stems P e Proj"(M, D) by means of A(D). We now make the classification explicit in the case where D e Dm(Z), using the isomorphism Endi>er(D) = Z/|G|. If M, N are modules in .F(Z[G]) such that M e Dffl(Z) and N e D,,(M), then Proj"(M, N) ^ 0. Thus, suppose that P e Proj"(M, A') is a projective «-stem and E is an arbitrary element of Ext" (M, AO, the universal property of projective modules leads to a commutative diagram of Z[G]-modules, thus
>
N
—> tLn—\
—> • • • —
We define the k-invariant of the transition, /t(P -> E) e Z/|G|, by *(P - • E) =
K(a+)
It is easy to verify the following properties: Proposition 34.1: If E e Ext"(M, AO and P, Q e Proj"(M, AO then: (i) fc(P ^ E) = £(P ^ Q)fc(Q ^ E); (ii) * ( P - » - P ) = l ; (iii) k{PMaking a specific choice of projective extension P e Proj"(M, A^), if E e Ext"(M, AO, then E e Proj"(M, AO <^=> k(P -»• E) e (Z/|G|)* and we obtain: Proposition 34.2: The correspondence E \-t k(P —*• E) defines a bijective mapping Proj"(M, N) -+ (Z/|G|)*. We observed in Section 26 that Ext"(Z, 7) is a left module over the ring Endp er (7) when J e Dn(Z) End Per (7) x Ext^(Z, J) -+ Ext^(Z, J) [a] * [E] = [«,(E)] We note the following obvious identity:
136
Stable Modules and the D(2)-Pmblem
Proposition 34.3: Let P e Proj"(Z, J) be a projective n-stem and let E e Ext"(Z, J); then for all a e Endper (7)
a,(E)) = K(a)k(P -* E )
k(P^
> Choose a lifting of the identity over Z P E
/ o - > y - 4 /»„_, -> • Po^ in '= l
0 4-Id
Z ^ 0
from which we calculate that fc(P —>• E) = ic(
a.(E) from
y
-•
J
Po- * Z - •
4- a o
\o -^
y
i —>
pi
1 .
0
I Id
E'o- > Z -> 0
which we calculate that J f c ( P - • a*(E0)
== K((X
o a '+)
) =
P - >E)
n
We previously observed, in Section 26, that the unit group A(J) = Autver(J) stabilizes the subcategoryProj"(Z, y). IfP € Proj"(Z, y)anda e A(J), then, since a*(P) is also a projective n-stem, the ^-invariant ^(a*(P) -»• E) is denned for any E e Ext"(Z, y). The ^-invariant is covariant with respect to the action of ^4(y) in the following sense: Proposition 34.4: Let P e Proj"(Z, y) be a projective n-stem and let E e Ext"(Z, y); then for all a e A{J) k(a,(P) - • a.(E)) = *(P - • E) Prao/- We have seen that Jt(P -> a»(E» =
D
k-invariants
137
35 The simplest stable modules to study are ^i(Z) and its dual Q_i(Z). From the exact sequence 0 -> I(G) -4 Z[G] - 4 z - » 0 defining I(G) = Ker(e) we see that the stable module ^i(Z) coincides with [I(G)]. Moreover, since rk z (I(G)) < rk z (Z[G]), we see that I(G) is a minimal representative of Q i (Z). Proposition 35.1: Let G be a finite group which has the cancellation property for free modules; then I(G) is the unique minimal representative of ^i(Z), so that S2i(Z) is straight. Proof: First note that, if 77 : Z[G] ->• Z is a surjective Z[G]-homomorphism, then ??(•) = ± 1 , so that r\ — ±e. In particular, Ker(??) — I(G). Suppose that there is a module isomorphism h : 7 © Z[G] = / © Z[G], and consider the extension 0 -* 7 © Z[G] -V Z[G] © Z[C] 4- Z -> 0 where y = (/ © 1) o h and where r) = € o p, with /? : Z[G] © Z[G] -> Z[G] denoting projection on to the first summand. We may factor out the free summand in Ker(rj) to get an exact sequence
where P = Z[C] © Z[G]//(0 © Z[G]). It follows that P is torsion free over Z. However, there is another exact sequence 0 -» Z[G] -* Z[G] © Z[G] - • P -> 0 which, since P is torsion free over Z, is an exact sequence in ^-"(Z[G]). From the relative injectivity of free modules it follows that P © Z[G] = Z[G] © Z[G], so that P is stably free over Z[G]. By hypothesis on G, P is free, necessarily having vkz[c\{P) = 1. Since 77*: P -» Z is surjective, then, by our initial • observation, 7 = 1 , and this completes the proof. In Chapter 9, we shall see the above conclusion fails in general when the cancellation property for free modules fails to hold. Since duality 7 i-> 7* defines a 1-1 correspondence £2i(Z) -o- Q_|(Z) we see from (35.1) that: Corollary 35.2: If the finite group G has the cancellation property for free modules, then I*(G) is the unique minimal representative of Q_i(Z), so that Q.-\(Z) is straight.
138
Stable Modules and the D(2)-Problem
Even without the hypothesis of free cancellation, it is possible to say something. Firstly, the dual augmentation sequence 0 -> Z 4- Z[G] ^ I*(G) -> 0 is still an exact sequence in .F(Z[G]). Moreover, €*( 1) is the the central element £ = J2geG e z t G ] - T n u s Im(e*) is the two-sided ideal of Z[G] generated by S, and I*(G) = Z[G]/(E). It follows that: Proposition 35.3: I*(G) has a ring structure with respect to which/* : Z[G] ->• I*(G) is a ring homomorphism. This has a useful consequence: Proposition 35.4: I*(G) is monogenic as a module over Z[G], being generated by the image of 1 under /*. This extends to give a criterion for recognizing I(G)* within its stability class: Proposition 35.5: Let J be stably equivalent to I(G)*; then J = Z [ C ] I(G)* if and only if there exists a surjective Z[G]-homomorphism
k-invariants
139
36 Swan modules Module extensions of the form 0 ->• I(G) -> M -> Z -> 0 are classified by Ext'(Z, I(G)) = Ext'(Z, £2i(G)) = Z/|G|. Here we give explicit models, first introduced by Swan in [58] (see also [63]), for the modules corresponding to the various ^-invariants. Since I = I(G) is a submodule of Z[G], the representation to which it corresponds, denoted by g v-*- k(g), is simply the restriction of the regular representation of G to I. For each r e Z we may construct a representation g i->- kr(g) of G on the abelian group I(G) © Z . , ,
(Kg)
rg-r\
We denote by (I, r) the Z[G]-module whose underlying abelian group is I(G) © Z and whose associated representation is kr. In passing, we note that, in terms of the standard interpretation of cohomology using the 'bar resolution' ([34], Chapter IV), the function g i->- rg — r is a 1-cocycle on the Z[G]-module Z taking values in I(G). This observation, however instructive, is not necessary in what follows. The projection from (I, r) onto Z is a Z[G]-homomorphism, and defines an extension £(r) = (0 ->• I -> (I, r) -» Z - • 0) A. i is equivalent to the regular representation, so that £(1) is simply the defining extension of the augmentation ideal £{\) = (0 - • I -> Z[G]) -4 Z -> 0) with the consequence that (I, 1) is isomorphic to the group ring Z[G]. Consequently we may, and do, take £(1) as the reference extension in calculating ^-invariants. With this convention we have: Proposition 36.1: £{r) is the extension with ^-invariant given by k{£{\) -> £(/•)) = r Proof: First note that there is a commutative diagram Z[G]4
z->
0
4- x r 4. x r 4, x r Z[C]4
z->
0
140
Stable Modules and the D(2)-Problem
which gives us K(1 —-> I) = r where K is the unique ring isomorphism K : Endj>er(I) -»• Z/|G|. However, there is a commutative diagram, thus £(\)
/O-* 1 ^
4-(pr= £{r)
(I, l)-»>Z-». 0s
J.XC \ 0 - » . I-*-
4-^r
4-
(I,r)-
where, relative to the cocycle description of £(r), <pr is described by the matrix
By definition, £(£(1) -> £(>)) = AT(I ^> I), giving the value k(£(\) -> £(r)) = D r by the above. This completes the proof. It is useful to have an alternative description of the modules (I, r). Observe that e~'(/-Z) is a two-sided ideal in Z[C]: Proposition 36.2: (I, r) = e /- Let ??r : e'V^Z) -> Z be the homomorphism ?r() Then r\r is surjective. Let T denote the exact sequence T - (0 - • I - • e-'(rZ) ^> Z - • 0) We have a commutative diagram
= r. It follows that D
Since Ext2[G](Z, I) = Z/|G|, the congruence class of £(r) is entirely determined by the residue class of r (mod|G|); that is: Proposition 36.3: £(r) % £{r') <=*> r = r' mod|G|. Proof: We describe an explicit congruence
irm : £(r) ^
£(r + m\G\)
k-invariants
141
by means of Id m ( E - | G | ) \ Id
D
37 Swan's isomorphism criterion In Section 36, we gave a complete description of the congruence classes of Swan modules. We now turn to the more general problem of deciding when two Swan modules are isomorphic. Since I = I(G) e S2|(Z), it follows quite generally that EndOer(I) = ) and there is a natural surjective ring homomorphism vl : Endzicid) ~> Z / | C | We show how the monogenicity of I* allows for a direct calculation of v1. First consider the 'reduced norm' factorization in our context. Over any ring A, a short exact sequence of A-modules
is said to be. fully invariant when each A-endomorphism q> : M —> M preserves the exact sequence; that is, when each
making the following commute 0-> K -4 M A Q^ I
I
0
o^/c-4M-4g^o Proposition 37.1: The extensions £(1) and £(1)* are fully invariant. Proof: Ifw e Z[G], we denote by X« : Z[G] ->• Z[G] the Z[G]-endomorphism given by A.1((x) = MX Clearly every Z[G] endomorphism of Z[G] has this form. However e is a ring homomorphism, so that, if x e I(G), then e(A.B(x)) = e(A.H)6(x) = 0
142
Stable Modules and the D(2)-Problem
and so A.,,(I) C I. In particular, the following diagram commutes 0 -> I(G) 3 Z[G] -4 Z -> 0 I K\i
IK
I €(u)
0 -> I(G) -3 Z[C] -4 Z -). 0 that is, £(l) is fully invariant. Full invariance of £(1)* follows by duality.
•
Any endomorphism X of Z[G] has the form X = Xu, and we write det(w) = ,, : Z[G] -> Z[G]). Put
By duality, N(u) is also the determinant of the Z-linear mapping I* -»• I* induced by Xu on I*. Full invariance of £(1) or £(1)* shows that (37.2)
det(w) = e(u)N(u)
To proceed, it is technically simpler to deal with £{\)*. By (35.4), the dual augmentation module I*(G) is monogenic as a Z[G]-module; it follows that every Z[G]-endomorphism / : P(G) ->• I*(G) has the form / = ixu for some u e Z[G]; again by duality: Proposition 37.3: Every Z[G]-endomorphism cp : I(G) -*• I(G) has a representation in the form
IK
;o 'I I*(G) -* 0
Since the map at the right-hand end is zero, it follows that the Z[G]-endomorphism e(w) : Z -> Z factors through the free module Z[G]. By (33.1), e(u) — n\G\ for some integer n. It follows that if u\,U2 e Z[G] represent the same endomorphism of I(G) then e(«i)-e(ii2)s0mod(|G|) If [ ] : Z —*• Z/|G| denotes the natural epimorphism, a chase of definitions shows that:
k-invariants
143
Proposition 37.4: If u e Z[G] is any element such that
= [€(«)]
If cp = XU|! belongs to the unit group of Endz[c](I(G)), then e(«) = vl(
(I, r) X
Z -> 0
0 -* I -> (I, rs) -^ Z -> 0 Now Xu : I —>• I is an isomorphism by the hypothesis follows from the Five Lemma.
./V(M)
= ± 1, so the result D
We put T(G) = \m(vl) C (Z/|G|)*. Observe that T(G) acts on (Z/|G|)* by left translation
T(G) x (Z/IGD* (M,
[s])
»
[rs}
Theorem 37.7: Let [r], [^] e Z/|G|; the extensions £(r), f (^) are isomorphic if and only if there exists rj 6 F(G) such that r = ^5; that is, if and only if [r], [s] belong to same orbit of T(G) in Z/|G|. Proof: ( = ^ ) Let
l
4, Id
144
Stable Modules and the D(2)-Problem
Choose u e Z[G] such that A.,,|; = (p0 and consider the following diagram
0 ^ 1 ^ I
(I, s) 4- Z -* 0 -1
|
.
, TJ €_
where ?j = e(t<). Composing the down arrows, we obtain a congruence
Id
Id
o/ It follows immediately that rjs = r mod|G| and proves ( converse is just (37.6), so the proof is complete.
>•)• However, the D
As a consequence, we get the following theorem of Swan which completely describes the isomorphism types of the modules (I, s). In fact, Swan proves the dual result (see (37. 11) and (37.12) below). Theorem 37.8: For [r], [s] e Z/\G\, (I, r) = (I, s) if and only if there exists r] G T(C) such that [r] = rj[s]. Proof: Let Q denote the trivial Q-module of dimension 1; then Q occurs in Q[G] with multiplicity 1. In particular, it does not occur in the rational augmentation ideal IQ. Consequently, any Q[G]-homomorphism ijs : IQ ->• Q must be trivial, a fortiori, any Z[G]-homomorphism iff : I —>• Z is also trivial. Suppose that (p : (I, r) -> (I, s) is an isomorphism; it follows from the above remarks that the restriction of | o cp to I is trivial. Hence the following diagram commutes
0
I->• (I, r ) - t 4,
\. (p
± Id 0/
and
8{s). The conclusion
a
k-invariants
145
Let r, s be positive integers; we observed in (36.2) above that (I, r) = e~'(rZ). Let ns : Z -» Z/s denote the canonical surjection, and let r]rs : (I, r) -» Z/s denote the composite e r We see that for any r, s there is an exact sequence 0 -> (I, rs) -> (I, r) ^ Z/s -* 0 As a consequence, we have: Theorem 37.9: If r is coprime to |G|, then, for any positive integer s, there is an isomorphism: (I, rs) 8 Z[C] = (I, r) © (I, s) Proof: As a special case of (37.8) we get an exact sequence 0 - > ( I , s ) ^ (I, l ) ^ Z / s - » O whilst the general case of (37.8) gives the exact sequence 0 -> (I, rs) ->• (I, r) ^> Z/s -^ 0 (I, 1) is free, hence projective; since r is coprime to \G\, (I, r) is also projective. Applying Schanuel's Lemma to the above pair of exact sequences gives an isomorphism (I, rs) 0 (I, 1) = (I, r) ® (I, s) The result follows since (I, 1) = Z[G].
D
We consider briefly the corresponding results for I*(G) under duality. Since Z/1G | is commutative, vl can equally well be regarded as a ring homomorphism vl : End Z [ C ] (ir" -»> Z/|G| However, the correspondence a \-> a* induces an isomorphism of rings End Z [ C | (I)' w = End Z [ C ] (r) so that there is a surjective ring homomorphism v* : End zlC) (F) -+ Z/|G| defined (implicitly) by the formula v*(a*) =
v\a)
146
Stable Modules and the D(2)-Problem
A straightforward chase of definitions shows that (37.10)
v = v*
Since I* = I*(G) has a natural ring structure there is a further simplification; the epimorphism ix : Z[G] -*• Endz[c](I*); w i-> ixu induces a natural ring isomorphism I* = Endz[G](I*X and hence an isomorphism of groups U(V) 4 Aut Z[ c](T) Composition with v* gives a homomorphism of groups r\ : U(l*) ->• (Z/|G|)*, and a chase of definitions shows that: Proposition 37.11: Let a = i*(v) e U(I*(G)) where t; e Z[G]; ^(a) 6 Z/|G| is computed as r,(a) = [€(«)] Swan's own expression of the above isomorphism criterion ([58], Lemma (6.3)) is: (37.12) (I, r) = Z [ C ] (I, r')&r
= r\(a)r' for some unit a e I*(G).
In particular: (37.13) (I, r) is free « = ^ there exists a unit a 6 P(G) such that r = rj{a).
38 Congruence classes and Swan modules We begin by recalling a basic fact about the pushout construction. This is implicit in the results of Chapter 4, but, for clarity, we spell out the details in a particular case. Lemma 38.1: Let E, F e Ext"(Z, J), and let a : J ->• J be a Z[G]homomorphism. If there is an elementary congruence h : E -> F, then there is an elementary congruence ti : a*(E) ->• a*(F). Proof: Decompose E as a Yoneda product E = E + o E_ where
E+ = ( 0 ^ y 4 £„_, ->. A: -^ 0); E_ = (0 -^ # -> £ n _ 2 ->
• £o -^ Z -» 0)
^-invariants
147
Likewise, decompose F = F + oF_ and suppose that h : E ->• F is an elementary congruence, thus £ 0 -> Z -> 0 '
E |ld
;/.„_,
I/j0
I Id 0
Fo We may split h under Yoneda product 0
E+ F+
0\
0 By general properties of pushouts, any morphism a : J —> J induces a morphism on pushouts as below /0->- 7-3
a»(E+)
ih'+ a,(F+)
=
| Id \ 0 - » 7-s
| r\ L ->• (
where £„_, = ( 7 8 £ B -i)/(« x - i ) , FB_, = ( 7 8 F«_i)/(a x - 7 ) and A;_, is the map induced by the matrix Id 0 0 hnBy definition, a*(E) = a*(E+)oE_,anda*(F) = a t (F+)oF_. Since the righthand end map of h'+ is still r\, and the left-hand end map is still the identity on y, h'+ and /i_ glue together to give the required elementary congruence y -* F;,_, - » — > E O - * Z ^ O \ lid
;/«„_,
|/i
y -> F,;_, - > — > F
O
0
|Id
n
^ Z ^ O /
Suppose that 7 e Dn(Z); by (26.1), the set Proj"(Z, 7) of projective «-stems (0 - • 7 - > PB_, - •
>• Po - • Z -»> 0 )
148
Stable Modules and the D(2)-Problem
is nonempty. We wish to find a convenient parametrization of the set of congruence classes in Proj"(Z, J). First observe that J e D,,_i(I(C)), so that, by (26.1), we may choose a projective (n - l)-stem Q e Proj"~'(I(G), J) which, by re-indexing, we may write for convenience, thus Q = (0 -> J -> g,,_, - • . . . _ • Q, - £ I(G) - * 0 The Yoneda product Qo£(t)
takes the form
Q o S(t) = (0 -+ J - » 2 « - i - • • • • - » > Gi ' ^ (/, 0 - ^ Z -> 0) From a straightforward chase of definitions we get
(38.2)
k(Q o £{\) -> Q o 5(0) = [r] e (Z/|C|)*
Proposition 38.3: Let J e Dn(Z) where n > 2, and let Q be any projective (n - l)-stem in Proj"-'(I(G), J); then: (i) for V e Proj"(Z, y) there exists a unique t e (Z/|G|)* such that V * Q o £(t); furthermore (ii) if a : J —> J is any Z[G]-homomorphism, then a*CP) ^ Q o £{tK{a)). Proof: To prove (i) choose Q e Proj""'(I(G), 7) and put [1] = Q o £"(1) e Proj"(Z, y). Suppose k{[\] -+ V) = t e (Z/|G|)*, then by (38.2), t = k{[\] —*• Q o £(t)), so that P % Qo £{t). Uniqueness is clear. To prove (ii), let va : V -> a*CP) be the canonical morphism associated with the pushout construction; va takes the form /
0 ->• y ->• P n -i ->
^ P o -» Z -> 0"
4-0? -|, v;I_i
^
UQ
4- Id
o^y^p; , ^ —>po^z->o and tautologouslyfc(P->• a*CP)) = transition k([\] -> a*(P)) thus
K(<X).
We calculate the ^-invariant of the
However, [r]/c(a) = k([\] -> Q o £(tic(a)), so that a*(P) % Q o £(tic(a)), completing the proof. • 39 The Tate ring Let G be a finite group and let
k-invariants
149
be a resolution of Z by finite generated projectives P,. Then the dual sequence
is also exact. For convenience, we write v*
= ( Q ^
Z
£ >
P_XH
p_2H
•••%
p_nh
^
••)
where P_n = /*„*_, and 3_n = 3*. Splicing V and P* together gives a so-called 'complete resolution'
If A is a Z[G]-module, the Tate cohomology groups Hk(G, A) are defined by H"(G, A) = Ker(3,f+1)/Im(9,f) where, as usual, 3^ : Hora A (/ ) n -i, A) ->• HomA(f,,, /I) is the induced map 3,f (a) = a o dn. When A is a Z[C]-lattice, we have: Theorem 39.1: For all n e Z H"(C, A) ^ HomDer(D,,(Z), /I) Proo/- By (25.7), it suffices to show that ft-"(G, A) = Hom Per (D_ n (Z), A) when « > 0. Take a truncated resolution
and its (re-indexed) dual
^ = ( o ^ z ^ , ^ p_2 ^ ... 3 V p_w _ y*-> o) Here 7/v e -^*(Z[G]) represents DN(Z), and so D_W(Z) is represented by 7^. Assume thatn < Af;thenH-"(G, A) = Ext^-^y*, /I) = Homu e r(DAr- n (^), /I), and hence ft-"(G, A) = HomI,er(DW-nD_A,(Z), A) = Homx3er(D_n(Z), A) as claimed. D We now specialize to the case A = Z. Then, for all m, n 6 Z, there are dimension shifting isomorphisms (depending on the choice of the complete resolution V) Hom Per (D n(Z), Z) = Hom Per(Dn+,,,(Z), Dm(Z))
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Stable Modules and the D(2)-Problem
Moreover, composition (f,g)^fog
gives bilinear maps for any a, b,c e Z.
o: HomCer(Dfc(Z), DC(Z)) x HonWD^Z), D 6(Z))-> HomOer(Da(Z), DC(Z)) Re-interpreting the Tate groups H*(G, Z) via the dimension shifting isomorphisms above, we get H"(G, Z) = Hom 0er (D n+m (Z), D,,,(Z)) and hence bilinear pairings o : H*(G, Z) x H; (G, Z) -+ H*+/(G, Z) It is known that the product is anti-commutative; that is
for x, e H*'(G, Z) It follows from standard universal coefficient theorems that ft"*(G, Z) = HomAb(H*(G, Z); Z/|G|) Moreover the pairing ft"*(G, Z) x ft*(G, Z) - • H°(G, Z) = Z/|G| is equivalent to the evaluation pairing HomAb(H*(G, Z), Z/|G|) x H*(G, Z) - • Z/|G| Finally, for any Z[G]-lattice A, the Tate groups H*(G, A) possess the structure of a graded module over the Tate ring. In fact, writing Dn = D,,(Z), composition Homper(Dp, A) x Homo er (D p+9 , Dp) ->• Hom 0er (D p+9 , A) becomes the pairing H p (G, A) x H ? (G, Z) -^ Hp+
Chapter 7 Groups of periodic cohomology
For a finite group G there are a priori two possibilities; within the derived category of Z[G]: either (i) the derived modules (D,,(Z)),,<=z a r e isomorphically distinct or (ii) Dn(Z) = Dm(Z) for some m, n eZ with m =fi n. In this chapter, we show that the categorization of finite groups within the two types depends only upon the Sylow subgroup structure. In particular, 'most' finite groups are in case (i), and G is of type (ii) when for each odd prime p, the Sylow /7-subgroup is cyclic and the Sylow 2-subgroup is either cyclic or generalized quaternion. Zassenhaus [84] has classified all finite soluble groups satisfying these restrictions, and his classification has been extended by Suzuki [56] to include nonsoluble groups. In a final section we give a brief outline of their results.
40 Periodicity conditions We say that n > 0 is a cohomologicalperiod of G if Dll+k(Z) = Ac(Z) for all k e Z. There are a number of ways of characterizing this possibility. Consider the following conditions which can be placed on a finite group C: V\(n): Dn+k(Z) = Dk(Z) for all integers k; V2(n): Dn+ic(Z) = Dk(Z) for at least one nonzero integer k; P 3 (n): There exists an exact sequence in J"(Z[G]) of the form 0 ->• Z ->• P n _ , - »
where each P, isfinitelygenerated projective. VA(n):Y?'(G-Z)^Z/\G\. 151
> Po ->• Z ->• 0
152
Stable Modules and the D(2)-Problem
Theorem 40.1: If G is a finite group and n is a positive integer, then the conditions V\{n), Vi{n), V^n), V^in) are equivalent. Proof: The implication V\(n) =*• Viin) is clear. To prove that Vj,{n), suppose that Dn+k(Z) = Dk(Z) for some nonzero integer k. Since D_k is a self-equivalence of the derived category, we see that there are isomorphisms in the derived category A,(Z) = D_kDn+k(Z) = D.kDk(Z)
= Z
Using the description of isomorphism types in the derived category given in Section 20 we can express this in terms of exact sequences as follows. There is an exact sequence 0 -> M -* Pn_x -*
> Po -» Z - • 0
where each P, isfinitelygenerated projective, and where M®Q\ = Z © 02 for some finitely generated projectives Q\, Qi- We may modify the above exact sequence in stages: firstly, to an exact sequence of the form 0 -* M © Q\ - • Pn-\ © Q\ - • Pn-2 - •
>• ^o - • Z -»• 0
since A/ © 2 i = Z © Q2. we then obtain an exact sequence of the form 0 -+ Z ©
^ Po -» Z -> 0
Finally, from the relative injectivity of projectives, we see easily that the obvious quotient P' = (Pn-\ © 2 0 / 6 2 is projective. Hence we get an exact sequence 0 - • Z -+ />' ->• P n _ 2 -»-
^ Po -+ Z ->• 0
in which Po • • • Pn-2» P' are all finitely generated projective, and V-}(n) is satisfied. To prove V^(n) =$• V\(n), suppose that V^in) is true, then there is an isomorphism (in the derived category) Dn(Z) = Z. Since each Dk is a selfequivalence of the derived category, we see that Dn+k(Z) = Dk(Z) for all integers k, and this is condition V\(n). We have now shown that conditions V\(n),V2(n),V-i(n) are equivalent. However, if Dn(Z) is isomorphic in the derived category to Z, we see that H"(G, Z) = Homx,er(Dn(Z), Z) = Homper(Z, Z) = Z/|G| and this proves that V\(n) = To complete the proof, it suffices to show that V^(n) =>• V\(n). The pairing o : ft-"(G, Z) x H"(G; Z) -^ H°(G, Z) = Z/|G|
Groups of periodic cohomology
153
is equivalent to the evaluation pairing HomAb(H"(G, Z), Z/|G|) x H"(G, Z) -»• Z/|G| Since n > 0, there is no distinction between H"(G; Z) and H"(G; Z), so that, if u e H"(G, Z) = Z/|G| is a generator, then there exists u~l e H"(G, Z) such that M~'M = 1 e //°(G, Z) = Z/|G|. For any Z[G]-lattice A, H*(G, A) is a module over the Tate ring H*(G; Z), thus o : Hk(G, A) x H"(G;Z) -* H*+"(G, A) If £ € H"(G, Z), we denote by p^ the Z-homomorphism /j f : H*(G, A) -> H*+"(Z, A);
p f (u) = i; o £
If M e H"(G, Z) = Z/|G| is a generator, then p« : H*(Z, A) -^ H*+n(Z, A) is an isomorphism with inverse p~l for all Z[G]-lattices A. Make a choice
>• F o - • Z -> 0
where each F, is finitely generated free over Z[G].
