PRIMITIVES AND CENTRAL DETECTION NUMBERS IN GROUP COHOMOLOGY

NICHOLAS J. KUHN

Abstract. Fix a prime p. Given a finite group G, let H^{∗}(G) denote its mod p cohomology. In the early 1990’s, Henn, Lannes, and Schwartz introduced two invariants d_{0}(G) and d_{1}(G) of H^{∗}(G) viewed as a module over the mod p– Steenrod algebra. They showed that, in a precise sense, H^{∗}(G) is respectively detected and determined by H^{d}(C_{G}(V)) for d ≤ d_{0}(G) and d ≤ d_{1}(G), with V running through the elementary abelian p–subgroups of G.

The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^{∗}(G) to H^{∗}(C), where C is the maximal central elementary abelian p–subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^{∗}(C) ⊗_{H^{∗}(G)} 𝔽_{p}, a number that tends to be quite easy to calculate.

Our results are complete when G has a p–Sylow subgroup P in which every element of order p is central. Using Benson–Carlson duality, we show that in this case, d_{0}(G) = d_{0}(P) = e(P), and a similar exact formula holds for d_{1}. As a bonus, we learn that H^{e(G)}(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.

In general, we are able to show that d_{0}(G) ≤ max{e(C_{G}(V)) | V < G} if certain cases of Benson’s Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p–rank of G and the depth of H^{∗}(G) is at most 2. When we look at examples with p = 2, we learn that d_{0}(G) ≤ 14 for all groups with 2–Sylow subgroup of order up to 64, with equality realized when G = SU(3, 4).

Enroute we study two objects of independent interest. If C is any central elementary abelian p–subgroup of G, then H^{∗}(G) is a H^{∗}(C)–comodule, and we prove that the subalgebra of H^{∗}(C)–primitives is always Noetherian of Krull dimension equal to the p–rank of G minus the p–rank of C. If the depth of H^{∗}(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen–Macauley in a certain sense, and prove related structural results.

1. Introduction

Fix a prime p, and let H^{∗}(G) denote the mod p cohomology ring of a finite group G. The p–elementary abelian subgroups of G have had a featured role in the study of group cohomology since D.Quillen’s famous work [Q] in the late 1960’s. In particular, these subgroups become the objects in a category A(G) having morphisms the homomorphisms generated by subgroup inclusion and conjugation by elements in G. The inclusions V < G then induce a map

H^{∗}(G) ⟶^{λ_{0}} lim_{V ∈A(G)} H^{∗}(V) ⊆ ∏_{V} H^{∗}(V),

Date: December 5, 2006. 2000 Mathematics Subject Classification. Primary 20J06; Secondary 55R40. This research was partially supported by a grant from the National Science Foundation.

and λ_{0} is shown to have kernel and cokernel that are nilpotent in an appropriate sense.

Viewing H^{∗}(G) as the mod p cohomology of the classifying space BG makes it evident that H^{∗}(G) is an object in K and U, the categories of unstable algebras and modules over the mod p Steenrod algebra A. The 1980’s and 1990’s saw a revolution in our understanding of these categories, and the 1995 paper of H.- W.Henn, J.Lannes, and L.Schwartz [HLS1] revisited Quillen’s approximation of H^{∗}(G) from this new perspective.

For each d ≥ 0, the group homomorphisms V × C_{G}(V) → G induce a map of unstable algebras

H^{∗}(G) → ∏_{V} H^{∗}(V) ⊗ H^{≤d}(C_{G}(V)),

where M^{≤d} denotes the quotient of a graded module M^{∗} by all elements of degree more than d. The image of this map lands in an evident subalgebra of ‘compatible’ elements which Henn, Lannes, and Schwartz show can be naturally identified with L_{d}H^{∗}(G), where L_{d} : U → U is localization away from the localizing subcategory generated by (d+1)–fold suspensions of unstable modules. Thus Quillen’s map can be viewed as the just the bottom of a tower of localizations of H^{∗}(G) associated to the nilpotent filtration of U:

\xymatrix{ & ⋮ \ar[d] \\  & L_{2}H^{∗}(G) \ar[d]^{p_{2}} \\  & L_{1}H^{∗}(G) \ar[d]^{p_{1}} \\ H^{∗}(G) \ar[r]^{λ_{0}} \ar[ur]^{λ_{1}} \ar[uur]^{λ_{2}} & L_{0}H^{∗}(G),}

where we have

H^{∗}(G) ⟶^{λ_{d}} L_{d}H^{∗}(G) ⊆ ∏_{V} H^{∗}(V) ⊗ H^{≤d}(C_{G}(V)).

This caused the authors of [HLS1] to introduce two new invariants of G: d_{0}(G) and d_{1}(G) are the smallest d’s such that H^{∗}(G) is respectively detected by, and isomorphic to, L_{d}H^{∗}(G). Alternatively, d_{0}(G) is the smallest d such that H^{∗}(G) contains no (d + 1)–fold suspensions of a nontrivial unstable module, and d_{1}(G) is the smallest d such that also Ext_{A}^{1}(Σ^{d+1}N, H^{∗}(G)) = 0 for all N ∈ U.

These invariants satisfy a few easily verified nice properties: d_{0}(G×H) = d_{0}(G)+ d_{0}(H), d_{1}(G × H) = max{d_{1}(G) + d_{0}(H), d_{0}(G) + d_{1}(H)}, and d_{i}(G) ≤ d_{i}(P) if P is a p–Sylow subgroup of G. However, they are not well behaved under taking subgroups, quotient groups, and extensions; e.g., every G embeds in a symmetric group Σ_{n} and d_{0}(Σ_{n}) = 0. Rough upper bounds for d_{0}(G) and d_{1}(G) were found in [HLS1]; e.g., d_{0}(G) is bounded by n^{2} if a p–Sylow subgroup of G admits a faithful n dimensional complex representation. However, in all but a few examples, these bounds seem far from optimal. Up to now, what determines these group invariants has remained mysterious, and they have not been connected to other work in group cohomology.

A main goal of this paper is to present a way to calculate the number d_{0}(G), and, in some cases, d_{1}(G). Our finding is that these numbers seem to be controlled by the restriction of cohomology to maximal central p–elementary abelian subgroups. Our results are complete when G has a Sylow subgroup that is p–central, i.e. a group in which every element of order p is central. For example, when p = 2, we compute that d_{0}(SU(3, 4)) = 14 and d_{1}(SU(3, 4)) = 18, where, by constrast, the estimates from [HLS1] yield only that d_{0}(SU(3, 4)) ≤ 64 and d_{1}(SU(3, 4)) ≤ 120.

Our method is to combine U–technology, in the spirit of [HLS1], with duality results as in the work of D.Benson and J.Carlson [BC1]. We ultimately connect a conjectured upper bound for d_{0}(G) to Benson’s Regularity Conjecture [Be], known to hold if the p–rank of G and the p–rank of Z(G) differ by at most 2. This is the case for all 2-groups of order 64 or less, and, using cohomology calculations from [CTVZ], we’ve been able to verify by hand that d_{0}(G) ≤ 14 for all such groups.

A number of side results of independent interest come up in our investigations. We are led to study carefully the cohomology of central extensions, in particular

the structure of associated algebras of primitives. One outcome of this is a new proof of A.Adem and D.Karagueuzian’s theorem [AK] that p–central p–groups have nonzero essential cohomology. We show that in an explicit degree there is a nonzero cohomology class that is simultaneously essential and annihilated by all Steenrod operations of positive degree.

Deriving our general estimate of d_{0}(G) involves a careful study of the depth essential cohomology of Carlson, et. al. [CTVZ] in the important special case that the depth of H^{∗}(G) equals the rank of the center. We prove that then the depth essential cohomology is both nonzero - reproving the main theorem of [G1] without D.Green’s hypothesis that G be a p–group - and Cohen–Macauley.

In the next section we describe our results in more detail.

2. Main Results

2.1. The cohomology of central extensions. Suppose we have a central extension of finite groups

C ⟶^{i} G ⟶^{q} Q,

where C is p–elementary abelian of rank c.

We define various objects associated to this situation.

The extension corresponds to an element τ ∈ H^{2}(Q; C). Since H^{2}(Q; C) = Hom(H_{2}(Q), C), the extension can also be considered as corresponding to a homomorphism τ : H_{2}(Q) → C, or, equivalently, its dual τ^{#} : C^{#} → H^{2}(Q).

Let {E_{r}^{∗,∗}} denote the Serre spectral sequence associated to the extension, converging to H^{∗}(G), and with E_{2}^{∗.∗} = H^{∗}(Q) ⊗ H^{∗}(C). Under the identification C^{#} = H^{1}(C), it is standard that τ^{#} corresponds to d_{2} : E_{2}^{0,1} → E_{2}^{2,0}.

Let I_{τ} ⊂ H^{∗}(Q) be the ideal generated by A · im(τ_{G}^{#}), so that H^{∗}(Q)/I_{τ} is an unstable algebra. It is easy to see that I_{τ} is contained in the kernel of inflation q^{∗} : H^{∗}(Q) → H^{∗}(G).

Call a subalgebra A of H^{∗}(G) a (G, C)–Duflot subalgebra, if the composite A ⊆ H^{∗}(G) ⟶^{i^{∗}} im(i^{∗}) is an isomorphism, where i^{∗} : H^{∗}(G) → H^{∗}(C) is the restriction. As we will describe more precisely in §2.5, as an algebra, the Hopf algebra im(i^{∗}) ⊆ H^{∗}(C) will necessarily be free graded commutative on c polynomial generators, possibly tensored with an exterior algebra on some generators in degree 1, if p is odd. It follows that Duflot subalgebras exist and have the same form. Let Q_{A}H^{∗}(G) denote the graded algebra^{1} of A–indecomposables H^{∗}(G) ⊗_{A} 𝔽_{p}, or, equivalently, the quotient of H^{∗}(G) by the ideal generated by the positive degree elements of A.

H^{∗}(C) is a Hopf algebra, and the multiplication map m : C × G → G induces a map of unstable algebras

m^{∗} : H^{∗}(G) → H^{∗}(C) ⊗ H^{∗}(G)

making H^{∗}(G) into a H^{∗}(C)–comodule. We define the associated algebra of primitives to be

\begin{aligned}P_{C}H^{∗}(G) & = \{x ∈ H^{∗}(G) | m^{∗}(x) = 1 ⊗ x\} \\ & = Eq \{H^{∗}(G) \begin{matrix}m^{∗} \\ ⟶ \\ ⟶ \\ π^{∗} \end{matrix} H^{∗}(C × G)\}, \end{aligned}

where π : C × G → G is the projection. It is easy to check that P_{C}H^{∗}(G) is an unstable algebra that contains the image of the inflation map. Thus q^{∗} : H^{∗}(Q) → H^{∗}(G) refines to a map of unstable algebras

q_{τ} : H^{∗}(Q)/I_{τ} → P_{C}H^{∗}(G).

Theorem 2.1. With the notation as above, the following are true.

(a) H^{∗}(G) is a free A–module. Moreover {E_{r}^{∗,∗}} is a spectral sequence of free E_{∞}^{0,∗}– modules, and applying Q_{E_{∞}^{0,∗}} to the spectral sequence yields a spectral sequence converging to Q_{A}H^{∗}(G) with E_{2}–term Q_{E_{∞}^{0,∗}}H^{∗}(C) ⊗ H^{∗}(Q).

(b) The composite P_{C}H^{∗}(G) ↪ H^{∗}(G) ↠ Q_{A}H^{∗}(G) is monic.

(c) Both P_{C}H^{∗}(G) and Q_{A}H^{∗}(G) are finitely generated H^{∗}(Q)–modules.

(d) The map q_{τ} : H^{∗}(Q)/I_{τ} → P_{C}H^{∗}(G) is an F –isomorphism^{2}, and the rings H^{∗}(Q)/I_{τ}, im(q^{∗}), P_{C}H^{∗}(G), and Q_{A}H^{∗}(G), are all Noetherian of Krull dimension equal to (the p–rank of G) - (the rank of C).

Let C(G) < G be the p–elementary abelian part of Z(G). If C = C(G), the first part of statement (a) recovers J. Duflot’s result [D] that the depth of H^{∗}(G) is at least as great as the rank of C(G).^{3} We will call a (G, C(G))–Duflot subalgebra of H^{∗}(G) simply a Duflot subalgebra.

The p–rank of G equals the rank of C exactly when C = C(G) and G is p–central, and we have the following corollary.

Corollary 2.2. If G is p–central, and C = C(G), then the rings H^{∗}(Q)/I_{τ}, im(q^{∗}), P_{C}H^{∗}(G), and Q_{A}H^{∗}(G) all have Krull dimension zero and so are finite dimensional 𝔽_{p}–algebras.

1Q_{A}H^{∗}(G) will not necessarily be an unstable algebra, as A need not be closed under Steenrod operations.

