ON CARLSON’S DEPTH CONJECTURE IN GROUP COHOMOLOGY

DAVID J. GREEN

Abstract. We establish a weak form of Carlson’s conjecture on the depth of the mod-p cohomology ring of a p-group. In particular, Duflot’s lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups. The proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map. We demonstrate the existence of systems of parameters (so-called polarised systems) which are particularly well adapted to this comodule structure.

Introduction

Let G be a finite p-group and C its greatest central elementary abelian subgroup. Write k for the prime field 𝔽_{p}. Cohomology will always be with coefficients in k. Denote by r the p-rank of G, and by z the p-rank of C. Following Broto and Henn [3] we shall exploit the fact that the multiplication map μ: G × C → G, (g, c) ↦ g.c is a group homomorphism. The main result of this paper is as follows: Theorem 0.1. Suppose that G is a p-group whose centre has p-rank z. Then the following statements are equivalent:

(1) The mod-p cohomology ring H^{∗}(G) is not detected on the centralizers of its rank z + 1 elementary abelian subgroups.

(2) There is an associated prime 𝔭 such that H^{∗}(G)/𝔭 has dimension z.

(3) The depth of H^{∗}(G) equals z. This is a special case of a conjecture due to Carlson [4], reproduced here as Conjecture 3.2. Recall that Duflot proved in [6] that z is a lower bound for the depth. So this result characterises the cases where Duflot’s lower bound is tight.

Theorem 0.1 is proved as Theorem 8.1. The proof rests upon the existence of polarised systems: homogeneous systems of parameters for H^{∗}(G) which are particularly well adapted to the H^{∗}(C)-comodule structure. There are two extreme types of behaviour which a cohomology class x ∈ H^{∗}(G) can demonstrate under the comodule structure map μ^{∗}: one extreme is that Res_{C}(x) is nonzero, and so μ^{∗}(x) = 1 ⊗ Res_{C}(x) + other terms. The other extreme is that x is primitive, meaning that μ^{∗}(x) = x ⊗ 1. Roughly speaking, a polarised system of parameters is one consisting solely of elements which each exhibit one or the other of these extreme kinds of behaviour. The precise definition, which ensures that each such system is a detecting sequence for the depth of H^{∗}(G), is given in Definition 2.2. Polarised systems of parameters always exist, as is proved in Theorem 7.6.

In addition to Theorem 0.1 we also prove a weak form of the general case of Carlson’s conjecture. This is done in Theorem 5.1, which includes the following statement: Theorem 0.2. Let ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} be a polarised system of parameters for H^{∗}(G). Then H^{∗}(G) has depth z + s_{𝔞}, where s_{𝔞} ∈ {0, …, r − z} is defined by

s_{𝔞} = max{s ≤ r − z | κ_{1}, …, κ_{s} is a regular sequence in H^{∗}(G)} .

Date: 5 June 2002.

1991 Mathematics Subject Classification. Primary 20J06.

1. Primitive comodule elements

Group multiplication μ: G × C → G is a group homomorphism. As observed by Broto and Henn [3], this means that H^{∗}(G) inherits the structure of a comodule over the coalgebra H^{∗}(C).

Recall that x ∈ H^{∗}(G) is called a primitive comodule element if μ^{∗}(x) = x ⊗ 1 ∈ H^{∗}(G×C) ≅ H^{∗}(G)⊗_{k} H^{∗}(C). As the comodule structure map μ^{∗} is simultaneously a ring homomorphism, it follows that the primitives form a subalgebra PH^{∗}(G) of H^{∗}(G). As the quotient map G → G/C coequalises μ and projection onto the first factor of G × C, it follows that the image of inflation from H^{∗}(G/C) is contained in PH^{∗}(G). Lemma 1.1. Suppose that I is a homogeneous ideal in H^{∗}(G) which is generated by primitive elements. Then

μ^{∗}(I) ⊆ I ⊗_{k} H^{∗}(C) ,

and so μ^{∗} induces a ring homomorphism

λ: H^{∗}(G)/I → H^{∗}(G)/I ⊗_{k} H^{∗}(C)

which induces an H^{∗}(C)-comodule structure on H^{∗}(G)/I.

Proof. For x ∈ I and y ∈ H^{∗}(G) one has μ^{∗}(xy) = μ^{∗}(x)μ^{∗}(y) = (x ⊗ 1)μ^{∗}(y) ∈ I ⊗_{k} H^{∗}(C). □

Lemma 1.2. Suppose that ζ_{1}, …, ζ_{t} is a sequence of homogeneous elements of H^{∗}(G) whose restrictions form a regular sequence in H^{∗}(C), and suppose that I is an ideal in H^{∗}(G) generated by primitive elements. Then ζ_{1}, …, ζ_{t} is a regular sequence for the quotient ring H^{∗}(G)/I.

Proof. Carlson’s proof for the case I = 0 (Proposition 5.2 of [5]) generalises easily. Denote by R the polynomial algebra k[ζ_{1}, …, ζ_{t}]. The map λ of Lemma 1.1 induces an R-module structure on H^{∗}(G)/I ⊗_{k} H^{∗}(C), and λ is a split monomorphism of R-modules, the splitting map being induced by projection onto the first factor G×C → G. So as an R-module H^{∗}(G)/I is a direct summand of H^{∗}(G)/I ⊗_{k}H^{∗}(C). The result will therefore follow if we can show that H^{∗}(G)/I ⊗_{k} H^{∗}(C) is a free Rmodule.