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Stable Modules and the D(2)-Problem
Using the stable module 'constructions' M (->• fi/t(M) rather than the functors M \-> Dk(M), a similar argument to (40.1), with only slightly different justifications, gives: Proposition 40.3: If G is a finite group and n is a positive integer, then the conditions J-\(n), J~2(n), J-^(n) are equivalent. The relation between the notions of cohomological period and free cohomological period will be considered again in Chapter 8; in particular, we will show that, if k is a cohomological period of the finite group G, then, for some positive integer N, Nk is a free cohomological period of G (c.f. Section 46 infra). For the moment, we are content to consider some examples.
41 Examples By (40.2), the smallest possible non-trivial cohomological period is k = 2. This is realized in the case of cyclic groups. (i) Cyclic groups: In fact, taking the cyclic group Cn in its balanced presentation
there is a free resolution of period 2 0 -* Z 4 Z[C n] *-+ Z[C n] -4 Z -* 0 Swan has shown that the converse statement is also true, namely, if k = 2 is a cohomological period of G then G must be cyclic ([60] Lemma (5.2), p. 205). (ii) Dihedral groups of order = 2 mod(4): The dihedral group D4n+2 of order An + 2 can be defined by means of the balanced presentation D 4 n + 2 = (x, y U 2 " + 1 = y2, yx" = x"+*y) and D<\n+2 admits a free resolution of period 4 (this is false for the dihedral groups of order An) 0 ->• Z 4- Z[D 4n+2 ] - i Z[D 4, 1+2 ] 2 4 Z[D 4 ,, +2 ] 2 4 Z[D4,,+2] 4 z ^ 0 Here 8=1
y
),
32~
;
3, ~ ( J C - l . y - 1 )
Groups of periodic cohomology
155
where Zx = ]+X +
\-x2";
Y:y=l+y;
0, = 1 +x + • • • +x"~l; and
02 = 1 + x + • • • + x" = 0, + x" whilst e is the augmentation map, and e* is its dual. (iii) The quaternion groups Q(4n) (n > 2): For each n > 2, the quaternion group Q(4n) is defined by the balanced presentation \ x" =y2,xyx
Q(4n) = (x,y
= y)
It is well known that there is a faithful imbedding i : D2n -> 50(3) of the dihedral group D2n in 50(3), the (proper) rotation group of Euclidean 3-space. For example, if
0 -> Z 4 Z[Q(4n)] -4 Z[Q(4n)]2 % Z[Q(4n)]2 A Z[Q(4n)] 4 Z -> 0 where
where E^ = 1 + x + • • • + its dual.
JC"~',
whilst e is the augmentation map, and e* is
Proposition 41.1: Q(4n) has free period 4 for n > 2. In particular: Proposition 41.2: Q(2 m+2 ) has free period 4 for m > 1. (iv) Groups of order /?<7 where p, 9 are distinct primes: Let p, q be distinct primes, and suppose without loss of generality that q < p. If q does not divide p — 1, then there is a unique group of order pq, namely the cyclic group Cpq. When q does divide p — 1, there is, in addition, a unique non-abelian group Q(pq) of order pq, which is described as a semidirect product
=< cp y \ cq
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Stable Modules and the D(2)-Problem
where the generator of Cq acts on Cp as an automorphism of order q. It can be shown that: Proposition 41.3: Let p,q be distinct primes with q < p, and such that q divides p — 1; then Q{pq) has cohomological period 2q. The case of the dihedral group 2p is the first case of this. Beyond that, the next case is the non-abelian group of order 21 which has cohomological period 6. (v) Cp x Cp where p is prime: By contrast, the product Cp x Cp is not periodic. To see this, let n : Cp x Cp -» Cp be the projection n(x\, X2) = x\, and let j : Cp ->• Cp x C p be the inclusion homomorphism j(x) = (x, 1); then JT o j = Id : C p , ->• Cp, and so, taking induced maps in cohomology ; • o 7i* = Id* : H"(C P ;Z) -* H«(CP; Z) In particular, for each q, W{CP; Z) is a direct summand of H^(CP x Cp; Z). If Cp x C p were periodic, then for some n > 0 H2"(Cp x C p ; Z ) = Z / p 2 This is a contradiction since H 2"(Cp;Z) = Z/p is not a direct summand of
Zip2. 42 Subgroup structure at odd primes From the Shapiro Lemma it follows that: Proposition 42.1: Let G be a finite group; if G has cohomological period In, then so does every subgroup. Since Cp x Cp is not periodic, it now follows that: Proposition 42.2: Let G be a finite abelian group; then G has periodic cohomology if and only if G is cyclic. Likewise: Proposition 42.3: Let G be a finite group of periodic cohomology, and let H be a subgroup of order p2 where p is a prime; then H is cyclic. Let p be a prime, and let G be a group of order p" where (n > 1); if n — 1, then G is cyclic and so has periodic cohomology. Thus suppose that n > 2; as is well known, every group of order p2 is abelian, so that, by (42.3), every subgroup of order p2 must be cyclic. In fact, this necessary condition is also sufficient as we shall see. We first establish some preliminary results.
Groups ofperiodic cohomology
157
Let G be a group and let x, y e; we denote by [x, y] € G the commutator [x,y]
l
=
l
and we denote by [G, G] the commutator subgroup of G, that is, the subgroup generated by all commutators. We denote by Z(G) the centre of G. We say that G satisfies condition Z when [G, G] C Z{G). Proposition 42.4: Let G satisfy Z; then for each x, y e G [x,y]" = [x*,y] Proof: By induction on n; for n = 1 there is nothing to prove. Suppose proved for/r; then [x" + \ y] = x[xn,
y]yxax-{n+l)y-1
=
[x",y][x,y]
=
lx,y]n[x,y]
= [x, y]n+l
•
Put
t(n) = Proposition 42.5: Let G satisfy Z; then for each x, y e G
(xyf = [y,x]'wxny". Proof: By induction on n. For n = 1 there is nothing to prove. Suppose proved for n; then (xy)"+l =xy[y,xfn)x"yn
= [x, y]yx[y,
x]'wx"y"
and, since [y, x] e Z(G) and [x, y] = [y, x]~l, we get (xy)n+l = [y,x]'(n)-lyx"+iyn
= [y,xfn)-l[y,
x"+x}xn+ly"+x
However, by (42.4), [y, xn+l] = [y, x]n+l, so that (xv)" +1 = [y
x]'^+nxn+]y"+i
the result follows since t(n) + n = tin + 1). For the rest of the discussion, p will denote an odd prime.
•
158
Stable Modules and the D(2)-Problem
Theorem 42.6: Let G be a non-abelian group of order p" where p is an odd prime; then G has a subgroup isomorphic to Cp x Cp. Proof: Put K = Z(G), and let n : G -> G/K be the projection; K is an abelian group of order pa for some a > 1. If K is not cyclic then
for some m > 2 and some e,- > 1, so that K, and hence G, has a subgroup isomorphic to Cp x Cp. Thus we may assume that K is cyclic, K = Cpm say, so that G/K is a group of order pN for some N > 1, and Z{G/K) is a non-trivial abelian group of order / / where 1 < b < N. Moreover, it cannot then be true that Z ( G / / O is also cyclic, so that Z(G/K) contains a subgroup Q = CpxCp. Putting H = n~l(Q), we have a central extension 1 -* K -> H -4 Q -> 1 where A" S Cpm and Q = Cpx.Cp. It follows easily now that W satisfies condition Z, and, by (42.4), (42.5), the map H -+ H defined by cp(x) = xp gives a homomorphism (p : H -*• K. For x G 2 w e denote by (x) the subgroup of Q generated by x, and put G(x) — TT~\{X)). By centrality of the extension, it is clear that each G(x) is abelian. From the classification of abelian groups we see that for a given x £ Q, either (i) G(x) = Cr+\ or (ii) G(x) = Cpm x Cp. If each G(x) = Cp™+i, then it is straightforward to see that Ker(
Groups of periodic cohomology
159
43 Subgroup structure at the prime 2 For notational convenience, we denote the cyclic group of order 2" by C(2") rather than C2». If G has periodic cohomology then every abelian subgroup is cyclic. In the case where \G\ = 2" then Z(G) = C(2m) for some m > 1; in particular, G has a unique central subgroup Z = C(2). In fact, this is the only subgroup of G isomorphic to C(2), for suppose that G has another subgroup W = C(2), then W fl Z = {1} and Z centralizes W so that WZ is a subgroup of G isomorphic to C(2) x C(2). This is a contradiction since WZ is then a subgroup of a group with periodic cohomology, but does not itself have periodic cohomology. Thus we have shown: Proposition 43.1: Let G be a group of order 2" for some n > 1; if G has periodic cohomology, then G has a unique subgroup isomorphic to C(2). In this section, we show the converse is true by giving Zassenhaus' classification of groups of order 2" having periodic cohomology. Thus suppose that \G\ — 2" and that G has a unique subgroup Z of order 2, then it is clear that Z is central in G. Let A be a maximal abelian normal subgroup of G. By classification of abelian groups of order 2N, we have A = A i x • • • x A,,, where each A,- is cyclic, of order 2°', say. However Z is the unique subgroup of A of order 2, so that in — 1 and A = A, is cyclic. Consider the 'operator homomorphism' c : G/A -» Aut(A) given by
Since A is a maximal normal abelian subgroup, it follows easily that c is injective. Since A is cyclic, Aut(A) is abelian, and we get: Proposition 43.2: Let G be a group of order 2", and suppose that G contains a unique subgroup of order 2; then G occurs as an extension 1 -* A -> G^
Q^
1
where A is cyclic and Q is a subgroup of Aut(A); in particular, Q is abelian. We may therefore put A = C(2") = {1, x,..., of special cases:
x 2 "" 1 ); there are a number
(i) Q = (1); in this case G — A is cyclic, (ii) « = 1; then Aut(A) = {1} so that Q — (1) and again G = A is cyclic.
160
Stable Modules and the D(2)-Problem
(iii) n = 2 and Q ^ {1}; in this case, Aut(A) = C(2) and the non-trivial element of Aut(A) is the involution r(x) = x~l. Consider the extension 1 -> A -* G 4 C(2) -> 1 and let y e G be such that 7i(y) — r, so that yxy~x = x~'. Since 7r(;y2) = 1, then y2 is one of 1, x, x2, x 3 . If y2 is either x or x 3 , then ord()>) = 8.and G = C(8), which contradicts the fact that yxy~x = x~]. Thus y2 is either 1 or x2. However, if y2 = 1, then the extension splits, contradicting the assumption that G has a unique subgroup of order 2. Thus the only possibility we are left with is y 2 = x2; that is, G is generated by two elements {x, v}, satisfying the conditions A
9
x = 1;
9
x = y ;
_ i
yxy
_ 1
= JC
and in this case G = 2(8), the quaternion group of order 8. We can now consider the general case where n > 3 and Q ^ {1}. Then Aut(A) = C(2"~2) x C(2); where the factor C(2) is generated by x where z(x) = x~l, and the factor C(2"~2) is generated by a where a{x) = x5. It follows that G contains a subgroup H of the form 1 -+ A -+ H 4- * ->. 1 where ^ C Aut(/\) is a subgroup of order 2. Let \jr e vy denote a generator; there are precisely three candidates for i/*", namely: (i) iA = T; (ii) i/r =a 2 "" 2 ; (iii) i/f = ra2" . In fact neither (ii) nor (iii) can occur because the corresponding extension splits, contradicting the assumption that G has a unique subgroup of order 2. Since cases (ii) and (iii) can only occur when there is a non-trivial projection of Q on to the factor C(2"~2) generated by a, the only possibility left for G is as an extension
Let y e G be such that n(y) = r, so that yxy""' = x ~ \ let B denote the subgroup of A generated by y2. Observe that B ^ A, since otherwise y2 would be a generator of A, contradicting the relation yxv" 1 = x" 1 . It follows that y2 is already a square in A, say y2 = x2*. The relation yxy~* = x~' can be
Groups of periodic cohomology
161
written xyx = y, from which we see that x2kyx2k = y Since x2k commutes with y, we now have xAk — 1. Hence 4k — 2" or, more conveniently, 2k = 2""', so that G is generated by elements x, y subject to the relation that x2" ' = y2;
xyx = y
and so G = Q(2"+l). Thus to summarize: Theorem 43.3: Let G be a group of order 2'" for some m > 1; then the following conditions on G are equivalent: (i) (ii) (iii) (iv)
G has periodic cohomology; G has a unique subgroup isomorphic to C(2); Every subgroup of order 4 is cyclic; G is isomorphic either to 2(2"') or to C(2m).
Evidently, the case G = Q(2m) can only occur when m > 3.
44 The Artin-Tate Theorem Let G be a finite group, let p be a prime dividing \G\, and write |G| = kp" where k is coprime to p; G(p) will denote a Sylow /^-subgroup of G, and H m (G; A)(p) will denote the p-primary component of H'"(G; A). When m > 0, the image of the transfer map
/ :Hm(G(py,Z)-> H'"(G;Z) lies in the /^-primary subgroup Hm(G; Z\p) (or we could use Tate cohomology and allow any value of m). With this notation, we have: Proposition 44.1: For each m > 0, there is an isomorphism H'"(G(/7); Z) S Hm(G; Z) ( r t 8 Ker(r) Proof: Let / : G(p) C G be the inclusion of the Sylow p-subgroup; then the composite toi* :H m (G;Z)-» H'"(G;Z) has the effect of multiplying by k — [G; G(p)], and, since k is coprime to p, toi* gives an automorphism
162
Stable Modules and the D(2)-Problem
of the p-primary part. Since H'"(G; Z)(p) isfinite,some power of toi* restricted to H'"(G; Z)(P) is the identity, say (t o i*)e+] = Id
whene > 0. Now put y = <*o(f o('*) e .Then?oj is the identity on H'"(G;Z) (p) , and j : H m (G; Z) (p) -> W(G(p); Z) is a right splitting for the surjection
t:H'"{G(p);Z)^Hm(G;Z)(p) The result follows.
•
Corollary 44.2: Suppose that H'"(G(p);Z) = Z/p" for some a > 0; if the transfer map t : U'"(G(p);Z) -> Hm(G;Z)(/,) is nonzero then W(G;Z\p) = Z/pk, and t : H'"(G(p);Z) -> U'"(G;Z)(p) is an isomorphism. Proof: We have seen that tis surjective, and that Hm(G(/?);Z) = H'"(G;Z\p)® Ker(f). Since, by hypothesis, U"'(G(p); Z) = Z/pa, then either H'"(G; Z\p) = 0 or Ker(/) = 0. If H m(G; Z) (p) ^ 0 then Ker(0 = 0, and so t :H is both injective and surjective.
•
Proposition 44.3: Suppose that Hm(G(p); Z) = Z/p"; then there exists e > 1 such that for any multiple d of e
is an isomorphism. Proof: By (44.2), it suffices to show that there exists e > 1 such that for any multiple d of e
t :W"n{G(p);Z) ^ Y\"\G;Z\p) is nonzero. Let e be the exponent of Aut(Z/p") multiple of e
= (Z/p")*; then if d is a
for all § e (Z/pn)*. G acts as a group of automorphisms of H'"(G(p);Z) = Z/p" via the Eckmann-Shapiro identity Bm(G(p); Z) = H m(G; Z ®Z[G(P)] Z[G]). Let <: e H'"(G(p);Z) be a generator. If c* is the conjugation automorphism of H'"(G(p);Z) = H'"(G(p);SZ) induced by g € G, then we may
Groups of periodic cohomology
163
write
for some %g e (Z/p")*. When d is a multiple of e, we have, by naturality of the G-action
cl(z") = c*(z? = ^ V = z" e Hrf'"(G(p);Z) Thus zrf is invariant under the action of G, so that i*t{zd) = kzd where k = [G : G(p)]. Since k is coprime to p, then f(zd) ^ 0, and the result follows. D p^ be the distinct primes Corollary 44.4: Let G be a finite group, let p\,..., dividing | G |; then G has periodic cohomology if and only if G(p,) has periodic cohomology for each /. Proof: We may write the prime factorization of \G\ in the form \ G \ = P \ ••• P
N
If G has periodic cohomology, then so does every subgroup. In particular, so also does G(pi). Suppose, conversely, that G(/?,) has periodic cohomology; then, for some m,, H'"'(G(p,-);Z) ^ Z/p"'. Then by (44.3) there exists e, > 1 such that, for every nonzero, positive multiple d\ of e,- the transfer map
is an isomorphism, each side being isomorphic to Z/p"1 Thus when m is a multiple of each e,m,
is an isomorphism for each i. It follows easily that
and so G has periodic cohomology, by (40.1).
D
From our previous discussion, in Sections 42—43 of the periodicity conditions for groups of prime power order, we get the Artin-Tate-Zassenhaus characterization of groups of periodic cohomology: pN be the distinct Theorem 44.5: Let G be a finite group, and let p\,..., primes dividing |G|; then G has a finite cohomological period if and only if G(2) is either cyclic or generalized quaternionic and G{p,) is cyclic whenever Pi is odd.
164
Stable Modules and the D(2)-Problem
Alternatively expressed, G has periodic cohomology if and only if each subgroup of order p2 is cyclic. 45 The Zassenhaus-Suzuki classification The Zassenhaus-Suzuki Theorem classifies all finite groups whose odd Sylow subgroups are cyclic and whose Sylow 2-subgroup contains a cyclic subgroup of index 2. This is slightly larger than the class of groups of periodic cohomology, since it allows the possibility of C^ x Ci as a Sylow 2-subgroup. However, the subclass of groups of periodic cohomology is easily retrieved from this classification. We treat the soluble case first. Wolf's reworking of Zassenhaus gives four such classes ([82] tabulated on p. 179). We adapt his treatment to suit our purpose: A(m, n; r): Let C,, = (Y : Y" — 1) be the cyclic group of order n > 2, and let r > 1 be an integer coprime to n; then there is an automorphism <pr : C,, —»• C,, defined by <pr(Y°) = Y" When ord(<pr) divides m, that is, when r"' = 1 mod n), there is a split extension 1 ->• Cn -> A(m, n;r) -* Cm -> 1
of Cn by Cm, so that A(m, n; r) is a semidirect product A(m,n;r)^Cny\
Cm
where the generator X of Cm acts by conjugation on C,, as
XYX~1 = <pr(Y) = r We define this formally by the presentation A(m,n;r)=
(X, Y : Xm = Yn = 1;
XXX"1 = Yr)
where the parameters m, n, r are subject to the conditions: (i) m, n are coprime, with m odd; and (ii) rm = 1 (mod n). In order for A(m, n;r) to satisfy the p2 conditions it is necessary and sufficient that m and n be coprime. Moreover, since Aut(C2«) is a 2-group, any extension 1 - • C2» -> G -* Cv -+ 1
Groups of periodic cohomology
165
in which v is odd is necessarily a direct product G = Ci« x Cv. Thus in the semidirect product
A(m,n;r) = Cny\ Cm (r)
we may suppose, without loss of generality, that the 2-torsion of G lies in the factor Cm; that is, we may suppose that n is odd. B(2"+2, v, n; r, s, t): Put m = 2"+1ii and form the nonsplit extension 1 - * A(m,n;r)^
B(2u+2,v,n;r,s,t)^
C2 - * 1
where a generator /{which projects on to the generator of Ci acts by conjugation on the factors of the metacyclic subgroup A(m, n; r) by RXR-X = Xs;
RYR-1 = Y1,
and satisfies R2 — X^. To achieve this, in addition to the conditions required for A(m, n\r), we also require: (iii) u > 1 and v is odd; (iv) s = — 1 mod 2" and s 2 = 1 mod m; (v) t2 = rs~l = 1 rnodn. In this case, the Sylow 2-subgroup is Q(2"+2) and the cohomological period is > LCM(4, ord(<pr). C(8; m, n;r): Take the quaternion group of order 8 in its standard presentation 2(8) = , Q:P2
=
Q2;PQP=Q).
Let a : g(8) ->• 2(8) be the automorphism of order 3 given by
a(P) = PQ; a(Q) = P~x and take the split extension
C(8;m,n;r) = (e(8)xC,,)Xl
C» («,r)
in which a generator X of Cm acts on 2(8) x Cn by means of XPX~l - PQ;
XQX'1 = P~l;
XYX~{ = Yr
The conditions that the parameters must satisfy here are: (i) (ii) (iii) (iv)
m = 0 (mod 3); rm = 1 (mod n); m, n are coprime; n is odd.
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Conditions (i) and (ii) are required to construct the extension; conditions (iii) and (iv) are then necessary and sufficient to guarantee the p2 conditions. The Sylow subgroup is 2(8), and the cohomological period is LCM(4, ord(^ r ). V(\6;m, n; r, s, t): We construct the nonsplit extension 1 -> C(8; m, n;r) -»• P(16; m, n; r, s, t) -> C2 ->• 1 where a generator R which projects on to the generator of Ci acts by conjugation on C(8;m, n;r) by RPR~l = PQ-
RQR~] - / > - ' ;
RXR~l = Xs;
RYR~l = Y'
and satisfies R2 = P2. The conditions required to guarantee this are: (i) m = 0 (mod 3); (ii) (ii) rm = 1 (mod n); (iii) m, « are coprime; (iv) n is odd; (v) s 2 = 1 (mod m); (vi) s = - 1 (mod 3); (vii) r ' - ' = i 2 = 1 (mod n). The Sylow 2-subgroup is 2(16) and the cohomological period is > LCM(4, ord(<pr). Zassenhaus' classification of soluble groups satisfying the p2 conditions can be stated in two parts; the first, when the Sylow 2-subgroup is cyclic, is actually due to Burnside: Theorem 45.1: Let G be a soluble finite group in which all Sylow subgroups are cyclic ; then G = A(m, n; r) for some suitable m, n, r. Theorem 45.2: Let G be a soluble finite group in which the Sylow 2-subgroup is generalized quaternion and all odd Sylow subgroups are cyclic; then G is isomorphic to one of the groups B(2"+2, v, n;r, s, t), C(8;m, n\r), V(\6;m, n;r,s,t).
We now restrict attention to groups of period four. The natural analysis is in terms of the structure of the Sylow 2-subgroup, beginning with the case where it is trivial. At a number of points, we refer to the notation of Milnor's paper [42], where the question of which finite groups G can act freely on S" is considered. Such a group must have n + 1 as a cohomological period, and Milnor showed in addition that either G is of odd order or else contains a unique element of order 2, which must necessarily be central. In connection with the question for n = 3, Milnor listed all such groups of period four.