2In the sense of Quillen [Q]: ker(q̄^{∗}) is nilpotent, and for all x ∈ P_{C}H^{∗}(G), there exists a k so that x^{p^{k}} ∈ im(q̄^{∗}).

3We are claiming no originality in the proof of this, which is similar to all proofs of Duflot’s theorem following [BrH]. The spectral sequence refinement seems to be a new observation.

2.2. Quillen’s category and functors involving primitives. Our algebras of primitives arise in two formulae associated to H^{∗}(G), viewed as an object in K. To describe these, we need to introduce some notation.

Given a small category C, we let C^{#} denote the associated twisted arrow category: the objects of C^{#} are the morphisms of C, and a morphism α ⇝ β from α : A_{1} → A_{2} to β : B_{1} → B_{2} is a commutative diagram in C

\xymatrix{A_{1} \ar[d] \ar[r]^{α} & A_{2} \\ B_{1} \ar[r]^{β} & B_{2}. \ar[u]}

The functor assigning H^{∗}(V) to V ∈ A(G) is contravariant, while the assignment of H^{∗}(C_{G}(V)) is covariant. Now observe that the assignment of P_{α(V_{1})}H^{∗}(C_{G}(V_{2})) to α : V_{1} → V_{2} can be viewed as defining a contravariant functor of A(G)^{#}.

Let A_{C}(G) denote the full subcategory of A(G) having as objects the V containing C(G). If G is p–central, then A_{C}(G) has a single object and morphism.

2.3. A formula for the locally finite part of H^{∗}(G). If M is an unstable A– module, we define M_{LF}, the locally finite part of M, by

M_{LF} = {x ∈ M |Ax ⊂ M is finite}.

This is again an unstable module, and is an unstable algebra if M is.

Theorem 2.3. There is a natural isomorphism of unstable algebras

H^{∗}(G)_{LF} ≃ lim_{V_{1}⟶^{α}V_{2}} P_{α(V_{1})}H^{∗}(C_{G}(V_{2})),

where the limit is over A_{C}(G)^{#}.

Corollary 2.4. If G is p–central, then H^{∗}(G)_{LF} = P_{C(G)}H^{∗}(G).

2.4. A formula for R̄_{d}H^{∗}(G). An unstable module M ∈ U has a canonical ‘nilpotent’ filtration [S1, K1, HLS1]:

⋯ ⊆ nil_{2}M ⊆ nil_{1}M ⊆ nil_{0}M = M.

In general, nil_{d}M/nil_{d+1}M = Σ^{d}R_{d}M, where R_{d}M is reduced, i.e. has no nontrivial submodules that are suspensions. We let R̄_{d}M denote the nilclosure L_{0}R_{d}M of R_{d}M.

The module nil_{d}M identifies with the kernel of λ_{d} : M → L_{d−1}M, and a bit of diagram chasing will show that Σ^{d}R̄_{d}M is isomorphic to the kernel of L_{d}M → L_{d−1}M: see Proposition 3.1. Thus d_{0}(G) is the length of the filtration of H^{∗}(G), and also is the biggest d such that R̄_{d}H^{∗}(G) ≠ 0.

Theorem 2.5. There is a natural isomorphism of unstable modules

R̄_{d}H^{∗}(G) ≃ lim_{V_{1}⟶^{α}V_{2}} H^{∗}(V_{1}) ⊗ P_{α(V_{1})}H^{d}(C_{G}(V_{2})),

where the limit is over A_{C}(G)^{#}.

Corollary 2.6. If G is p–central, then there is an isomorphism of unstable modules

R̄_{d}H^{∗}(G) ≃ H^{∗}(C(G)) ⊗ P_{C(G)}H^{d}(G).

2.5. Invariants of restriction to C(G). If i : C < G is a central p–elementary abelian of rank c, then

H^{∗}(C) ≃ \begin{cases}𝔽_{2}[x_{1}, …, x_{c}] & if p = 2 \\ Λ(x_{1}, …, x_{c}) ⊗ 𝔽_{p}[y_{1}, …, y_{c}] & if p is odd, \end{cases}

where |x_{i}| = 1 and y_{i} = β(x_{i}), and is a Hopf algebra in the usual way.

In §6, we will see that, after a change of basis for H^{1}(C), the image of the restriction homomorphism i^{∗} : H^{∗}(G) → H^{∗}(C) will be a sub Hopf algebra of H^{∗}(C) of the form

im(i^{∗}) = \begin{cases}𝔽_{2}[x_{1}^{2^{j_{1}}} , …, x_{c}^{2^{jc}}] & if p = 2 \\ 𝔽_{p}[y_{1}^{p^{j_{1}}} , …, y_{b}^{p^{j_{b}}} , y_{b+1}, …, y_{c}] ⊗ Λ(x_{b+1}, …, x_{c}) & if p is odd, \end{cases}

with the j_{i} forming a sequence of nonincreasing nonnegative integers^{4}. Now suppose that C = C(G). We will say that G has type [a_{1}, …, a_{c}] where

(a_{1}, …, a_{c}) = \begin{cases}(2^{j_{1}}, …, 2^{j_{c}}) & if p = 2 \\ (2p^{j_{1}}, …, 2p^{j_{b}}, 1, …, 1) & if p is odd. \end{cases}

The type of G has the form [1, …, 1] if and only if G = C × H, where Z(H) has order prime to p. In all other cases, a_{1} = 2p^{k} for some k ≥ 0.

Define e(G) and h(G) by

e(G) = ∑_{i=1}^{c}(a_{i} − 1),

and

h(G) ≃ \begin{cases}2p^{k−1} & if a_{1} = 2p^{k} with k ≥ 1 \\ 1 & if a_{1} = 2 \\ 0 & if a_{1} = 1. \end{cases}

For example, Q_{8} × ℤ/4 has type [4, 2] when p = 2, so that e(Q_{8} × ℤ/4) = 4, and h(Q_{8} × ℤ/4) = 2.

Remark 2.7. The careful reader will observe that the type of G is just the list of the degrees of the unstable A–algebra generators of im(i^{∗}), listed in decreasing order, e(G) is the top nonzero degree of the finite dimensional Hopf algebra

H^{∗}(C) ⊗_{H^{∗}(G)} 𝔽_{p} = H^{∗}(C)/(im(i^{∗>0})),

and h(G) is the top nonzero degree of the module A · H^{1}(C) projected into this Hopf algebra.

2.6. P_{C(G)}H^{∗}(G), d_{0}(G), and d_{1}(G) when G is p–central. If G is p–central with C = C(G), then P_{C}H^{∗}(G) is a finite dimensional unstable algebra. We identify its top degree and more.

Theorem 2.8. Let G be p–central, C = C(G), and A be a Duflot subalgebra of H^{∗}(G). Then both P_{C}H^{∗}(G) and Q_{A}H^{∗}(G) are zero in degrees greater than e(G), and one dimensional in degree e(G). Furthermore, P_{C}H^{e(G)}(G) is annihilated by all positive degree elements of the Steenrod algebra, and, if G is a p–group, consists of essential^{5} cohomology classes.

4In the odd prime case, c − b will be the rank of the largest subgroup of C splitting off G as a direct summand.

The last statement implies the main result of [AK]: p–central p–groups have nonzero essential cohomology.

This theorem, combined with Corollary 2.6 and related results, leads to the following calculation.

Theorem 2.9. Let G be p-central. Then d_{0}(G) = e(G) and d_{1}(G) = e(G) + h(G). Furthermore, if G is a finite group with p–Sylow subgroup P, and P is p–central, then d_{0}(G) = d_{0}(P) and d_{1}(G) = d_{1}(P).

Corollary 2.10. If G is p–central, and H < G, then d_{0}(H) ≤ d_{0}(G) and d_{1}(H) ≤ d_{1}(G).

Examples 2.11. (a) As Q_{8} is 2–central of type [4], d_{0}(Q_{8}) = 3 and d_{1}(Q_{8}) = 3 + 2 = 5, in agreement with [HLS1, (II.4.6)].

(b) The hypotheses of p–centrality is needed in the last part of the theorem: as observed in [HLS1, II.4.7], if G = GL_{2}(𝔽_{3}) and P = SD_{16}, then P is the 2– Sylow subgroup of G, but d_{0}(G) = 0 < 2 = d_{0}(P), and d_{1}(G) = 2 < 4 = d_{1}(P). Similarly, G needs to be p–central in the corollary: if H = Z/4 < D_{8} = G, then d_{0}(H) = 1 > 0 = d_{0}(G) and d_{1}(H) = 2 > 0 = d_{1}(G). The example H = ℤ/4 < ℤ/8 = G shows that the inequalities of the corollary can be equalities, even when H is a proper subgroup of a p–central p–group G.

(c) The 2-Sylow subgroup P of the simple group SU(3, 4) is 2–central of type [8, 8]. Thus d_{0}(SU(3, 4)) = d_{0}(P) = 14 and d_{1}(SU(3, 4)) = d_{1}(P) = 18. Similarly, the 2-Sylow subgroup Q of the simple group Sz(8) is 2–central of type [4, 4, 4]. Thus d_{0}(Sz(8)) = d_{0}(Q) = 9 and d_{1}(Sz(8)) = d_{1}(Q) = 11. We will see that P and Q have the largest d_{0} of all 2–groups of order dividing 64. For more about the SU(3, 4) example, see §9.

(d) In [AKM], the authors associate a 2-central Galois group G_{𝔽} to every field 𝔽 of characteristic different from 2 that is not formally real. (They call this the W –group of 𝔽 because of its connections to the Witt ring W 𝔽 [MiS].) From their construction it is easy to deduce that G_{𝔽} has type [2, …, 2]. Thus d_{0}(G_{𝔽}) = r and d_{1}(G_{𝔽}) = r + 1, where G_{𝔽} has rank r. In particular, the universal group W –group W (n) has d_{0}(W (n)) = \binom{n+1}{2} and d_{1}(W (n)) = \binom{n+1}{2} + 1. For more about this example, see §9.

2.7. Central essential cohomology. Our calculation of d_{0}(G) when G is p– central relies on Corollary 2.6. To understand d_{0}(G) for general G, one needs to use the more complicated formula given in Theorem 2.5. Using some analysis of this already done by us in our companion paper [K3], we are led to a formula^{6} for d_{0}(G) that makes use of the following variant of essential cohomology.

5Recall that x ∈ H^{∗}(G) is essential if it restricts to zero on all proper subgroups.

6Thus far, we have not found an analogous formula for d_{1}(G).

We define Cess^{∗}(G), the central essential cohomology of G, to be the kernel of the restriction map

H^{∗}(G) → ∏_{C(G)<U} H^{∗}(C_{G}(U)),

where the product is over p–elementary abelian subgroups U of G that are strictly bigger than C(G).

This product is over the empty set if G is p–central, so the concept is really only interesting when this is not the case. Furthermore, a theorem of Carlson [Ca1] implies that Cess^{∗}(G) is nonzero only if the rank of C(G) equals the depth of H^{∗}(G): see Theorem 2.13 below for the converse. If this is the case, Cess^{∗}(G) is precisely the depth essential cohomology of [CTVZ].

Note that Cess^{∗}(G) has the following structure, compatible in the usual ways: it is an ideal in H^{∗}(G), an unstable module, and an H^{∗}(C(G))–comodule. We have the following general structural results.

Theorem 2.12. If A is a Duflot subalgebra of H^{∗}(G), then the following hold.

(a) Cess^{∗}(G) is a finitely generated free A–module.

(b) The composite P_{C}Cess^{∗}(G) ↪ Cess^{∗}(G) ↠ Q_{A}Cess^{∗}(G) is monic.

(c) The sequence 0 → Q_{A}Cess^{∗}(G) → Q_{A}H^{∗}(G) → ∏_{C(G)<U} Q_{A}H^{∗}(C_{G}(U)) is

exact.

Statement (a) implies that Cess^{∗}(G) is a Cohen–MacCauley module, and thus can be viewed as a variant of D.Green’s theorem [G3] about the essential cohomology Ess^{∗}(G). Statement (b) will have application below. Statement (c) gets us most of the way towards proving the following theorem.

Theorem 2.13. Cess^{∗}(G) ≠ 0 if and only if the depth of H^{∗}(G) is the rank of C(G).

The ‘only if’ statement here is just Carlson’s theorem. The ‘if’ statement is a special case of Carlson’s Depth Conjecture, and has been previously proved by D.Green under the extra hypothesis that G is a p–group [G1].

2.8. d_{0}(G) for general G. Thanks to Theorem 2.12, we can make the following definitions. If Cess^{∗}(G) is nonzero, define e′(G) to be the largest d such that Q_{A}Cess^{d}(G) is nonzero, and e′′(G) to be the largest d such that P_{C}Cess^{d}(G) is nonzero. If Cess^{∗}(G) = 0, we let e′′(G) = e′(G) = −1. Note that Theorem 2.12(b) implies that e′′(G) ≤ e′(G).

