Depth and the Cohomology of Wreath Products

Jon F. Carlson^{1}  and  Hans-Werner Henn^{2}

Department of Mathematics Mathematisches Institut

University of Georgia Universität Heidelberg

Athens, GA 30602 Im Neuenheimer Feld 288

USA D-69120 Heidelberg GERMANY

1. Introduction.

Suppose that G is a finite group and that k is a field of characteristic p > 0. The cohomology ring, H^{∗}(G, k), is a finitely generated, graded-commutative kalgebra and hence is subject to the usual ring-theoretic scrutinies of commutative algebra. Our knowledge of the ideal spectra and associated varieties is reasonably advanced thanks to the work of Quillen and others (see [1] or [7] for general reference). Questions about depth and associated primes seem more mysterious. The depth of H^{∗}(G, k) is defined to be the length of the longest regular sequence in the ring. The most general result on depth is probably the theorem of Jeanne Duflot ([5], see also [2]) which says that the depth of H^{∗}(G, k) is at least equal to the p-rank of the center of a Sylow p-subgroup of G. This is truely a satisfying result in that it relates a cohomological structure directly to structure of the underlying group. However it does not tell us what the depth is, and there are many examples where the depth exceeds Duflot’s lower bound. The study of the associated primes for H^{∗}(G, k) is in even greater disarray. It is known that they must be invariant under the action of the subalgebra generated by the reduced power operations of the Steenrod algebra [9], which implies that they must be the radicals of the restrictions to elementary abelian p-subgroups. But little else is known beyond the only partially verified assertions of the relation to depth in [3].

In this paper we consider the behavior of the depth and of the associated

Typeset by AMS-TEX

1Partially supported by a grant from NSF.

2Supported by a Heisenberg grant from DFG.

primes of the cohomology ring H^{∗}(G, k) under the wreath product operation: G → G≀ℤ/p. In particular, we show that the depth increases by one and also that the minimal dimension of the associated primes increases by at most one. This allows us to determine completely the depths of the mod-p cohomology rings of the symmetric groups. The example of an n-fold wreath product (⋯ (ℤ/p ≀ ℤ/p) ⋯) ≀ ℤ/p shows that the depth can exceed the p-rank of the center of a p-group by an arbitrarily large amount. The results give some validity to the question in [3] on the relation of depth and the dimensions of associated primes. It seems also to be connected to question of detectability of cohomology (see [4], [8]).

The first author would like to thank the Mathematisches Institut of the Universität Heidelberg for its kind hospitality and support during the period when the results of this paper were discovered. Both authors would like to thank the referee for many helpful suggestions.

2. The cohomology of wreath products.

We begin this section by outlining the structure of the ring H^{∗}(G ≀ ℤ/p, k). We rely on the fundamental result of Nakaoka, which expresses the ring in terms of the related spectral sequence. Specifically, there is the extension

1 → G^{p} → G ≀ ℤ/p → ℤ/p → 1.

The associated Lyndon-Hochschild-Serre spectral sequence has E_{2}-term

E_{2}^{∗,∗} = H^{∗}(ℤ/p, H^{∗}(G^{p}, k)).

Nakaoka’s Theorem [11] says that the spectral sequence collapses at the E_{2}-term and that E_{2}^{∗,∗}, which is a ring and a k-algebra, is isomorphic to H^{∗}(G ≀ ℤ/p, k) as a k-algebra.

Now consider first the terms along the fiber

E_{2}^{0,∗} = H^{0}(ℤ/p, H^{∗}(G^{p}, k)) ≅ H^{∗}(G^{p}, k)^{ℤ/p}.

This subring is the ring of invariants of H^{∗}(G^{p}, k) ≅ (H^{∗}(G, k))^{⊗p} under the permutation σ of order p which permutes the factors cyclically. Because the spectral sequence collapses, this may also be identified with the image of the restriction homomorphism H^{∗}(G ≀ ℤ/p, k) → H^{∗}(G^{p}, k). Let I be the kernel of restriction. To describe I more concretely we proceed as follows.