Groups of periodic cohomology
167
Odd order groups of period 4 An odd order group G with periodic cohomology must be of the form G = A(m, n; r) where both m and n are odd; moreover, the order of q>r must also be odd. Since the minimal period of A(m, n;r)h2 ord(<pr), it must then be true that (pr = Id. Thus we may take r = 1 and write G = A(m, n;l) = Cm xC,, = Cmn. We obtain: Proposition 45.3: A finite group of odd order and period < 4 is necessarily cyclic. Groups ofperiod four with cyclic Sylow 2-subgroup In general, a finite group G with periodic cohomology and cyclic Sylow 2subgroup has the form G = .4(2"; v, n;r) where D, n are both odd; such a group has a subgroup of the form H = A(v, n;s). If G has period < 4 so also does H, and so H is cyclic, by (45.3). Without loss of generality, we may describe G in the form G = A(2" ,2b+\;t) where the order of t mod 2b + 1 is of the form 2C where 0 < c < a. In fact, it is easy to calculate that the minimal period of G = .4(2", 2b + 1; t) is 2 C+I . Thus the condition that G has period < 4 forces c to be either 0 or 1. In the case c — 0, G is necessarily cyclic. In t h e c a s e c = l,weseethatG =A(2a,2b+ l;f)wheref 2 = 1 mod 2b + 1. Let i/v : CN —> CN denote the canonical involution TN(X) = x~'. An easy argument using prime decomposition shows that, if 6 : C2b+\ -*• C26+1 is a non-trivial automorphism of order 2 then either 9 = i2b+i, or else 6 is equivalent to
for some factorization (2r + l)(2s + 1) = (2b + 1). We now see easily that: Proposition 45.4: Let G be a finite group of period 4; if the Sylow 2-subgroup of G is cyclic, then either (i) G is cyclic = C2* where k > \,or (ii) G = A(2\ 2r + 1; - 1 ) x C2s+\ for some k > 1, r > 1 and s > 0. The special cases k = 1 and k — 2 merit further attention. In the case k = 1, recall from Section 41, that the dihedral group D4n+2 = {$,r,: | 2 " + l = i,2 = 1; ^ i , " 1 = r ' ) has period 4. D4n+2 occurs as -4(2, 2n + 1; —1); the 'quasi-dihedral' groups D 4 n + 2 x C2m+i also occur as .4(2(2m + 1), 2n + 1;— 1). Observe that D4n
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does not have periodic cohomology, since it has a subgroup {1, £", rj, % isomorphic to C2 x C2. In the case £ = 2, we have seen in Section 41 that the quaternion group
has period 4. When m is odd, the Sylow 2-subgroup is C4, and it occurs in the above list as .4(4, m; — 1). For Ic > 3, there does not seem to be any generally agreed name for the groups A(2k, 2m + 1; - 1 ) . Milnor ([42]) calls them 'D2*(2m+i)' but this clashes with the now standard nomenclature for dihedral groups. Groups with quatemionic Sylow 2-subgroup of order > 32 The groups Q(4m) when m is odd are more accurately described as binary dihedral groups. Here we consider the true quaternion group G(2"+ 2 ) = (P,Q:
P2" = Q2;QPQ~1
=
P~l)
of order 2" +2 . This occurs as B(2U+2, 1, 1; 1, 1, 1) under a notation change in which the ' P ' of Q(2U+1) is actually the lR' ofB(2u+2, 1, 1; 1, 1, 1). The groups Q(2u+2;m, n): when m, n are odd of coprime order, we describe Cm x Cn = Cmn in the form CmxCn
= {X,Y
:Xm
= Y" = 1;
XY=YX)
and let Q{2u+2;m, n) denote the split extension 1 -+ Cm x C -+ Q(2u+2;m, n) -* 0(2"+ 2 ) - • 1 where the action of Q(2U+2) on Cm x Cn is given by " 1 = X~l\
PYP~{ = 7;
QXQ~] = X;
g y g " 1 = Y~l
It is easy to see that this group occurs as B(2U+2, m, n; 1, —1, —1). Q(2u+2;m, ri) has period 4. Observe that the abelianization g(2" + 2 ) a b is isomorphic to C2" x Ci where P projects on to the first factor and Q on to the second. In principal, the generator P can act with any order 2k for 0 < k < u. However, if 2 < k the resulting extension has period > 4. More generally, if k is odd and coprime to both m and n, then the direct product Q(2"+2; m, n) x Ck has period 4, and occurs in the above list as S(2" +2 ,m,kn;\,—\,t) where t2 = 1 (mod kn), t = 1 (mod k), and f = — 1 (mod n); moreover, this is the most general group of period 4 with quatemionic Sylow subgroup of order > 32.
Groups of periodic cohomology
169
Groups with Sylow 2-subgroups of type Q(8) Within the general class of periodic groups (2(8) can arise as a Sylow 2-subgroup in two ways, as either (I) B(23,v,n;r,k,l),or (II) C(8;/n,n,r). In period 4, groups of type (I) take the form Q(%\m,n) x C* for some suitable odd k, m, n. In type (II) one case is of special interest, namely the split extension l - > e(8)->C(8;3*, l ; l ) - > C y - > 1 where a generator Z of C3* acts on (2(8) by means of ZPZ~[=PQ;
ZQZTX = />-'
In Milnor's notation, C(8;3*, 1; 1) = P g ' 3t . More generally, when n is odd and coprime to 3, the product Pg',3i x Cn = C(8;3*,rc; 1) = C(8;3V 1; 1) has period 4; this exhausts the groups of period 4 which have Q(8) as Sylow 2-subgroup.
Groups with Sylow 2-subgroups of type Q(16) Within the general class of periodic groups 2(16) can arise as a Sylow 2subgroup in two ways, as either (III) B(24,v,n;r,k,l),or (IV) V(l6;m,n;r,s,t) In period 4, groups of type B(24, v, n; r, k, 1) take the form 2(16; in, n) x C^ for some suitable odd k, m, n. Again following Milnor, if n is odd and coprime to 3, we write P"6#.n = £>(16;3\ n; 1, 1, - 1 ) . Then the most general group period 4 group of type V is P^n x Cm, where m is odd and coprime to both 3 and n; we note that P%gn x Cm can be described in the form X>(16; 3*, mn; 1, 1, t), where t = 1 mod (m), t = — 1 mod («), and ; 2 = 1 mod {mn).
Nonsoluble groups ofperiod 4 Suzuki has classified of nonsoluble groups in which the /?2-conditions are satisfied [56]. In this case, the condition of nonsolubility implies that the Sylow 2-subgroup cannot be cyclic, and so is forced to be generalized quaternion; then we have:
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Theorem 45.5: Let G be a nonsoluble finite group in which the Sylow 2subgroup is generalized quaternion and all odd Sylow subgroups are cyclic; then G has a subgroup Go of index < 2 such that
G0 =
A(m,n;r)xSL2(Fp)
for some prime p and suitable m, n, r; here Fp is the field with p elements. From this we get one completely new example of a group of period 4, namely SL2(F5) which has order 120; it arises geometrically as the binary icosahedral group I*, that is, the spin double covering of the simple group A5, of order 60. Note that the groups SL2(F2), SL2(F3) are soluble and already accounted for, whilst the groups SL2(FP) for p > 7 have period > 4. In addition, when m is odd and coprime to 3,5 the groups Cm x I* also have period 4. Milnor's list can be retrieved by eliminating the dihedral and quasi-dihedral groups from the above discussion.
Chapter 8 Algebraic homotopy theory
In this chapter, we encounter homotopy theory in both its geometric and algebraic aspects. We begin by reviewing the homotopy theory of projective chain complexes, as outlined, for example, by Wall in [69]. To ensure consistency of notation, we start from elementary considerations. One of our primary aims is to introduce the finiteness obstruction x of Swan [58] and Wall [68] which, viewed algebraically, detects when a projective chain complex is homotopy equivalent to a free chain complex. If X is a CW complex we say that X is reduced when it has a single 0-cell. Clearly a reduced complex is automatically connected. If X is a connected CW complex and T C X^ is a maximal tree then the quotient X/T is a reduced CW complex which is homotopy equivalent to X. Without further mention we assume that in a reduced complex the basepoint is the unique 0-cell, and allow ourselves to write n\(X) rather than Tt\(X, *). We shall fix a finitely presented group G, described in some specific way, and consider all CW complexes with the fundamental group isomorphic to G. In working with distinct spaces whose fundamental groups though isomorphic are nevertheless distinct, it is necessary to keep track of the different ways elements in the various fundamental groups can be identified. We do this by assuming at the outset that each CW complex X under consideration is given a specific isomorphism (sometimes called a polarization), px : TT\(X, *) -» G. If / : X -»• Y is a cellular map of based complexes, there is a natural induced map / , : TC\(X, *) —*• rt\{Y, *). We say that /* is a mapping over G when py o /» = jix- In cases where the nature of the identifications px and PY is clear, we shall abuse notation and say that / induces the identity on n\.
171
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Stable Modules and the D(2)-Problem 46 Chain complexes, homotopy and homology
Let A be a ring; by a A-chain complex we mean a sequence C* = (Cr, 9 r c ) re z where (C r ) r6 z is a graded A-module, and for each r, 9rc : Cr -»• Cr-\ is a A-homomorphism, the rth boundary map such that dr 9 , | = 0 If C* = (C r , 3 r c ) r , D* = (£>r, 9rD)r are A-chain complexes, by a chain homomorphism over A, / = (fr): C* ->• D*, we mean a graded A-homomorphism /* : C, —> Z)* such that the following diagram commutes for each r <-H+I
4- /n + l
—>•
Ln
4- fn
A,+i -* DB We say that a chain complex C* is bounded below when, for some N e Z, Cr — 0 for all r < A'. Likewise, we say that a chain complex C* is bounded above when, for some N eZ,Cr = 0 for all A' < r. Without further mention, all chain complexes will henceforth be bounded below. We denote by Chain(A) the category whose objects are A-chain complexes C* which are bounded below and whose morphisms are chain homomorphisms over A. If C* is a A-chain complex we denote by Zn(C) = Ker(9f) the module of n-cycles, and by Bn(C) = Im(9^+|) the module of n-boundaries. Since dfdf^ = 0 it follows that Bn(C) C Zn(C) A A-chain complex C* gives rise to graded A-module //*(C) = (//,,(C)),,gz, the homology of C*, by means of Hn{C) =
Zn(C)/Bn(C)
The correspondence C» i-> //*(C) is functorial; given a chain mapping / : C* —*• D*, for each n e Z there is an induced A-homomorphism //„(/) : // n (C) - • // n (D) given by
where z e Z n (C). Fix a short exact sequence of chain complexes
Algebraic homotopy theory
173
For each n, there is a A-linear mapping S : Zn+\(C) -» Hn(A) defined by the formula
where [a] denotes the class in H"(A) of a e Z(A); 8 vanishes on Bn+\(C), and so induces a homomorphism, S : Hn+](C) -*• H"(A). Proposition 46.1: With this notation, the sequence Hn+X{B) ""4.(P) Hn+l(C) 4 . Hn(A) "4° H,,(B) "^
Hn(C)
is exact for all n. If / , g : C* —> D* are chain maps, we say that / is homotopic to g (written / ~ g) when there are A-homomorphisms rjn : Cn -> Dn+\ such that, for each n fn-gn
= r\n-\dn + 3,,+l»7n
Proposition 46.2: Let / , g : C* -> D* be chain maps; if / ~ g, then //„(/) — Hn(g) for All n.
Proof: It suffices to show that for all z 6 Z,,(C)
We know that fniz) - gn(z) = dn+ir]n(z) + r)n-\3n(z) for all z e Cn. If z e Zn(C), that is, if d,,(z) = 0, then
fn(z)-gn(z) - dn+\1n(z) so that f,,(z) - gn(z) € Bn(D), and the desired conclusion follows.
D
Example: The cellular chains on a CW complex For any, not necessarily finite, group G, we denote by CW G the category whose objects are pairs (X, px), where X is afinitereduced CW complex with n\ (X) = G, and where px '• 7t\(X) = G is a specific isomorphism, and whose morphisms are cellular maps which induce the identity on n\. For any n > 1, we denote by CW'c the full subcategory of CW G consisting of complexes of dimension < n. If K is a connected CW complex with it\{K) = G, we denote by C*(K) the complex of free Z[G]-modules constructed from the universal covering K.
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Stable Modules and the D(2)-Problem
Formally, we construct C*{K) from standard singular homology in the manner of Milnor [44] : take Cn(K) to be the relative singular homology group
The boundary operator of the triple {K{n\ K(n~ x\ K{"-2)) gives a natural boundary map 3n : Cn{K) -»• Cn-](K). On taking the homology of this complex in the standard way one simply retrieves the singular homology of K
However, the covering action of G on K extends to an action of Z[G] on each Cr{K), under which Cr{K) becomes a free module over Z[G] with basis elements of Cr(K) corresponding to liftings to K of r-cells in K. Moreover, the boundary maps of C^K) are equivariant with respect to the covering action of G on K, so that C*{K) can be regarded as a chain complex over Z[G]. Since K is simply connected, then H\(K;Z) = 0, and the sequence C2(K) X CX(K) X C0(K) is exact at C\(K). Since K is its own universal covering, we may equally write C^K) = C^K), though there is a pedantic difference in that Cr{K) is now a free module over Z[G], not because G is the fundamental group of K, but rather because G is a group which acts freely on K. We say that a chain complex C* is contmctible when Idc, — 0. If C* is contractible, then from the identity \dpn = 3«+i??n + r]n-idn it follows easily that Zn(C) = Bn{C); hence Hn(C) = 0 for each n, and C* is exact. When C* is projective, the converse is true; thus suppose C* = (•••—> Cn -4- Cn_i -^> • • • —> C\ -» Co —*• 0) is an exact sequence of A-modules in which each C,- is projective. We construct a contraction for C* as follows: since d\ is surjective, then by the universal property of projective modules there exists a homomorphism r)o : Co —> Ci making the following diagram commute
id
Algebraic homotopy theory
175
that is, 3irj0 = Idq,. Taking, conventionally, C_i = 0 and rj-\ : C-\ -» Co 30 : Co -> C_ ] to be the zero maps, we see that Id Co = Suppose inductively that there are maps r\r : C r ->• C r + i such that H C r = dr+lVr + rir-\dr for r < n. In particular We,., =3«»?n-l +/?,,-29,,_l so that
However, 3 n _|3,, = 0, so that 3 n = 3 n jj n _i3 n , whence 3,,(IdCn — »?,i-i3,,) = 9,, — 9,,TJ,,_I9,,. By exactness, it follows that Im(Id Cn — ry,,_i3,,) C Im(3,, + i). Thus by the universal property for the projective module C,,, there exists a homomorphism r\n : Cn -> Cn+\ making the following diagram commute P,,
ldPn - /?,,_i
tin
3n+l
Pn+\
"
Im(3 n + 1 )
that is, Idc n — t]n-\dn = dn+irjn or, equivalently
This completes the induction, and the construction of the contraction r\ = ('Jr)reN. and we have shown: Proposition 46.3: If C* is a projective chain complex, then C* is contractible < = • //*(C) = 0 If / : C* —>• D* is a chain mapping, we define a chain complex M(f\, so-called mapping cone of f
the
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Stable Modules and the D(2)-Problem
where we represent the elements of Cn_i © Dn as column vectors
Vd where cn_] e Cn_i andrfn e D n . Proposition 46.4: Let /* : C* ->• £)» be a chain mapping; if M{f) is chain contractible, then there exists a chain mapping g* : D* -> C* such that ^ o / ~ Idc a n d / ° g — Wofrao/- Any A-linear mapping tyn : M(f)n matrix
-> M(f)n+\
is represented by a
\Yn-i
of A-linear mappings 5n — 1 • ^ n — 1
^n
:Dn
^
^«?
-»• C,,;
y n _ i : C n _ i ->• A , + i ; ?;„
: Dn
->• D,,+i.
A straightforward calculation gives the following matrix for *„_] 9,f + 9,^
If ^ is a chain contraction, then for each n
so that, equating entries in the (1, 2) position, we get
that is, g* : D* -^ C» is chain mapping. Equating entries in the (1, 1) position gives
that is, f = {%„)„ gives a chain homotopygo/ -» Idc,. Finally, equating entries in the (2, 2) position gives
Algebraic homotopy theory
177
that is, 77 = {i)n)n gives a chain homotopy g o / -> Idc,. This completes the proof. • For any chain complex C*, we denote by C*(-l) the chain complex given by n(-1)
= CB_,;
f!-l
A chain mapping / : C* -> D* gives rise to an obvious 'mapping cone' exact sequence
Proposition 46.5: If / : C» ->• D* is a chain mapping, the long exact sequence in homology of the mapping cone sequence takes the following form
• • • -• Hn+l(M(f)) 4 Hn(C) A Hn(D) \ 4
Hn(M(f))
4. //„_,(/))•••
/- In view of the identity # n ( C ( - l ) ) = //,,_i(C), the long exact sequence obtained from the mapping cone sequence has the form
• • • 4 Hn+l(M(f)) 4 tfB(C) - i Hn(D) X Hn{M(f)) 4
f/
However, an easy calculation shows the connecting homomorphism 5: Hn+i(C{— 1)) = // n (C) ->• Hn(D) coincides with the homomomorphism /* : // n (C) - • Hn(D) induced by / : C, - • D*. D Theorem 46.6: Let / : C* ->• D* be a chain map between (nonnegative) projective chain complexes, then the following statements are equivalent (i) (ii) (iii) (iv)
//*(/) is an isomorphism for each n; M(f) is exact; M ( / ) is chain contractible; there exists a chain mapping g : D* -*• C* such that / o g ~ Id and g o / ~ Id.
Proof: It suffices to show that (i) => (ii) =>• (iii) =>• (iv) =>• (i). Since C* and £>* are assumed to be projective, so is M(f), and the implications (ii) =• (iii) and (iii) => (iv) are simply restatements of (46.3) and (46.4). To show that (i) => (ii), note that, if //„(/) is an isomorphism for each n, then from the exact sequence of (46.5) it follows that H,,(M(f)) = 0 for each
178
Stable Modules and the D(2)-Problem
n, and so M(f) is exact. Finally, the implication (iv) =>• (i) is an immediate consequence of functoriality. D A chain complex C* is said to be offinite type when C* is bounded both above and below, and, in addition, each Cr is finitely generated. To any projective chain complex Pt of finite type we associate its generalized Euler characteristic (or Wall obstruction) defined by
The following is usually known as 'Whitehead's Trick' [69]. Proposition 46.7: Let P* be a projective chain complex of finite type, and suppose that P* is chain contractible; then 0 r P2r+\ = 0 r Pir. Proof: If Qa is projective then any exact sequence of the form
splits and Q\ = 0o©AMf Q\ isprojective, then so also is K. A straightforward induction, using cutting and splicing, shows that if 0 -> K - • 0,,_, -*
• 0 i - • 0o -» 0
is an exact sequence where each 0 r is projective, then so also is K. Now suppose that Pt is a contractible projective chain complex of finite type. By dimension shifting, if necessary, we may write P* in the form p. = (o - • pn X • • • % />, X
P0
-> o)
Since P* is chain contractible this sequence is exact. It suffices to show that
0
\ Pn-2k = ^ 0
Pn-U-l
The proof goes by induction on n. When n = 1, d\ : P\ -> P$ is the desired isomorphism. In general, suppose true for n — 1, and split the above exact sequence thus
0-+ P,A Pn-\ ^ and Pi = (0 -» ImO,,_,) C Pn-2
Algebraic homotopy theory
179
By the observation above, Im(3 n _i) is projective, and by induction
\
\\
I
However, again since Im(3,,_i) is projective, the sequence
splits and It follows that P
" © \W
P
\\
"~2k = P «- 1 © K& P"-2k~l \\
)
/
which is the required result.
•
Theorem 46.8: Let C*, D* be projective chain complexes of finite type, and let / : C* —> D* be a chain map which induces an isomorphism on homology in each dimension; then
Proof: Let M* denote the mapping cone of / . Clearly A/* is projective of finite type, and M* is chain contractible, by (46.6). It follows that Mik+\ = 0
0
However, Mn=C,,-\@ Dn, so that ) C2k e I ^ D2k+] 1 = 1 ^ C a + i \0<*
/
\0<*
/
and hence, taking formal differences in
\0
/
\0
we obtain = 0
so that x(C») = x(D») as required.
•
Example: Cohomological period and free cohomological period Recall that if G is a finite group then the reduced projective class group K0(Z[G]) is finite. This has the following consequence.
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Stable Modules and the D(2)-Problem
Proposition 46.9: If k is a cohomological period of the finite group G, then Nk is a free period of G for some positive integer N. Proof: If A: is a cohomological period of G, then there exists an exact sequence P* = (0 -> Z -* P*_, ->
• P o -> Z -* 0)
where each P, is finitely generated projective. Consider the Wall obstruction X(P*) e #o(Z[G]). Since K0(Z[G]) is finite there exists a positive integer AT such that Nx(P*) = 0. However Nx(P*) is just the Wall obstruction of
AT
the N-fold Yoneda product of P* with itself. It follows from (30.2) that P ... P N
can be replaced up to congruence by an exact sequence of the form 0 -> Z -> FAK-I ->•
> F o ->• Z -> 0
where each F,- is finitely generated free over Z[G]; that is, Nk is a free period of G.
47 Two-dimensional complexes We now restrict attention to complexes of dimension < 2; observe that for K 6 CW£, the kernel of d2 : C2(K) -+ C{{K) is precisely H2(K; Z) and so the chain complex 0 - • C2(K) X Ci(A-) \
C0(K) -». 0
extends to an exact sequence of Z[G]-modules 0 -»• W 2 (^ : Z) -* C2(AT) -^> Ci(AT) -H> CO(AT) -)> / / 0 ( ^ : Z) -> 0 Since ^ is connected then Ho(K;Z) = Z. Moreover there is a canonical identification coming from the augmentation map e : Co(K) ->• Z obtained by projecting to the base point in K. If AT is a finite reduced 2-complex with 7ri(A") = G we choose, once and for all, a lifting of the basepoint to a 0-cell *K e K. The map to the basepoint K ->• * induces an augmentation e : CQ(K) = Z[G] -•• Z. If L is another such complex, by an augmented
Algebraic homotopy theory
181
chain map
=\
i
1
C2(L) -> Ci(I) -»• Z[G] Such a commutative diagram automatically induces a Z[G]-homomorphism (pt: n2{K) —> 7t2(L) where we make the identifications TT2(K) = Ker(C2(K) —>
CX(K)), m(L) = Ker(C2(L) - • C,(L)). Furthermore, since ^ is simply connected, Hi{K\Zi) may, by the Hurewicz Theorem, be identified with the second homotopy group ni{K) = TtiiK). The inclusion njiK) -> Ci{K) is the co-augmentation, and it is useful to describe C*{K) as an exact sequence of Z[G]-modules displaying both augmentation and co-augmentation, thus
= (0 -> w2(^) - • C2(A") 4 C,(^) X C0{K) -4 Z We wish to give a purely algebraic approximation to two-dimensional geometric homotopy. We use the above as a model. To start with a two-dimensional chain complex
over Z[G] would be too general. The first restriction we impose is that Wo(E) = Z, corresponding to connectivity; the second is exactness at E\, corresponding to simple connectivity of K. In the case of a cellular chain complex C*(K), the Cn(K) are free over Z[G]. As we shall see, this is slightly too restrictive, and we allow £,• to be stably free. Thus, by an algebraic 2-complex over G we mean a sequence of Z[G] modules and homomorphisms
for which: (i) H0(E) = Z and (ii) Ker(31) = Im(a2). (iii) Ei is finitely generated stably free.
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Stable Modules and the D(2)-Problem
We denote by Alg c the full subcategory of Chain(Z[C]) whose objects are algebraic 2-complexes; that is, morphisms take the form E
/ 0 - > - E2 -*• Ei -»• Eo -* 0 \ 4, /i 2
4-^i
4-^o
0-> F2 ->• F, -> Fb -» 0 / Clearly the correspondence A" h-> C*(A") defines a functor C* : CWg —» AlgG. By analogy with the geometrical case, it is useful to think of H2(E) as an 'algebraic n2 , and we put 7T2(E) = Ker(32 : E2 -> £,) enabling us to write E in augmented, co-augmented form as an exact sequence E = (0 -> 7T2(E) -» E2 % Ex ^ Eo -> Z -> 0\ When so written, the appropriate notion of homotopy equivalence is weak homotopy equivalence; if E, F are algebraic 2-complexes, a chain mapping h : E -> F is said to be a weak homotopy equivalence when the induced maps h* : //o(E) —>• //o(F) and /J* : 7r2(E) —*• n2(F) are isomorphisms. By hypothesis, // ( (E) = //,(F) = 0 so the isomorphism /z* : //i(E) ->• //j(F) is automatic. From (46.6) it follows that: Proposition 47.1: Let h : E —> F be a morphism of algebraic 2-complexes over G; then h is a weak homotopy equivalence •<=>• /i is a chain homotopy equivalence. Thus the question of whether or not to include augmentation and co-augmentation in the description is essentially a matter of taste. It also follows directly from (46.6) that: Proposition 47.2: Weak homotopy equivalence is an equivalence relation on AIg G .
This has the practical consequence that when h : P, -> P 2 is a morphism of algebraic 2-complexes inducing isomorphisms hr : Hr(P\) —> Hr(P2), then there exists a morphism g : P 2 -*• P\ inducing gr =h~x : Hr(P2) ->• // r (Pi). There is the generalization which it is occasionally necessary to consider; denote by Proj c the full subcategory of Chain(Z[G]) whose objects are projective 2-complexes P =
Algebraic homotopy theory
183
in which, apart from the previous conditions Coker(9i) = Z and Ker(9j) = Im(92), we require also that each P, be finitely generated projective. Likewise there is a specialization, Freec, in which objects
must have F,- finitely generated. One sees easily that any algebraic 2-complex is congruent, hence homotopy equivalent, to a free 2-complex, whereas a projective 2-complex
is homotopy equivalent to an algebraic 2-complex if and only if x(P) = 0. Let E = be an algebraic 2-complex over G. We say that E is geometrically realizable when there is an object X e CW^ and a weak homotopy equivalence q> : C*(X) —> E. By symmetry, this is equivalent to the existence of a weak homotopy equivalence i]/ : E ->• C*(X). Writing an algebraic 2-complex in augmented and co-augmented form E =
TT2(E) -> E2 X £ | X Eo -* Z -* o)
gives (at least in those cases which come within the scope of our previous definitions, that is, when G is finite), an element of Stab3(Z, K2(E)). In the case of a geometrical cellular chain complex C.(ff) = (0 -> TT2(K) - • C2(K) X C(A-) 4 . Co(ff) -4 Z -» 0) we get an element of Free3(Z, TT2(K)). 48 Cayley complexes The above discussion can be carried out in the context of group presentations; to any finite presentation Q = (x\,..., xg | W\,..., Wr) of a group C, we associate a canonical two-dimensional complex Kg with Tt\ (Kg) = G by regarding the generators x, as 1 -cells and the relators Wj as 2-cells. For any such presentation Q, we write n2(Q) for the second homotopy group n2(Kg). Put K — Kg, and let K denote the universal cover; then n2(Q) = TT2(K) — 7i2(K) is a module over Z[G] via the action of G on K. We have a free algebraic 2-complex o-»• m(Q)-+
c2(Kg)
the algebraic Cayley complex of the presentation. When G is finite, ^ ( Z ) is defined, and we obtain:
184
Stable Modules and the D(2)-Problem
Proposition 48.1: If Q is a finite presentation of G, then n2(Q) e Again, when G is finite, K is a finite 2-complex and, by simple connectivity, we have K ~ S2 v • • • v S2 A simple argument using the multiplicativity of the Euler characteristic on finite coverings allows us to calculate the Z-rank of 7T2(<5): Proposition 48.2: Let Q = (x\,..., xg; W\,..., Wr) be a finite presentation of the finite group G and K = Kg the canonical 2-complex associated with Q; then K2(G) is a free abelian group of rank (r — g + 1)|G| — 1. This yields a well-known fact: Proposition 48.3: Let Q = (x\,..., of the finite group G; then g < r.
xg; W\,..., Wr) be a finite presentation
From the Hurewicz Theorem we see that niiKg) = // 2 (^g;Z). Thus calculation of 7i2(Kg), and hence of ^ ( Z ) , comes down to being able to give an effective description of Ker(32). This is possible in proportion to the extent that we can describe G effectively. Fox gave a formal method, free differential calculus [19], which we illustrate by an example. The quaternion group of order 16 We take 2(16) in its standard presentation Q(16)=(x,y:x4
= y2
The two generators x, y give rise to two 1-cells €\, €2, which, when lifted to the universal covering, we may portray as follows [1]
€l
M
[1]
€l
[y]
If ei is orientated by starting from [1] and ending at [x], then its boundary is given by
Likewise 3(«2) = y - 1 Observe that 3(ei) and 3(62) belong to, and indeed together generate, the augmentation ideal I(<2(16)).