Using the formula for R̄_{d}H^{∗}(G) given in Theorem 2.5, we will prove the following.

Theorem 2.14. d_{0}(G) = max{e′′(C_{G}(V)) | V < G}.

Corollary 2.15. d_{0}(G) ≤ max{e′(C_{G}(V)) | V < G}.

When computing these maxima, one can restrict to the p–elementary abelian groups V which satisfy V = C(C_{G}(V)).^{7} The next proposition says that if one has some a priori computation of the depth of H^{∗}(G), one may be able to cut down even more on the V ’s to be checked.

7Since C_{G}(V) = C_{G}(U), where U = C(C_{G}(V)).

Proposition 2.16. Assuming that V = C(C_{G}(V)), Cess^{∗}(C_{G}(V)) = 0 unless the rank of V is at least equal to the depth of H^{∗}(G).

This will be proved by combining Carlson’s theorem with some U–technology. This leads to the following generalization of our calculation of d_{0}(G) for p–central G.

Corollary 2.17. If H^{∗}(G) is Cohen–Macauley, then

d_{0}(G) = max{e(C_{G}(V)) | V < G is maximal}.

This follows from the above, as C_{G}(V) is p–central when V is maximal. Here we have used that, by Theorem 2.8, when G is p–central, e′′(G) = e′(G) = e(G).

Conjecture 2.18. e′(G) < e(G) if G is not p–central.

As will be explained in §8, Benson’s Strong Regularity Conjecture [Be] asserts that certain local cohomology groups H_{H̃^{∗}(G)}^{i,j}(H^{∗}(G)) vanish. We connect our conjecture to his.

Proposition 2.19. For a fixed finite group G, Conjecture 2.18 is implied by the Strong Regularity Conjecture.

Let G have p–rank r and C(G) have rank c with c < r. Benson [Be] has shown that his conjecture is true if r − c ≤ 2. We deduce the next corollary.

Corollary 2.20. Let G have p–rank r, and let d be the depth of H^{∗}(G). If r−d ≤ 2, then

d_{0}(G) ≤ max{e(C_{G}(V)) | V < G}.

The hypothesis of this corollary applies to all 2–groups of order dividing 64.

2.9. Calculations when p = 2. The Appendix has various tables of values of d_{0}(G), d_{1}(G), e(G), e′(G), and e′′(G) for 2–groups of order dividing 64. The tables were compiled by hand using the calculations in [CTVZ]. Their calculations let one immediately determine if Cess^{∗}(G) ≠ 0, and, when this is the case, one can read off the values of e(G) and e′(G), and sometimes e′′(G).

From our tables, one learns the following about d_{0}(G) when p = 2:

Theorem 2.21. Let G be a finite group with 2–Sylow subgroup P of order dividing 64. Then d_{0}(G) ≤ 7 unless P is isomorphic to either the Sylow subgroup of SU(3, 4), in which case d_{0}(G) = 14, or the Sylow subgroup of Sz(8), in which case d_{0}(G) = 9.

2.10. Organization of the paper. The rest of the paper is organized as follows. The nilpotent filtration of U is reviewed in §3, along with basic properties of the functors R̄_{d}, and the invariants d_{0} and d_{1}. Starting from results in [HLS1], in §4 we then deduce the formulae given in Theorem 2.3 and Theorem 2.5. In §5, we prove Theorem 2.1 with a careful analysis of the Lyndon–Serre spectral sequence associated to the group extension C → G → G/C, heavily using that the spectral sequence is a spectral sequence of H^{∗}(C)–comodules. In the p–central case, we also input Carlson and Benson’s theorem that if H^{∗}(G) is Cohen–Macauley then it is Gorenstein: this leads to proofs of Theorem 2.8 and Theorem 2.9 in §7. Using an analysis of the formula in Theorem 2.5 done by us in [K3], Theorem 2.14 is proved in §8, which then continues with our results about Cess^{∗}(G) and the conjectured inequality e′(G) ≤ e(G). Though short examples occur throughout, some longer examples that illustrate the general theory make up §9.

2.11. Acknowledgements. Early versions of some of these results were presented at a talk in Oberwolfach in September, 2005: see [K2]. Some of this research was done during a visit to the Cambridge University Pure Mathematics Department in the first half of 2006, and the author thanks DPMMS for its hospitality. Conversations with Dave Benson about the local cohomology spectral sequence have been enlightening, as have conversations with David Green.

3. The nilpotent filtration of U

The nilpotent filtration of U was introduced in [S1], and its main properties were developed in [K1, S2, HLS1, BrZ1]. Here we collect the results that we need.^{8}

3.1. The definition of L_{d}, R_{d}, and R̄_{d}. For d ≥ 0, let N il_{d} ⊂ U be the localizing subcategory generated by d–fold suspensions of unstable A–modules, i.e. N il_{d} is the smallest full subcategory containing all d–fold suspensions of unstable modules that is closed under extensions and filtered colimits. Associated to the descending filtration

⋯ ⊂ N il_{2} ⊂ N il_{1} ⊂ N il_{0} = U,

there is a natural localization tower for M ∈ U,

\xymatrix{ & ⋮ \ar[d] \\  & L_{2}M \ar[d] \\  & L_{1}M \ar[d] \\ M \ar[r]^{λ_{0}} \ar[ur]^{λ_{1}} \ar[uur]^{λ_{2}} & L_{0}M,}

where L_{d} : U → U is localization away from N il_{d+1}.^{9} The natural transformation λ_{d} : M → L_{d}M is characterized by the following properties:

(a) L_{d}M is N il_{d+1}–closed, i.e. Ext_{U}^{s}(N, L_{d}M) = 0 for s = 0, 1 and N ∈ N il_{d+1},

(b) λ_{d} is a N il_{d+1}–isomorphism, i.e. ker λ_{d} and coker λ_{d} are both in N il_{d+1}.

A module M ∈ U admits a natural filtration

⋯ ⊆ nil_{2}M ⊆ nil_{1}M ⊆ nil_{0}M = M,

where nil_{d}M is the largest submodule in N il_{d}. For d > 0, nil_{d}M = ker λ_{d−1}.

An unstable module M is called reduced if nil_{1}M = 0. As observed in [K1, Prop.2.2], nil_{d}M/nil_{d+1}M = Σ^{d}R_{d}M, where R_{d}M is a reduced unstable module. (See also [S2, Lemma 6.1.4].) Then R̄_{d}M is defined to be the N il_{1}–closure of R_{d}M. Thus R_{d}M ⊆ L_{0}R_{d}M = R̄_{d}M.

8This section necessarily overlaps with the presentation in our recent preprint [K3].

9What we are calling L_{d} here was called L_{d+1} in [HLS1].

We have the following useful alternative definition of R̄_{d}M. (Compare with [HLS1, I(3.8.1)].)

Proposition 3.1. There is a natural isomorphism

Σ^{d}R̄_{d}M ≃ ker {L_{d}M → L_{d−1}M}.

The functors L_{d} and L_{d−1} are left exact, as they are localizations, and thus we conclude

Corollary 3.2. R̄_{d} : U → U is left exact.

Proof of Proposition 3.1. Let c_{d+1}M = coker {λ_{d} : M → L_{d}M}. Then c_{d+1}M ∈ N il_{d+1}, and there is an exact sequence

0 → nil_{d+1}M → M → L_{d}M → c_{d+1}M → 0.

Diagram chasing then shows that there is a natural short exact sequence

0 → nil_{d}M/nil_{d+1}M → ker {L_{d}M → L_{d−1}M} → ker {c_{d+1}M → c_{d}M} → 0.

As the middle module here is N il_{d+1}–closed, and the right module is in N il_{d+1}, we see that the left map identifies with λ_{d}. Recalling that nil_{d}M/nil_{d+1}M = Σ^{d}R_{d}M, this says that there is a natural isomorphism

L_{d}(Σ^{d}R_{d}M) ≃ ker {L_{d}M → L_{d−1}M}.

The proof of the proposition is then completed by observing that L_{d}(Σ^{d}R_{d}M) ≃ Σ^{d}R̄_{d}M, a consequence of the next proposition. □

Proposition 3.3. There is a natural isomorphism L_{c+d}(Σ^{d}M) ≃ Σ^{d}L_{c}M, for all M ∈ U.

Proof. We need to check that the map Σ^{d}λ_{c} : Σ^{d}M → Σ^{d}L_{c}M satisfies the two properties characterizing localization away from N il_{c+d+1}.

That ker(Σ^{d}λ_{c}) and coker(Σ^{d}λ_{c}) are both in N il_{c+d+1} is clear, as ker(λ_{c}) and coker(λ_{c}) are both N il_{c+1}, and the d–fold suspension of a module in N il_{c+1} will be in N il_{c+d+1}.

To see that the range of Σ^{d}λ_{c} is N il_{c+d+1}–closed, we check that if M ∈ U is N il_{c+1}–closed then Σ^{d}M is N il_{c+d+1}–closed. This follows from the following characterization of N il_{c+1}–closed modules: M ∈ U is N il_{c+1}–closed if and only if it fits into an exact sequence of the form

0 → M → ∏_{α} H^{∗}(V_{α}) ⊗ M_{α} → ∏_{β} H^{∗}(W_{β}) ⊗ N_{β},

with all the modules M_{α} and N_{β} concentrated in degrees between 0 and c. See [BrZ2, Prop.1.15]. □

3.2. Further properties of L_{d}, R_{d}, and R̄_{d}. We need to recall some notation and terminology. If V is an elementary p–group, T_{V} : U → U is defined to be the left adjoint to H^{∗}(V) ⊗ , as famously studied by Lannes [L1, L3]. Given a Noetherian unstable algebra K ∈ K, K_{f.g.} − U is defined to be the category studied in [HLS1, I.4] whose objects are finitely generated K–modules M whose K–module structure map K ⊗ M → M is in U, and morphisms are K–module maps in U.

Proposition 3.4. The functor L_{d} : U → U satisfies the following properties.

(a) There are natural isomorphisms L_{0}(M ⊗ N) ≃ L_{0}M ⊗ L_{0}N.

(b) There are natural isomorphisms T_{V} L_{d}M ≃ L_{d}T_{V} M.

(c) If K ∈ K, then L_{d}K ∈ K, and K → L_{d}K is a map of unstable algebras. If K is also Noetherian, and M ∈ K_{f.g.} − U, then L_{d}K ∈ K_{f.g.} − U, and thus is Noetherian, and L_{d}M ∈ L_{d}K_{f.g.} − U.

Property (b) can be deduced from properties of T_{V} as follows. First, to see that T_{V} L_{d}M is N il_{d+1}–closed, we compute, for s = 0, 1 and N ∈ N il_{d+1}:

Ext_{U}^{s}(N, T_{V} L_{d}M) = Ext_{U}^{s}(H^{∗}(V) ⊗ N, L_{d}M) = 0,

since H^{∗}(V) ⊗ N will be in N il_{d+1} if N is. Second, T_{V} λ_{d} : T_{V} M → T_{V} L_{d}M is a N il_{d+1}–isomorphism, as the kernel and cokernel are in N il_{d+1}, since T_{V} is exact and sends N il_{d+1} to itself.

See [HLS1, I.4] and [BrZ1] for more detail about properties (a) and (c).

Proposition 3.5. The functors R_{d} : U → U satisfy the following properties.

(a) There are a natural isomorphisms R_{∗}(M ⊗ N) ≃ R_{∗}M ⊗ R_{∗}N of graded objects in U.

(b) There are natural isomorphism T_{V} R_{d}M ≃ R_{d}T_{V} M.

(c) If K ∈ K, then R_{0}K ∈ K, and K → R_{0}K is a map of unstable algebras. If K is also Noetherian, and M ∈ K_{f.g.} − U, then R_{0}K is also a Noetherian unstable algebra, and R_{d}M ∈ R_{0}K_{f.g.} − U, for all d.

For the first two properties, see [K1, §3], and the last follows easily from the first.

Proposition 3.6. The functors R̄_{d} : U → U satisfy the following properties.

(a) There are natural isomorphisms R̄_{∗}(M ⊗ N) ≃ R̄_{∗}M ⊗ R̄_{∗}N of graded objects in U.

(b) There are natural isomorphisms T_{V} R̄_{d}M ≃ R̄_{d}T_{V} M.

(c) If K ∈ K, then R̄_{0}K ∈ K, and K → R̄_{0}K is a map of unstable algebras. If K is also Noetherian, and M ∈ K_{f.g.} − U, then R̄_{0}K is also a Noetherian unstable algebra, and R̄_{d}M ∈ R̄_{0}K_{f.g.} − U, for all d.

This, of course, follows from the previous two propositions.

