FUNCTIONAL EQUATIONS FOR POINCARÉ SERIES IN GROUP COHOMOLOGY

David J. Benson and Jon F. Carlson^{1}

Mathematical Institute Department of Mathematics 24-29 St. Giles University of Georgia Oxford OX1 3LB Athens, GA 30602

1. Introduction.

Suppose that k is a field of prime characteristic p and that G is a finite group. The Poincaré series for the cohomology ring H^{∗}(G, k) is p_{G}(t) = ∑_{i≥0} Dim H^{i}(G, k)t^{i}. A classical result says that p_{G}(t) is a rational function in t. In [3] we showed that when H^{∗}(G, k) is Cohen-Macaulay then p_{G}(t), as a rational function, satisfies the equation

(1.1)  p_{G} (\frac{1}{t}) = (−t)^{r}p_{G}(t)

where r is the p-rank of G. At the time when that paper was written we speculated that the converse might also be true: that is, the functional equation (1.1) might imply that the ring is Cohen-Macaulay. Shortly thereafter we succeeded in proving the converse in the case that the depth of H^{∗}(G, k) (i.e., the length of the longest regular sequence) is at least r − 1. It turns out that this result is best possible, at least for the general problem. One of Rusin’s calculations [8] shows that for p = 2 there is a group of order 32 whose cohomology ring is not Cohen-Macaulay but has Poincaré series satisfying (1.1).

In this paper is presented what we hope is the complete story. In Section 2 and 3 we give a proof that Cohen-Macaulay is implied when the Poincaré series satisfies (1.1) and the depth is at least r − 1. We show in the case of depth r − 1, that

p_{G} (\frac{1}{t}) − (−t)^{r}p_{G}(t) = \frac{(−t)^{r−1}(1 + t)p(K, t)}{∏_{i=1}^{r−1}(1 − t^{n_{i}})}

1 Partially supported by a grant from NSF.

Typeset by AMS-TEX

where K is the stable kernel of multiplication by ζ_{r} and its Poincaré series p(K, t) satisfies

t^{c}p(K, \frac{1}{t}) = p(K, t)

where c = ∑_{i=1}^{r−1}(n_{i} − 1). The result is actually a corollary of a more general theorem on the Poincaré series of a certain spectral sequence. Included, for the Cohen-Macaulay case, is a proof, which is uncluttered by the extra agenda of [3], that (1.1) holds. In Section 4 we expose the example of Rusin and also list without proof some facts about the ring structure of H^{∗}(G, k) which are required when the functional equation holds. In total, the results present a picture, however fuzzy, of some of the ways in which the ring structure is revealed by the form of the Poincaré series. It would seem that there are several mysteries still worthy of investigation here.

Part of this paper was written while the second author was visiting the University of Warwick. He would like to thank the people at the Mathematics Institute for their hospitality and the S.E.R.C. for financial support.

2. A sequence of spectral sequences.

We assume throughout that G is a finite group and that r is the p-rank of G for p a prime. Let k be a field of characteristic p. Then the cohomology ring H^{∗}(G, k) has Krull dimension r, and the maximal ideal spectrum, V_{G}(k), of H^{∗}(G, k) is an affine variety of dimension r. By a homogeneous set of parameters for H^{∗}(G, k), we mean a set ζ_{1}, …, ζ_{r} of homogeneous elements of H^{∗}(G, k) such that H^{∗}(G, k) is a finitely generated module over the polynomial subring k[ζ_{1}, …, ζ_{r}] ⊆ H^{∗}(G, k). Each ζ_{i} is homogeneous in the sense that it has a degree, n_{i} = deg ζ_{i}, i.e. ζ_{i} ∈ H^{n_{i}}(G, k). The cohomology ring H^{∗}(G, k) is finitely generated over k[ζ_{1}, …, ζ_{r}] if and only if ⋂_{i=1}^{r} V_{G}(ζ_{i}) = {0} where V_{G}(ζ_{i}) is the subvariety of V_{G}(k) corresponding to the principal ideal generated by ζ_{i}. See [1] or [7] for a general reference on this material.