185
Algebraic homotopy theory
The first relation x4 = y2 gives a 2-cell Ei, which, when lifted to the universal covering, becomes a 6-sided polygon bounded by translates of the basic 1-cells as follows M
ei
[l]
E,
4
= y2]
Expressing the boundary of the 2-cell Ei in terms of the translates of the basic 1 -cells we obtain
or, in matrix terms 3(Ei) =
l+X+X2+X3
-d+y)
Similarly, the second relation xyx — y gives a 2-cell E 2 , which, when lifted to the universal covering, becomes the following 4-sided polygon
€2
X
[yx]
[xyx = y]
186
Stable Modules and the D(2)-Problem
The boundary E2 is then expressed as
or, in matrix terms 3(E 2) =
{\+yx\ \-\+x)
The algebraic Cayley complex of the presentation is therefore 0 -> Ker(32) -* Z[g(16)] 2 4- Z[2(16)] 2 4 Z[Q(16)] 4- Z -* 0 where / 1 -I-x -I-x 2 4-x 3
vx -I- 1 \
and where e is the augmentation map. Normally speaking, it is still a step to go from a description of 32 to a description of TT2(£) = Ker(32). In this case we are aided by the fact that we can extend the Cayley complex to a complete resolution of period 4 (compare the complete resolution given in Section 41 and also [12] Chapter XII) as follows
Z[g(16)] 4 Z[<2(16)]2 3- Z[0(16)]2 4 Z[g(16)] where x- 1 1 — yx This identifies 7r2(Q(16)) with I*, the dual of the augmentation ideal. Every finite presentation determines a reduced two-dimensional cell complex, and vice versa. The viewpoints are equivalent and occasionally complementary. We observed above that for a finite presentation of a finite group G, n2(G) € £*3(Z). It is perhaps worth pointing out here what this does not mean. It does not of itself establish that each element of ^ ( Z ) can be realized in the form 7r2(5) where Q is a finite presentation of G. This point is crucial, and we shall return to it later. Evidently, an algebraic 2-complex £ e AlgG is geometrically realizable when there is a finite presentation of G
and a (weak) homotopy equivalence h : Ct(Q) -*• £.
Algebraic homotopy theory
187
When G is finite, the general fact that Tt2{Q) e ^3(Z) means that, although the isomorphism class of JT2(G) in !F(Z[G]) depends upon the particular presentation Q, its stable class depends only upon G. In fact, this is true for all finitely presented groups, as was first demonstrated by Tietze [66]. Tietze's Theorem shows that any two finite presentations of G are related by a finite chain of transformations of the following type (or their inverses): (I) Add a new generator x and a new relator R of the form xw~], where w is a word in the existing generators. (II) Add a relator R which is a word in the existing relators. If Q = {x\,... ,xg | W\,..., Wr) is a finite presentation of G, then under a Tietze transformation of type (I), iz2{Kg) remains the same, whilst under a Tietze transformation of type (II), itiiKg) adds on a direct summand Z[G]. It follows that: Theorem 48.4: Let Qx = {*,,..., xs | / ? , , . . . , Rr) and Q2 = (yu ..., yh | S\,..., Ss) be finite presentations for G; then 7T2(Kgl) and TTi{Kg2) are stably equivalent. We point out that Q,2{TJ) also plays a role in free differential calculus; Ker(9|) (sometimes called the second Fox ideal, despite the fact that it is not normally an ideal) is a representative of ^ ( Z ) .
49 Algebraicity of two-dimensional homotopy The main result of this section applies to all finitely presented groups, not merely finite groups. Throughout this section, until further notice, G will denote a finitely presented group. The functor C* : CW^ ->• Alg c transforms geometric homotopy into chain homotopy; that is, if K, L e CWg and / : K -> L is a homotopy equivalence over the identity in G, then C(/)* : C*(K) -»• C*(L) is a chain homotopy equivalence. If K, L e CW2G, and / : K -> L is a cellular homotopy equivalence which induces the identity on G = n\(K) = TZ\{L), then the induced maps /* : H0(K) -»• H0(L) and /* : 7T2(K) ->• TT2(L) are isomorphisms, so that, by (46.6), C*(K) and C*(L) are weakly homotopy equivalent; that is: Proposition 49.1: If K, L e CW2C, then K~GL=*
C.(K) ~ C*(L)
The converse to this is also true, as we now proceed to show.
188
Stable Modules and the D(2)-Problem
We can make some simplifying transformations. Let g = (xi,...,xg
|
Wu...,Wr)
be a finite presentation for G, and let y — y(x\,..., generators. Consider the presentation G(y) = {xu...,xg+l
xg) be a word in the
\Wu...,Wr+])
where Wr+i = x~lxy. Then Q(y) is also a presentation of G, the passage from G to Q{y) being effected a Tietze transformation of type (I). If K = Kg is the Cayley complex of Q and K{y) is that of Q{y) then K(y) is homotopy equivalent to K by a simple homotopy equivalence over the identity on n\ = G. Proposition 49.2: Let K, L be finite reduced 2-complexes with 7ri(AT) = 7t\{L) = G; then there exist finite reduced 2-complexes K\, L\ such that (i) K\ ~ G K and L\ ~ G L; (ii) K\n ^ L[[\ Proof: Let [x\,..., xm] (resp. {y\,..., yn}) be the generating sets of G corresponding to the 1-cells of K (resp. L), and regard y,- as a word in{x\,..., xm}. We can regard K as the Cayley complex of some presentation Q. Start with the generating set {x\,... ,xm}; by adding the elements yi,... ,yn successively, we obtain a sequence of presentations G, G ( y \ ) , G ( y \ , y 2 ) , . . . , G ( y i , y i , - - - , y n ) and a sequence of (simple) homotopy equivalences over the identity on it\ = G K C % ) C K(yu y2) C • • • C K(yi, y2,...,
yn) = K{
Likewise, regarding L as the Cayley complex £ = Cu of some presentation 7i, and adding the elements x\,...,xm successively, starting with the generating we obtain a sequence of presentations set {y\,...,yn}, H, H(xi),
H(xux2),...,
H(xi,x2,...,xm)
and a sequence of (simple) homotopy equivalences again over the identity on G L C L ( x \ ) C L ( x i , x 2 ) C ••• C L ( x i , x 2 , . . . , x m ) = L] Then in some ordering, the 1-cells of K\ and those of L\ correspond to the generating elements (x\, x2,..., xm, y\, y2,..., yn); that is, we may identify l) K[ with L™. a
Algebraic homotopy theory
189
Proposition 49.3: Let K, L be finite reduced 2-complexes with n\{K) = n\{L) = G, and suppose that Kw = L (1). If u : C*(K) -> CCL) is an augmented Z[G]-chain map; then v is chain homotopic to a chain map /x which restricts to the identity on C\{K) = C\(L). Proof: Begin with the chain map
d
'c2(K) i( < ) - >Z[G] C.(tf) lv = I v2 1 Vo I V\ C*(L)
\ C2(L) % n
t
,Id
> Z[G]
where, by hypothesis, Ci(L) = Ci(Ar)and31' r = d[ = 3i.ThenIm(/rf-v 0 ) C Ker(e) = Im(3i). Choose a Z[G]-homomorphism ho : Z[G] ->• C,(L) such that 3]/io = Id — \>o and put A.i = /IQ3I + vi, X2 = v2. We obtain a chain map
/
c2(/s:) -^ d(if) X z[G] -4 z
o\
i,C 2 (L)^t C , ( L ) ^ Z[G] Clearly Im(/^i - A.,) c Ker(3]) = I m ^ ) . We can now choose a Z[G]homomorphism h\ : C\{K) ->• C 2 (i) such that 92^1 = '^1 - A.|. Put \i2 =
= 32-^,32
The commutative diagram C2(K) | Ai 2
C2(L)
C\{K) X Z[G] -4 Z ->• 0
4- W
J.W lid
Ci(L) 4 Z[G] 4 Z ^ O
defines the desired chain map /z, and /i = (h\, ho) is a chain homotopy from \x to v. •
190
Stable Modules and the D(2)-Problem
If (p : C*(K) -> C*(L) is an algebraic chain map, by a realization of (p we mean a cellular map / : K -» L such that C*(/) =
C,(L)
First choose some cellular map k : K -*• L with the property that k extends the identity of K^ = L^ and induces the identity on n\. This can be done, for example, by first attaching cells of dimension > 3 to L to form a space BG of homotopy type K(G, 1). By construction, B^ — L. Then the identity map on Km extends to a map k : K —»• Be which classifies the universal G-bundle K —>• K and induces the identity on n\. Without loss of generality, the cellular approximation theorem allows us to choose k to be cellular without altering its definition on Kw where it already is cellular. In particular k : K ->• k{K) c B^ = L. Now the chain map C*(A.) induced by k makes the following diagram commute C2(,A ) —*• L-l^/v )
4. k2
||
C2(L) % C,(L) In particular
By the Hurewicz Theorem, Ker(3f) = H2(L;Z) can be identified with n2(L),
Algebraic homotopy theory
191
so that cp2 — ^2 gives a Z[G]-homomorphism
n - k2 : C2(K) -+ n2(L) Let [E\,..., £V} be an enumeration of the 2-cells of K. After lifting to K, we get a Z[G]-basis {E\,..., Er] of C2(K). For each i, let /z, : S2 —> L be a map whose homotopy class [hi] satisfies [hi] = (
h)(Ei)
Let E denote the standard 2-cell £ = { x e R 2 : | | x | | < 1} and put
a = JxeR 2 :||x|| = l j . The quotient E/a is homeomorphic to the one point union E v S2, so that we have a 'pinch map' c : £ -> £ v 5 2 , obtained by collapsing a to a point. Doing this for each of the 2-cells of K gives a collection of pinch maps c, : Ei —> £,• v 5?. Now we may write
where ' ~ ' makes identifications in the boundaries of the £,. Put
where ' ~ ' makes the same identifications as before; observe that E, and £,• v5, 2 have the same boundary. One may think of X as formed from K by attaching a copy of S2 at a single interior point within each 2-cell of K. Put A., = A.|£. and define
and put / = 4< o c : K —*• L. f is a basepoint preserving map; it is straightforward to see that / extends the identity on Km and induces the identity on 7T\. Furthermore, one calculates easily that, if /,, : C,,(K) —> Cn{L) denotes the induced map on n-chains, then
where we identify [ht] e n2(L) with its image in C2(L). However, by choice
192
Stable Modules and the D(2)-Pwblem
Thus
and so f2 =
D
Theorem 49.5: Let G be a finitely presented group, and let K, L e suppose that there exists an algebraic chain map cp : C*(K) -> C*(L) which is a chain homotopy equivalence, then there exists a cellular homotopy equivalence / : K -¥ L whose induced map on the fundamental group is the identity. Proof: Let
V. = C.(y) o<po C,(8) : C,(Ki) - • C.(Lfi Then fx is an algebraic chain homotopy equivalence inducing the identity on HoBy (49.3), we may further assume that /x restricts to the identity on Cr{K\) = Cr(L\) for r = 0, 1. Applying (49.4), we obtain a cellular map g : K\ -*• L\ inducing the identity on the fundamental group and such that g* = fx : H2{K\\ Z) ->• H2(L\\Z). In particular, g* : H2{K\\Z) ->• H2(L\;Z) is an isomorphism, so that g is a homotopy equivalence, by Whitehead's Theorem. Then / = / 6 o g o a : A ' - » L i s t h e desired cellular homotopy equivalence. D Corollary 49.6: If K, L e CW2G, then
K~G L^-
C,(K) ~ C*(L)
When Ti is a category which possesses a suitable notion of homotopy, we will denote by H its class of homotopy types. Then (49.5) can be paraphrased as saying: (49.7) The mapping induced by the cellular chain functor
is injective.
Chapter 9 Stability theorems
In this chapter we prove three of the six theorems stated in the Introduction. We have seen that stabilization induces a tree structure on the stable module £23(Z). Two-dimensional homotopy types possess an analogous tree structure; homotopy types at the bottom level are said to be minimal. We say that G has the realization property when all algebraic 2-complexes over Z[G] are geometrically realizable. The first of our results is then: Theorem II: The finite group G has the realization property if and only if all minimal algebraic 2-complexes are realizable. Theorem II is a general condition on homotopy types. It is more useful to have a criterion for realizability in terms of homotopy groups. The second result of this Chapter gives a sufficient condition on the minimal modules in Qi(Z) for G to possess the realization property. We say that a module J e fi3(Z) is realizable when for some finite 2-complex K e CWg there is an isomorphism of Z[G]modules J = ni{K); J is said to be full when the natural map Auiz[G](J) —> Ker(SJ) C Autp er(7) is surjective; we also prove: Theorem III: If each minimal module J e ^(Z) then G has the realization property.
is both realizable and full,
We proceed to both results via stabilization; at the geometric level this is simply the operation X
H>
X V S2
At the algebraic level, the straightforward analogue of geometric stabilization simply adds a copy of Z[G] to the 'algebraic 7^', replacing
193
194
Stable Modules and the D(2)-Pwblem
by 0 -> J ® Z[G] ' ^ d E2 © Z[G] %°]
E A E O ^ Z ^ O
Algebraically, the situation is slightly more complicated, in that it is also necessary to consider 'internal stabilizations' which do not alter congruence classes. This is simply the 'linear algebra' version of altering a group presentation by introducing a new generator and a cancelling relation. Finally, we give the classification of minimal two-dimensional homotopy types over a finite group G of period 4. It assumes its simplest form when G has free period four. Then there is a natural 1-1 correspondence between algebraic homotopy types and stably free modules; that is: Theorem IV: Let G be a finite group of free period 4; there is an isomorphism of directed graphs Alic <—> SF(Z[G]) At the minimal level, this gives a garametrization of minimal homotopy types by stably free modules of rank 1, Alggin <—> SF\{Z[G]). To describe the general case, we must first re-interpret the finiteness obstruction as a map, the 'Swan map' 5 : (Z/ICD* -> K0(Z[G]) Given a projective resolution of period 4 P = (0 -* Z -» P3 -* P2 -» P\ -* Po -* Z -> 0) the class of x(P) e K0(Z[G]) is not in itself an invariant of G. Rather, if P' = (0 - • Z ->• P3' -> P'2 -» P,' ->
PQ
->• Z -^ 0)
is another projective resolution of period 4, the difference x(P) — x(P') lies in Im(5), and allows us to define an absolute invariant x(C) € K0(Z[G])/Im(S). In the general case, minimal homotopy types are parametrized by rank 1 projectives within any stable class representing x(C). Without exception, throughout this chapter, G will denote a finite group. 50 Algebraic stabilization Fix a finite group G, put A = Z[G], and let M, N e ,F(A). If K is also a module in .F(A) there are functors 5 : Ext'(M, A0 ^ : Ext'(M, A^) -> Ext'(M, A^ ©
Stability theorems
195
defined on objects thus
and
where
( ° ) "(c) = (c 0)
()
and where E ^ , £+ act on morphisms in the obvious way. More generally, an extension £ = (o -* N - i An 4 • • • X Ao A M -+ o) e Ext n + I (M, N) can be decomposed as a Yoneda product of short exact sequences £ = £n o f,,_i o • • • o £\ o £o where, for 0 < /• < n £r = (0^
Ker(3r) -^ Ar X Ker(9r_,)
with
£„ = (o -> N -4 A,, 4 Ker(3n_,) -» and f0 = (0 - • Ker(/7) -> Ao 4- M ->• 0) For 1 < r < «, we define E*(f) = £„ o • • • o £r+l o Lf (5 r ) o E*(£,_,) o 5 r _ with the special cases at the ends given by Z*(£)
= £no---o£i
oE
and
X«(£)
= Xf.(£n)o£n_]
o---o£lO£0
With obvious action on morphisms, there are 'external addition' functors E5 : Ext"+1(JW, N) -+ E\f+[(M +1
+I
© K, N);
E f : Ext" (iW, N) ->• Ext" (M, N © K)
196
Stable Modules and the D(2)-Problem
determined respectively by the correspondences £ !->• E+(£)and£ i-> Ef(£). By contrast the correspondence £ H> E,5 (£) for 0 < r < n gives an 'internal addition' functor E * : Ext n+1 (M, AT) -* Ext" +1 (M, N) Internal addition does not alter the congruence class, as we now see: Proposition 50.1: If £ e Ext"+1(M, N), then
E*(£) « £ for all r in the range 1 < r < n. Proof: If £ = (0 ->• N -4 ,4, 0 /iT 4- Ao 8 # 4- M ->• 0), then Ef(f) is the extension (0^-N ^> /I, 8 / ( T ^ ^ o 0 A " ^ M -> 0). For r = 1, 2, the inclusion maps jr : Ar ^ Ar@ K,
ah> (a, 0)
induce an elementary congruence j : £ —> T,f{£), the formal inverse n : >• £ being induced by the projection maps nr : Ar 0 /if -¥• Ar,
{a, A.) i-^- a
The general case follows by dimension shifting.
•
In the special case where K = A*, the internal addition functor E^* is said to be an internal stabilization functor. Likewise E^ , E^ are called external stabilization functors, and to simplify notation we write
If £ e Ext" + '(M, A') then by a internal stabilization of £ we mean an object of the form
where 1 < r < n. Recall that Stab"+1(M, AO is the full subcategory of Ext" +1 (M, N) consisting of extensions of the form £ = (o _>
N
-4 En X • • • X E0 4 M - •
where each Z?r is finitely generated stably free. The internal stabilization functors E* preserve Stab"+1(M, A0 and, by (50.1), leave congruence classes
197
Stability theorems
unchanged. We allow the trivial case where each v, = 0, so that £ can be regarded as a trivial stabilization of itself. More generally, if S is stably free, the functors Er s , Ef, E+ will be referred to as quasi-stabilizations. In this case, internal quasi-stabilization maps Stab"+I (M, N) to itself and preserves congruence classes.
51 Stable isomorphism of stably free extensions Recall the notion of isomorphism of extensions. Let M, M', N, N' e let £ e E\tn+l(M, N) and £' e Ext" +1 (M', N'), and let cp : M -> M' be an isomorphism. We say that a morphism of the following type
0-> N £'
->•
A
0
o
A'o~*
0
M' ->
0
is a lifting of ^>. Moreover, when
0->£'
V
->•
M
N'
M' ^ 0
in which (i) both rows are exact; (ii)
0
defining K — Ker(ipo) splits. In particular there exists a homomorphism p : E -> K which splits the sequence on the left; that is, p o k — Id^. Let V'o : E —*• Q © K be the associated isomorphism, ijfo(e) — (
198
Stable Modules and the D(2)-Problem
commutative diagram 0-> N -+ E 4- M -> 0 4. f+
i fQ
\
defines a morphism \jr : £ -» £+(£'). Since
N -4 F 4- M -^ 0
£ \,
0,
Since /M > v(N') we may choose a surjective homomorphism r\ : AM -> N'. Let
and let
0 _ , iv © A " ^ F © A" ^
0 ->
A/'
-4
F'
M -• 0
4- A/' -> 0
Stability theorems
199
Since q>+ is surjective, then by (51.1),
Proof: By induction on n; note that the case n = 0 follows directly from (51.2), since we may take V = £ and Q = £'. Thus suppose that n > 1 and that the statement is true for n — 1, and decompose £ and £' as Yoneda products C
C
C
C
£?'
£?'
where £, = (0 -> AT -» Fn -+ fi -> 0);
and £[ = (0 -> W' -> F,; -> Q' -> 0);
Put I = u(fi')- By induction, there exists a stably free module T, an internal stabilization V,, of £„, and a internal quasi-stabilization Q,, of £'n such that ^> lifts to an isomorphism of extensions rjrn : S+(P,,) ^ ^ S+(Q,,) Put T = Sl(£i) and £ = Sl(£j) and put V = E 1 ( £ , ) O E^P,,) = T o E^; (P n ) and Q = Sl(£( o S^(Q,,) = ? o S|(Q n ). Then T3 is an internal stabilization off, and Q is an internal quasi-stabilization o f f . Now T, Q take the following forms . F = (()—>• N —*• Fn © A" —> Q ® Av —> 0);
g = (o ->. A'' -^ F ' © r -> £2' © r ->• o)
200
Stable Modules and the D(2)-Problem
The isomorphism rjrn : E+CPn) -> £+(<2n) restricts at the left-hand end to an isomorphism
f+: Q e A ? -> n' e r By (51.2), when /x > v(N'), ^+ lifts to a n isomorphism of extensions
for some stably free module S. \jr\ and \j/n glue together by Yoneda product to give an isomorphism of extensions
However
and
so that (p lifts to an isomorphism of extensions
as claimed.
•
52 Eventual stability of homotopy types Wefixafinitegroup G and put A = Z[G]; a{G) will denote the greatest integer s for which there is a stably free module S e ^"(A) such that S
Stability theorems
201
v(J) = 2. Moreover, in this case, rkz(7) = rkz(I*) = \G\ — 1 for all minimal modules J e ^ ( Z ) . Choose N, A" e Qn+i(M), and let £ e Stabn+](M, N), £' e Stab" +I (M', A'')- In particular, neither N nor A'' is projective. In the special case n — 0, weak homotopy equivalence is equivalent, by the Five Lemma, to 'isomorphism over IdM': Proposition 52.1: Let N, N' e £2i(M) be minimal modules, and let £ e Stab'(M, A'), £' e Stab'(M, N'); then
when \x > max{a(G) + 1, V\{M)}. In particular, if Z[G] has the property that stably free modules are free, then
provided fj, > v\(M). Proof: Put v = v\ (M). Since isomorphism over Id w is an equivalence relation, we may, without loss of generality, suppose that v(N') — v. Write £ = (0^A'-^F-^M^0) and £' = (0^iV'4F'4M->0) By (51.2), Id w lifts to an isomorphism of extensions
for some stably free module S e F(A). Since N, N'aie both minimal, rkz(A') = rkzCA''), and a straightforward rank calculation using the exactness of £, £' shows that F = F'. Exactness of £+(£) and £+(£') now gives F 0 A v = F 0 5 . If v = ui(Af) > 2, it follows by the Swan-Jacobinski Theorem that S = A v , and there is an isomorphism
The result for /x > v(A/) follows after further stabilization by E^~". If y = vi(M) = 1 and a(G) = 1 then 5 © A**-" = A^ for \x - v > 1, and the result follows on applying E^~" to f : ££(£) ->• S^(£'). a When n > 0, we can no longer equate weak homotopy equivalence with isomorphism over the identity. We begin by comparing homotopy types at the minimal level.
202
Stable Modules and the D(2)-Problem
Proposition 52.2: Let n > 0 and let N, N' e Qn+\(M) be minimal modules. If £ e Stab" +I (M, N), £' e Stab" +1 (M, N') then there is a homotopy equivalence
provided yu > max{cr(G)-|-1, v,,+\(M)}. In particular, if Z[C] has the property that stably free modules are free, then there is a homotopy equivalence
provided //, > v,,+)(A/). Proo/: By (51.3) there is an isomorphism of extensions over
where v = v(W), V is an internal stabilization of £, Q is an internal quasistabilization of £', v = v(N') and S is a stably free module with the property that 5
However £ « V and Q « f; since congruence (%) implies homotopy equivalence (~), we see in particular for v > 2 that
The result for /K. > v > 2 now follows by further stabilization. It remains to consider the case v = 1. Then (51.3) gives an isomorphism over Id^
where S is a stably free module such that S ® Q = Q[G]. In the case where o{G) = 0, that is when stably free modules over A are free, S = A and we have E | ( £ ) « S | ( P ) = E|(Q) « E | ( f ' ) giving a homotopy equivalence E^(f) ~ E | ( f ' ) whilst for fi > 1, the homotopy equivalences
Stability theorems
203
are obtained by stabilization. In general, we cannot assume that A has the cancellation property for free modules. Nevertheless, we still have S © A = A2 and we obtain E 2 (£) «
S 2 (??)