A Noetherian unstable algebra K has a finite Krull dimension dim K. We have an addendum to Proposition 3.4.

Proposition 3.7 ([K3, Prop.4.10]). If an unstable algebra K is Noetherian, then dim K = dim L_{0}K.

Another special property of L_{0} that we will need goes as follows.

Proposition 3.8 ([L1, Lem.4.3.3]). Let f : M → N be a map in K. Then

L_{0}f : L_{0}M → L_{0}N

is an isomorphism if and only if, for all p–elementary abelian groups V, the induced map

f^{∗} : Hom_{K}(N, H^{∗}(V)) → Hom_{K}(M, H^{∗}(V))

is a bijection.

As in the introduction, given M ∈ U, M_{LF} denotes the submodule of locally finite elements: x ∈ M such that Ax ⊆ M is finite.

Proposition 3.9. There is a natural isomorphism (R_{d}M)^{0} = (R̄_{d}M)^{0} ≃ (M_{LF})^{d}.

See [K1, §3] for a proof.

Finally, Henn [H] proved the following important finiteness result.

Proposition 3.10. Let K ∈ K be Noetherian, and M ∈ K_{f.g.} − U. Then the M is N il_{d}–local for d >> 0. In particular, the nilpotent filtration of M has finite length.

3.3. Properties of d_{0}M and d_{1}M. The authors of [HLS1] define d_{0}M and d_{1}M as follows.

Definition 3.11. Let M be an unstable module.

(a) Let d_{0}M be the smallest d such that λ_{d} is monic, or ∞ if no such d exists. Equivalently, d_{0}M is the smallest d such that Hom_{U}(N, M) = 0 for all N ∈ N il_{d+1}, or the smallest d such that nil_{d+1}M = 0. If M is nonzero, d_{0}M is also the largest d such that R_{d}M is nonzero, or the largest d such that R̄_{d}M is nonzero.

(b) Let d_{1}M be the smallest d such that λ_{d} is an isomorphism, or ∞ if no such d exists. Equivalently, d_{1}M is the smallest d such that Ext_{U}^{s}(N, M) = 0 for s = 0, 1 and all N ∈ N il_{d+1}.

As fundamental examples, we have that d_{0}H^{∗}(V) = d_{1}H^{∗}(V) = 0 for all elementary abelian p–groups V.

Proposition 3.12. Let M and N be unstable modules.

(a) For s = 0, 1, d_{s}(M ⊕ N) = max{d_{s}M, d_{s}N}.

(b) If M and N are nonzero, d_{0}(M ⊗ N) = d_{0}M + d_{0}N and d_{1}(M ⊗ N) = max{d_{1}M + d_{0}N, d_{0}M + d_{1}N}.

(c) For s = 0, 1, d_{s}T_{V} M = d_{s}M.

(d) If M is nonzero, for s = 0, 1, d_{s}(Σ^{n}M) = d_{s}M + n.

For properties (a) and (b) see [HLS1, Prop.I.3.6]. Using the exactness of T_{V}, property (c) follows from Proposition 3.4(b). Property (d) follows from Proposition 3.3.

Proposition 3.13. Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence in U.

(a) For s = 0, 1, d_{s}M_{2} ≤ max{d_{s}M_{1}, d_{s}M_{3}}. Furthermore, if d_{s}M_{3} < d_{s}M_{1}, then d_{s}M_{2} = d_{s}M_{1}.

(b) d_{0}M_{1} ≤ d_{0}M_{2} and d_{1}M_{1} ≤ max{d_{1}M_{2}, d_{0}M_{3}}. Furthermore, if d_{1}M_{2} < d_{0}M_{3}, then d_{1}M_{1} = d_{0}M_{3}.

This is proved with straightforward use of the long exact Ext^{∗} sequence associated to a short exact sequence. Compare with [HLS1, Prop.I.3.6].

Corollary 3.14. If M ∈ U is reduced, then d_{1}M = d_{0}(L_{0}M/M).

This follows by applying Proposition 3.13(b) to 0 → M → L_{0}M → L_{0}M/M → 0.

3.4. Basic properties of d_{0}(G) and d_{1}(G). By abuse of notation, if G is a finite group, for s = 0, 1, we write d_{s}(G) for d_{s}H^{∗}(G). For example, d_{0}(V) = d_{1}(V) = 0 for all elementary abelian p–groups V.

The properties of d_{0}M and d_{1}M presented above have the following immediate consequences for d_{0}(G) and d_{1}(G)

Proposition 3.15. Let G and H be finite groups.

(a) d_{0}(G × H) = d_{0}(G) + d_{0}(H).

(b) d_{1}(G × H) = max{d_{1}(G) + d_{0}(H), d_{0}(G) + d_{1}(H)}.

(c) If P is a p–Sylow subgroup of G, then d_{s}(G) ≤ d_{s}(P) for s = 0, 1.

(d) If V is a p–elementary abelian subgroup of G, then d_{s}(C_{G}(V)) ≤ d_{s}(G) for s = 0, 1.

Properties (a) and (b) follow from Proposition 3.12 (b). As the unstable module H^{∗}(G) is a direct summand of H^{∗}(P) if P is a p–Sylow subgroup, property (c) follows from Proposition 3.12 (a). Similarly property (d) follows from Proposition 3.12 (c), as H^{∗}(C_{G}(V)) is a direct summand of T_{V} H^{∗}(G) [L2]^{10}.

4. Formulae for H^{∗}(G)_{LF} and R̄_{d}(H^{∗}(G))

In this section we prove the formulae for H^{∗}(G)_{LF} and R̄_{d}H^{∗}(G) given in §2.

4.1. A formula for L_{d}H^{∗}(G). The starting point for all of these are the following constructions. Given a morphism α : V_{1} → V_{2} in A(G), there are maps

α^{∗} : H^{∗}(V_{2}) → H^{∗}(V_{1}),
α_{∗} : H^{∗}(C_{G}(V_{1})) → H^{∗}(C_{G}(V_{2})), and
m_{α}^{∗} : H^{∗}(C_{G}(V_{2})) → H^{∗}(V_{1}) ⊗ H^{∗}(C_{G}(V_{2})).

10It is unfortunate that this much referenced elegant 1986 preprint has never been published.

Here α_{∗} is induced by conjugation by g^{−1} where g ∈ G is any element^{11} chosen so that conjugation by g induces α, and

m_{α} : V_{1} × C_{G}(V_{2}) → C_{G}(V_{2})

is the homomorphism sending (x, y) to α(x)y. We also let m_{V} : V × V → V denote multiplication in an elementary abelian group V.

To state one of the formulae from [HLS1], we recall two other bits of notation from §2. Given an unstable module M, we let M^{≤d} denote M modulo degrees greater than d. Given a category C, we let C^{#} denote the associated twisted arrow category: the objects of C^{#} are the morphisms of C, and a morphism α ⇝ β from α : A_{1} → A_{2} to β : B_{1} → B_{2} is a commutative diagram in C

\xymatrix{A_{1} \ar[d] \ar[r]^{α} & A_{2} \\ B_{1} \ar[r]^{β} & B_{2}. \ar[u]}

[HLS1, Formula I(5.5.1)] now reads

Theorem 4.1. The homomorphisms V_{1} × C_{G}(V_{2}) ⟶^{m_{α}} C_{G}(V_{2}) ⊂ G induce an isomorphism of unstable algebras from L_{d}H^{∗}(G) to

lim_{V_{1}⟶^{α}V_{2}} Eq \{ \xymatrix{H^{∗}(V_{1}) ⊗ H^{≤d}(C_{G}(V_{2})) \ar[r]^{μ(α)} \ar[r]_{ν(α)} & H^{∗}(V_{1}) ⊗ (H^{∗}(V_{1}) ⊗ H^{∗}(C_{G}(V_{2})))^{≤d}} \},

where μ(α) is induced by 1 ⊗ m_{α}^{∗}, ν(α) is induced by m_{V_{1}}^{∗} ⊗ 1, and the limit is over A(G)^{#}.

4.2. A formula for R̄_{d}H^{∗}(G). Recall our notation from §2: if W is a central elementary abelian p–subgroup of Q, then P_{W} H^{∗}(Q) denotes the algebra of primitives in the H^{∗}(W)–comodule H^{∗}(Q).

Proposition 4.2. As unstable modules, R̄_{d}H^{∗}(G) is naturally isomorphic to

lim_{V_{1}⟶^{α}V_{2}} H^{∗}(V_{1}) ⊗ P_{α(V_{1})}H^{d}(C_{G}(V_{2})),

where the limit is over A(G)^{#}.

Proof. Recall that Σ^{d}R̄_{d}M is the kernel of L_{d}M → L_{d−1}M. As kernels commute with limits and equalizers, it follows from the previous theorem that R̄_{d}H^{∗}(G) is naturally isomorphic to

lim_{V_{1}⟶^{α}V_{2}} Eq \{ \xymatrix{H^{∗}(V_{1}) ⊗ H^{d}(C_{G}(V_{2})) \ar[r]^{μ(α)} \ar[r]_{ν(α)} & H^{∗}(V_{1}) ⊗ (H^{∗}(V_{1}) ⊗ H^{∗}(C_{G}(V_{2})))^{d}} \},

where μ(α) is induced by 1 ⊗ m_{α}^{∗} and ν(α) is induced by m_{V_{1}}^{∗} ⊗ 1. But now we observe that the equalizer in this formula is precisely H^{∗}(V_{1}) ⊗ P_{α(V_{1})}H^{d}(C_{G}(V_{2})). For ν(α) is the composite

H^{∗}(V_{1}) ⊗ H^{d}(C_{G}(V_{2})) ⟶^{m_{V_{1}}^{∗} ⊗1} H^{∗}(V_{1}) ⊗ H^{∗}(V_{1}) ⊗ H^{d}(C_{G}(V_{2}))

⟶^{truncate} H^{∗}(V_{1}) ⊗ H^{0}(V_{1}) ⊗ H^{d}(C_{G}(V_{2})),

11This is well defined as any two choices will differ by an element of C_{G}(V_{1}), and so will agree on cohomology.

and this identifies with

H^{∗}(V_{1}) ⊗ H^{d}(C_{G}(V_{2})) ⟶^{1⊗π^{∗}} H^{∗}(V_{1}) ⊗ (H^{∗}(V_{1}) ⊗ H^{∗}(C_{G}(V_{2})))^{d},

where π : V_{1} × C_{G}(V_{2}) → C_{G}(V_{2}) is the projection. □

4.3. A formula for H^{∗}(G)_{LF}.

Proposition 4.3. As unstable algebras, H^{∗}(G)_{LF} is naturally isomorphic to

lim_{V_{1}⟶^{α}V_{2}} P_{α(V_{1})}H^{∗}(C_{G}(V_{2})),

where the limit is over A(G)^{#}.

Proof. As there are no nonzero locally finite elements in H̃^{∗}(V_{1})⊗H^{∗}(C_{G}(V_{2})), the composite H^{∗}(G)_{LF} ⊂ H^{∗}(G) → H^{∗}(C_{G}(V_{2})) has image in P_{α(V_{1})}H^{∗}(C_{G}(V_{2})) for any α : V_{1} → V_{2} in A(G). Thus one gets a natural map of unstable algebras

H^{∗}(G)_{LF} → lim_{V_{1}⟶^{α}V_{2}} P_{α(V_{1})}H^{∗}(C_{G}(V_{2})).

That this is an isomorphism follows from Proposition 4.2, recalling that Proposition 3.9 said that there is a natural isomorphism (R̄_{d}M)^{0} ≃ (M_{LF})^{d}. □

4.4. Replacing A(G) with A_{C}(G). Recall that C(G) denotes the maximal central p–elementary abelian subgroup of G, and A_{C}(G) denotes the full subcategory of A(G) with objects C(G) ≤ V < G.

Theorem 4.4. One can take the limit over A_{C}(G)^{#}, rather than A(G)^{#} in Theorem 4.1, Proposition 4.2, and Proposition 4.3.

This will follow quite formally from the following simple observations. Let C = C(G). Given V < G, let CV < G be the subgroup generated by C and V. This induces an evident functor C : A(P) → A_{C}(G). Furthermore, the natural inclusion V → CV induces an identification C_{G}(CV) = C_{G}(V).

Given α : V_{1} → V_{2}, let α_{C} : V_{1} → CV_{2} be the evident map, and then let

\xymatrix{α & α_{C} \ar[l]_{f_{α}} \ar[r]^{g_{α}} & Cα,}

morphisms in A(G)^{#}, correspond to the diagram in A(G)

[figure: V_{1}; α; V_{1}; α_{C}; CV_{1}; Cα; V_{2}; CV_{2}; CV_{2}.]

