COMMUTATIVE ALGEBRA FOR COHOMOLOGY RINGS OF VIRTUAL ...

COMMUTATIVE ALGEBRA FOR COHOMOLOGY RINGS OF VIRTUAL DUALITY GROUPS. D. J. BENSON AND J. P. C. GREENLEES Abstract. Cohomology rings of nite groups have strong duality properties, as shown

by Benson and Carlson [2], and Greenlees [13]. We prove here that cohomology rings of virtual duality groups have a ring theoretic duality property, which combines the duality properties of nite groups with the cohomological duality of the subgroup of nite index. The formal behaviour of the local cohomology theorem is precisely analogous to that for a compact Lie group [4], except that the dimension appears to be negative.

1. Introduction We shall be discussing cohomology rings of certain discrete groups ?, and we work throughout with a Noetherian commutative ring k of coecients. We shall usually omit notation for coecients in k, and for tensor products over k. We begin by introducing certain notation and terminology that is used throughout. Recall that a discrete group N is a duality group of dimension n if there is a dualizing kN module I so that there are isomorphisms H i (N ; L) = Hn?i(N ; I L) for all kN -modules L. It turns out that if N is a duality group of dimension n and dualizing module I then H s (N ; kN ) = 0 if s 6= n and I = H n(N ; kN ). Conversely if k has a nite resolution by nitely generated projective kN -modules and H s(N ; kN ) = 0 for s 6= n then N is a duality group of dimension n. A group ? is a virtual duality group of virtual dimension n if it has a subgroup N of nite index which is a duality group of dimension n. If ? is a virtual duality group of virtual dimension n then every torsion free subgroup of nite index is a duality group of dimension n; indeed, by Shapiro's lemma H s(N ; kN ) = H s(?; k?). A duality group N is said to be a Poincare duality group (or PD group) if the dualizing module is free of rank one over k, and a PD group is said to be orientable if the N -action on the dualizing module is trivial. An orientable virtual PD group is a virtual PD group ? with the property that ? acts trivially on H d (?; k?). Note that this is a stronger condition than asking for an orientable PD subgroup of nite index. For further discussion of these notions, see the books of Bieri [5] and Brown [8]. In this article we consider the cohomology ring H (?) of a virtual duality group ?, and show that it has strong ring theoretic duality properties. This melds the duality theorem for the duality group of nite index with the duality enjoyed by the nite quotient group; in particular our duality results generalize both these types of duality. 1

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D. J. BENSON AND J. P. C. GREENLEES

Duality properties for nite groups have been studied by Benson{Carlson [2], and Greenlees [13]. The method of [2] uses nite Koszul complexes, and therefore depends on the choice of a system of parameters. The method of [13] uses stable Koszul complexes, which are in nite but essentially independent of choices. The invariant approach also allows an implementation through the use of highly structured ring and module spectra, and this extends to the case of compact Lie groups [4]. Here we give two approaches to duality for the cohomology of virtual duality groups: a purely algebraic one in the spirit of [2] and [13], and one using highly structured ring and module spectra as in [4]. Each method has advantages. In fact the algebraic method applies to all virtual duality groups, and over an arbitrary commutative Noetherian ring, whereas the topological one is restricted to virtual duality groups (such as arithmetic groups and mapping class groups) where the duality is realized topologically. On the other hand the topological proof applies to groups with a normal duality group and a compact Lie quotient, and extends to generalized cohomology theories, such as K -theory. To state our main theorems we let J denote the ideal of positive degree elements in the ring H (?). One of our theorems is stated in terms of Grothendieck's J -local cohomology HJ (M ) of a module M over the ring H (?) [15]; a summary of its properties in a convenient form is given in [14].

Theorem 1.1. If ? is a virtual duality group of virtual dimension n and with dualizing module I , then for any ? module M there is a spectral sequence E2; = HJ;(H (?; M )) =) nH (?; I M ) with dierentials dr : Ers;t ?! Ers+r;t?r+1.

This is a duality theorem, which melds the duality theorem for the duality group of nite index with the type of duality enjoyed by nite groups and has both these as special cases. The theorem is based on the commutative algebra of the graded ring H (B ?), and its ideal J of elements with positive cohomological degree. It is essential to be clear about grading conventions, and for clarity all graded objects are cohomologically graded. In particular we shall use the cohomological suspension de ned by (nM )s = M s?n . We view the homology groups H (B ?) as being concentrated in negative cohomological degrees, so that (n H(B ?; I ))s = Hn?s(B ?; I ): Theorem 1.1 is proved as 4.1. A topological proof is given in 5.1, for groups, such as arithmetic groups and mapping class groups, in which the algebraic duality is realized by an action on a suitable manifold. Notice that if ? is an orientable PD group the spectral sequence calculates H(?) with a dimension shift. If, in addition, H (?) is Cohen{Macaulay of dimension r then the spectral sequence collapses to show that HJr (H (?)) = nH (?). If k is a eld this shows that H (?) is Gorenstein. To state the analogue of the results of [2] we need to choose a particular system of radical generators of J . In fact we choose 1; : : : ; r 2 H (G), with i of codegree ni so that H (?) is nitely generated as a module over the subring generated by the in ations of the i . For a virtual duality group one may construct a connected nite complex B^ of nitely generated projectives so that H (B^ ) is a nitely generated free k-module with trivial action. For the purposes of the following statement this should be thought of as a close approximation to k = H0(B^ ).

