LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY J.P.C.GREENLEES Abstract. The article describes the role of local homology and cohomology* * in under- standing the equivariant cohomology and homology of universal spaces. Thi* *s brings to light an interesting duality property related to the Gorenstein condition. The * *phenomena are studied and illustrated in several rather different families of examples.* * Both topology and commutative algebra benefit from the connection, and many interesting que* *stions remain open. Contents 1. Overview 2 Part I. Equivariant K-theory. 2 2. The complex representation ring * * 2 3. Free G-spaces. * * 4 4. The Atiyah-Segal completion theorem and its dual. * * 5 5. Duality for the representation ring. * * 7 6. A proof for p-groups. * * 8 7. Variations and extensions. * * 11 Part II. Ordinary cohomology and graded connected k-algebras. 13 8. Cohomology of some classes of groups. * * 13 8.1. Finite groups. * * 13 8.2. Compact Lie groups. * * 15 8.3. Arithmetic groups. * * 16 8.4. p-adic Lie groups. * * 16 9. The local cohomology theorem for group cohomology. * * 17 10. An algebraic proof of the local cohomology theorem for ordinary cohomolo* *gy. 18 11. Structural implications of the local cohomology theorem. * * 20 Part III. Examples related to bordism. 22 12. The theorem for MU-modules. * *22 13. The proof. * *23 13.1. G-spectra. * * 23 13.2. Commutative algebra with G-spectra. * * 24 13.3. The map A is an equivalence. * * 25 13.4. The map B is an equivalence. * * 25 14. Homotopically Gorenstein rings. * * 26 15. The chromatic case. * * 27 16. Connective K theory. * * 27 1 2 J.P.C.GREENLEES References * *29 1. Overview The usefulness of local cohomology in equivariant topology is not just a supe* *rficial phe- nomenon. It arises because similar structures occur in both contexts. The aim* * of these lectures is to explain one particular connection that I am especially familiar * *with, showing the common structures in the process. This connection is useful in both directi* *ons. I hope to expose some interesting algebraic structures and recommend them for further * *study, and to display some topological phenomena that may be amenable to methods of commut* *ative algebra. The main ingredients on the topological side are a group G and a G-equivarian* *t cohomol- ogy theory E*G(.). On the commutative algebra side we begin with an augmented k* *-algebra S, and the augmentation ideal J = ker(S -! k); in the topological examples S = * *E*Gis the equivariant coefficient ring and k = E* is the non-equivariant coefficient ring* *. Aside from the common structures, the reason for studying these phenomena is that the ring* *s S have an interesting duality property generalizing the notion of Gorenstein rings. The paper is organized around families of examples of rings S arising from eq* *uivariant cohomology theories. We start in Part I with a rather simple instance (K theor* *y and the complex representation ring), where quite complete results are available in ver* *y concrete terms. This should give some life to the ideas. We then summarize the other e* *xamples we want to discuss, before considering each in more detail: these subsequent ex* *amples all correspond to complete rings. First in Part II we consider the case of ordinary* * cohomology, where S is a complete graded ring over a field k. In this simpler situation it * *is possible to give general geometric results of some substance. In Part III we turn to chroma* *tic examples, where S is finite over k; the geometry is more complicated here, so we restrict* * attention to rather crude structural phenomena and to some very special cases. During the discussion, there will always be a group G in the background. Most* * results are interesting even for the group of order 2, and the reader may want to concentra* *te on this case to begin with. A few sections treat groups which are not finite, and reade* *rs will lose little by ignoring them. Part I. Equivariant K-theory. In Part I, we concentrate on a particular example which is not only very clos* *e to algebra, but can also be made completely explicit. Most of the phenomena we are concerne* *d with occur in this case in very concrete forms. 2. The complex representation ring For further details of this section see [51, 52]. Let G be a compact Lie group and X a G-space (i.e. a topological space with c* *ontinuous left G-action). A complex G-equivariant vector bundle over X is a continuously* * varying family of complex vector spaces, parametrized by X. More precisely, it is a G-m* *ap ss : -! X so that for all x 2 X, the fibre x := ss-1(X) is a complex vector space, and * *for each g 2 G, the translation g : x -! gx is linear. We also require that is locally * *trivial: for LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 3 each point x 2 X there is an open neighbourhood U over which ss is projection i* *n the sense that there is a commutative diagram ~= ss-1(U)___________//GU x Cn GGssGG wwww GG wwprojw GG## --ww U We let VectG(X) denote the set of isomorphism classes of vector bundles on X.* * Direct sum of vector spaces extends to vector bundles, making VectG(X) into a monoid. The * *equivariant K-theory of X is obtained by adjoining inverses to form an abelian group K0G(X) = AbGp(VectG(X))=([ j] = [] + [j]): Tensor product of vector spaces extends vector bundles, and this makes K0G(X) i* *nto a com- mutative ring. Note that if X is a point, a vector bundle over X is just a vector space with* * linear G action, which is simply a complex representation: K0G(pt) = R(G) where R(G) is the complex representation ring. Indeed, tensor product with a re* *presentation makes K0G(X) into an R(G)-module, so from the algebraic point of view we are di* *scussing the ring R(G) and modules over it. Example 2.1. (i) If G is cyclic of order n we may choose a faithful one dimensi* *onal rep- resentation ff and R(G) = Z[ff]=(ffn - 1). This is one dimensional, and using * *cyclotomic polynomials, we see that the irreducible components of spec(R(G)) correspond to* * the sub- groups of G. (ii) If G is dihedral of order 8, character theory shows R(D8) = Z[a; b; oe]=(a2 = b2 = 1; aoe = boe = oe; oe2 = 1 + a + b + ab* *): For an arbitary compact Lie group G, R(G) is Noetherian. Segal has shown [51]* * that its dimension is 1+rank(G) where rank(G) is the dimension of the maximal torus in G* *, and the irreducible components of spec(R(G)) correspond to the conjugacy classes of top* *ologically cyclic subgroups of G. Now in fact the functor K0Gextends to an equivariant cohomology theory K*G: G* *-spaces-! AbGp . We will not explain in detail what this means, but it is a contravarian* *t functor with good exactness properties (analogous to those of functors which take exact sequ* *ences of modules to long exact sequences of cohomology groups), and which takes sums to * *products. The extension can be given by stating that K*Gis 2-periodic in the sense that K* *nG= Kn+2G and the odd part is given by K-1G(X) = ker(K0G(S1 x X) -! K0G({1} x X)). The fa* *ct that this gives a suitably exact functor is Bott periodicity. For example K*G(pt) = R(G)[u; u-1] where u is a unit of degree 2; the value K*G(X) on a G-space X is a module over* * K*G(pt), and multiplication by u gives the periodicity. Henceforth we adopt the convenient a* *bbreviation K*G= K*G(pt) for the coefficient ring. The other consequence of having a cohomology theory is that there is an assoc* *iated ho- mology theory KG*(X). This is more complicated to define, so we will be content* * to say that 4 J.P.C.GREENLEES it is related to cohomology in such a way that a form of Poincare duality holds* * when X is a manifold. The most trivial example of this is that KiG(pt) = KG-i(pt), so th* *at, with the usual convention for relating upper and lower indexing, we have K*G= KG*= R(G)[u; u-1]: Where necessary we refer to lower indices as degrees and upper indices as codeg* *rees; for example u 2 KG2= K-2Gis of degree 2 and codegree -2. 3. Free G-spaces. For further details of this section see [52, 27]. The simplest sort of G-spaces are those with a free action (i.e. so that the* * identity element of the group is the only element fixing anything). In particular if X * *is a free G- space then equivariant vector bundles -! X over X correspond to non-equivarian* *t bundles j -! X=G over the quotient: given we take j = =G and given j we take to be the pullback of j along the quotient map X -! X=G. This passes to K theory to say t* *hat if X is free K*G(X) = K*(X=G): For finite groups there is a similar statement KG*(X) = K*(X=G) for homology. However this is less elementary, and involves transfer arguments.