In the ring of invariants we find the traces. If x_{1}, …, x_{p} ∈ H^{∗}(G, k) then the trace of x_{1} ⊗ ⋯ ⊗ x_{p} is

τ(x_{1} ⊗ ⋯ ⊗ x_{p}) = ∑_{i=0}^{p−1} σ^{i}(x_{1} ⊗ ⋯ ⊗ x_{p}),

which is invariant under the permutation σ. The set of all traces T r(H^{∗}(G^{p}, k)) forms an ideal inside the invariants (H^{∗}(G^{p}, k))^{ℤ/p}. The quotient by this ideal may be described as follows.

The map ϕ : H^{∗}(G, k) → (H^{∗p}(G^{p}, k))^{ℤ/p}, defined on homogeneous elements by ϕ(x) = (x⊗⋯⊗x) for x ∈ H^{∗}(G, k) has the property that ϕ(xy) = ϵϕ(x)ϕ(y) where ϵ = (−1)^{deg(x)deg(y)p(p−1)/2}. Hence ϕ is multiplicative if p is 2 or if p is is congruent to 1 modulo 4. On the other hand if p ≡ 3 mod 4 then we can also make ϕ into a multiplicative map by the introduction of a sign convention. Specifically let

ϕ(x) = μ(deg(x))(x ⊗ ⋯ ⊗ x) for x ∈ H^{∗}(G, k)

where deg(x) is the degree of x and

μ(n) = \{ \begin{matrix}−1 & if n ≡ 1 or 2 mod 4 \\ 1 & if n ≡ 0 or 3 mod 4. \end{matrix}

In either case, ϕ induces a ring homomorphism from H^{∗}(G, k) to the quotient of H^{∗}(G^{p}, k)^{ℤ/p} by T r(H^{∗}(G^{p}, k)), and ϕ multiplies degrees by p. So we have that the quotient is isomorphic to ΦH^{∗}(G, k) where for a graded vector space M we define the graded vector space ΦM by

(ΦM)^{n} = \{ \begin{matrix}M^{n/p} & if n ≡ 0 mod p \\ 0 & if n ≢ 0 mod p . \end{matrix}

The element m ∈ M^{n} will be denoted by Φm if it is considered as element in (ΦM)^{np}. If M is a graded algebra then the formula Φ(m)Φ(n) = (−1)^{deg(m)deg(n)p(p−1)/2}Φ(mn) defines an algebra structure on Φ(M), and with this algebra structure the quotient is isomorphic to Φ(H^{∗}(G, k)) as a graded algebra.

The set of all traces form an ideal also in H^{∗}(G ≀ ℤ/p, k), where it is the annihilator of the ideal I, the kernel of the restriction to H^{∗}(G^{p}, k). Furthermore the quotient ring H^{∗}(G ≀ ℤ/p, k)/T r(H^{∗}(G^{p}, k)) can be identified with H^{∗}(ℤ/p, k) ⊗ ΦH^{∗}(G, k) and I embeds into this quotient as H̃^{∗}(ℤ/p, k)⊗ΦH^{∗}(G, k), where H̃ denotes reduced homology. Summing up we have the following exact sequence of modules over H^{∗}(G ≀ ℤ/p, k)

(2.1) 0 → H̃^{∗}(ℤ/p, k) ⊗ ΦH^{∗}(G, k) → H^{∗}(G ≀ ℤ/p, k) → (H^{∗}(G^{p}, k))^{ℤ/p} → 0

With this preparation we are ready to prove our main theorem which reads as follows.

Theorem 2.1. Suppose that the depth of H^{∗}(G, k) is d. Then the depth of H^{∗}(G ≀ ℤ/p, k) is d + 1.

The proof uses the homological criterion for depth given in Theorem 16.7 of [10]. In the case that M is a finitely generated, graded A = H^{∗}(G, k)-module, it says that the depth of M is the least integer n such that Ext_{A}^{n}(k, M) ≠ 0. From sequence (2.1) and the long exact sequence on Ext, the theorem is a consequence of the following two results.

Lemma 2.2. The depth of H̃^{∗}(ℤ/p, k) ⊗ Φ(H^{∗}(G, k)) as an H^{∗}(G ≀ ℤ/p, k)- module is d + 1.

Lemma 2.3. The depth of (H^{∗}(G^{p}, k))^{ℤ/p} as an H^{∗}(G ≀ ℤ/p, k)-module is at least d + 1.