COHOMOLOGY RINGS OF VIRTUAL DUALITY GROUPS

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Theorem 1.2. If ? is a virtual duality group and M is a k?-module, and s = n + Pri=1(ni ? 1), then there is a spectral sequence E2pq = Extpk?(Hq (B^ ); M ) =) Hs?p?q ((M I ) k? B^ ):

At least in the Cohen{Macaulay case, it is easy to obtain a conclusion independent of the choice of elements.

Theorem 1.3. Suppose that ? is an oriented virtual PD group of dimension n, and k is a eld.

If H (?; k) is Cohen{Macaulay, then the Poincare series

p?(t) =

X i0

ti dimk H i(?; k)

satis es the functional equation

p?(1=t) = t?n (?t)r p?(t): Here, r = rp (?) is the maximal rank of a nite elementary abelian p-subgroup of ? if Char(k) = p is a prime, and r = 0 if Char(k) = 0. The algebraic proof is covered in Sections 2, 3 and 4. The topological proof is covered in Sections 5 and 6. These are completely independent of each other, and there will be readers wishing to ignore one of the methods entirely. We conclude with a brief discussion of examples.

Part A: The algebraic method 2. The algebraic construction A common method for getting control on in nite discrete groups is to attempt to construct a nite dimensional space on which they act with nite stabilizers. However the existence of such a space is a stringent condition on a group. In this section we construct a suitable algebraic counterpart for arbitrary groups satisfying an obviously necessary niteness condition. We need to use several standard niteness conditions. A k?-module M is said to be of type FP1 if it has a resolution by nitely generated projective k?-modules. If this resolution may be chosen to be of nite length (i.e. eventually zero), then M is said to be of type FP . Replacing projective by free in this de nition gives the de nition of type FL (\ ni libre"). The group ? is said to be of type FP1 , FP or FL over k if the trivial k?-module k is of type FP1 , FP or FL respectively. The group ? is said to be virtually of type FP1 , FP or FL (written V FP1 , V FP , V FL) if it has a normal subgroup of nite index of the respective type.

Lemma 2.1. Suppose that ? has type FP1 over k. If X is a nonnegatively indexed chain complex

?! X2 ?! X1 ?! X0 ?! 0 of projective k?-modules, and each Hi (X ) is a nitely generated free k-module with trivial ?action, then X is chain homotopy equivalent to a nonnegatively indexed chain complex X^ with each X^ i a nitely generated projective k?-module. If Xi = 0 for i large enough then X^ may be chosen with X^ i = 0 for i large enough.

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Proof: We prove by induction on n that any such X is equivalent to a complex with Xi a nitely generated projective k?-module for i n. Suppose that this is true for all such complexes for i n ? 1 (for n = 0 this is vacuously true, so the induction begins). We may choose a resolution P of H0(X ) by nitely generated projective k?-modules. Then there is a map of complexes P ?! X inducing an isomorphism in degree zero homology. The algebraic mapping cone M of

this map of complexes has no homology in degree zero. So it is isomorphic to a direct sum of an exact complex of projectives of the form = ?! 0 ?! P10 ?! P00 ?! 0 ?! and a complex M0 with M00 = 0. By the inductive hypothesis, M0 (and hence also M ) is equivalent to a complex M^ 0 with M^ i0 a nitely generated projective k?-module for i n. The algebraic mapping cone on the map M^ 0 ?! P is chain homotopy equivalent to X , and consists of nitely generated projective k?-modules in degrees i n + 1, so applying ?1 completes the inductive step of the proof. If Xi is already nitely generated for i n ? 1 then the eect of this construction is to add on an exact sequence of nitely generated projectives in these degrees. These may be stripped o again so that X is unaected in degrees i n ? 2. It follows that we may perform this construction in nitely often, and it will \converge". Finally, suppose that Xi = 0 for i m. Replace the complex X^ by the truncated complex obtained by deleting the terms from X^ m+1 onwards and replacing X^ m by Ker(X^ m?1 ?! X^ m?2 ). With this new X^ , the algebraic mapping cone of X ?! X^ is a nite acyclic complex in which all terms except possibly one is projective. It follows that the remaining term is also projective, which shows that X^ now has the required properties. Lemma 2.2. Let ? be a group of type V FP over k. Then a k?-module M of type FP1 is of type FP if and only if its restriction to each nite subgroup is projective. Proof: See for example Theorem A0 of Kropholler [18]. Lemma 2.3. Let ? be a group and N a subgroup of nite index, and let k be a commutative ring of coecients. If M is a k?-module of nite projective dimension whose restriction to N is projective, then M itself is projective. Proof: De ne Y by the short exact sequence 0 ?! k ?! k[?=N ] ?! Y ?! 0: Tensoring with M and iterating, we obtain the exact sequence 0 ?! M ?! k[?=N ] M ?! k[?=N ] Y M ?! ?! k[?=N ] Y (r?1) M ?! Y r M ?! 0: Now, since M restricted to N is projective, we nd k[?=N ] Y i M is projective for all i, so the terms in the sequence are projective, except perhaps the rst and last. If M has projective dimension at most r, then so does Y r M , and so, by the extended form of Schanuel's lemma, M is projective as required. If M is a k?-module, we write k?(M ) for the kernel of a map from a projective module onto M . This is well de ned up to adding and subtracting projective summands (Schanuel's lemma). Then we inductively de ne 1k?(M ) = k?(M ) and nk?(M ) = k?( kn??1(M )).