* * Accordingly, if G is a general compact Lie group the statement must be modified by inserting* * a suitable kind of twist. The other thing about free G-spaces is that there is a terminal free G space * *EG in the homotopy category. This means that for any free G-space X there is a G-map X : * *X -! EG, unique up to homotopy. In fact EG is characterized by two properties: it is* * free and non-equivariantly contractible. For example if G is cyclic we may view G as a s* *ubgroup of the unit complex numbers, and then EG = S(1C) is the unit sphere in the direct * *sum 1C of infinitely many copies of C. This may be more familiar to some through the * *quotient BG = EG=G, called the classifying space of G: for instance BC2 = RP 1. Note in particular that for any free G-space X the universal map X induces a * *diagram K0G(X)Ooo____K0G(EG)OOO ~=|| ~=|| | | K0(X=G) oo___K0(BG): Thus a knowledge of K0G(EG) gives canonical characteristic classes in K0G(X) = * *K0(X=G). These can be useful invariants for distinguishing different G-spaces with the s* *ame quotient X=G. For example if G = C2 we may have X1 = G x RP 3whilst X2 = S(2C) (i.e. t* *he 3-sphere with the antipodal action). In both cases the quotient is projective s* *pace RP 3, but the different characteristic classes distinguish them. A similar motivation for studying KG*(EG) can be given, but a more convincing* * one can be given because quite powerful torsion invariants of free G-manifolds belong t* *o this group [55]. LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 5 4.The Atiyah-Segal completion theorem and its dual. For further details of this section see [3, 24]. In this section we describe the calculation of K*G(EG) and the associated hom* *ology KG*(EG). We present this in a quasi-historical way, as a series of theorems st* *arting very concrete and proceeding to greater generality at the cost of some additional ma* *chinery. Remembering the periodicity of K theory, we may work with the degree 0 part o* *f the coefficient ring S = R(G), and it suffices to describe the 0th and 1st K groups* *. The ring S = R(G) is augmented over k = R(1) by the map S = R(G) -! R(1) = Z taking the virt* *ual dimension, and we need to consider the augmentation ideal J = ker(R(G) -! R(1) * *= Z) of elements of virtual dimension 0. Theorem 4.1. (Atiyah (1961) [1]) The equivariant K theory of EG is K0G(EG) = K0(BG) = R(G)^J and K1G(EG) = K1(BG) = 0: Example 4.2. (i) When G = C2, we have seen R(C2) = Z[ff]=(ff2 - 1), and since 1* * and ff are one dimensional representations, we see J = (1 - ff). Now a short calculati* *on allows us to identify the completion: (1 - ff)2 = 1 - 2ff + ff2 = 2(1 - ff), so that (1 - ff)n+1 = 2n(1 - ff); and R(C2)^J= Z Z^2: (ii) More generally, the Artin induction theorem easily shows that if G is a p-* *group R(G)^J= Z J^p: (iii)It is not hard to see that for finite groups, J-completion is injective if* * and only if G is a p-group. There is a more conceptual way of expressing the phenomena. We consider the p* *rojection map EG -! pt and the induced map K*G(X) -! K*G(EG x X) = K*(EG xG X) in K-theory. Theorem 4.3. (Atiyah-Segal (1969) [3]) For any finite G-space X the natural map K*G(X) -! K*G(EG x X) = K*(EG xG X) is completion at the augmentation ideal J of R(G). We pause for some remarks. Firstly, Theorem 4.1 is the special case X = pt, * *since we have seen that K*G(pt) = R(G)[u; u-1]. Next, we comment that the codomain K*G(E* *G x X) is a cohomology theory of X, so that for the statement to be plausible, it is n* *ecessary that K*G(X)^Jis a cohomology theory on finite complexes X. However, for each i* *, the R(G)-module KiG(X) is Noetherian provided X is a finite G-complex. This is cle* *ar for K0G(G=H) = K0H(pt) = R(H), and the general case follows by induction on the num* *ber of cells, using the exactness properties of a cohomology theory. Accordingly, the* * Artin-Rees lemma implies that J-completion is exact, and so K*G(X)^Jis a cohomology theory* * on finite 6 J.P.C.GREENLEES complexes X. Since J-completion is definitely not exact for arbitrary R(G)-mod* *ules, the statement must be modified if it is to cover arbitrary complexes X. Atiyah and * *Segal used the device of pro-groups, but for our purposes we prefer an alternative solutio* *n. Theorem 4.4. (Greenlees-May [31]) For any G-space X there is a natural short ex* *act se- quence 0 -! LJ1(K*G(X)) -! K*G(EG x X) -! LJ0(K*G(X)) -! 0; where LJiis the ith left derived functor of J-completion. The Artin-Rees lemma implies that for a Noetherian module M, we have LJ0M = M* *^Jand LJiM = 0 for i > 0, so that that if we take X to be a finite complex, we obtain* * the previous versions of the theorem. It is a general phenomenon in topology that statements in homology have bette* *r finiteness properties than in cohomology, so it is natural to seek statements about K homo* *logy. Finally, local cohomology is about to make an appearance. Theorem 4.5. (Greenlees (1993) [24]) For any finite group G and any G-complex X* *, there is a spectral sequence H*J(KG*(X)) ) KG*(EG x X) = K*(EG xG X); where J is the augmentation ideal of R(G). We outline a proof for p-groups in Section 6, and a rather different proof fo* *r all groups in Part III (Section 13). The next step is the major input from commutative algebra, and appears quite * *magical from the topological point of view. Grothendieck's vanishing theorem states tha* *t local coho- mology vanishes above the dimension; since R(G) is one dimensional, the spectra* *l sequence collapses. Corollary 4.6. (Greenlees (1993) [24]) For any finite group G and any G-complex* * X, there is a short exact sequence 0 -! H1J(KGi+1(X)) -! KGi(EG x X) -! H0J(KGi(X)) -! 0: In particular if X = pt KG0(EG) = H0J(R(G)) = Z and KG1(EG) = H1J(R(G)): Example 4.7. If G = C2 we may make the calculation completely explicit. We wor* *k in R(C2) = Z[ff]=ff2- 1, and J = (1 - ff). Since (1 - ff)(1 + ff) = 1 - ff2 = 0, i* *nverting (1 - ff) kills 1 + ff. Similarly, since (1 - ff)2 = 2(1 - ff), inverting 1 - ff inverts * *2. Thus the stable Koszul complex R(C2) -! R(C2)[1=(1 - ff)] becomes Z Z -! Z[1=2]; showing H0J(R(C2)) = Z and H1J(R(C2)) = Z=21 . The calculation of KG*(EG) is quite explicit in general. Indeed, H0J(R(G)) is* * easily cal- culated in terms of characters: J consists of representations whose characters * *vanish at the identity element e 2 G. Since characters separate conjugacy classes, H0J(R(G)) * *consists of representations whose characters vanish except at e. This consists of the integ* *er multiples of the regular representation. LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 7 It is not hard to calculate H1J(R(G)) either. Indeed, for any p-group, Artin* *'s induction theorem shows that the p-adic and J-adic topology on J coincide, and hence H1J(R(G)) = R(G)=Z Z=p1 : Remark 4.8. The theorem is also true for an arbitrary compact Lie group G. For * *this one must use the proof in Part III. The crucial ingredient is that the representing* * spectrum of K theory is a highly structured ring spectrum. Until recently this was only kno* *wn for finite groups, but M.Joachim [43] tells me that he has proved the result for arbitrary* * compact Lie groups. Since the dimension of R(G) is 1 + rank(G), the spectral sequence will not no* *rmally collapse. However it still collapses when KG*(X) has depth rank(G). Remark 4.9. An alternative way of motivating this whole connection is that by d* *efinition J acts as 0 on K*G(G) = K*(pt). By induction J acts nilpotently on K*G(X) and KG** *(X) if X is finite and free. Since EG is a direct limit of finite free complexes, we expect* * K*G(EG) to be the simplest possible "inverse limit" of J-power torsion complexes (hence it is* * calculated by the left derived functors of J-completion), and KG*(EG) to be the simplest poss* *ible "direct limit" of J-power torsion complexes (hence it is calculated by the right derive* *d functors of J-power torsion). Here `simplest possible' has a rather complicated meaning* *, requiring homotopy invariance, functoriality and exactness. 5. Duality for the representation ring. In this section we introduce one of the central properties of equivariant coh* *omology rings; in this case we are considering the ring S = R(G) augmented over k = R(1) = Z. * * For the representation ring, this property appears rather commonplace, but at least* * it is visible to inspection. The idea is to combine the local cohomology theorems with the u* *niversal coefficient theorem to obtain a duality statement. We are going to use the form appropriate to complete rings, and the completio* *n theorem in the non-equivariant form: K0(BG) = R(G)^Jand K1(BG) = 0: Similarly, we use the non-equivariant form of the local cohomology theorem; ass* *uming G is finite, this takes the form of the identifications K0(BG) = H0J(R(G)) and K1(BG) = H1J(R(G)): We have used the non-equivariant forms since there is a Universal Coefficient* * Theorem [2] for calculating K*(X) from K*(X). This is a short exact sequence 0 -! Ext1Z(Ki+1(X); Z) -! Ki(X) -! Hom Z(Ki(X); Z) -! 0: We apply this to X = BG and substitute the algebraic expressions for its homolo* *gy and cohomology to obtain a short exact sequence 0 -! Ext1Z(H1J(R(G)); Z) -! R(G)^J-! Hom Z(H0J(R(G)); Z) -! 0: This no longer mentions K theory: it is just a statement about the augmented * *Z-algebra R(G). It is reminiscent of local duality for Gorenstein rings, but of course R(* *G) is not local or even Cohen-Macaulay. It is also notable that local cohomology has separated the* * Z-torsion free H0Jfrom the torsion part H1J. The former gives the uncompleted part of R(G* *)^J(namely 8 J.P.C.GREENLEES multiples of the regular representation), and the latter gives the completed pa* *rt (namely J^(p) in the case of a p-group). 