Proof of Lemma 2.2. First we notice that that the depth of I ≅ H̃^{∗}(ℤ/p, k) ⊗ Φ(H^{∗}(G, k)) as an H^{∗}(G≀ℤ/p, k)-module agrees with the depth of H̃^{∗}(ℤ/p, k)⊗ Φ(H^{∗}(G, k)) as an ideal in H^{∗}(ℤ/p, k) ⊗ Φ(H^{∗}(G, k)). Now let x_{1}, …, x_{d} be a regular sequence of maximal length in H^{∗}(G, k) and let μ be a generator of H^{2}(ℤ/p, k). Because Φ(H^{∗}(G, k)) and H^{∗}(G, k) are isomorphic it is straigthtforward to see that the sequence μ ⊗ 1, 1 ⊗ Φ(x_{1}), …, 1 ⊗ Φ(x_{d}) is a regular sequence of maximal length on H̃^{∗}(ℤ/p, k) ⊗ Φ(H^{∗}(G, k)) and hence the depth is d + 1.

Proof of Lemma 2.3. Consider a regular sequence x_{1}, …, x_{d} of homogeneous elements in H^{∗}(G, k). If P is the polynomial algebra with polynomial generators x_{1}, …, x_{d} then H^{∗}(G, k) is a free graded module over P and we can write H^{∗}(G, k) ≅ P ⊗ V where V is a graded vector space with basis corresponding to the generators of H^{∗}(G, k) as P -module (see Proposition 10.3.4 of [7]). We claim that the elements ϕ(x_{1}), …, ϕ(x_{d}), τ(x_{1} ⊗ 1 ⊗ ⋯ ⊗ 1) form a regular sequence for (H^{∗}(G^{p}, k))^{ℤ/p}.

To see this we proceed as follows. We pick a vector space basis B′ of V and consider the monomial basis M of P. We choose any order on B′ and consider the lexicographical order on M induced by declaring that x_{1} > x_{2} > … > x_{d}. A vector space basis of H^{∗}(G, k) ≅ P ⊗V is then given by the set of elements m⊗v with m ∈ M and v ∈ B′. We order this basis via the induced lexicographical order on the set B := M × B′. Finally we consider the induced lexicographical order on the set B^{p}. With this order a basis of (H^{∗}(G^{p}, k))^{ℤ/p} is given by the elements b ⊗ … ⊗ b with b ∈ B together with the elements τ(b_{1} ⊗ b_{2} … ⊗ b_{p}) where each b_{i} is in B, not all of the b_{i} are equal and (b_{1}, …, b_{p}) is larger than any of its cyclic permutations in B^{p}.

That the first d elements in the sequence above form a regular sequence is easy to see. The action of ϕ(x_{i}) on basis elements is given by

ϕ(x_{i})(b ⊗ … ⊗ b) = ±x_{i}b ⊗ x_{i}b ⊗ … ⊗ x_{i}b

respectively by

ϕ(x_{i})τ(b_{1} ⊗ b_{2} ⊗ … ⊗ b_{p}) = ±τ(x_{i}b_{1} ⊗ x_{i}b_{2} ⊗ … ⊗ x_{i}b_{p})

and hence after dividing out ϕ(x_{1}), …, ϕ(x_{k}) the quotient has a vector space basis consisting of elements b ⊗ … ⊗ b and τ(b_{1} ⊗ b_{2} ⊗ ⋯ ⊗ b_{p}) as above with the additional condition that b, or respectively the gcd of b_{1}, … b_{p}, is not divisible by x_{i} for all i ≤ k. In particular ϕ(x_{k+1}) is still regular on this quotient (for k + 1 ≤ d).

Now the action of τ(x_{1} ⊗ 1 ⊗ … ⊗ 1) on basis elements is given by

τ(x_{1} ⊗ 1 ⊗ … ⊗ 1)(b ⊗ … ⊗ b) = τ(x_{1}b ⊗ b ⊗ … ⊗ b)

or by

τ(x_{1} ⊗ 1 ⊗ … ⊗ 1)τ(b_{1} ⊗ b_{2} ⊗ … ⊗ b_{p}) = Σ_{i}τ(b_{1} ⊗ … ⊗ x_{1}b_{i} ⊗ … ⊗ b_{p}) .