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Proposition 2.4. Let ? be a group of type V FP over k. Then there exists a nite complex B^ of nitely generated projective k?-modules with H0(B^ ) = k, and such that each Hi (B^ ) is a nitely generated free k-module with trivial ?-action.

Remark: This seems to be related to some theorems of Stark [21]. Proof: Let N be a normal subgroup of ? of nite index, of type FP over k, and set G = ?=N . By Evens' nite generation theorem, we may choose elements 1; : : : ; r with i 2 H n (G; k) whose in ations Inf(1); : : : ; Inf(r ) 2 H (?; k) generate a subring over which the whole of H (?; k) is nitely generated as a module. For each i , choose a corresponding cocycle ^i : nkG (k) ?! k, and n i

i

make sure it is surjective by enlarging kG(k) by a suitable projective if necessary. Write L for the corresponding kernel. Then we can form a pushout diagram for the beginning of a resolution of k by nitely generated projective kG-modules as follows: i

0

# L #

i

0

# = L # 0 ?! nkG(k) ?! Pn ?1 ?! ?! P0 ?! k ?! 0 # # k k 0 ?! k ?! Pn ?1=L ?! ?! P0 ?! k ?! 0 # # i

i

i

i

i

0

i

0

We truncate the bottom row of this diagram by removing the copy of k at the beginning and the end, to make a complex C whose homology consists of a copy of k in degrees zero and ni ? 1 with trivial G-action. We then form the complex B = C1 k k C , which is a complex of projective modules by the theory of varieties [2]. We in ate to regard this as a complex of k?-modules. Next, we take a nite resolution of k by projective kN -modules, and tensor induce to give a complex Q of k?-modules. Note that, even if we started o with nitely generated modules, we will have lost this property after tensor induction. However, Q does have nite length, and is exact except in degree zero, where its homology is k. Let B0 = B Q . Then the restriction of B0 to any nite subgroup is a complex of projective modules, so B0 consists of modules of type FP . It follows from Lemma 2.3 that B0 is in fact a complex of projective modules. Furthermore, its homology consists of a nite direct sum of copies of k with trivial ?-action. So by Lemma 2.1, B0 is chain homotopy equivalent to a complex B^ with each B^i a nitely generated projective k?-module. i

r

Remark 2.5. We note that the argument of the proposition applies to show that for any complex

C of nitely generated projective kG modules the complex C0 = C Q is a complex of projective k?-modules, and chain equivalent to a complex C^ of nitely generated projective k?-modules.

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3. The algebraic duality theorem. In this section we consider a virtual duality group ?. It is necessarily of type VFP, and we denote by B^ and Q complexes resulting from the construction described in the proof of Proposition 2.4. Proposition 3.1. If ? is a virtual duality group over k with dualizing module I and dualizing degree n, then H j+n(Homk?(B^ ; k?)) = Homk (Hj (B^ ); I ):

Proof: Let P be a projective resolution of k as a k?-module, and compare the two spectral sequences of the double complex Homk?(P B^ ; k?). If we take homology using the P dierential rst, the E1 page consists of Homk?(B^ ; k?) along the edge, and the E2 page consists of H (Homk?(B^; k?)). On the other hand, if take homology using the dierential coming from B^ rst, then the E1 page is Homk?(H(B^ ); Homk (P ; k?)) = Homk (H (B^ ); Homk?(P ; k?)) (since the modules in B^ are projective and the ?-action on H (B^ ) is trivial) and the E2 page consists of Homk (H (B^ ); I ) in the nth column and zero elsewhere. In fact, this cohomology equivalence comes from a chain homotopy equivalence, as we shall show. We regard the complex Homk?(Q ; k?) (the dual of Q ) as a chain complex with negative degrees. Then we have the following duality statement. Lemma 3.2. There is a map of chain complexes of k?-modules : I k Q ?! nHomk?(Q ; k?) which is a homology equivalence. Proof: Since Q restricted to N is a projective kN -resolution of k, we have a homology equivalence of chain complexes of projective kN -modules I k Q ?! nHomkN (Q; kN ): Taking the adjoint of this map gives a map of chain complexes of kN -modules I k Q k Q ?! nkN: By Frobenius reciprocity, there is a corresponding map of chain complexes of k?-modules I k Q k Q ?! nk?; whose adjoint I k Q ?! nHomk?(Q ; k?) is the desired homology equivalence. The fact that it is a homology equivalence is easily seen by restricting back to N and using the Eckmann{Shapiro isomorphism. Proposition 3.3. There is a chain homotopy equivalence of complexes of k?-modules ^ : I k B^ ?! sHomk?(B^ ; k?) P where s = n + ri=1 (ni ? 1).


