DICKSON INVARIANTS, REGULARITY AND COMPUTATION IN GROUP COHOMOLOGY

DAVE BENSON

Abstract. In this paper, we investigate the commutative algebra of the cohomology ring H^{∗}(G, k) of a finite group G over a field k. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo–Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler.

As a by-product of this investigation, we give a new proof of the Bourguiba– Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra.

Finally, we describe an improvement of Carlson’s algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson’s algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

1. Introduction

The purpose of this paper is to investigate the cohomology ring H^{∗}(G, k) of a finite group G over a field k of characteristic p from the point of view of commutative algebra. In particular, we formulate the following conjecture, which is one of the central focuses of the paper.

Conjecture 1.1. Let G be a finite group, and let k be a field of characteristic p. Then the Castelnuovo–Mumford regularity of the cohomology ring is equal to zero:

Reg H^{∗}(G, k) = 0.

The author is partly supported by a grant from the NSF. During the course of the preparation of this paper, he was a guest of the Mathematical Sciences Research Institute in Berkeley, California.

The approach is a synthesis of many recent threads in representation theory and cohomology of finite groups, and is inspired in part by ideas of Carlson [13], Greenlees [21], and Okuyama and Sasaki [22]. This paper is designed to be read in conjunction with the survey paper [6], but may also be read independently. In particular, we refer the reader to the last section of that paper for examples, and generalizations of the above conjecture to the context of the cohomology of compact Lie groups, orientable virtual Poincaré duality groups, and p-adic Lie groups.

Our main theorems are as follows. Recall from §2 of [9] that H^{∗}(G, k) always contains a (possibly dilated) copy of the Dickson invariants as a homogeneous set of parameters. See also §8 of this paper, and Wilkerson’s primer [26]. The following theorem is a consequence of Theorems 8.5 and 9.6.

Theorem 1.2. The Dickson invariants form a filter-regular system of parameters in H^{∗}(G, k).

This allows us to give a new proof (see Theorem 9.9) of the Bourguiba–Zarati theorem [10] relating the depth to the Dickson invariants, in the particular case of finite group cohomology, without using the machinery of unstable algebras over the Steenrod algebra.

Theorem 1.3 (Bourguiba–Zarati). Let G be a finite group and let k be a field of characteristic p. Suppose that H^{∗}(G, k) has depth d. Then the first d Dickson invariants, in ascending order of degree, form a regular sequence in H^{∗}(G, k).

In the language of quasi-regular sequences, introduced in [8] and further investigated in [12, 13], the following is a consequence of Corollary 4.7 and Theorems 8.5 and 9.6.

Theorem 1.4. Let G be a finite group and k be a field of characteristic p. Then the following are equivalent.

(i) There exists a quasi-regular sequence in H^{∗}(G, k) (see Definition 3.2). (ii) The Dickson invariants form a quasi-regular sequence in H^{∗}(G, k). (iii) Every homogeneous system of parameters which satisfies the rank-restric-

tion condition (see Definition 8.3) is a quasi-regular sequence.

(iv) Every filter-regular sequence in H^{∗}(G, k) is a quasi-regular sequence.

(v) For 0 ≤ i ≤ r, we have a_{𝔪}^{i}H^{∗}(G, k) < 0.

The proof gives something stronger, which we formulate using strengthened versions of the condition of quasi-regularity. In the language of Definition 3.2, the above theorem is still true with quasi-regular replaced by strongly quasiregular or by very strongly quasi-regular, but with stronger bounds in condition (v). Namely, for strongly quasi-regular, the inequality in (v) is replaced by a_{𝔪}^{i}H^{∗}(G, k) ≤ −i for 0 ≤ i ≤ r, and for very strongly quasi-regular, it is replaced by a_{𝔪}^{i}H^{∗}(G, k) < −i for 0 ≤ i ≤ r − 1 and a_{𝔪}^{r}H^{∗}(G, k) ≤ −r.

We also prove that the equivalent conditions of even the very strongly quasiregular version always hold if the difference between the Krull dimension and the depth of H^{∗}(G, k) is at most two. This improves a theorem of Okuyama and Sasaki [22] for quasi-regular sequences. The proof of the following theorem can be found in §9.

Theorem 1.5. If the Krull dimension and the depth of H^{∗}(G, k) differ by at most two, then there is a very strongly quasi-regular sequence in H^{∗}(G, k). In particular, Reg H^{∗}(G, k) = 0.

In §10, we use the machinery we have developed to give an improvement of Carlson’s method [12, 13] for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. Carlson’s algorithm depends on some conjectures in the cohomology of finite groups. The conjectures are verified for the group in question during the course of the computation, so there is no uncertainty about the answer. But the method is not guaranteed to succeed, if one of the conjectures should fail for the group in question. Our improved version avoids the use of these conjectures, and gives improved bounds on how much must be computed before the computation can be deemed to be complete.

Theorem 10.1 is a slightly strengthened version of the following statement. Suppose that we’ve computed up to degree N, and that it’s at least far enough so that we’ve at least got a homogeneous sequence of parameters ζ_{1}, …, ζ_{r} in H^{∗}(G, k) (this can be checked by restriction to elementary abelian p-subgroups). Consider the ring τ_{N}H^{∗}(G, k) defined by the generators and relations we’ve found so far. If ζ_{1}, …, ζ_{r} are filter-regular in τ_{N}H^{∗}(G, k) and

N > max{0, Reg τ_{N}H^{∗}(G, k)} + ∑_{j=1}^{r}(|ζ_{j}| − 1)  (1.6)

then the obvious map τ_{N}H^{∗}(G, k) → H^{∗}(G, k) is an isomorphism. The given condition is guaranteed to hold for large enough N, and in the presence of Conjecture 1.1, the bound (1.6) becomes N > ∑_{j=1}^{r}(|ζ_{j}| − 1).

The inequality > in (1.6) can be replaced by ≥ if the p-rank of the center of a Sylow p-subgroup of G is at least two, or if we have some other a priori way of knowing that the depth of the cohomology ring is at least two.

Acknowledgement. I would like to thank Rodney Sharp for his comments on an earlier draft of this paper.

2. Notation and conventions

We shall need to use the theory of varieties for modules, so we begin by introducing the notation. For background, see for example [1, 20]. Let G be a finite group, and k be a field of characteristic p. We write V_{G} for the maximal ideal spectrum of H^{∗}(G, k). This is a homogeneous affine variety whose dimension is equal to the p-rank r_{p}(G), which we shall denote by r.

If M is a finitely generated kG-module, we write V_{G}(M) for the subvariety of V_{G} defined by the kernel of the action of H^{∗}(G, k) on Ext_{kG}^{∗}(M, M). Then

V_{G}(M_{1} ⊕ M_{2}) = V_{G}(M_{1}) ∪ V_{G}(M_{2})
V_{G}(M_{1} ⊗ M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2}).

A kG-module M is projective if and only if V_{G}(M) = {0}. If H is a subgroup of G, the restriction map res_{G,H}: H^{∗}(G, k) → H^{∗}(H, k) gives rise to a map res_{G,H}^{∗}: V_{H} → V_{G}.

If ζ is a nonzero element of H^{n}(G, k), then we write L_{ζ} for the kernel of a representative cocycle ζ̂: Ω^{n}k → k. This gives a short exact sequence

0 → L_{ζ} → Ω^{n}k → k → 0.  (2.1)

The variety V_{G}(L_{ζ}) is equal to the hypersurface V_{G}⟨ζ⟩ in V_{G} defined by regarding ζ as an element of the coordinate ring. If ζ_{1}, …, ζ_{r} is a homogeneous set of parameters in H^{∗}(G, k) then

V_{G}(L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}}) = V_{G}(L_{ζ_{1}}) ∩ ⋯ ∩ V_{G}(L_{ζ_{r}})
= V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{r}⟩ = {0}

and so L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r}} is projective.

Similarly, the variety of L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{r−1}} is one dimensional. So this module is periodic.

We work throughout this paper with graded rings satisfying the following conditions.

Condition 2.2. Let k be a field of characteristic p (possibly p = 0). The ℤgraded ring H is finitely generated by positive degree elements as an algebra over H^{0} = k, and is graded commutative in the sense that ab = (−1)^{|a||b|}ba. The maximal ideal 𝔪 is generated by the positive degree elements, so that

H = H^{0} ⊕ H^{>0} = k ⊕ 𝔪.

If M is a graded module over H, we use the notation M[i] for the shifted module with M[i]^{n} = M^{i+n}. The element of M[i] corresponding to m ∈ M is written σ^{i}m. The action of x ∈ H^{j} is given by x σ^{i}m = (−1)^{ij}σ^{i}(xm).

3. Filter-regular sequences

We shall need the following condition on a sequence of integers.

Condition 3.1. The sequence of integers d_{0}, …, d_{r} satisfies (a) d_{i} ≥ d_{i−1} − 1 for 1 ≤ i ≤ r,

(b) d_{i−1} ≥ d_{i} for 1 ≤ i ≤ r, and

(c) d_{0} ≥ −1.