In particular we see that τ(x_{1} ⊗ 1 ⊗ … ⊗ 1)τ(b_{1} ⊗ b_{2} ⊗ … ⊗ b_{p}) ≡ τ(x_{1}b_{1} ⊗ b_{2} ⊗ … ⊗ b_{p}) modulo terms of smaller lexicograghical order.

So in order to see that τ(x_{1}⊗1⊗…⊗1) is regular on τ((P ⊗V)^{⊗p})/(ϕ(x_{1}), ..., ϕ(x_{d})), it suffices to show that for the remaining basis elements in τ((P ⊗ V)^{⊗p})/(ϕ(x_{1}), ..., ϕ(x_{d})) the elements τ(x_{1}b_{1} ⊗ b_{2} ⊗ … ⊗ b_{p}) are nontrivial basis elements in this quotient, i.e. the gcd of x_{1}b_{1}, b_{2}, …, b_{p} is not divisible by x_{i} for all i ≤ k. This is obvious if x_{1} divides b_{1}. And if x_{1} does not divide b_{1}, then x_{1} does not divide any of the b_{i} because b_{1} ≥ b_{i} for all i, and hence τ(x_{1}b_{1} ⊗ b_{2} ⊗ ⋯ ⊗ b_{p}) is again a nontrivial basis element. Hence we have proved Lemma 2.3 and the Theorem.

3. Dimensions, Associated Primes and Symmetric Groups. It is well known that the depth of a commutative noetherian local ring is

at most equal to the minimum of the dimensions of the associated primes in the ring. The same is true for the graded commutative ring H^{∗}(G, k). By the dimension of a prime 𝔭 in a cohomology ring H^{∗}(G, k), we mean the Krull dimension of the quotient ring H^{∗}(G, k)/𝔭, or the dimension of its maximal ideal spectrum V_{G}(𝔭). Thus for cohomology rings the conclusion of the first sentence of the paragraph can be established by the following argument. If x_{1}, …, x_{d} is any regular sequence of homogeneous elements for H^{∗}(G, k), then as noted in the last section H^{∗}(G, k) is a free module over the polynomial subring generated by x_{1}, …, x_{d} (see also (10.3.4) of [7]). In particular it is a torsion free module. So suppose that there is an element a in H^{∗}(G, k) whose annihilator is a prime ideal 𝔭 of dimension s. In the terminology of ([13], top of page 193), H^{∗}(G, k)/𝔭 is a finite integral domain which has transcendence degree s over k. In particular, it cannot contain a set of more than s algebraically independent elements. If we were to have that d > s, then some nonzero polynomial in x_{1}, …, x_{d} would have to be in 𝔭, and hence would annihilate a. However the torsion free condition makes this impossible. So s ≥ d.

The minimal primes in H^{∗}(G, k) are always among the associated primes (see [10], (6.5)), and they are in one-to-one correspondence with the components of the maximal ideal spectrum V_{G}(k). By Quillen [12] , the components of the variety are in one-to-one correspondence with the conjugacy classes of maximal elementary abelian p-subgroups of G. That is, if E is a maximal elementary abelian p-subgroup of G, then the radical of the restriction to E is a minimal prime ideal in H^{∗}(G, k) and its dimension is the rank of E. It follows that the depth of H^{∗}(G, k) never exceeds the minimum of the ranks of the maximal elementary abelian p-subgroups of G.

With regards to associated primes, it was speculated in [3] that the depth of any mod-p cohomology ring H^{∗}(G, k) might coincide with the minimum of the dimensions of the associated primes. The following analysis and examples support the speculation, even though the evidence is not strong.

Proposition 3.1. (a) If a is the minimum of the dimensions of the associated primes of H^{∗}(G, k), then the minimum of the dimensions of the associated primes of H^{∗}(G ≀ ℤ/p, k) is at most a + 1.

(b) Suppose that m is the minimum of the dimensions of the components of the variety of H^{∗}(G, k). Then m + 1 is the minimum of the dimension of the components of the maximal ideal spectrum of H^{∗}(G ≀ ℤ/p, k).