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Proof: It is shown in Section 5 of [2] that there are maps of chain complexes of k[?=N ]-modules C ?! n ?1Homk (C ; k) inducing an isomorphism in homology. Actually, this is only proved i

i

i

there in the case of a eld of coecients, but it holds more generally. Hence we have a homology isomorphism C1 k k C ?! (n ?1)Homk (C1 ; k) k k Homk (C ; k): Tensor this with I Q and follow it with 0 = 1 : Homk (C1 ; k) k k Homk (C ; k) k ?nI Q ?! Homk(C1 ; k) k k Homk (C ; k) k Homk?(Q; k?) where is the map of Lemma 3.2. Next, the natural map 00 : Homk (C1 ; k) k k Homk(C ; k) k Homk?(Q ; k?) ?! Homk?(C1 k k C k Q; k?) is an isomorphism of chain complexes of k?-modules. To see this, it is enough to check that it is an isomorphism of abelian groups which preserves the ?-action. To check that it is an isomorphism of abelian groups, it suces to notice that each C is a nite chain complex of nitely generated free abelian groups. The ?-action is easy to check. The codomain of 00 is Homk?(B0 ; k?) in the notation of 2.4; since we have a chain homotopy equivalence B^ ?! B 0 of complexes of projectives, the codomain of 00 is equivalent to the codomain in the statement. Composing these maps we obtain a homology isomorphism of bounded below projective complexes which is therefore a chain homotopy equivalence as required. Corollary 3.4. If M is a k?-module, then there is a chain homotopy equivalence of chain complexes of k-modules (M I ) k? B^ ?! sHomk?(B^ ; M ): Proof: This follows by applying M k? ? to the map given in the proposition. Corollary 3.5. If k is a eld and ? is an oriented virtual PD group, then the Poincare series X f (t) = ti dimk Hi (B^) r

i

i

r

r

r

r

r

i

satis es the functional equation where s = n +

Pr (n ? 1). i=1 i

i0

f (t) = tsf (t?1 );

Proof: The space Hi(B^) is the k-vector space dual of H?iHomk?(B^; k), which by the previous

corollary (with M = k) is isomorphic to Hs?i(B^ ). So the coecient of ti in f (t) is equal to the coecient of ts?i . Theorem 3.6. If ? is a virtual duality group over k and M is an k?-module, and s = n + Pr (n ? 1), then there is a spectral sequence i i=1 E2pq = Extpk?(Hq (B^); M ) =) Hs?p?q ((M k I ) k? B^ ):

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Proof: Consider the double complex

Homk?(P k B^ ; M ) where P is a projective resolution of k as an k?-module. In the spectral sequence in which we take homology using the P dierential rst, the E1 page consists of Homk?(B^ ; M ) along the edge, and the E2 page consists of H (Homk?(B^ ; M )). By Corollary 3.4, this is isomorphic to Hs? ((M k I ) k? B^ ). It follows that the other spectral sequence, in which we take homology using the B^ dierential rst, converges to the same thing. The E1p;q term is Homk?(Pp k Hq (B^ ); M ), and so the E2p;q term is Extpk?(Hq (B^ ); M ) as required. The dierentials in this spectral sequence can be described in the same way as in [2]. In particular, there are dierentials corresponding to multiplication by the in ations of the elements i used to construct the complex B^ . So just as in [2], we obtain the following statement about Poincare series:

Theorem 3.7. Suppose that ? is an oriented virtual PD group of dimension n, and k is a eld. If H (?; k) is Cohen{Macaulay, then the Poincare series p?(t) =

X i0

ti dimk H i(?; k)

satis es the functional equation

p?(1=t) = t?n (?t)r p?(t): Here, r = rp (?) is the maximal rank of a nite elementary abelian p-subgroup of ? if Char(k) = p is a prime, and r = 0 if Char(k) = 0.

Proof: Take M = k in the spectral sequence of the last theorem, and choose the number of i to

be as small as possible, namely r. Since we also have I = k, it becomes: E2p;q = Extpk?(Hq (B^ ); k) =) Hs?p?q (k k? B^ ): Since the cohomology is Cohen{Macaulay, the i form a regular sequence in H (?; k), and so the E1 page is H (?; k)=(1; : : : ; r ) concentrated along the bottom row. The Poincare series of this quotient therefore satis es the functional equation of Corollary 3.5. Since

Yr

p?(t) = f (t)= (1 ? tn ); i

i=1

the required functional equation for p? (t) follows. 4. The local cohomology theorem. In this section we prove the local cohomology theorem stated in the introduction.

Theorem 4.1. If ? is a virtual duality group of virtual dimension n and with dualizing module I , then for any ? module M there is a spectral sequence E2; = HJ (H (?; M )) =) nH (?; I M )


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The idea is to apply the methods of [13] to the resolutions constructed in Section 2, principally using Proposition 3.3 as a substitute for the self-duality of the group ring of a nite group. We are thus able to prove the local cohomology theorem for all virtual duality groups. There is a little work to be done, because of the fact that the complex B 0 has to be replaced by the complex B^ of nitely generated projectives. First we begin by recalling the construction from [13], and to begin with we work entirely with kG-modules, where G = ?=N . We view the complex

B = C1 C

r

Zr ,

of kG modules as graded over and non-zero in the cuboidal box with vertices at (1 (n1 ? 1); : : : ; r (nr ? 1)) with i = 0 or 1. Of course we may splice together two copies of C since H0(C ) = k = Hn?1(C ), and thus formr a complex C2 . Tensoring with the other complexes C we may view this as stacking boxes in Z . Notice that there is a co bre sequence i

i

i

j

i

n?1C ?! C ?! C2 ; i

i

i

i

and similarly for boxes. Next we let R denote the complex obtained by stacking boxes to ll the non-negative orthant. More generally for any subset f1; 2; : : :rg we let R[ ] be the complex obtained by stacking boxes so as to ll the region with gradings (i1; : : : ; ir ) with ij 0 if j 62 . In particular we let S denote the complex R[f1; 2; : : : ; rg] in which the whole of Zr has been lled with boxes. Now form the dual stable Koszul complex