Definition 3.2. Let H be a graded ring satisfying Condition 2.2, let M be a finitely generated graded H-module of Krull dimension r, and let d_{0}, …, d_{r} be integers satisfying Condition 3.1(a) and (b). Let ζ_{1}, …, ζ_{r} be a homogeneous system of parameters in H for M (i.e., M/(ζ_{1}, …, ζ_{r}) has finite length) with |ζ_{i}| = n_{i}. We say that ζ_{1}, …, ζ_{r} is a filter-regular system of parameters of type (d_{0}, …, d_{r}) for M if for each i = 0, …, r − 1, the map

(M/(ζ_{1}, …, ζ_{i}))^{j} → (M/(ζ_{1}, …, ζ_{i}))^{j+n_{i+1}}  (3.3)

induced by multiplication by ζ_{i+1} is injective for j > n_{1} + ⋯ + n_{i} + d_{i}, and (M/(ζ_{1}, …, ζ_{r}))^{j} = 0 for j > n_{1} + ⋯ + n_{r} + d_{r}. If there exist values of d_{0}, …, d_{r} for which this condition holds, then we say that ζ_{1}, …, ζ_{r} is filter-regular for M. The existence of a filter-regular sequence is guaranteed by the standard method of prime avoidance. The smaller the values of the d_{i}, the stronger the condition becomes. If M = H and d_{i} = −1 for 0 ≤ i ≤ r then the sequence is quasi-regular, in the terminology of [8].

We introduce two strengthened versions of the above quasi-regularity condition. If d_{i} = −i for 0 ≤ i ≤ r, then we say that the sequence is strongly quasiregular. We shall see that the existence of a strongly quasi-regular sequence is equivalent to Reg H = 0, so that Conjecture 1.1 states that the cohomology of a finite group has a strongly quasi-regular sequence.

If d_{i} = −i − 1 for 0 ≤ i ≤ r − 1 and d_{r} = −r, then we say that the sequence is very strongly quasi-regular. This is the strongest of these conditions for which the method of §7 works. There is quite a lot of evidence that the cohomology ring of a finite group always has a very strongly quasi-regular sequence. We shall prove this in the case where the Krull dimension and depth differ by at most two.

Lemma 3.4. Suppose that M has depth d. Then the first d terms of any filterregular sequence form a regular sequence.

Proof. (See also exercise 18.3.8 of Brodmann and Sharp [11]). Let ζ_{1}, …, ζ_{r} be a filter-regular sequence. We prove by induction on i that for 0 ≤ i ≤ d − 1, multiplication by ζ_{i+1} is injective on M/(ζ_{1}, …, ζ_{i}). It follows from the inductive hypothesis that M/(ζ_{1}, …, ζ_{i}) has depth d − i > 0. The existence of a regular element in this quotient means that if multiplication by ζ_{i+1} has a kernel then the kernel has elements of arbitrarily large degree. So ζ_{i+1} is regular, and the proof by induction is complete. □

Filter-regularity can be interpreted in a number of cohomological ways, and this is the subject of the next few sections. We begin by interpreting it in terms of the Koszul complex. We write 𝒦^{∗}(M; ζ_{1}, …, ζ_{i}) for the doubly graded Koszul complex of M with respect to parameters ζ_{1}, …, ζ_{i}. In order to fit with the indexing of the spectral sequences of §6, we regard the degree coming from M as vertical, and we make the Koszul degree horizontal and negative. So 𝒦^{0} is M with no shift, and 𝒦^{−i} is M shifted upwards in degree by ∑_{j=1}^{i} |ζ_{j}|.

Lemma 3.5. Let d_{0}, …, d_{r} be integers satisfying Condition 3.1(a) and (b). If ζ_{1}, …, ζ_{r} is a filter-regular sequence of type (d_{0}, …, d_{r}) for M with |ζ_{i}| = n_{i} then

H^{−s,t}(𝒦^{∗}(M; ζ_{1}, …, ζ_{r})) = 0 for t > n_{1} + ⋯ + n_{s} + d_{r−s}.

Proof. Write _{i}H^{−s,t} for H^{−s,t}(𝒦^{∗}(M; ζ_{1}, …, ζ_{i})). Then the short exact sequence of complexes

0 → 𝒦^{∗}(M; ζ_{1}, …, ζ_{i−1}) → 𝒦^{∗}(M; ζ_{1}, …, ζ_{i}) → 𝒦^{∗+1}(M; ζ_{1}, …, ζ_{i−1})[−n_{i}] → 0

gives rise to a long exact sequence

⋯ → _{i−1}H^{−s,t} → _{i}H^{−s,t} → _{i−1}H^{−s+1,t−n_{i}} ⟶^{ζ_{i}} _{i−1}H^{−s+1,t} → ⋯

This allows us to show by induction on i, using the inequality d_{i} ≥ d_{i−1} − 1, that _{i}H^{−s,t} = 0 provided s > 0 and t > n_{1} + ⋯ + n_{s} + d_{i−s}. If s = 0 then rH^{0,t} = (M/(ζ_{1}, …, ζ_{r}))^{t} is zero for t > n_{1} + ⋯ + n_{r} + d_{r}. □

Sometimes it is more convenient to work in terms of the stable Koszul complex, or in other words, in terms of local cohomology. This is the subject of the next section.

4. Local cohomology

Recall that for graded modules over a graded ring, local cohomology has two gradings. We write H_{𝔪}^{i,n}M for the degree n part of the ith local cohomology module of M. So the first grading gives local cohomological degree, and the second keeps track of internal degree.

The definition of Castelnuovo–Mumford regularity is usually only considered in the context of strictly commutative graded rings generated in degree one. The following definition from Eisenbud and Goto [18] seems to work well in our more general context.

Definition 4.1. Let H be a graded ring satisfying Condition 2.2. If M is a finitely generated graded H-module, we set

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

(or a_{𝔪}^{i}(M) = −∞ if H_{𝔪}^{i}M = 0). The Castelnuovo–Mumford regularity of M is defined to be

Reg M = max_{j≥0}{a_{𝔪}^{j}(M) + j}.

Remarks concerning the history and geometric significance of this definition can be found in §20.5 of Eisenbud [17].

Theorem 4.2. Let G be a finite group of p-rank r and let k be a field of characteristic p. Then Reg H^{∗}(G, k) ≥ 0.

Proof. The “last survivor” described in theorem 1.3 of [8] and reinterpreted in terms of local cohomology in theorem 4.1 of [5] (see also §5 below) says that H_{𝔪}^{r,−r}H^{∗}(G, k) ≠ 0. So a_{𝔪}^{r}(H^{∗}(G, k)) ≥ −r. □

The proof of the following lemma is similar to arguments from Trung [24, 25].

Lemma 4.3. Let H be a graded ring satisfying Condition 2.2, and let M be a finitely generated graded H-module. If ζ ∈ H^{n} is such that the kernel of ζ on M is 𝔪-torsion, then

a_{𝔪}^{i+1}(M) + n ≤ a_{𝔪}^{i}(M/ζM) ≤ max(a_{𝔪}^{i}(M), a_{𝔪}^{i+1}(M) + n).  (4.4)

Proof. The short exact sequence

0 → Ker(ζ) → M → M/Ker(ζ) → 0

shows that for i > 0 we have H_{𝔪}^{i}M = H_{𝔪}^{i}(M/Ker(ζ)). So for i ≥ 0 the exact sequence

0 → M/Ker(ζ)[−n] ⟶^{ζ} M → M/ζM → 0

gives a long exact sequence

⋯ → H_{𝔪}^{i,j}M → H_{𝔪}^{i,j}(M/ζM) → H_{𝔪}^{i+1,j−n}M ⟶^{ζ} H_{𝔪}^{i+1,j}M → ⋯

The element of highest degree in H_{𝔪}^{i+1}M is in the kernel of ζ, so this gives the desired inequalities. □

Theorem 4.5. Let H be a graded ring satisfying Condition 2.2, let M be a finitely generated graded H-module of Krull dimension r, and let d_{0}, …, d_{r} be integers satisfying Condition 3.1(a) and (b). Then the following are equivalent.

(i) There exists a filter-regular sequence ζ_{1}, …, ζ_{r} of type (d_{0}, …, d_{r}) for M. (ii) Every filter-regular sequence ζ_{1}, …, ζ_{r} for M has type (d_{0}, …, d_{r}). (iii) For 0 ≤ i ≤ r we have a_{𝔪}^{i}(M) ≤ d_{i}.

If these equivalent conditions hold, then

Reg M ≤ max_{0≤i≤r}{d_{i} + i}.

Proof. (i) ⇒ (iii): Suppose that ζ_{1}, …, ζ_{r} is a filter-regular sequence of type (d_{0}, …, d_{r}) for M with |ζ_{i}| = n_{i}. We use downward induction on s, beginning with the case s = r, to show that for all i ≥ 0 and 0 ≤ s ≤ r we have

a_{𝔪}^{i}(M/(ζ_{1}, …, ζ_{s})) ≤ n_{1} + ⋯ + n_{s} + d_{s+i}.  (4.6)

The case i = 0 comes from the definition of filter-regular sequence of type (d_{0}, …, d_{s}). The case i > 0 comes from the inductive hypothesis and the left half of the inequality (4.4). The case s = 0 gives (iii).