Let ζ_{1}, …, ζ_{r} be a homogeneous set of parameters for H^{∗}(G, k). Let (P, ∂) be a kG-projective resolution of k. Then for each i = 1, …, r we have a commutative

FUNCTIONAL EQUATIONS FOR POINCARÉ SERIES IN GROUP COHOMOLOGY 3

diagram

(2.1) \begin{aligned}… & ⟶ & P_{n_{i}} & ⟶ & P_{n_{i}−1} & ⟶ & P_{n_{i}−2} & ⟶ & … & ⟶ & P_{1} & ⟶ & P_{0} & ⟶^{ε} & k & ⟶ & 0 \\ & & ↓ζ̂_{i} & & ↓ & & ∥ & & & & ∥ & & ∥ & & ∥ \\ 0 & ⟶ & k & ⟶ & L_{i} & ⟶ & P_{n_{i}−2} & ⟶ & … & ⟶ & P_{1} & ⟶ & P_{0} & ⟶^{ε} & k & ⟶ & 0 \\ & & & & ∥ & & ∥ & & & & ∥ & & ∥ \\ C^{(i)} : & & 0 & ⟶ & C_{n_{i}−1}^{(i)} & ⟶ & C_{n_{i}−2}^{(i)} & ⟶ & … & ⟶ & C_{1}^{(i)} & ⟶ & C_{0}^{(i)} & ⟶ & 0 \end{aligned}

where ζ̂_{i} is a cocycle representing the cohomology element ζ_{i}, the middle row is the pushout along ζ̂_{i} of the upper row, and the complex C^{(i)} in the bottom row is defined by the diagram. In particular, C^{(i)} is constructed by truncating the ends of the exact sequence in the middle row. Thus it is clear that H_{j}(C^{(i)}) ≅ k if j = 0, n_{i} − 1 and H_{j}(C^{(i)}) = 0, otherwise. The module L_{i} ≅ Ω^{−1}(L_{ζ_{i}}) in the notation of [2]. It should be noted that its variety V_{G}(L_{i}) ≅ V_{G}(ζ_{i}), the variety of the ideal generated by ζ_{i}.

Now let D^{(0)} = k be the complex with D_{0}^{(0)} = k in degree 0 and with 0-modules, D_{j}^{(0)} = 0, in all other degrees. Let D^{(1)} = C^{(1)} and for i > 1, let D^{(i)} = C^{(1)} ⊗⋯⊗ C^{(i)}. The tensor product is over k, and D^{(i)} is made into a complex of kG-modules by letting G act diagonally as usual. The homology of D^{(i)} is given by the Künneth formula

H_{m}(D^{(i)}) = ∑_{j_{1}+···+j_{i}=n_{i}} H_{j_{1}}(C^{(1)}) ⊗ ⋯ ⊗ H_{j_{i}}(C^{(i)}).

So H_{m}(D^{(i)}) is either a sum of copies of k or is zero. Now

D_{t}^{(i)} = ∑_{j_{1}+···+j_{i}=t} C_{j_{1}}^{(1)} ⊗ ⋯ ⊗ C_{j_{i}}^{(i)}

is a projective module if t ≠ ∑_{j=1}^{i}(n_{j} −1). This is because whenever M and N are kGmodules and one of them is projective then the tensor product M⊗_{k}N is a projective kG-module. If t = ∑_{j=1}^{i}(n_{j} − 1) then D_{t}^{(i)} = L_{1} ⊗ ⋯ ⊗ L_{i} is projective if and only if

i = r. The last statement is a consequence of the fact that V_{G}(D_{t}^{(i)}) = ⋂_{j=1}^{i} V_{G}(ζ_{j}) (see [2] or [3]).

Again let (P, ∂) be a kG-projective resolution of k. We define

Ext_{kG}^{m}(D^{(i)}, k) = H^{n}(Hom_{kG}(P ⊗ D^{(i)}, k)).

We have that Lemma 2.2. For m > ∑_{j=1}^{i}(n_{j} − 1) = b_{i},

Ext_{kG}^{m}(D^{(i)}, k) = Ext_{kG}^{m−b_{i}}(L_{1} ⊗ ⋯ ⊗ L_{i}, k).

Proof. Associated to the double complex P ⊗ D^{(i)} is a spectral sequence whose E_{0} term is

E_{0}^{s,t} = Hom_{kG}(P_{t} ⊗ D_{s}^{(i)}, k).

