ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

DAVID BENSON

Abstract. Let K be a field of characteristic p and let G be a finite group of order divisible by p. The regularity conjecture states that the Castelnuovo– Mumford regularity of the cohomology ring H^{∗}(G, K) is always equal to zero. We prove that if the regularity conjecture holds for a finite group H then it holds for the wreath product H ≀ ℤ/p. As a corollary, we prove the regularity conjecture for the symmetric groups Σ_{n}. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most two, the regularity conjecture is already known to hold by previous work.

For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement, namely that the dimensions of the cohomology groups are Polynomial On Residue Classes (PORC) in the sense of Graham Higman.

1. Introduction

Let K be a field of characteristic p, and let G be a finite group. Then the regularity conjecture [2, 3] states that the Castelnuovo–Mumford regularity of the cohomology ring Reg H^{∗}(G, K) is always equal to zero. Briefly, the definitions are as follows. We write 𝔪 for the maximal ideal of positive degree elements in H^{∗}(G, K). If M is a graded H^{∗}(G, K)-module, we define Γ_{𝔪}M to be the 𝔪torsion in M, namely {x ∈ M | ∃ n > 0, 𝔪^{n}x = 0}. Then Γ_{𝔪} is left exact but not right exact, and its right derived functors give the local cohomology of M and are written H_{𝔪}^{i}M. Since there is also an internal grading, we write H_{𝔪}^{i,j}M for the jth graded piece. We define a_{𝔪}^{i}(M) = max{j ∈ ℤ | H_{𝔪}^{i,j}M ≠ 0} (±∞ is allowed) and the Castelnuovo–Mumford regularity is Reg M = max{a_{𝔪}^{i}M + i}. This is a measure of how far you have to go along a resolution before “regular behaviour” sets in.

There are various motivations for studying regularity of cohomology rings. One is that it gives a priori bounds for how much of a projective resolution you have to compute before you can be sure that you have all the generators and relations in the cohomology ring. See Theorem 10.1 of [3] for further details. Another motivation is that it gives more precise information about the duality developed by Benson and Carlson [4]. In terms of the definitions of that paper, a regularity of zero ensures that the “last survivor” really is last. A further consequence of the regularity conjecture is that regular behaviour for the dimensions of the cohomology groups dim_{K} H^{n}(G, K) begins straight away, as we prove in Theorem 5.6.

This research was partly supported by NSF grant DMS-0242909.

In Theorem 1.5 of [3], it was proved that as long as the Krull dimension and the depth of H^{∗}(G, K) differ by at most two, the regularity conjecture holds. Unfortunately, the vast majority of the examples for which the regularity conjecture has been checked satisfy this bound; for example, this is the case for all 2-groups of order at most 64 (Carlson et al. [5]) as well as various other classes of finite groups studied in [11, 13, 17, 20]. So to gain more confidence in the conjecture, it is desirable to check families of examples where the difference is greater than two.

One good way of producing examples where the difference between the Krull dimension and the depth is large is to look at wreath products. The goal of this article is to provide further evidence for the regularity conjecture by examining wreath products. In particular, we shall prove that the conjecture holds for the cohomology of the finite symmetric groups, where the difference between the Krull dimension and the depth is arbitrarily large. This follows from Quillen’s stratification theorem [18, 19]: Σ_{p^{n}} has p-rank p^{n−1}, so this is the Krull dimension; but there is also a conjugacy class of elementary abelian p-subgroups of rank n, so that the depth is at most n.

Let H be a finite group and consider the wreath product

H ≀ ℤ/p = H^{p} ⋊ ℤ/p.

Our main theorem is the following, whose proof can be found in Section 3. Theorem 1.1. Suppose that Reg H^{∗}(H, K) = 0. Then Reg H^{∗}(H ≀ ℤ/p, K) = 0.

Our proof involves using the structure of the cohomology of wreath products, as described in Nakaoka [15]. Using the fact that the Sylow p-subgroups of Σ_{n} are direct products of wreath products, we obtain the following corollary. Corollary 1.2. If Σ_{n} ≥ G ≥ Syl_{p}(Σ_{n}) then Reg H^{∗}(G, K) = 0.