~
E 2 (Q)
^
^2+{£l)
and hence homotopy equivalences
for all /it, > 2. This completes the proof.
•
We obtain the following 'eventual stability' theorem: Corollary 52.3: (Eventual stability of homotopy types) Let N, N' e Q,,+\(M) and £ e Stab"+'(M, N), £' € Stab"+I(M, N') where n > 0; then there exist integers a, b > 1 such that
Proof: By (51.3), there is an internal stabilization V of £, an internal quasistabilization Q of v(N') such that Id^ lifts to an isomorphism
Since 5 is stably free then by the Swan-Jacobinski Theorem there exists an integer \x in the range 0 < \i < 1 such that S © A"- = Afe. Hence
is an isomorphism over Id^, so that E£ + "(£) % E^ + "(P n ) = S* (&,) « E* (f') Putting a = /x + y we have a homotopy equivalence £+(£) — %+(£') as claimed. D 53 The Swan map Let E be a stably free «-stem E = (0 ->• J -> £•„_, ->
>• £ 0 ->• Z -^ 0)
we have a unique ring isomorphism K-7 : Endper(-/) ->• Z/|G| and a bijection (26.5) e E : A(J) -> Proj"(Z, 7)
204
Stable Modules and the D(2)-Problem
given by e E (a) = a*(E). We define the Swan map SE : (Z/|G|)* -+ K0(Z[G]) by
where / : Proj"(Z, J) - • K0(Z[G]) is the Wall obstruction X(O -* J -* />„_, -* • • • -* Po -> Z -* 0) = 5 E depends ostensibly upon the particular stably free n-stem E. In fact, we show, in Theorem (53.7) below, that all Swan mappings are the same. As in Section 36, for any t e (Z/|G|)* we denote by £{t) = (0 ->• I(G) -> (/, t) \
Z -* 0)
the defining extension of the (projective) Swan module (/, t). In particular £(l) = (0 -> I(G) -4 Z[G] - 4 z ^ 0 ) is the defining extension of the augmentation ideal. We put S = S£m S is called the canonical Swan mapping (more properly, in view of the uniqueness property we shall establish, it should perhaps be called the canonical form of the Swan mapping). By a straightforward chase of definitions we get: Proposition 53.1: S : (Z/|G|)* ->• K0(Z[G]) is the mapping given by S([r]) = [(I, r)] It follows immediately from (37.9) that: (53.2) The canonical Swan map is ahomomorphism,S':(Z/1G I)* -» AT0(Z[G]). Example: The finiteness obstruction of a projective n-stem Let G be a finite group and let V\, V2 be projective n-stems in Proj"(Z, J) where J e D n(Z); then
To see this observe that, by (38.3), we may suppose that P; % Q o S{t{) for some fixed projective (n — l)-stem Q e Proj"~'(I(G), / ) . However, the Wall obstruction / is invariant under homotopy equivalence, and hence, for proj ective stems, also under congruence (46.8). Thus xCP,) = [(/, £,•)] — x(Q), so that XCP\) ~ XCP2) = [(/, ttj] - [(/, t2)] = S(tit2l) e Im(S) as claimed.
Stability theorems
205
It follows that for J 6 Dn(Z), thefinitenessobstruction gives a well-defined mapping X : Proj"(Z, J) - • £ 0 (Z[G])/Im(S) We proceed to show that SE = S for any stably free n-stem E. Proposition 53.3: Let E, F be stably free n-stems; if E % F then SE = SF. Proof: Write E = (0 -> J -> F n _! -*
>• £ 0 ~* Z -> 0)
F = (0 -* 7 -> F n _, -»•
• Fo -* Z -> 0)
and
and let [r] e (Z/1G |)*. Since E is a projective n-stem, the Z[G]-homomorphism Z —>• Z lifts to a self-morphism of E, thus ••-• F
o
^
4,
Z ^
4-
(T
x r
• • -)• Fo ->• Z -^ 0, Moreover, since Z —> Z is an isomorphism in the derived category, so also is a, and from the definition of the ring homomorphism K we have K[<*]
= [r]
Since a is an isomorphism in the derived category, a*(E) is a projective n-stem by (26.3). It follows directly from the definition that SE([r]) = x (a*(E)). Since projectives are relatively injective in !F{Z[G]), a : J ->• J extends to a self-morphism of F, thus
\. a
I
y-^Fn_,^
4-
i
x
^ F o ^ Z ^
so that, from the independence of K upon the choice of projective n-stem
In particular, [.$] = [r], so that 5F([r]) = x(«*(F ))- It suffices to show that x(a*(E)) = x(«*(F)). However, by hypothesis, there exists an elementary congruence E -> F, so that by (38.1), there exists an an elementary congruence v : a*(E) ->• a*(F).
206
Stable Modules and the D(2)-Pwblem
Since a is an isomorphism in the derived category, a*(E) and a*(F) are both projective n-stems, so u is a homotopy equivalence. Since the Wall obstruction X is invariant under homotopy equivalence, we have
as desired, and this completes the proof.
•
As before, let £{t) denote the standard projective 1-stem 0->I-»-(U)-»-Z->0 Proposition 53.4: If (I, t) is stably free, then Sm = S. Proof: Let r be coprime to \G\, so that x r : Z —> Z is an isomorphism in the derived category. We may lift x r t o a self morphism of £{t), thus 0 -* I - * (I, t) -+ Z ^ 0 I r ,|, Ar 4- x r 0 -^ I -> (I, t) -+ Z ^ 0 where, to distinguish the mapping from the formula, we write Xr : Z[G] -> Z[G] for the mapping A.r(x) = rx, and r for its restriction r = A.r|] : I —> I. However, it is straightforward to verify that r*(£(f)) = f (rt), so that
However, x((I, t)) - 0, by hypothesis, so that S£(t)([r]) — 5([r]) as claimed. D Proposition 53.5: If E is a stably free 1-stem, then SS+(E) = SE. Proof: Write T = (0 -*• J 4 F 4 Z -^ 0), so that E+CF) = ( o ^ ; e z[G] 4 F $ Z[G] 4 z -+ where
M and p = (p,0)
Stability theorems
207
Let r be coprime to \G\, so that x r : Z —> Z is an isomorphism in the derived category, and lift x r to a self morphism of T thus 0 - > 7 -> F -»• Z -> 0 | a 4- «o I x r
o^y
-+ F - » z
->• o
where, since x r : Z ->• Z is an isomorphism in the derived category, so also is a. Then a*(JF) is a projective cover by (23.2), and moreover
However, x r lifts to £+(.F), thus 0 ^ 7 ® Z[G] ->• F © Z[G] -> Z -> 0 0 -> 7 © Z[G] -»• F © Z[G] -^ Z - • 0 where, for )8 = a, 0 Id We see easily that a»(E+(^")) % E+Ca^J")), and so
which completes the proof.
D F
Corollary 53.6: If F is a stably free 1-stem, then S = S. Proof: The proof is essentially a repetition of the proof of the 'Eventual Stability Theorem' in a very easy case. Write F in the form
Comparing F with £( 1) we get J © Z[G] = I © F Moreover, since F is stably free, we have y © Z[Gf = I © Z[G] 6
208
Stable Modules and the D(2)-Problem
for some a, b. Write [t] = ifc(E+(£(1)) ->- E+(F)). However, it is clearly true that [t] = *(X!*(£(1)) -+ E*(£(0)), so that
Hence 5£J(F) _ by (53.3). However, by (53.5), we have SF = 5 s j m
and
so that
From the congruence £+(F) % T
=S
by (53.4), and the result follows.
• F
Theorem 53.7: If F is a stably free n-stem, then S = S. Proof: From (53.6) it suffices to take n > 2. Write F in the form F = (0 -> J -* F n _, ->
• F o -> Z -* 0)
where each F,- is stably free. As / e £2n(Z) = £2n_i(I(G)), we may choose a projective (n - l)-stem Q G Proj"~1(I(G), / ) Q = (0 -»• 7 ->• J2«-I ->
>• 6 i - ^ I(G) ->• 0)
By (38.3), F % Q o f(r) for some f e (Z/|G|)*. Suppose that [r] also belongs to (Z/|G|)*, and lift the Z[G] homomorphism Z -> Z to a self morphism of F, thus
—>•
J
—>
r
n
- l
—*••••—*
tQ
—>
A
—»•
Since Z —>• Z is an isomorphism in the derived category, so also is a, and from the definition of the ring homomorphism K we have K[a] = [r]
Stability theorems
209
Also a*(F) is a projective n-stem, and by definition SF([r]) = X («.(*)) However, by (38.3), a*(F) « Q o £(ic(a)t), and thus
= X(QoE(rr))
Since the Wall obstruction is invariant under homotopy equivalence, and thereby under congruence, we have x(£(0 ~ X(Q)) = X(F) = 0, so that, finally, SF([r]) = x(£(r)) = S([r]) as desired. D The definition of 'Swan map' can be extended to all projective, rather than merely stably free, n-stems P, by means of 5 P = x ° cp OK~1 However, Sp is, in general, no longer a homomorphism, but rather one has (53.8)
p
We leave the verification to the reader. Finally, if J € Qn(Z), we can define a Swan map SJ directly on A(J) SJ
=S OKJ
where, as usual, KJ : A(J) ->• (Z/|G|)* is the unique ring isomorphism. We can re-phrase (53.7) as follows: Proposition 53.9: Let J e £2n(Z); then SJ(a) n-stem F e Stab"(Z, J).
x(«*(F))forany stably free
Corollary 53.10: If J e £2n(Z), then under the action of A(J) on Proj"(Z, J), the stabilizer of Stab"(Z, J) is Ker(5). More generally, by virtue of (53.8), we have: Corollary 53.11: Let c e ko(Z[G]); if 7 6 D^(Z); then under the action of A(J) on Proj"(Z, J), the stabilizer of Proj"(Z, J;c) is Ker(S). (53.10) is the special case of (53.11) obtained by taking c = 0.
210
Stable Modules and the D(2)-Problem 54 Module automorphisms and ^-invariants
Let J e Dn(Z); we have seen, in Chapter 7, how the ring Endx>er(7) of endomorphisms of J in the derived category, and in particular the unit group A(J) = Aut£>er(7), acts on Ext"(Z, 7), and thereby on ^-invariants. Here, instead, we consider the action of the group Autz[C](V) ofmodule automorphisms. We denote by [ ] : Endz[G](7) -»• End£>er(7) the natural ring homomorphism, and by v the composite v = K O [ ] : Endz[G](7) - • Endper(Z); clearly both [ ] and v are surjective. Regarding Autz[cj(7) as the group of units of Endz[G](-/). v restricts to a group homomorphism v:Aut Z [ C ](y)-»(Z/|C|)* The significant change is that, whereas K : A{J) —>• (Z/|G|)* is a group isomorphism, and Proj"(Z, J) corresponds to a single orbit, v : Autz[G](J) —*• (Z/|G|)* is, in general, no longer surjective, and Proj"(Z, J) decomposes into several orbits. It is helpful to describe the action of Autz[C](-^) o n Ext"(Z, J) explicitly. Thus let E e Ext"(Z, J) be an n-fold extension
E
_
/ n
.
j _>
r-
""r'
"• c
Ji rj
For a e AutZ[C](7) we define an extension a • E «.E=(O^7^I£,,_,^I ...^£O AZ We note that 7 -4 £ n -i 4- a J, Id 7
—>• £,,_i
is a pushout diagram so that we may identify a • E with the 'pushout extension' a»(E) (Section 24). Moreover, there is an obvious homomorphism of extensions E
la \ 0
—>• 7 —>
I Id
4-Id 4. Id
En-] —*• • • • —*• ^ o —* Z —>
It follows immediately from the results of Section 34 that (54.1)
k(P -+ a • E) = v(a)k(P -+ E)
Stability theorems (54.2)
211
k(a • P -> a • E) = k(P -> E)
where P 6 Proj"(Z, 7) is a projective «-stem, E 6 Ext"(Z, 7) and a e Autz[G](-/)- Though tautologous, the following observation is nevertheless invaluable: Proposition 54.3: For a e Autz[G](^) the natural homomorphism a» : E —»• a • E is an isomorphism over Id z . Let a : 7 -> 7 be a Z[G] homomorphism which defines an isomorphism in Der, and let P be a projective n-stem; for each a e Autz[G|(^) the natural homomorphism a* : P —> a • P is a homotopy equivalence. In fact, we have: Proposition 54.4: lfa:J-*-J
is a Z[G]-homomorphism, then
P ~ a,(P) <^=> [a] e Im(u y ) /!• Suppose that takes the form P ; * = «*(P)
: P —> a*(P) is a homotopy equivalence, so that ^ 7 -> P,,_, -H
4, Id 4- Id V - ) • Qn-\
~
where i/r : J -> 7 is an isomorphism of Z[C] modules and v2 : Pi ->• 22 is the pushout map. Then K-7([i/r]) = fc(P ->• a*(P)). However, tautologously, one has KJ([a]) = k(P -+ a»(P)). Thus KJ([a]) = KJ([\JS]), and so [a] = [f]. But [i/f] is, by definition, v/(Vf); that is, [a] e ImCv7) as required. Conversely, suppose that [a] e ImtV), and choose an isomorphism of Z[G] modules x/r : J —> J such that a ^ \fr. Then a^CP) is congruent to ifr*(P). However, ijr defines a tautologous homotopy equivalence P -> ^r»(P) /»«-! Id Pn-\
P0-+ Z - > 0 X 4, Id 4, Id P o - » Z ^ 0,/
Thus i/r*(P) ~ P. Since ot«(P) is congruent to, and hence homotopy equivalent, D to i/f*(P), we see that a*(P) ~ P, as claimed. This completes the proof. In particular, we have: Proposition 54.5: If P is a projective n-stem, then for each a e Autz[c](7)
212
Stable Modules and the D(2)-Pwblem
It follows that, if J e £2n(Z) and F is a stably free n-stem, then, for each a e Autz[G](J), x(a • F) = 0. This may be re-phrased: if J e £2n(Z), and [ ] : Autz[G](J) ->• A(J) is the identification map, then Im([ ]) c Ker(Sy) moreover, from (53.10) and (54.4), we see directly that: Theorem 54.6: If J e £2n(Z) then extensions in Stab" +1 (Z, J) are classified up to homotopy equivalence by Ker(S y )/Im(i/). J e Qn(Z) is said to befull when the natural map [ ] : Autz[G](-/) -> Ker(5 7) is surjective. Theorem 54.7: Let J e fin(Z[G]); if J is full, then J 8 Z[G] is also full. Proof: There is a ring homomorphism Ez[G] Z[G]) defined by means of stabilization, viz. a 0
:
Endz[c](7) —> Endz[G](J ©
0 Id
This formula also gives a ring isomorphism Ej>er : Endp er(7) —>• Endoer(^ © Z[G]). By uniqueness of the ring isomorphisms KJ , icJeZ[G], it follows firstly that KJ = KJenG] 0 E r e r
and by composition with the Swan map, defined on the unit groups, also that
induces an isomorphism making the following diagram commute
Aut Z[G] (7)
Aut Z(G] (7 © Z[G])
Ker(SJ)
Ker(S yeZ[G1 )
Stability theorems
213
It follows that, if [ ] : Autz[G](^) -*• Ker(5 y ) is surjective, then [ ] : Aut Z [ C ] (/ 8 Z[G]) -> Ker(5 yeZ[G1 ) is also surjective. This completes the proof.
•
55 Realization Theorems There is a natural geometric stabilization process within CWQ : for X e we define S(X) = X V S2. More generally, we write Sm(X) = Xv S2 v • • • v S2 The correspondence X t-> S(X) is functorial with respect to based continuous maps; moreover, the following functorial square commutes up to natural equivalence CWG
Stab| [ C ] (Z,
-±>
TT 2 (-))
-A
CW G
Stab| [ C ] (Z,
JT2(-)
© Z[G])
Recall that an algebraic 2-complex E over Z[G] is said to be geometrically realized when there is a finite 2-complex C with n\{C) — G, and a homotopy equivalence
214
Stable Modules and the D(2)-Problem
show that, if N' e £23(Z) has height h > 1, and £' e Stab 3 (Z, W')> then £' is geometrically realizable. By the Swan-Jacobinski Theorem we may write
N' = N®Ah where N e £23(Z) is any module of minimal height. Let (£(r))r be a complete set of representatives of congruence classes of extensions in Stab3(Z, N), where r runs through the elements of Ker(S : (Z/|G|)* -* K^(Z[G])). By Yoneda's Theorem, £' is congruent to E+£(r) for some unique r e Ker(S). By hypothesis, each £(r) is homotopy equivalent to some finite 2-complex K(r). Thus there is a homotopy equivalence T,^_£(r) ~ Sh(K(r)) where KvS2v---vS2
S'\K)=
for any finite 2-complex K. By Yoneda's Theorem, £' is congruent to £+£(r) for some unique r 6 Ker(S), and so £' ~ In particular, £' is geometrically realizable, and this completes the proof.
•
The 'minimal realization' theorem just proved is a general condition on homotopy types. It is more useful to have a criterion for realizability in terms of homotopy groups. Say that a module J e ^3(Z) is realizable when there exists a finite presentation Q such that J = ^(C?). By means of the Tietze operation, which adds a redundant relation, we see that realizability is preserved under the stabilization operation / H-> J © Z[G]; that is: Proposition 55.1: Let G be a finite group and suppose that J e ^ ( Z ) ; if J is realizable, then J © Z[G] is also realizable. A module J e ^ ( Z ) is said to be strongly realizable when it is both realizable and full. From (54.7) and (55.1), we obtain: Theorem 55.2: Let G be a finite group and suppose that / e ^ ( Z ) ; if J is strongly realizable, then J © Z[G] is also strongly realizable. Strongly realizable modules have the following property:
Stability theorems
215
Lemma 55.3: Let G be a finite group and suppose that J e ^ ( Z ) is strongly realizable; then any algebraic 2-complex of the form
is geometrically realizable. Proof: By hypothesis, J is realized in the form J = TiiiQ) for some finite presentation
of G. Let [ 1 ] denote the algebraic Cayley complex of Q [1] = (0 -> J -» Z[G] r -> Z[C]« -+ Z[G] - • Z -» 0) If £ is an algebraic 2-complex of the form O^y^F2->F|^Fo-»Z-»O then by (38.3), £ is congruent to an extension of the form £(s) = (0 -» J -» Z[C] r -> Z[G]« -> (/, s) - • Z -» 0) wherei = /t([l] ->• £)) e (Z/|G|)*.Since£:isstablyfree,/(f) = 0. However, X is invariant under congruence (46.8), so that xU, s) = 0 and s e Ker(S : (Z/|G|)* -> ^ 0 (Z[G]). By hypothesis, there exists u € Aut Z(G |(./) such that v(u) = s where v : Autz[G](^) —> (Z/|G|)* is the natural map. Let w = « - ' e Aut Z[G] (y), so that v(w) = s~l e (Z[G]/|G|)*. The /c-invariant of the transition [1] —> w*£(s) is given by
Thus w*£(s) is congruent to [1], and hence w*£(s) ~ [1]. However, vv» : £(s) -> w*£(s) is an isomorphism, as in (54.3), a fortiori, a homotopy equivalence. Since £ « £(s), then £ ~ £(s), and we see finally that £ ~ [I]. The result follows, since [1] is the algebraic Cayley complex of the presentation Q. D
216
Stable Modules and the D(2)-Problem
We say that the stable module Sli(L) is strongly realizable when each J e £23(Z) is strongly realizable. From (55.1) we obtain: Theorem 55.4: The stable module ^ ( Z ) is strongly realizable if and only if each minimal module J e ^ ( Z ) is both realizable and full. From (55.3) and (55.4) we now get: Theorem III: If each minimal module J e ^ ( Z ) is both realizable and full, then G has the realization property.
56 Augmentation sequences Following Swan [63], for any finite group G we denote by SF(Z[G]) the isomorphism types of finitely generated stably free projective modules over Z[G]. When G is understood we write this as SF. Moreover, we write SF\(Z[G]) (= SF[) for the class of isomorphism types of stably free modules of rank = 1. Also following Swan, we write LF(Z[G]) (= LF) for the class of all finitely projective (= locally free) modules over Z[G]. In (16.1) we pointed out the result of Swan that, if P is a projective module over Z[G], then P
Stability theorems
217
where rj is an augmentation. Clearly any augmentation sequence is an exact sequence in T(Z[G]). The augmentation ideal I = I(G) is defined by the standard augmentation sequence 0 -> I -> Z[G] -4 Z ->• 0 where e : Z[G] —>• Z is the standard augmentation map given by e(g) — 1 for all g e G. Proposition 56.3: Let £* = (0 —> J^ —> F^ —> Z —>• 0) be augmentation sequences for £ = 1,2; then the natural map Hom(£i, £2) ->• Hom(Fi, F2) is surjective. /V0077 Let (p : F\ -> F2 be a homomorphism over Z[G]. Then (p2o<po 7',) (g> Id : Di
I 9 i
U —> J2 -^ F2 ~* L ~^ U
'-'
Recalling from Section 37 the notion of full invariance, we have: Corollary 56.4: Any augmentation sequence is fully invariant. With the same notation, we have: Corollary 56.5: If £\, £2 are augmentation sequences then £\ = £2 *! ?> F\ = F2
Proof: The implication (=>) is clear. To prove (•<=), observe that, if cp : F\ -> Fi is an isomorphism, cp can, by (56.3), be completed to a commutative diagram of the form F, 4-
fi
- •
VJ
1 ->• A J2
F2
z 0 ;
P2
(pz : Z -*• Z is clearly surjective and so is an isomorphism. It follows that tpj is an isomorphism. D
218
Stable Modules and the D(2)-Pwblem
If F is a stably free rank 1-projective, we write *([F]) = [J] when there exists a sequence of the form 0 -» J -*• F -> Z ->• 0. On the face of it, 4* is only a multi-valued (surjective) relation with domain SF] and codomain Q™'"(Z). In fact, the situation is simpler: Proposition 56.6: ^ is a surjective mapping * : SFi ->• fi',"'"(Z). Proof: Given augmentation sequences £,. = (o -* Ji -> Fj.4- Z -* o) for < = 1, 2, it suffices to show that F\ = F2 = > J\ = h- However, by (56.5), F\ = F2 => £\ = £2, whilst it is clear that £\ = £2 => J\ = h• For any augmentation module J, there is a unique ring isomorphism K = KJ : End Per (y) ->• EndDer(Z) = Z/|G| Theorem 56.7: If J is an augmentation module, then there is a 1 — 1 correspondence * - ' ( 7 ) ^ ^ Ker(5- / )/Im(v / ) Proof: Augmentation sequences 0—> J —> F —>• Z -» 0 are simply elements of Stab1 (Z, J). By the ^-invariant classification they are classified up to congruence by Ker(5y ). In general, from (54.6), homotopy types of extensions in Stab" +l (Z, J) are classified by Ker(5 y )/Im(v / ); however, for Stab'(Z, J), homotopy equivalence of such extensions is, by the Five Lemma, identical with isomorphism over Id z . In fact, the qualification can be dispensed with, since Aut(Z) = {±Idz), and given an isomorphism of augmentation sequences over —Idz, thus 0
0
J
->•
4-
VJ
F| ->
J -> F 2 ^
Z
—>•
0
; -W z
z _>.
0
the sequences become isomorphic over Idz on multiplying through by — 1. Thus the isomorphism classes of extensions in Stab'(Z, J) are classified by Ker(SJ)/lm(vJ), and the conclusion follows from (56.5) Q It follows from the Swan-Jacobinski Theorem that:
Stability theorems
219
Proposition 56.8: SF is a fork with \SF\ | prongs. The Swan-Jacobinski Theorem also shows that, if F is a stably free module of rank r > 2, then F = Z[G] r , and together with Schanuel's Lemma implies that, if r] : Z[G] r -> Z is surjective for r > 2, then
One sees that * extends to give a mapping of directed graphs (in this case, both forks)
using the same definition, namely ^([F]) = [J] when there exists a sequence of the form O - ^ Z - ^ ^ - ^ Z - ^ - O . In fact, we have: Theorem 56.9: For any finite group G, 4* : SF —> ^i(Z) is a surjective level preserving mapping of directed graphs with the property that, for each J e fii(Z), the multiplicity of the fibre * - ' ( . / ) is \Ker(SJ)\/\lm(vJ)\. Proof: The only detail left to be checked is that, when J e ^i(Z) is nonminimal, the formal multiplicity, |Ker(S" / )|/|Im(v / )|, is the actual multiplicity in this case, namely 1. This follows from the /c-invariant classification together with the Swan-Jacobinski Theorem and Schanuel's Lemma. • One can generalize the above discussion to arbitrary projectives: for c e K0(Z[G]), we put LF" = {PeLF:[P]
= c€
K0(Z[G])}
C
and denote by LF^ the subset of LF consisting of projectives of rank k. As a directed graph, with edges defined, as usual, by stabilization, LF is the disjoint union of its connected components LF°, where c runs through ATo(Z[G]), and each LF° is a fork, in which LF\ is the set of minimal elements. By a quasi-augmentation wemeanaZ[G]-homomorphismoftheform)j : P —>• Z where P is a projective module of rank 1, not assumed to be stably free; a Z[G]-module J is said to be a quasi-augmentation module when it has the form J = Ker(rj) where r\ is an augmentation. The previous discussion for augmentation modules goes through almost unchanged for quasi-augmentation modules; * extends to give a mapping of directed graphs
>• Df (Z)
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Stable Modules and the D(2)-Problem
using the same definition as before, namely *([P]) = [J] when there exists a sequence of the form
In fact, we have: Theorem 56.10: * : LF° -> D'j(Z) is a surjective level preserving mapping of directed graphs with the property that, for each J e D'j(Z), the multiplicity of the fibre * - ' ( 7 ) is |Ker(5 y )|/|Im(v / )|. 57 Classification over groups of period 4 For any finite group G, the correspondence F i-> ^ ( F ) determines a level preserving surjective mapping of directed graphs n-i : AlgG —>• £23 (Z). We begin by finding the size of the fibres of this map. Consider the natural action of Ker^ 7 ) on Stab3(Z, J), where J e ^3(Z).By (54.4), for F e Stab 3 (Z, J) the stabilizer of the homotopy type [F] is Im(v y ). Thus we get: Theorem 57.1: If G is a finite group, then the set of homotopy types within Stab 3 (Z, J) for J e £23(Z) is in 1-1 correspondence with Ker(5 v )/Im(v / ). In particular, for any finite group G and any J e ^ ( Z ) : (57.2)
K ' ( - / ) | = |Ker(S y )|/|Im(v y )|
We are now in a position to parametrize two-dimensional algebraic homotopy types over a finite group of free period 4. Theorem IV: Let G be a finite group of free period 4; there is an isomorphism of directed graphs Alic ^
SF,(Z[C])
In particular, the set of minimal two-dimensional algebraic homotopy types over G is faithfully parametrized by the set SF\(Z[G]) of stably free modules of rank 1. Proof: By (56.9) there is a level preserving map X : SF(Z[G]) —>• £2i(Z) with the property \k~l(J)\ = |Ker(5 y )|/v(7), where v(J) = |Im(v ; )|. Moreover, there is an isomorphism of directed graphs given by duality S : ^i(Z) -» n_i(Z),S(7) = J*, and itis straightforward to check that v(S(7))j= v(7).Thus putting fj,\ = 8 o X we see that (A\ is a level preserving map AlgG -» £2_i(Z) with the property that ^[(J) = |Ker(5 y )|/v(y) for all J e Q_i(Z).