0 1 M M M M L = @ R[] ?! R[] ?! ?! R[] ?! R[]A jj=r

jj=r?1

jj=1

jj=0

in which the dierentials are alternating sums of projections. It is clear that at each point (i1; : : : ; ir ) of Zr this complex is obtained by tensoring the reduced chain complex of the simplex = f j i < 0g whose vertices index the negative coordinates, with the projective kG-module occupying the point in S . Therefore the homology of L is only non-zero in degree r where it is concentrated in the negative orthant. More precisely, if we let R! denote the complex obtained by stacking boxes to ll the negative orthant, we have R! if i = r Hi (L) = 0 otherwise: We now want to repeat the construction of Section 2. We begin by regarding the discussion so far as ?-equivariant by in ation, and then form the complex L0 = L Q of projective modules. We then replace it by a chain equivalent complex L^ of nitely generated projectives. However there are three reasons for care: rstly, the complexes R[ ] are not bounded below, secondly, we want to carry the Koszul structure maps along in the construction and, nally, we want to retain the periodicities given by multiplication by i . We have just observed that the process of stacking one box on top of another can be achieved as a mapping cone. More generally, if B [i1 ; : : : ; ir ] denotes a copy of the box complex with its bottom vertex at (i1n1 ; : : : ; ir nr ) we lter the whole page lled with boxes by i1 + i2 + + ir . Thus we let Sj be the quotient of S = R[f1; 2; : : : ; rg] in which the subcomplex of all boxes with i1 + + ir < j has been factored out, and we let Sjn = Ker(Sj ?! Sn+1 ). Thus Sjn=Sjn?1 is the

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sum of boxes with total coordinate n, and we have a co bre sequence M ?1 B [i1 ; : : : ; ir ] ?! Sjn?1 ?! Sjn: i1 ++ir =n

We regard this as a method for constructing S01; S02; S03; : : : . in succession: we may construct S0 = S01 by passage to the direct limit. At each stage we may form the approximation by choosing an arrow in the diagram L ?1 i1++i =n+1 B^ [i1; : : : ; ir ] > S^jn

L ?1

# n+1 n i ++i =n+1 B [i1 ; : : : ; ir ] ?! Sj ?! Sj r

1

#

r

(as we may do since B^ is projective), and taking the mapping cone in the top row. Again we may pass to direct limits and obtain S^0 , and again the limit is achieved in each degree, since B^ is bounded below. On the other hand we know that S?s is simply a desuspension of S0, and S is the inverse limit of S?s under the quotient maps. Now, because the above construction of S^0 also gave the ltration, we have counterparts of the quotient maps, still with sums of boxes as the kernels. We may therefore form S^ = lims S^?s, and since B^ is bounded above, this inverse limit is also achieved in each degree. We therefore obtain an approximation of S which has approximations of the various complexes R[] as quotients, and we may form the approximation L^ simply by taking these quotients and making the dierentials from alternating sums of the quotient maps. We are now ready to give the proof. Consider the double complex Homk?(L^ ; M ); we compare the two spectral sequences for calculating its cohomology. If we take homology in the Koszul direction rst, then because of the projectivity of boundaries and cycles we obtain E1 term Homk?(H L^ ; M ): Now Hi (L) = 0 for i < r and Hr (L) = R! . Furthermore Hr (L ) is the inverse limit of suitable box quotients of the complexes Sj . Therefore Hi (L^ ) = 0 for i < r and Hr (L^ ) is the corresponding inverse limit of the corresponding quotients of S^j . It is reasonable to call this limit R^! , and in particular it is a complex of nitely generated projectives, it is bounded above, has homology equivalent to R! and has a ltration with quotients which are nite sums of copies of B^ . Now by naturality of 3.3 for box quotients we nd ^ Homk?(R^! ; M ) = n (M I ) k? R: Its homology is therefore nH (?; M I ). Q To understand the second spectral sequence, let = i2 i . Now, note that because limits are achieved in each degree, we may calculate ^ H (Homk?(R^[]; M )) = H (Hom k?(lim(R;^ ); M )) = lim ! H (Homk?(R; M )); = H (?; M )[1=]:


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Thus if we take projective cohomology of Homk?(L^ ; M ) rst, then the E1 term is the stable Koszul complex

H (?; M ) ?!

M

jj=1

H (?; M )[1=] ?! ?!

M

jj=r

H (?; M )[1=]

for calculating the local cohomology of H (?; M ) with respect to the ideal generated by the in ations of the elements 1; : : : ; r . Since we have remarked that this ideal has the the augmentation ideal J as its radical, the homology of this complex is the J -local cohomology and the theorem is proved.

Part T: The topological approach

5. Statement of the theorem. In this part we prove the local cohomology theorem by topological methods for groups ? which are virtual duality groups for geometric reasons. It is entirely independent of Part A, but many of the steps in one proof have obvious counterparts in the other. Two classes of virtual duality groups of particular interest are the arithmetic groups and the mapping class groups. For these we have the extra geometric information that the present topological approach seems to require. We hope that by more careful analysis the topological approach can be extended to a wider class of virtual duality groups, such as the S -arithmetic groups.