6. A proof for p-groups. In this section I will outline a proof of the local cohomology theorem for p-* *groups. The reason for restricting to this case is that on the one hand it is a convincingl* *y large family of examples, whilst on the other, it is possible to give a rather concrete vers* *ion of the proof in which the correspondence between topological and commutative algebraic struc* *tures is highlighted. In Part III we will see one method for going beyond p-groups using some quite* * sophisticated machinery. For the case of K-theory, and for finite groups, there is the more * *elementary approach used in [24]: the Burnside ring A(G) of finite G-sets is a good approx* *imation to the representation ring. Indeed the permutation representation homomorphism A(G) -!* * R(G) has the property that p ______________ J(A(G)) . R(G) = J(R(G)) so that we may replace the representation ring by the Burnside ring in the proo* *f, considering all R(G)-modules (such as R(G) itself) as modules over A(G). For topological r* *easons, manipulations with the Burnside ring are much more elementary. For infinite com* *pact Lie groups this method fails since A(G) is still 1-dimensional whilst the dimension* * of R(G) is equal to the rank of G. For the present we return to a method applying to p-gr* *oups and complex orientable theories. We introduce notation by summarizing standard constructions in the commutativ* *e algebra of a ring S. The unstable Koszul complex is the cochain complex K(x) = (S - x!* * S) concentrated in codegrees 0 and 1. This is the desuspension of the mapping cone* * of x: K(x) = -1M(x) -! S -x! S: To form the stable Koszul complex, we may now do the analogous construction for* * powers of x, assemble them into a direct system and pass to limits: K(x) = -1M(x) -! S -x! S # # # 2 K(x2) = -1M(x2) -! S -x! S # # # 3 K(x3) = -1M(x3) -! S -x! S # # # .. . . . .. .. # # # K(x1 ) = -1M(l) -! S -l! S[1=x] Now given an ideal J = (x1; x2; : :;:xr) in S we may form the stable Koszul c* *omplex K(J1 ) = K(x11) K(x12) . . .K(x1r): The notation is reasonable since the complex is independent of generators up to* * quasi- isomorphism (exercise). This allows us to define local cohomology by H*J(M) := H*(K(J1 ) M); LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 9 and its dual local homology by HJ*(M) := H*(Hom (P K(J1 ); M)); where P K(J1 ) is a complex of projectives quasi-isomorphic to K(J1 ). These d* *efinitions become interesting by Grothendieck's theorem that local cohomology calculates r* *ight derived functors of J-power torsion H*J(M) = R*J(M) where J(M) = {x 2 M | JN x = 0 forN >> 0}; and the fact [32] that local homology calculates left derived functors of compl* *etion HJ*(M) = L*J(M) where J(M) = lim M=JkM: k Note also that Grothendieck vanishing and the evident universal coefficient the* *orem E*;*2= Ext*S(H*J(S); M) ) HJ*(M) shows that the LiJ(M) = 0 for i > dim(S). This explains why only the zeroth and* * first derived functors entered into the Completion Theorem 4.4. Now in the topological context, the analogue of the stable Koszul complex is * *the universal free space EG. Let us start with the case when G is a cyclic group, G C*. We n* *ote that for each k the group G acts freely on the unit sphere S(kC) in the k-fold direct su* *mSkC. Since S(kC) is a (2k - 1)-sphere (and hence (2k - 2)-connected), the union S(1C) = * *kS(kC) is contractible. Thus EG = S(1C) in this case. Remark 6.1. For the best analogy with the above commutative algebra we need to * *work with G-spaces equipped with a G-fixed basepoint, so we need to describe the rou* *tine trans- lation between the based and unbased context. For cohomology, it is in terms of the reduced cohomology: if Y is a G-space w* *ith G-fixed basepoint y0 we have maps {y0} -! Y - ! {y0} so that if we define the reduced c* *ohomology by eK*G(Y ) = ker(K*G(Y ) -! K*G(y0)), we have K*G(Y ) = eK*G(Y ) K*G(y0). If X is an unbased G-space we form the based space X+, where the subscript + * *denotes the addition of a disjoint basepoint fixed by G. Thus eK*G(X+) = K*G(X): We need one more construction. In addition to the unit sphere S(V ) of a fini* *te dimensional inner product space V , we may form the one point compactification SV . A usefu* *l feature of the based context is that SV is the mapping cone of the map S(V )+ -! pt+ takin* *g S(V ) to the non-base point. Consequently we may also say that S(V )+ is the desuspen* *sion of the mapping cone of the inclusion S0 -! SV . 10 J.P.C.GREENLEES Now we have the ingredients to construct a diagram exactly analogous to the o* *ne for the stable Koszul complex [x] C EG(1)+ = S(C)+ -! S0 -! S # # # [x2] 2C EG(3)+ = S(2C)+ -! S0 -! S # # # [x3] 3C EG(5)+ = S(3C) -! S0 -! S # # # .. . . . .. .. # # # EG+ = S(1C)+ -! S0 -l! S1C : To complete the analogy, I should explain why it is reasonable to consider S0 a* *nd SkC as analogues of the same object S, and why we have written [xk] for the inclusion * *map. For this we may as well allow G to be arbitrary. The point is that equivariant Bott* * periodicity provides a specific isomorphism eK0G(SV ) ~=eK0G(S0) = R(G) for any complex rep* *resentation V . We then have a diagram eKG0(S0)[x(V/)]/_eKG0(SV ) = || ~=Bott|| fflffl| fflffl| R(G) K eKG0(S0) KKK | KKK |= .O(V )K%%KKfflffl| R(G) This defines the element O(V ) 2 R(G) called the Euler class of V , and by tran* *sitivity of Bott periodicity isomorphisms, O(V W ) = O(V )O(W ). Thus the map marked [xk] * *above is O(kC) = O(C)k. For an arbitrary group G and representation V , the construct* *ion of the Bott map means that O(V ) is the alternating sum of exterior powers of V . It f* *ollows that when we apply eKG*(.) to S(1V )+ -! S0 -! S1V we obtain Ke*G(S(1V )+) -! R(G) -! R(G)[1=O(V )] and it is in this sense that KeG*(S(1V )+) is the analogue of the stable Koszul* * complex K(O(V )1 ). Now, just as most ideals are not principal, so most groups are not cyclic. H* *owever, we can deal with this in the same way that we constructed a stable Koszul complex * *for an ideal with several generators. For example, if G = C1x C2x . .x.Cr is a product of cy* *clic groups, we have EG = S(1C1) x S(1C2) x . .x.S(1Cr) where Ci is the natural representation of the cyclic group Ci on C. It is not * *hard to use character theory to shown that if G is a p-group (or more generally supersolubl* *e) then it has LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 11 complex representations V1; V2; : :;:Vr for some r so that G acts freely on S(V* *1) x S(V2) x . .x.S(Vr), and hence EG = S(1V1) x S(1V2) x . .x.S(1Vr): (Equally it is not hard to show that most groups do not have such representatio* *ns; the smallest example is A4, but see [49] for much more detailed information). Suppo* *sing that we do have such a model for EG, we may immediately outline a proof of the local co* *homology theorem. Outline of proof of 4.5. The theorem was stated with an unbased space X, but the proof works with base* *d spaces, like X+. The translation is routine as described in Remark 6.1. In particular E* *G+ ^ X+ = (EG x X)+ and eKG*(X+) = KG*(X). We apply eKG*to the filtered space EG+ ^ X+. Because the filtration precisely* * modelled that of the stable Koszul complex, we immediately find a spectral sequence E*;*1= K(E1 ) R(G)KG*(X) ) KG*(EG+); where E = (O(V1); O(V2); : :;:O(Vr)) is the ideal generated by the Euler classes. Again, the correspondence between * *topology and commutative algebra makes clear that d1 is the Koszul differential, so that E*;*2= H*E(KG*(X)): Finally we need to explain why E may be replaced by J. First, the inclusion S0 * *-! SV is obviously null-homotopic non-equivariantly if V 6= 0, so E J. In the cyclic ca* *se we had equality, and for a general p-group we have p__ E = J: This can be observed explicitly for the representation ring, since the primes a* *re known by [51], but in fact it follows from some finiteness assumptions [28]. Since local cohomology only depends on the radical it proves H*J(M) = H*E(M) * *as re- quired. As far as any applications we have made are concerned, there is no particular* * advantage to using the augmentation ideal J rather than the Euler class ideal E. It is wh* *en comparing different groups that it comes into its own. 7. Variations and extensions. There are two directions to develop the above ideas. The first replaces free * *G-spaces (i.e. spaces with all isotropy groups trivial) by spaces with isotropy in some other * *family F of subgroups. On the ring theoretic side, the ideal J is replaced by the ideal J(F* *) of elements restricting to zero in R(H) for all subgroups H in F. For equivariant K theory* * this goes smoothly. However there is no comparable expression in terms of the non-equivar* *iant case, and hence no duality statement. We will not pursue this variation any further h* *ere. 12 J.P.C.GREENLEES The second variation is to replace K theory by another cohomology theory; on * *the ring theoretic side this replaces the augmented ring R(G) by another. If K theory i* *s replaced by E-theory then R(G) is replaced by the coefficient ring S = E*G(or its degree* * zero part E0G), R(1) is replaced by k = E* (or its degree zero part E0) and J is replaced* * by the kernel J = ker(E*G-! E*) of the map forgetting equivariance. Context 7.1. The correspondence between commutative algebra and topology is via o S = E*G o k = E* o J = ker(E*G-! E*). The remainder of this article is structured round a number of these. In this* * section we give an overview. In the following table the first row summarizes the properties of K theory di* *scussed above, and each subsequent row is an analogous example. To avoid discussion of special* * cases, the information refers only to finite groups G. The final two columns refer to the* * question of whether the homology and cohomology of EG can be calculated using local cohomol* *ogy and completions as was the case for K theory. ______________________________________________________________________________* *__|||| |_Cohomology_theory__|CoefficientsDimension_Augmented_over__|Homology___Cohom.