To get the E_{1} page we take the cohomology with respect to the coboundary induced by ∂ ⊗ 1 on P ⊗ D^{(i)}. For fixed s, P_{∗} ⊗ D_{s}^{(i)} is a projective resolution of D_{s}^{(i)}. Therefore the cohomology is E_{1}^{s,t} = Ext_{kG}^{t}(D_{s}^{(i)}, k) and E_{1}^{s,t} = 0 unless either t = 0 or s = b_{i}. This is because D_{s}^{(i)} is projective if s ≠ b_{i}. The d_{1}-differential takes E_{1}^{s,t} to E_{1}^{s+1,t} is the zero map if t > 0. Likewise, for any u > 1, the E_{u} page of the spectral sequence has E_{u}^{s,t} = 0 except when either t = 0 and 0 ≤ s ≤ b_{i} or s = b_{i}. Hence the d_{u} differential which takes E_{u}^{s,t} to E_{u}^{s+u,t−u+1} is always zero. It follows from that for m > b_{i}, Ext_{kG}^{m}(D^{(i)}, k) = E_{∞}^{b_{i},m−b_{i}} = E_{1}^{b_{i},m−b_{i}} ≅ Ext_{kG}^{m−b_{i}}(D_{b_{i}}^{(i)}, k) as desired.

From another direction we can, similarly, form a whole sequence of spectral sequences E_{∗}^{∗,∗}(i) i = 1, …, r. We use the fact that D^{(i)} = D^{(i−1)} ⊗ C^{(i)}. For any i the E_{0} term is given by

E_{0}^{s,t}(i) = Hom_{kG}((P ⊗ D^{(i−1)})_{s} ⊗ C_{t}^{(i)}, k).

So we may notice that the total complex for which this is a spectral sequence is Hom_{kG}(P ⊗ D^{(i)}, k). Therefore the spectral sequence will converge to the cohomology of the total complex which is Ext_{kG}^{∗}(D^{(i)}, k). The E_{1} page of the spectral sequence is obtained by taking the cohomology with respect to coboundary map induced by the boundary on C^{(i)}. Because (P ⊗ D^{(i−1)})_{s} is projective we get that

E_{1}^{s,t}(i) = Hom_{kG}((P ⊗ D^{(i−1)})_{s} ⊗ H_{t}(C^{(i)}), k).

Now recall that H_{t}(C^{(i)}) = k if i = 1, n_{i} − 1 and is zero otherwise. So

E_{1}^{s,t}(i) = Hom_{kG}((P ⊗ D^{(i−1)})_{s}, k)

if t = 0 or n_{i} − 1. Now taking the cohomology with respect to the map induced by the boundary on P ⊗ D^{(i−1)}, we get the E_{2} term

E_{2}^{s,t}(i) = Ext_{kG}^{s}(D^{(i−1)} ⊗ H_{t}(C^{(i)}), k).

FUNCTIONAL EQUATIONS FOR POINCARÉ SERIES IN GROUP COHOMOLOGY 5

As with the E_{1} page, the E_{2} page has only two rows with t = 0, n_{i} − 1. Hence all further differentials are zero except for d_{n_{i}}. That is E_{2}^{s,t} = ⋯ = E_{n_{i}}^{s,t} and E_{n_{i}+1}^{s,t} = ⋯ = E_{∞}^{s,t} for all s and t. We need to notice several facts about the spectral sequence.

Lemma 2.3. For each i, Ext_{kG}^{∗}(D^{(i)}, k) is a module over H^{∗}(G, k). Moreover with the natural action of H^{∗}(G, k) on the E_{2} page of the spectral sequence E_{∗}^{∗,∗}(i), the differential d_{n_{i}} is a homomorphism of H^{∗}(G, k)-modules.

Proof. There exists a diagonal approximation μ : P → P ⊗ P which lifts the identity on k and provides the cup product on H^{∗}(G, k). Then the tensor product μ ⊗ 1 : P ⊗ D^{(i)} → (P ⊗ P) ⊗ D^{(i)} ≅ P ⊗ (P ⊗ D^{(i)}) gives a map

Hom_{kG}(P, k) ⊗ Hom_{kG}(P ⊗ D^{(i)}, k) ⟶^{(μ⊗1)^{∗}} Hom(P ⊗ D^{(i)}, k)

which defines the H^{∗}(G, k)-action. The diagonal approximation commutes with the boundary on C^{(i)} and hence the action on the E_{2} page of E(i) is the same as the action of H^{∗}(G, k) on Ext_{kG}^{∗}(D^{(i−1)}, k), the rows of the E_{2} page. The last statement can be deduced in a similar fashion from an analysis of the d_{n_{2}} differential.

Lemma 2.4. The edge homomorphism

θ_{i} : Ext_{kG}^{∗}(D^{(i−1)}, k) ≅ E_{2}^{∗,0}(i) → E_{∞}^{∗,0} ⊆ Ext_{kG}^{∗}(D^{(i)}, k)

is a homomorphism of H^{∗}(G, k)-modules.