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221
However, as we have seen in (57.2), there is a surjective level preserving map n2 : Algc -> £23(Z) with the property that |jr2"~'(y)l = \Ke.x(SJ)\/v{J). However, since G has free period 4, we have equality Q3 (Z) = £2_ 1 (Z), and it follows easily now that the tree structures on Alg c and SF\ (Z[G]) are isomorphic. O In particular, we have: Proposition 57.3: If G has free period 4, AIgG contains at least two minimal homotopy types precisely when Z[C] admits a stably free module which is not free. Notice that this occurs for all the generalized quaternion groups 2(2") with n > 5. [63], [29]. Now consider the more general case where the group G admits a projective resolution of period 4. As in Section 56, when c 6 Ko(Z[G]), we put
LF° = {P eLF: [P]=c] Again, by the Swan-Jacobinski Theorem each LF° is a fork in which the minimal level is the set LF*j of isomorphism types of projective modules P of rank = 1 with [P] — c. Suppose Z admits a complete resolution over Z[C] of the form
where each /-",• is finitely generated projective. By a sequence of elementary congruences, we may suppose the extension is congruent to one of the form 0 - > Z - * / > - > - F 2 - > - F , ->• F o -»• Z -> 0 where P is finitely generated and each F,- is finitely generated. It was shown by Milgram ([40], [41]) that for some groups G (for example, G = 2(8, 3, 11) [4]) P cannot be chosen to be stably free; that is, the finiteness obstruction is necessarily nonzero. By appealing to (56.10) instead of (56.9), Theorem IV generalizes as follows: Theorem 57.4: Let G be a finite group which admits a finitely generated projective resolution P of period 4; there is an isomorphism of directed graphs <—• LF=(Z[G})
where c = / ( P ) e K0(Z[G]). Swan [60] showed that the finiteness obstruction x(P) is not an absolute invariant of G, but can vary arbitrarily within a coset of Im(5). This implies
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Stable Modules and the D(2)-Problem
that the isomorphism type of the directed graph Lf r (Z[G]) remains constant as y r u n s through c + lm(S). It follows that: Corollary 57.5: Let G be a finite group which admits a finitely generated projective resolution P of period 4; then the set AlgG of homotopy classes of algebraic 2-complexes over G is fork, and has a unique homotopy type at each non-minimal level. The results just proved are rather stronger than a more direct application of the methods developed to achieve them might suggest. To see this, let G be an arbitrary finite group, and revisit the stability theorems in the case of extensions in Stab3(Z, J) for J e Q3(Z). We know, from Chapter 5, that £23(Z) is a fork, with a unique isomorphism type Jh at each height h > 1. In general, there are many choices of module at height 0. We choose Jo at height 0 to have the property that v(J0) — min{v(J) : height(y) = 0);thatis, y(7o) = ^ 3(Z). Then, by the Swan-Jacobinski Theorem, Jh = Jo® Z[G]h when h > 1. Moreover, we also make an arbitrary choice of a stably free 3-stem £(1) € Stab3(Z, Jo). Then we have: Proposition 57.6: Let G be a finite group, and let £ e Stab3(Z, 7/,); then there is a homotopy equivalence
provided h > max{or(G) + 1, v3(Z)}. Proof: Without loss we may write Jh = Jo © Z[G]'\ We use £(1) as the reference extension to parametrize Stab3(Z, Jo). Let (£(r)) r be a complete set of representatives of congruence classes of extensions in Stab3(Z, Jo), where r runs through the elements of Ker(S : (Z/|G|)* ->• K0(Z[G])), so that r =it(£(l)->- E(r)). Put 5 = E*.(£(l)). Then we may use S as a reference extension to parametrize Stab3(Z, Jh). In particular, for some unique r e Ker(S), r = k(S —>• £). However it is straightforward to see that r = k(S —*• E^.(£(/•)). It follows that there is a congruence E^_(£(r)) % £. Now by (52.2), when h > max{
2h+(£(r))
Composition gives the desired homotopy equivalence E^(£(l)) —> £.
D
This is not best possible, and in his thesis [9], Browning proved: Theorem 57.7: (Browning's Stability Theorem) Let G be a finite group, let Jo € ^3(Z) be any module of minimal height, and let £Q e Stab3(Z, JQ) be
Stability theorems
223
any stably free 3-stem at the minimal level; then for each h > 1, Stab3(Z, Jh) contains a unique homotopy type, namely that of E Browning's proof is a reworking of the Swan-Jacobinski Theorem in the context of extensions, extending earlier work of Williams [81]; see also [20], [21], [33]. For groups of period 4, Corollary (57.5) gives the same conclusion as Browning by a different route.
Chapter 10 The D(2)-problem
We finally turn to consider the D(2)-problem. We state it thus: D(2)-problem: Suppose that X is a finite three-dimensional cell complex such that H3(X;Z) = H3(X; B) = 0 for all local coefficient systems B on X. Is X homotopy equivalent to a finite two-dimensional complex? We show that, when G is finite, the D(2)-problem is equivalent to the Realization Problem; that is: Theorem I: The D(2) property holds for the finite group G if and only if each algebraic 2-complex over G is geometrically realizable. It then follows that Theorem III, already proved in Chapter 9, gives a sufficient condition for the D(2)-property to hold. Moreover, for groups of period 4 using the parametrization of homotopy types of algebraic 2-complexes given by Theorem IV and Theorem (57.4), and by applying the computations of Swan [63], we are able, in Section 62, to verify the D(2)-property in some specific cases, as well as to identify some potential counterexamples. 58 Cohomologically two-dimensional 3-complexes We begin with an elementary observation: Proposition 58.1: Let X be a finite 3-complex with n\(X) = G, and put K = X<2); then there is a canonical exact sequence of Z[G]-modules 0 -» H3(X;Z) -> C3(X) -^> 7t2(K) -> TT2(X) -> 0 Proof: We have exact sequences
0 -* Z3(X) -+ C3(X) 4 Im(33) -+ 0 224
The D(2)-pmblem
225
and 0 -> Im(33) -> Z 2(X) ->• H2(X;Z) -* 0 Splicing these together gives 0 -* Z 3(X) -> C3(X) 4 Z2(X) -> H2(X;Z) -> 0 However, since A' = X(2) we have
whilst
so that we have an exact sequence 0 -> H3(X;Z) -* C3(X) 3 - A/2(A-;Z) -+ tf2(X;Z) -+ 0 The result follows from the Hurewicz Theorem, as 7T2(K) = 7T2(k) = and 7T2(X) = TT2(X) = H2(X; Z).
D
Let X be a C W complex of dimension 3; we say that is cohomologically twodimensional when H^iX; Z) = H3(X; B) — 0 for any local coefficient system B on X. In the case when n\(X) = G is finite, there is a useful simplification. Recall that Cn{X) and Cn(X) have the same underlying abelian group, namely Hn(X(n), X("-l); Z). We take Cn(X) simply to be an abelian group, whilst Cn(X) is a module over Z[G], from the covering action of G on X. Since Cn{X) is a free abelian group of finite rank, the Eckmann-Shapiro Lemma gives an isomorphism Hom z (C(X);Z) = Hom ztC] (C n (X);Z[G]) and so we may interpret
H"(X;Z)= H"(X;Z[G]) Since X is a finite complex of dimension < 3, by the Universal Coefficient Theorem for Z-homology ([53], Chapter 5) Tor(//3(X; Z)) = Tor(// 4 (X; Z)) = 0 and Hi(X;Z) is a free abelian group whose Z-rank is the same as that of H\X;Z) = H3(X; Z[G]) = 0. Thus, when JT,(X) = G is finite, the condition ) = 0' is redundant and we obtain:
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Stable Modules and the D(2)-Problem
Proposition 58.2: Let X be a finite 3-complex in which n\ (X) is finite; then X is cohomologically two-dimensional if and only if H3(X; B) = 0 for any local coefficient system B on X. Observe that the above argument fails in general, since then the EckmannShapiro argument gives only an inclusion Hom Z[G] (C n (X);Z[G]) c Hom z (C n (*);Z) which is never an isomorphism when G is infinite (cf. [12], p. 358). Henceforth, unless stated explicitly to the contrary, G will denote afinitegroup. As a consequence, we have: Proposition 58.3: Let X be a finite 3-complex in which X\{X) = G is finite; if X is cohomologically two-dimensional, then: (i) (ii) (iii) (iv)
Im(33) is isomorphic to the free Z[G]-module C 3(X); Im(93) is a direct summand of JT2(K); and n2(X) ^ 7r2(A:)/Im03), n2(X) = n2(K) 0 Im(33).
Proof: By hypothesis // 3 (X;Z) = 0. Thus 33 : C3(X) -» Im(33) is an isomorphism. This proves (i). Furthermore, the exact sequence of (58.1) reduces to 0 -»• C 3(X) % m(K)
-> JT2(X) -»• 0
However Tor(7T2(X)) = Tor(// 2 (X;Z)) = Tor(// 3 (X;Z)) so that, again by the cohomology assumption on X, n2(X) is torsion free. Thus we have a short exact sequence in J-(Z[G]) 0 -> C3(X) X 7T2(K) -S. 7T2(X) ->• 0 in which C3(X) = Im(33) is Z[G]-free (of rank n say) and hence relatively injective. Thus n2(K) £ jr2(X) 0 Im(33) = 7r2(X) © Z[Gf This proves (ii), (iii) and (iv) simultaneously.
•
Since Im(33) is finitely generated and free over Z[G], it follows immediately from (58.3) that:
The D(2)-problem
227
Corollary 58.4: Let X be a finite 3-complex with 7T](X) = G; if X is cohomologically two-dimensional, then TT2(X) e
59 The virtual 2-complex Continuing with the notation above, G will denote a finite group, X will denote a cohomologically two-dimensional finite 3-complex with n\(X) = G, and K will denote the 2-skeleton of X. Up to homotopy type, there is no loss of generality in assuming that X is also reduced. Note that C*(X) = (0 -> C3(X) -> C2(X) - * C,(X) - * C0(X) -> Z - * 0) the cellular chain complex of X, fails in general to be exact at C2. We may refine the analysis of (58.3) slightly. Firstly, since // 3 (X; Z) = 0 the boundary map 33 : C3(X) -> C2(X) is injective. Moreover, we have: Proposition 59.1 TT2(X)
= KerO2 : C2(X)/lm(d3) -> C,(X)
Proof: This follows directly from the Hurewicz Theorem given that 7r2(X) = H2(X;Z). O Proposition 59.2: Im(33) is a direct summand of C2(X) and C2(X)/Im(33) is stably free over Z[G]. Proof: We have an exact sequence 0 -»• 7i2(X) -»• C 2 (X)/Im03) 4 C,(X) in which JT 2 (X) being a representative of ^ ( Z ) , is torsion free over Z. However, C\(X) is free over Z[G] and hence free over Z. Thus C2(X)/Im(33) is also torsion free over Z. The result follows from the injectivity of the free module Im(33) relative to JF(Z[G]), given that C2(X) is also free. • Let j : JT 2 (X) ->• C 2(X)/Im (33) be the inclusion obtained by making the identifications TT2(X)
= 7t2{K)/\m(h)\
C2{X) = C2(K).
We obtain an algebraic 2-complex (X) e AlgG by 0 -»• 7V2(X) -4 C2(X)/Im(33) -»• C,(X) - • C0(X) - • Z -» 0 Observe that (X) is functorial in the cell structure of X. We may think of (X) as being a 'virtual' two-dimensional homotopy type representing X. There is
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Stable Modules and the D(2)-Problem
a natural Z[G]-chain map
-> Ci(X)-»- C 0 ( X ) ^ Z - > 0
-* C 2 (X)/Im0 3 ) -+ C,(X) -»• C0(X) -* Z -* 0
where, since X is reduced, we may identify the epimorphism Co(X) —>• Z with the augmentation map. The 'algebraic n2 of <X> is simply 7r2((X» = 7T2(K)/lm(d3). Identifying ?r2(X) with H2(X; Z), by means of the Hurewicz Theorem, it is straightforward to check that: Proposition 59.3:
->
C2(K)
-* C i ( ^ ) - y Co(K) -
I v2 0
n2(K)/lm(d3)
-> C2(X)/Im(a3) -» Ci(X)
--
> z->- 0 ;w ;w Co(X) -> z ^0
Theorem 59.4: Let G be a finite group, and let X be a finite cohomologically two-dimensional 3-complex with ^i(X) = G; then X is homotopy equivalent (over the identity on n\ = G) to a finite 2-complex if and only if the virtual 2-complex (X) is geometrically realizable. Proof: Suppose that L is a finite 2-complex for which there exists a homotopy equivalence i/r : L -> X over the identity on G, then \/r : iv2{L) ->• n2(X) is an isomorphism. Hence q> o fa : ^ ( L ) -^ n- 2 «^)) is an isomorphism; thus (p o fa : C*(L) -> (X) is a weak homotopy equivalence, and so (X) is geometrically realizable.
The D(2)-pwblem
229
Conversely, suppose that (X) is geometrically realizable; let / : (X) -> C*(L), be a weak homotopy equivalence, where L is a finite 2-complex. As usual, let K denote the 2-skeleton of X; without loss of generality, we may suppose that: (i) Lw = Kw, and that (ii) fr = Id : Cr(K) = Cr{L) for r < 1. Furthermore, assuming X has TV cells of dimension 3, we may write X = KUai £<3) U«2 £<3) • • • Uow 4 3 ) Observe that / induces an isomorphism 3 - TT2(L)
The chain map / o n : C*(A") —*• C*(L) has the property that ( / o v)r = Id : Cr{K) = Cr(L) for r < 1. By (49.4), there exists a cellular map g : tf -> L such that: (i) U=Id: 7i\{K) -+ n,(L), and (ii) g* = f* ° ^* : In particular, since /* is an isomorphism, we see that Ker(g* : n2(K) -> n2{L)) = Ker(v,) = Im(93 : C3(X) - • 7r2(AT)) = Z[G]N The classes of the attaching maps a 7 all belong to • TZ2{L))
so that g : K -± L extends over the 3-cells of X to a map /i : X —>• L such that: (iii) K=ld:
nx(X) -s> TT,(L) and
(iv) A. = / , : w2(X) - • TT2(L).
Since Hr(X;Z) = Hr(L;Z) = 0 for 2 < r, it follows from Whitehead's Theorem that A is a homotopy equivalence as required. • 60 Reduction to two-dimensional data Theorem (59.4) allows us to trace the fate of an individual cohomologically twodimensional 3-complex X with finite fundamental group G; it will be homotopy equivalent to a finite 2-complex L precisely when the two-dimensional chain
230
Stable Modules and the D(2)-Problem
complex (X) is homotopy equivalent to the two-dimensional chain complex C*(L). We proceed to prove a result which treats the D(2)-problem in its entirety. We first give a criterion which allows us to construct 3-complexes which are cohomologically two-dimensional. We adopt the following notation: K: a finite 2-complex with Ti\{K) = G; m: an integer > 1; a: a Z[G]-homomorphism Z[G]'" -+ n2(K); {€j}\<j<m: the canonical Z[G]-basis for Z[G] m ; ay. a map S2 -> K in the homotopy class a(e,-). Furthermore, X{a) will denote the 3-complex obtained by attaching 3-cells to K by means of ot\,a2,... ,am; that is X(a) = K Uai £<3) Ua2 Ef • • • UO Proposition 60.1: Let A" be a finite 2-complex with 7i\{K) = G, and let a : Z[G] m ->• 7i2(K) be an injective Z[G]-homomorphism such that ^2(^')/Ini(Q;) is torsion free over Z; then H3(X(a); B) = 0 for all local coefficient systems B on X(a). Proof: We can make the following identifications C3{X&)) ^
Z[G] m
C2{X{a)) ^
C2{K)
iz2{K) ^
Z2{K)
Moreover, if i : n2(K) = Z2(K) c C2(K) denotes the inclusion, we may further make the identifications - • C2{X\a)) ^ i o a : H\X{a);B)
Z[G]"! - • C2(K)
<—* (Z[G]'")* ® B/Im((i oaf
Since a is injective and n2(K)/lm(a) is Z-free, all modules in the exact sequence 0 -»• Z[G]'" A n2(K) - • 7T2(/O/Im(a:) -* 0 are in .F(Z)[G]; hence its dual sequence is also exact, and a* : n2(K)* —> (Z[G]m)* is surjective. Similarly 0 ->• TT2(K) -> C2{K) -> C|(A") ->• C0(k) -> Z ->• 0 is an exact sequence within J r (Z)[G], and its dual sequence is likewise exact. Thus i* : C2{K)* -> TZ2{K)* is surjective and (a o i)* : C2{K)* -+ (Z[G]'")*
The D(2)-problem
231
is surjective. Thus (i o a)*
B) <—• (Z[G]'")* ® B/Im((i o a)* ® 1B)
we see that H3(X(a); B) = 0 for all local coefficient systems S.
D
We can use this to construct cohomologically two-dimensional 3-complexes corresponding to any two-dimensional Z[G]-free chain complex. Theorem 60.2: Let £ e AlgG; then there exists a finite 3-complex X such that (i) 7r,(X) = G; (ii) // 3 (X; B) = 0 for all local coefficient systems B; and (iii) (X) is homotopy equivalent to £. Proof: Represent the weak homotopy type £ e AlgG by £ = (0 -+ J -+ Z[G]X 4 Z[Gf 4 Z[G] 4 Z - > 0 ) Let L be a finite 2-complex with n\(L) = G; then by (52.3)
for some n, w > 1. However, £™(C*(L)) = C,,(L V m5 2 ). Put AT = L V mS2, and let
J@Z[G]n
If \jr : £ + ( £ ) —>• ^ is the natural chain projection, then f o y : C*(X) —>• f has the property that Ker(i/f o q> : 7T2(K) ->• 7 is isomorphic to Z[G]". Choose a Z[G] basis a i , a%,..., am for K e r ( / ) , and form X(a) by attaching 3-cells E\\ E2 ,..., E^ho K by means of a\, a2, • • •, am; that is X(a) = K Uai E? Ua2 E? • • • UOm E™
It follows from (60.1) that Hr(X(a); B) = 0 for all r > 3, and it is straightforward to see that {X) is homotopy equivalent to £. a Finally we obtain: Theorem I: The D(2) property holds for the finite group G if and only if each algebraic 2-complex over G is geometrically realizable.
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Stable Modules and the D(2)-Problem
Proof: Suppose the D(2)-property holds for G. If E is an algebraic 2-complex over G, then by (60.2) there exists a finite 3-complex X which is cohomologically two-dimensional and for which E ~ (X). Since we are assuming that the D(2)-property holds for G, then, by (59.4), there is a finite 2-complex K with iti(K) = G such that (X) is homotopy equivalent to C*(K). Thus E ~ C*(K), and E is geometrically realized. Conversely, if each algebraic 2-complex is geometrically realized, then, for any finite cohomologically two-dimensional complex X, {X) is geometrically realized. By (59.4), X is homotopy equivalent to a finite 2-complex, and the D(2)-property holds. • From Theorems I and III, we get: Corollary 60.3: Let G be a finite group; if each minimal module J e is both realizable and full, then the D(2)-property holds for G. 61 Group presentations and Theorem III shows that strong realizability of all modules J € £23 (Z) is a sufficient condition for the D(2)-property to hold. Though we shall not pursue the point here, the case of finite abelian groups shows that it is not necessary for minimal modules to be strongly realizable in order for the homotopy tree AlgG to be geometrically realizable. This is essentially a result of Browning generalizing earlier work of Metzler [39]. Latiolais [32] gives a rather more general account. Consequently, strong realizability is not a necessary condition for the D(2)-property to hold. On the other hand, realizability is a necessary condition, since ^ ( Z ) is the stable class of any module of the form nqiQ) where Q is a finite presentation of G. Since any J e ^ ( Z ) is the 'algebraic Hi of some algebraic 2-complex, we see immediately from Theorem I that: Proposition 61.1: If the finite group G satisfies the D(2) property, then is realizable. The question of the realizability of ^ ( Z ) is related to the classical question of minimal presentations of (finite) groups ([60]). We saw previously, (29.1), that the rational isomorphism type of the stable module £2n(Z) is given by (61.2)
I nB(Z)®Q = nB(Q)= \
n(Q)=
[Q] ,
if n is even .. . ,.
[ [IQ(G)]
if n is odd
n
\ *
If J e Q-}(Z) we define the module height hmoA{J) of J ®Q = I Q (G) © Q[G]k
The D(2)-pwblem
233
We define an integer valued invariant a>3(G) of G, thus co3(G) =
hmod(J)
for any minimal module J e ^ ( Z ) . Clearly 0 < co^G); moreover, for any N e n 3 (Z), co3(G) < hmod(N). We can approach this from a more geometric point of view. In Chapter 8 (48.3), we saw, for any finite presentation of the finite group G g = (xi,...,Xg\
wi,...,wr)
that g < r. The difference r — g is called the excess of the presentation, written exc(<5)-We define exc(G), the excess of G, to be the minimum of exc(^) as Q runs through all finite presentations of G. Let Kg denote the Cayley complex associated with Q and let C*(G) denote the the cellular chain complex of K.q\ then we get an exact sequence of Z[G]-modules of the form
o -* TT2(0 -* c2(ic) 4- c,(£) 4. co(£) -• z -• o Making the obvious identifications C2(iC) = Z[G] r ;
C,(iC) = Z[G]«;
Co(/C) = Z[G]
we calculate that (61.3)
hmod(n2(G))
= r-g=
exc(g)
It follows immediately that (61.4)
co3(G) < exc(G)
(61.4) is called Swan's Inequality. In different notation, it was observed by Swan in [60]. In all cases where both sides of the inequality are known, they are actually equal. As an illustration, we note that, if D2N is the dihedral group of order 2N, then CDT,{D2N) — exc(D 2^) = 0 when N is odd, and CD^DJN) — exc(D2/v) = 1 when N is even (see, for example, [78]). The question of whether the inequality is always an equation is both fundamental and difficult; &>3(G) is a 'linear invariant' and, for a particular finite group G, one might have reasonable hope that it can be calculated exactly. By contrast, the task of calculating exc(G), defined by minimizing over a nonrecursive set, is exceedingly difficult if not hopeless. The answer is still not known for many quite small and familiar groups. As Swan points out in [60], in the absence of any other method, the only real hope, at present, of finding the exact value of exc(G) is to try to find a presentation with excess equal to the computed value of U>T,{G).
