COMMUTATIVE ALGEBRA FOR COHOMOLOGY RINGS OF VIRTUAL DUALITY GROUPS.

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

Abstract. Cohomology rings of finite 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 finite groups with the cohomological duality of the subgroup of finite 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 Γ, and we work throughout with a Noetherian commutative ring k of coefficients. We shall usually omit notation for coefficients 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 kNmodule I so that there are isomorphisms

H^{i}(N; L) ≅ H_{n−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 ≠ n and I ≅ H^{n}(N; kN). Conversely if k has a finite resolution by finitely generated projective kN-modules and H^{s}(N; kN) = 0 for s ≠ n then N is a duality group of dimension n.

A group Γ is a virtual duality group of virtual dimension n if it has a subgroup N of finite index which is a duality group of dimension n. If Γ is a virtual duality group of virtual dimension n then every torsion free subgroup of finite index is a duality group of dimension n; indeed, by Shapiro’s lemma H^{s}(N; kN) = H^{s}(Γ; kΓ).

A duality group N is said to be a Poincaré 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 Γ with the property that Γ acts trivially on H^{d}(Γ; kΓ). Note that this is a stronger condition than asking for an orientable PD subgroup of finite 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^{∗}(Γ) of a virtual duality group Γ, and show that it has strong ring theoretic duality properties. This melds the duality theorem for the duality group of finite index with the duality enjoyed by the finite quotient group; in particular our duality results generalize both these types of duality.

Duality properties for finite groups have been studied by Benson–Carlson [2], and Greenlees [13]. The method of [2] uses finite Koszul complexes, and therefore depends on the choice of a system of parameters. The method of [13] uses stable Koszul complexes, which are infinite 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^{∗}(Γ). One of our theorems is stated in terms of Grothendieck’s J-local cohomology H_{J}^{∗}(M) of a module M over the ring H^{∗}(Γ) [15]; a summary of its properties in a convenient form is given in [14].

Theorem 1.1. If Γ is a virtual duality group of virtual dimension n and with dualizing module I, then for any Γ module M there is a spectral sequence

E_{2}^{∗,∗} = H_{J}^{∗,∗}(H^{∗}(Γ; M)) ⟹ Σ^{n}H_{∗}(Γ; I ⊗ M)

with differentials d_{r} : E_{r}^{s,t} ⟶ E_{r}^{s+r,t−r+1}.

This is a duality theorem, which melds the duality theorem for the duality group of finite index with the type of duality enjoyed by finite groups and has both these as special cases.

The theorem is based on the commutative algebra of the graded ring H^{∗}(BΓ), 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 defined by (Σ^{n}M)^{s} = M^{s−n}. We view the homology groups H_{∗}(BΓ) as being concentrated in negative cohomological degrees, so that

(Σ^{n}H_{∗}(BΓ; I))^{s} = H_{n−s}(BΓ; 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 Γ is an orientable PD group the spectral sequence calculates H_{∗}(Γ) with a dimension shift. If, in addition, H^{∗}(Γ) is Cohen–Macaulay of dimension r then the spectral sequence collapses to show that H_{J}^{r}(H^{∗}(Γ)) = Σ^{n}H_{∗}(Γ). If k is a field this shows that H^{∗}(Γ) 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 ζ_{1}, …, ζ_{r} ∈ H^{∗}(G), with ζ_{i} of codegree n_{i} so that H^{∗}(Γ) is finitely generated as a module over the subring generated by the inflations of the ζ_{i}. For a virtual duality group one may construct a connected finite complex B̂_{∗} of finitely generated projectives so that H_{∗}(B̂_{∗}) is a finitely 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 = H_{0}(B̂_{∗}).

Theorem 1.2. If Γ is a virtual duality group and M is a kΓ-module, and s = n + ∑_{i=1}^{r}(n_{i} − 1), then there is a spectral sequence

E_{2}^{pq} = Ext_{kΓ}^{p}(H_{q}(B̂_{∗}), M) ⟹ H_{s−p−q}((M ⊗ I) ⊗_{kΓ} B̂_{∗}).

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 Γ is an oriented virtual PD group of dimension n, and k is a field. If H^{∗}(Γ, k) is Cohen–Macaulay, then the Poincaré series

p_{Γ}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(Γ; k)

satisfies the functional equation

p_{Γ}(1/t) = t^{−n}(−t)^{r}p_{Γ}(t).

Here, r = r_{p}(Γ) is the maximal rank of a finite elementary abelian p-subgroup of Γ 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 infinite discrete groups is to attempt to construct a finite dimensional space on which they act with finite 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 finiteness condition.

We need to use several standard finiteness conditions. A kΓ-module M is said to be of type F P_{∞} if it has a resolution by finitely generated projective kΓ-modules. If this resolution may be chosen to be of finite length (i.e. eventually zero), then M is said to be of type F P. Replacing projective by free in this definition gives the definition of type F L (“fini libre”). The group Γ is said to be of type F P_{∞}, F P or F L over k if the trivial kΓ-module k is of type F P_{∞}, F P or F L respectively. The group Γ is said to be virtually of type F P_{∞}, F P or F L (written V F P_{∞}, V F P, V F L) if it has a normal subgroup of finite index of the respective type.