To see that H^{∗}(G)/I ⊗_{k} H^{∗}(C) is indeed a free R-module set

F_{i} := ∑_{j≥i} H^{j}(G)/(I ∩ H^{j}(G))

and observe that F_{i} ⊗_{k} H^{∗}(C) is an R-submodule of H^{∗}(G)/I ⊗_{k} H^{∗}(C) = F_{0} ⊗_{k} H^{∗}(C). Projection G × C → C makes F_{i} ⊗_{k} H^{∗}(C)/F_{i+1} ⊗_{k} H^{∗}(C) a free H^{∗}(C)- module. Now for x ∈ F_{i}, y ∈ H^{∗}(C) and θ ∈ R we have θ.(x⊗y) ∈ x⊗(Res_{C}^{G} θ).y + F_{i+1} ⊗_{k} H^{∗}(C). So the R-module structure on F_{i} ⊗_{k} H^{∗}(C)/F_{i+1} ⊗_{k} H^{∗}(C) is induced by the restriction map R → H^{∗}(G) → H^{∗}(G)/I → H^{∗}(C) from the free H^{∗}(C)-structure. But H^{∗}(C) is a free R-module by Theorem 10.3.4 of [7], because the restrictions of ζ_{1}, …, ζ_{t} form a regular sequence. So F_{i}⊗_{k} H^{∗}(C)/F_{i+1}⊗_{k} H^{∗}(C) is a free R-module for all i, whence it follows that F_{0} ⊗_{k} H^{∗}(C)/F_{i} ⊗_{k} H^{∗}(C) is a free R-module for all i. As the degree of each homogeneous element of F_{i} ⊗_{k} H^{∗}(C) is at least i, it follows that H^{∗}(G)/I ⊗_{k} H^{∗}(C) is itself a free R-module. □

Corollary 1.3. Let G be a p-group whose centre has p-rank z. Suppose that there is a length s regular sequence in H^{∗}(G) which consists entirely of primitive elements. Then the depth of H^{∗}(G) is at least z + s.

Proof. Let I be the ideal generated by the primitive elements in the regular sequence. Let ζ_{1}, …, ζ_{z} be elements of H^{∗}(G) whose restrictions form a homogeneous system of parameters for H^{∗}(C): it is well known that such sequences exist. Now apply Lemma 1.2.

One concrete example of such classes ζ_{i} is obtained as follows. Let ρ_{G} be the regular representation of G, and ζ_{i} the Chern class c_{p^{n}−p^{n−i}}(ρ_{G}) for 1 ≤ i ≤ z, where p^{n} is the order of G. Then ρ_{G} restricts to C as |G : C| copies of the regular representation ρ_{C}, whence Res_{C}(ζ_{i}) = c_{p^{z}−p^{z−i}}(ρ_{C})^{|G:C|}. But the c_{p^{z}−p^{z−i}}(ρ_{C}) are (up to a sign) the Dickson invariants. See the proof of Theorem 7.6 for more details. □

2. Polarised systems of parameters

We shall now give the definition of a polarised system of parameters, the key definition of this paper. In fact we shall introduce two closely related concepts: the axioms for a polarised system (Definition 2.2) are easily checked in practice, whereas the special polarised systems of Definition 2.5 have precisely the properties we shall need to investigate depth. Lemma 2.6 shows that the two concepts are more or less interchangeable. Definition 2.1. Let G be a p-group of p-rank r whose centre has p-rank z. Denote by C the greatest central elementary abelian subgroup of G, and set

A^{C}(G) := {V ≤ G | V is elementary abelian and contains C} ,
A_{d}^{C}(G) := {V ∈ A^{C}(G) | V has p-rank d}
H_{d}^{C}(G) := {C_{G}(V) | V ∈ A_{d}^{C}(G)} .

So A^{C}(G) is the disjoint union of the A_{z+s}^{C}(G) for 0 ≤ s ≤ r − z. Definition 2.2. Let G be a p-group of p-rank r whose centre has p-rank z. Recall that inflation map Inf : H^{∗}(V/C) → H^{∗}(V) is a split monomorphism for each V ∈ A^{C}(G), and so its image Im Inf is isomorphic to H^{∗}(V/C).

A system ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} of homogeneous elements of H^{∗}(G) shall be called a polarised system of parameters if the following four axioms are satisfied.

(PS1): Res_{C}(ζ_{1}), …, Res_{C}(ζ_{z}) is a system of parameters for H^{∗}(C).

(PS2): Res_{V} (κ_{j}) lies in Im Inf for each 1 ≤ j ≤ r−z and for each V ∈ A^{C}(G). (PS3): For each V ∈ A^{C}(G), the restrictions Res_{V} (κ_{1}), …, Res_{V} (κ_{s}) constitute a system of parameters for Im Inf. Here z + s is the rank of V.

(PS4): Res_{V} (κ_{j}) = 0 for V ∈ A_{z+s}^{C}(G) with 0 ≤ s < j ≤ r − z. Remark 2.3. Polarised systems of parameters always exist, as we shall see in Theorem 7.6. Observe that Axiom (PS1) involves only the ζ_{i}, whereas the remaining axioms involve only the κ_{j}. Basically Axiom (PS1) says that ζ_{1}, …, ζ_{z} is a regular sequence which can be detected on the centre, (PS2) says that the κ_{j} are primitive after raising to suitably high pth powers, and (PS3) says that the κ_{j} together with the ζ_{i} will form a detecting sequence for the depth of H^{∗}(G). Axiom (PS4) is a more technical condition which we shall only use once: it is needed in Lemma 2.6 to show that, after raising to a suitably high pth power, each κ_{j} is a sum of transfer classes as required by Axiom (PS5) below. Lemma 2.4. Polarised systems of parameters for H^{∗}(G) are indeed systems of parameters.

Proof. Let V ∈ A_{z+s}^{C}(G). The restrictions of ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{s} constitute a system of parameters for H^{∗}(V) by (PS1) and (PS3). Hence ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} are algebraically independent over k, for we may choose V to have p-rank r.

Now let γ be a homogeneous element of H^{∗}(G). For V ∈ A^{C}(G) there is a monic polynomial f_{V} (x) with coefficients in k[ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z}] such that f_{V} (γ) has zero restriction to V. Taking the product of all such polynomials one obtains a polynomial f(x) such that f(γ) has zero restriction to each maximal elementary abelian subgroup of G. So f(γ) is nilpotent by a well-known result of Quillen. □

Definition 2.5. Let G be a p-group of p-rank r whose centre has p-rank z. A system ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} of homogeneous elements of H^{∗}(G) shall be called a special polarised system of parameters if it satisfies the following five axioms: (PS1), (PS3), (PS4) and

(PS2′): κ_{j} is a primitive element of the H^{∗}(C)-comodule H^{∗}(G) for each

1 ≤ j ≤ r − z. (PS5): κ_{j} lies in ∑_{H∈H_{z+i}^{C}(G)} Tr_{H}^{G}(H^{∗}(H)) for each 1 ≤ i ≤ j ≤ r − z.

Lemma 2.6. Axiom (PS2′) implies Axiom (PS2), and so every special polarised system is a polarised system of parameters. Conversely for each polarised system ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} there is a nonnegative integer N such that ζ_{1}, …, ζ_{z}, κ_{1}^{p^{N}} , …, κ_{r−z}^{p^{N}} is a special polarised system of parameters for H^{∗}(G).

Proof. Let V ∈ A^{C}(G). Restriction from H^{∗}(G) to H^{∗}(V) is a map of H^{∗}(C)- comodules and so sends primitive elements to primitive elements. But the subalgebra of primitive elements of H^{∗}(V) coincides with the image of inflation from V/C. So (PS2′) implies (PS2).

Now suppose ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} is a polarised system for H^{∗}(G). For each 1 ≤ j ≤ r − z the restriction of κ_{j} to each V ∈ A^{C}(G) is primitive by (PS2). Hence the element μ^{∗}(κ_{j})−κ_{j} ⊗1 of H^{∗}(G)⊗H^{∗}(C) has zero restriction to every maximal elementary abelian subgroup and is therefore nilpotent.

For fixed 1 ≤ i ≤ r −z, denote by K the set consisting of those subgroups K of G such that C_{G}(K) is not G-conjugate to any subgroup of any H ∈ H_{z+i}^{C}. Following Carlson (Proof of Corollary 2.2 of [4]), observe that every K ∈ K has p-rank less than z + i. Moreover every K ∈ K is contained in K_{C} = ⟨K, C⟩, and K_{C} itself lies in K. So Res_{K_{C}} (κ_{j}) = 0 for all j ≥ i by (PS4). Hence each such κ_{j} lies in the radical ideal \sqrt{J′}, where J′ is the ideal ⋂_{K∈K} ker Res_{K}. So by Benson’s result on the image of the transfer map (Theorem 1.1 of [1]) the κ_{j} also lie in \sqrt{J}, where J is the ideal ∑_{H∈H_{z+i}^{C}(G)} Tr_{H}^{G}(H^{∗}(H)). □

Remark 2.7. The above proof is the only time we shall make use of the Axiom (PS4). In particular, the results of §4 do not depend on (PS4). I do not know whether or not (PS4) is a consequence of the remaining axioms for a special polarised system of parameters.

Axiom (PS5) will be used in Lemma 4.2 to prove the existence of an associated prime ideal with desirable properties. Lemma 2.8. Suppose that ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} is a polarised system of parameters for H^{∗}(G). Let 0 ≤ s ≤ r − z. Then the sequence ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{s} is regular in H^{∗}(G) if and only if the sequence κ_{1}, …, κ_{s} is regular in H^{∗}(G).

Proof. Recall that regular sequences may be permuted at will. Moreover, replacing one element of a sequence by its pth power has no effect on whether the sequence is regular or not. Hence by Lemma 2.6 it suffices to prove the result for special polarised systems.

So we may assume that the given sequence is a special polarised system of parameters. Let I be the homogeneous ideal I in H^{∗}(G) generated by κ_{1}, …, κ_{s}. By Axiom (PS2′) this ideal is generated by primitive elements. Also, the restrictions of ζ_{1}, …, ζ_{z} form a regular sequence in H^{∗}(C) by Axiom (PS1). Therefore Lemma 1.2 tells us that ζ_{1}, …, ζ_{z} constitute a regular sequence for H^{∗}(G)/I. □

3. Three depth-related numbers

In this section we shall introduce three numbers τ_{H}, τ_{𝔞} and τ_{𝔞,H}, each of which is an approximation to the depth τ of H^{∗}(G). Definition 3.1. Let G be a p-group of p-rank r whose centre has p-rank z. Write τ for the depth of H^{∗}(G) and set

τ_{H} := max{d ∈ {z, …, r} | The family H_{d}^{C}(G) detects H^{∗}(G)} .

In [4] Carlson formulates the following conjecture: Conjecture 3.2 (Carlson). The number τ_{H} coincides with the depth τ of H^{∗}(G). Moreover, H^{∗}(G) has an associated prime 𝔭 such that dim H^{∗}(G)/𝔭 = τ. In fact, Carlson formulates the conjecture not just for p-groups, but for arbitrary finite groups. In this article however we only consider p-groups. In Theorem 8.1 we shall prove a special case of this conjecture, after deriving a partial result for the general case in Theorem 5.1. For this we need two more depth-related numbers. Definition 3.3. Let G be a p-group of p-rank r whose centre has p-rank z, and let 𝔞 = (ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z}) be a polarised system of parameters for H^{∗}(G).

Define τ_{𝔞} to be z + s_{𝔞}, where s_{𝔞} is the largest s ∈ {0, …, r − z} such that κ_{1}, …, κ_{s} is a regular sequence in H^{∗}(G).

Let S_{𝔞,H} be the subset of {0, …, r − z} such that s lies in S_{𝔞,H} if and only if the restriction map

H^{∗}(G)/(κ_{1}, …, κ_{s−1}) → ∏_{H∈H_{z+s}^{C}} H^{∗}(H)/(Res_{H} κ_{1}, …, Res_{H} κ_{s−1})

is injective^{1}. Define s_{𝔞,H} := max S_{𝔞,H} and τ_{𝔞,H} := z + s_{𝔞,H}. Lemma 3.4. Let 𝔞 = (ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z}) be a polarised system of parameters for H^{∗}(G). If s > 0 lies in S_{𝔞,H} then the family H_{z+s}^{C}(G) detects H^{∗}(G) and s − 1 lies in S_{𝔞,H}. Therefore τ_{H} ≥ τ_{𝔞,H} and S_{𝔞,H} = {0, …, s_{𝔞,H}}. For the proof we shall need an elementary fact about regular sequences. Lemma 3.5. Suppose that R, S are connected graded commutative k-algebras and that f: R → S is an algebra homomorphism which respects the grading. Suppose further that ζ_{1}, …, ζ_{d} is a family of homogenous positive-degree elements of R satisfying the following conditions:

(1) f(ζ_{1}), …, f(ζ_{d}) is a regular sequence in S.

(2) The induced map f_{d}: R/(ζ_{1}, …, ζ_{d}) → S/(f(ζ_{1}), …, f(ζ_{d})) is an injection. Then f: R → S is an injection.

Proof. It suffices to prove the case r = 1. Write ζ for ζ_{1}. Let a ≠ 0 be an element of ker(f) whose degree is as small as possible. Since f(a) = 0 in S/(f(ζ)) it follows that there is an a′ ∈ R with a = a′ζ. Since f(a) = 0 and f(ζ) is regular it follows that a′ ∈ ker(f), contradicting the minimality of deg(a). □

Proof of Lemma 3.4. Apply Lemma 3.5 to the family κ_{1}, …, κ_{s−1} with R = H^{∗}(G), S = ∏_{H_{z+s}^{C}} H^{∗}(H) and f the product of the restriction maps. Because s ∈ S_{𝔞,H} the induced map of quotients is an injection. By Axiom (PS3) the restrictions of κ_{1}, …, κ_{s−1} form a regular sequence in H^{∗}(V) for each V ∈ A_{z+s}^{C}(G), and so by [5, Prop. 5.2] they form a regular sequence in H^{∗}(H) for each H ∈ H_{z+s}^{C}. Hence the restrictions form a regular sequence in S, and so the family H_{z+s}^{C} detects H^{∗}(G).

1So S_{𝔞,H} always contains 0, and 1 lies in S_{𝔞,H} if and only if the family H_{z+1}^{C} detects H^{∗}(G).

If instead we just invoke the first step in the proof of Lemma 3.5, we see that the H^{∗}(H)/(Res κ_{1}, …, Res κ_{s−2}) with H ∈ H_{z+s}^{C} detect H^{∗}(G)/(κ_{1}, …, κ_{s−2}). □

4. Depth and special polarised systems

The following fact from commutative algebra is well known. Lemma 4.1. Let A be a graded commutative ring and M a Noetherian graded Amodule. Suppose that 𝔭 is an associated prime of M, and that ζ_{1}, …, ζ_{d} is a regular sequence for M. Then M/(ζ_{1}, …, ζ_{d})M has an associated prime 𝔮 containing 𝔭.

Proof. It suffices to prove the case d = 1. Write ζ for ζ_{1}. Pick x ∈ M with Ann_{A}(x) = 𝔭. If x lies in ζM then there is an x′ ∈ M with ζx′ = x. As ζ is regular it follows that Ann_{A}(x′) = 𝔭 too, so replace x by x′. This can only happen finitely often, as M is Noetherian and Ax is strictly contained in Ax′. So we may assume that x does not lie in ζM, which means that the image of x in M/ζM is nonzero and annihilated by 𝔭. □

Lemma 4.2. Let 𝔞 = (ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z}) be a special polarised system of parameters for H^{∗}(G), and I the ideal generated by the κ_{j} with j ≤ s_{𝔞,H}. Then the H^{∗}(G)-module H^{∗}(G)/I has an associated prime 𝔭 which contains κ_{1}, …, κ_{r−z}.

Proof. Set s = s_{𝔞,H} + 1. The result is trivial if s_{𝔞,H} = r − z, so we may assume that s ≤ r − z. By definition of s_{𝔞,H}, the family H = H_{z+s}^{C} does not detect the quotient H^{∗}(G)/I. Pick a class x ∈ H^{∗}(G) which does not lie in the ideal I, but whose restriction to each H ∈ H does lie in the ideal H^{∗}(H). Res_{H}(I). Let A be the ideal of classes in H^{∗}(G) which annihilate the image of x in the quotient H^{∗}(G)/I.

For any j ≥ s we have κ_{j} ∈ ∑_{H∈H} Tr_{H}^{G} H^{∗}(H) by Axiom (PS5), say κ_{j} = ∑_{H} Tr_{H} γ_{H}. So κ_{j}x = ∑_{H} Tr_{H}(γ_{H} Res_{H}(x)) by Frobenius reciprocity. Now by assumption Res_{H}(x) lies in the ideal generated by Res_{H}(κ_{1}), …, Res_{H}(κ_{s−1}); and this by a second application of Frobenius reciprocity means that κ_{j}x lies in the ideal I. So κ_{j} ∈ A for all j ≥ s, which means that the H^{∗}(G)-module H^{∗}(G)/I has an associated prime 𝔭 containing κ_{1}, …, κ_{r−z}. □

Theorem 4.3. Let G be a p-group of p-rank r whose centre has p-rank z, and let 𝔞 = (ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z}) be a special polarised system of parameters for H^{∗}(G). Then the numbers τ_{𝔞} and τ_{𝔞,H} both coincide with the depth τ of H^{∗}(G).

Proof. We shall prove that τ ≥ τ_{𝔞} ≥ τ_{𝔞,H} = τ. Each κ_{j} is primitive by Axiom (PS2′), so τ ≥ τ_{𝔞} by Corollary 1.3.

τ_{𝔞} ≥ τ_{𝔞,H}: Suppose that s ∈ S_{𝔞,H} and κ_{1}, …, κ_{s−1} is a regular sequence in H^{∗}(G). If κ_{1}, …, κ_{s} is not a regular sequence then there is some nonzero x ∈ H^{∗}(G)/(κ_{1}, …, κ_{s−1}) annihilated by κ_{s}. Since s ∈ S_{𝔞,H} there is some H ∈ H_{z+s}^{C} such that Res_{H}(x) is nonzero in H^{∗}(H)/(Res_{H} κ_{1}, …, Res_{H} κ_{s−1}). But this is a contradiction since (as in the proof of Lemma 3.4) the restrictions of κ_{1}, …, κ_{s} form a regular sequence in H^{∗}(H). By induction on s we deduce that τ_{𝔞} ≥ τ_{𝔞,H}.

τ_{𝔞,H} = τ: Set s = s_{𝔞,H} and denote by I the ideal (κ_{1}, …, κ_{s}) of H^{∗}(G). By Lemma 4.2 the H^{∗}(G)-module H^{∗}(G)/I has an associated prime 𝔭 which contains κ_{1}, …, κ_{r−z}. As τ_{𝔞} ≥ τ_{𝔞,H} the sequence κ_{1}, …, κ_{s} is regular in H^{∗}(G), so by Lemma 2.8 the sequence ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{s} is regular in H^{∗}(G). Therefore the sequence ζ_{1}, …, ζ_{z} is regular for H^{∗}(G)/I, and so (by Lemma 4.1) the H^{∗}(G)-module H^{∗}(G)/(ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{s}) has an associated prime 𝔮 containing all elements of the homogeneous system of parameters ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} for H^{∗}(G). So the depth of this quotient module is zero. But every regular sequence in H^{∗}(G) can be extended to a length τ regular sequence (see [2, §4.3–4], for example). So τ = z + s = τ_{𝔞,H}. □

5. Depth and polarised systems

In this section we shall remove the requirement in Theorem 4.3 that the polarised system be special.

Theorem 5.1. Let G be a p-group of p-rank r whose centre has p-rank z, and let 𝔞 = (ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z}) be a polarised system of parameters for H^{∗}(G). Then the numbers τ_{𝔞} and τ_{𝔞,H} both coincide with the depth τ of H^{∗}(G).

For the proof we shall need one further fact about regular sequences.

Lemma 5.2. Suppose that R, S are connected graded commutative k-algebras and that f: R → S is an algebra homomorphism which respects the grading. Suppose further that ζ_{1}, …, ζ_{d} is a family of homogenous positive-degree elements of R satisfying the following conditions:

(1) ζ_{1}, …, ζ_{d} is a regular sequence in R.

(2) The induced map f\underline{n}: R/(ζ_{1}^{n_{1}}, …, ζ_{d}^{n_{d}}) → S/(f(ζ_{1})^{n_{1}}, …, f(ζ_{d})^{n_{d}}) is an injection for certain positive integers n_{1}, …, n_{d}.

Then the induced map f\underline{1} : R/(ζ_{1}, …, ζ_{d}) → S/(f(ζ_{1}), …, f(ζ_{d})) is an injection.

Proof. Pick x ∈ R with f\underline{1}(x) = 0. Then f\underline{n}(ζ_{1}^{n_{1}−1} … ζ_{d}^{n_{d}−1}x) = 0 and so

(5.3)  ζ_{1}^{n_{1}−1} … ζ_{d}^{n_{d}−1}x lies in the ideal (ζ_{1}^{n_{1}}, …, ζ_{d}^{n_{d}}) of R.

Set ζ′ := ζ_{1}^{n_{1}−1} … ζ_{d−1}^{n_{d−1}−1}. Then there are a_{1}, …, a_{d} ∈ R such that ζ′ζ_{d}^{n_{d}−1}x = ζ_{1}^{n_{1}}a_{1} + ⋯ + ζ_{d}^{n_{d}}a_{d}, whence ζ_{d}^{n_{d}−1}(ζ′x − ζ_{d}a_{d}) ∈ (ζ_{1}^{n_{1}}, …, ζ_{d−1}^{n_{d−1}}). As the sequence ζ_{1}^{n_{1}}, …, ζ_{d−1}^{n_{d−1}}, ζ_{d}^{s} is regular in R for s ≥ 1 we deduce that ζ′x ∈ (ζ_{1}^{n_{1}}, …, ζ_{d−1}^{n_{d−1}}, ζ_{d}). So we have reduced Eqn. (5.3) to the case n_{d} = 1 without altering the remaining n_{t}. As regular sequences may be permuted at will we deduce that x ∈ (ζ_{1}, …, ζ_{d}). □

Proof of Theorem 5.1. Arguing exactly as in the proof of Theorem 4.3 one shows that τ_{𝔞} ≥ τ_{𝔞,H}. By Lemma 2.6 there is an integer N ≥ 0 such that the system of parameters 𝔟 = (ζ_{1}, …, ζ_{z}, κ_{1}^{p^{N}} , …, κ_{r−z}^{p^{N}}) is special polarised. From the definition one sees that τ_{𝔞} = τ_{𝔟}. So by Theorem 4.3 it suffices to prove that τ_{𝔞,H} ≥ τ_{𝔟,H}. But this is a consequence of Lemma 5.2 applied to the sequence κ_{1}, …, κ_{s_{𝔟,H}−1} with R = H^{∗}(G), S = ∏_{H} H^{∗}(H) and f the restriction map, where H = H_{z+s_{𝔟,H}}^{C}(G) and n_{i} = p^{N} for all i. □

6. Dickson invariants

Let V be an m-dimensional k-vector space. We shall make extensive use of the Dickson invariants, the polynomial generators of the ring of GL(V)-invariants in k[V]. See Benson’s book [2] for proofs of the properties of these invariants.

Denote by f_{V} the polynomial in k[V][X] defined as follows:

(6.1)  f_{V} (X) = ∏_{v∈V} (X − v) .

Recall that Dickson proved there are homogeneous polynomials D_{s}(V) for 1 ≤ s ≤ m such that

(6.2)  f_{V} (X) = ∑_{s=0}^{m}(−1)^{s}D_{s}(V)X^{p^{m−s}} ,

where D_{0}(V) = 1. The sequence D_{1}(V), …, D_{m}(V) is regular in k[V], and the invariant ring k[V]^{GL(V)} is the polynomial algebra k[D_{1}(V), …, D_{m}(V)].

If π: V → U is projection onto a codimension ℓ subspace, then the induced map k[V] → k[U] sends

(6.3)  D_{s}(V) ↦ \begin{cases}D_{s}(U)^{p^{ℓ}} & if s ≤ dim(U), \\ 0 & otherwise. \end{cases}

7. Existence of polarised systems

For each elementary abelian p-group V we shall embed k[V^{∗}] in H^{∗}(V) by identifying V^{∗} with the image of the Bockstein map H^{1}(V) → H^{2}(V).

7.1. A construction using the norm map. Let G be a p-group of order p^{n} and p-rank r whose centre has p-rank z. We shall only be interested in the case r > z. Let U_{1}, …, U_{K} be representatives of the G-orbits in A_{z+1}^{C}(G), which is a G-set via the conjugation action. For each U ∈ A_{z+1}^{C}(G) pick a basis element x_{U} for the one-dimensional subspace Ann(C) of U^{∗}, and observe that x_{U}^{p−1} is independent of the basis element chosen. As before, view U^{∗} as a subspace of H^{2}(U).

Define Θ ∈ H^{∗}(G) by

(7.1)  Θ = ∏_{i=1}^{K} N_{U_{i}}^{G} (1 + x_{U_{i}}^{p−1})^{|G:N_{G}(U_{i})|} .

Now consider the restriction Res_{V} (Θ) for V ∈ A^{C}(G). By the Mackey formula

Res_{V} (Θ) = ∏_{i=1}^{K} ∏_{g∈U_{i}\G/V} N_{U_{i}^{g}∩V}^{V} g^{∗} Res_{U_{i}∩^{g}V}^{U_{i}} (1 + x_{U_{i}}^{p−1})^{|G:N_{G}(U_{i})|} .

The intersection U_{i} ∩^{g}V always contains C, the largest central elementary abelian subgroup of G. Conversely the intersection equals C (and x_{U_{i}} therefore restricts to zero) unless U′ = U_{i}^{g} lies in V, in which case g^{∗}x_{U_{i}}^{p−1} = x_{U′}^{p−1}. Moreover, the number of double cosets U_{i}gV leading to this U′ is |N_{G}(U_{i})|/|V | and every U′ in A_{z+1}^{C}(V) := {U ∈ A_{z+1}^{C}(G) | U ≤ V } occurs for some i. So

(7.2)  Res_{V} (Θ) = ∏_{U′∈A_{z+1}^{C}(V)} N_{U′}^{V}(1 + x_{U′}^{p−1})^{|G:V |} .

In particular for U ∈ A_{z+1}^{C}(G) one has

(7.3)  Res_{U}(Θ) = 1 + x_{U}^{(p−1)p^{n−(z+1)}} .

Let η ∈ H^{2(p−1)p^{n−(z+1)}}(G) be the homogeneous component of Θ in this degree. As the norm map from H^{∗}(U′) to H^{∗}(V) is a ring homomorphism (see [7, Proposition 6.3.3]), we deduce from Eqn. (7.2) that

(7.4)  Res_{V} (η) = η̂^{|G:V |} for η̂ = ∑_{U′∈A_{z+1}^{C}(V)} N_{U′}^{V}x_{U′}^{p−1} .

Denote by W = W (V) the subspace Ann(C) of V^{∗}. Then η̂ lies in k[W], since N_{U′}^{V}(x_{U′}) is the product of all ϕ ∈ V^{∗} with Res_{U′}(ϕ) = x_{U′}. Moreover η̂ is by construction a GL(W)-invariant, so a scalar multiple of D_{1}(W) for degree reasons. By considering the restriction to any U ∈ A_{z+1}^{C}(V), we deduce from Eqn. (6.3) that

(7.5)  Res_{V}^{G}(η) = D_{1}(W)^{|G:V |} .

7.2. The existence proof.

Theorem 7.6. Let G be a p-group of order p^{n} and p-rank r whose centre has prank z. For 1 ≤ i ≤ z define ζ_{i} ∈ H^{∗}(G) by ζ_{i} = c_{p^{n}−p^{n−i}}(ρ_{G}), a Chern class of the regular representation of G.

If z < r define η ∈ H^{∗}(G) as above and homogeneous elements κ_{1}, κ_{2}, …, κ_{r−z} ∈ H^{∗}(G) as follows:

κ_{j} := P^{p^{n−z+j−4}} ⋯ P^{p^{n−z−1}}P^{p^{n−z−2}}(η) ∈ H^{2(p^{n−z}−p^{n−z−j})}(G)

for 1 ≤ j ≤ r − z. Then for each 1 ≤ j ≤ r − z and for each V ∈ A_{z+s}^{C}(G) one has

(7.7)  Res_{V}^{G}(κ_{j}) = \begin{cases}D_{j}(W)^{|G:V |} & if j ≤ s, and \\ 0 & otherwise. \end{cases}

Here, W is the subspace of V^{∗} which annihilates C.

Then ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} is a polarised system of parameters for H^{∗}(G). So H^{∗}(G) has both polarised and special polarised systems of parameters.

Proof. Equation (7.7) holds for κ_{1} = η by Eqn. (7.5). The general case of Eqn. (7.7) follows from Eqn. (6.3) and the action of the Steenrod algebra on the Dickson invariants (see [12]).

Axioms (PS2) and (PS4) follow immediately from Eqn. (7.7). Axiom (PS3) holds because the Dickson invariants form a regular sequence. Observe that ρ_{G} restricts to C as p^{n−z} copies of the regular representation ρ_{C}. So the total Chern class c(ρ_{G}) restricts to C as c(ρ_{C})^{p^{n−z}}, meaning that Res_{C}(ζ_{i}) = c_{p^{z}−p^{z−i}}(ρ_{C})^{p^{n−z}} for 1 ≤ i ≤ z. In view of Eqn. (6.1) one has c(ρ_{C}) = f_{C^{∗}}(1) and hence c_{p^{z}−p^{z−i}}(ρ_{C}) = (−1)^{i}D_{i}(C^{∗}) by Equation (6.2), a well known observation due originally to Milgram. So Res_{C}(ζ_{i}) = (−1)^{i}D_{i}(C^{∗})^{|G:C|}, which means that Axiom (PS1) is satisfied and so a polarised system of parameters has been constructed. Hence special polarised systems of parameters exist too, by Lemma 2.6. □

8. Tightness of Duflot’s lower bound

Recall that Duflot’s Theorem [6] states that the depth of H^{∗}(G) is at least z.

Theorem 8.1. Let G be a p-group of p-rank r whose centre has p-rank z. Then the following statements are equivalent:

(1) The mod-p cohomology ring H^{∗}(G) is not detected on the family H_{z+1}^{C}(G).

(2) H^{∗}(G) has an associated prime 𝔭 such that the dimension of H^{∗}(G)/𝔭 is z.

(3) There is a polarised system of parameters ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} for H^{∗}(G) such that κ_{1} is a zero divisor in H^{∗}(G).

(4) If ζ_{1}, …, ζ_{z}, κ_{1}, …, κ_{r−z} is a polarised system of parameters for H^{∗}(G) then κ_{1} is a zero divisor in H^{∗}(G).

(5) The depth of H^{∗}(G) equals z.

Proof. Carlson proved in [4] that (1) implies (5). A standard commutative algebra argument shows that (2) implies (5). We saw in Lemma 2.8 that (5) implies (4), and (3) follows from (4) by the existence result Theorem 7.6.

Now let 𝔞 be a polarised system of parameters satisfying Statement (3), which is equivalent to τ_{𝔞} = z. So τ_{𝔞,H} = z by Theorem 5.1. As in the proof of that theorem there is a special polarised system of parameters 𝔟 which satisfies τ_{𝔟,H} = τ_{𝔞,H} = z, so (2) follows by Lemma 4.2. Finally consider the definition of τ_{𝔞,H}. If τ_{𝔞,H} = z then H_{z+1}^{C}(G) does not detect H^{∗}(G), yielding (1). □

9. An example

Let G be the extraspecial p-group p_{+}^{1+2n} with n ≥ 1. This group has order p^{2n+1} and p-rank n + 1. Its centre has p-rank 1. If p is odd G has exponent p. The mod-p cohomology ring H^{∗}(G) is Cohen-Macaulay for p = 2 by Theorem 4.6 of [11], for Quillen computes the cohomology ring as the quotient of a polynomial algebra by a regular sequence. Also, Milgram and Tezuka showed in [9] that the cohomology ring is Cohen-Macaulay for G = 3_{+}^{1+2}.

From now on assume that p is odd, with n ≥ 2 if p = 3. Then by a result of Minh [10], there are essential classes. For such groups the centre C is cyclic of order p, and the set H_{2}^{C}(G) of centralisers coincides with the set of maximal subgroups. Consequently H_{z+1}^{C}(G) does not detect H^{∗}(G), and so H^{∗}(G) has depth 1 by the part of Theorem 8.1 proved by Carlson in [4].

Now let V be a rank n + 1 elementary abelian subgroup of G, and let ρ̂ be a one-dimensional ordinary representation of V whose restriction to C is not trivial. Let ρ be the induced representation of G. Then ρ is an irreducible representation of degree p^{n}, and its character restricts to each U ∈ A_{n+1}^{C}(G) as the sum of all degree one characters whose restrictions to C coincide with the restriction of the character of ρ̂. Set ζ_{1} := c_{p^{n}}(ρ) and κ_{j} := c_{p^{n}−p^{n−j}}(ρ) for 1 ≤ j ≤ n. Then ζ_{1}, κ_{1}, …, κ_{n} satisfies the axioms for a polarised system of parameters for H^{∗}(G). So combining Minh’s result with Theorem 8.1 one deduces that κ_{1} has nontrivial annihilator in H^{∗}(G). Conversely a direct proof of this fact would yield a new proof of Minh’s result. If n = 1 and p > 3 it is known (see [8]) that c_{2}(ρ) is a nonzero essential class which annihilates κ_{1}.

References

[1] D. J. Benson. The image of the transfer map. Archiv Math., 61:7–11, 1993.

[2] D. J. Benson. Polynomial invariants of finite groups. London Math. Soc. Lecture Note Series, vol. 190. Cambridge University Press, Cambridge, 1993.

[3] C. Broto and H.-W. Henn. Some remarks on central elementary abelian p-subgroups and cohomology of classifying spaces. Quart. J. Math. Oxford Ser. (2), 44(174):155–163, 1993.

[4] J. F. Carlson. Depth and transfer maps in the cohomology of groups. Math. Z., 218(3):461– 468, 1995.

[5] J. F. Carlson. Problems in the calculation of group cohomology. In P. Dräxler, G. O. Michler, and C. M. Ringel, editors, Computational methods for representations of groups and algebras (Essen, 1997), pages 107–120. Birkhäuser, Basel, 1999.

[6] J. Duflot. Depth and equivariant cohomology. Comment. Math. Helv., 56(4):627–637, 1981.

[7] L. Evens. The cohomology of groups. Oxford Univ. Press, Oxford, 1991.

[8] I. J. Leary. The integral cohomology rings of some p-groups. Math. Proc. Cambridge Philos. Soc., 110(1):25–32, 1991.

[9] R. J. Milgram and M. Tezuka. The geometry and cohomology of M_{12}. II. Bol. Soc. Mat. Mexicana (3), 1(2):91–108, 1995.

[10] P. A. Minh. Essential cohomology and extraspecial p-groups. Trans. Amer. Math. Soc., 353(5):1937–1957, 2000.

[11] D. Quillen. The mod-2 cohomology rings of extra-special 2-groups and the spinor groups. Math. Ann., 194:197–212, 1971.

[12] C. Wilkerson. A primer on the Dickson invariants. In H. R. Miller and S. B. Priddy, editors, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), Contemporary Math., vol. 19, pages 421–434, Providence, RI, 1983. Amer. Math. Soc.

Dept of Mathematics, Univ. of Wuppertal, D–42097 Wuppertal, Germany

E-mail address: green@math.uni-wuppertal.de