(iii) ⇒ (ii): If a_{𝔪}^{i}(M) ≤ d_{i} for all i ≥ 0 and ζ_{1}, …, ζ_{r} is filter-regular, then we use (upward) induction on s to prove that (4.6) holds for all i ≥ 0 and all 0 ≤ s ≤ r. The case s = 0 is the given condition. For s > 0, using the right half of inequality (4.4), we have

a_{𝔪}^{i}(M/(ζ_{1}, …, ζ_{s})) ≤ max(a_{𝔪}^{i}(M/(ζ_{1}, …, ζ_{s−1})), a_{𝔪}^{i+1}(M/(ζ_{1}, …, ζ_{s−1})) + n_{s})
≤ max(n_{1} + ⋯ + n_{s−1} + d_{s+i−1}, n_{1} + ⋯ + n_{s} + d_{s+i}).

Condition 3.1(a) ensures that

n_{1} + ⋯ + n_{s−1} + d_{s+i−1} ≤ n_{1} + ⋯ + n_{s} + d_{s+i}

since n_{s} ≥ 1. So by induction we obtain inequality (4.6) for all i ≥ 0 and 0 ≤ s ≤ r. The case i = 0 gives (ii).

The implication (ii) ⇒ (i) is obvious. Finally, if (iii) holds then the last statement of the theorem is immediate from the definition of regularity. □

We spell out several particular cases of this theorem.

Corollary 4.7. Suppose that H is a graded ring satisfying Condition (2.2). Then

(a) The following are equivalent.

(i) There is a quasi-regular sequence for H,

(ii) Every filter-regular sequence for H is quasi-regular,

(iii) For all i ≥ 0 we have a_{𝔪}^{i}(H) < 0. Under these conditions we have Reg H < r.

(b) The following are equivalent.

(i) There is a strongly quasi-regular sequence for H,

(ii) Every filter-regular sequence for H is strongly quasi-regular,

(iii) Reg H ≤ 0 (i.e., for all i ≥ 0 we have a_{𝔪}^{i}(H) ≤ −i).

(c) The following are equivalent.

(i) There exists a very strongly quasi-regular sequence for H,

(ii) Every filter-regular sequence for H is very strongly quasi-regular, (iii) For 0 ≤ i ≤ r − 1 we have a_{𝔪}^{i}(H) < −i, and a_{𝔪}^{r}(H) ≤ −r.

Under these conditions, we have Reg H ≤ 0. □

Corollary 4.8. There exists a strongly quasi-regular sequence for H^{∗}(G, k) if and only if Reg H^{∗}(G, k) = 0.

Proof. This follows from Theorem 4.2 and Corollary 4.7. □

5. Free resolutions

In this section we relate the invariants a_{𝔪}^{i}(M) to some invariants defined in terms of free resolutions over polynomial algebras. The theorems of this section generalize some of the theorems of Eisenbud and Goto [18], with proofs similar to the outline given in exercise 15.3.7 of Brodmann and Sharp [11].

Suppose that H satisfies Condition 2.2, and that M is a finitely generated graded H-module of Krull dimension r. Using Noether normalization, we choose a polynomial ring R = k[ζ_{1}, …, ζ_{r}] with |ζ_{i}| = n_{i} and a homomorphism of graded rings R → H so that M is a finitely generated graded R-module, and ζ_{1}, …, ζ_{r} is filter-regular for M. Let

0 → F_{r} → ⋯ → F_{0} → M → 0

be the minimal resolution of M over R. Then we define β_{j}^{R}(M) to be the largest degree of a generator of F_{j} as an R-module (or β_{j}^{R}(M) = −∞ if F_{j} = 0). This is the same as the largest value of ℓ such that Tor_{j,ℓ}^{R}(M, k) ≠ 0.

Theorem 5.1. β_{0}^{R}(M) ≤ max_{i}{a_{𝔪}^{i}(M) + i} + ∑_{j=1}^{r}(n_{j} − 1).

Proof. We prove by downward induction on s that

β_{0}^{R}(M/(ζ_{1}, …, ζ_{s})) ≤ max_{i}{a_{𝔪}^{i}(M/(ζ_{1}, …, ζ_{s})) + i} + ∑_{j=s+1}^{r}(n_{j} − 1). (5.2)

This is clearly true for s = r, so suppose s < r. By the left hand inequality of (4.4), we have

a_{𝔪}^{i+1}(M/(ζ_{1}, …, ζ_{s})) + i + 1 + ∑_{j=s+1}^{r}(n_{j} − 1)

≤ a_{𝔪}^{i}(M/(ζ_{1}, …, ζ_{s+1})) + i + ∑_{j=s+2}^{r}(n_{j} − 1).

Now given a minimal set of generators for M/(ζ_{1}, …, ζ_{s+1}), we can lift to elements of M/(ζ_{1}, …, ζ_{s}) and adjoin a set of 𝔪-torsion elements to obtain a set of generators for M/(ζ_{1}, …, ζ_{s}). So we have

β_{0}^{R}(M/(ζ_{1}, …, ζ_{s})) ≤ max{a_{𝔪}^{0}(M/(ζ_{1}, …, ζ_{s})), β_{0}^{R}(M/(ζ_{1}, …, ζ_{s+1}))}

≤ max{a_{𝔪}^{0}(M/(ζ_{1}, …, ζ_{s})) + ∑_{j=s+1}^{r}(n_{j} − 1), β_{0}^{R}(M/(ζ_{1}, …, ζ_{s+1}))}

So if (5.2) is true for s + 1 then it is true for s. □

Corollary 5.3. Let d_{0}, …, d_{r} be integers satisfying Condition 3.1(a). Then

β_{0}^{R}(M) − d_{r} ≤ max_{i}{a_{𝔪}^{i}(M) − d_{i}} + ∑_{j=1}^{r} n_{j}.

Proof. This follows from the theorem together with the inequality

−d_{r} ≤ −d_{i} − i + r,

which in turn follows from Condition 3.1(a). □

Lemma 5.4. Let F be a finitely generated free graded R-module (i.e., a finite direct sum of shifts of copies of R). Then a_{𝔪}^{r}(F) = β_{0}^{R}(F) − ∑_{j=1}^{r} n_{j}.

Proof. A direct computation using the Koszul complex shows that a_{𝔪}^{r}(R) = − ∑_{j=1}^{r} n_{j}. The module F consists of a direct sum of shifts of copies of R, with top generator in degree β_{0}^{R}(F). □

The following is the main theorem of this section.

Theorem 5.5. Let d_{0}, …, d_{r} be integers satisfying Condition 3.1(a) and (b), and suppose that M, H and R are as described at the beginning of this section. Then

max_{i≥0}{a_{𝔪}^{i}(M) − d_{i}} = max_{i≥0}{β_{r−i}^{R}(M) − d_{i}} − ∑_{j=1}^{r} n_{j}.  (5.6)

Proof. We prove this by induction on the projective dimension of M as an Rmodule. In the case of projective dimension zero, the theorem follows directly from Lemma 5.4. So suppose that M has projective dimension at least one. Let 0 → N → F → M → 0 be the first stage of a minimal resolution of M over R. Set d′_{i} = d_{i−1} if i > 0 and d′_{0} = d_{0}. Then the sequence of integers d′_{0}, …, d′_{r} satisfies Condition 3.1(a) and (b). We have H_{𝔪}^{i}F = 0 for i ≠ r, and a_{𝔪}^{r}(F) = β_{0}^{R}(M) − ∑_{j=0}^{r} n_{j} by Lemma 5.4. So the long exact sequence in local cohomology

⋯ → H_{𝔪}^{i−1}N → H_{𝔪}^{i−1}F → H_{𝔪}^{i−1}M → H_{𝔪}^{i}N → H_{𝔪}^{i}F …

shows that for 0 < i < r we have a_{𝔪}^{i}(N) = a_{𝔪}^{i−1}(M), so that

a_{𝔪}^{i}(N) − d′_{i} = a_{𝔪}^{i−1}(M) − d_{i−1},

while a_{𝔪}^{0}(N) = −∞. Furthermore,

a_{𝔪}^{r−1}(M) ≤ a_{𝔪}^{r}(N) ≤ max{a_{𝔪}^{r−1}(M), β_{0}^{R}(M) − ∑_{j=1}^{r} n_{j}},

so that using Condition 3.1(b), we have

a_{𝔪}^{r−1}(M) − d_{r−1} ≤ a_{𝔪}^{r}(N) − d′_{r}

≤ max{a_{𝔪}^{r−1}(M) − d_{r−1}, β_{0}^{R}(M) − d_{r−1} − ∑_{j=1}^{r} n_{j}}
≤ max{a_{𝔪}^{r−1}(M) − d_{r−1}, β_{0}^{R}(M) − d_{r} − ∑_{j=1}^{r} n_{j}},

and

a_{𝔪}^{r}(M) ≤ a_{𝔪}^{r}(F) = β_{0}^{R}(M) − ∑_{j=1}^{r} n_{j}.