Lemma 2.1. Suppose that Γ has type F P_{∞} over k. If X_{∗} is a nonnegatively indexed chain complex

⋯ ⟶ X_{2} ⟶ X_{1} ⟶ X_{0} ⟶ 0

of projective kΓ-modules, and each H_{i}(X) is a finitely generated free k-module with trivial Γaction, then X_{∗} is chain homotopy equivalent to a nonnegatively indexed chain complex X̂_{∗} with each X̂_{i} a finitely generated projective kΓ-module. If X_{i} = 0 for i large enough then X̂_{∗} may be chosen with X̂_{i} = 0 for i large enough.

Proof: We prove by induction on n that any such X_{∗} is equivalent to a complex with X_{i} a finitely generated projective kΓ-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 H_{0}(X) by finitely generated projective kΓ-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 ⟶ P′_{1} ⟶^{≅} P′_{0} ⟶ 0 ⟶ ⋯

and a complex M′_{∗} with M′_{0} = 0. By the inductive hypothesis, M′_{∗} (and hence also M_{∗}) is equivalent to a complex M̂′_{∗} with M̂′_{i} a finitely generated projective kΓ-module for i ≤ n. The algebraic mapping cone on the map M̂′_{∗} ⟶ ΣP_{∗} is chain homotopy equivalent to ΣX_{∗}, and consists of finitely generated projective kΓ-modules in degrees i ≤ n + 1, so applying Σ^{−1} completes the inductive step of the proof.

If X_{i} is already finitely generated for i ≤ n − 1 then the effect of this construction is to add on an exact sequence of finitely generated projectives in these degrees. These may be stripped off again so that X_{∗} is unaffected in degrees i ≤ n − 2. It follows that we may perform this construction infinitely often, and it will “converge”.

Finally, suppose that X_{i} = 0 for i ≥ m. Replace the complex X̂_{∗} by the truncated complex obtained by deleting the terms from X̂_{m+1} onwards and replacing X̂_{m} by Ker(X̂_{m−1} ⟶ X̂_{m−2}). With this new X̂_{∗}, the algebraic mapping cone of X_{∗} ⟶ X̂_{∗} is a finite acyclic complex in which all terms except possibly one is projective. It follows that the remaining term is also projective, which shows that X̂_{∗} now has the required properties.

Lemma 2.2. Let Γ be a group of type V F P over k. Then a kΓ-module M of type F P_{∞} is of type F P if and only if its restriction to each finite subgroup is projective.

Proof: See for example Theorem A′ of Kropholler [18].

Lemma 2.3. Let Γ be a group and N a subgroup of finite index, and let k be a commutative ring of coefficients. If M is a kΓ-module of finite projective dimension whose restriction to N is projective, then M itself is projective.

Proof: Define Y by the short exact sequence

0 ⟶ k ⟶ k[Γ/N] ⟶ Y ⟶ 0.

Tensoring with M and iterating, we obtain the exact sequence

0 ⟶ M ⟶ k[Γ/N] ⊗ M ⟶ k[Γ/N] ⊗ Y ⊗ M ⟶ ⋯

⟶ k[Γ/N] ⊗ Y^{⊗(r−1)} ⊗ M ⟶ Y^{⊗r} ⊗ M ⟶ 0.

Now, since M restricted to N is projective, we find k[Γ/N] ⊗ Y^{⊗i} ⊗ M is projective for all i, so the terms in the sequence are projective, except perhaps the first 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 kΓ-module, we write Ω_{kΓ}(M) for the kernel of a map from a projective module onto M. This is well defined up to adding and subtracting projective summands (Schanuel’s lemma). Then we inductively define Ω_{kΓ}^{1}(M) = Ω_{kΓ}(M) and Ω_{kΓ}^{n}(M) = Ω_{kΓ}(Ω_{kΓ}^{n−1}(M)).

Proposition 2.4. Let Γ be a group of type V F P over k. Then there exists a finite complex B̂_{∗} of finitely generated projective kΓ-modules with H_{0}(B̂_{∗}) ≅ k, and such that each H_{i}(B̂_{∗}) is a finitely generated free k-module with trivial Γ-action.

Remark: This seems to be related to some theorems of Stark [21].

Proof: Let N be a normal subgroup of Γ of finite index, of type F P over k, and set G = Γ/N. By Evens’ finite generation theorem, we may choose elements ζ_{1}, …, ζ_{r} with ζ_{i} ∈ H^{n_{i}}(G, k) whose inflations Inf(ζ_{1}), …, Inf(ζ_{r}) ∈ H^{∗}(Γ, k) generate a subring over which the whole of H^{∗}(Γ, k) is finitely generated as a module. For each ζ_{i}, choose a corresponding cocycle ζ̂_{i} : Ω_{kG}^{n_{i}}(k) ⟶ k, and make sure it is surjective by enlarging Ω_{kG}^{n_{i}}(k) by a suitable projective if necessary. Write L_{ζ_{i}} for the corresponding kernel. Then we can form a pushout diagram for the beginning of a resolution of k by finitely generated projective kG-modules as follows:

0  0
↓  ↓
L_{ζ_{i}}  =  L_{ζ_{i}}
↓  ↓
0 ⟶ Ω_{kG}^{n_{i}}(k) ⟶ P_{n_{i}−1} ⟶ ⋯ ⟶ P_{0} ⟶ k ⟶ 0
↓  ↓  ∥  ∥
0 ⟶  k  ⟶ P_{n_{i}−1}/L_{ζ_{i}} ⟶ ⋯ ⟶ P_{0} ⟶ k ⟶ 0
↓  ↓
0  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_{ζ_{i}} whose homology consists of a copy of k in degrees zero and n_{i} − 1 with trivial G-action. We then form the complex B = C_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} C_{ζ_{r}}, which is a complex of projective modules by the theory of varieties [2]. We inflate to regard this as a complex of kΓ-modules.

Next, we take a finite resolution of k by projective kN-modules, and tensor induce to give a complex Q_{∗} of kΓ-modules. Note that, even if we started off with finitely generated modules, we will have lost this property after tensor induction. However, Q_{∗} does have finite length, and is exact except in degree zero, where its homology is k. Let B′_{∗} = B_{∗} ⊗ Q_{∗}. Then the restriction of B′_{∗} to any finite subgroup is a complex of projective modules, so B′_{∗} consists of modules of type F P. It follows from Lemma 2.3 that B′_{∗} is in fact a complex of projective modules. Furthermore, its homology consists of a finite direct sum of copies of k with trivial Γ-action. So by Lemma 2.1, B′_{∗} is chain homotopy equivalent to a complex B̂_{∗} with each B̂_{i} a finitely generated projective kΓ-module.

Remark 2.5. We note that the argument of the proposition applies to show that for any complex C_{∗} of finitely generated projective kG modules the complex C′_{∗} = C_{∗} ⊗Q_{∗} is a complex of projective kΓ-modules, and chain equivalent to a complex Ĉ_{∗} of finitely generated projective kΓ-modules.

3. The algebraic duality theorem.

In this section we consider a virtual duality group Γ. It is necessarily of type VFP, and we denote by B̂_{∗} and Q_{∗} complexes resulting from the construction described in the proof of Proposition 2.4.

Proposition 3.1. If Γ is a virtual duality group over k with dualizing module I and dualizing degree n, then

H^{j+n}(Hom_{kΓ}(B̂_{∗}, kΓ)) ≅ Hom_{k}(H_{j}(B̂), I).

Proof: Let P_{∗} be a projective resolution of k as a kΓ-module, and compare the two spectral sequences of the double complex Hom_{kΓ}(P_{∗} ⊗ B̂_{∗}, kΓ). If we take homology using the P_{∗} differential first, the E_{1} page consists of Hom_{kΓ}(B̂_{∗}, kΓ) along the edge, and the E_{2} page consists of H^{∗}(Hom_{kΓ}(B̂_{∗}, kΓ)). On the other hand, if take homology using the differential coming from B̂_{∗} first, then the E_{1} page is

Hom_{kΓ}(H_{∗}(B̂), Hom_{k}(P_{∗}, kΓ)) ≅ Hom_{k}(H_{∗}(B̂), Hom_{kΓ}(P_{∗}, kΓ))

(since the modules in B̂_{∗} are projective and the Γ-action on H_{∗}(B̂) is trivial) and the E_{2} page consists of Hom_{k}(H_{∗}(B̂), 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 Hom_{kΓ}(Q_{∗}, kΓ) (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 kΓ-modules

δ : I ⊗_{k} Q_{∗} ⟶ Σ^{n}Hom_{kΓ}(Q_{∗}, kΓ)

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_{∗} ⟶ Σ^{n}Hom_{kN}(Q_{∗}, kN).

Taking the adjoint of this map gives a map of chain complexes of kN-modules

I ⊗_{k} Q_{∗} ⊗_{k} Q_{∗} ⟶ Σ^{n}kN.

By Frobenius reciprocity, there is a corresponding map of chain complexes of kΓ-modules

I ⊗_{k} Q_{∗} ⊗_{k} Q_{∗} ⟶ Σ^{n}kΓ,

whose adjoint

I ⊗_{k} Q_{∗} ⟶ Σ^{n}Hom_{kΓ}(Q_{∗}, kΓ)

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 kΓ-modules

δ̂: I ⊗_{k} B̂_{∗} ⟶ Σ^{s}Hom_{kΓ}(B̂_{∗}, kΓ)

where s = n + ∑_{i=1}^{r}(n_{i} − 1).

Proof: It is shown in Section 5 of [2] that there are maps of chain complexes of k[Γ/N]-modules C_{ζ_{i}} ⟶ Σ^{n_{i}−1}Hom_{k}(C_{ζ_{i}}, k) inducing an isomorphism in homology. Actually, this is only proved there in the case of a field of coefficients, but it holds more generally. Hence we have a homology isomorphism

C_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} C_{ζ_{r}} ⟶ Σ^{Σ_{i}(n_{i}−1)}Hom_{k}(C_{ζ_{1}}, k) ⊗_{k} ⋯ ⊗_{k} Hom_{k}(C_{ζ_{r}}, k).

Tensor this with I ⊗ Q_{∗} and follow it with

δ′ = 1 ⊗ δ : Hom_{k}(C_{ζ_{1}}, k) ⊗_{k} ⋯ ⊗_{k} Hom_{k}(C_{ζ_{r}}, k) ⊗_{k} Σ^{−n}I ⊗ Q_{∗} ⟶ Hom_{k}(C_{ζ_{1}}, k) ⊗_{k} ⋯ ⊗_{k} Hom_{k}(C_{ζ_{r}}, k) ⊗_{k} Hom_{kΓ}(Q_{∗}, kΓ)

where δ is the map of Lemma 3.2.

Next, the natural map

δ′′ : Hom_{k}(C_{ζ_{1}}, k) ⊗_{k} ⋯ ⊗_{k} Hom_{k}(C_{ζ_{r}}, k) ⊗_{k} Hom_{kΓ}(Q_{∗}, kΓ) ⟶ Hom_{kΓ}(C_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} C_{ζ_{r}} ⊗_{k} Q_{∗}, kΓ) is an isomorphism of chain complexes of kΓ-modules. To see this, it is enough to check that it is an isomorphism of abelian groups which preserves the Γ-action. To check that it is an isomorphism of abelian groups, it suffices to notice that each C_{ζ_{i}} is a finite chain complex of finitely generated free abelian groups. The Γ-action is easy to check.

The codomain of δ′′ is Hom_{kΓ}(B′_{∗}, kΓ) in the notation of 2.4; since we have a chain homotopy equivalence B̂ ⟶ B′ of complexes of projectives, the codomain of δ′′ 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 kΓ-module, then there is a chain homotopy equivalence of chain complexes of k-modules

(M ⊗ I) ⊗_{kΓ} B̂_{∗} ⟶ Σ^{s}Hom_{kΓ}(B̂_{∗}, M).

Proof: This follows by applying M ⊗_{kΓ} − to the map given in the proposition.

Corollary 3.5. If k is a field and Γ is an oriented virtual PD group, then the Poincaré series

f(t) = ∑_{i≥0} t^{i} dim_{k} H_{i}(B̂_{∗})

satisfies the functional equation

f(t) = t^{s}f(t^{−1}),

where s = n + ∑_{i=1}^{r}(n_{i} − 1).

Proof: The space H_{i}(B̂_{∗}) is the k-vector space dual of H_{−i}Hom_{kΓ}(B̂_{∗}, k), which by the previous corollary (with M = k) is isomorphic to H_{s−i}(B̂_{∗}). So the coefficient of t^{i} in f(t) is equal to the coefficient of t^{s−i}.

Theorem 3.6. If Γ is a virtual duality group over k and M is an kΓ-module, and s = n + ∑_{i=1}^{r}(n_{i} − 1), then there is a spectral sequence

E_{2}^{pq} = Ext_{kΓ}^{p}(H_{q}(B̂_{∗}), M) ⟹ H_{s−p−q}((M ⊗_{k} I) ⊗_{kΓ} B̂_{∗}).

Proof: Consider the double complex

Hom_{kΓ}(P_{∗} ⊗_{k} B̂_{∗}, M)

where P_{∗} is a projective resolution of k as an kΓ-module. In the spectral sequence in which we take homology using the P_{∗} differential first, the E_{1} page consists of Hom_{kΓ}(B̂_{∗}, M) along the edge, and the E_{2} page consists of H^{∗}(Hom_{kΓ}(B̂_{∗}, M)). By Corollary 3.4, this is isomorphic to H_{s−∗}((M ⊗_{k} I) ⊗_{kΓ} B̂_{∗}). It follows that the other spectral sequence, in which we take homology using the B̂_{∗} differential first, converges to the same thing. The E_{1}^{p,q} term is Hom_{kΓ}(P_{p} ⊗_{k} H_{q}(B̂_{∗}), M), and so the E_{2}^{p,q} term is Ext_{kΓ}^{p}(H_{q}(B̂_{∗}), M) as required.

The differentials in this spectral sequence can be described in the same way as in [2]. In particular, there are differentials corresponding to multiplication by the inflations of the elements ζ_{i} used to construct the complex B̂_{∗}. So just as in [2], we obtain the following statement about Poincaré series:

Theorem 3.7. Suppose that Γ is an oriented virtual PD group of dimension n, and k is a field. If H^{∗}(Γ, k) is Cohen–Macaulay, then the Poincaré series

p_{Γ}(t) = ∑_{i≥0} t^{i} dim_{k} H^{i}(Γ; k)

satisfies the functional equation

p_{Γ}(1/t) = t^{−n}(−t)^{r}p_{Γ}(t).

Here, r = r_{p}(Γ) is the maximal rank of a finite elementary abelian p-subgroup of Γ 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 ζ_{i} to be as small as possible, namely r. Since we also have I = k, it becomes:

E_{2}^{p,q} = Ext_{kΓ}^{p}(H_{q}(B̂_{∗}), k) ⟹ H_{s−p−q}(k ⊗_{kΓ} B̂_{∗}).

Since the cohomology is Cohen–Macaulay, the ζ_{i} form a regular sequence in H^{∗}(Γ, k), and so the E_{∞} page is H^{∗}(Γ, k)/(ζ_{1}, …, ζ_{r}) concentrated along the bottom row. The Poincaré series of this quotient therefore satisfies the functional equation of Corollary 3.5. Since

p_{Γ}(t) = f(t)/ ∏_{i=1}^{r}(1 − t^{n_{i}}),

the required functional equation for p_{Γ}(t) follows.

4. The local cohomology theorem.

In this section we prove the local cohomology theorem stated in the introduction.

Theorem 4.1. If Γ is a virtual duality group of virtual dimension n and with dualizing module I, then for any Γ module M there is a spectral sequence

E_{2}^{∗,∗} = H_{J}^{∗}(H^{∗}(Γ; M)) ⟹ Σ^{n}H_{∗}(Γ; I ⊗ M)

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 finite 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′ has to be replaced by the complex B̂ of finitely generated projectives.

First we begin by recalling the construction from [13], and to begin with we work entirely with kG-modules, where G = Γ/N. We view the complex

B = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}}

of kG modules as graded over ℤ^{r}, and non-zero in the cuboidal box with vertices at (ϵ_{1}(n_{1} − 1), …, ϵ_{r}(n_{r} − 1)) with ϵ_{i} = 0 or 1. Of course we may splice together two copies of C_{ζ_{i}} since H_{0}(C_{ζ_{i}}) = k = H_{n−1}(C_{ζ_{i}}), and thus form a complex C_{ζ_{i}^{2}}. Tensoring with the other complexes C_{ζ_{j}} we may view this as stacking boxes in ℤ^{r}. Notice that there is a cofibre sequence

Σ^{n−1}C_{ζ_{i}} ⟶^{ζ_{i}} C_{ζ_{i}} ⟶ C_{ζ_{i}^{2}},

and similarly for boxes.

Next we let R denote the complex obtained by stacking boxes to fill the non-negative orthant. More generally for any subset σ ⊆ {1, 2, … r} we let R[σ] be the complex obtained by stacking boxes so as to fill the region with gradings (i_{1}, …, i_{r}) with i_{j} ≥ 0 if j ∉ σ. In particular we let S denote the complex R[{1, 2, …, r}] in which the whole of ℤ^{r} has been filled with boxes. Now form the dual stable Koszul complex

L_{•} = ( ⨁_{|σ|=r} R[σ] ⟶ ⨁_{|σ|=r−1} R[σ] ⟶ ⋯ ⟶ ⨁_{|σ|=1} R[σ] ⟶ ⨁_{|σ|=0} R[σ])

in which the differentials are alternating sums of projections. It is clear that at each point (i_{1}, …, i_{r}) of ℤ^{r} this complex is obtained by tensoring the reduced chain complex of the simplex σ = {α | i_{α} < 0} 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 fill the negative orthant, we have

H_{i}(L_{•}) = \{ \begin{matrix}R^{!} & if i = r \\ 0 & otherwise. \end{matrix}

We now want to repeat the construction of Section 2. We begin by regarding the discussion so far as Γ-equivariant by inflation, and then form the complex L′_{•} = L_{•} ⊗ Q_{∗} of projective modules. We then replace it by a chain equivalent complex L̂_{•} of finitely generated projectives. However there are three reasons for care: firstly, the complexes R[σ] are not bounded below, secondly, we want to carry the Koszul structure maps along in the construction and, finally, we want to retain the periodicities given by multiplication by ζ_{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[i_{1}, …, i_{r}] denotes a copy of the box complex with its bottom vertex at (i_{1}n_{1}, …, i_{r}n_{r}) we filter the whole page filled with boxes by i_{1} + i_{2} + ⋯ + i_{r}. Thus we let S_{j} be the quotient of S = R[{1, 2, …, r}] in which the subcomplex of all boxes with i_{1} + ⋯ + i_{r} < j has been factored out, and we let S_{j}^{n} = Ker(S_{j} ⟶ S_{n+1}). Thus S_{j}^{n}/S_{j}^{n−1} is the sum of boxes with total coordinate n, and we have a cofibre sequence

⨁_{i_{1}+···+i_{r}=n} Σ^{−1}B[i_{1}, …, i_{r}] ⟶ S_{j}^{n−1} ⟶ S_{j}^{n}.

We regard this as a method for constructing S_{0}^{1}, S_{0}^{2}, S_{0}^{3}, . . .. in succession: we may construct S_{0} = S_{0}^{∞} by passage to the direct limit. At each stage we may form the approximation by choosing an arrow in the diagram

Σ^{−1} ⨁_{i_{1}+···+i_{r}=n+1} B̂[i_{1}, …, i_{r}] ⋯ > Ŝ_{j}^{n}
↓  ↓
Σ^{−1} ⨁_{i_{1}+···+i_{r}=n+1} B[i_{1}, …, i_{r}] ⟶ S_{j}^{n} ⟶ S_{j}^{n+1}

(as we may do since B̂ is projective), and taking the mapping cone in the top row. Again we may pass to direct limits and obtain Ŝ_{0}, and again the limit is achieved in each degree, since B̂ is bounded below.

On the other hand we know that S_{−s} is simply a desuspension of S_{0}, and S is the inverse limit of S_{−s} under the quotient maps. Now, because the above construction of Ŝ_{0} also gave the filtration, we have counterparts of the quotient maps, still with sums of boxes as the kernels. We may therefore form Ŝ = lim_{←}_{s} Ŝ_{−s}, and since B̂ 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[σ] as quotients, and we may form the approximation L̂_{•} simply by taking these quotients and making the differentials from alternating sums of the quotient maps.

We are now ready to give the proof. Consider the double complex

Hom_{kΓ}(L̂_{•}, M);

we compare the two spectral sequences for calculating its cohomology. If we take homology in the Koszul direction first, then because of the projectivity of boundaries and cycles we obtain E_{1} term

Hom_{kΓ}(H_{∗}L̂, M).

Now H_{i}(L) = 0 for i < r and H_{r}(L) = R^{!}. Furthermore H_{r}(L_{•}) is the inverse limit of suitable box quotients of the complexes S_{j}. Therefore H_{i}(L̂_{•}) = 0 for i < r and H_{r}(L̂_{•}) is the corresponding inverse limit of the corresponding quotients of Ŝ_{j}. It is reasonable to call this limit R̂^{!}, and in particular it is a complex of finitely generated projectives, it is bounded above, has homology equivalent to R^{!} and has a filtration with quotients which are finite sums of copies of B̂. Now by naturality of 3.3 for box quotients we find

Hom_{kΓ}(R̂^{!}, M) = Σ^{n}(M ⊗ I) ⊗_{kΓ} R̂.

Its homology is therefore Σ^{n}H_{∗}(Γ; M ⊗ I).

To understand the second spectral sequence, let ζ_{σ} = ∏_{i∈σ} ζ_{i}. Now, note that because limits are achieved in each degree, we may calculate

H^{∗}(Hom_{kΓ}(R̂[σ], M)) = H^{∗}(Hom_{kΓ}(lim_{←}(R̂, ζ_{σ}^{∗}), M))
= lim_{→} (H^{∗}(Hom_{kΓ}(R̂, M)), ζ_{σ})
= H^{∗}(Γ; M)[1/ζ_{σ}].

Thus if we take projective cohomology of Hom_{kΓ}(L̂_{•}, M) first, then the E_{1} term is the stable Koszul complex

H^{∗}(Γ; M) ⟶ ⨁_{|σ|=1} H^{∗}(Γ; M)[1/ζ_{σ}] ⟶ ⋯ ⟶ ⨁_{|σ|=r} H^{∗}(Γ; M)[1/ζ_{σ}]

for calculating the local cohomology of H^{∗}(Γ; M) with respect to the ideal generated by the inflations of the elements ζ_{1}, …, ζ_{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 Γ 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 Γ 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^{∗}(BΓ) of elements with positive cohomological degree then there is a spectral sequence

E_{2}^{s,t} = H_{J}^{s,t}(H^{∗}(BΓ)) ⟹ H_{n−s−t}(BΓ; I)

with differentials d_{r} : E_{r}^{s,t} ⟶ E_{r}^{s+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 Λ 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

H_{J}^{∗}(H^{∗}(BΛ)) ⟹ Σ^{−d}H_{∗}(BΛ).

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 Λ of dimension d as the quotient. This takes the form of a spectral sequence

H_{J}^{∗}(H^{∗}(BΓ)) ⟹ Σ^{n−d}H_{∗}(BΓ; I ⊗ ϵ)

where I and ϵ are both representations of the component group; I is analogous to the dualizing module, and ϵ is the action of Γ on H^{d}(S^{Ad(Λ)}; k). However we do not know of any particular interest in this case.

6. Topological proof of the theorem.

We begin the discussion without assumption on Γ, introducing restrictions as they become necessary.

We suppose given an extension

1 ⟶ N ⟶ Γ ⟶ G ⟶ 1

with G finite. The subgroup N will be torsion free and have other good properties. We use the usual trick for constructing a classifying space BΓ; we assume that Γ acts on a space X so that the restriction of the action to N is free. Letting Γ act on EG via the quotient map, we have

EΓ = EG × X,

and thus

BΓ = EG ×_{G} (X/N).

The idea is that, when N is sufficiently nice, the geometry of the G-space X/N will be rather simple, and we can understand BΓ by doing a little G-equivariant homotopy theory on X/N.

Now

H^{∗}(BΓ) = 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 defined by

b_{G}^{∗}(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 coefficients, so that

H^{∗}(BΓ) = b_{G}^{∗}(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 first assumption on X/N: we ask that it be finite dimensional. The rings which concern us are H^{∗}(BΓ) and H^{∗}(BG), and inflation gives a ring homomorphism H^{∗}(BG) ⟶ H^{∗}(BΓ), which allows us to regard any H^{∗}(BΓ)-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(Γ). Notice that the ideal of H^{∗}(BΓ) generated by the image of J(G) in H^{∗}(BΓ) has radical J(Γ) because X/N is finite dimensional. Thus the J(Γ) local cohomology of any H^{∗}(BΓ)-module M is equal to its J(G)-local cohomology:

H_{J(Γ)}^{∗}(M) = H_{J(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 E_{∞}-ring spectrum, by equivariant infinite 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: firstly it extends to the case of compact Lie groups, and secondly there is no published reference for equivariant infinite 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 Γ_{J}m, and the construction makes it obvious that there is a spectral sequence

E_{2}^{∗,∗} = H_{J}^{∗}(m_{∗}^{G}) ⟹ π_{∗}^{G}(Γ_{J}m).

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 ≃ Γ_{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

Γ_{J}m ≃ m ∧ EG_{+}.

Hence there is a spectral sequence

E_{2}^{∗,∗} = H_{J}^{∗}(m_{∗}^{G}) ⟹ π_{∗}^{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^{∗}(BΓ), Theorem 6.1 gives a spectral sequence whose E_{2}-term is the J-local cohomology of H^{∗}(BΓ). We now need to relate the equivariant homotopy of EG_{+} ∧ F (X/N_{+} ∧ EG_{+}, H) to H_{∗}(BΓ). 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 finite dimensional there is a spectral sequence

H_{J}^{∗}(H^{∗}(BΓ)) ⟹ π_{∗}^{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 finite index.

We still have to relate π_{∗}^{G}(EG_{+}∧F (X/N_{+}, H)) to the homology of BΓ; 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 finite 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

π_{∗}^{G}(EG_{+} ∧ F (X/N_{+}, H))) = π_{∗}^{G}(EG_{+} ∧ DX/N_{+} ∧ H) = H̃_{∗}(EG_{+} ∧_{G} DX/N_{+}).

Finally, since X/N_{+} is finite, 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, ∞), V × {0}) for some representation V of G. Now, if ν is the normal bundle of X/N in V ⊕ 1 and ν′ is the normal bundle of ∂X/N in V, Atiyah duality states

Σ^{V ⊕1}DX/N_{+} ≃ (X/N)^{ν}/(∂X/N)^{ν′},

where superscripts denote Thom spaces. In particular, if X/N has empty boundary,

Σ^{V ⊕1}DX/N_{+} ≃ (X/N)^{ν}.

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.

Suppose then that Γ is arithmetic in a connected semisimple algebraic group λ over ℚ; then if K is a maximal compact subgroup of λ(ℝ) then the symmetric space X′ = λ(ℝ)/K is diffeomorphic to ℝ^{d} for some d. Borel and Serre show that X′ is the interior of a contractible Γ-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 λ. The case with empty boundary (i.e. l = 0) is covered by the usual convention that the (−1)-sphere is the empty space. Thus Γ is a virtual duality group with virtual dimension d − l and dualizing module I = H^{d−l}(Γ; ℤΓ) = H_{c}^{d−l}(X) ≅ H_{l}(X, ∂X) = H̃_{l−1}(∂X). As usual, the reduced homology of the empty space is ℤ in 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 ℤ is ξ we should more precisely say

I ≅ H̃_{l−1}(∂X) ⊗ ξ.

Note that this is the dualizing module for any group Γ arithmetic in λ, so that in particular Γ is a PD group if and only if the boundary ∂X is empty, if and only if λ is of ℚ-rank 0. In that case Γ is cocompact in λ(ℝ) and the virtual dimension of Γ is d.

Example 6.3. Let Γ be the infinite dihedral group. This is an arithmetic subgroup of SL_{2}, and occurs as an extension

1 ⟶ ℤ ⟶ Γ ⟶ C_{2} ⟶ 1.

We may take X = ℝ and X/N = S^{1}, and we find I = ℤ̃. This shows that the tangent bundle need not be orientable.

In fact H^{∗}(Γ) = 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 H_{J}^{0}(H^{∗}(Γ)) ≅ ℤ and H_{J}^{1}(H^{∗}(Γ)) is additively ℤ/2 in codegree 0 and ℤ/2 ⊕ ℤ/2 in each negative even degree. The local cohomology spectral sequence necessarily collapses to calculate ΣH_{∗}(Γ; ℤ̃) as required.

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 Hérault 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 γ ∈ λ(ℚ) acts smoothly on X (essentially by [7, 5.6(2)]), and since the action preserves corners and their indices, composition with γ would give a second smooth structure on the manifold with boundary. Therefore γ acts smoothly on the manifold with boundary. It follows that we may replace X by a smooth manifold with boundary, and the action of Γ 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.

H̃_{∗}(EG_{+} ∧_{G} DX/N_{+}) = Σ^{d−l}H_{∗}(BΓ; I)

Proof: By Atiyah duality we have the cofibre sequence

(∂X/N)^{ν′} ⟶ (X/N)^{ν} ⟶ Σ^{V ⊕1}DX/N_{+}.

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 tangent bundles need not be orientable, but the same representation ξ 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, H̃_{∗}(EG_{+} ∧_{G} DX/N_{+}) = Σ^{v+1}H̃_{∗}(EG_{+} ∧ Σ^{V ⊕1}DX/N_{+}). Now, both ν and ν′ have fibre dimension v + 1 − d, and so the Thom isomorphisms give

H̃_{∗}(EG_{+} ∧_{G} (∂X/N)^{ν′}) ≅ Σ^{d−v−1}H_{∗}(EG_{+} ×_{G} ∂X/N; ξ) = Σ^{d−v−1}H_{∗}(EΓ ×_{Γ} ∂X; ξ)

and

H̃_{∗}(EG_{+} ∧_{G} (X/N)^{ν}) ≅ Σ^{d−v−1}H_{∗}(EG_{+} ×_{G} X/N; ξ) = Σ^{d−v−1}H_{∗}(EΓ ×_{Γ} X; ξ).

Finally the Serre spectral sequence of the fibration ∂X ⟶ EΓ ×_{Γ} ∂X ⟶ BΓ, 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_{∗}(EΓ ×_{Γ} X, EΓ ×_{Γ} ∂X; ξ) = Σ^{−l}H_{∗}(BΓ; I)

as required. Adding up the relevent suspensions we reach d − l = (v + 1) + (d − v − 1) − l as asserted.

Part D: Discussion

7. Examples

Example 7.1. Although the example of SL_{2}(ℤ) may seem trivial, there are features to note. Since SL_{2}(ℤ) = C_{4} ∗_{C_{2}} C_{6} we easily calculate that H^{∗}(SL_{2}(ℤ)) = ℤ[c]/(12c), and hence

H_{J}^{0}(H^{∗}(SL_{2}(ℤ))) ≅ ℤ

whilst H_{J}^{1}(H^{∗}(SL_{2}(ℤ))) is additively ℤ/12 in each even negative degree. This means that the spectral sequence appears to converge to H_{∗}(SL_{2}(ℤ)) without suspension. However, we have proved it actually converges to ΣH_{∗}(SL_{2}(ℤ); I). In this case the components of ∂X correspond to the rational cusps P^{1}(ℚ) of the action of SL_{2}(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_{∗}(SL_{2}(ℤ); M) = 0. It would be interesting to have a geometric construction of such a module M.

Example 7.2. The integral cohomology of SL_{3}(ℤ) has been calculated by Soulé [20]. He shows that the cohomology only has 2 and 3 torsion, that H^{∗}(SL_{3}(ℤ))_{(3)} = ℤ_{(3)}[a_{4}]/(3a_{4})∨ℤ_{(3)}[b_{4}]/(3b_{4}), and H^{∗}(SL_{3}(ℤ))_{(2)} = ℤ_{(2)}[c_{3}, d_{3}, c_{4}, d_{4}, c_{5}, c_{6}, d_{6}]/I, where I declares that c_{4} and d_{4} are of additive order 4, all the other generators are of additive order 2; in addition

c_{3}d_{6} = d_{4}d_{6} = c_{5}d_{6} = c_{6}d_{6} = c_{3}d_{3} = d_{3}d_{4} = d_{3}c_{5} = d_{3}c_{6} = 0,

d_{6}(d_{6} + d_{3}^{2}) = c_{4}d_{4} + c_{3}c_{5} = c_{4}(c_{6} + c_{3}^{2}) = c_{4}c_{6} + c_{5}^{2} = 0

and

c_{3}c_{6} + d_{4}c_{5} = c_{4}d_{4}^{2} + c_{6}^{2} = c_{5}c_{6} + c_{5}c_{3}^{2} = 0.

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

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Department of Mathematics, University of Georgia, Athens, GA 30602, USA.

E-mail address: djb@byrd.math.uga.edu

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

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