* *__||||| ||K-theory || R(G)0 1 ^ R(1) = Z | [24]| [3] * * || ||En-theory ||En(BG) n Zp[[u1; : :;:un-1]][|24,|20, 22]T * * || ||Stable cohomotopy |A(G)|+*nilp 1 A(1)*= Z ||FALSE! [15]* * || ||Ordinary ||H*(BG) rankp(G) H* = Fp || [26] T * * || |_Connective_K-theory|ku_(BG)________2_________ku__=_Z[u]____|[24,_20,_22]T___* *__| Remark 7.2. (i) There are various ways of grouping these examples. First, K-the* *ory and En-theory are periodic, so we may work with ungraded rings. In the cohomotopy e* *xample the Burnside ring A(G) is in degree 0, and all other elements are nilpotent (Nishid* *a's nilpotence theorem); accordingly we may work over the degree zero part A(G) in this case t* *oo. The other examples are graded. (ii) More significant is the fact that apart from K-theory and cohomotopy, all * *the examples are complete for the augmentation ideal. This means that the completion theore* *m is a tautology for finite complexes (hence the entries T for `true' and `tautology')* * and concentrates attention on local cohomology and the duality statement. This is no disadvantage for present purposes, but in topology it is a major p* *roblem to identify natural rings of which these are completions. To see why it is valuab* *le, one may imagine we only knew the completion of R(G). We may then seek to refine it to R* *(G) itself, hoping thereby to invent representation theory. (iii) In the first three examples, the ring is essentially finite over its none* *quivariant counter- part: in the other two we have finitely generated algebras over the non-equivar* *iant coeffi- cients. (iv) All the examples for which the local cohomology theorem is known to be tru* *e can be proved for p-groups using the method of Part I. They can be proved for finite g* *roups by the method described in [24, Appendix], using [28] to establish finiteness properti* *es. However, LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 13 the technology necessary for implementing the strategy was not introduced until* * [20]; it was shown to apply to these cases in [22]. (v) Stable cohomotopy is quite exceptional. The fact that local cohomology of s* *s*Gdoes not calculate ss*(BG) means that the type of proof for the completion theorem we wi* *ll give in the complex orientable case (by formally deducing it from the local cohomology * *theorem) cannot apply. Carlsson's proof of the completion theorem ("the Segal conjecture* *") is not a simple formality like the proofs we give here, and it seems that the substantia* *l calculational input is necessary. Other examples of this type occur in algebraic K theory, bu* *t we shall not be concerned with them further. In Part II (Sections 8 to 11) we discuss ordinary cohomology: here the topolo* *gical technol- ogy is at a minimum, but the commutative algebra is correspondingly increased. * *In Section 12 we discuss a family of examples arising from the topological theory of manif* *olds: this has the advantage of being a rather extensive family from the ring-theoretic po* *int of view. Finally, we look at two special cases of this. In Section 15 we discuss the En-* *theory example because an interesting interaction of Cousin filtrations occurs. In Section 16* * we turn to connective K theory; the interest here is that although it is very close to exa* *mples where rather complete results are available, some rather intricate commutative algebr* *a appears. Part II. Ordinary cohomology and graded connected k-algebras. We spend Sections 8 to 11 considering ordinary cohomology. This has several * *special attractions, especially to algebraists. First, the cohomology of a point can be* * described in purely algebraic terms as an Ext algebra relevant to representation theory. Sec* *ond, the ring S is a connected commutative graded algebra over a field k, so it is especially* * easy to apply the methods of commutative algebra. 8. Cohomology of some classes of groups. For the duration of Part II, S is an N-cograded algebra over a field k, and S* *0 = k. Properly speaking, S is graded commutative in the sense that ab = (-1)deg(a)deg* *(b)ba, and all the results we need can be proved in this context. Those not wishing to wo* *rry about graded commutativity can restrict attention to the case that char(k) = 2. For m* *ost of the discussion it suffices to replace S by its even-graded commutative subring. In this section the augmentation ideal J = ker(S -! k) is the unique maximal * *ideal of the graded local ring S. To emphasize this we write m for it. 8.1. Finite groups. The first class of examples we discuss are the rings S = H*(G; k) := Ext*kG(k; k) for a finite group G, where kG denotes the group ring. General references for t* *his subsection are [5, 11]. One of the principal reasons for studying this is representation * *theory. For example the rate of growth of a minimal resolution of k is exactly captured by * *the Hilbert series X h(G; k) := dimk(Hi(G; k))ti; i0 for Noetherian rings this power series is easily seen to be the expansion of a * *rational function about t = 1. The cohomology ring H*(G; k) is known explicitly for a large numbe* *r of groups, and we tabulate some examples to bring the discussion down to earth. All the ex* *amples are 14 J.P.C.GREENLEES 2-groups, and k is a field of characteristic 2. We use the convention that a s* *ubscript on a generator indicates cohomological degree. The Cohen-Macaulay defect for a lo* *cal ring S is defined by CM-defect(S) = dim(S) - depth(S). The groups are elementary ab* *elian, quaternion of order 8, dihedral of order 8, semidihedral of order 16, a certain* * group of order 32, and the extra-special 2-groups. _______________________________________________________________________________* *_________||*| |_Group_G__|____________H_(G;_k)_____________dim__depth__defect_______Hilbert_s* *eries____||r|r| || (C2) || k[x(1)1; x(2)1;2. .;.x(r)1]23r3 r 0 1=(12- t* *) |2 | || Q8 ||k[x1; y1; z4]=(x + xy + y ; x ; y1) 1 0 (1 + t + t )=(1* * -2t)(1 + t|) | || D8 || k[x1; y1; u2]=(xy) 2 2 0 1=(1 -2t* *) 2 | | || SD16 || See below 2 1 1 1=(1 - t) (* *1 + t ) | | || 7a21+n|| 3 1 2 * * || |___2______|k[x(1)1;_x(2)1;_:_:;:x(n)1]=I__k[i2n-r]rr______0___________________* *_________| Remark 8.1. (i) There is a copious supply of further examples on J.F.Carlson's * *webpages [14] (the cohomology of all but 5 of the 267 groups of order 64 are there, toge* *ther with all 2-groups of smaller order). (ii) Quillen [46] has shown that dim (H*(G; k)) = rankp(G); where the p-rank rankp(G) of G is the rank of the largest elementary abelian p-* *subroup (Cp)r in G. (iii) For p-groups of small order, the Cohen-Macaulay defect tends to be quite * *small. Indeed, Duflot [16] has shown that depth(H*(G; k)) rankp(Z(G)) where char(k) = p, and that in any case the depth is at least 1 [17]. (iv) The cohomology ring of the semidihedral group is k[x1; y1; z3; t4]=(xy + y2; y3; yz; x3z + x3y + y2t + z2): The cohomology ring of 7a2 has 8 generators and 18 relations, but the relevant * *information is well summarized in [6]. (v) The case of the extra special groups of order 21+2m is included because Qui* *llen's calcu- lation [47] is so elegant (see also [5, 5.5] for a brief account). The ideal I * *is generated by a regular sequence. To be more precise, consider an extension 1 -! C2 -! G -! E - ! 1 where E is elementary abelian of rank n. If C2 is the commutator subgroup and the centre o* *f G then G is called extraspecial, n = 2m is even and there are only two isomorphism types* * of extraspecial groups of this order, but the cohomology calculation applies to any extension o* *f the given form. As usual, r is the dimension of the cohomology ring, or equivalently the * *rank of the largest elementary abelian subgroup. To describe the ideal, we use the fact that any such extension is classified * *by the map q : E -! C2 given by squaring the preimage in G of an element of E. This q is a* * quadratic form with associated bilineary form b. Now if e1; : :;:en is a basis of E as an* * F2-vector space and v = ieixithen I is generated by the sequence q(v); q(v; F (v)); : :;:q(v; F n-r(v)); LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 15 where F is the Frobenius map, and the sequence turns out to be regular. The cod* *egrees are 2; 4; 8; : :;:2n-r+1, so it is easy to write down the Hilbert series. 