The corollary is obtained in Section 3, where we also treat the alternating groups A_{2^{n}} in characteristic two.

For more general wreath products, similar techniques probably work. But the technical details become much harder. Instead, we show that a numerical consequence of regularity does hold for more general wreath products. Namely, in Theorem 5.6, we show that one consequence of the regularity conjecture is that the dimension function i ↦ dim_{K} H^{i}(G, K) is a PORC (Polynomial On Residue Classes) function in the sense of Higman [12]:

Theorem 1.3. If |G| is divisible by the characteristic of K and if Reg H^{∗}(G, K) = 0, then there exist an integer d and polynomials f_{0}, …, f_{d−1} such that for all i ≥ 0 dim_{K} H^{i}(G, K) = f_{j}(i), where j is the remainder on dividing i by d.

The fact that this regular behaviour occurs for large enough i is a simple consequence of finite generation. The interesting consequence of the regularity conjecture is that the eventual behaviour begins straight away. We prove the following theorem in Section 5.

Theorem 1.4. Let K be a field of characteristic p. Let H be a finite group and let G be a permutation group on a finite set Ω. Let H ≀ G be the wreath product, where G permutes a product of copies of H indexed by Ω. Suppose that p divides the order of H, the function i ↦ H^{i}(H, K) is PORC, and that for every subgroup J of G of order divisible by p, the function i ↦ H^{i}(J, K) is PORC. Then the function i ↦ H^{i}(H ≀ G, K) is PORC.

2. Castelnuovo–Mumford regularity

Let R = ⨁_{j≥0} R_{j} be a Noetherian graded commutative K-algebra. Here graded commutative means that for homogeneous elements x and y we have yx = (−1)^{|x||y|}xy, where |x| denotes the degree of x. We shall assume that R is connected, meaning that R_{0} = K. For example, these conditions hold in case R is the cohomology ring of a finite group, R = H^{∗}(G, K). Let 𝔪 be the maximal ideal spanned by the homogeneous elements of positive degree, 𝔪 = ⨁_{j>0} R_{j}. If M is a graded R-module (we allow positive and negative grading), then the local cohomology is doubly graded, H_{𝔪}^{i,j}M. The first grading is the local cohomological degree, and the second is the internal degree coming from the grading on R and M. We define the a-invariants of M to be

a_{𝔪}^{i}(M) = max{j ∈ ℤ | H_{𝔪}^{i,j}M ≠ 0},

a_{𝔪}^{i}(M) = −∞ if H_{𝔪}^{i,j}M = 0 for all j, and a_{𝔪}^{i}(M) = ∞ if H_{𝔪}^{i,j}M ≠ 0 for arbitrarily large values of j. Note that a_{𝔪}^{i}(M) = −∞ unless i lies between the depth and the Krull dimension of M, by a theorem of Grothendieck.

The Castelnuovo–Mumford regularity is defined to be

Reg M = max_{i≥0}{a_{𝔪}^{i}(M) + i}.

See §4 of [3] for further details, and §20.5 of Eisenbud [9] for the history and geometric significance of this definition for strictly commutative K-algebras whose generators have degree equal to one. See also Maclagan and Smith [14] for a more general context for this definition, and Chipalkatti [7] for a closely related definition. The significance of regularity in group cohomology is the following conjecture from [2, 3].

Conjecture 2.1. Let G be a finite group. Then Reg H^{∗}(G, K) = 0.

We mention that it is proved in Theorem 4.2 of [3] that the inequality

Reg H^{∗}(G, K) ≥ 0  (2.2)

holds.