Theorem 5.1. If ? is an arithmetic group or a mapping class group of virtual dimension n and

with dualizing module I , and if J is the ideal in H (B ?) of elements with positive cohomological degree then there is a spectral sequence

E2s;t = HJs;t(H (B?)) =) Hn?s?t(B?; I ) with dierentials dr : Ers;t ?! Ers+r;t?r+1. We prove this in the following section. This result should be contrasted with the case of a compact Lie group when there is a similar spectral sequence with a suspension of the opposite sense. More precisely, if is a compact Lie group of dimension d we have proved in [4] that when the adjoint representation is orientable there is a spectral sequence HJ (H (B)) =) ?d H (B): The proof we give below clearly extends to the case of topological groups in which there is a normal duality subgroup N with a compact Lie group of dimension d as the quotient. This takes the form of a spectral sequence

HJ (H (B?)) =) n?d H(B?; I ) where I and are both representations of the component group; I is analogous to the dualizing module, and is the action of ? on H d (S Ad(); k). However we do not know of any particular interest in this case.

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6. Topological proof of the theorem. We begin the discussion without assumption on ?, introducing restrictions as they become necessary. We suppose given an extension 1 ?! N ?! ? ?! G ?! 1 with G nite. The subgroup N will be torsion free and have other good properties. We use the usual trick for constructing a classifying space B ?; we assume that ? acts on a space X so that the restriction of the action to N is free. Letting ? act on EG via the quotient map, we have E ? = EG X; and thus B? = EG G (X=N ): The idea is that, when N is suciently nice, the geometry of the G-space X=N will be rather simple, and we can understand B ? by doing a little G-equivariant homotopy theory on X=N . Now H (B?) = H (EG G X=N ) = H (EG G X=N+; EG G ): More generally the reduced Borel cohomology of a based G-space Y is de ned by bG(Y ) = H (EG G Y; EG G ); and this theory is represented by the G-spectrum b = F (EG+; H ), where H represents ordinary equivariant cohomology with constant coecients, so that H (B?) = bG(X=N+) = [X=N+ ; b]G = [X=N+ ; F (EG+; H )]G = [S 0; F (X=N+ ^ EG+; H )]G: We now wish to do a little commutative ring theory, and for this we need our rst assumption on X=N : we ask that it be nite dimensional. The rings which concern us are H (B ?) and H (BG), and in ation gives a ring homomorphism H (BG) ?! H (B?), which allows us to regard any H (B ?)-module as a module over H (BG). Now let J (G) denote the ideal of positive codegree elements in H (BG), and similarly for J (?). Notice that the ideal of H (B ?) generated by the image of J (G) in H (B ?) has radical J (?) because X=N is nite dimensional. Thus the J (?) local cohomology of any H (B?)-module M is equal to its J (G)-local cohomology: HJ(?)(M ) = HJ(G) (M ): We may therefore safely write J for J (G) without fear of causing confusion. Moving to the topological counterpart, we recall that b = F (EG+ ; H ) is equivalent to commutative algebra over the equivariant sphere spectrum (i.e., it is a highly structured ring Gspectrum). In fact H is an E1-ring spectrum, by equivariant in nite loop space theory, and therefore F (EG+ ; H ) is; the result follows by [10, II.3.6]. An alternative is to quote the theorem of Elmendorf{May [9]. The latter is preferable for two reasons: rstly it extends to the case of compact Lie groups, and secondly there is no published reference for equivariant in nite loop space theory. For the remainder of the section we work with highly structured modules over b. Now for any b-module G-spectrum m we may form the (derived) J -power torsion spectrum ?J m, and the construction makes it obvious that there is a spectral sequence E2; = HJ (mG ) =) G(?J m):


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This is the construction which gives rise to the possibility of a geometric proof of a local cohomology theorem. Applying () ^b m to the equivalence EG+ ^ b ' ?J b of [4] we obtain the basic result. Theorem 6.1. [4] For any b-module G-spectrum m there is an equivalence of b-module G-spectra ?J m ' m ^ EG+ : Hence there is a spectral sequence E2; = HJ (mG ) =) G(m ^ EG+): We wish to apply this to the b-module m = F (X=N+ ^ EG+; H ) and thereby obtain our local cohomology theorem for virtual duality groups. Since we have seen that the equivariant homotopy of F (X=N+ ^ EG+; H ) is H (B ?), Theorem 6.1 gives a spectral sequence whose E2-term is the J -local cohomology of H (B?). We now need to relate the equivariant homotopy of EG+ ^ F (X=N+ ^ EG+ ; H ) to H(B?). Indeed, since the projection EG+ ?! S 0 is a nonequivariant equivalence it induces a G-equivalence EG+ ^ F (X=N+ ^ EG+; H ) ' EG+ ^ F (X=N+ ; H ): Summary 6.2. Provided X=N is nite dimensional there is a spectral sequence HJ (H (B?)) =) G(EG+ ^ F (X=N+ ; H )): Of course, this applies without hypothesis on X=N provided J is replaced by J (G), but this is not very satisfactory since it depends on the choice of torsion free subgroup of nite index. We still have to relate G (EG+ ^ F (X=N+ ; H )) to the homology of B ?; for this we use functional duality. Thus DX := F (X; S 0) denotes the functional dual of a G-spectrum X in the usual way. To make the analysis possible we assume that X=N+ is a nite G-space; this means that X=N+ is strongly dualizable and hence EG+ ^ F (X=N+ ; H ) ' EG+ ^ DX=N+ ^ H: In this case, since H is split, we have G(EG+ ^ F (X=N+ ; H ))) = G(EG+ ^ DX=N+ ^ H ) = H~ (EG+ ^G DX=N+): Finally, since X=N+ is nite, the functional dual may be calculated as the Spanier-Whitehead dual. We must relate DX=N+ to X=N+ , and for this we assume that X=N is a compact Gmanifold with boundary @X=N , so that we can apply Atiyah duality. This is best known in the smooth case, but works equally well in the piecewise linear setting. In this case we may embed (X=N; @X=N ) properly in (V [0; 1); V f0g) for some representation V of G. Now, if is the normal bundle of X=N in V 1 and 0 is the normal bundle of @X=N in V , Atiyah duality states V 1 DX=N+ ' (X=N ) =(@X=N ) ; where superscripts denote Thom spaces. In particular, if X=N has empty boundary, V 1 DX=N+ ' (X=N ) : From the work of Borel and Serre [7] we know that our assumptions cover all arithmetic groups, and from the work of Harvey [17] and Harer [16] that it covers all mapping class groups. We concentrate on the former, noting that for mapping class groups one must work instead in the piecewise linear category. 0