Proof. This map is induced by the chain homomorphism C^{(i)} → D^{(0)} ≅ k of degree zero which induces the isomorphism H_{0}(C^{(i)}) ≅ k. This is the map ε in the middle row of (2.1). Clearly it commutes with the diagonal approximation.

Lemma 2.5. In the spectral sequence E(i) the d_{n_{i}} differential

d_{n_{i}} : E_{2}^{∗,n_{i}−1} ≅ Ext_{kG}^{∗}(D^{(i−1)}, k) → E_{2}^{∗,0} ≅ Ext_{kG}^{∗}(D^{(i−1)}, k)

is multiplication by ζ_{i} ∈ H^{∗}(G, k).

Proof. For convenience let n_{i} = n. Because ζ_{i} is not nilpotent, we know that n is even if p is odd. Given η ∈ E_{2}^{m,n−1}, we choose a representative η_{0} ∈ E_{0}^{m,n−1}. Since η transgresses, we have elements x_{j} ∈ E_{0}^{m+j,n−1−j} (1 ≤ j ≤ n − 1), satisfying d_{1}η_{0} = d_{0}x_{1}, d_{1}x_{j} = d_{0}x_{j+1} (1 ≤ j ≤ n − 2) and [d_{1}x_{n−1}] = d_{n}η ∈ E_{n}^{m+n−1,0}.

Let x_{0} be the image of η_{0} under the map induced by P_{n−1} → L_{i}. Since the composite of this map with P_{n} → P_{n−1} factors as

P_{n} ⟶^{η̂} k → L_{i},

we have d_{0}x_{0} = η̂ ⊗ ζ̂ as elements of Hom_{kG}((P ⊗ D)_{m} ⊗ P_{n}, k). Thus the cochain

x = x_{0} − x_{1} + ⋯ − x_{n−1} ∈ Hom_{kG}(((P ⊗ D) ⊗ P)_{m+n−1}, k)

satisfies

dx_{i} = d_{0}x + d_{1}x = d_{0}x_{i_{0}} − d_{1}x_{n−1} = η̂ ⊗ ζ̂ − d_{1}x_{n−1}.

It follows that η̂ ⊗ ζ̂ is cohomologous to d_{1}x_{n−1}, so that d_{n}η = ζη as required.

3. Poincaré Polynomials.

In this section we explore some of the relationships among the Poincaré series for the complexes in Section 2. We rely heavily on the freedom which we are allowed in choosing the system of parameters. In particular, if we make a “generic” choice of the parameters then we get “generic” Poincaré series. The term “generic” for the choice means that each parameter should have large degree and be in a sort of general position relative to a set of associated primes. Then for such a choice the Poincaré series depend only on the degrees of the parameters. We begin with some definitions.

Suppose we are given a set ζ_{1}, …, ζ_{r} of homogeneous parameters for H^{∗}(G, k). We construct the complexes C^{(i)} and D^{(i)} as in the previous section. For each i, let

f_{i}(t) = ∑_{j≥0} Dim (Ext_{kG}^{j}(D^{(i)}, k)) t^{j}

be the Poincaré series for the cohomology of D^{(i)}. notice that D^{(0)} = k and hence f_{0}(t) is the Poincaré series for the cohomology ring of G. That is, f_{0}(t) = ∑_{j≥0} Dim(H^{j}(G, k))t^{j}. At the other end we have D^{(r)} is homotopy equivalent to a Poincaré duality complex of projective kG-modules [3]. The duality is in degree b_{r} = ∑_{i=1}^{r}(n_{i} − 1), for n_{i} = deg(ζ_{i}). Therefore we have the following.

Proposition 3.1. The Poincaré series f_{r}(t) is a polynomial of degree b_{r}, and it satisfies the functional equation

t^{b_{r}}f_{r} (\frac{1}{t}) = f_{r}(t).

FUNCTIONAL EQUATIONS FOR POINCARÉ SERIES IN GROUP COHOMOLOGY 7

Proof. The projectivity of all terms in D^{(r)} is a consequence of the fact that V_{G}(L_{1} ⊗ ⋯ ⊗ L_{r}) = ⋂_{i=1}^{r} V_{G}(ζ_{i}) = {0}. Hence f_{r}(t) is a polynomial by Lemma 2.2. A more detailed analysis of D^{(r)} and a proof of the duality can be found in [3].

Now let A_{i,j} denote the kernel of multiplication by ζ_{i} on Ext_{kG}^{j}(D^{(i−1)}, k). Let

p_{i}(t) = ∑_{j≥0}(DimA_{i,j})t^{j}.

This is the Poincaré series of the annihilator in Ext_{kG}^{∗}(D^{(i−1)}, k) of ζ_{i}. Lemma 3.2. For a reasonable choice of the parameters ζ_{1}, …, ζ_{n}, the Poincaré series p_{1}(t), …, p_{r}(t) are polynomials. Proof. (cf. Section 7 of [5]). First of all, we may note that Ext_{kG}^{∗}(D^{(0)}, k) = H^{∗}(G, k) contains a regular element. Hence if we let ζ_{1} be any regular homogeneous element of H^{∗}(G, k) then p_{1}(t) = 0. Then Ext_{kG}^{∗}(D^{(1)}, k) ≅ H^{∗}(G, k)/ζ_{1}H^{∗}(G, k) by Lemma 2.5. If it has a regular element then we may assume that p_{2}(t) = 0 and continue. In particular, if d is the depth of H^{∗}(G, k) then we may assume that ζ_{1}, …, ζ_{d} is a regular sequence, p_{1}(t) = ⋯ = p_{d}(t) = 0 and Ext_{kG}^{∗}(D^{(d)}, k) ≅ H^{∗}(G, k)/(ζ_{1}, …, ζ_{d}).

For i ≥ d, let

W_{i} = {x ∈ Ext_{kG}^{∗}(D^{(i)}, k) | Dim H^{∗}(G, k) · x < ∞}.

Notice that W_{i} has finite dimension over k, because Ext_{kG}^{∗}(D^{(i)}, k) is a finitely generated module over the noetherian ring H^{∗}(G, k). If y ∈ Ext_{kG}^{∗}(D^{(i)}, k) is not in W_{i}, then the annihilator of y in H^{∗}(G, k) is contained in one of the associated primes of the H^{∗}(G, k)-module Ext_{kG}^{∗}(D^{(i)}, k)/W_{i}. However there are only a finite number of the associated primes and none of them are maximal. Hence any homogeneous element ζ_{i+1} which is not in any of these associated primes must have the property that

{y ∈ Ext_{kG}^{∗}(D^{(i)}, k) | ζ_{i+1}y = 0} ⊆ W_{i}.

Of course ζ_{i+1} must also be chosen so that Dim(V_{G}(ζ_{1})∩⋯∩V_{G}(ζ_{i+1})) = r −i−1. But because the associated primes of Ext_{kG}^{∗}(D^{(i)}, k)/W_{i} are not maximal, there is no problem in finding elements ζ_{i+1} which satisfy this condition.

Finally we notice that if the degree ζ_{i+1} is large enough then ζ_{i+1}w = 0 for all w ∈ W_{i}, A_{i+1,∗} = W_{i} and the polynomial f_{i+1}(t) is precisely the Poincaré polynomial for W_{i}. In particular, if for each i

deg(ζ_{i+1}) > max{deg(w) | w ∈ W_{i}, w ≠ 0, w homogeneous}

then the edge homomorphism

θ_{i+1} : Ext_{kG}^{∗}(D^{(i)}, k) → Ext_{kG}^{∗}(D^{(i+1)}, k)

is injective on W_{i} (see Lemma 2.4). Therefore for i > d, p_{i}(t) ≠ 0, because

ζ_{i} · [θ_{i} ∘ ⋯ ∘ θ_{d+1}(W_{d})] = 0

while θ_{1} ∘ ⋯ ∘ θ_{d+1}(W_{d}) ≠ 0. So we have proved the following

Proposition 3.3. For a reasonable choice of the parameters ζ_{1}, …, ζ_{r} we have that p_{1}(t) = ⋯ = p_{d}(t) = 0 where d is the depth of H^{∗}(G, k) (cf. [6]). The Poincaré series p_{d+1}(t), …, p_{r}(t) are nonzero polynomials. Moreover, for each i, the same series p_{1}(t), …, p_{i}(t), up to the ith term is obtained if the sequence of parameters is replaced by the sequence ζ_{1}, …, ζ_{i−1}, ζ_{i}^{m}, …, ζ_{r} for any m > 1.

Definition 3.4. A system of parameters is said to be generic if it satisfies all of the conditions in the conclusion of Proposition 3.3.

The term “generic” is justified by the fact that such a system of parameters is obtained by choosing elements of large degree and by avoiding a finite set of prime ideals at each stage in the choice.

Proposition 3.5. Let ζ_{1}, …, ζ_{r} be any system of homogeneous parameters for H^{∗}(G, k). Let f_{i}, p_{i} be the Poincaré series defined as above. Let n_{i} = deg(ζ_{i}) for each i. Then

f_{i}(t) = t^{n_{i}−1}(1 + t)p_{i}(t) + (1 − t^{n_{i}})f_{i−1}(t).

Moreover

f_{r}(t) = t^{n_{r}−1}(1 + t)p_{r}(t) + (1 − t^{n_{r}})t^{n_{r−1}−1}(1 + t)p_{r−1}(t)

+ ⋯ + ∏_{j=2}^{r}(1 − t^{n_{j}})t^{n_{1}−1}(1 + t)p_{1}(t)

+ ∏_{j=1}^{r}(1 − t^{n_{j}})f_{0}(t).

Proof. From the last section we know that the spectral sequence E(i) converges to the cohomology Ext_{kG}^{∗}(D^{(i)}, k). The E_{∞} term of this spectral sequence has two rows and hence

Dim Ext_{kG}^{m}(D^{(i)}, k) = Dim E_{∞}^{m,0} + Dim E_{∞}^{m−n_{i}+1,n_{i}−1}.

FUNCTIONAL EQUATIONS FOR POINCARÉ SERIES IN GROUP COHOMOLOGY 9

But E_{∞}^{∗,n_{i}−1} is the kernel of multiplication ζ_{i} on Ext_{kG}^{∗}(D^{(i−1)}, k). Because of the degree shift its Poincaré series is t^{n_{i}−1}p_{i}(t). Also E_{∞}^{m,0} is the cokernel of multiplication by ζ_{i}. Let its Poincaré series be denoted by q_{i}(t). Then we have that

f_{i}(t) = t^{n_{i}−1}p_{i}(t) + q_{i}(t).

On the other hand we have for each m ≥ 0 an exact sequence

0 → A_{i,m} → Ext_{kG}^{m}(D^{(i−1)}, k) ⟶^{ζ_{i}} Ext_{kG}^{m+n_{i}}(D^{(i−1)}, k) → B_{m+n_{i}} → 0

where the middle map is multiplication by ζ_{i}, A_{i,m} is the kernel and B_{m+n_{i}} is the cokernel. Hence the alternating sum of the dimensions of the terms of the sequence is zero. This implies the equation

−t^{n_{i}}p_{i}(t) + t^{n_{i}}f_{i−1}(t) − f_{i−1}(t) + q_{i}(t) = 0

on Poincaré series. Thus

q_{i}(t) = (1 − t^{n_{i}})f_{i−1}(t) + t^{n_{i}}p_{i}(t).

We get the desired result by substituting the last relation into the equation for f_{i}(t), obtained above. The last equation in the Proposition is a consequence of the previous ones.

If H^{∗}(G, k) is Cohen-Macaulay then we may assume that p_{i}(t) = 0 for all i. Hence Propositions (3.1) and (3.5) imply the following.

Corollary 3.6. If H^{∗}(G, k) is Cohen-Macaulay then its Poincaré series satisfies (1.1).

In order to develop a functional equation for p_{r}(t) we need the following technical result.

Proposition 3.7. Let ℝ be the real numbers. Suppose that p_{1}, …, p_{n} ∈ ℝ[t] and f ∈ ℝ[t_{1}, …, t_{n+1}] satisfy the equation φ(t, a) = f(p_{1}, …, p_{n}, t^{a}) = 0 identically as a function of t for infinitely many values of a ≥ 0. Then φ(t, a) = 0 for all a ≥ 0.

Proof. We may write

φ(t, a) = α_{0} + α_{1}t^{a} + ⋯ + α_{m}t^{ma}

10 DAVID J. BENSON AND JON F. CARLSON

where α_{i} ∈ ℝ[t] for i = 0, …, m. Now for any fixed t > 0, φ(t, a) vanishes for infinitely many values of a ≥ 0. Hence, so also does ∂φ/∂a. Therefore for any fixed t > 0, t ≠ 1, we have that

\frac{1}{t^{a} log t} \frac{∂φ}{∂a} = α_{1} + 2α_{2}t^{a} + ⋯ + mα_{m}t^{(m−1)a}

vanishes for infinitely many values of a. By repeating the argument, it follows that m!α_{m} = 0 for all t > 0, t ≠ 1. Thus α_{m} = 0 identically and φ(t, a) = 0 for all a ≥ 0, as desired.

Corollary 3.8. The Proposition remains true if the functions p_{1}, …, p_{n} are allowed to be rational functions of t.

Proof. Let p_{i}(t) = q_{i}(t)/d_{i}(t). Now multiply the function φ(t, a) = f(p_{1}, …, p_{n}, t^{a}) by a sufficiently high power of d(t) = d_{1}(t) … d_{n}(t) to clear all of the denominators. We have that

φ′(t, a) = (d(t))^{n}φ(t, a) = 0

identically as a function of t for infinitely many values of a ≥ 0. The same proof as in the above Proposition can be applied.

Theorem 3.9. Suppose that ζ_{1}, …, ζ_{r} is a generic set of parameters for H^{∗}(G, k) (see (3.4)). Let c = b_{r−1} = ∑_{i=1}^{r−1}(n_{i} − 1) where n_{i} = deg(ζ_{i}) for i = 1, …, n. Then the polynomial p_{r}(t) satisfies the functional equation

t^{c} · p_{r} (\frac{1}{t}) = p_{r}(t).

Proof. For the system of parameters ζ_{1}, …, ζ_{r−1}, ζ_{r}^{m} we have Poincaré series f_{i}^{(m)}(t) for Ext_{kG}^{∗}(D^{(i)}, k) and p_{i}^{(m)}(t) for the kernel of ζ_{i} on Ext_{kG}^{∗}(D^{(i−1)}, k). From Proposition 3.3 we see that p_{i}^{(m)}(t) = p_{i}(t) for all i = 1, …, n and all m. But it is also clear that f_{i}^{(m)}(t) = f_{i}(t) for 1 ≤ i ≤ r − 1 and all m. So let

h(t) =t^{n_{r−1}−1}(1 + t)p_{r−1}(t) + …

+ ∏_{j=2}^{r−1}(1 − t^{n_{j}})t^{n_{j}−1}(1 + t)p_{1}(t)

+ ∏_{j=1}^{r−1}(1 − t^{n_{j}})f_{0}(t).

FUNCTIONAL EQUATIONS FOR POINCARÉ SERIES IN GROUP COHOMOLOGY 11

Then from Proposition 3.5 and the above, we have that

f_{r}^{(m)}(t) = t^{mn_{r}−1}(1 + t)p_{r}(t) + (1 − t^{mn_{r}})h(t).

By Proposition 3.1, f_{r}^{(m)}(t) satisfies the functional equation

t^{a_{m}}f_{r}^{(m)} (\frac{1}{t}) = f_{r}^{(m)}(t)

where a_{m} = ∑_{i=1}^{r−1}(n_{i} − 1) + mn_{r} − 1 = c + mn_{r} − 1. By substitution we get that

(t^{c−1}p_{r} (\frac{1}{t}) − t^{mn_{r}−1}p_{r}(t)) (1 + t) − (1 − t^{mn_{r}})(h(t) + h (\frac{1}{t})) = 0

for all m > 0. Then by the previous result (3.8), with a = m, we have that the equation holds for m = 0. Thus

(t^{c−1}p_{r} (\frac{1}{t}) − t^{−1}p_{r}(t)) (1 + t) = 0

and the Theorem is proved.

Corollary 3.10. Suppose that the cohomology ring H^{∗}(G, k) has dimension r and depth r−1. Then the Poincaré series p_{G}(t) = f_{0}(t) satisfies the functional equation

p_{G} (\frac{1}{t}) − (−t)^{r}p_{G}(t) = \frac{(−t)^{r−1}(1 + t)}{∏_{i=1}^{r−1}(1 − t^{n_{i}})}p_{r}(t).

In particular, it does not satisfy equation (1.1).

Proof. By Proposition 3.3, p_{1}(t) = ⋯ = p_{r−1}(t) = 0 and we have that

f_{0}(t) = \frac{f_{r}(t)}{∏_{j=1}^{r}(1 − t^{n_{j}})} − \frac{t^{n_{r}−1}(1 + t)p_{r}(t)}{∏_{j=1}^{r}(1 − t^{n_{j}})}

by Proposition 3.5. The functional equation is obtained in a straightforward fashion using Proposition 3.1 and Theorem 3.9. The last statement of the corollary is obvious from the fact that p_{r}(t) ≠ 0.

4. Larger codepth.

In this section we show that the result in Corollary 3.10 is best possible. That is, there are counterexamples to the corollary if the depth is more than one smaller than the dimension. The actual example has codepth 2. We should note that the direct product of this group with itself will be a counterexample with codepth 4. It should also be pointed out that we know of no example in odd characteristic.

12 DAVID J. BENSON AND JON F. CARLSON

Proposition 4.1. Let k be a field of characteristic 2. If G is the group Γ_{7}a_{2}, then H^{∗}(G, k) has dimension r = 3, depth 1, and its Poincaré series satisfies the equation

(4.2)  p_{G} (\frac{1}{t}) = (−t)^{r}p_{G}(t).

All of the necessary calculations for the proof are found in [8]. See also [4] for a different analysis of the cohomology of G. A few notes on the structure of the cohomology ring seem to be in order. In the notation of [8], a system of parameters for H^{∗}(G, k) can be taken to be the elements z, x and s in degrees one, two and four respectively. The Poincaré series (not in reduced form) for H^{∗}(G, k) is

f_{0}(t) = p_{G}(t) = \frac{1 + t + t^{3} + t^{4}}{(1 − t)(1 − t^{2})(1 − t^{4})}.

The element s is regular whereas x and z are not. If we let ζ_{1} = s, ζ_{2} = x and ζ_{3} = z (a generic sequence), then in the notation of the last section we have the partial Poincaré series:

f_{1}(t) = \frac{1 + t + t^{2} + t^{3}}{(1 − t)(1 − t^{2})},  p_{1}(t) = 0,
f_{2}(t) = \frac{1 + t + t^{2} + t^{3}}{1 − t} ,  p_{2}(t) = t,
f_{3}(t) = 1 + 2t + 2t^{2} + 2t^{3} + t^{4},  p_{3}(t) = t + t^{3}

Consider the function f_{1}(t), corresponding to H^{∗}(G, k)/s · H^{∗}(G, k). This looks like the Poincaré series for a free k[z, x]-module with generators in degrees 0, 1, 3 and 4. However, the generator y in degree 1 is annihilated by both z and x. And the elements which might otherwise be zy and xy are really new generators, called w and u in [8]. The relation xw = zu shows that the submodule generated by w and u is isomorphic to the ideal of k[z, x] consisting of all elements of positive degree. A careful analysis shows that a situation like this is necessary in order to achieve the conditions of the proposition. In fact, we can prove the following. Proposition 4.3. Suppose that H^{∗}(G, k) has dimension r and depth r − 2. Let ζ_{1}, …, ζ_{n} be a generic homogeneous system of parameters for H^{∗}(G, k), defining Poincaré series f_{i}(t) and p_{i}(t) as in the last section. If the Poincaré series f_{0}(t) = p_{G}(t) satisfies the functional equation (4.2) then p_{r−1}(t) and p_{r}(t) satisfy the equations

(1) p_{r−1}(t) = t^{b_{r−2}−1}p_{r−1} (\frac{1}{t}) and

(2) p_{r}(t) = (1 + t^{n_{r−1}}) p_{r−1}(t).

Moreover the Massey product ⟨ ; ζ_{r−1}, ζ_{r}⟩ is not zero on some element of positive

degree in H^{∗}(G, k).

The proofs of the two equations are similar to the proofs of the last section.

The statement about Massey products require an investigation into the spectral

sequence in [3]. We refer the reader to Section 11 of that paper for the needed

definitions and constructions.

References

1. D. J. Benson, Representations and Cohomology II: Cohomology of Groups and Modules, Cambridge Studies in Advanced Mathematics 31 (1991), Cambridge University Press, Cambridge.

2. D. J. Benson and J. F. Carlson, Complexity and multiple complexes, Math. Zeit. 195 (1987), 221-238.

3. , Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. (to appear).

4. , Products in negative cohomology, J. Pure Appl. Algebra (to appear).

5. J. F. Carlson, Projective resolutions and degree shifting for cohomology and group algebras, Representations of Algebras and Related Topics (H. Tachikawa and S. Brenner, eds.), London Math. Soc. Lecture Note Ser, vol. 168, Cambridge University Press, Cambridge, 1992, pp. 80– 126.

6. J. Duflot, Depth and equivariant cohomology, Comm. Math. Helvetici 56 (1981), 627-637.

7. L. Evens, The Cohomology of Groups (1991), Oxford University Press, New York.

8. D. Rusin, The cohomology of groups of order 32, Math. Comp. 53 (1989), 359-385.