Regularity can be reformulated in terms of free resolutions as follows. Let ζ_{1}, …, ζ_{r} be a homogeneous system of parameters in R, and let

0 → F_{r} → ⋯ → F_{1} → F_{0} → M → 0

be a minimal free resolution of M over the polynomial ring K[ζ_{1}, …, ζ_{r}]. Define β_{i}^{R}(M) to be the largest degree of a generator of F_{i} as an R-module (or β_{i}^{R}(M) = −∞ if F_{i} = 0). Then it is proved in Corollary 5.7 of [3] that

Reg M = max_{i≥0}{β_{i}^{R}(M) − i} − ∑_{j=1}^{r}(|ζ_{j}| − 1).  (2.3)

In particular, if M is finitely generated then Reg M is finite, and hence the a_{𝔪}^{i}(M) are either finite or equal to −∞.

In order to prove the main theorem, we begin with some general properties of regularity.

Proposition 2.4. Suppose that

0 → M_{1} → M_{2} → M_{3} → 0

is a short exact sequence of R-modules. Then we have

Reg M_{2} ≤ max{Reg M_{1}, Reg M_{3}}.

If Reg M_{1} ≤ Reg M_{3} then Reg M_{2} = Reg M_{3}.

Proof. This follows from the long exact sequence in local cohomology

⋯ → H_{𝔪}^{i,j}(M_{1}) → H_{𝔪}^{i,j}(M_{2}) → H_{𝔪}^{i,j}(M_{3}) → H_{𝔪}^{i+1,j}(M_{1}) → ⋯

If Reg M_{1} ≤ Reg M_{3}, choose i and j so that i + j = Reg M_{3} and H_{𝔪}^{i,j}(M_{3}) ≠ 0. Then H_{𝔪}^{i+1,j}(M_{1}) = 0 and so H_{𝔪}^{i,j}(M_{2}) ≠ 0. □

Corollary 2.5. Suppose that

M = M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{n} = {0}

is a filtration of an R-module M. Then

Reg M ≤ max_{0≤i<n}{Reg M_{i}/M_{i+1}}.

If for 0 < i < n we have Reg M_{i}/M_{i+1} ≤ Reg M_{0}/M_{1} then Reg M = Reg M_{0}/M_{1}.

Proof. This follows from Proposition 2.4 and induction on n. □

Proposition 2.6. If R_{1} and R_{2} are graded K-algebras satisfying the hypotheses of this section, with graded modules M_{1} and M_{2} respectively, then we may regard M_{1} ⊗_{K} M_{2} as an R_{1} ⊗_{K} R_{2}-module, and its regularity is

Reg (M_{1} ⊗_{K} M_{2}) = Reg M_{1} + Reg M_{2}.

Proof. If ζ_{1}, …, ζ_{r} is a homogeneous system of parameters for R_{1} and η_{1}, …, η_{s} is one for R_{2}, then

ζ_{1} ⊗ 1, …, ζ_{r} ⊗ 1, 1 ⊗ η_{1}, …, 1 ⊗ η_{s}  (2.7)

is a homogeneous system of parameters for R_{1} ⊗_{K} R_{2}. If F_{∗} and F′_{∗} are minimal free resolutions of M_{1} and M_{2} over the respective polynomial subrings of R_{1} and R_{2}, then F_{∗} ⊗_{K} F′_{∗} is a minimal free resolution of M_{1} ⊗_{K} M_{2} over the polynomial subring generated by the parameters (2.7). It follows that

β_{i}^{R_{1}⊗R_{2}}(M_{1} ⊗ M_{2}) = max_{j+k=i}{β_{j}^{R_{1}}(M_{1}) + β_{k}^{R_{2}}(M_{2})}

and so

β_{i}^{R_{1}⊗R_{2}}(M_{1} ⊗ M_{2}) − i = max_{j+k=i}{(β_{j}^{R_{1}}(M_{1}) − j) + (β_{k}^{R_{2}}(M_{2}) − k)}.

Now use the formula (2.3) for regularity. □

Next, we write R^{[p]} for the graded ring whose homogeneous elements are symbols x^{[p]} with x a homogeneous element of R, and with |x^{[p]}| = p|x|. Since (−1)^{p|x|·p|y|} is equal to (−1)^{|x||y|} in K (check separately for p = 2 and p odd), R^{[p]} is again a Noetherian graded commutative K-algebra. Similarly, if M is a graded R-module, we write M^{[p]} for the corresponding graded R^{[p]}-module with homogeneous elements m^{[p]}.