8.2. Compact Lie groups. If G is a compact Lie group the ring S = H*(BG; k) is the cohomology ring of the classifying space of G that we met in Section 3. * *One reason for studying this is exactly parallel to the motivation we gave for K theory: i* *t tells us about characteristic classes. One often thinks of compact Lie groups such as the orth* *ogonal and unitary groups, but of course finite groups are perfectly good examples, and us* *ing the bar construction for BG one sees that it can be constructed with cells correspondin* *g to copies of kG in the bar resolution: thus H*(BG; k) = Ext*kG(k; k), and our notation is* * consistent. We may give a good supply of examples here too. _______________________________________________________________________________* *____||*| |_Group_G__|_______H_(BG;_k)__________dim__depth__defect________Hilbert_series_* *____||r|2r| || (U(1)) |k[x(1)2;|x(2)2; . .;.x(r)2]r r 0 1=(1 - t4) * * || || SU(2) || k[x4] 1 1 0 21=(1 - t4) * * |2n | || U(n) || k[c1; c2; : :;:cn] n n 0 1=(1 - t4)(1 - t8) . .(* *.1 - t4n) || || O(2n) || k[p1; p2; : :;:pn] n n 0 1=(1 - t )(1 - t ) . .(* *.1 -|t ) | |__Spin(n)_k|[w2;_w3:_:;:wn]=I__k[i2n-r]r____r______0__________________________* *____| Remark 8.2. (i) By force of precedent, the Chern class ci is of cohomological d* *egree 2i and the Pontrjagin class pi is of cohomological degree 4i. Otherwise we have re* *tained the convention that subscripts refer to codegrees. (ii) The field k can be of any characteristic except that char(k) 6= 2 for O(2n* *) and char(k) = 2 for Spin(n). (iii) The case O(2n) is included because its behaviour is slightly more complic* *ated than its polynomial cohomology ring suggests (see Example 9.5(iii)). (iv) The cohomology of Spin(n) is also calculated in [47], and was Quillen's mo* *tivation for considering the extraspecial 2-groups. The ideal I is generated by the regular * *sequence n-r 2n-r-1 2 1 w2; Sq1w2; : :;:Sq2 Sq . .S.q Sq w2; where Sqi is the ith Steenrod square. Quillen deduces this calculation by compa* *rison with the calculation for the extraspecial subgroup of Spin(n). One other feature will be important. The group G acts on itself smoothly by c* *onjugation. Since it preserves the identity element e 2 G, a group element gives a self-map* * of the tangent space TeG, and we obtain the adjoint representation ad : G -! GL(TeG); and hence the k-orientation representation sign * G -ad!GL(TeG) det-!R* -! {+1; -1} -! k : If the image is trivial we say that ad is orientable over k. Evidently if char(k) = 2, the adjoint representation is orientable for every * *group G. By continuity, each connected component maps to the same point, so that if G is co* *nnected, or if it has an odd number of components then ad is orientable over any field. * *However if 16 J.P.C.GREENLEES char(k) 6= 2 then ad(O(2n)) is not orientable over k (this is easy to check for* * n = 1, and follows in general). 8.3. Arithmetic groups. For an arbitrary discrete group G we may consider the c* *ohomol- ogy ring S = H*(G; k) = Ext*kG(k; k): This can be arbitrarily unpleasant unless we place a restriction on G. We are i* *nterested in arithmetic groups, such as SLn(Z) and On(Z). However the appropriate level of g* *enerality is a little wider: virtual duality groups. A general reference for this subsect* *ion is [11]. Before reaching virtual duality groups we should discuss duality groups. The * *reader may like to construct an analogy with Gorenstein local rings. A Gorenstein ring can* * be character- ized as one having finite injective dimension over itself, but perhaps the reas* *on Gorenstein rings are so important is their duality. Similarly, if we restrict attention to* * groups G with the finiteness condition that k admits a resolution by finitely generated projectiv* *e kG modules, we may characterize duality groups as torsion free groups G so that Hi(G; kG) := ExtikG(k; kG) = 0 unlessi = n for some n. The number n is called the dimension of G and the module I = Hn(G; kG) is called the dualizing module. Such groups then have a duality isomorphism Hi(G; M) ~=Hn-i(G; M I): We say that G is a Dn-group; if I is one dimensional over k we say G is a Poinc* *are duality group (P Dn-group), and if in addition G acts trivially on I we say it is orien* *table. A group G is said to be a virtual duality group if it has a subgroup G0of fin* *ite index which is a duality group. The virtual dimension of G is the dimension of G0(and is w* *ell defined) and the dualizing module for G is I = Hn(G0; kG0) ~=Hn(G; kG) where the isomorphism is Shapiro's lemma and this shows I is a G-module. Example 8.3. (i) Borel and Serre [13] show that any torsion free arithmetic gro* *up is a duality group, and a general arithmetic group is a virtual duality group. (ii) Fundamental groups of a knot complement are D3-groups. (iii) Mapping class groups are virtual duality groups [41, 40]. (iv) Automorphism groups of free groups are virtual duality groups [10]. Thus the class of virtual duality groups is very extensive. However there ar* *e very few non-trivial examples where the cohomology ring is known explicitly. 8.4. p-adic Lie groups. It is possible to transpose the entire previous section* * into the category of profinite groups. If G is a profinite group we may consider module* *s for its (discrete) finite quotients. However it is important to be able to discuss inv* *erse limits of these (compact G-modules) and direct limits (discrete G-modules). It is possibl* *e to put both classes of modules into a reasonably well behaved category [54], and make most * *of the usual constructions of homological algebra in this category. In particular the comple* *te group ring k[[G]] behaves like a free module and the cohomology ring may be defined by S := H*(G; k) = Ext*k[[G]](k; k): LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 17 Just as in the discrete case we may define duality groups and virtual duality* * groups. Virtual duality groups have a virtual dimension and a (compact) dualizing modul* *e I = Hn(G; k[[G]]). Example 8.4. Lazard shows that any p-adic Lie group is a profinite virtual dual* *ity group [44]. As in the discrete case, the interesting examples that come up are automo* *rphism groups of suitable structures. Familiar examples are the groups SLn(Z^p), and another * *class of exam- ples of considerable interest to topologists are the Morava stabilizer groups (* *automorphism groups of certain formal groups). For all these more general classes of groups, the ring S is known to share wi* *th the case of finite groups the properties that it is a finitely generated k-algebra and o* *f dimension rankp(G) [46]. 9. The local cohomology theorem for group cohomology. We return to the local cohomology theorem, treated in Part I for the represen* *tation ring. Now we consider what it says for ordinary cohomology, which is to say for the a* *ugmented rings k-algebras S discussed in Section 8. This case is much more interesting b* *ecause S can be of higher dimension, and exhibits a wide variety of behaviours as the group * *varies. We are writing m for the augmentation idea: it is the maximal ideal of positi* *ve codegree elements. We write DM = Hom S(M; I(k)) = Hom k(M; k) for the Matlis dual, where I(k) is the injective envelope of k. Theorem 9.1. [26] If S = H*(G; k) for a finite group G there is a spectral sequ* *ence E*;*2= H*m(S) ) DS; with differentials dr : Es;tr-! Es+r;t-r+1r. This is a spectral sequence of S-* *modules in the sense that dr : Es;*r-! Es+r;*ris a map of S-modules for all r and s. We will outline the proof of the result in Section 10 below, but first we dis* *cuss some special cases. Example 9.2. (i) If G is an elementary abelian 2-group of rank r and k is of ch* *aracteristic 2 then S = k[x1; : :;:xr]. It is easy to calculate the local cohomology from th* *e stable Koszul complex, and we deduce H*m(S) = Hrm(S) = -rDS: More generally, whenever S is Cohen-Macaulay of dimension r, we find H*m(S) = Hrm(S) and hence the spectral sequence collapses to give Hrm(S) = -rDS; and hence S is Gorenstein. This recovers a theorem of Benson and Carlson [7]. (ii) The simplest non Cohen-Macaulay example is the semidihedral group of order* * 16, of dimension 2 and depth 1. The spectral sequence again collapses since the E2-ter* *m is con- centrated in two adjacent columns. (iii) The simplest example with Cohen-Macaulay defect 2 is the group 7a2. The c* *ohomology is calculated by Rusin [50], and the relevant features are highlighted by Benso* *n and Carlson 18 J.P.C.GREENLEES [7]. One may show that the spectral sequence does not collapse, and one may ide* *ntify the differential. Turning next to compact Lie groups, recall the conjugation of G induces the a* *djoint representation G -! k*, and let k(ad) denote k with this action. Theorem 9.3. [9] If S = H*(BG; k) for a compact Lie group G of dimension d, the* *n there is a spectral sequence H*m(S) ) -dH*(BG; k(ad)); of S-modules where k(ad) is the coefficient