Proof. Suppose that ζ ∈ H^{∗}(G, k) is an element whose annihilator is a prime 𝔭 with H^{∗}(G, k)/𝔭 of minimal dimension a. Then we claim that (in the notation of the last section) the annihilator 𝔭′ of ϕ(ζ)μ has dimension a + 1. This is because T r(H^{∗}(G^{p}, k)) annihilates μ and hence

H^{∗}(G ≀ ℤ/p, k)/𝔭′ ≅ Φ(H^{∗}(G, k)/𝔭) ⊗ H^{∗}(ℤ/p, k).

This proves (a).

For (b) we use Quillen’s correspondence of the components of V_{G}(k) with the maximal elementary abelian p-subgroups of G. Let y ∈ G ≀ ℤ/p be an element of order p in G ≀ ℤ/p that cyclically permutes the factors of G^{p} = G × ⋯ × G. Then the centralizer of y is the direct product ΔG× < y > where ΔG = {(g, ⋯, g)|g ∈ G} ≅ G is the diagonal subgroup. Let E be a maximal elementary abelian p-subgroup of rank m in G. Then ΔE× < y > is contained in no larger elementary abelian p-subgroup of G ≀ ℤ/p.

On the other hand, suppose that E ⊆ G≀ℤ/p is a maximal elementary abelian p-subgroup. If E ⊆ G^{p} then the intersection of E with each of the factors

G_{i} = {(g_{1}, …, g_{p}) | g_{j} = 1, if j ≠ i} ⊆ G^{p}

is a maximal elementary abelian p-subgroup of G_{i}. In this case it is clear that the rank of E is greater than m. So we may assume that E ⊈ G^{p}, and E is in the centralizer of an element of order p of the form x = (g_{1}, …, g_{p})y for y as above. Then x^{p} = 1 implies that g_{1} ⋯ g_{p} = 1 and that

E ∩ G^{p} ≅ {(u, g_{1}^{−1}ug_{1}, …, g_{p−1}^{−1} ⋯ g_{1}^{−1}ug_{1} ⋯ g_{p−1}) | u ∈ U}

where U is a maximal elementary abelian p-subgroup of G. Again the rank of E is greater than m.

Finally we consider the case of the symmetric groups S_{n}. For notation let W_{1} = ℤ/p and inductively define the n-fold wreath product W_{n} = W_{n−1} ≀ ℤ/p.

Corollary 3.2. Suppose that n has p-adic expansion n = b_{0} + b_{1}p + ⋯ + b_{t}p^{t}. Then

depth(H^{∗}(S_{n}, k)) = b_{1} + 2b_{2} + ⋯ + tb_{t}.

This number is equal to the minimum of the dimensions of the components of H^{∗}(S_{n}, k) and also the minimum of the dimensions of the associated primes.

Proof. Let P be a Sylow p-subgroup of S_{n}. Now the depth of H^{∗}(G, k) is at least equal to the depth of H^{∗}(P, k) by known arguments (see [7], proof of 10.3.1). The subgroup P is isomorphic to

W_{1}^{b_{1}} × W_{2}^{b_{2}} × ⋯ × W_{t}^{b_{t}}

In general for finite groups A and B we have that H^{∗}(A × B, k) ≅ H^{∗}(A, k) ⊗_{k} H^{∗}(B, k) with the usual sign convention for multiplying elements of odd degree. So it can be seen, using a freeness argument as in Lemma 2.3 that the depth of H^{∗}(A × B, k) is greater than or equal to the sum of the depths of H^{∗}(A, k) and H^{∗}(B, k). An induction argument on Theorem 2.1 shows that depth(H^{∗}(W_{j}, k)) = j and hence we have that

depth(H^{∗}(G, k)) ≥ depth(H^{∗}(P, k)) ≥ b_{1} + 2b_{2} + ⋯ + tb_{t}.

On the other hand, by induction on Proposition 3.1(b), we get that H^{∗}(W_{j}, k) has a component of dimension j and hence a maximal elementary abelian psubgroup of rank j. So P has a maximal elementary abelian p-subgroup of rank b_{1} + ⋯ + tb_{t}. It follows that H^{∗}(G, k) has a minimal prime (and an associated prime) of that dimension. This gives the reverse inequality.

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