Proposition 2.8. We have

p · Reg M − (p − 1)Dim(M) ≤ Reg M^{[p]} ≤ p · Reg M − (p − 1)Depth(M).

Proof. Let d be the depth of M and s its Krull dimension. We have

a_{𝔪}^{i}(M^{[p]}) = p · a_{𝔪}^{i}(M),

and so

Reg M^{[p]} = max_{d≤i≤s}{p · a_{𝔪}^{i}(M) + i} = max_{d≤i≤s}{p(a_{𝔪}^{i}(M) + i) − (p − 1)i}.  □

3. Proof of the main theorem

It was proved by Nakaoka (Theorem 3.3 of [15]; see also the end of Section 4.1 of [1]) that there is a ring isomorphism

H^{∗}(H ≀ ℤ/p, K) ≅ H^{∗}(ℤ/p, H^{∗}(H^{p}, K)).

We write t for the generator of ℤ/p, and we use multiplicative notation, so that ℤ/p = ⟨t | t^{p} = 1⟩.

As a representation of ℤ/p, H^{∗}(H^{p}, K) ≅ H^{∗}(H, K)^{⊗p} decomposes as a direct sum of two pieces. One piece is spanned by the elements x ⊗ ⋯ ⊗ x, as x runs over a vector space basis of H^{∗}(H, K), with trivial ℤ/p-action; the other piece is spanned by tensors involving more than one basis element, and the ℤ/p-action on this summand is free. Notice that this decomposition depends on a choice of homogeneous basis for H^{∗}(H, K), and is therefore not canonical.

Let T ⊆ H^{∗}(H ≀ ℤ/p, K) be the image of the transfer from H^{∗}(H^{p}, K). Then T lies inside H^{0}(ℤ/p, H^{∗}(H^{p}, K)) = H^{∗}(H^{p}, K)^{ℤ/p}, and consists of the invariants in the free summand described in the previous paragraph. Furthermore,

H^{∗}(H ≀ ℤ/p, K)/T ≅ H^{∗}(ℤ/p, K) ⊗ H^{∗}(H, K)^{[p]}  (3.1)

where [p] indicates as in §2 that the degrees have been multiplied by a factor of p. The elements of H^{∗}(H, K)^{[p]} in this isomorphism are spanned by the images x^{[p]} of the elements x ⊗ ⋯ ⊗ x.

We regard everything in sight as a module over H^{∗}(H ≀ℤ/p, K) for the purpose of computing regularity. We may do this by the following proposition.

Proposition 3.2. Let H be a subgroup of a finite group G. Then the regularity of an H^{∗}(H, K)-module is the same whether regarded as an H^{∗}(H, K)-module or as an H^{∗}(G, K)-module via restriction.

Proof. By a theorem of Evens [10], H^{∗}(H, K) is finitely generated as a module over H^{∗}(G, K) via the restriction map. So if M is an H^{∗}(H, K)-module, then the local cohomology of M is the same whether computed as an H^{∗}(H, K)-module or as an H^{∗}(G, K)-module. □

Lemma 3.3. We have

Reg H^{∗}(H ≀ ℤ/p, K)/T = Reg H^{∗}(H, K)^{[p]} ≤ p · Reg H^{∗}(H, K) − p + 1.

Proof. The equality follows from the isomorphism (3.1) and Proposition 2.6. The depth of H^{∗}(H, K) is at least one by a theorem of Duflot [8], so the inequality follows from Proposition 2.8. □

Lemma 3.4. As an H^{∗}(H ≀ ℤ/p, K)-module, H^{∗}(H, K)^{⊗p} has a filtration with p + 1 filtered quotients, which consist of p − 1 copies of T followed by one copy of H^{∗}(H, K)^{[p]}, and finally another copy of T.