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Stable Modules and the D(2)-Pwblem
One may avoid the difficulty of Swan's Inequality by restricting attention to groups G where exc(C) takes the minimum value 0; that is, when G has a presentation =
{Xu...,xg
in which r = g. Such a presentation is said to be balanced. In this case, by Swan's Inequality, one has: Proposition 61.5: Let G be a finite group having a balanced presentation
then
TCI(Q)
is a minimal element
62 Verifying the D(2)-property By Theorem I, the finite group G has the D(2)-property if and only if the tree Algc of two-dimensional algebraic homotopy types is geometrically realizable. This is equivalent to requiring that every minimal algebraic homotopy 2-type is homotopy equivalent to the Cayley complex of a presentation. If G is a finite group having a free resolution of period 4, it is easy to see that G has a minimal algebraic 2-complex of the form F = (0 -* J -> F2 -> Fx -> Fo -+ Z -> 0) where rkz(7) = \G\ — 1. One can see this by first observing that there is a minimal complete resolution of the form
where P is stably free of rank 1, and taking J = Im(P ->• Fi). Thus, if F is realizable, it must be realizable by a balanced presentation. In view of Theorem IV, we see that, if G has cancellation property for free modules, then F is the unique minimal algebraic 2-type. We obtain: Theorem 62.1: Let G be afinitegroup which admits a free resolution of period 4; if G has the free cancellation property, then G satisfies the D(2)—property <=>• G admits a balanced presentation. We now consider some examples:
The D(2)-problem
235
(i) Cyclic groups The cyclic group C,, evidently admits a balanced presentation Cn = (x | x") The cyclic groups are unique in admitting a free resolution of period 2
o ^ z - i z[cn]*-+z[cn] 4z->o which can be repeated to give a free resolution of period 4 0 -• Z 4 Z[CB] "->' Z[C,,] 4- Z[C,,] *-+ Z[CB] 4 z ^ 0 by taking E = 1+JCH
|-x"~'. Since Z[C,,] has the Eichler property, it has
the cancellation property for free modules. Again by Theorem IV we have (62.2) The cyclic groups Cn satisfy the D(2)-property. (ii) Dihedral groups of order = 2 mod(4) The dihedral group D4n+2 of order An + 2 can be defined by means of the balanced presentation
D4«+2 = (/ vx , y,. I| x,.2/1
+1
,,\ = ,,2 y , ,y, „x« — „x" + 'y)
and £>4n+2 admits a free resolution of period 4 (this is false for the dihedral groups of order An) 0 - • Z 4- Z[D 4n+2 ] -4 Z[D 4 n + 2 ] 2 ^ Z[D 4 n + 2 ] 2 ^ Z[D 4,,+2] 4 z ^ 0 Here (5=
;
^ -X+X"y
)>
92~
;
y_Zy
x
3i ~ (x — 1, y — 1)
»-l)'
where E, = l + x + - - - + x 2 " ;
Er = l + y ;
6>i = 1 + x + • • • + x n ~ ' and
6>2 = 1 + x + • • • + x" = 6», + x" whilst e is the augmentation map, and e* is its dual. On general grounds, one can show that Z[D 4n+2 ] has the Eichler property, and so satisfies free cancellation. Thus, again from Theorem IV, we get: (62.3) The dihedral groups Z)4n+2 satisfy the D(2)-property.
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Stable Modules and the D(2)-Problem
(iii) Direct products D2n x Cm(n,m odd and coprime) It can be shown along the lines of [46] that, when n, m are odd and coprime, the 'quasi-dihedral' groups G = Din x Cm admit balanced presentations with two generators and two-relations, and that the corresponding algebraic Cayley complexes can be extended to give a periodic resolution of the form
0 -* Z -> Z[C] -»- Z[G] 2 -• Z[G] 2 -> Z[G] -* Z -• 0 Once again, by virtue of the Eichler property, Din x Cm has free cancellation and we have: (62.4) The 'quasi- dihedral' groups Din x Cm, n, m odd and coprime, satisfy the D(2)-property. By a celebrated theorem of Milnor [42], no dihedral or quasi-dihedral group can be the fundamental group of a closed 3-manifold. The next few examples, however, do all occur as fundamental groups of closed 3-manifolds. In this connection, one should note on general grounds, using Morse Theory, that, if the finite group G is the fundamental group of a closed 3-manifold, then G admits a finite free resolution of period 4 of the form 0 -»• Z -»• Z[G] -* Z[G]« -*• Z[GF -* Z[G] - • Z -» 0 where g is the number of generators/relations. We will pursue this point further in Chapter 11. (iv) The spin Euclidean groups We consider first the symmetry groups of the standard Euclidean solids; that is, denote by T, O, I c SO(3) respectively the tetrahedral, octahedral, and icosahedral groups. These groups do not have periodic cohomology. However, their liftings T, O, I C Spin(3) are fundamental groups of closed 3-manifolds, and so, by the general considerations above, do have both resolutions of free period 4, and balanced presentations. In fact, explicit balanced presentations were given by Kenne in his thesis [31] (see also [46]). These groups do not satisfy the Eichler property. Nevertheless, Swan [63] has shown they have free cancellation. From Theorem IV, we see that: (62.5) The spin Euclidean groups T, O, I satisfy the D(2)-property. (v) The exceptional quaternion groups Q{4n) (2 < n < 5) For each n > 2, the quaternion group Q(4n) is defined by the balanced presentation Q(4n) = (x,y
\ x" = y2, xyx = y)
The D(2)-problem
237
Q(4n) can be regarded as the spin double cover of the dihedral group Din. We saw in Chapter 8 that g(4n) has finite free resolution of period 4 O ^ Z 4 Z[G(4n)] -4 Z[0(4n)] 2 X Z[g(4n)] 2 X Z[Q(4n)] - 4 z ^ 0 obtained by extending the algebraic Cayley complex of the above presentation one step to the left. As can be seen from the Wedderburn calculations in Section 12, the quaternion groups all fail to have the Eichler property. Nevertheless, the first four examples, 0(8), <2(12), 2(16), and 2(20) are exceptional in that they do possess the cancellation property for free modules; thus we get: (62.6) The exceptional quaternion groups 2(8), Q(12), 2(16), 2(20) satisfy the D(2)-property. A similar argument shows: (62.7) The direct product 2(8) x C3 satisfies the D(2)-property. (vi) The generic quaternion group Q(4n) (n > 6) Now consider the generic quaternion groups Q(4n) (n > 6). By Theorem IV, the minimal homotopy types are parametrized by SF\. As we have already pointed out, Q(4n) possesses a balanced presentation and a free resolution of period 4. However, the situation is now much more problematic, since for all n > 6, 2(4«) has stably free modules which are not free, so that Q(4n) has at least two minimal algebraic homotopy 2-types, and, of these, only one, the Cayley complex of the standard presentation, is as yet known to be realizable. In the first few cases, the precise numbers of stably free modules, and hence minimal 2-types, can be retrieved from the calculations of Swan ([63], Theorems III and IV). They are: |SF,(Q(24))|=3; |SF,(G(28))| = 2; |5F,(6(32))| = 2; |SF,(e(36))|=4; |SF,(G(40))|=8. (vii) The groups Q(8n; p, q) The group Q(8n; p, q) can be defined as the pull back extension 1 -> Cp x Cq -> 2(8«; p, q) -»• Q(in) ->• 1 arising from the canonical imbedding C2 x C2 ->• Aut(Cp) x Aut(C9), by projecting Q(Sn) on to C2 x Cj. When n, p, q are mutually coprime, the groups
238
Stable Modules and the D(2)-Problem
have period 4. From the point of view of the D(2)-property, they are genuinely more complicated than the generic quaternion groups. Milgram has shown([40], [41], see also [4]) that in some cases (for example Q(8; 3, 11)) they do not have free period 4. In all except the degenerate case where p = q = 1, <2(8, p, q) admits a generic quaternion group as quotient. Thus Swan's Theorem [63] shows that the cancellation property for free modules fails for all except the degenerate case. In fact, from Swan's results, one sees slightly more: cancellation fails in every stable class of KQ except in the case 2(24) = Q(8; 3, 1). It follows that cancellation fails in the stable class of the finiteness obstruction, in every nondegenerate case. This includes Q(8; 3, 1), since there cancellation fails in the stably free class, which in this case is also the class of the (zero) finiteness obstruction. Hence we see: (62.8) Over <2(8, p, q), the fork of two-dimensional algebraic homotopy types has at least two prongs except in the degenerate case p = q = 1.
Chapter 11 Poincare 3-complexes
A finite Poincare complex M of dimension n is said to be of standard form when it can be described thus M = K Ua • A' is a continuous map. As we described in the Introduction, Wall showed in [72] that, when n > 4, every finite Poincare ^-complex is homotopy equivalent to one in standard form. Since the question in dimension 3 was the genesis of the D(2)problem, it is appropriate to conclude our investigation of the D(2)-problem for finite fundamental groups by asking how far it takes us in the direction of obtaining standard forms in dimension 3. We prove: Theorem V: Let G be a finite group; then G has a standard Poincare 3-form if and only if there is a finite presentation Q of G with K2(G) = I*(C). Moreover, G then necessarily has free period 4, and the presentation Q is automatically balanced. If G is a finite group of free period 4, which also has the D(2)-property, it is straightforward to see that there exists afinitePoincare 3-complex M of standard form such that n\(M) = G. However, the intransigence of the D(2)-problem suggests the possibility that Poincare 3-complexes might need to have more than one 3-cell. To an extent, this is supported by a consideration of classical examples. On general grounds, it can be shown that smooth 3-manifolds admit cellular representations with just one top-dimensional cell. Nevertheless, their natural representations are often more complicated. For example, if we regard the 3-sphere S3 as the group of unit quaternions, then S3 admits a cell structure, given by the Schlafli symbol {3, 3,4}, which is equivariant under the action of the integral quaternion group 2(8), and which has two orbits of 3-cells under 239
240
Stable Modules and the D(2)-Problem
the Qs-action [15]. In other words, the natural geometric cell structure on the quotient S3/Q(%) has two top-dimensional cells, and so, in Wall's scheme of things, is nonstandard. Moreover, the associated presentation of Q(8) is unbalanced. Now of course, in this case, since S3/Q% is a closed 3-manifold, it also has another cell structure, with a single top-dimensional cell, and for which the associated presentation is balanced. Yet, at the algebraic level, this is not immediately obvious, and it requires at least some effort to go from the unbalanced presentation to a balanced one. That being so, for certain finite groups of period 4, we show that the connection between the standard form problem and the D(2)-problem is one of equivalence. Theorem VI: Let G be a finite group which admits a free resolution of period 4; if G has the free cancellation property, then G satisfies the D(2)—property «=>• G admits a balanced presentation. These results were published in [26].
63 Attaching 3-cells to a presentation Let G denote a finite group, and let M be a closed connected 3-manifold with 7i\{M) = G. By Morse theory, M admits a handle structure with one 0-handle, one 3-handle, and equal numbers of 1- and 2-handles. Let Mo denote the bounded manifold obtained by removing the 3-handle, so that (63.1)
M = Mo U ei
With this notation, we have: Proposition 63.2: There is an isomorphism of Z[G]-modules jr2(M0) = V(G) Proof: We denote by Mo the universal covering of A^o, and by 9 Mo the (disconnected) covering of the boundary 9 Mo induced from the covering map Mo —*• MQ. Taking Z-coefficients throughout, we get the following exact sequence of relative cohomology H°(M0;dM0) -*• H°(M0) -> H°(dM0) -*• Hl(M However, H' (Mo) = 0, whilst Lefschetz Duality gives
Poincare 3-complexes
241
Moreover H°(Mo) = Z. Let e* : Z -> Z[G], denote the dual augmentation map e*(l) = J2g 8- I t ' s c ' e a r t n a t dM0 is a disjoint union of \G\ copies of S2, and under the covering action of G, we may make the identification H°(dMo) = Z[G]. Moreover, the mapping H°(M0) ->• H°(dMo) then corresponds to e*, and the above exact sequence takes the form 0 ^ z 4
Z[G] - • H\M0;dM0)
-* 0
from which we make the identification
However Lefschetz Duality gives an isomorphism so that, finally, the result follows from the Hurewicz Theorem H2(M0)
= 7i2(M0) = TT 2 (MO)
This completes the proof
•
Suppose that the finite group G is the fundamental group of a closed 3manifold M. As in (63.1), we write M — A/o U e3, noting that the presentation of G given by this handle decomposition is balanced. However, Mo collapses on to a two-dimensional complex K with just one 0-cell and equal numbers of 1- and 2-cells, so that, for some mapping a : 3e3 — (63.3)
M~K(2) Ua ee3
and K{2)Uae3 is afinitePoincare 3-complex with 7T\(KmUae3) = nx{M) = G. Moreover, niiG) = xi{K) = 7T2(Mo), SO from (63.2) we see that (63.4)
n2(G) = I*(G).
We shall show, as a partial converse, that a finite group G has the property that 7T2(G) = [I*(G)] precisely when there exists a finite connected Poincare complex Xc with TT\(XG) = G. First we prove some preparatory results. By the Hurewicz Theorem, any simply connected Poincare 3-complex is homotopy equivalent to S3. Thus a finite complex XQ with finite fundamental group G is a Poincare complex if and only if its universal cover Xc is homotopy equivalent to S3. Suppose that Q = {x\,..., xg; W\,..., Wr) is a finite presentation of afinitegroup G, and consider the effect of attaching a collection of 3-cells to Kg. Let a : Z[G](N+X) -> n2(Q) be a homomorphism of Z[G]modules, and let as : S2 ->• Kg be a map in the homotopy class a(€j), where
242
Stable Modules and the D(2)-Pwblem
{tj}\<j
M(«) = Kg U K ) . ] J e) Choose liftings of,- : S2 —> /^ of the ay to the universal covering of /w and let a* : Z[G] -*• 7T2(5) be the homomorphism of Z[G]N+] determined by sending €j to the homotopy class [2,]. For each g e G and each index;, 1 < j < N+\, \etS2 ; (resp. e\ ,) denote a copy of 5 2(resp. e 3), and let ga,- : S2 t, ->• A" denote the translate of ay byg.Puta = JJ 7 g®j '• ]Jg y 52y -> /^. Then the universal covering A/(a) of M(a) can be described as a pushout
The Mayer-Vietoris Theorem for the pushout
gives an exact sequence, with Z-coefficients 0 -> H3(M) - • H2 (\}S2gj
) ( ^ 0) H2{K) 0 W2 ( \ \ e 3 g j
Proposition 63.5: If Q — (x\,...,xg;W\,..., Wr) is a finite presentation of a finite group G, and a : Z[G](-N+[^ —>• 7T2(
Jfc = 0
Ker(a),
it = 3
Proof: The proof is entirely straightforward once it is appreciated that we may identify H2(Ugj S2gJ) with Z[G] W+1 , and (a, 0 ) : H2(UgJ S2gJ) - • H2(k) ® D HiiUgj 4 ; > w i t h a Corollary 63.6: Let Q — {x\, ...,xg;W\,..., Wr) be a finite presentation ( +1) of a finite group G, and a : Z[G] " -> 7T2(£) be a homomorphism of
Poincare 3-complexes
243
Z[G]-modules; then M{a) is a Poincare complex if and only if a is surjective and n — r — g.
Proof: M{a) is a Poincare complex if and only if M(a) is homotopy equivalent to S3. However, M(a) is a 1-connected complex of dimension 3, so that, by the Whitehead Theorem, M{a) is homotopy equivalent to S3 if and only if H2{M; Z) = 0 and H3(M; Z) = Z. By (63.5), we see that H2(M; Z) = 0 if and only if a is surjective. A straightforward rank calculation shows in this case that
so that, when a is surjective, H^(M; Z) = Z if and only if n = r — g. In fact, in this case we have seen, in (63.5), that Ker(a) is the trivial Z[G]-module with Z-rank equal to 1. D
64 A characterization of groups of period 4 We can now prove: Theorem 64.1: The following conditions on the finite group G are equivalent: (i) [;r2(G)] = [I*(G)] € Stab(Z[G]); (ii) G has a (finite) free resolution of period 4; (iii) G has a (finite) free resolution of period 4 of the following type;
o -» z -> ztG]" -> c2(k) X cx(k) X co(k) -* z -* o where
C2(AT) 4. C,(£) 4 C o (£) -> Z -> 0 is the partial resolution afforded by some finite presentation C? for G. (iv) there exists a finite connected Poincare complex XQ with TT\{XQ) = G. Proof: (i) = > (iv) Suppose that n2{G) = [I*(G)], and let W = ( x , , . . . , xg \ Wx,..., Ws) be a finite presentation for G; then we have the following exact sequence involving
0 -»• JI2(H) -»• Z[G] r
-> Z[G]«
- • Z[G]
- • Z -> 0
The hypothesis n2(G) = [I*(G)] implies that, for some m, n > 0 0 Z[G]ffl ^ I*(G) © Z[Gf
244
Stable Modules and the D(2)-Problem
Putr = s+m.Ifm = 0, we put Q- 7i;ifm > Owe modify H. by adding m trivial relations Ws+\, • •., Wr to obtain a presentation £ — (x\,... ,xg \ W\,... ,Wr) with
and n = r — g. Either way, we produce an exact sequence of Z[G]-modules 0 -> m -»• Z[G] r -> Z[G]« -> Z[G] - • Z -* 0 in which 7r2 = n2(G) = I*(G) © Z[G]". Since I*(G) is an epimorphic image of Z[G], we may choose a surjective homomorphism of Z[G]-modules a : Z[G] n + 1 ->• TT2. Put X G = M(a) = ^
Uao e^ Ua, e\--- Uffn e^
Then, since n = r — g and a is an epimorphism, it follows, by (63.6), that XG is a finite Poincare 3-complex with TT\(XG) = G. (iv) = • (iii) If Xg is a finite Poincare 3-complex with it\{Xc), then the cellular chain complex 0 - • Z -> C3(X^) -> C2(X^) 4 . C,(X^) 4 Co(X^) -»• Z - • 0 is a finite free resolution of period 4 in which
C 2 (£;) \ CX(XG) X
CO(XG)
-• Z - • 0
is the partial resolution afforded by some finite presentation Q for G, namely that given by the 2-skeleton of XG(iii) = > (ii) follows from (63.5) and (63.6). (ii) =>• (i) As we observed in Chapter 7, the condition that G has free period 4 is equivalent to requiring that ^ ( Z ) = S2_i(Z). However, ^ ( ^ ) is a representative of ^ ( Z ) , and I*(G) is a representative of £2_i(Z). • For each integer n > 0 introduce a class 'P(n) of finite groups as follows: V(n): A finite group G belongs to V(n) when there exists a finite Poincare 3-complex XG for which 7Ti(XG) = G and which has exactly n + 1 cells of dimension 3; that is, X<j has a cell structure XG = K(2) U e^ U • • • U e3n
Poincare 3-complexes
245
Let M denote the class of finite groups G for which there exists a (smooth) closed connected 3-manifold M with Ti\{M) = G; these classes are related by a sequence of inclusions M C V(0) C 7>(1) C • • • C V(n) C V(n + 1) C • • • The inclusion V(n) c V(n + 1) is obtained by taking the one point union X V e 3 of X € P(n) with a 3-cell e 3, attached at a point * e 3e 3 . The inclusion .M C V(0) is obtained in the usual way, by taking a handle decomposition of a closed connected 3-manifold with a unique 3-handle. The condition for membership of V(n) can be formulated algebraically; recall that to any finite presentation Q = (x\,... ,xg | W\,..., Wr) of the group G we associate a canonical two-dimensional complex Kg with n\(Kg) = G by regarding the generators x, as 1 -cells and the relators Wj as 2-cells. If V denotes the class of finite groups of free period 4, then evidently V — Un>o ^(n) by Theorem B. The filtration on V can then expressed by observing that the finite group G belongs to V{n) precisely when the trivial module Z admits a finitely generated free resolution of period 4 over Z[G] of the type 0 -»• Z -+ Z[G] n+1 -» C2{K) X Ci(K) X C0(K) -»• Z -+ 0 and where Jf) 4 Ci(iP) 4- Co(AT) -> Z -> 0 is the partial resolution afforded by some finite presentation Q for G.
65 Poincare 3-complexes of standard form We begin with an algebraic result: Lemma 65.1: Let A be a finitely generated module over Z[G] which is stably equivalent to I*(G); if there exists an epimorphism
Proof: If
246
Stable Modules and the D(2)-Pmblem
is the trivial one-dimensional module over Q[G]. Hence we have a short exact sequence of Z[G]-modules 0 -* Z -> Z[G] -^ A -> 0 where Z = Ker(
0 -> A* -> Z[G] -»• Z -*• 0
We also have another short exact sequence (II)
0 -> I(G) -> Z[G] -> Z -» 0
Applying Schanuel's Lemma to (I) and (II), we obtain an isomorphism of Z[G]modules
Let n : I(G) © Z —> Z be the projection. Since A* and I(G) are rationally equivalent, A* admits no non-trivial mapping to the trivial one-dimensional Q[G] module. Hence n o 4* : A* -> Z is trivial, so that *(A*) C I(G). Reversing the argument, we see that *~'(I(G)) C A*. That is, * : A* © Z -> I(G) © Z restricts to an isomorphism 4> : A* -*• I(G). Dualizing, we see that A =z[cj I*(G). • Now suppose that the finite group G possesses a standard three-dimensional form, that is, a Poincare 3-complex XG with just one top-dimensional cell and which satisfies TI\(XG) = G. Let Q be the presentation of G afforded by the 2-skeleton of XQ- From Theorem B and the algebraic interpretation of the filtration on V, we see that there is a finitely generated free resolution of period 4 for Z over Z[G] which takes the form 0 -* Z -* Z[G] -> C2 (K) A Ci(AT) A C 0(tf) -> Z -»• 0 where C2V,J^ ) —^ C i ^ A J —> L,Q\rL ) —?• A —* U
is the partial resolution afforded by Q. In particular, we have an epimorphism ofZ[G]-modules
Z[G] -
^ 4
^
Poincare 3-complexes
247
and we have seen, in (65.1), that this implies that 7i2(Q)) = I*(G). Since C0(K) = Z[G]
Cx{K) = Z[Gf
C0(K) = Z[G]r
then the resolution becomes 0 -» Z -» Z[G] -* Z[G] r 4- Z[G]4' 4 Z[G] -* Z -> 0 from which it follows that r — g; that is, the presentation Q is balanced. Suppose, conversely, that Q = (x\,..., xg; W\,..., Wg) is a balanced presentation for G such that it^iQ) = I*(G) as modules over Z[G]. Choose an epimorphism of Z[G]-modules, a : Z[G] ->• 7T2(^). On applying the construction of Section 63 we see that XQ = M(a) = A"(2)Uo.e3 is the required Poincare 3-complex of standard form. We have proved: Theorem V: Let G be a finite group; then there exists a Poincare 3-complex of standard form XG = K(1) Ua e3 with n\ (M) = G if and only if G admits a presentation Q such that 7i2(Q) = I*(G), the dual module of the augmentation ideal. Moreover, G then necessarily has free period 4, and the presentation Q is automatically balanced. We note that the existence of a balanced presentation is not by itself enough to ensure the existence of an isomorphism ni{Q~) = I*(G) over Z[G]: the following example of Mennicke [38] H=(x,y\x9
= y3,
y~x xy = xA)
is a finite group of order 27, but is not of free period 4, nor indeed, since it has order 3 3 but is not cyclic, does it have any finite cohomological period.
66 Relationship with the D(2)-problem In this section we link the standard form problem with D(2)-problem. We say that a finite group G is Class 1, II, III according to the following: Class I: Every minimal module J e Qi(Z) is realized. Class II: fi3(Z) contains at least two isomorphically distinct minimal modules one of which is realized, the other not. Class III: No minimal module J e ^ ( Z ) is realized. It is clear that the classes are mutually disjoint. Note that Class I is nonempty; in fact we have:
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Stable Modules and the D(2)-Problem
Proposition 66.1: Class I contains all finite groups which possess the cancellation property for free modules and occur as the fundamental group of a closed 3-manifold. Proof: Let M be a closed connected 3-manifold with finite fundamental group G, and let MQ be the open manifold obtained by removing a small closed 3-disc e 3 . It follows immediately from (63.2) that I(G)* is realized as TT2(K) for any finite 2-complex K, which is a deformation retract of Mo. The hypothesis that G has the cancellation property for free modules now guarantees that I(G)* is the unique minimal representative of £23(Z). D By contrast, it is presently unknown whether either Class II or Class III is non-empty. On the basis of present knowledge, which is rather meagre, it is still reasonable to hope that both classes are empty. The question for Class III is essentially classical and is closely related to Swan's Inequality. It follows immediately from Theorem I that: Proposition 66.2: Let G be a finite group; if G is either Class II or Class III, then the D(2) property fails for G. To focus the discussion, we restrict our attention to the case where G has free period 4. If G further has the cancellation property for free modules, then £23(Z) is straight by (35.1), so that G is either Class I or Class III. When G has free period 4,1(G)* is a minimal module for &3(Z), so that any minimal module J has rk z (y) — \G\ — 1. A straightforward calculation of ranks shows that a finite group of free period 4 is of class III precisely when it has no balanced presentation. For groups of free period 4, I(G)* is a distinguished minimal module in £23(Z) so we can split Case II into two further subcases. Class II(a): I(G)* is realized, but some other minimal module is not. Class II(b): I(G)* is not realized, but some other minimal module is realized. It is evident from the above discussion that, if G is a finite group of free period 4, then G has a standard Poincare 3-form if and only if G is Class I or Class II(a); that is: Theorem VI: Let G be a finite group which admits a free resolution of period 4; if G has the free cancellation property then G satisfies the D(2)-property «=>• G admits a balanced presentation.
Poincare 3-complexes
249
If the existence of groups in Class III seems very difficult, the question of the existence of groups in Class II is slightly more clear cut, since, from the results of Swan [63] outlined at the end of Chapter 10, there are many balanced groups of free period 4, for example, the generalized quaternion groups 2(2") with n > 5, which Jo possess more than one minimal module in ^ ( Z ) . The question is then whether all of them can be realized. If at least one is not realized, then the D(2)-property fails. Interesting examples to study in this context are the groups g(8; p, q). It can be shown that, when p, q are distinct primes, Q(8; p, q) has the balanced presentation Q(8; p, q) = (x,y\
yx"y
= x",
xyx =
y2"-])
As we have already observed, Milgram [40], [41], and also Madsen and Bentzsen [4] have shown that, whilst many of the groups Q(8; p, q) have free period 4, some, at least, do not. By Swan's computational results [63], none of them has free cancellation. It is conjectured, but not yet proved, that 2(8; p, q) cannot be the fundamental group of a closed 3-manifold when both p ^ 1 and Likewise, for n > 4 the groups Q(2"; p, q) are also of interest. They have cohomological period 4, though very little appears to be known on the question of when this can be improved to free period 4. However, Charles Thomas [65] has shown that except in trivial cases, they cannot occur as the fundamental group of any closed 3-manifold. Furthermore, except in the degenerate case n = 4; p = q = 1 they do not possess the free cancellation property.