Lemma 4.5. Let F : A(G)^{#} → 𝔽_{p}-vector spaces be a contravariant functor such that for all α : V_{1} → V_{2}, F (f_{α}) : F (α) → F (α_{C}) is an isomorphism. Then the natural map

Ψ : lim_{α∈A(G)^{#}} F (α) → lim_{α∈A_{C}(G)^{#}} F (α)

is an isomorphism.

Note that both F (V_{1} ⟶^{α} V_{2}) = H^{∗}(V_{1}) and F (V_{1} ⟶^{α} V_{2}) = H^{∗}(C_{G}(V_{2})) satisfy the hypothesis of the lemma. Theorem 4.4 then follows from the lemma, as the relevant F ’s are built from these two examples by constructions that preserve isomorphisms.

Proof of Lemma 4.5. We define Φ : lim_{α∈A^{C}(G)^{#}} F (α) → lim_{α∈A(P)^{#}} F (α), an inverse

to Ψ, as follows. Given x = (x_{β}) ∈ lim_{β∈A^{C}(G)^{#}} F (β), let Φ(x) = (Φ(x)_{α}) ∈

∏_{α∈A(G)^{#}} F (α), where Φ(x)_{α} = F (f_{α})^{−1}F (g_{α})(x_{Cα}). One then checks that Φ(x) ∈ lim_{α∈A(G)^{#}} F (α), Ψ ∘ Φ = 1, and Φ ∘ Ψ = 1. □

4.5. Rewriting the formulae. If C is a small category, and

F : C^{#} → 𝔽_{p}-vector spaces

is a contravariant functor, there is a canonical isomorphism

lim_{C^{#}} F = Eq \{ ∏_{C∈obC} F (1_{C}) \begin{matrix}μ \\ ⟶ \\ ⟶ \\ ν \end{matrix} ∏_{α∈morC} F (α)\} ,

where, given α : C_{1} → C_{2}, the α-component of μ and ν are induced by applying F to the canonical morphisms in C^{#} from α to 1_{C_{1}} and 1_{C_{2}} respectively.

Thus, for example, R̄_{d}H^{∗}(G) will be naturally isomorphic to

Eq \{∏_{V} H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(V)) \begin{matrix}μ \\ ⟶ \\ ⟶ \\ ν \end{matrix} ∏_{α:V_{1}→V_{2}} H^{∗}(V_{1}) ⊗ P_{α(V_{1})}H^{d}(C_{G}(V_{2}))\} ,

where μ and ν are induced by

1 ⊗ α_{∗} : H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{G}(V_{1})) → H^{∗}(V_{1}) ⊗ P_{α(V_{1})}H^{d}(C_{G}(V_{2}))

and

α^{∗} ⊗ i : H^{∗}(V_{2}) ⊗ P_{V_{2}}H^{d}(C_{G}(V_{2})) → H^{∗}(V_{1}) ⊗ P_{α(V_{1})}H^{d}(C_{G}(V_{2}))

for each α : V_{1} → V_{2}. (i is the evident inclusion.)

Morphisms in A(G) factor as inclusions followed by isomorphisms induced by the inner automorphism group Inn(G), so this last formula rewrites as follows.

Proposition 4.6. R̄_{d}H^{∗}(G) is naturally isomorphic to

Eq \{[∏_{V} H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(V))]^{Inn(G)} \begin{matrix}μ \\ → \\ → \\ ν \end{matrix} ∏_{V_{1}<V_{2}} H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{G}(V_{2}))\} ,

where μ and ν are induced by

1 ⊗ η_{∗} : H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{G}(V_{1})) → H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{G}(V_{2}))

and

η^{∗} ⊗ i : H^{∗}(V_{2}) ⊗ P_{V_{2}}H^{d}(C_{G}(V_{2})) → H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{G}(V_{2}))

for each inclusion η : V_{1} < V_{2} in A_{C}(G).

Otherwise said,

x = (x_{V}) ∈ [∏_{V} H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(V))]^{Inn(G)}

is in R̄_{d}H^{∗}(G) exactly when the components are related by

(1 ⊗ η_{∗})(x_{V_{1}}) = (η^{∗} ⊗ i)(x_{V_{2}}),

for each inclusion η : V_{1} < V_{2} in A_{C}(G).

Similarly, we have Proposition 4.7. H^{∗}(G)_{LF} is naturally isomorphic to

Eq \{[∏_{V} P_{V} H^{∗}(C_{G}(V))]^{Inn(G)} \begin{matrix}μ \\ ⟶ \\ ⟶ \\ ν \end{matrix} ∏_{V_{1}<V_{2}} P_{V_{1}}H^{∗}(C_{G}(V_{2}))\} ,

where μ and ν are induced by

η_{∗} : P_{V_{1}}H^{∗}(C_{G}(V_{1})) → P_{V_{1}}H^{∗}(C_{G}(V_{2}))

and

i : P_{V_{2}}H^{∗}(C_{G}(V_{2})) ⊆ P_{V_{1}}H^{∗}(C_{G}(V_{2}))

for each inclusion η : V_{1} < V_{2} in A_{C}(G).

5. The cohomology of central extensions

Let C be a central p–elementary abelian subgroup of a finite group G, and let Q = G/C. This is the first of two sections in which we study the rich structure of the Lyndon–Hochschild–Serre spectral sequence {E_{r}^{∗,∗}(G, C)} associated to the central extension:

C ⟶^{i} G ⟶^{q} Q.

Enroute, we will prove Theorem 2.1.

To begin, we recall the following standard facts. The spectral sequence is a spectral sequence of differential graded algebras, converging to H^{∗}(G), and with E_{2}^{∗,∗} = H^{∗}(Q) ⊗ H^{∗}(C). Furthermore, E_{r}^{∗,∗} = E_{∞}^{∗,∗} for r >> 0 [E].

Recall that the extension corresponds to an element τ ∈ H^{2}(Q; C), or equivalently a homomorphism τ : H_{2}(Q) → C. Under the identification C^{#} = H^{1}(C), its dual τ^{#} : C^{#} → H^{2}(Q) corresponds to d_{2} : E_{2}^{0,1} → E_{2}^{2,0}.

5.1. H^{∗}(C)–comodule structure of the spectral sequence. As C is central, multiplication m : C × G → G is a group homomorphism. The induced algebra map

m^{∗} : H^{∗}(G) → H^{∗}(C) ⊗ H^{∗}(G),

makes H^{∗}(G) into a H^{∗}(C)–comodule. The restriction i^{∗} : H^{∗}(G) → H^{∗}(C) is both an algebra and comodule map, and it follows that E_{∞}^{0,∗} = im(i^{∗}) is a subHopf algebra of E_{2}^{0,∗} = H^{∗}(C).

One can strengthen these last observations to statements about the whole spectral sequence. A good functorial model for BG, say the reduced bar construction, shows that BC is an abelian topological group, BG is a BC–space equipped with proper free action via Bm : BC × BG → BG, and BG → BQ is the associated principal BC–bundle. The Serre spectral sequence arises from the pullback to BG of the skeletal filtration of BQ. This will be a filtration of BG by BC–subspaces, and we conclude the following.

Lemma 5.1. For all k and r, E_{r}^{k,∗} is an H^{∗}(C)–comodule, such that the maps

d_{r} : E_{r}^{k,∗} → E_{r}^{k+r,∗}

and

E_{r}^{i,∗} ⊗ E_{r}^{j,∗} → E_{r}^{i+j,∗}

are maps of H^{∗}(C)–comodules. In particular, E_{r}^{0,∗} is a subHopf algebra of E_{2}^{0,∗} = H^{∗}(C).

5.2. A handy Hopf algebra lemma. We now digress to state and prove a handy statement about (connected graded) Hopf algebras that we can apply to the situation of the previous subsection.

We need some notation. Let H be a graded connected Hopf algebra over a field 𝔽. There is a canonical splitting of vector spaces H = 𝔽 ⊕ I(H), where I(H) is the augmentation ideal. If M is a right H–module, let the module of indecomposables be defined by Q_{H}M = M ⊗_{H} 𝔽 = M/MI(H). Dually, if M is a right H–comodule, let the module of primitives be defined by

\begin{aligned}P_{H}M & = Eq \{M \begin{matrix}Δ \\ ⟶ \\ ⟶ \\ i \end{matrix} M ⊗ H\} \\ & = ker\{Δ̄ : M → M ⊗ I(H)\}, \end{aligned}

where Δ : M → M ⊗ H is the comodule structure, i is the inclusion induced by the unit 𝔽 → H, and Δ̄ is the composite M ⟶^{Δ} M ⊗ H → M ⊗ I(H).

Lemma 5.2. Let K be a subHopf algebra of a Hopf algebra H. Suppose M is simultaneously an H–comodule and K–module such that the K–module structure map M ⊗ K → M is a map of H–comodules. Then

(a) M is a free K–module, and

(b) the composite P_{H}M ↪ M ↠ Q_{K}M is monic.

Remark 5.3. To put this in perspective, the lemma has long been known if K = H, and, in this case, P_{H}M ≃ Q_{H}M [Sw, Thm.4.1.1]. Our proof is very similar to the proofs of Proposition 1.7 and Theorem 4.4 of Milnor and Moore’s classic paper [MM]. Compare also to Green’s lemma [G3, Lem2.1].

Before proving the lemma, we note the following consequence. Given H and K as in the lemma, let K −H −Mod be the category of M as in the lemma: an object is a vector space M that is simultaneously an H–comodule and K–module such that the K–module structure map M ⊗ K → M is a map of H–comodules. Morphisms are linear maps that are both K–module and H–comodule maps. K − H − Mod is an abelian category in the obvious way.

Corollary 5.4. (a) Every short exact sequence 0 → M_{1} → M → M_{2} → 0 in K − H − Mod is split as a sequence of K–modules.

(b) The functor sending M to Q_{K}(M) is exact on K − H − Mod.

Proof of Lemma 5.2. Choose a section s : Q_{K}M → M of the quotient π : M → Q_{K}M, and let m_{s} : Q_{K}M ⊗ I(K) → MI(K) be the epimorphism given by m_{s}(x, k) = s(x)k. Statement (a) is asserting that m_{s} is an isomorphism.

Let Δ_{K} : MI(K) → M ⊗ I(H) be the composite

MI(K) ⊂ M ⟶^{Δ̄} M ⊗ I(H),

Statement (b) asserts that P_{H}M ∩ MI(K) = {0}, i.e. that Δ_{K} is monic. Thus both statements will follow from the following claim:

Δ_{K} ∘ m_{s} : Q_{K}M ⊗ I(K) → M ⊗ I(H)

is monic.

To prove this claim, let F_{n}M be the K–submodule of M generated by elements of degree up to n. Given x ∈ (Q_{K}M)^{n}, and k ∈ I(K), let Δ(s(x)) = ∑ y′ ⊗ h′, and Δ(k) = ∑ k′ ⊗ k′′. Then

\begin{aligned}Δ_{K}(m_{s}(x, k)) & = Δ̄(s(x)k) \\ & ≡ s(x) ⊗ k \end{aligned}

modulo terms of the form y′k′ ⊗ h′k′′ with either |y′| < |s(x)| = n, or k′ ∈ I(K). Otherwise said,

Δ_{K}(m_{s}(x, k)) ≡ s(x) ⊗ k mod (F_{n−1}M + I(K)M) ⊗ I(H).

Thus

π(Δ_{K}(m_{s}(x, k))) ≡ x ⊗ k mod (Q_{K}M)^{<n} ⊗ I(H),

and so both π ∘ Δ_{K} ∘ m_{s} and Δ_{K} ∘ m_{s} are monic. □

5.3. Proof of statements (a) and (b) of Theorem 2.1. Let B_{r}^{∗,∗} ⊆ Z_{r}^{∗,∗} ⊆ E_{r}^{∗,∗} denote the r–boundaries and r–cycles of the spectral sequence.

We can apply Lemma 5.2 to our spectral sequence by letting H = H^{∗}(C), K = E_{∞}^{0,∗}, and M any of E_{r}^{k,∗}, Z_{r}^{k,∗}, B_{r}^{k,∗}. We deduce

Proposition 5.5. For all k and r, we have

(a) E_{r}^{k,∗}, Z_{r}^{k,∗}, and B_{r}^{k,∗} are free E_{∞}^{0,∗}–modules, and

(b) the composite P_{H^{∗}(C)}E_{r}^{k,∗} ↪ E_{r}^{k,∗} ↠ Q_{E_{r}^{0,∗}}E_{r}^{k,∗} is monic.

It follows that the short exact sequences of E_{∞}^{0,∗}–modules

0 → Z_{r}^{∗,∗} → E_{r}^{∗,∗} → B_{r}^{∗,∗} → 0,

and

0 → B_{r}^{∗,∗} → Z_{r}^{∗,∗} → E_{r+1}^{∗,∗} → 0,

are all split as E_{∞}^{0,∗}–modules. Thus the spectral sequence remains a spectral sequence after applying Q_{E_{∞}^{0,∗}}.