Proof. Since the image of the restriction map H^{∗}(H ≀ ℤ/p, K) → H^{∗}(H, K)^{⊗p} consists of ℤ/p-invariants, we have commuting actions of H^{∗}(H ≀ ℤ/p, K) and Kℤ/p on H^{∗}(H, K)^{⊗p}. Consider the action of (1 − t) ∈ Kℤ/p. As mentioned above, H^{∗}(H, K)^{⊗p} has only trivial and free summands as a Kℤ/p-module. So we have a filtration of H^{∗}(H ≀ ℤ/p, K)-modules

H^{∗}(H, K)^{⊗p} ⊇ Ker(1 − t)^{p−1} ⊇ Ker(1 − t)^{p−2} ⊇ ⋯

⋯ ⊇ Ker(1 − t)^{2} ⊇ Ker(1 − t) ⊇ Im(1 − t)^{p−1} ⊇ 0.

All except one of these filtered quotients come from the free summand, and are isomorphic to T. The remaining filtered quotient

Ker(1 − t)/Im(1 − t)^{p−1}

is isomorphic to the trivial Kℤ/p-summand, namely to H^{∗}(H, K)^{[p]}.

We remark that by switching from kernels to images earlier in the filtration, the copy of H^{∗}(H, K)^{[p]} as a filtered quotient may be placed anywhere except at the beginning or the end. □

Proposition 3.5. If Reg H^{∗}(H, K) = 0 then as an H^{∗}(H ≀ ℤ/p, K)-module we have Reg T = 0.

Proof. By Lemma 3.3, we have Reg H^{∗}(H, K)^{[p]} < 0, and by Proposition 2.6 we have Reg H^{∗}(H, K)^{⊗p} = 0. Applying Corollary 2.5 to the filtration of H^{∗}(H, K)^{⊗p} described in Lemma 3.4, we deduce that Reg T = 0. □

Proof of Theorem 1.1. If Reg H^{∗}(H, K) = 0 then by Lemma 3.3 we have

Reg H^{∗}(H ≀ ℤ/p, K)/T < 0,

and by Proposition 3.5 we have Reg T = 0. It follows from Proposition 2.4 that Reg H^{∗}(H ≀ℤ/p, K) ≤ 0. Combining this with the inequality (2.2), this completes the proof of Theorem 1.1. □

4. Examples

The most obvious example where we can apply our main theorem is the Sylow p-subgroups of the symmetric groups. These are direct products of iterated wreath products ℤ/p ≀ ℤ/p ≀ ⋯ ≀ ℤ/p, and so by Theorem 1.1 and Proposition 2.6, the cohomology in characteristic p of these Sylow p-subgroups satisfies the regularity conjecture. To complete the proof of Corollary 1.2, we apply the following proposition.

Proposition 4.1. Let K be a field of characteristic p and let S be a Sylow psubgroup of a finite group G. Then Reg H^{∗}(G, K) ≤ Reg H^{∗}(S, K). In particular, if Reg H^{∗}(S, K) = 0 then Reg H^{∗}(G, K) = 0.

Proof. A standard argument using the transfer map (see for example §XII.10 of Cartan and Eilenberg [6]) shows that H^{∗}(G, K) is a direct summand of H^{∗}(S, K) as an H^{∗}(G, K)-module via the restriction map. Combining this with Proposition 3.2 proves the first statement. For the second statement, we use the inequality (2.2). □

There are other interesting finite groups whose Sylow p-subgroups are direct products of iterated wreath products in a similar manner. For example, if G = GL_{n}(𝔽_{q}) with q coprime to p then the Sylow p-subgroups of G have this form (except in certain cases when p = 2). However, for these groups the computations of Quillen [20] show that H^{∗}(G, K) is Cohen–Macaulay, so that the regularity conjecture holds (for example by Theorem 1.5 of [3]).

A more interesting class of examples is the alternating groups in characteristic two.