67 Terminus: the limits of this book To see how much remains to be done one only has to pose the fundamental question: Does the D(2)-property hold for all finitely presented groups? If one believes the answer to be 'Yes' (and on present evidence this still seems the more likely outcome), then, even in the 'easy' case of groups of period 4 ('easy' because one has an easily understood model for ^ ( Z ) ) , the task still seems formidable. For example, in the case of the groups of type <2(8; p, q) where one knows there is more than one minimal module in ^ ( Z ) , one is faced with the task of finding enough balanced presentations to exhaust the minimal homotopy types. This seems like a very interesting problem, though perhapsat the borders of computational intractability (c.f. [4], [40], [41], [63]).
250
Stable Modules and the D(2)-Problem
We note that, in the case of finite abelian groups, enough minimal presentations have been found to exhaust the minimal homotopy classes. This was done by Browning, generalizing Metzler [39]; see Latolais's clear account in [32]. Incidentally, it follows from this that the D(2)-property holds for finite abelian groups. For a general finite group, one lacks an understanding of ^ ( Z ) . Indeed, the only non-periodic non-abelian finite group that the author knows to possess the D(2)-property is the dihedral group of order 8, and unfortunately the proof of this seems, at present, to require cancellation properties somewhat stronger than the Swan-Jacobinski Theorem, and which do not hold in general. Secondly, the original motivation for considering the D(2)-problem for finite groups of period 4 in isolation arose from a desire to examine the structure of Poincare 3-complexes, and it may fairly be argued that we have not considered the structure of Poincare 3-complexes either generally, allowing for infinite fundamental groups, or specifically, by relating general Poincare 3-complexes to manifolds. Both questions are vexed. In the case of finite fundamental groups, it is not known which Poincare 3-complexes correspond to manifolds. The conjecture is that the linear spherical space forms (see [82]) are the only closed 3-manifolds with finite fundamental groups. At the time of writing this still seems to be unproved, although strenuous attempts have been made towards it in recent years. For the question of the general structure, the literature seems rather sparse, the most relevant references in the literature being [17], [24], [67]. In [17], Epstein gave some necessary conditions for a group to be the fundamental group of a closed 3-manifold, the most obvious restriction, which we noted in the text using a Morse Theory argument, being that the group must possess a balanced presentation. This is also observed by Turaev [67]. The detail in Turaev's paper is rather meagre, but it contains a sketch of a duality theory for Fox ideals which seems to coincide with that of Hempel [24]. Finally, it will be pointed out that we have not considered any case of the D(2)-problem with an infinite fundamental group. In general, aspects of the problem which seem clear in the case of finite groups become far more problematic in the case of infinite fundamental groups, whilst the fundamental problem of finding enough minimal presentations remains at least as intractable as before. However, in any particular case there may be compensating advantages, and to demonstrate this, and make some small amends for this neglect of infinite groups we show in an Appendix that the
Poincare 3-complexes
251
D(2)-property holds for non-abelian free groups. Comparing the solution for free groups with those we have obtained in the finite case, it should be apparent, if it were not already so, that the details of a successful solution to any D(2)problem will, at the very least, depend heavily on the module theory of the particular fundamental group under consideration.
Appendix A: The D(2)-property for free groups
We sketch the proof of the following: Theorem: The D(2)-property holds for all non-abelian free groups of finite rank. Throughout we denote by G(n) the non-abelian free group of rank n; then the following is known [2] (also [3] p. 213): Theorem A.I: (Bass-Seshadri) If P is a finitely generated projective module over Z[G(n)], then P is free. Now fix G = G(n), and consider an algebraic 2-complex over Z[G] E = (o -+ J 4 . E2 ^ E, 4 £ 0 -* Z -> 0\ that is, we are assuming that £ 0 , E\, Ei are finitely generated and stably free. Then by (A.I), £o. E{, Ei are all free. Furthermore, we have: Proposition A.2: Im(9i) is free. Proof: Consider the standard presentation Q(n) = {xi,..., x,,) for G = G(n) with n generators and no relations; the algebraic Cayley complex is the following exact sequence 0 -> Z[G]" 4- Z[G] -> Z -> 0 where, if ( e i , . . . , en) is the canonical basis for Z[G]", 5(e,) = x, — 1. Comparing this with the exact sequence 0 —• Im(3|) -*• Eo -% Z -> 0, we see from Schanuel's Lemma that Im(3|)©Z[G] = It follows that Im(3i) is stably free, and hence free, by (A.I).
•
It now follows that the exact sequence 0 -*• Im(32) ->• E\ - I Im^,) ->• 0 splits, so that Im(3i)eim(3 2 ) = £, Thus Im(92) is projective and so, again by (A.I): Proposition A.3: Im(32) is free. 252
The D(2)-property for free groups
253
Iterating the argument, the sequence 0 -> J -*• E2 -* Im(92) -> 0 splits, and so: Proposition A.4: J is free. It follows that: Proposition A.5: If K is a finite 2-complex with Ji\(K) = G(n), then n2(K) is a free module over Z[G(n)]. Now suppose that X is a cohomologically two-dimensional finite 3-complex with ji\ (X) = G(n), and consider the cellular chain complex C,(X) = (c 3 (X) 4- C2(X) 4 C,(X) 4- C0(X) -> Z -»• 0) Since dim(X) < 3 it follows that // 3 (X; Z) = Ker(33 : C3(X) ->• C2(X)), and from the hypothesis // 3 (X; Z) = 0 we see that: Proposition A.6: 93 : C3(X) -> C2(X) is injective. Thus we may form the virtual 2-complex (X) = (0 - * TT2(X) -> C 2 (X)/Im(a3 ) 4 C,(X) 4 C0(X) -+ Z ->
It is not immediately apparent that this is a genuine algebraic 2-complex. To check that it is we must show that C2(X)/Im(d3) is a free module. We do this in three stages. Put A = Z[G(n)]; first observe that: Proposition A.7: ExtA(7r2(X); B) = 0 for any A-module B. Proof: Put B2 = lm(33 : C3(X) -» C2(X)), and Z2 = Ker(32 : C2(X) -+ C,(X)). Then B2 C Z2 C C2(X). Let ; : B2 C Z 2 / : Z 2 C C2(X),
i o j : B2 C C2(X)
denote the various inclusions. Observe that, by the Hurewicz Theorem n2(X) = H2(X) = Z2/B2 For any coefficient module B, we have induced maps (/ o j)* : HomA(C2 (X), B) -> Hom A (S 2 , B);
j * : Hom A (Z 2 , B) -> Hom A (5 2 , B)
and since (i o j)* = j * o /'*, we have a filtration Im((/ o j)*) c lm(j*) C HomA(fl2. B), and a short exact sequence (*)
0 ->• Im(y*)/Im((i o j)*) -> flf/Im((i o j)*) -> B 2 e /Im0*) -> 0
where Bf = HomA(B2 , B). Now B2 is free, since 93 : C3(X) -> B2 is an isomorphism. Moreover, putting A" = X<2), we see by the Hurewicz Theorem that Z2 = JT2(K), SO that Z2 is free by (A.5). Hence applying HomA(—, B) to the exact sequence 0 ->• B2 4- Z2 ->• JT 2 (X) -> 0
we get an exact sequence in cohomology 0 -> HomA(7r2(X), B) -> Hom A(Z 2, B) 4 Hom A(B2, B) -> Ext'(^ 2 (X), B) ->• 0 which shows that HomA(B2, B)/Im(7*) = Ext'(7T2(X), B). However, 93 : C3(X) -> B2 is an isomorphism, and induces an isomorphism HomA(B2, B)/Im((i o y)*) S HomA(C3(X), B)/Im(33*) = // 3 (X, B)
254
Appendix A
Thus the exact sequence (*) becomes 0 ->• Im(./*)/Im((( o ;)*) -> H\X, B) -» Ext'(^ 2 (X), B) -» 0
(**) 3
Since // (X, £) = 0 it follows that Ext1 (7T2(X), B) = 0 as claimed.
D
Theorem A.8: TZ2(X) is a free module. Proof: As above, put A" = X(2); then we have an exact sequence (I)
0 -* C3(X) -^ 7T2(/O -^ n2(X) -> 0
Here we are identifying n2(X) with H2(X;Z), and p is the natural surjection p : n2(K) -* jr2 (/O/Im(3 3 ) = Ker(32)/Im(33) = tf2(X;Z) In the first instance, (I) gives rise to an exact sequence in cohomology 0 ^ HomA(7T2W, B)^HomA(n2(K),
B) % HomA(C 3(X), B) -* ExtA(7r2(X);B)
However, by (A.7), ExtA(;r2(X); B) = 0, and we are reduced to a short exact sequence (II)
0 -> HomA(jr2(X), B) -> HomA(^2(A-), B) % HomA(C3(X), B) -»• 0
Taking B = C3 (X), if r e HomA(7r2(/Q, C3(X)) is such that 33*(r) = Idc3, we see that r splits (I) on the left. Thus
so that JT 2 (X) is projective and hence free by (A. 1).
•
As a consequence, we obtain: Corollary A.9: C2(X)/Im(33) is a free module. Proof: We have an exact sequence 0 —> 7T2(X) -> C2(X)/Im(33) -4- Im(32) ->• 0 in which Im(32) is free, by the argument of (A.3) applied to the cellular chain complex
Ct(K). Hence C2(X)/lm(3 3) S ;r2(X) 0 Im(32) and the result follows, again by (A. 1).
D
It follows that if X is a cohomologically two-dimensional finite 3-complex with jri(X) = G(/i), the virtual 2-complex (X) is a bonafide algebraic 2-complex. Now the geometric Cayley complex K(n) of the standard presentation Q(n) is just a wedge of n-copies of the 1-sphere K(n) = S1 V • • • v S'. We put K(n, 0) = K{n) and n
K(n,m) = K(n)v S2 v • • • v S2 whenever in > 1. Let C»(n, m) denote the cellular chain complex of K(n, m) CM, m) = (0 ->• 7t2(K(n, m)) 4- Z[C(n)]'" -^> Z[C(n)]" ^ Z[G(n)] ->• Z -» 0) Clearly w2(AT(n, m)) = Z[G(n)]'". Now let E = (0 - > 7 - > E2 - > £ , ->• £ 0 ->• Z -»• 0 )
The D(2)-property for free groups
255
be an algebraic 2-complex over Z[G(n)]. By (A.5), J is free. Suppose J = Z[G(n)] m . Since G(n) has cohomological dimension 1 Ext3(Z, J) = Ext3(Z, n2(K(n, in))) = 0 so that the extensions C»(n, m) and E are necessarily congruent, and, since they are projective complexes, also homotopy equivalent; that is: Proposition A.10: Every algebraic 2-complex over Z[G(«)] is geometrically realizable. It follows that, if X is a finite cohomologically two-dimensional 3-complex with 7T](X) = G(n) then the virtual 2-complex {X) is geometrically realizable by some finite 2-complex K(n, in). Hence X is homotopy equivalent to K(n, m), as in the easy half of (59.4). We have shown: Theorem A.ll: The D(2)-property holds for G(n). Generalizations of this argument will be considered in [30].
Appendix B: The Realization Theorem
The proof of the Realization Theorem (Theorem I) given in the text relies heavily on the fact that, over a finite group G, finitely generated projectives are injective relative to the class of Z[G]-lattices. In this appendix, we give a proof of the Realization Theorem which avoids this property. Let G be a finitely presented group; by an algebraic 2-complex over G we mean an exact sequence F = where FQ, F\, FI are finitely generated stably free modules over Z[G]. We denote by Alg c the category of algebraic 2-complexes over Z[G]. In general, the module J = 7T2(F) need not be finitely generated over Z[G]. We say that a finitely presented group G is of type FL(3) when there is at least one algebraic 2-complex for which JT 2 (F) is finitely generated. We prove: Realization Theorem: Let G be a finitely presented group of type FL(3); then the D(2)-property holds for G if and only each algebraic 2-complex E 6 Alg c admits a geometric realization. Without the FL(3) hypothesis, the argument holds in one direction, namely: Weak Realization Theorem: If each algebraic 2-complex over Z[G] admits a geometric realization then the D(2)-property holds for G. The property FL(3) has a slightly stronger formulation, since it follows easily from Schanuel's Lemma that: Proposition B.I: If the finitely presented group G is of type FL(3) then 7r2(F) is finitely generated for every algebraic 2-complex F over Z[G]. For groups of type FL(3), the proof of the Eventual Stability Theorem given in the text (52.3) extends easily to give: Proposition B.2: Let G be a finitely presented group of type FL(3), and let E, E' be algebraic 2-complexes over Z[G]; then for some jLt, v > 0 there is a chain homotopy equivalence
If E' is geometrically realizable, then so is E^(E'). Hence we see:
256
The Realization Theorem
257
Proposition B.3: If G is a finitely presented group of type FL(3), then for each algebraic 2-complex E e AIgc there exists n > 0 such that S"E admits a geometric realization. The following proposition compensates for the fact that, in general, projectives are not relatively injective. Proposition B.4: Let A be a ring and let
be an exact sequence of A-modules in which Q is projective; then p* : Ext'(A7, B) J=> Ext'(M, B) is surjective for all B. Moreover, the following conditions are then equivalent: (i) S splits; (ii) p* : Ext'(N, B) 4- Ext'(M, B) is an isomorphism for all B; (iii) p* : Ext'(A', Q) -^ Ext'(M, Q) is an isomorphism. Proof: Writing AB = Hom(A, B), we have an exact sequence
0 _> NB £ MB h QB -X Ert\N, B) 4- Ext'(M, B) -> 0 the final '0' arising Ext'(Q, B) = 0 since Q is projective. In particular p* : Ext1 ( # , £ ) - » Ext'(A/,B) is surjective as claimed. We now show (i) ==> (ii) = > (iii) ^=> (i). (i) =$• (ii): Suppose that S splits. Then there exists a A-homomorphism r : M -> Q such that r oi = Id e . Hence '* or* = IdHom(e.B) B
B
In particular, /* : M -^ Q is surjective. Thus
8: QB ^
En\N,B)
is identically zero, since Ker(i5) = Imi* = \\om(Q, B). Since Ker(/;*) = Im<5 = 0, we have an injective map p* : Ext'(N, B) —> Ext1 (A/, B). Hence p * : E x t ' ( ^ , B ) 4 - Ext'(M,S) is an isomorphism, since surjectivity is already established, (ii) = > (iii) is clear. (iii) = > (i): If p* : Ext'(yV, B) 4- Ext'(M, B) is an isomorphism, then S : Hom(g, Q) —>• Ext^A7, £?) is necessarily zero, by exactness. Then the sequence 0 -> Hom(/V, Q) -5- Hom(/W, Q) 4 Hom(g,
0 ^ 0
is exact. Choosing, r 6 Hom(A/, Q) such that /*(r) = Id e , then r o i = IdG, and r is a left splitting of S. Thus (iii) =>• (i) as claimed. • Now fix a finitely presented group G. Let X be finite, cohomologically twodimensional 3-complex with Ji\(X) = G. and put K = X(2). Then we can decompose the cellular chain complex C.(X) as a pair of exact sequences 0 -> C3(X) -4 ^2(A-) -4 n-2(X) -> 0
258
Appendix B
and 0 -* n2(K) -4 C2(X) 4- C,(X) X C0(X) 4 Z ^ O where / o j = 93. We form a new exact sequence by taking the quotient by the images of C3(X) thus (X) = (O -> TT2(X) -4 C 2(X)/Im(93 ) -4- C,(X) -H> C O (X) 4 Z ^
where i, 92 are the obvious homomorphisms induced by i and 92. (X) is called the virtual 2-complex of X. We first prove: Proposition B.5: The sequence o ->• c 3 ( x ) -4 JI2(K)
-4 TT2(X) -> o
splits; in consequence 7r2(X) € ft3(Z). Proof: Since C3(X) is free, it is enough, by (B.4), to show that p* : E x t W X ) , B) 4 . E x t ' ^ C O , B) is an isomorphism for all coefficient modules B. Thus consider the exact sequence (I)
7T2(Kf 4- C 3 (X) e -U Ext'(;r2 (X), B) -^ Ext'(w2(/C), B) -> 0.
Clearly there are inclusions Im(i o ; ) • C Imy* C C 3 (X) e since (i o 7 )* = 7* o i- : C 2 (X) e -> C 3 (X)B giving an exact sequence (II)
0 -> ImiVIm(/ o ;)* -^ C 3 (X) e /Im(; o j)* -> ^ ( X f / I m ; * 1 -> 0.
However, 1 0 7 = 93 so that C 3 (X) e /Im(/ o j)* = H3(X, B). By hypothesis, H3(X, B) = 0. It follows from (II) that C 3 (X) e /Im7* = 0; that is, j * : n2{Kf ->• C 3 (X) e is surjective. Thus in the exact sequence (I) it follows that S = 0, and so
p* : ExtWX), B) 4 ExtV2(/O, B) is an isomorphism as required.
•
Let X be a finite cohomologically two-dimensional 3-complex with nx(X) S G. In addition to the exact sequence (Hi)
0 -> C3(X) -4 m(K) A n2(X) -> 0
we also have (IV)
0 -+ Tt2{K) -U C2(X) % Im(32) ->• 0
where 33 = /' o j . Together these give rise to an exact sequence (V)
0 -> TT2(X) -4 C2(X)/Im(93) % Im(92) ->• 0
The Realization Theorem
259
where i, §2 are induced by i , d2 respectively. Applying Hom(—, B) to (V) gives a long exact sequence 0 -» Im(32)B X (C 2 (X)/Im(3 3 ))e -4 n2(Xf
(VI)
X Ext'(Im(3 2), B) X • • •
% Ext"(Im(32), B) X Ext"(C2(X)/lm(93 ), B) X Ext"(n2(X), B) X • • • Proposition B.6: There is a natural surjection v : n2(K)B -» Ext'(Im(32), B) and natural isomorphisms v : Ext"(n2(K), B) -+ Ext" +'(Im(92), B) for n > 1. Proof: Consider the long exact sequence obtained by applying Hom(—, B) to (IV), (VII)
• • • 4- TZ2(K)B 4- Ext'(Im(92), B) X Ext'(C 2 (X), B) • • •
% Ext"(Im(32), B) X Exl"(C2(X), B) X Exf{n2{K), B)^- Ext"+1(Im(92), B) • • • where v denotes the natural coboundary operator. Since C2 = C2{X) is Z[G]-free, we see that Ext"(C2(X), B) = 0 for all n > 1. The result follows immediately. • Proposition B.7: Ext"(C2(X)/Im(33), B) = 0 for n > 2 and all coefficient modules B. Proof: For n > 1, the following triangle commutes by naturality
Ext"+l(Im(9 2),i3)
Ext"(n2(X), B)
Ext"(7T 2 (A:), B)
where S is the coboundary map of the long exact sequence of (V), and u is the coboundary map of the long exact sequence of (IV). In the proof of (B.4), we saw that p* : E x t ' t ^ W , B) 4- Ext'(^2 (A'), B) is an isomorphism for all B. Since v : Ext"(7r2(/O, B) -> Ext" +l(Im(32), B) is an isomorphism for n > 1 (by (B.5), we see that S : Ext"fc(X), B) -> Ext"+1(Im(32), B) is an isomorphism for n > 1. In the exact sequence (VI) 4- Ext"+1(Im(32), B) X Ext" +l (C 2 (X)/Im(33 ), B) X Ext"+](jT2(X), B) X S is an isomorphism; thus it follows that Ext" +1(C2(X)/Im(3 3), B) = Ofor/? > 1; equivalent!^ Ext"(C2(X)/Im(33), B) = 0 for n > 2.
•
260
Appendix B
Slightly more delicate to prove is: Proposition B.8: Ext'(C 2 (X)/Im(9 3 ), B) = 0 for all coefficient modules B. Proof: The following triangle also commutes by naturality. S Hom(;r2(X), B)
-
Ext'(Im(32), B)
We have seen that the exact sequence
(ill)
o -+ c3(X) -4 n2(K) 4- n2(X) -> o
splits. If r : n2(K) -> C3(X) is a splitting on the left of (III), then the mapping (p*, r*) : Hom(7r2(X), B) © Hom(C3(X), B) ->• Wom{n2(K), B) is an isomorphism. Thus 5 factorizes as follows:
Ext'(Im(32),B)
B)
vo(p*,r")
Hom(7T2(X), B) © Hom(C3, B) where c : Hom(7r2(X), B) -> Hom(7r2(X), B) © Hom(C3(X), B) is the canonical inclusion c(a) = (a, 0). However, since d2 o ( o j = 92 o 93 = 0 it is straightforward to check that v o (/?*, r*)|Hom(C3.s) = 0. Since v is surjective and (p*, r*) is an isomorphism, it follows that S = v o (p*, r*)|HOm(W2(x).e) : Hom(jr2(X), B) -+ Ext'(Im3 2, B) is surjective. Now from the following portion of (VI) 7T2(Xf A Ext1(Im(32), B) X Ext'(C 2(X)/Im(9 3 ), B) -+ Ext1(7r2(X), B) \
Ext2(Im(32), B)
since S : n2(X)B -+ Ext'(Im(92), B) is surjective, and S : E\t](n2(X), B) -+ Ext2(Im(32), B) is an isomorphism, we see that Ext'(C2 (X)/Im(3 3 ), B) = 0 as claimed.
D
The Realization Theorem
261
We can now proceed to: Proof of Weak Realization Theorem: Let G be a finitely presented group, and let X be a finite cohomologically two-dimensional 3-complex with Ji[(X) = G. We have an exact sequence (VIII)
0 -> Im(33) -»• C2(X) -+ C2(JSC)/Im(93) -* 0
Since C2(X) is free then Ext"(C2(X)), B) = 0 for all « > 2 and all coefficient modules B. By (B.7) and (B.8), we see that Ext"(C2(X)/Im(a 3), B) = 0 for all n > 1 and all coefficient modules 5. Thus the natural map C2(X) -> C2(X)/Im(33) induces isomorphisms Ext"(C2(Ar)/Im(33), B) 4- Ext"(C2(X), B) = 0 for all n > 1. Since Im(33) = C3(X) is free, hence projective, we see that (VIII) splits, by (B.4). Hence C2(Ar)/Im(93) is stably free, since Im(33) and C3(X) are both free. In particular, the virtual 2-complex {X) = (0 -> ^ ( X ) -» C2(X)/Im(33) 4- C,(X) is a bonafide algebraic 2-complex. By hypothesis, there exists a finite 2-complex K with 7t,(K) = G such that {X) ~ C»(AT). It then follows, as in the proof of (59.4), that X is homotopy equivalent to K, and so the D(2)-property holds for G. • Proof of Realization Theorem: Suppose that G is of type FL(3). By the Weak Realization Theorem, just proved, it suffices to show that, if the D(2)-property holds for G, then each algebraic 2-complex is geometrically realizable. Thus let E € AlgG E = (o -+ J -* ZfGf % Z[Gf 4 Z[G] - 4 Z - * o ) By the stable realization theorem, (B.3), if L is any finite 2-complex with JC\{L) = G for some n,m> 1. However, S™(C*(L)) = C,(L v mS2). Put A" = L v mS 2 , and let
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Index
algebra group 38 nondegenerate 26 quaternion 33 semisimple 20 algebraic 2-complex 181 Artinian condition 25 augmentation mapping 216 module 216 sequence 216 bad quaternionic factor 58 balanced presentation 234 Baer sum 89 cancellation strong 56 weak 53, 55 Cayley complex 183 cellular chain complex 173 co-augmentation 181 co-index 31 co-isomorphic 60 congruence 88, 97 contractible 174 Corepresentation Formula 78 crow's foot 121 D(2)-problem 3,224 derived category 69 functor 74 dimension shifting 113 discriminant 28 reduced 30 Eckmann-Shapiro relations 129, 131 Eichler condition 57
excess 233 extension of scalars 125 fibre product 52 FL(3)-condition 256 fork 55 full module 193 invariance 141 geometrically realizable 183 good quaternionic factor 58 group algebra 38 dihedral 47 quasi-dihedral 236 quaternion 48 representation 38 spin Euclidean 236 homotopy 109, 173 hyper-stability 5, 84 injectivity relative 69 strong relative 70 ^-invariant 135 lattice 53 Lefschetz Duality 240 lifting 99 Mennicke's group 247 Milnor's list 170 module augmentation 216 full 193 finitely semisimple 18 isotypic 18
266
Index minimal, within stability class 53 projective 21 realizable 193 semisimple 18 simple 13 stable 5 Swan 139 minimal algebraic 2-complex 213 nilpotent ideal 23 order 26 maximal 31 period cohomological 151 free 153 Poincare 3-complex 239 Poincare n-complex 3 pre-Eichler lattice 57 projective cover 72 module 21 resolution 108 n-stem 98 property S 14 pullback 87, 95 pushout 87, 96 quasi-dihedral group 236 quaternion algebra 33 factor 58 group 48 exceptional quaternion group 236 generic quaternion group 237 radical 23 realizable module 193 Realization Theorem 5, 256 reduced complex 171
Representation Formula 78 resolution 108 free 108 projective 108 restriction of scalars 125 Schanuel's Lemma 53 Schur's Lemma 13 Schlafli symbol 239 Shapiro's Lemma 129, 131 simple ring 16 Spin(3) 34 spin Euclidean groups 236 stable module 5 stability relation 5, 53 stabilization functor 196 stably free extension 123 Swan inequality 233 isomorphism criterion 144 mapping 204 module 139 tame class 69 Tate cohomology 149 Theorem Artin-Tate-Zassenhaus 163 Bass-Seshadri 252 Browning 9, 222 Maschke 40 Milnor 236 Swan-Jacobinski 57 Tietze 187 Yoneda 103 virtual 2-complex 228 Wall finiteness obstruction 178,204 weak cancellation 53, 55 weak homotopy equivalence 182 Wedderburn decomposition 20 Yoneda product 97 Zassenhaus-Suzuki classification 164
267