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Suppose then that ? is arithmetic in a connected semisimple algebraic group over Q; then if K is a maximal compact subgroup of (R) then the symmetric space X 0 = (R)=K is dieomorphic to Rd for some d. Borel and Serre show that X 0 is the interior of a contractible ?-manifold X with corners, and that X=N is a compact G-manifold with corners. Furthermore the boundary @X has the non-equivariant homotopy type of a bouquet of (l ? 1)spheres where l is the Q-rank of the algebraic group . The case with empty boundary (i.e. l = 0) is covered by the usual convention that the (?1)-sphere is the empty space. Thus ? is a virtual duality group with virtual dimension d ? l and dualizing module I = H d?l (?; Z?) = Hcd?l (X ) = Hl (X; @X ) = H~ l?1 (@X ). As usual, the reduced homology of the empty space is Zin degree ?1. The isomorphism is Lefschetz duality, and only depends on the choice of orientation of the contractible space X . However the action of G must be taken into account at this point, and if the orientation representation of G on Zis we should more precisely say I = H~ l?1 (@X ) : Note that this is the dualizing module for any group ? arithmetic in , so that in particular ? is a PD group if and only if the boundary @X is empty, if and only if is of Q-rank 0. In that case ? is cocompact in (R) and the virtual dimension of ? is d. Example 6.3. Let ? be the in nite dihedral group. This is an arithmetic subgroup of SL2, and occurs as an extension 1 ?! Z?! ? ?! C2 ?! 1: ~ . This shows that the tangent bundle We may take X = R and X=N = S 1 , and we nd I = Z need not be orientable. In fact H (?) = Z[c]=(2c) _ Z[d]=(2d) where c and d have codegree 2; the element c + d generates an ideal with J as its radical. Hence HJ0 (H (?)) = Zand HJ1 (H (?)) is additively Z=2 in codegree 0 and Z=2 Z=2 in each negative even degree. The local cohomology spectral sequence necessarily ~ ) as required. collapses to calculate H (?; Z Now to apply the stated form of Atiyah duality we need to know that we can apply the standard method of [19, II.5] to our G-manifold X=N with corners. In fact Douady and Herault prove in the appendix to [7] that the smooth manifold X with corners may be replaced by a homeomorphic smooth manifold with boundary, so that the smooth structure is only altered on proper corners. Furthermore, they prove that the smooth structure with this property is unique. Now any element 2 (Q) acts smoothly on X (essentially by [7, 5.6(2)]), and since the action preserves corners and their indices, composition with would give a second smooth structure on the manifold with boundary. Therefore acts smoothly on the manifold with boundary. It follows that we may replace X by a smooth manifold with boundary, and the action of ? is smooth. Thus X=N is a smooth manifold with boundary on which G acts smoothly; this is precisely what is required to apply the cited form of Atiyah duality.

Lemma 6.4.

H~ (EG+ ^G DX=N+) = d?l H (B?; I )

Proof: By Atiyah duality we have the co bre sequence (@X=N ) ?! (X=N ) ?! V 1 DX=N+ : 0

Next, it is clear that by using a complex representation V we may assume V 1 is orientable. The orientability of the normal bundles is then equivalent to that of the tangent bundles. The


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tangent bundles need not be orientable, but the same representation of G describes the failure of orientability in both cases. Let v be the dimension of V , so that, recalling the convention that all algebraic suspensions refer to cohomological degrees, H~ (EG+ ^G DX=N+ ) = v+1 H~ (EG+ ^ V 1 DX=N+ ). Now, both and 0 have bre dimension v + 1 ? d, and so the Thom isomorphisms give H~ (EG+ ^G (@X=N ) ) = d?v?1H (EG+ G @X=N ; ) = d?v?1H (E ? ? @X ; ) and H~ (EG+ ^G (X=N )) = d?v?1H (EG+ G X=N ; ) = d?v?1H (E ? ? X ; ): Finally the Serre spectral sequence of the bration @X ?! E ? ? @X ?! B ?, together with the fact that @X is equivalent to a bouquet of (l ? 1)-spheres shows that the inclusion of the boundary induces an injective map with cokernel H(E ? ? X; E ? ? @X ; ) = ?l H (B?; I ) as required. Adding up the relevent suspensions we reach d ? l = (v + 1) + (d ? v ? 1) ? l as asserted. 0