Proposition 4.2. The Sylow 2-subgroups of A_{2^{n}} are of the form

ℤ/2 ≀ ℤ/2 ≀ ⋯ ≀ ℤ/2 ≀ (ℤ/2 × ℤ/2).

Proof. This interesting and subtle fact was communicated to me by Mark Feshbach. The proof can be found in a paper of Warren Wong [22]. The essential observation is that if S is a Sylow 2-subgroup of A_{2m} then V^{m} ⋊ S is a Sylow 2-subgroup of A_{4m}. Here, V is a Klein four group acting regularly on four points, and each pair of points permuted by S is replaced by a copy of V with an outer automorphism of order two. As abstract groups, V^{m} ⋊ S is isomorphic to ℤ/2 ≀ S. This is an example of Bernhard Neumann’s twisted wreath products [16]. □

So an analysis of the case of A_{2^{n}} involves understanding wreath products with ℤ/2 × ℤ/2.

Theorem 4.3. Let K be a field of characteristic two. If Reg H^{∗}(H, K) = 0 then Reg H^{∗}(H ≀ (ℤ/2 × ℤ/2), K) = 0.

Proof. We can mimic the proof for the wreath product of H with ℤ/p as follows. Let V = ℤ/2 × ℤ/2 = {1, a, b, c}. Let T_{1} be image of transfer from H^{∗}(H^{4}, K) to H^{∗}(H ≀V, K). Then T_{1} lies inside H^{0}(V, H^{∗}(H^{4}, K)), namely inside the invariants of V on H^{∗}(H, K)^{⊗4}. We write T_{2} for the image of transfer from H^{∗}(H^{2}, K) to H^{∗}(H ≀ ℤ/2, K), so that there is a direct sum of three copies of H^{∗}(ℤ/2, K) ⊗ T_{2}^{[2]} inside H^{∗}(H ≀ V, K)/T_{1}, one for each subgroup of order two in V. These three copies of T_{2}^{[2]} can be seen as the subspace of H^{∗}(H, K)^{⊗4} spanned by elements like x⊗x⊗y ⊗y where the tensor factors occur in two pairs. Finally, the quotient of H^{∗}(H ≀ V, K)/T by these summands is isomorphic to H^{∗}(V, K) ⊗ H^{∗}(H, K)^{[4]}. We shall show that each of these pieces has regularity ≤ 0.

We filter H^{∗}(H, K)^{⊗4} in such a way that the filtered quotients consist of four copies of T_{1}, two copies of T_{2}^{[2]} for each of the three subgroups of V of order two, and one copy of H^{∗}(H, K)^{[4]}. The top and bottom filtered quotients are copies of T_{1}. By Proposition 3.5, we have Reg T_{2} = 0, and so by Proposition 2.8 we have Reg T_{2}^{[2]} < 0. Also by Proposition 2.8 we have Reg H^{∗}(H, k)^{[4]} < 0. Since Reg H^{∗}(H, K)^{⊗4} = 0 by Proposition 2.6, it follows from Corollary 2.5 that Reg T_{1} = 0.

Again applying Proposition 2.6, we have

Reg H^{∗}(ℤ/2, K) ⊗ T_{2}^{[2]} < 0 and Reg H^{∗}(V, K) ⊗ H^{∗}(H, K)^{[4]} < 0.

Finally, we apply Corollary 2.5 to the filtration described in the first part of the proof to deduce that H^{∗}(H ≀ V, K) has regularity ≤ 0. Combining this with the inequality (2.2) completes the proof of the theorem. □

Presumably the same method of proof works for a wreath product with any p-group, but it seems hard to set up the details of the filtrations.

Corollary 4.4. Let K be a field of characteristic two. If A_{2^{n}} ≥ G ≥ Syl_{2}(A_{2^{n}}) then Reg H^{∗}(G, K) = 0. □

5. PORC functions

A function i ↦ c_{i} from nonnegative integers to integers is said to be PORC (Polynomial On Residue Classes; the terminology comes from Higman [12]) if there exists a positive integer d (the modulus) and polynomials f_{0}, …, f_{d−1} such that for all i ≥ 0 we have c_{i} = f_{j}(i), where j is the unique integer satisfying 0 ≤ j < d and j ≡ i (mod d). The function i ↦ c_{i} is said to be almost PORC if this condition holds for all large enough i.