system in which each element g of t* *he group acts on k by +1 or -1 according to whether conjugation by g preserves or reverses th* *e orientation in a neighbourhood of the identity. Remark 9.4. When G is orientable the spectral sequence is just like the finite * *case with a shift H*m(S) ) -dH*(BG; k); Example 9.5. (i) All the finite group examples discussed before are also exampl* *es here. (ii) If G is the circle group (of dimension d = 1) then S = k[x2] and if G = SU* *(2) (of dimension 3) then S = k[x4]. Both groups are connected, so that k(ad) = k. Thes* *e examples show how the dimension shift -d comes into play. (iii) We have already commented that the adjoint representation of O(2n) is non* *-trivial if char(k) 6= 2. This accounts for the fact that the local cohomology of H*(BO* *(2n)) = k[p1; : :;:pn] is a copy of its dual shifted by -2n(n + 1), whereas the dimensi* *on shift for an orientable group of the same dimension (namely n(2n - 1)) and rank (namely n) w* *ould be -n - n(2n - 1). The difference of 2n is explained by the non-trivial adjoint ac* *tion. Finally, we turn to virtual duality groups. Theorem 9.6. [8] If S = H*(BG; k) for a (discrete or profinite) virtual duality* * group G of dimension n there is a spectral sequence H*m(S) ) nH*(G; I); of S-modules where I is the dualizing module. Remark 9.7. When G is an orientable Poincare duality group the spectral sequenc* *e is just like the finite case with a shift H*(S) ) nH*(BG; k); We will discuss the implications of these theorems for the structure of the r* *ing S in the Section 11, and in Section 10 we sketch a proof. 10. An algebraic proof of the local cohomology theorem for ordinary cohomology. In this section we outline the proof of the local cohomology theorem in the c* *ase of finite groups. This is the proof given in [26]; it is possible to give topological pro* *ofs as in Part I or Part III and another proof is given in [19]. In fact we will prove a more general result for cohomology of a kG-module M. * *This states that there is a spectral sequence H*m(H*(G; M)) ) H*(G; M): LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 19 The case M = k recovers the theorem. Similarly there are versions of the result* *s for other classes of groups with modules of coefficients. The simplest case is when G has periodic cohomology. This is known to be a ve* *ry restrictive assumption, but once this case is clear we should be able to describe the proof* * in general. The assumption means that there is a codegree n and an element x 2 S of codegre* *e n so that multiplication by x gives an isomorphism Sm ~=Sm+n whenever m 0. Thus S k[x].* * Now we may view x as an extension class x 2 ExtnkG(k; k) and represent it by an exa* *ct sequence ___________________________________ 0 -! k -! ||Cn-1_-!_Cn-2_-!_._.-.!_C1_-!_C0__||-! k -! 0: If we begin a minimal resolution the C0; C1; : :;:Cn-2 are projective by constr* *uction, and Cn-1 is projective since Hn(G) ~=H0(G) = k. We thus consider the complex ___________________________________ C(x) = ||Cn-1_-!_Cn-2_-!_._.-.!_C1_-!_C0__|| and form a projective resolution P of k by concatenating copies of C(x), as we * *may do because Hn-1(C(x)) = k = H0(C(x)). We thus let h ___________________________________ P = : :-:! ||Cn-1_-!_Cn-2_-!_._.-.!_C1_-!_C0__||-! ___________________________________ ___________________________________i | Cn-1 -! Cn-2 -! . .-.! C1 -! C0 |-! | Cn-1 -! Cn-2 -! . .-.! C1 -! C0 |: |__________________________________| |__________________________________| Now we may write the stable Koszul complex S -! lim 1=xkS = S[1=x]: ! k It may help to visualize the term 1=xkS as a copy of S, displayed vertically, s* *o that the bottom is in degree -kn, and the maps in the direct system just include each co* *lumn in the next. We model this at the level of resolutions, at least after reversing arrow* *s, to obtain P = L0 - L1 = lim L[-kn; 1) = lim -knP: k k Again, one may view L[-kn; 1) as the chain complex L, displayed vertically, wit* *h the bottom is in degree -kn, and the maps in the inverse system just project each c* *olumn onto the previous one. Thus Lo = (L0 - L1) is a double complex, and the proof proce* *eds by considering the double complex X = Hom kG(Lo; M) There are two spectral sequences for calculating the cohomology of the double c* *omplex. The one in which we take Koszul cohomology first collapses to show H*(X) = H*(H*Koszul(Hom kG(Lo; M))) = H*(Hom kG(HKoszul*(Lo); M)) since HKoszul*(Lo) = H1(Lo) = P !; where P != L1=P is the part of L1 in negative codegrees. Now Hom k(P !; M) = P k M so that H*(X) = H*(G; M): Taking the spectral sequence arising from the other filtration we obtain H*Koszul(H*(Hom kG(Lo; M))) ) H*(X) = H*(G; M): 20 J.P.C.GREENLEES This is the required spectral sequence. Indeed, by definition we have H*(Hom kG* *(L0; M)) = H*(G; M), and for L1 we calculate H*(Hom kG(L1; M)) = H*(Hom kG(lim L[-kn; 1); M)) n = lim H*(Hom kG(L[-kn; 1); M)) ! n = lim 1=xnH*(G; M)) ! n = H*(G; M))[1=x]; where the second equality used the fact that the limit is achieved in each degr* *ee. Thus we obtain the stable Koszul complex, (H*(Hom kG(Lo; M))) = (H*(G; M) -! H*(G; M)[1=x]): The cohomology of this complex is the local cohomology of H*(G; M). When G does not have periodic cohomology, Noether normalization shows that S * *is finite over a polynomial subring k[x1; : :;:xr]. We replace the single complex C(x) by* * C = C(x1) . . .C(xr), which we can view as an r-dimensional box. This time the top module* *s in each complex C(xi) are not projective, but by the theory of support varieties [5, Ch* *apter 5], because S is finite over the polynomial subring, their tensor product is projec* *tive. Now the proof is directly analogous. We mimic the construction of the multigraded Koszu* *l complex by stacking boxes. For example with r = 2 we have (L0 - L1 - L2) = (P ([0; 1)x[0; 1)) - P ((-1; 1)x[0; 1))P ([0; 1)x(-1; 1)) - P ((-1; 1)x(-1; 1* *))); where P ([0; 1) x [0; 1)), for instance, is the result of stacking boxes in the* * first quadrant. 11.Structural implications of the local cohomology theorem. This section summarizes the contents of [30], outlining the implications for * *a finitely generated k-algebra S of the existence of a spectral sequence H*m(S) ) aDS: Duality is exact in the context so it is equivalent to say there is a spectral * *sequence DH*m(S) ) -aS: We say that S has a local cohomology theorem with shift a, or that it is an LCT* * aring. First we repeat the immediate observation that if S is Cohen-Macaulay of dime* *nsion r then the spectral sequence collapses to give an isomorphism DHrm(S) = r-aS; so that S is Gorenstein. Next, consider the case that S has Cohen-Macaulay defect 1 (in the sense that* * its depth is one less than its dimension). We write = DHrm(S) for the canonical module,* * and = DHr-1m(S) for the subcanonical module. Thus the spectral sequence collapses* * to the short exact sequence 0 -! a-r+1 -! S -! a-r -! 0: Proposition 11.1. If S is an LCT aring with Cohen-Macaulay defect 1 then o has depth r and Hrm() = r-aD, and LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 21 o has depth r - 1 and Hr-1m() = r-1-aD. It is natural to say that a ring with Cohen-Macaulay defect 1 is almost Cohen* *-Macaulay and to say that it is almost Gorenstein if it satisfies the conclusion of the p* *roposition. Proof: Note first that the depth statements are equivalent to the vanishing of * *local coho- mology up to the top degree. To give the idea of proof, we will just explain th* *e vanishing of lower local cohomology groups. The isomorphisms for the top local cohomology* * groups follow by extending the analysis one more step. Grothendieck's spectral sequence Ep;q2= Hpm(DHr-qm(S)) ) DS (which applies to an arbitrary commutative ring) gives Him() = Hi+2m() for i r* * - 3. Applying the local cohomology to the short exact sequence from the local cohomo* *logy the- orem gives Hi-1m() ~=Him() for i r - 2. Combining these gives the statements a* *bout depth. Remark 11.2. It is worth recording some consequences of these results for an ar* *bitrary LCT aring. First, we recall that Grothendieck defined the dual localization fun* *ctor L"M = D(D(M)"), which is useful for Artinian modules like local cohomology modules. I* *t is exact, and an easy consequence of local duality is the fact that if " is a prime with * *S=" of dimension d then L"Him(M) = Hi-dm(M"). Accordingly, we may apply L" to see that if S is a* *n LCT a ring then S" is an LCT a-dring. Hence we conclude that for any minimal prime ",* * the ring S" is Gorenstein (one says S is Gorenstein in codimension 0), and that if " is * *of height 1, the ring S" is almost Gorenstein (one says S is almost Gorenstein in codimensio* *n 1). On a more concrete level, if S is a finitely generated k-algebra in codegrees* * 0, we may obtain conditions on its Hilbert series [S](t) = idim k(Si)ti. Corollary 11.3. (i) (Stanley [53]) If S is a Cohen-Macaulay LCT aring then [S](1=t) = t-a(-t)r[S](t): (ii) If S is an almost Cohen-Macaulay LCT aring then [S](1=t) - t-a(-t)r[S](t) = (-1)r-1(1 + t)[](t) and [](1=t) = ta(-t)1-r[](t): Remark 11.4. In [30] the functional equation for is misstated: two signs are n* *egated, one in the statement and one in the proof. Proof : To give the idea, we present the proof of Part (i) in a way that sugges* *ts that of Part (ii), referring the reader to [30] for further details. We first observe t* *hat the result is elementary for a polynomial ring. Now by Noether normalization we may find a po* *lynomial subring "S S over which S is a finitely generated module. We may work entirely* * with S"-modules in the rest of the proof. By the Auslander-Buchsbaum formula, since * *S is Cohen- Macaulay, S = F0 k "Sas "S-modules, where F0 is a finite dimensional graded vec* *tor space. Thus for Hilbert series, [S] = [S"][F0]. Now calculate [S](1=t) = [F0_][S"](1=t) = (-1)r[F0_][DHrm(S")] = (-1)r[DHrm(S)] = (-1)rt* *r-a[S]; 22 J.P.C.GREENLEES where the last equality is the local cohomology theorem. The proof in the almos* *t Cohen- Macaulay case uses exactly the same ingredients, now working with a short exact* * sequence 0 -! F1 k "S-! F0 k "S-! S -! 