Now let A be a (G, C)–Duflot subalgebra of H^{∗}(G) as defined in §2: a subalgebra

such that the composite A ↪ H^{∗}(G) ↠^{i^{∗}} im(i^{∗}) = E_{∞}^{0,∗} is an isomorphism^{12}. We check the first two parts of Theorem 2.1: (a) H^{∗}(G) is a free A–module so that the spectral sequence {Q_{E_{∞}^{0,∗}}E_{r}^{∗,∗}} converges to Q_{A}H^{∗}(G), and (b) P_{H^{∗}(C)}H^{∗}(G) → Q_{A}H^{∗}(G) is monic.

Let F_{k}BG be the inverse image of the k–skeleton of BQ under the projection BG → BQ, and then let F^{k} be the image of H^{∗}(BG) → H^{∗}(F_{k}BG). Then F^{0} = E_{∞}^{0,∗}, and for k ≥ 1, there are short exact sequences

(5.1)  0 → E_{∞}^{k,∗} → F^{k} → F^{k−1} → 0

of objects that are simultaneously A–modules and H^{∗}(C)–comodules. Proposition 5.5(a) and induction on k show these sequences split as A–modules.

12We still need to show that such subalgebras exist.

Now consider the induced diagram

\xymatrix{0 \ar[r] & P_{H^{∗}(C)}E_{∞}^{k,∗} \ar[d] \ar[r] & P_{H^{∗}(C)}F^{k} \ar[d] \ar[r] & P_{H^{∗}(C)}F^{k−1} \ar[d] \\ 0 \ar[r] & Q_{A}E_{∞}^{k,∗} \ar[r] & Q_{A}F^{k} \ar[r] & Q_{A}F^{k−1} \ar[r] & 0.}

Here the top sequence is exact as indicated, as is the bottom, as (5.1) is split as A–modules. The left vertical arrow is monic by Proposition 5.5(b), as is the right vertical arrow, by induction on k, and it follows the middle arrow is also.

Thus we have proved that, for all k, F^{k} is a free A–module and P_{H^{∗}(C)}F^{k} → Q_{A}F^{k} is monic. As the connectivitity of the maps H^{∗}(G) → F^{k} goes to infinity as k goes to infinity, we conclude that the same is true for H^{∗}(G).

Remark 5.6. The decreasing filtration of H^{∗}(G) induces a filtration on P_{C}H^{∗}(G) and Q_{A}H^{∗}(G). We have shown that Q_{E_{∞}^{0,∗}}E_{∞}^{∗,∗} is the bigraded algebra associated to the filtration of Q_{A}H^{∗}(G). By contrast, we can only conclude that the associated bigraded algebra of P_{C}H^{∗}(G) embeds in P_{H^{∗}(C)}E_{∞}^{∗,∗}.

5.4. Finite generation. Statement (c) of Theorem 2.1 says that both P_{C}H^{∗}(G) and Q_{A}H^{∗}(G) are finitely generated H^{∗}(Q)–modules. Our proof of this is similar to arguments used by L.Evens in [E].

We first note that i^{∗} : H^{∗}(G) → H^{∗}(C) makes H^{∗}(C) into a finitely generated H^{∗}(G)–module. Otherwise put, E_{2}^{0,∗} is a finitely generated module over the ring E_{∞}^{0,∗}, which is Noetherian.

It follows that E_{2}^{∗,∗} = H^{∗}(Q) ⊗ E_{2}^{0,∗} is a finitely generated module over the Noetherian ring H^{∗}(Q) ⊗ E_{∞}^{0,∗}. By induction on r, we conclude that, for all r ≥ 2, E_{r}^{∗,∗} is a finitely generated H^{∗}(Q) ⊗ E_{∞}^{0,∗}–module.

Passing to E_{∞}^{0,∗}–indecomposables, it follows that Q_{E_{∞}^{0,∗}}E_{∞}^{∗,∗} is a finitely generated H^{∗}(Q)–module, and thus the same is true for Q_{A}H^{∗}(G), P_{H^{∗}(C)}E_{∞}^{∗,∗}, and P_{H^{∗}(C)}H^{∗}(G).

5.5. The image of inflation. The quotient map q : G → G/C induces the inflation homomorphism q^{∗} : H^{∗}(G/C) → H^{∗}(G). Its image, im(q^{∗}), is an unstable subalgebra of H^{∗}(G) and also identifies with E_{∞}^{∗,0} in the spectral sequence.

One approach to understanding im(q^{∗}) is to try to understand ker(q^{∗}). Recall that the classifying homomorphism τ^{#} : C^{#} → H^{2}(G/C) corresponds to d_{2} : E_{2}^{0,1} → E_{2}^{2,0}. Thus ker(q^{∗}) is an ideal that is closed under Steenrod operations, and contains im(τ^{#}). As in §2, we let I_{τ} ⊂ H^{∗}(G/C) be the smallest ideal with these properties. Thus there is an epimorphism of unstable algebras

H^{∗}(G/C)/I_{τ} ↠ im(q^{∗}),

which in many cases is an isomorphism.

As has already been said, im(q^{∗}) is contained in the subalgebra P_{C}H^{∗}(G), but it seems worthwhile, at this point, to explicitly explain why. The diagram

C × G \begin{matrix}m \\ ⟶ \\ ⟶ \\ π \end{matrix} G ⟶^{q} G/C

is a coequalizer diagram in the category of groups, i.e., a group homomorphism f : G → H satisfies f ∘ m = f ∘ π if and only if f factors uniquely through q.

Applying cohomology, we have that q^{∗} ∘ m^{∗} = q^{∗} ∘ π^{∗}, and so

im(q^{∗}) ⊆ Eq{m^{∗}, π^{∗}} = P_{C}H^{∗}(G).

In degree 1, inflation is as nice as possible.

Lemma 5.7. q^{∗} : H^{1}(G/C) → P_{C}H^{1}(G) is an isomorphism.

Proof. The exact sequence arising from the corner of the spectral sequence,

0 → H^{1}(G/C) ⟶^{q^{∗}} H^{1}(G) ⟶^{i^{∗}} H^{1}(C),

can be viewed as the degree 1 part of a sequence of H^{∗}(C)–comodules, if H^{∗}(G/C) is given the trivial comodule structure. Taking primitives yields an exact sequence

0 → P_{C}H^{1}(G/C) ⟶^{q^{∗}} P_{C}H^{1}(G) ⟶^{i^{∗}} P_{C}H^{1}(C)

which identifies with

0 → H^{1}(G/C) ⟶^{q^{∗}} P_{C}H^{1}(G) → 0,

as P_{C}H^{1}(G/C) = H^{1}(G/C) and P_{C}H^{1}(C) = 0. □

In higher degrees, the inclusion im(q^{∗}) ⊆ P_{C}H^{∗}(G) certainly may be proper: see Example 9.1. However, we now show that the N il_{1}–closures of each of the maps

H^{∗}(G/C)/I_{τ} ↠ im(q^{∗}) ↪ P_{C}H^{∗}(G)

is an isomorphism. Equivalently, the composite is an F –isomorphism, as asserted in statement (d) of Theorem 2.1.

We prove this in two steps.

Proposition 5.8. L_{0}(im(q^{∗})) ≃ L_{0}(P_{C}H^{∗}(G)).

Proof. By Proposition 3.8, we need to show that, for all U, there are bijections

Hom_{K}(P_{C}H^{∗}(G), H^{∗}(U)) ≃ Hom_{K}(im(Inf_{G/C}^{G}), H^{∗}(U)).

Using that H^{∗}(U) is injective in K, Hom_{K}(P_{C}H^{∗}(G), H^{∗}(U)) identifies with

Coeq \{Hom_{K}(H^{∗}(C × G), H^{∗}(U)) \begin{matrix}m_{∗} \\ ⟶ \\ ⟶ \\ π_{∗} \end{matrix} Hom_{K}(H^{∗}(G), H^{∗}(U))\},

and Hom_{K}(im(q^{∗}), H^{∗}(U)) identifies with the image of

Hom_{K}(H^{∗}(G), H^{∗}(U)) ⟶^{q^{∗}} Hom_{K}(H^{∗}(G/C), H^{∗}(U)).

Lannes showed [L1, Prop.4.3.1] that Rep(U, G) ≃ Hom_{K}(H^{∗}(G), H^{∗}(U)), where Rep(U, G) is set of orbits of Hom(U, G) under the conjugation action of G. Thus the next lemma is equivalent to the proposition.

□

Lemma 5.9. The diagram of sets

Rep(U, C × G) \begin{matrix}m_{∗} \\ ⟶ \\ ⟶ \\ π_{∗} \end{matrix} Rep(U, G) ⟶^{q_{∗}} Rep(U, G/C)

is exact in the following sense: given homomorphisms α, β : U → G, q ∘ α = q ∘ β if and only if there exists γ : U → C × G such that α = m ∘ γ and β = π ∘ γ.

Proof. Such a γ is equivalent to a pair (δ, β), where δ : U → C is a homomorphism satisfying δ(u)β(u) = α(u) for all u ∈ U. Now suppose given α, β : U → G such that q ∘ α = q ∘ β. Then the function δ : U → G defined by δ(u) = α^{−1}(u)β(u) will take values in C because q ∘ α = q ∘ β, and will be a homomorphism because C is central in G. □

Proposition 5.10. L_{0}(H^{∗}(G/C)/I_{τ}) ≃ L_{0}(im(q^{∗})).

Proof. In this case, Hom_{K}(H^{∗}(G/C)/I_{τ} , H^{∗}(U)) identifies with the set

{α ∈ Rep(U, G/C) | α^{∗}(τ) = 0 ∈ H^{2}(G/C; C)},

while Hom_{K}(im(q^{∗}), H^{∗}(U)) can be viewed as the set

{α ∈ Rep(U, G/C) | α factors through q : G → G/C }.

But these sets are the same, because α^{∗}(τ) represents the top extension in the pullback diagram

[figure: C; G(α); U; α; C; G; q; G/C,]

and this extension is trivial if and only if α factors through q. □

We have a formula for L_{0}(im(q^{∗})) analogous to the formula

L_{0}(H^{∗}(G)) = lim_{V ∈A(G)} H^{∗}(V).

Let A(G, C) be the full subcategory of A(G) with objects the V ∈ A(G) containing C, and note that H^{∗}(V/C) = P_{C}H^{∗}(V) for all such V.

Proposition 5.11. L_{0}(im(q^{∗})) ≃ lim_{V ∈A(G,C)} H^{∗}(V/C).

Proof. As H^{∗}(V/C) is N il_{1}–closed, so is lim_{V ∈A(G,C)} H^{∗}(V/C). Arguing as in the

previous proofs, the proposition is equivalent to the statement that, for all U, the image of Rep(U, G) ⟶^{q_{∗}} Rep(U, G/C) identifies with colim_{V ∈A(G,C)} Hom(U, V/C). This

is easily checked: details are left to the reader. □

These propositions allow us to quickly prove the last statement of Theorem 2.1: the Krull dimension of any of H^{∗}(G/C)/I_{τ}, im(q^{∗}), P_{C}H^{∗}(G), or Q_{A}H^{∗}(G) equals (the p–rank of G) - (the rank of C).

To begin with, Proposition 3.7 said that dim K = dim L_{0}K if K is a Noetherian unstable algebra. Thus

dim H^{∗}(G/C)/I_{τ} = dim im(q^{∗}) = dim P_{C}H^{∗}(G) = dim lim_{V ∈A(G,C)} H^{∗}(V/C).

The second equality also follows from the fact that P_{C}H^{∗}(G) is a finitely generated H^{∗}(G/C)/I_{τ} –module, and similarly dim H^{∗}(G/C)/I_{τ} = dim Q_{A}H^{∗}(G) is true.

As ∏_{V ∈A(G,C)} H^{∗}(V/C) is a finitely generated H^{∗}(G/C)–module, it certainly is

also finitely generated over lim_{V ∈A(G,C)} H^{∗}(V/C). Thus we conclude

dim lim_{V ∈A(G,C)} H^{∗}(V/C) = dim ∏_{V ∈A(G,C)} H^{∗}(V/C)

= max_{V ∈A(G,C)}{rank of V/C}

= (the p–rank of G) - (the rank of C).

6. Transgressions and the structure of E_{r}^{0,∗}

In this section we continue our examination of the spectral sequence associated to the central extension C ⟶^{i} G ⟶^{q} Q, where C is p–elementary abelian of rank c. We carefully describe the form of the differentials

d_{r} : E_{r}^{0,r−1+∗} → E_{r}^{r,∗},

and prove that for all r, the Hopf algebra E_{r}^{0,∗} ⊂ H^{∗}(C) must be a free commutative algebra^{13} of a standard form. In particular, im(i^{∗}) = E_{∞}^{0,∗} is free, and so a subalgebra of H^{∗}(G) generated by any lift of a miniminal set of generators of im(i^{∗}) will be a (G, C)–Duflot subalgebra.

To begin our analysis, we know that τ^{#} : C^{#} → H^{2}(Q) corresponds to the transgression d_{2} : E_{2}^{0,1} → E_{2}^{2,0}.

6.1. The cokernel of τ. The cokernel of τ has a group theoretic meaning. The proof of the next lemma is left to the reader.

Lemma 6.1. Let f : C → C_{0} be the cokernel of τ : H_{2}(Q) → C. Then f factors through C ⟶^{i} G and is the universal homomorphism from C to a p–elementary abelian group with this property.

As f is split epic, the lemma tells us that (G, C) is isomorphic to a pair of the form C_{0} × (G_{1}, C_{1}), where no factor of C_{1} splits off G_{1}. The spectral sequences will be related by E_{r}^{∗,∗}(G, C) ≃ H^{∗}(C_{0}) ⊗ E_{r}^{∗,∗}(G_{1}, C_{1}), and d_{2} : E_{r}^{0,1}(G_{1}, C_{1}) → E_{r}^{2,0}(G_{1}, C_{1}) will be monic. Thus in our analysis, we can assume that d_{2} : E_{2}^{0,1} → E_{2}^{2,0} is an inclusion if we wish.

6.2. Properties of d_{r}. Using that E_{r}^{∗,∗} is free over E_{r}^{0,∗}, we will see that the differentials we are interested in are ‘almost’ determined by two standard properties.

The first is that d_{r} is a derivation.

The second is the transgression theorem: recall that E_{r}^{0,∗} is an unstable subalgebra of H^{∗}(C) and E_{r}^{∗,0} is an unstable quotient algebra of H^{∗}(Q). Given x ∈ E_{r}^{0,r−1} and a ∈ A, ax ∈ E_{r+|a|}^{0,r+|a|−1} and d_{r+|a|}(ax) is represented by ad_{r}(x).

13In the usual graded sense, if p is odd.

6.3. The kernel of the DeRham and Koszul derivations. Our differentials are modelled by two standard derivations.

Let S^{∗}(V) be the symmetric algebra on a graded 𝔽_{p}–vector space V, with V concentrated in even degrees if p is odd. The DeRham derivation is the derivation

d_{V} : S^{∗}(V) → S^{∗}(V) ⊗ ΣV

determined by letting d_{V} (v) = 1 ⊗ σv for v ∈ V.

We will need to know its kernel. Let Φ : S^{∗}(V) → S^{∗}(V) denote the p^{th} power map.

Lemma 6.2. The kernel of d_{V} is S^{∗}(Φ(V)).

This is a special case of a result due to Cartier [Car]. For completeness, we sketch an elegant argument we learned from [FLS, proof of Prop.3.3]. The kernel of d_{V} is H^{0} of the DeRham complex Ω^{∗}(V) = (S^{∗}(V) ⊗ Λ^{∗}(ΣV), d_{V}). Since Ω^{∗}(V ⊕ W) ≃ Ω^{∗}(V) ⊗ Ω^{∗}(W), the Kunneth theorem allows one to reduce to the case when V is one dimensional, where is it easily checked.

Let Λ^{∗}(V) be the exterior algebra on a graded 𝔽_{p}–vector space V, with V concentrated in odd degrees. The Koszul derivation is the derivation

δ_{V} : Λ^{∗}(V) → Λ^{∗}(V) ⊗ ΣV

determined by letting δ_{V} (v) = 1 ⊗ σv for v ∈ V. This is the bottom of the Koszul complex (Λ^{∗}(V) ⊗ S^{∗}(ΣV), δ_{V}), which is acyclic, and we have

Lemma 6.3. The kernel of δ_{V} is 𝔽_{p}, i.e. δ_{V} is monic in positive degrees.

6.4. The structure of E_{r}^{0,∗} when p = 2. If p = 2, E_{2}^{0,∗} = S^{∗}(C^{#}). As the squaring operation Φ in degree n corresponds to the Steenrod operation Sq^{n}, the image of Φ^{k−1} : E_{2}^{0,1} → E_{2}^{0,2^{k}} lands in the subspace E_{2^{k}+1}^{0,2^{k}}. Thus we can define an increasing filtration of C^{#},

C_{0}^{#} ⊆ C_{1}^{#} ⊆ C_{2}^{#} ⊆ …

by letting C_{k}^{#} be the kernel of the composite

E_{2}^{0,1} ⟶^{Φ^{k}} E_{2^{k}+1}^{0,2^{k}} ⟶^{d_{2^{k}+1}} E_{2^{k}+1}^{2^{k}+1,0}.

Theorem 6.4. The only possible nonzero differentials

d_{r} : E_{r}^{0,∗} → E_{r}^{r,∗}

are d_{2^{k}+1} with k = 0, 1, 2, . . .. For each such k, E_{2^{k}+2}^{0,∗} = ker d_{2^{k}+1} equals

S^{∗}(C_{0}^{#} + Φ(C_{1}^{#}) + ⋯ + Φ^{k}(C_{k}^{#}) + Φ^{k+1}(C^{#})).

This polynomial algebra is noncanonically isomorphic to

S^{∗}(V_{0} ⊕ Φ(V_{1}) ⊕ ⋯ ⊕ Φ^{k}(V_{k}) ⊕ Φ^{k+1}(C^{#}/C_{k}^{#})),

where V_{i} = C_{i}^{#}/C_{i−1}^{#}. In particular, one can choose a basis for C^{#} such that E_{∞}^{0,∗} has the form described in §2.5.

Remark 6.5. In practical terms, the filtration of C^{#} is often determined by the extension homomorphism τ : H_{2}(Q) → C together with the action of the Steenrod operations on H^{∗}(Q).

Note that C_{k}^{#} will also be the kernel of the composite

E_{2}^{0,1} ⟶^{d_{2}} E_{2}^{2,0} ⟶^{Sq^{1}} … ⟶^{Sq^{2^{k−1}}} E_{2}^{2^{k}+1,0} ↠ E_{2^{k}+1}^{2^{k}+1,0},

where the last map is the evident quotient. Equivalently, C_{k}^{#} is the kernel of

C^{#} ⟶^{τ_{G}^{#}} H^{2}(Q) ⟶^{Sq^{1}} … ⟶^{Sq^{2^{k−1}}} H^{2^{k}+1}(Q) ↠ E_{2^{k}+1}^{2^{k}+1,0}.

Let I_{τ} (k) ⊂ H^{∗}(Q) be the ideal generated by A(k − 1) · im(τ^{#}), where A(k) ⊂ A is the subalgebra generated by Sq^{1}, …, Sq^{2^{k}}. Then the quotient map H^{∗}(Q) ↠ E_{2^{k}+1}^{∗,0} factors

H^{∗}(Q) ↠ H^{∗}(Q)/I_{τ} (k − 1) ↠ E_{2^{k}+1}^{∗,0},

and the second map is often an isomorphism.

Proof of Theorem 6.4. By definition, Φ^{k}(C_{k}^{#}) is the kernel of the transgression d_{2^{k}+1} : Φ^{k}(C^{#}) → E_{2^{k}+1}^{2^{k}+1,0}, and so consists of permanent cycles. It follows that, for each k, the subalgebra S(k) = S^{∗}(C_{0}^{#} + Φ(C_{1}^{#}) + ⋯ + Φ^{k}(C_{k}^{#})) is also contained in E_{∞}^{0,∗}.

By induction, we now show that

E_{2^{k}+1}^{0,∗} = S^{∗}(C_{0}^{#} + Φ(C_{1}^{#}) + ⋯ + Φ^{k−1}(C_{k−1}^{#}) + Φ^{k}(C^{#})).

When k = 0, this just says that E_{2}^{0,∗} = S^{∗}(C^{#}), and so is certainly true. Now we assume the statement for k and prove it with k replaced by k + 1.

Let V = Φ^{k}(C^{#})/Φ^{k}(C_{k}^{#}). Then E_{2^{k}+1}^{0,∗} ≃ S(k) ⊗ S^{∗}(V), ΣV identifies with the

image of d_{2^{k}+1} : Φ^{k}(C^{#}) → E_{2^{k}+1}^{2^{k}+1,0}, and we have a commutative diagram

\xymatrix{E_{2^{k}+1}^{0,∗} \ar[d]^{≀} \ar[r]^{d_{2^{k}+1}} & E_{2^{k}+1}^{2^{k}+1,∗} \\ S(k) ⊗ S^{∗}(V) \ar[r]^{1⊗d_{V}} & S(k) ⊗ S^{∗}(V) ⊗ ΣV. \ar[u]}

Here the right vertical map is induced from the inclusion ΣV ⊆ E_{2^{k}+1}^{2^{k}+1,0} using the

E_{2^{k}+1}^{0,∗}–module structure on E_{2^{k}+1}^{2^{k}+1,∗}.

We have reached the key point in our proof: as in our proof of Theorem 2.1, Lemma 5.2 shows that this module structure is free, and thus the right vertical map is monic. It follows that the kernel of the top map identifies with ker(1 ⊗ d_{V}) which equals S(k) ⊗ S^{∗}(Φ(V)), by Lemma 6.2. Otherwise said,

E_{2^{k}+2}^{0,∗} = S^{∗}(C_{0}^{#} + Φ(C_{1}^{#}) + ⋯ + Φ^{k}(C_{k}^{#}) + Φ^{k+1}(C^{#})).

Finally, we note that once we know that E_{2^{k}+2}^{0,∗} has this form, E_{2^{k+1}+1}^{0,∗} = E_{2^{k}+2}^{0,∗} follows by the transgression theorem. □

6.5. The structure of E_{r}^{0,∗} when p is odd. When p is odd,

E_{2}^{0,∗} = Λ^{∗}(C^{#}) ⊗ S^{∗}(β(C^{#})).

As the p^{th} power operation Φ in degree 2n corresponds to the Steenrod operation P^{n}, the image of Φ^{k} ∘ β : E_{2}^{0,1} → E_{2}^{0,2p^{k}} lands in the subspace E_{2p^{k}+1}^{0,2p^{k}}. Thus we can define an increasing filtration of C^{#},

C_{0}^{#} ⊆ C_{1}^{#} ⊆ C_{2}^{#} ⊆ …

by letting C_{0}^{#} be the kernel of E_{2}^{0,1} ⟶^{d_{2}} E_{2}^{2,0}, and, for k ≥ 0, C_{k+1}^{#} be the kernel of the composite

E_{2}^{0,1} ⟶^{Φ^{k}∘β} E_{2p^{k}+1}^{0,2p^{k}} ⟶^{d_{2p^{k}+1}} E_{2p^{k}+1}^{2p^{k}+1,0}.

Theorem 6.6. The only possible nonzero differentials

d_{r} : E_{r}^{0,∗} → E_{r}^{r,∗}

are d_{2} and d_{2p^{k}+1} with k = 0, 1, 2, . . .. Furthermore,

E_{3}^{0,∗} = Λ^{∗}(C_{0}^{#}) ⊗ S^{∗}(β(C^{#})),

and, for each k ≥ 0,

E_{2p^{k}+2}^{0,∗} = Λ^{∗}(C_{0}^{#}) ⊗ S^{∗}(β(C_{1}^{#}) + Φβ(C_{2}^{#}) + ⋯ + Φ^{k}β(C_{k+1}^{#}) + Φ^{k+1}β(C^{#})).

This free commutative algebra is noncanonically isomorphic to

Λ^{∗}(V_{0}) ⊗ S^{∗}(β(V_{0}) ⊕ β(V_{1}) ⊕ ⋯ ⊕ Φ^{k}β(V_{k+1}) ⊕ Φ^{k+1}β(C^{#}/C_{k+1}^{#})),

where V_{i} = C_{i}^{#}/C_{i−1}^{#}. In particular, one can choose a basis for C^{#} such that E_{∞}^{0,∗} has the form described in §2.5.

Remark 6.7. Similar to the case when p = 2, C_{k+1}^{#} will be the kernel of

C^{#} ⟶^{τ^{#}} H^{2}(Q) ⟶^{β} H^{3}(Q) ⟶^{P^{1}} … ⟶^{P^{p^{k−1}}} H^{2p^{k}+1}(Q) ↠ E_{2p^{k}+1}^{∗,0},

and the quotient map H^{∗}(Q) ↠ E_{2p^{k}+1}^{∗,0} factors as

H^{∗}(Q) ↠ H^{∗}(Q)/I_{τ} (k) ↠ E_{2p^{k}+1}^{∗,0}

where I_{τ} (k) ⊂ H^{∗}(Q) is the ideal generated by A(k − 1) · im(τ^{#}). Here A(k) ⊂ A is the subalgebra generated by β, P^{1}, P^{p}, …, P^{p^{k−1}}.