It is well known that a function is almost PORC if and only if the Poincaré series f(t) = ∑_{i=0}^{∞} c_{i}t^{i} is the power series expansion of a rational function of the form

f(t) = \frac{p(t)}{∏_{i=1}^{r}(1 − t^{n_{i}})}  (5.1)

where p(t) is a polynomial with integer coefficients and where n_{1}, …, n_{r} are positive integers. The modulus d can be taken to be any positive number divisible by all the n_{i}. So f(t) could be rewritten in the form p(t)/(1 − t^{d})^{r} by multiplying the numerator and denominator of the rational function (5.1) by suitable polynomials.

Example 5.2. The Poincaré series of the cohomology of an elementary abelian p-group E of rank r is given by

∑_{i≥0} dim_{K} H^{i}(E, K) = \frac{1}{(1 − t)^{r}} .

The coefficients are PORC if and only if r > 0.

The following lemma summarizes some obvious properties of PORC functions.

Lemma 5.3. The set of Poincaré series of PORC functions is closed under the following operations:

(i) addition and subtraction,

(ii) multiplication,

(iii) multiplication by integers (nonzero constant functions are not PORC, so this is not a special case of (ii)),

(iv) division by t if c_{0} = 0 (but not multiplication by t),

(v) division by a nonzero integer, if each of the c_{i} is divisible by that integer, (vi) replacing t by t^{n} for some positive integer n. □

Theorem 5.4. Let

f(t) = ∑_{i=0}^{∞} c_{i}t^{i} = \frac{p(t)}{∏_{i=1}^{r}(1 − t^{n_{i}})}

be the Poincaré series of an almost PORC function i ↦ c_{i}. Then the function i ↦ c_{i} is PORC if and only if the degree of p(t) is strictly less than ∑_{i=1}^{r} n_{i}.

Proof. The hypothesis is unchanged if we multiply the top and bottom of the rational function by a suitable polynomial so that all the n_{i} are equal, say n_{i} = d.

First we suppose that deg p(t) < ∑_{i=1}^{r} n_{i}, and we’ll prove that i ↦ c_{i} is PORC. Since ℤ-linear combinations of PORC functions are PORC by (i) and (iii) of Lemma 5.3, it suffices to treat the case where p(t) = t^{j} for some j < dr. So f(t) takes the form

f(t) = \frac{t^{j}}{(1 − t^{d})^{r}} = t^{j} ∑_{i=0}^{∞} \binom{i + r − 1}{i}t^{id} = ∑_{i=0}^{∞} \binom{i + r − 1}{r − 1}t^{j+id}.

Since j < dr, for values of i with j + id ≥ 0, we have i + r > 0. So the binomial coefficient \binom{i+r−1}{r−1} makes sense and is a polynomial function of i which takes the value zero for −r < i < 0.

Conversely, suppose that f(t) is the Poincaré series of a PORC function i ↦ c_{i}. If deg p(t) ≥ dr then using the fact that t^{j}/(1 − t^{d})^{r} is PORC for j < dr, we may repeatedly add PORC functions and divide by t (using parts (i) and (iv) of Lemma 5.3) until only the term of highest degree remains, and p(t) is a nonzero multiple of

t^{dr}/(1 − t^{d})^{r} = t^{dr} ∑_{i=0}^{∞} \binom{i + r − 1}{r − 1}t^{id} = ∑_{k=r}^{∞} \binom{k − 1}{r − 1}t^{kd}.