Part D: Discussion 7. Examples Example 7.1. Although the example of SL2(Z) may seem trivial, there are features to note. Since SL2(Z) = C4 C2 C6 we easily calculate that H (SL2(Z)) = Z[c]=(12c), and hence HJ0(H (SL2(Z))) =Z whilst HJ1 (H (SL2(Z))) is additively Z=12 in each even negative degree. This means that the spectral sequence appears to converge to H (SL2(Z)) without suspension. However, we have proved it actually converges to H (SL2(Z); I ). In this case the components of @X correspond to the rational cusps P 1 (Q) of the action of SL2(Z) on the upper half-plane. Thus I is the augmentation ZP 1 (Q) ?! Z. The coincidence would be explained if there were an exact sequence 0 ?! Z ?! M ?! I ?! 0, where H (SL2(Z); M ) = 0. It would be interesting to have a geometric construction of such a module M . Example 7.2. The integral cohomology of SL3(Z) has been calculated by Soule [20]. He shows that the cohomology only has 2 and 3 torsion, that H (SL3(Z))(3) = Z(3) [a4]=(3a4) _ Z(3) [b4]=(3b4), and H (SL3(Z))(2) = Z(2)[c3; d3; c4; d4; c5; c6; d6]=I , where I declares that c4 and d4 are of additive order 4, all the other generators are of additive order 2; in addition c3d6 = d4d6 = c5d6 = c6d6 = c3 d3 = d3d4 = d3c5 = d3c6 = 0; and

d6(d6 + d23) = c4 d4 + c3c5 = c4(c6 + c23) = c4c6 + c25 = 0 c3c6 + d4c5 = c4d24 + c26 = c5c6 + c5c23 = 0:

The 3-local part of the local cohomology is similar to the local cohomology of the in nite dihedral group. However, the 2-local bit seems harder, although it is only two dimensional and is Cohen{ Macaulay.

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D. J. BENSON AND J. P. C. GREENLEES References

1. D. J. Benson and J. F. Carlson. \Complexity and multiple complexes." Math. Z. 195 (1987), 221{238. 2. D. J. Benson and J. F. Carlson. \Projective resolutions and Poincare duality complexes." Trans. American Math. Soc. 342 (1994), 447{488. 3. D. J. Benson and J. F. Carlson. \Functional equations for Poincare series in group cohomology." To appear in Bull. London Math. Soc. 4. D. J. Benson and J. P. C. Greenlees. \Commutative algebra for cohomology rings of classifying spaces of compact Lie groups" Preprint (1995). 5. R. Bieri. Homological dimension of discrete groups. Queen Mary College Mathematics Notes, 1976. 6. A. Borel. \Sur la cohomologie des espaces bres principaux et des espaces homogenes de groupes de Lie compacts." Ann. Math. 57 (1953), 115{207. 7. A. Borel and J. P. Serre. \Corners and arithmetic groups." Comm. Math. Helv. 48 (1973) 436-491 8. K. Brown. Cohomology of groups. Graduate texts in Mathematics 87, Springer-Verlag (1982) 9. A. Elmendorf and J. P. May. \Algebras over equivariant sphere spectra." Preprint (1995). 10. A. Elmendorf, I. Kriz, M. Mandel and J. P. May. \Rings, modules and algebras in stable homotopy theory." Preprint (1995) (available by ftp from Hopf.math.purdue.edu) 11. A. Elmendorf, I. Kriz, M. Mandel and J. P. May. \Modern foundations for stable homotopy theory." Handbook of Topology (ed. I. M. James) North Holland (1995). 12. J. P. C. Greenlees. \The K -homology of universal spaces and local cohomology of the representation ring." Topology 32 (1993), 295{308. 13. J. P. C. Greenlees. \Commutative algebra in group cohomology." J. Pure and Applied Algebra 98 (1995) 151-162 14. J. P. C. Greenlees and J. P. May. \Completions in algebra and topology." Handbook of Topology (ed. I. M. James), North Holland (1995). 15. A. Grothendieck (notes by R. Hartshorne). \Local cohomology." Lecture Notes in Mathematics 41, Springer-Verlag (1967). 16. J.L.Harer \The virtual cohomological dimension of the mapping class group of an orientable surface." Inventiones Math. 84 (1986) 157-176 17. W.Harvey \Boundary structure for the modular group." Riemann surfaces and related topics: proceedings of the 1978 Stony Brook conference. Annals of Maths. studies 97 (1978) 245-251 18. P. Kropholler. On groups of type (F P )1. J. Pure and Applied Algebra 90 (1993) 55-67 19. L. G. Lewis, J. P. May and M. Steinberger (with contributions by J. E. McClure). \Equivariant stable homotopy theory." Lecture Notes in Mathematics 1213, Springer-Verlag (1986). 20. C. Soule. \The cohomology of SL3 (Z)." Topology 17 (1978) 1-22 21. C. W. Stark. \Homologically trival actions on chain complexes and niteness properties of in nite groups." J. Pure & Applied Algebra 94 (1994), 217{228. Department of Mathematics, University of Georgia, Athens, GA 30602, USA.

E-mail address : [email protected]

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK.

E-mail address : j.greenlees@sheeld.ac.uk