On the other hand, for 0 < i ≤ r we have β_{r−i}^{R}(N) = β_{r−i+1}^{R}(M), so that

β_{r−i}^{R}(N) − d′_{i} = β_{r−i+1}^{R}(M) − d_{i−1},

and β_{r}^{R}(N) = −∞.

Combining these statements, we obtain

max_{i≥0}{a_{𝔪}^{i}(M) − d_{i}} ≤ max{max_{i≥0}{a_{𝔪}^{i}(N) − d′_{i}}, a_{𝔪}^{r}(M) − d_{r}}

≤ max{max_{i≥0}{a_{𝔪}^{i}(N) − d′_{i}}, β_{0}^{R}(M) − d_{r} − ∑_{j=1}^{r} n_{j}}.

Combining this with Corollary 5.3, we obtain

max_{i≥0}{a_{𝔪}^{i}(M) − d_{i}} = max{max_{i≥0}{a_{𝔪}^{i}(N) − d′_{i}}, β_{0}^{R}(M) − d_{r} − ∑_{j=1}^{r} n_{j}}.

By the inductive hypothesis, the theorem is true for N. So we have

max_{i≥0}{a_{𝔪}^{i}(M) − d_{i}} = max{max_{i≥0}{β_{r−i}^{R}(N) − d′_{i}}, β_{0}^{R}(M) − d_{r}} − ∑_{j=1}^{r} n_{j}

= max_{i≥0}{β_{r−1}^{R}(M) − d_{i}} − ∑_{j=1}^{r} n_{j}.

This completes the inductive proof of the theorem. □

Note that the summation term in part (ii) of the following corollary is usually omitted in the literature because of the assumption that the generators are in degree one.

Corollary 5.7. We have

(i) max_{i≥0}{a_{𝔪}^{i}(M)} = max_{i≥0}{β_{i}^{R}(M)} − ∑_{j=1}^{r} n_{j},

(ii) Reg M = max_{i≥0}{a_{𝔪}^{i}(M) + i} = max_{i≥0}{β_{i}^{R}(M) − i} − ∑_{j=1}^{r}(n_{j} − 1). □

Corollary 5.8. (i) There exists a quasi-regular sequence in H if and only if for all i ≥ 0 we have β_{i}^{R}(H) < ∑_{j=1}^{r} n_{j}.

(ii) There exists a strongly quasi-regular sequence in H if and only if for all i ≥ 0 we have β_{i}^{R}(H) ≤ i + ∑_{j=1}^{r}(n_{j} − 1).

(iii) There exists a very strongly quasi-regular sequence in H if and only if β_{0}^{R}(H) ≤ ∑_{j=1}^{r}(n_{j} − 1), and for 1 ≤ i ≤ r, β_{i}^{R}(H) < i + ∑_{j=1}^{r}(n_{j} − 1). □

6. The last survivor

In this section, we construct spectral sequences closely related to those of Benson and Carlson [8], Carlson [12] and Greenlees [21]. We describe the “last survivor” from §7 of [8] in terms of these spectral sequences.

Let ζ_{1}, …, ζ_{r} be a homogeneous system of parameters in H^{∗}(G, k) with |ζ_{i}| = n_{i}. then the last survivor is a nonzero element of H^{∗}(G, k) of degree ∑_{i=1}^{r}(n_{i}−1), not in the ideal generated by the parameters, whose existence is guaranteed from the spectral sequence. This discussion will be needed in the proof of Theorem 7.1, and again in §10. The proof given here for the existence of the last survivor is significantly simpler than the one to be found in [8].

Let L_{ζ_{i}} be the modules defined by (2.1). Set

M_{0} = k, M_{i} = L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{i}} (i ≥ 1).

Note that M_{r} is projective, and M_{r−1} is periodic, by the argument given in §2.

Tensoring the short exact sequence (2.1) for L_{ζ_{i}} with M_{i−1} gives a short exact sequence

0 → M_{i} → Ω^{n_{i}}M_{i−1} ⊕ (projective) → M_{i−1} → 0.  (6.1)

To build the spectral sequence, for each parameter ζ_{i} we truncate the sequence (2.1) to form a cochain complex

0 → Ω^{n_{i}}k ⟶^{ζ̂_{i}} k → 0  (6.2)

where k is in degree zero and Ω^{n_{i}}k is in degree −1. Let C_{i}^{∗} be the tensor product of these complexes for ζ_{1}, …, ζ_{i}. Let P̂_{∗} be a complete resolution of k as a kG-module, and set _{i}E_{0}^{−s,t} = Hom_{kG}(P̂_{t}, C_{i}^{−s}). Then

iE_{1}^{−s,t} = Ĥ^{t}(G, C_{i}^{−s}) ≅ 𝒦^{∗}(Ĥ^{∗}(G, k); ζ_{1}, …, ζ_{i}),

the Koszul complex for Ĥ^{∗}(G, k) with respect to the parameters ζ_{1}, …, ζ_{i}. The d_{1} differential is the same as the Koszul differential. This _{i}E_{1} page is indexed so that it lives in columns −i to 0 in the plane, with each column consisting of a number of copies of Tate cohomology, suitably shifted. The copy in degree zero is unshifted. Using the other spectral sequence of this double complex, we see that the above spectral sequence converges to Ĥ^{t−s+i}(G, M_{i}). So we have

iE_{1}^{∗∗} = 𝒦^{∗}(Ĥ^{∗}(G, k); ζ_{1}, …, ζ_{i}) ⇒ Ĥ^{∗+i}(G, M_{i}).

In particular, if i = r, we have

rE_{1}^{∗∗} = 𝒦^{∗}(Ĥ^{∗}(G, k); ζ_{1}, …, ζ_{r}) ⇒ 0.  (6.3)

This spectral sequence satisfies a duality which combines the Poincaré duality of [8] with Tate duality, as explained in §6 of [7]. We recall that if we dualize the complex (6.2), we obtain a complex of the form

0 → k ⟶^{±ζ̂_{i}} Ω^{−n_{i}}k → 0

where the sign is (−1)^{n_{i}(n_{i}+1)/2} (see §5 of [8]). In particular, L_{ζ_{i}}^{∗} ≅ Ω^{−n_{i}−1}L_{ζ_{i}}, M_{i}^{∗} ≅ Ω^{−n_{1}−···−n_{i}−i}M_{i}, and Ĥ^{j}(G, M_{i}^{∗}) ≅ Ĥ^{j+n_{1}+···+n_{i}+i}(G, M_{i}).

Combining this with Tate duality, we obtain a duality between _{i}E_{1}^{−s,t} and iE_{1}^{s−i,n_{1}+···+n_{i}−t−1}. Since this duality is induced by a homomorphism of double complexes at the _{i}E_{0} level, it commutes with the differentials in the spectral sequence and passes down to a duality on each page. The duality on _{i}E_{∞}^{∗∗} is a graded version of the duality between Ĥ^{t−s+i}(G, M_{i}) and

Ĥ^{(s−i)+(n_{1}+···+n_{i}−t−1)+i}(G, M_{i}) ≅ Ĥ^{s−t−i−1}(G, M_{i}^{∗}).

If H^{∗}(G, k) is Cohen–Macaulay, then ζ_{1}, …, ζ_{r} is a regular sequence. In this case, provided r ≥ 2, there are no nonzero products in Tate cohomology from negative to positive degrees, by lemma 2.1 and theorem 3.1 of [7]. It follows that _{r}E_{2}^{∗∗} is concentrated in the −r and 0 columns. So d_{j} = 0 for 2 ≤ j < r, and d_{r} is an isomorphism from the −r column to the 0 column.

If E is an elementary abelian p-subgroup of G of maximal rank (i.e., rank r), then H^{∗}(E, k) is Cohen–Macaulay. Restriction from G to E induces a map of the above spectral sequences. It follows that the identity element in _{r}E_{1}^{0,0} is not hit by any differential until the _{r}E_{r} page. So there is a uniquely defined element of _{r}E_{r}^{−r,r−1} which is sent by d_{r} to the identity element. The image of

this element under the map _{r}E_{r}^{−r,r−1} ⟶^{⊆} _{r}E_{1}^{−r,r−1} ⟶^{≅} Ĥ^{−n_{1}−···−n_{r}+r−1}(G, k) is

written γ(ζ_{1}, …, ζ_{r}) (see §7 of [8]). This element, or an element of positive Tate cohomology with nonzero pairing with this element, is called the “last survivor.” Note that the element of negative degree is well defined, whereas the element of positive degree is really a coset. It is also observed in [8] that since γ(ζ_{1}, …, ζ_{r}) has nonzero restriction to every elementary abelian p-subgroup of maximal rank, and restriction is Tate dual to transfer, the last survivor in positive degree can be taken to be a transfer from any given elementary abelian p-subgroup of maximal rank.

Now there are maps of cochain complexes

\xymatrix{0 \ar[r] & 0 \ar[r] \ar[d] & k \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ω^{n_{i}}k \ar[r]^{ζ̂_{i}} & k \ar[r] & 0.}

So if 1 ≤ i < r, then there is a map of complexes C_{i−1}^{∗} → C_{i}^{∗} and hence a map of spectral sequences _{i−1}E_{∗}^{∗∗} → _{i}E_{∗}^{∗∗}. Composing, we get maps of spectral sequences from each _{i}E_{∗}^{∗∗} to _{r}E_{∗}^{∗∗}, which do the obvious thing at the E_{1} page. Since the identity element in _{r}E_{1}^{0,0} is the image under d_{r} of the element coming from the last survivor, it follows that for i < r, the identity element in _{i}E_{1}^{0,0} survives to _{i}E_{∞}^{0,0}. Applying the duality, _{i}E_{1}^{−i,n_{1}+···+n_{i}−1} is a one dimensional space isomorphic to Ĥ^{−1}(G, k), and this one dimensional space survives to _{i}E_{∞}^{−i,n_{1}+···+n_{i}−1} provided i < r.

7. Module theoretic interpretation

In this section, we interpret filter-regularity of a sequence of parameters in H^{∗}(G, k) in module theoretic terms.

Theorem 7.1. Let d_{0}, …, d_{r} be integers satisfying Condition 3.1. Then a sequence of parameters ζ_{1}, …, ζ_{r} for H^{∗}(G, k) is filter-regular of type (d_{0}, …, d_{r}) if and only if for each i = 0, …, r − 1, multiplication by ζ_{i+1} is injective on H^{j}(G, M_{i}) for j > n_{1} + … n_{i} + i + d_{i}.

Under these circumstances, if 1 ≤ i ≤ r − 1 and j > n_{1} + ⋯ + n_{i} + d_{i}, then we have

H^{j+i}(G, M_{i}) ≅ (H^{∗}(G, k)/(ζ_{1}, …, ζ_{i}))^{j}.

Proof. For j ≥ n_{1} + ⋯ + n_{i}, this theorem is an easy inductive argument, using the long exact sequence in cohomology coming from the short exact sequence (6.1). If some of the d_{i} are negative, the improvement in the bound comes from a careful investigation of Tate cohomology in degree −1.

We begin the induction as follows. The long exact sequence in Tate cohomology coming from the short exact sequence (2.1) for ζ_{1} is

⋯ → Ĥ^{j+n_{1}}(G, M_{1}) → Ĥ^{j}(G, k) ⟶^{ζ_{1}} Ĥ^{j+n_{1}}(G, k) → Ĥ^{j+n_{1}+1}(G, M_{1}) → ⋯

By a theorem of Duflot [16], the depth of H^{∗}(G, k) is at least one. So by Lemma 3.4, every filter-regular sequence begins with a regular element. It follows that in the above sequence, the map given by multiplication by ζ_{1} is injective for j ≥ 0. So we have H^{j+1}(G, M_{1}) ≅ (H^{∗}(G, k)/(ζ_{1}))^{j} for j > n_{1} − 1. Provided r > 1, multiplication by ζ_{1} is the zero map on Ĥ^{−1}(G, k), and so H^{n_{1}}(G, M_{1}) ≅ H^{n_{1}−1}(G, k) as required.

Next, the long exact sequence in Tate cohomology coming from the short exact sequence (6.1) for ζ_{2} is

⋯ → Ĥ^{j+n_{2}}(G, M_{2}) → Ĥ^{j}(G, M_{1}) ⟶^{ζ_{2}} Ĥ^{j+n_{2}}(G, M_{1}) → Ĥ^{j+n_{2}+1}(G, M_{2}) → ⋯

Provided j ≥ n_{1}, we have

\xymatrix{(Ĥ^{∗}(G, k)/(ζ_{1}))^{j−1} \ar[r]^{ζ_{2}} \ar[d]^{≅} & (Ĥ^{∗}(G, k)/(ζ_{1}))^{j+n_{2}−1} \ar[d]^{≅} \\ Ĥ^{j}(G, M_{1}) \ar[r]^{ζ_{2}} & Ĥ^{j+n_{2}}(G, M_{1})}

and so ζ_{2} is injective on Ĥ^{j}(G, M_{1}) for j > n_{1} +d_{1} if and only if it is injective on (H^{∗}(G, k)/(ζ_{1}))^{j} for j > n_{1} + d_{1} − 1, except possibly in the case where d_{1} = −2 (its smallest possible value) and j = n_{1} −1. We also deduce, for the same values of j, that

H^{j+n_{2}+1}(G, M_{2}) ≅ (H^{∗}(G, M_{1})/(ζ_{2}))^{j+n_{2}} ≅ (H^{∗}(G, k)/(ζ_{1}, ζ_{2}))^{j+n_{2}−1}.

In the case where d_{1} = −2 and j = n_{1} − 1, we must work harder. In this case, we have the following commutative diagram.

[figure: Ĥ^{−2}(G, k); ζ_{1}; ζ_{2}; H^{n_{1}−2}(G, k); ζ_{2}; H^{n_{1}−1}(G, M_{1}); Ĥ^{−1}(G, k); 0; 0; d_{2}; H^{n_{2}−2}(G, k); ζ_{1}; H^{n_{1}+n_{2}−2}(G, k); H^{n_{1}+n_{2}−1}(G, M_{1}); 0; H^{n_{2}−1}(G, k); H^{n_{1}+n_{2}}(G, M_{2})]

Now Ĥ^{−1}(G, k) is one dimensional. The issue is whether an element x of H^{n_{1}−1}(G, M_{1}) with nonzero image α in Ĥ^{−1}(G, k) can make H^{n_{1}+n_{2}}(G, M_{2}) too small. Lift ζ_{2}x ∈ H^{n_{1}+n_{2}−1}(G, M_{1}) to an element y of H^{n_{1}+n_{2}−2}(G, k). This is well defined modulo (ζ_{1}, ζ_{2}), and is in fact the Massey product ⟨ζ_{1}, α, ζ_{2}⟩. The crucial observation is that this element is d_{2}(α) ∈ H^{∗}(G, k)/(ζ_{1}, ζ_{2}), where d_{2} is the differential in the spectral sequence _{2}E_{2}^{∗∗} constructed in the last section. Provided r ≥ 3, we showed that this differential is zero. It follows that y can be adjusted by an element in the image of ζ_{1} so that y = ζ_{2}z with z ∈ H^{n_{1}−2}(G, k). The image u of z in H^{n_{1}−1}(G, M_{1}) has the property that ζ_{2}u = ζ_{2}x in H^{n_{1}+n_{2}−1}(G, M_{1}). This shows that

H^{n_{1}+n_{2}}(G, M_{2}) ≅ (H^{∗}(G, k)/(ζ_{1}, ζ_{2}))^{n_{1}+n_{2}−2}

as required.

The inductive case is similar, but it is clear that it is more efficient to express the last step directly in terms of the spectral sequence. The long exact sequence

⋯ → Ĥ^{j+n_{i}}(G, M_{i}) → Ĥ^{j}(G, M_{i−1}) ⟶^{ζ_{i}} Ĥ^{j+n_{i}}(G, M_{i−1}) → Ĥ^{j+n_{i}+1}(G, M_{i}) → ⋯

deals with everything, except in the case where d_{i} = −i−1 (its smallest possible value), and then there is a question as to what happens in the lowest degree. In terms of the spectral sequence, the information we have is that _{i}E_{2}^{s,t} = 0 provided s < 0 and s + t ≥ n_{1} + ⋯ + n_{i} − i − 1, except for _{i}E_{2}^{−i,n_{1}+···+n_{i}−1}, which is isomorphic to the one dimensional space Ĥ^{−1}(G, k). Provided i < r, this element survives to_{i}E_{∞} by the discussion at the end of the previous section. The inductive step then follows from the fact that this spectral sequence converges to Ĥ^{∗+i}(G, M_{i}).

At the end of the induction, notice that the equivalent condition does not mention d_{r}. This is because M_{r} is a projective module (see §2), and so H^{j}(G, M_{r}) = 0 for all j > 0. So ζ_{r}: H^{j}(G, M_{r−1}) → H^{j}(G, M_{r−1}) is an isomorphism for j > 0, and it follows that (H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}))^{j} = 0 for j > n_{1} +⋯+n_{r} +d_{r−1}. Since d_{r} ≥ d_{r−1}, the last condition for filter-regularity holds, and we are done. □

Corollary 7.2. Suppose that ζ_{1}, …, ζ_{r} is a filter-regular system of parameters of type (d_{0}, …, d_{r}) for H^{∗}(G, k). Then ζ_{1}, …, ζ_{r} is also filter-regular of type (d_{0}, …, d_{r−2}, d_{r−2} − 1, d_{r−2} − 1).