The polynomial \binom{k−1}{r−1} takes value (−1)^{r−1} at k = 0, which does not agree with the value of c_{0}, contradicting the statement that f(t) is PORC. So we conclude that deg p(t) < dr. □

Proposition 5.5. Let K be a field, and suppose that R = ⨁_{i≥0} R_{i} is a connected graded commutative Noetherian K-algebra (see the comments at the beginning of Section 2). Let M = ⨁_{i≥0} M_{i} be a finitely generated nonnegatively graded R-module. Then i ↦ c_{i} = dim_{K} M_{i} is a PORC function provided that the ainvariants satisfy a_{𝔪}^{i}(M) < 0 for all i ≥ 0.

Proof. Let ζ_{1}, …, ζ_{r} be a homogeneous system of parameters in R with deg ζ_{i} = n_{i} and set S = K[ζ_{1}, …, ζ_{r}]. Using Theorem 5.5 of [3], if a_{𝔪}^{i}(M) < 0 for all i ≥ 0 then β_{i}^{S}(M) < ∑_{j=1}^{r} n_{j} for all i ≥ 0. Let

0 → F_{r} → ⋯ → F_{0} → M → 0

be the minimal free resolution of M. The Poincaré series of each F_{j} has the form (polynomial of degree < ∑_{i=1}^{r} n_{i})/ ∏_{i=1}^{r}(1 − x^{n_{i}}). The Poincaré series for M is the alternating sum of those for the F_{i}, so it has the same form. Now apply Theorem 5.4. □

Theorem 5.6. Let G be a finite group, and let K be a field of characteristic p dividing |G|. Among the following statements, we have the implications (a) ⇒ (b) ⇒ (c):

(a) Reg H^{∗}(G, K) = 0,

(b) For all i ≥ 0 we have a_{𝔪}^{i}H^{∗}(G, K) < 0,

(c) The function i ↦ dim_{K} H^{i}(G, K) is a PORC function.

Proof. By a theorem of Duflot [8], since p divides |G|, the depth of H^{∗}(G, K) is at least one. So a_{𝔪}^{0}H^{∗}(G, K) = −∞. Condition (a) states that a_{𝔪}^{i}H^{∗}(G, K) ≤ −i for all i ≥ 0, so it follows that (a) implies (b). The statement that (b) implies (c) is contained in Proposition 5.5. □

Remark 5.7. It is shown in Corollary 4.7 of [3] that condition (b) in Theorem 5.6 is equivalent to the existence of a quasi-regular sequence in H^{∗}(G, K) in the sense of Benson and Carlson [4].

Proof of Theorem 1.4. Using Nakaoka’s formula [15] for the cohomology of the wreath product, and some counting arguments of Burnside, Webb [21] (Theorem 3.1) gives the following formula for the Poincaré series of the cohomology of the wreath product:

∑_{i≥0} t^{i} dim_{K} H^{i}(H ≀ G, K) = \frac{1}{|G|} ∑_{J′≤J} μ(J′, J) |J′| g_{J′}(t) f_{J}(t).  (5.8)

Here, the sum ranges over all pairs of subgroups J′ ⊆ J of G. For each subgroup J of G, f_{J}(t) denotes f(t^{|Ω_{1}|}) ⋯ f(t^{|Ω_{n}|}) where f(t) = ∑_{i≥0} t^{i} dim_{K} H^{i}(H, K) and where Ω = Ω_{1} ∪ ⋯ ∪ Ω_{n} is the decomposition of Ω into orbits of J. The function g_{J}(t) is the Poincaré series ∑_{i≥0} t^{i} dim_{K} H^{i}(J, K).

It follows from Theorem 5.6 and parts (ii) and (vi) of Lemma 5.3 that the functions f_{J}(t) have PORC coefficients. We divide the terms in the sum (5.8) into two types. The terms where |J| is not divisible by p have PORC coefficients by part (iii) of Lemma 5.3, whereas for the terms where |J| is divisible by p we need to use parts (ii) and (iii) of the lemma. Finally, the quotient by |G| has PORC coefficients by part (v) of the lemma. □

Acknowledgement. The author acknowledges with thanks the hospitality and support of the Centre Bernoulli, EPFL, Lausanne, where this work was carried out. He also thanks Srikanth Iyengar and Peter Webb for conversations about the material of this paper.

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Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE