DEPTH AND TRANSFER MAPS IN

THE COHOMOLOGY OF GROUPS

Jon F. Carlson^{∗}

Department of Mathematics University of Gerogia Athens, Georgia 30602

1. Introduction.

Suppose that G is a finite group and that k is a field of characteristic p > 0 dividing the order of G. There has been a lot of recent work that has focused on the algebraic structure of the mod-p cohomology ring H^{∗}(G, k) ≃ k⊗_{𝔽_{p}}H^{∗}(G, 𝔽_{p}), and the relation to the structure of the group itself. Quillen’s Dimension Theorem [ ] showed the correspondence of the components of the maximal ideal spectrum with the conjugacy classes of maximal elementary abelian p-subgroups via restriction maps. Duflot [ ] proved that the depth (length of the longest regular sequence) of the cohomology ring is at least equal to the prank of the center of G. The associated primes are known to be invariant under the action of the Steenrod algebra ([ ], [ ]) which means basically that they are the radicals of the kernels of restrictions to certain subgroups. Regular elements are supported on elementary abelian p-subgroups whose centralizers detect all of the cohomology ([ ], [ ]).

What is missing from this is the idea that the transfer maps may play a determining role in some of the ring structure such as associated primes and depth. For example, it can be easily seen that in order for the cohomology ring to be Cohen-Macaulay it is necessary that the components of the maximal ideal spectrum have the same dimension and hence that the maximal elementary abelian p-subgroups have the same rank. However the semidihedral 2-group and, for p odd, the extraspecial group of order p^{3} and exponent p^{2} are examples of groups which have only one class of maximal elementary abelian p-subgroups but whose mod-p cohomology rings are not Cohen-Macaulay. Previously we knew these facts only by direct calculation of the cohomology rings ([ ], [ ]). However an easy proof can be constructed using transfer maps. Step one is to notice that in these cohomology rings any system of parameters would consist of two (the Krull dimension) elements, and that it’s possible to choose a system of parameters in which one of the elements is a transfer from an elementary abelian subgroup [ ]. If the cohomology ring were Cohen-Macaulay then the system of paramenters would have to be a regular sequence and each element would be a regular element. But this is impossible. The reason is that there is a maximal subgroup which contains all elements of order p, and hence there is an element in H^{1}(G, k) whose

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∗ Partially supported by a grant from NSF

restriction to the elementary abelian subgroup vanishes. By Frobenius reciprocity this element must be annihilated by the parameter which is a transfer.

In this paper we generalize the argument of the previous paragraph. The main theorem is that if there exists an element in H^{∗}(G, k) whose restrictions to the centralizers of elementary abelian p-subgroups of rank t vanish then the depth of H^{∗}(G, k) is less than t, and moreover H^{∗}(G, k) has an associated prime of dimension less than t, height at least r − t where r = p − rank(G) is the Krull dimension of H^{∗}(G, k) . The primary ingredient in the proof is Benson’s generalization, [ ], of [ ] which tells us the images of the transfer maps from the centralizers of the elementary abelian p-subgroups. The point is that the dimension of the ideal generated by the transfers from the centralizers of all elementary abelian p-subgroups of rank s is always less than s.

In Section 3, we show that the theorem has a strong converse in the case that the p-rank of G is 2. Indeed, we know of no counterexamples to the converse in any rank. although the converse does not hold for all finitely generated graded-commutative k-algebras, there may be reasons for suspecting that it may be true for mod −p cohomology rings of finite groups.

Notations. Throughout the paper we let r denote the p-rank of G, the largest integer such that G has an elementary abelian p-subgroup, E ≃ (ℤ/(p))^{r} ⊆ G. Let A(G) be the collection of all elementary abelian p-subgroups and for 0 ≤ s ≤ r let A_{s}(G) be the set of all elements of A(G) of rank s. If H is a subgroup of G, let T r_{H}^{G} : H^{∗}(H, k) →H^{∗}(G, k) be the transfer or corestriction map and let res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k) be the restriction map. Recall that these two maps are related by Frobenius reciprocity:

T r_{H}^{G}(γ) · ζ = T r_{H}^{G}(γ · res_{G,H}(ζ))

where γ ∈ H^{∗}(H, k) and ζ ∈ H^{∗}(G, k) .

As a general reference for the cohomological results of this paper, we refer the reader to the books by Benson [ ] and Evens [ ].

2. Dimensions of transfer ideals.

We begin this section by quoting Benson’s theorem (Theorem 1.1 of [ ]). It should be noted that statement of the theorem in [ ] is slightly flawed in that it does not account for the possibility that the index in G of one of the subgroups in H is relatively prime to p. For in this case every element H^{∗}(G, k) is a transfer from this subgroup in H. However, if we adopt the following convention then Benson’s result remains correct. If H has an element whose index in G is relatively prime to p, then let K be the set consisting only of the trivial subgroup and assume that J = J′ is the ideal of all elements of positive degree. Then we have

Theorem 2.1 [ ]. Suppose that H is a non empty collection of subgroups of G. Let K be the set of all subgroups K of G such that the Sylow p-subgroup of the centralizer C_{G}(K) is not conjugate to a subgroup of any of the subgroups in H. Let J = ∑_{H∈H} T r_{H}^{G}(H^{∗}(H, k))

and let J′ = ⋂_{K∈K} ker res_{G,K}. Then \sqrt{J} = \sqrt{J′}.

The main use that we make of the theorem is in the special case of the following corollary. For notation, suppose that H_{s} = {C_{G}(E) | E ∈ A_{s}(G)} for 0 ≤ s ≤ r. Let J_{s} = ∑_{H∈H_{s}} T r_{H}^{G}( ∑_{n>0} H^{n}(H, k)). Let V_{G}(k) denote the maximal ideal spectrum of H^{∗}(G, k)

and let V_{G}(J_{s}) ⊆ V_{G}(k) be the (Zariski-) closed set of all maximal ideals containing J_{s}.

Corollary 2.2. For s > 0, dim V_{G}(J_{s}) < s.

Proof. Let z be the p-rank of the center Z(G) of G. We have two cases. If s ≤ z then there is an elementary abelian p-subgroup with E ∈ A_{s}(G) with E ⊆ Z(G). So |G : C_{G}(E)| is not divisible by p and as we noted J_{s} = ∑_{n>0} H^{n}(G, k). Hence V_{G}(J_{s}) = {0} has dimension

0.

Now suppoe that s > z. Then | G : H | is divisible by p for all H ∈ H_{s}. The set K_{s} corresponding to H_{s} in (2.1) consists of all subgroups K such that if S_{p}(K) is a Sylow p-subgroup of K then C_{G}(S_{p}(K)) ⊈ H for all H ∈ H_{s}. We claim that every such K ∈ K_{s} has p-rank less than s. For suppose that there exists E ∈ A_{s}(G) such that E ⊆ K ∈ K_{s}. Then we can assume that E ⊆ S_{p}(K) and hence

C_{G}(S_{p}(K)) ⊆ C_{G}(E) ∈ H_{s}.

Now let J′_{s} = ⋂_{K∈K_{s}} ker res_{G,K}. Then every maximal ideal containing J′_{s} is the pullback

of a maximal ideal in H^{∗}(K, k), under the restriction map, for some K ∈ K_{s}. It follows that dim V_{G}(J′_{s}) is equal to the maximum of the dimensions, dim V_{K}(k) for all K ∈ K_{s}. By Quillen’s Dimension Theorem and the previous paragraph, dim V_{G}(J′_{s}) < s. But by the Theorem (2.1), V_{G}(J_{s}) = V_{G}(J′_{s}).

We use the corollary to prove our main theorem:

Theorem 2.3. Suppose that H^{∗}(G, k) has an element ζ with the property that res_{G,C_{g}(E)}(ζ) = 0 for all E ∈ A_{s}(G). Then H^{∗}(G, k) has an associated prime 𝔭 with dim V_{G}(𝔭) < s. In this case depth(H^{∗}(G, k) ) < s.

Proof. Suppose that γ ∈ J_{s}, then γ = ∑_{H∈H_{s}} T r_{H}^{G}(γ_{H}) for γ_{H} ∈ ∑_{n>0} H^{∗}(H, k). By Frobenius

reciprocity

γ · ζ = ∑_{H∈H_{s}} T r_{H}^{G}(γ_{H} · res_{G,H}(ζ)) = 0

because res_{G,H}(ζ) = 0 for H ∈ H_{s}. Consequently there is an associated prime p such that

J_{s} ⊆ Ann_{H^{∗}(G,k)}(ζ) ⊆ 𝔭.

It follows that dim V_{G}(𝔭) ≤ dim V_{G}(J_{s}) < s by the corollary.

The statement about depth is standard commutative algebra (see [Serre], Prop. IV 7).

Now the two cases mentioned in the introduction should be clear. For if p = 2 and G is a semidihedral 2-group, or if p is odd and G is an extra special group of order p^{3} and exponent p^{2}, then there exists a subgroup H ⊆ G such that | G : H |= p and every element of order p is contained in H. Therefore the element in H^{1}(G, k) corresponding to the homomorphism G → G/H ≃ ℤ/(p) has the property that its restriction to C_{G}(E) ⊆ H is zero for every E ∈ A_{2}(G).

The other statement mentioned in the introduction can be made more explicit by the following.

Proposition 2.4. The cohomology ring H^{∗}(G, k) has a homogeneous set ζ_{1}, ⋯, ζ_{r} of parameters with the property that for every s

ζ_{s} ∈ ∑_{E∈H_{s}(G)} T r_{C_{G}(E)}^{G}(H^{∗}(C_{G}(E), k)).

Proof. Recall that a homogeneous set of parameters is a set ζ_{1}, ⋯, ζ_{r}, r = p − rank(G) =

dim V_{G}(k), such that H^{∗}(G, k)/(ζ_{q}, ⋯, ζ_{r}) is finite dimensional, or equivalently ⋂_{i=1}^{r} V_{G}(ζ_{i}) =

{0}. The proposition is proved by noticing that a homogeneous set of parameters can be chosen, inductively starting with r, to satisfy the conditions

(1)  V_{G}(ζ_{s+1} ⋯, ζ_{r}) = ∩_{i=s+1}^{r}V_{G}(ζ_{i}) = r − s

and

(2)  V_{g}(ζ_{s+1}, ⋯, ζ_{r}) ⊇ V_{G}(J_{r−s})

for all s. Condition (1) must be satisfied by any homogeneous set of parameters. Condition (2) is made psssible by the fact that

V_{G}(J_{1}) ⊆ ⋯ ⊆ V_{G}(J_{r−1}) ⊆ V_{G}(J_{r})

and the previous corollary that dim V_{G}(J_{r−s}) < r − s. It is standard commutative algebra that such choices can be made.

3. Some notes on the converse.

We wish to point out in this section that there are no known counterexamples to the converse of Theorem 2.3. Hence we ask the following question.

Question 3.1. Suppose that H^{∗}(G, k) has depth d. Is there an associated prime 𝔭 in H^{∗}(G, k) such that dim V_{G}(𝔭) = d?

The answer to the question for ordinary commutative or graded-commutative k-algebras would be “no”. As an example, consider the subring of the polynomial ring k[x, y] generated by x^{2}, x^{3}, xy and y. As a module over itself it has depth 1 inspite of the fact that the only associated prime is {0}. Some evidence for an affirmative answer is given in the following.

Proposition 3.2. Suppose that G has p-rank 2. Then H^{∗}(G, k) is Cohen-Macaulay (has depth 2) if and only if all associated primes have dimension 2.

Proof. The proof is given in [ ]. The point, in this case, is that if H^{∗}(G, k) is not Cohen-Macaulay then there must be nonzero products in negative cohomology. This, in turn, requires the existence of an element whose annihilator has dimension one.

Beyond that, there are some other reasons for suspecting that (3.1) might have an affirmative answer. We show below that an affirmative answer to (3.1) would be guaranteed by an affirmative answer to the follwoing question (3.3). It seems often to be the case that the images of restriction maps are generated, modulo nilpotent elements, by elements which have been “normed up”, by the Evens Norm Map from elements in the cohomology of intermediate subgroups. The norms are very well behaved in general, and give some hope that the question (3.3) might have a “yes” answer. However this is far from a proof.

Question 3.3. Suppose that H is a subgroup of group G. Is it true that

depth (res_{G,H}(H^{∗}(G, k)) ≥ depth H^{∗}(H, k)?

There are examples in which the above inequality is strict. The mod-2 cohomology ring of the Mathieu group, M_{11}, is Cohen-Macaulay (has depth 2) while its Sylow 2-subgroup is semi-dihedral and its cohomology ring has depth one. The restriction map to the Sylow p-subgroup of any group is injective. Thus we get the strict inequality. However, we know of no cases of strict inequality when G and H are both p-groups.

Theorem 3.4. Let e be a fixed integer 1 ≤ e ≤ r. Suppose that for all E ∈ A_{e}(G)

depth (res_{G,C_{G}(E)}(H^{∗}(G, k))) ≥ depth H^{∗}(C_{G}(E), k)

(which would be implied by an affirmative answer to (3.3)). Suppose also that

∏_{E∈A_{e}(G)} res_{G,C_{G}(E)} : H^{∗}(G, k) → ∏_{E∈A_{e}(G)} H^{∗}(C_{G}(E), k)

is injective. Then the depth of H^{∗}(G, k) is at least e.

Proof. By Duflot’s Theorem [ ], both H^{∗}(C_{G}(E), k) and also res_{G,C_{G}(E)}H^{∗}(G, k) have depth at least e. The proof of the theorem is based on the fact that any regular sequence on H^{∗}(G, k) may be extended to a regular sequence of maximal length (see [Serre], IV, Prop. 6 for a proof in the case of a semi-local ring). Here maximal length is the depth of H^{∗}(G, k) .

Choose a regular sequence ζ_{1}, ⋯, ζ_{e} inductively for H^{∗}(G, k) by the following method. Suppose that ζ_{1}, ⋯, ζ_{s} have been chosen. To get ζ_{s+1} it is only necessary to pick a nondivisor of zero in H^{∗}(G, k)/(ζ_{1}, ⋯, ζ_{s}), or equivalently to choose ζ_{s+1} ∈ H^{∗}(G, k) such that, for each E ∈ A_{e}(G), the element res_{G,C_{G}(E)}(ζ_{s+1}) is not a divisor of zero on the ring

R(E, s) = res_{G,C_{G}(E)}(H^{∗}(G, k))/(res_{G,C_{G}(E)}(ζ_{i}) | i = 1, ⋯, s).

But here it is only necessary that res_{G,C_{G}(E)}(ζ_{s+1}) not be in any of the associated prime ideals 𝔭 ∈ Ass(R(E, s)). Equivalently, we need only choose ζ_{s+1} so that it is not in any of the pullbacks res_{G,E}^{−1}(θ_{E}^{−1}(𝔭)) for E ∈ A_{e}(G) and 𝔭 ∈ Ass(R(E, s)), where

θ_{E} : res_{G,C_{G}(E)}(H^{∗}(G, k)) → R(E, s)

is the quotient map. Again, equivalently, we need that V_{G}(ζ_{s+1}) does not contain any of the subvarieties V_{G}(res_{G,C_{G}(E)}^{−1}(θ_{E}^{−1}(𝔭))). But finally we have that the statement about the depth guarantees that

dim V_{G}(res_{G,C_{G}(E)}^{−1}(θ_{E}^{−1}(𝔭))) ≥ e − s > 0

for all E ∈ A_{e}(G) and 𝔭 ∈ Ass (R(E, s)). Consequently the choice of ζ_{s+1} is possible.

A couple of remarks to tie this together are in order.

Remarks 3.5 (a) Suppose that ζ ∈ H^{∗}(G, k) and that 𝔞 = ann_{H^{∗}(G,k)}(ζ) is the annihilator ideal. If H^{∗}(G, k)/𝔞 has krull dimension s then res_{G,C_{G}(E)}(ζ) = 0 for all E ∈ A_{t}(G) with t > s. For suppose otherwise. Let 𝔞′ be the annihilator of res_{G,C_{G}(E)}(ζ) in H^{∗}(C_{G}(E), k). Then 𝔞 ⊆ res_{G,C_{G}(E)}^{−1}(𝔞′) and hence.

s = dimV_{G}(𝔞) ≤ dim V_{C_{G}(E)}(res_{G,C_{G}(E)}^{−1}(𝔞′)) = dim V_{C_{G}(E)}(𝔞′).

This says that H^{∗}(C_{G}(E), k) has an associated prime of dimension s or larger. However this is a contradiction to the fact that H^{∗}(C_{G}(E), k) has depth at least equal to p-rank (E) = t > s, by Duflot’s Theorem.

(b) It is well known that the depth of a finitely generated k-algebra is at most equal to the minimum of the dimensions of the components of the maximal ideal spectrum. A proof of this can be obtained from the above by noticing that there must be an element whose variety contains every component of the maximal ideal spectrum of H^{∗}(G, k) except for one of the components of minimal dimension. Then by Quillen’s theorem, this element or some power of it will vanish on restriction to every elementary abelian p-subgroup of p-rank larger than the minimum dimension of the components. In [ ] we show that for the Sylow p-subgroups of the symmetric groups, the question (3.1) has an affirmative answer. However this is a bit of a cheat because the depth in this case is precisely the minimum dimension of the components of the maximal ideal spectrum. Thus the result would follow from other considerations.

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