Proof. As mentioned in §2, the module M_{r−1} is periodic, and M_{r} is projective. So multiplication by ζ_{r} is an isomorphism from H^{j}(G, M_{r−1}) to H^{j+n_{r}}(G, M_{r−1}) for j > 0, and H^{j}(G, M_{r}) = 0 for j > 0. It follows that the condition on d_{r−1} in Theorem 7.1 is automatically satisfied. There is no condition on d_{r} in the theorem except d_{r} ≥ d_{r−1}. □

We shall improve on this corollary in Theorem 9.10.

8. Dickson invariants

In this section we review the construction of the Dickson invariants in the cohomology ring of a finite group, using the theory of Chern classes. We begin by reminding the reader that the cohomology ring of BU(n) is a polynomial ring on the Chern classes c_{1}, …, c_{n},

H^{∗}(BU(n); ℤ) = ℤ[c_{1}, …, c_{n}],

where |c_{i}| = 2i. If R is a commutative ring of coefficients then the universal coefficient theorem gives H^{∗}(BU(n); R) = R[c_{1}, …, c_{n}]. The total Chern class is defined to be the inhomogeneous element c = 1 + c_{1} + c_{2} + · · ·.

The maximal torus in U(n) is U(1)^{n}, whose cohomology is R[x_{1}, …, x_{n}] with |x_{i}| = 2. The restriction map H^{∗}(BU(n); R) → H^{∗}(BU(1)^{n}; R) takes c_{i} to the ith elementary symmetric function of x_{1}, …, x_{n}.

Let G be a finite group. A complex unitary representation ρ: G → U(n) gives rise to a restriction map in cohomology ρ^{∗}: H^{∗}(BU(n); R) → H^{∗}(BG; R).

Recall that the Chern classes of ρ are defined to be c_{i}(ρ) = ρ^{∗}(c_{i}) ∈ H^{∗}(BG; R) = H^{∗}(G, R). For example, if ρ is a one dimensional representation, there is only one Chern class c_{1}(ρ). It is the image of ρ under the map

Hom(G, U(1)) ≅ H^{1}(G, U(1)) ≅ H^{2}(G, ℤ) → H^{2}(G, R).

The total Chern class of a direct sum of representations is the product of the total Chern classes, c(ρ_{1} ⊕ ρ_{2}) = c(ρ_{1})c(ρ_{2}). We are interested in what happens when ρ = ρ_{G} is the regular representation.

If E is an elementary abelian group of order p^{r}, we write H^{•}(G, k) for the subring of H^{∗}(E, k) generated by the Bocksteins of the degree one elements. So H^{•}(E, k) is a polynomial ring on r generators in degree two. The regular representation of E decomposes as a direct sum of all the one dimensional representations, once each. So ρ_{E} factors as E → U(1)^{p^{r}} → U(p^{r}). As ρ runs over the one dimensional representations, c_{1}(ρ) runs over all the elements of

H^{2·}(E, 𝔽_{p}) = H^{2}(E, 𝔽_{p}) ∩ H^{•}(E, 𝔽_{p}) ⊆ H^{2}(E, k).

So c(ρ_{E}) = ∏_{w∈H^{2·}(E,𝔽_{p})}(1 + w), which means that c_{i}(ρ_{E}) is the ith elementary symmetric function of the elements of H^{2·}(E, 𝔽_{p}).

We recall from §8.1 of [2] that the invariants of Aut(E) = GL(r, 𝔽_{p}) on H^{•}(E, k) are given by

H^{•}(E, k)^{GL(r,𝔽_{p})} = k[c_{r,r−1}, …, c_{r,0}]

where the Dickson invariants c_{r,r−j} are defined by the equation

∏_{w∈H^{2·}(E,𝔽_{p})}(X − w) = X^{p^{r}} − c_{r,r−1}X^{p^{r−1}} + c_{r,r−2}X^{p^{r−2}} − ⋯ + (−1)^{r}c_{r,0}X.

In particular, we see from this that the ith elementary symmetric function of the elements of H^{2·}(E, 𝔽_{p}) vanishes except when i is of the form p^{r} − p^{r−j}, in which case it is the Dickson invariant c_{r,r−j}. Interpreting this in terms of Chern classes, we see that for an elementary abelian p-group of order p^{r}, the Chern class c_{i}(ρ_{E}) vanishes except when i is of the form p^{r} − p^{r−j}, in which case it is equal to the Dickson invariant c_{r,r−j} ∈ H^{2(p^{r}−p^{r−j})}(E, k), so that

c(ρ_{E}) = 1 + c_{r,r−1} + ⋯ + c_{r,0}.

Now if S is any p-group of order p^{n} and S′ is a subgroup of S of index p^{a}, then the restriction of ρ_{S} to S′ is a direct sum of p^{a} copies of ρ_{S′}. So res_{S,S′} c(ρ_{S}) = c(ρ_{S′})^{p^{a}}. In particular, if E is an elementary abelian subgroup of order p^{r} then

res_{S,E} c(ρ_{S}) = c(ρ_{E})^{p^{n−r}} = 1 + c_{r,r−1}^{p^{n−r}} + c_{r,r−2}^{p^{n−r}} + ⋯ + c_{r,0}^{p^{n−r}}.

So we define D_{j}(S) = c_{p^{n}−p^{n−j}}(ρ_{S}), and we have

res_{S,S′} D_{j}(S) = D_{j}(S′)^{|S:S′|},  (8.1)

and in particular

res_{S,E} D_{j}(S) = \begin{cases}c_{r,r−j}^{p^{n−r}} & j ≤ r \\ 0 & j > r. \end{cases}

If G is any finite group, let S be a Sylow p-subgroup of G, of order p^{n}. Then in the language of the Cartan–Eilenberg stable element method [15], D_{j}(S) is a stable element in H^{∗}(S, k) for the fusion in G. So it is an element of H^{∗}(G, k) ⊆ H^{∗}(S, k). So we write D_{j}(G) ∈ H^{2(p^{n}−p^{n−j})}(G, k) for these elements.

We have proved the following theorem on the existence of Dickson invariants in the cohomology ring of a finite group.

Theorem 8.2. Let G be a finite group and k a field of characteristic p. If |G|_{p} = p^{n}, then there are elements D_{j}(G) ∈ H^{2(p^{n}−p^{n−j})}(G, k) for j ≥ 0, with the following properties.

(i) If H is a subgroup of G then res_{G,H} D_{j}(G) = D_{j}(H)^{|G:H|_{p}}.

(ii) If E is an elementary abelian group of order p^{r} then D_{j}(E) is the Dickson invariant c_{r,r−j} if j ≤ r, and D_{j}(E) = 0 if j > r. □

We interpret the p-power powers in the exponents in the above theorem as saying that the Dickson invariants are dilated by these powers of p. Of course, there is often a copy of the Dickson invariants in H^{∗}(G, k) with much smaller dilation. For example, in [4] it is shown that the mod two cohomology of the sporadic Conway group Co_{3} contains a copy of the rank four Dickson invariants with no dilation, namely a polynomial ring on generators of degrees 8, 12, 14 and 15.

Definition 8.3. We say that a homogeneous system of parameters ζ_{1}, …, ζ_{r} in H^{∗}(G, k) satisfies the rank-restriction condition if, for 1 ≤ i ≤ r, ζ_{i} restricts to zero on every elementary abelian p-subgroup of G of rank less than i.

Lemma 8.4. Suppose that ζ_{1} …, ζ_{r} is a homogeneous set of parameters for H^{∗}(G, k) that satisfies the rank-restriction condition. If E is an elementary abelian p-subgroup of G of rank i then the restrictions of ζ_{1}, …, ζ_{i} to H^{∗}(E, k) form a homogeneous system of parameters, and ζ_{i+1}, …, ζ_{r} restrict to zero.

Proof. It is clear from the definition that ζ_{i+1}, …, ζ_{r} restrict to zero on H^{∗}(E, k). By Evens’ finite generation theorem [19], H^{∗}(E, k) is finitely generated as a module over the image of restriction from G. Since H^{∗}(G, k) is finitely generated as a module over k[ζ_{1}, …, ζ_{r}], it follows that H^{∗}(E, k) is finitely generated as a module over the subring generated by the restrictions of ζ_{1}, …, ζ_{i}. Since H^{∗}(E, k) has Krull dimension i, it follows that these restrictions form a homogeneous system of parameters for H^{∗}(E, k). □

Theorem 8.5. The Dickson invariants satisfy the rank-restriction condition.

Proof. It follows from Theorem 8.2 that if E is an elementary abelian p-subgroup of G of rank less than i then res_{G,E}(D_{i}(G)) = D_{i}(E)^{|G:E|_{p}} = 0. □

We shall make use of Theorem 8.5 in §9.

9. Virtual projectivity

The purpose of this section is to prove Theorem 9.6. The proof involves the notion of virtual projectivity introduced by Carlson, Peng and Wheeler [14]. We begin by recalling the basic definitions. Let W be a finitely generated kGmodule. A sequence

0 → M_{1} → M_{2} → M_{3} → 0  (9.1)

is said to be W -split if the sequence

0 → M_{1} ⊗ W → M_{2} ⊗ W → M_{3} ⊗ W → 0

obtained by tensoring with W splits.

A module M is said to be relatively W -projective if every homomorphism to the module M_{3} in a W -split sequence (9.1) lifts to M_{2}.

For any kG-modules M and N, there is a transfer map

Tr_{W} : Hom_{kG}(M ⊗ W, N ⊗ W) → Hom_{kG}(M, N).

For example, if W is the permutation module on the cosets of a subgroup H then the image of Tr_{W} agrees with the image of Tr_{H,G}. More generally, if W is the direct sum of such permutation modules as H runs over the elements of a collection ℋ of subgroups of G, then the image of Tr_{W} is equal to the sum of the images of Tr_{H,G} for H ∈ ℋ.

We write V_{G}(Tr_{W}) for the subvariety of V_{G} defined by the image of

Tr_{W} : Hom_{kG}(W, W) → Hom_{kG}(k, k).

Theorem 9.2. Let W be a finitely generated kG-module and let M be any kGmodule. Then the following are equivalent.

(i) M is relatively W -projective.

(ii) M is isomorphic to a direct summand of a tensor product of some module with W.

(iii) The identity map in Hom_{kG}(M, M) can be written as Tr_{W} (γ) for some γ ∈ Hom_{kG}(M ⊗ W, M ⊗ W).

Proof. See propositions 2.4 and 3.2 of [14]. □

It follows from this theorem that if M is relatively W -projective then every element of Ext_{kG}^{∗}(M, M) is in the image of Tr_{W}. A module M is said to be virtually W -projective if all elements of large enough degree in Ext_{kG}^{∗}(M, M) are in the image of Tr_{W}. An example is given in [14] of a module that is virtually W -projective but not relatively W -projective.

Theorem 9.3. Let W and M be finitely generated kG-modules such that

V_{G}(M) ∩ V_{G}(Tr_{W}) = {0}.

Then M is virtually W -projective.

Proof. See proposition 5.3 of [14]. □

Now let ℋ_{i} be the collection of centralizers C_{G}(E) of elementary abelian psubgroups E of rank i. Let W_{i} be the direct sum of the permutation modules corresponding to the subgroups in ℋ_{i}, so that the image of Tr_{W_{i}} is the sum of the images of Tr_{H,G} as H runs over the elements of ℋ_{i}.

Lemma 9.4. The subvariety V_{G}(Tr_{W_{i}}) is contained in the union of the images of res_{E,G}^{∗}: V_{E} → V_{G} as E runs over the set of elementary abelian p-subgroups of G of rank strictly less than i.

Proof. This follows from the transfer theorem of [3]. □

Theorem 9.5. Suppose that ζ_{1}, …, ζ_{r} is a homogeneous system of parameters for H^{∗}(G, k) that satisfies the rank-restriction condition. Then

M_{i−1} = L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{i−1}}

is virtually W_{i}-projective.

Proof. By Lemmas 8.4 and 9.4, we have

V_{G}(M_{i−1}) ∩ V_{G}(Tr_{W_{i}}) = V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{i−1}⟩ ∩ V_{G}(Tr_{W_{i}})

⊆ V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{i−1}⟩ ∩ ⋃_{r_{p}(E)<i} Im(res_{G,E}^{∗}) = {0}.

It now follows from Theorem 9.3 that M_{i−1} is virtually W_{i}-projective. □

Theorem 9.6. If ζ_{1}, …, ζ_{r} satisfy the rank-restriction condition, then they form a filter-regular sequence.

Proof. By Duflot’s theorem [16], if H = C_{G}(E) ∈ ℋ_{i} then the restrictions of ζ_{1}, …, ζ_{i−1} form a regular sequence in H^{∗}(H, k). So by Theorem 7.1, multiplication by ζ_{i−1} is injective on H^{j}(H, M_{i−2}) for j large enough.

Now consider the diagram

[figure: ⋯; H^{j+n_{i−1}}(G, M_{i−1}); H^{j}(G, M_{i−2}); ζ_{i−1}; H^{j+n_{i−1}}(G, M_{i−2}); ⋯; ⨁_{H∈ℋ_{i}}H^{j+n_{i−1}}(H, M_{i−1}); _{∑ Tr_{H,G}}; ⨁_{H∈ℋ_{i}}H^{j}(H, M_{i−2}); _{∑ Tr_{H,G}}; ζ_{i−1}; ⨁_{H∈ℋ_{i}}H^{j+n_{i−1}}(H, M_{i−2}); _{∑ Tr_{H,G}}]

(9.7)

We have just seen that the map marked ζ_{i−1} on the bottom row is injective for all large enough j. By Theorem 9.5, the left hand vertical map is surjective for all large enough j. A diagram chase shows that the map marked ζ_{i−1} on the top row is injective for all large enough j. □

Corollary 9.8. Let G be a finite group of p-rank r and let k be a field of characteristic p. Then the Dickson invariants D_{1}, …, D_{r} form a filter-regular sequence in H^{∗}(G, k).

Proof. This follows from Theorems 8.5 and 9.6. □

The following is the Bourguiba–Zarati theorem in the particular case of finite group cohomology. Their proof uses the classification of injective unstable modules over the Steenrod algebra.

Theorem 9.9. Let G be a finite group and let k be a field of characteristic p. Suppose that H^{∗}(G, k) has depth d. Then the Dickson invariants D_{1}, …, D_{d} form a regular sequence in H^{∗}(G, k).

Proof. This follows from Lemma 3.4 and Corollary 9.8. □

Theorem 9.10. Suppose that ζ_{1}, …, ζ_{r} is a filter-regular system of parameters of type (d_{0}, …, d_{r}) for H^{∗}(G, k). Then ζ_{1}, …, ζ_{r} is also filter-regular of type (d_{0}, …, d_{r−3}, d_{r−3} − 1, d_{r−3} − 2, d_{r−3} − 2).

Proof. Consider the diagram (9.7) with i = r, and recall from §2 that M_{r−1} is periodic. So the fact that ∑_{H∈ℋ_{i}} Tr_{H,G} surjects onto H^{j}(G, M_{r−1}) for all large enough values of j means that this holds for all positive values of j. Using Theorem 7.1, we see that we may replace d_{r−2} with d_{r−3} − 1. Corollary 7.2 now completes the proof of the theorem. □

The following is an improved version of a theorem of Okuyama and Sasaki [22]. Their version has quasi-regular instead of very strongly quasi-regular.

Corollary 9.11. Suppose that the difference between the depth and the Krull dimension of H^{∗}(G, k) is at most two. Then there is a very strongly quasi-regular sequence for H^{∗}(G, k).

Proof. Since the difference between the depth and the Krull dimension (which is equal to r) is at most two, we have a_{𝔪}^{i}(H^{∗}(G, k)) = −∞ for 0 ≤ i ≤ r − 3 (if there are any such values of i). By Theorem 4.5, it follows that there is a filter-regular sequence of type

(−1, −2, …, 2 − r, d_{r−2}, d_{r−1}, d_{r})

for some values of d_{r−2}, d_{r−1} and d_{r}. Applying Theorem 9.10, it follows that such a sequence is also filter-regular of type

(−1, −2, …, 2 − r, 1 − r, −r, −r),

namely a very strongly quasi-regular sequence. □

10. Computing group cohomology

In this section, we describe the relationship between the methods of this paper and those developed by Jon Carlson for computing group cohomology [12, 13]. The problem addressed in those papers is as follows. If we try to compute the cohomology of a finite group by beginning a projective resolution of the field of coefficients, then the problem is knowing when to stop. Carlson gives a method for bounding the degrees of the generators and relations of H^{∗}(G, k) in terms of information coming from an initial segment. The method depends on some conjectures in group cohomology, and these conjectures are verified along the way, so that there is no uncertainty in the answer. The conjectures hold in all examples computed so far, including all 2-groups of order at most 64. In this section, we present a method which is guaranteed to work, independently of any conjectures, and which will usually give slightly better bounds than Carlson’s. Even a slight improvement in the bound improves the time taken to make the computation by a substantial amount, and makes calculations accessible which were not before.

We begin with the notation. Suppose that we have computed the generators and relations for H^{∗}(G, k) in degrees up to and including N. Then we can build an approximation to H^{∗}(G, k) as follows. Choose a minimal set of generators x_{1}, …, x_{q} of degrees at most N, and let Q be the graded k-algebra generated by elements x̃_{1}, …, x̃_{q}, subject only to the relations x̃_{i}x̃_{j} = (−1)^{|x̃_{i}||x̃_{j}|}x̃_{j}x̃_{i}. Then there is a homomorphism of graded k-algebras Q → H^{∗}(G, k) taking x̃_{i} to x_{i}. Let J be the ideal in Q generated by the elements of the kernel of degree at most N. We write τ_{N}H^{∗}(G, k) for the quotient Q/J, namely the ring defined by the generators and relations of H^{∗}(G, k) of degree at most N. Then there is a uniquely defined homomorphism

τ_{N}H^{∗}(G, k) → H^{∗}(G, k)

taking x̃_{i} + J to x_{i}. In general, this homomorphism need not be injective or surjective, but it is an isomorphism up to and including degree N. If the degrees of all the generators of H^{∗}(G, k) are at most N then it is surjective, and if the degrees of the relations are also at most N then it is an isomorphism.