0. Example 11.5. We consider the semidihedral group SD16. It is of dimension 2 an* *d has Hilbert series f(t) = 1=(1 - t)2(1 + t2). We calculate t4 t2 t2 f(1=t) - (-t)2f(t) = ______________- ______________= -(1 + t)_____________: (1 - t)2(1 + t2)(1 - t)2(1 + t2) (1 - t)(1 + t2) It is then easy to check that with ffi(t) = t2=(1 - t)(1 + t2) we have ffi(1=t) = (-t)-1ffi(t) as required. It is interesting to note that f(t) also satisfies the single functional equa* *tion f(1=t) = t2(-t)2f(t) just as if it was the the Hilbert series of a Cohen-Macaulay LCT ring with shif* *t a = -2. Part III.Examples related to bordism. There is a rather general class of cohomology theories for which one can give* * a uniform treatment very close to ideas from commutative algebra. Curiously, these examp* *les arise from the study of a certain class of manifolds, but these geometric antecedents* * are not relevant here. Even in topology, Quillen's theorem [48] that there is an intim* *ate relation between bordism and the algebraic theory of formal groups means that much discu* *ssion of bordism is conducted in purely algebraic terms. 12. The theorem for MU-modules. There is an equivariant version of bordism constructed from certain manifolds* * with group action. The cohomology theory MU*G(.) is called equivariant (homotopical) compl* *ex cobor- dism. Its value MU*G(X) on a G-space X is a module over the coefficient ring S * *= MU*G. Theorem 12.1. (Greenlees-May [37]) If M*G(.) is module valued over MU*Gthen for* * any finite group there are spectral sequences H*J(MG*(X)) ) MG*(EG x X) = M*(EG xG X) and HJ*(M*G(X)) ) M*G(EG x X) = M*(EG xG X) where J = ker(MU*G-! MU*) is the augmentation ideal. Remark 12.2. (i) In fact MU*Gis not Noetherian, and J is not known to be of fin* *ite arithmetic rank. Accordingly there is work involved in showing the initial ter* *ms of the spectral sequences make sense: one shows that H*J0(.) is independent of J0 for * *all sufficiently large finitely generated ideals J0 J. (ii) We shall only be applying this in the case that M*Gis a Noetherian ring in* * its own right, and one may replace J with JM = ker(M*G- ! M*). We view this as an example in* * the form of Context 7.1 by replacing S with SM = M*G, and k with kM = M*. LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 23 (iii) The sense in which M*G(.) is required to be module valued will be explain* *ed in Section 13. Example 12.3. The easiest class of examples to describe are those which are J-c* *omplete in a suitable sense. Fortunately these are the ones we intend to discuss later. For this class we first define the non-equivariant theory. Indeed the coeffi* *cient ring of complex cobordism is a polynomial ring MU* = Z[x1; x2; x3; : :]: on infinitely many variables, where xihas degree 2i. For any flat module M* ove* *r MU* the definition M*(X) = MU*(X) MU* M* gives an exact functor of X, and is therefore a homology theory, and hence repr* *esented by a spectrum M. For topological reasons the flatness condition can be weakened to s* *omething much more easily verified (Landweber exact functor theorem). There are also man* *y module valued theories for which M* is not flat, but other means must be used to const* *ruct them. From any such non-equivariant theory one may form the complete theory M*G(X) = M*(EG xG X): One family of well known examples are the complete Johnson-Wilson theories M * *= En with E*n= Z^p[[u1; : :;:un-1]][u; u-1]; where the ui are of degree 0 and u is of degree 2. This may be constructed usi* *ng the Landweber exact functor theorem. One may show that for any finite group G, E*n(* *BG) is finite over E*n. Another example has ku* = Z[x1], and here ku*(BG) is only finite over a compl* *etion of ku*. The coefficient ring does not satisfy the hypotheses of the Landweber exa* *ct functor theorem, so other methods must be used to construct it. We will return to an algebraic investigation of these two last examples in Se* *ctions 15 and 16, but first we spend Section 13 explaining the idea of the proof of the the l* *ocal cohomology theorem for Noetherian MU-algebras. This is so closely analogous to commutative* * algebra in the derived category that it inspires a number of definitions of interest in* * algebra as well as topology, and we briefly introduce them in Section 14. 13.The proof. The main thing is that there is a good category to work in [20]. Working the* *re, the proof is essentially formal and just like working in an algebraic derived categ* *ory. This is the category of G-spectra, briefly introduced in Subsection 13.1. In Subsection 13.* *2 we transpose some commutative algebra into the category of G-spectra and in the final two su* *bsections we complete the proof. 13.1. G-spectra. A G-spectrum may be thought of as a generalized based G-space.* * The purpose of the generalization is to ensure that any equivariant cohomology theo* *ry E*G(.) is represented in the category of G-spectra. This means that there is a G-spectrum* * E so that for any unbased G-space X, E*G(X) = [X+; E]*G 24 J.P.C.GREENLEES where the expression on the right denotes G-homotopy classes of G-maps of G-spe* *ctra. Sufficiently well behaved cohomology theories (such as MU) are represented by r* *ing objects R in the category of G-spectra, and there is a category of module spectra over * *R. These rings R are analogous to differential graded algebras, and the homotopy categor* *y of modules over R is directly analogous to the derived category of differential graded mod* *ules over the differential graded ring. This homotopy category is where we work. Thus for a m* *odule M over R, and an unbased G-space X, M*G(X) = [X+; M]*G= [R ^ X+; M]*R;G and MG*(X) = [S0; X+ ^ M]G*= [R; X+ ^ M]R;G*; where the decoration R; G refers to equivariant homotopy classes of R-module ma* *ps. Warning 13.1. Even if R*G(.) is represented by a ring R, it is not automatic th* *at the representing G-spectrum M of a module valued cohomology theory M*G(.) is an R-m* *odule. However this is true in many cases, and specifically for the complete theories * *formed from En and ku discussed above. The proof of the the local cohomology theorem described in this section does * *not in fact apply to R = MU. It requires us to work with a cohomology theory R*G(.) represe* *nted by a ring G-spectrum R, with two properties. Firstly, the coefficient ring R*Gmust b* *e Noetherian (and for all subgroups H the modules R*Hmust be finitely generated), and second* *ly, it must be complex oriented (this is equivalent to saying that R is an MU-algebra up to* * homotopy, but it has a more concrete meaning that will be described at the appropriate po* *int). The coefficient ring of MU is not Noetherian, so the main obstacle is the construct* *ion of enough elements of the ideal J: this is interesting but not relevant to our applicatio* *ns. In fact the complete theories of both En and ku are both represented by ring spectra R to w* *hich the argument of this section does apply. 13.2. Commutative algebra with G-spectra. In the category of modules over R one can mimic most constructions in the derived category. For example we can const* *ruct the homotopy I-power torsion functor. If x 2 RG*= [R; R]G;R*we may form (x)R := fibre(R -! R[1=x]) and then if I = (x1; : :;:xn) we take IR := (x1)R ^R (x2)R ^R : :^:R(xn)R: Up to equivalence, IR does not depend on the generators used, and it only depen* *ds on the radical of the ideal I: this is an easy exercise exactly as in the algebraic de* *rived category of modules over a commutative ring (see Section 6). The case of a principal ideal * *is constructed as a fibre and so comes with a filtration of length 1, and hence IR has a filtr* *ation of length n. Because the construction is modelled on the stable Koszul comples, it is cle* *ar that the homotopy spectral sequence of the filtration takes the form E*;*2= H*I(M*G) ) [R; IM]R;G*: To prove the local cohomology theorem it therefore suffices to establish the * *two equiva- lences R ^ EG+ -A JR ^ EG+ -B! JR of equivariant R-modules. LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 25 We may deduce the local cohomology theorem by applying (M ^ X+) ^R (.) to obt* *ain the equivalence M ^ EG+ ^ X+ ' JR ^R M ^ X+: The homotopy of the left hand side is MG*(EG x X) and that of the right hand si* *de is calculated by a spectral sequence with E2 term H*J(MG*(X)). We may deduce the completion theorem by applying Hom R (. ^ X+; M) to obtain * *the equivalence Hom R(R ^ EG+ ^ X+; M) ' Hom R(JR ^ X+; M): The homotopy of the left hand side is M*G(EG x X), and the homotopy of the righ* *t hand side may be calculated by a spectral sequence with E2-term HJ*(M*G(X)). It remains to prove that the maps A and B are equivalences of equivariant R-m* *odules. 13.3. The map A is an equivalence. Since EG+ is free, it is built from free G-c* *ells Sn ^ G+, so it suffices to show that R ^ G+ - JR ^ G+ is an equivalence, which* * is to say that R - JR is a non-equivariant equivalence. But resG1J = 0 by definition, so resG1JR = resG1JR = 0R = R as required. 13.4. The map B is an equivalence. We show the mapping cone of B is contractibl* *e. In fact we may assume by induction on the group order that it is H-contractible fo* *r all proper subgroups H. This uses the fact that resGHJ has the same radical as J(H) = ker(* *R*H-! R*) as follows in the Noetherian setting from [28]. From this we deduce resGHJR = resGHJR = J(H)R: Now the mapping cone of B is JR ^ "EG where "EG is the mapping cone of EG+ -!* * S0. Because (E"G)G = S0, there is unique map E"G -! S1V which is the identity in G* *-fixed points, where V is the reduced regular representation. It suffices to show that* * JR ^ S1V is contractible. Indeed, if we define Q by the cofibre sequence E"G -! S1V -! Q; we see that Q is built from cells G=H+ for proper subgroups H. It follows from * *the inductive hypothesis JR ^ Q ' *. Since JR^S1V is obviously H-contractible for all proper subgroups H it suffi* *ces to show its G-homotopy is zero. Now for any R-module M we may define Euler classes O(V * *) exactly as was done for K-theory in Section 6. Indeed, it is immediate from the constru* *ction of MU that MU ^SV ' MU ^S|V,|(where |V | denotes V with the trivial G-action) so that* * we may take O(V ) to be the pullback of 1 2 MU|VG|= [S|V;|MU]*Gunder the inclusion S0 * *-! SV . From the definition we see that ssG*(M ^ S1V ) = lim ssG*(M ^ SnV) = ssG*(M)[1=O(V )]: ! n On the other hand, since the inclusion S0 -! SV is non-equivariantly nullhomoto* *pic, O(V ) 2 J so that inverting it kills J-local cohomology and therefore ssG*(JR ^ S1V ) =* * 0. This completes the proof. 26 J.P.C.GREENLEES 14.Homotopically Gorenstein rings. For further details see [29, 19]. To smooth the transition between topology and commutative algebra we now writ* *e k*(X) = E*(X) for the non-equivariant cohomology theory and S*(X) = E*G(X) for the equi* *variant one. In view of the completion theorem we may also write ^S*(X) = E*(EG xG X) f* *or the complete theory. We restrict attention to finite groups G, and therefore to th* *e case with shift a = 0 (in the notation of Section 11). We have obtained duality statements by combining the local cohomology theorem* * with the universal coefficient theorem. In the first instance these were spectral se* *quences H*J(k*(BG)) ) k*(BG) and Extk*(k*(BG); k*) ) k*(BG); but we have already reformulated the first as JF (EG+; k) ' k ^ EG+; and the second can be written Hom k(k ^ BG+; k) ' F (BG+; k); which we would like to think of as the fixed points of an equivariant equivalen* *ce Hom k(k ^ EG+; k) ' F (EG+; k): The analogy is with the algebraic derived category, so Hom corresponds to the * *total right derived functor of ordinary homomorphisms. Substituting in the local cohomology* * theorem expression for k ^ EG+ ' F (EG+; k) ^ EG+, and writing ^S= F (EG+; k) we find Hom k(JS^; k) ' ^S: More generally we can consider this condition on any ring ^Sup to homotopy equi* *pped with ring maps k -! ^S- ! k making it into a supplemented k-algebra. We may say tha* *t a supplemented ring ^Sis homotopically Gorenstein if there is such an equivalence* *. If k is a field we may take homotopy and deduce that ^S*is an LCT ring. If k is not a fie* *ld the left hand side is the composite of two functors. Each of these functors could be cal* *culated with a spectral sequence, but it is harder to extract information about their compos* *ite. However when k is of small injective dimension we can still get striking duality proper* *ties for ^S*. For complete rings it is quite often equivalent to consider the statement JS^' Hom k(S^; k); and this is better behaved for non-complete rings, so we may say that a supplem* *ented k- algebra S is homotopically Gorenstein if JS ' Hom k(S; k): It turns out [19] that under quite weak hypotheses this is equivalent to requir* *ing Hom S(k; S) ' k as modules over Hom S(k; k), which may be a more familiar form of the Gorenstei* *n condition. This is related to the notion of Gorenstein differential graded commutative alg* *ebras consid- ered by Avramov and Foxby [4], and to the ideas of Felix, Halperin and Thomas [* *23] but LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY 27 they only require the isomorphism as S-modules. This circle of ideas is investi* *gated further by Dwyer, Greenlees and Iyengar [18, 19]. 15.The chromatic case. For further details see [39]. We have explained that it is of interest to calculate the local cohomology mo* *dules H*J(S) where S = R*G= E*n(BG), because they give the E2-term of a spectral sequence fo* *r calcu- lating the more subtle invariant (En)*(BG). Playing this spectral sequence off * *against the universal coefficient theorem implies that the ring R*Ghas very special duality* * properties. Now here we have k = R* = Z^p[[u1; : :;:un-1]][u; u-1], and we may consider t* *he Landweber sequence of prime ideals 0 (p) (p; u1) (p; u1; u2) . . .(p; u1; : :;:un-1): It turns out that a natural ring of endomorphisms acts on R* and that the Landw* *eber sequence is the unique maximal sequence of invariant prime ideals. For any modu* *le M we may form the Cousin complex C(M) = (M[1=p] -! M=(p1 )[1=u1] -! M=(p1 ; u11)[1=u2] -! . .-.! M=(p1 ; u11; : * *:;:u1n-1): If (p; u1; : :;:un-1) is M-regular we say M is good; in this case M ' C(M). It * *is then natural to use this filtration to approach the calculation of local cohomology. Theorem 15.1. [39] If E*n(BG) is good, then local cohomology is trivial on pure* * chromatic strata in the sense that H*J(E*n(BG)) = H*(JC(E*n(BG)): Remark 15.2. (i) The module E*n(BG) is known to be good in many cases, includin* *g all abelian groups G, and all symmetric groups [42]. (ii) The complex JC(E*n(BG)) is highly non-trivial. Indeed, JC(E*n(BG)) is usua* *lly non- zero up to degree n: for example if G is abelian then in degree i it is |G|iCi(* *E*n). On the other hand if G is abelian of rank r, then J has arithmetic rank r, and hence * *the local cohomology is trivial above degree r. If n > r the exactness of the complex abo* *ve degree r must involve interesting differentials. (iii) The case of periodic K-theory gives something we have already seen. It st* *ates H*J(R(G)^p) = H*(JR(G)^p[1=p] -! JR(G)=p1 ) = H*(Z^p[1=p] -! R(G)=p1 ): This is very effective, and recovers the local cohomology calculations of Secti* *on 4. 16.Connective K theory. For further information see [12, Chapter 4]. In this section we present a very concrete example where the commutative alge* *bra is very intricate and very striking. The local cohomology theorem states there is a spe* *ctral sequence H*J(ku*(BG)) ) ku*(BG); for any finite group G, but we want to discuss a case where the entire behaviou* *r is understood. We take G = V an elementary abelian 2-group of rank r. In fact there is a sho* *rt exact sequence 0 -! T -! ku*(BV ) -! Q -! 0: 28 J.P.C.GREENLEES Here Q is the extended Rees ring for the J-completion ^R(V ) of the representat* *ion ring R(V ) = Z[ff1; : :;:ffr]=(ff21= . .=.ff2r= 1): This completion has the effect of 2-adic completion on J, and leaving alone the* * Z summand generated by the regular representation. Thus Q = ^R(V )[v; y1; y2; : :;:yr] ^R(V )[u; u-1] where yi= (1-ffi)=u, and we may take J to be generated by elements mapping to y* *1; : :;:yr. One may calculate H*J(Q) without too much difficulty: H0J(Q) = Z[u] and H1J(Q) * *is zero below degree -2r and the order of its Z-torsion increases with degree. On the other hand T is a bit more complicated. It is a P -submodule of F2[x1* *; : :;:xr] whereQP = F2[y1; : :;:yr] and x2i= yi. It turns out that it is generated by el* *ements the qS = i2Sxi(j;k2Sxjx3kas S ranges over subsets of {1; 2; : :;:r} with at least* * 2 elements. Now we can decompose T = T2 T3 . . .Tr, where Ti is generated by the qS with |S| = i. These P -modules Tishow some remarkable behaviour. Proposition 16.1. (i) Ti is of dimension r and depth i. (ii) Hjm(Ti) = 0 unless j = i or r. (iii) Him(Ti)_ is 1-dimensional if i < r. (iv) Hrm(Ti)_ = Tr-i+2(-r + 4) if i < r. (v) Hrm(Tr)_ = P (-r + 4). 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D* *urham Conference on pro-p-groups (to appear) [55]G.Wilson "K-theory invariants for unitary G-bordism." QJM 24 (1973) 499-526 Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: j.greenlees@sheffield.ac.uk