Theorem 10.1. Let r = r_{p}(G) > 1. Suppose that ζ_{1}, …, ζ_{r} ∈ τ_{N}H^{∗}(G, k) form a filter-regular homogeneous set of parameters with |ζ_{i}| = n_{i} ≥ 2. Suppose further that their images in H^{∗}(G, k) form a homogeneous system of parameters. Set

α = max_{0≤i≤r−2}{a_{𝔪}^{i}(τ_{N}H^{∗}(G, k)) + i}  (10.2)

(α = −∞ if τ_{N}H^{∗}(G, k) has depth at least r − 1). If

N > max{α, 0} + ∑_{j=1}^{r}(n_{j} − 1)  (10.3)

then the map τ_{N}H^{∗}(G, k) → H^{∗}(G, k) is an isomorphism.

Proof. Let 𝒦^{∗}(τ_{N}H^{∗}(G, k); ζ_{1}, …, ζ_{r}) be the Koszul complex, indexed in the same way as the _{r}E_{1} page of the spectral sequence (6.3).

For 0 ≤ ℓ ≤ r, set

d_{ℓ} = max{max_{0≤i≤ℓ}{a_{𝔪}^{i}(τ_{N}H^{∗}(G, k)) + i − ℓ}, −ℓ − 1}.

Then the sequence d_{0}, …, d_{r} satisfies Condition 3.1. Since a_{𝔪}^{ℓ}(τ_{N}H^{∗}(G, k)) ≤ d_{ℓ}, it follows from Theorem 4.5 that ζ_{1}, …, ζ_{r} is a filter-regular sequence of type (d_{0}, …, d_{r}) for τ_{N}H^{∗}(G, k).

If 0 ≤ i ≤ r − 2 then

N > α + ∑_{j=1}^{r}(n_{j} − 1) ≥ a_{𝔪}^{i}(τ_{N}H^{∗}(G, k)) + i + ∑_{j=1}^{r}(n_{j} − 1).

Since max_{0≤i≤ℓ}{d_{i} + i} = max{max_{0≤i≤ℓ}{a_{𝔪}^{i}(τ_{N}H^{∗}(G, k)) + i}, −1}, setting s = r − ℓ, we

find that for 2 ≤ s ≤ r we have

N > d_{r−s} + (r − s) + ∑_{j=1}^{r}(n_{j} − 1).

It follows that N + s > n_{1} + ⋯ + n_{s} + d_{r−s}, and so by Lemma 3.5, for 2 ≤ s ≤ r and t ≥ N + s we have

H^{−s,t}(𝒦^{∗}(τ_{N}H^{∗}(G, k); ζ_{1}, …, ζ_{r})) = 0.  (10.4)

There is an obvious map of Koszul complexes

𝒦^{∗}(τ_{N}H^{∗}(G, k); ζ_{1}, …, ζ_{r}) → _{r}E_{1}^{∗∗}  (10.5)

whose target is the E_{1} page of the spectral sequence _{r}E_{∗}^{∗∗} constructed in §6. Arguing by contradiction, let us suppose that τ_{N}H^{∗}(G, k) → H^{∗}(G, k) is not an isomorphism. Let N′ > N be the smallest degree in which τ_{N}H^{N′}(G, k) → H^{N′}(G, k) is not an isomorphism. So there is either a new generator or a new relation in degree N′, or both. Since N′ − 2 ≥ N − 1 ≥ ∑_{j=1}^{r}(n_{j} − 1), the map (10.5) is an isomorphism on elements of total degree t − s equal to N′ − 2 or N′ − 1. Moreover, since n_{i} ≥ 2, (10.5) is also an isomorphism for s > 0 and t−s = N′. And for t−s equal to N′−3, (10.5) is an isomorphism except possibly for s = r, t = −1 + ∑_{j=1}^{r} n_{j} in the case where N′ = 2 + ∑_{j=1}^{r}(n_{j} − 1). In this case, the map goes from the zero space to a one dimensional space isomorphic to Ĥ^{−1}(G, k), namely the element which is sent by d_{r} to the last survivor, as discussed in §6. In particular, this element is in the kernel of d_{1}.

Taking cohomology with respect to the horizontal differential, we see that

H^{−s,t}(𝒦^{∗}(τ_{N}H^{∗}(G, k); ζ_{1}, …, ζ_{r})) → _{r}E_{2}^{−s,t}

is an isomorphism for total degree t−s equal to N′ −2 or N′ −1, except possibly for s = 1, t = N′. Combining this with (10.4), we see that for 2 ≤ s ≤ r and t − s equal to N′ − 2 or N′ − 1 we have _{r}E_{2}^{−s,t} = 0.

rE_{1}

possibly the element of Ĥ^{−1} hitting the last survivor

circles where (10.5) is iso

rE_{2}

circles where rE_{2}^{s,t} = 0

— t = N′ —

detects extra generators detects extra relations

t s

It follows from the case t − s = N′ − 1 that there is no nonzero element that can hit a new generator in H^{0,N′}(𝒦^{∗}(Ĥ^{∗}(G, k); ζ_{1}, …, ζ_{r})). But the spectral sequence converges to zero, so this is a contradiction. So it follows that τ_{N}H^{N′}(G, k) → H^{N′}(G, k) is surjective. If it is not injective, let x ∈ τ_{N}H^{N′}(G, k) be in the kernel. Then in the Koszul complex, x is the image of some element y of vertical degree N′ in 𝒦^{−1}(τ_{N}H^{∗}(G, k); ζ_{1}, …, ζ_{r}). The image of y in _{r}E_{1}^{−1,N′} survives to give a nonzero element z ∈ _{r}E_{2}^{−1,N′}, The element z has to be killed somewhere in the spectral sequence. But again, there is no nonzero element in _{r}E_{2}^{−s,t} with t − s = N′ − 2 available to hit z, giving a contradiction. It follows that τ_{N} is an isomorphism as required. □

Remarks 10.6. (i) Given elements ζ_{1}, …, ζ_{r} ∈ τ_{N}H^{∗}(G, k), it is easy to tell whether their images form a homogeneous system of parameters in H^{∗}(G, k). Namely, for a representative of each conjugacy class of maximal elementary abelian p-subgroups E, we just check that H^{∗}(E, k)/(res_{G,E}(ζ_{1}), …, res_{G,E}(ζ_{r})) has finite length.

(ii) For N large enough, τ_{N}H^{∗}(G, k) → H^{∗}(G, k) is an isomorphism, and the right hand side of the inequality (10.3) is independent of N. So for N large enough, (10.3) holds. Since this inequality can be checked just from a knowledge of τ_{N}H^{∗}(G, k), the theorem gives a computable bound on the degrees of the generators and relations of H^{∗}(G, k), without referring to any conjectures.

(iii) Computing α in practise can be accomplished using Theorem 4.5.

(iv) The definition (10.2) of α implies that α ≤ Reg τ_{N}H^{∗}(G, k). So if Conjecture 1.1 holds, then we only ever have to compute up to degree

N = 1 + ∑_{j=1}^{r}(n_{j} − 1).

In particular, it follows from Corollary 9.11 that this is true if the depth and Krull dimension of H^{∗}(G, k) differ by at most two.

(v) If the Sylow p-subgroup of G has rank at least two, then by Duflot’s theorem [16], the depth of H^{∗}(G, k) is at least two. Then a theorem from [7] shows that there are no nonzero products from negative degrees to positive degrees in Tate cohomology. It follows that Theorem 10.1 can be improved in this case by replacing (10.3) by the inequality

N ≥ max{α, 0} + ∑_{j=1}^{r}(n_{j} − 1).

In the proof of the theorem, the diagram for _{r}E_{1} becomes

[figure: _{r}E_{1}; possibly the; element of Ĥ^{−1}; hitting the last; survivor; 0; — t = N′]

It is possible that there are unwanted elements of Ĥ^{−1}(G, k) appearing in the second column from the left, but they are hit from the leftmost column, so they do not damage the argument.

(vi) Dave Rusin [23] computed the cohomology of all the 2-groups of order up to 32, using the Eilenberg–Moore spectral sequence. This method becomes impractical for the groups of order 64. Jon Carlson has computed the cohomology of the groups of order 64 by computing the beginning of a resolution explicitly using the computer algebra package MAGMA, and using the methods of [13] to know when to stop computing. It turns out that for all the 2-groups of order at most 64, the depth and the Krull dimension differ by at most two, so that Reg H^{∗}(G, k) is guaranteed to be zero. In just one case, namely the Sylow 2-subgroup of P SL(3, 4), H_{𝔪}^{r,−r}H^{∗}(G, k) turned out to have dimension bigger than one. So the last survivor does not span this two dimensional space in this case.

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

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