Proof of Theorem 6.6. We just sketch the proof, as it follows along the lines of the proof of the p = 2 version of the theorem.

To compute E_{3}^{0,∗}, let V = C^{#}/C_{0}^{#}. Then ker d_{2} identifies with the kernel of

Λ^{∗}(C_{0}^{#}) ⊗ S^{∗}(β(C^{#})) ⊗ Λ^{∗}(V) ⟶^{1⊗1⊗δ_{V}} Λ^{∗}(C_{0}^{#}) ⊗ S^{∗}(β(C^{#})) ⊗ Λ^{∗}(V) ⊗ ΣV.

The formula for E_{3}^{0,∗} thus follows from Lemma 6.3.

To compute E_{2p^{k}+2}^{0,∗} for k ≥ 0, let V = Φ^{k}β(C^{#})/Φ^{k}β(C_{k+1}^{#}). Then the subalgebra

S(k) = Λ^{∗}(C_{0}^{#}) ⊗ S^{∗}(β(C_{1}^{#}) + Φβ(C_{2}^{#}) + ⋯ + Φ^{k}β(C_{k+1}^{#}))

is all permanent cycles. Using that E_{2p^{k}+1}^{2p^{k}+1,∗} is a free E_{2p^{k}+1}^{0,∗}–module, ker d_{2p^{k}+1} identifies with the kernel of

S(k) ⊗ S^{∗}(V) ⟶^{1⊗d_{V}} S(k) ⊗ S^{∗}(V) ⊗ ΣV,

and the formula for E_{2p^{k}+2}^{0,∗} follows from Lemma 6.2. □

7. p–central groups

In this section we prove our main theorems about p–central groups: Theorem 2.8 and Theorem 2.9.

We begin by recalling some notation from §2. If C = C(G) is the maximal central p–elementary abelian subgroup of a finite group G, we have shown that C^{#} = H^{1}(C) admits an ordered basis (x_{1}, …, x_{c}) so that, if y_{j} = β(x_{j}) for p odd,

Res_{C}^{G}(H^{∗}(G)) = \begin{cases}𝔽_{2}[x_{1}^{2^{j_{1}}} , …, x_{c}^{2^{jc}}] & if p = 2 \\ 𝔽_{p}[y_{1}^{p^{j_{1}}} , …, y_{b}^{p^{j_{b}}} , y_{b+1}, …, y_{c}] ⊗ Λ(x_{b+1}, …, x_{c}) & if p is odd, \end{cases}

with the j_{i} forming a sequence of nondecreasing nonnegative integers.

Then we say that G has type [a_{1}, …, a_{c}] where

(a_{1}, …, a_{c}) = \begin{cases}(2^{j_{1}}, …, 2^{j_{c}}) & if p = 2 \\ (2p^{j_{1}}, …, 2p^{j_{b}}, 1, …, 1) & if p is odd, \end{cases}

and we let e(G) = ∑_{i=1}^{c}(a_{i} − 1) and h(G) = \begin{cases}2p^{k−1} & if a_{1} = 2p^{k} with k ≥ 1 \\ 1 & if a_{1} = 2 \\ 0 & if a_{1} = 1. \end{cases}

We have the following lemma about products.

Lemma 7.1. Suppose G_{0} and G_{1} have maximal central p–elementary abelian subgroups C_{0} and C_{1}, and Duflot subalgebras A_{0} and A_{1}. Then the following hold.

(a) C_{0} × C_{1} = C(G_{0} × G_{1}), and

P_{C_{0}×C_{1}}H^{∗}(G_{0} × G_{1}) = P_{C_{0}}H^{∗}(G_{1}) ⊗ P_{C_{1}}H^{∗}(G_{1}).

(b) A_{0} ⊗ A_{1} will be a Duflot subalgebra for G_{0} × G_{1}, and

Q_{A_{0}⊗A_{1}}H^{∗}(G_{0} × G_{1}) = Q_{A_{0}}H^{∗}(G_{1}) ⊗ Q_{A_{1}}H^{∗}(G_{1}).

(c) e(G_{0} × G_{1}) = e(G_{0}) + e(G_{1}), and h(G_{0} × G_{1}) = max{h(G_{0}), h(G_{1})}.

Note that a subgroup H of a p–central group G is again p–central, and C(H) = C(G) ∩ H. The next lemma is easily deduced.

Lemma 7.2. Let G be p–central, and let A ⊂ H^{∗}(G) be a Duflot subalgebra. If j : H < G is a subgroup, then e(H) ≤ e(G), h(H) ≤ h(G), and j^{∗}(A) will be a Duflot subalgebra of H^{∗}(H).

Thanks to this lemma, Corollary 2.10 immediately follows from Theorem 2.9.

Remark 7.3. The example H = ℤ/4 < ℤ/8 = G shows that the inequalities of the lemma can be equalities, even when H is a proper subgroup of a p–group G.

7.1. Benson–Carlson duality. If G is p–central, then Q_{A}H^{∗}(G) will be a finite dimensional 𝔽_{p}–algebra if A is any Duflot subalgebra. Benson and Carlson tell us much more:

Theorem 7.4. If G is p–central and A is a Duflot subalgebra of H^{∗}(G), then Q_{A}H^{∗}(G) is a Poincaré duality algebra with top class in degree e(G).

Under the assumption that A is a polynomial algebra (always true if p = 2), this is an immediate application of the main theorem in [BC1]. The general case reduces to this one: G and A will admit decompositions G = C_{0} × G_{1} and A = H^{∗}(C_{0}) ⊗ A_{1}, with C_{0} p–elementary, G_{1} having no ℤ/p summands, and A_{1} a (necessarily polynomial) Duflot subalgebra of H^{∗}(G_{1}). Then Q_{A}H^{∗}(G) = Q_{A_{1}}H^{∗}(G_{1}), and e(G) = e(G_{1}).

7.2. Proof of Theorem 2.8. Let G be p–central, C = C(G), and A ⊂ H^{∗}(G) a Duflot subalgebra. We now prove the various parts of Theorem 2.8.

Firstly, Theorem 7.4 implies that Q_{A}H^{∗}(G) is zero in degrees greater than e(G), and one dimensional in degree e(G).

Now consider the Serre spectral sequence for C → G → G/C, as studied in Theorem 2.1. The bigraded algebra Q_{E_{∞}^{0,∗}}E_{∞}^{∗,∗} is the graded object associated to a decreasing filtration of the Poincaré duality algebra Q_{A}H^{∗}(G) with top degree e(G). This forces the following to be true: there is a largest s, s(G), such that E_{∞}^{s,∗} is nonzero, Q_{E_{∞}^{0,∗}}E_{∞}^{s(G),∗} will be one dimensional and concentrated in total degree

e(G), and nonzero classes in E_{∞}^{s(G),e(G)−s(G)} ⊂ H^{e(G)}(G) will be Poincaré duality classes.

These classes will also be H^{∗}(C)–comodule primitives, as E_{∞}^{s(G),∗} is a sub– H^{∗}(C)–comodule of H^{∗}(G), and everything in lowest degree must be primitive. As P_{C}H^{∗}(G) is contained in Q_{A}H^{∗}(G), we conclude that P_{C}H^{∗}(G) is also zero in degrees greater than e(G), and one dimensional in degree e(G).

By Corollary 2.4, P_{C}H^{e(G)}H^{∗}(G) is also be the top nonzero degree of H^{∗}(G)_{LF}, and so consists of classes annihilated by all positive degree Steenrod operations.

It remains to show that, under the additional assumption that G is a p–group, P_{C}H^{e(G)}H^{∗}(G) is essential cohomology. This we prove in the next subsection.

7.3. p–central p–groups and essential cohomology. Let P be a p–central p– group. We have shown that H^{e(P)}(P)_{LF} = P_{C(P)}H^{e(P)}(P) is a one dimensional subspace of H^{e(P)}(P).

Proposition 7.5. H^{e(P)}(P)_{LF} is essential.

Proof. As P is a p–group, maximal proper subgroups have the form j : Q < P, where Q is the kernel of a nonzero homomorphism x : P → ℤ/p. We need to show that j^{∗}(ζ) = 0 ∈ H^{∗}(Q) if ζ ∈ H^{e(P)}(P)_{LF} is nonzero.

The map j^{∗} : H^{∗}(P) → H^{∗}(Q) will take H^{e(P)}(P)_{LF} to H^{e(P)}(Q)_{LF}. If e(Q) < e(P), we are done: j^{∗}(ζ) will be an element of a zero group.

If e(Q) = e(P), we reason as follows. Let A be a Duflot subalgebra of H^{∗}(G), so that j^{∗}(A) is a Duflot subalgebra of H^{∗}(Q). If j^{∗}(ζ) ≠ 0, it will project to a nonzero element in Q_{j^{∗}(A)}H^{∗}(Q). We show that this is impossible. Regard x as an nonzero element in H^{1}(P). By construction, j^{∗}(x) = 0 ∈ H^{1}(Q). By Poincaré duality, there exists y ∈ H^{∗}(P) such that ζ = xy ∈ Q_{A}H^{∗}(P). But then j^{∗}(ζ) = j^{∗}(x)j^{∗}(y) = 0 ∈ Q_{j^{∗}(A)}H^{∗}(Q). □

Let A(P, P) be the two sided Burnside ring over 𝔽_{p}: the 𝔽_{p}–algebra with basis given by equivalence classes of diagrams P ≥ Q ⟶^{α} P, and multiplication defined using the double coset formula^{14}. If J is the ideal generated by all such diagrams with α not an isomorphism, then A(P, P)/J ≃ 𝔽_{p}[Out(P)], the group ring of the outer automorphism group.

14There are more elegant descriptions, but this is better for our purposes.

Using transfers (a.k.a. induction), A(P, P) acts on H^{∗}(P), with a basis element

[P ≥ Q ⟶^{α} P] inducing H^{∗}(P) ⟶^{α^{∗}} H^{∗}(Q) ⟶^{T r_{Q}^{P}} H^{∗}(P). As these are unstable A– module maps, it follows that H^{e(P)}(P)_{LF} is a one dimensional A(P, P)–submodule.

Corollary 7.6. The ideal J acts trivially on H^{e(P)}(P)_{LF}.

Proof. The previous proposition shows that if a homomorphism α : Q → P is not onto, then α^{∗}(H^{e(P)}(P)_{LF}) = 0. □

It follows that the A(P, P)–module H^{e(P)}(P)_{LF} is the pullback of a one dimensional representation of Out(P) over the prime field 𝔽_{p}. We let ω(P) denote this representation. Clearly ω(P) will be trivial if p = 2, but this need not be the case when p is odd.

Example 7.7. Let p = 3. Then ω(ℤ/9) = H^{1}(ℤ/9) is nontrivial, as −1 : ℤ/9 → ℤ/9 induces multiplication by −1 on H^{1}(ℤ/9).

7.4. d_{0}(G) when G has a p–central p–Sylow subgroup. We prove the parts of Theorem 2.9 involving d_{0}.

Firstly, if G is p–central, then Corollary 2.6 says that

R̄_{d}H^{∗}(G) ≃ H^{∗}(C(G)) ⊗ P_{C(G)}H^{d}(G).

Since d_{0}(G) is the largest d such that R̄_{d}H^{∗}(G) ≠ 0, it follows that d_{0}(G) will equal the top nonzero degree of P_{C(G)}H^{∗}(G), which we have computed to be e(G).

Now suppose that G is not necessarily p–central, but has a p–central p–Sylow subgroup P. We show that then d_{0}(G) = d_{0}(P).

We need to show that the largest d such that R̄_{d}H^{∗}(G) ≠ 0 is d = d_{0}(P) = e(P). Let e_{1} ∈ A(P, P) be an idempotent chosen so that A(P, P)e_{1} is the projective cover of ϵ, the trivial 𝔽_{p}[Out(P)]–module, pulled back to A(P, P). Standard arguments show that there are inclusions

e_{1}R̄_{d}H^{∗}(P) ⊆ R̄_{d}H^{∗}(G) ⊆ R̄_{d}H^{∗}(P).

Thus it suffices to show that e_{1}R̄_{e(P)}H^{∗}(P) ≠ 0. Otherwise said, it suffices to show that ϵ is a composition factor in the A(P, P)–modules R̄_{e(P)}H^{∗}(P).

If p = 2, we are done: by Corollary 7.6, R̄_{e(P)}H^{e(P)}(P) = H^{e(P)}(P)_{LF} ≃ ϵ. As a bonus, we learn that H^{∗}(G)_{LF} is one dimensional in degree e(P).

When p is odd, more care (and maybe luck) is needed. Recall that R̄_{e(P)}H^{∗}(P) = H^{∗}(C(P)) ⊗ H^{e(P)}(P)_{LF}. The fact that J acts as 0 on H^{e(P)}(P)_{LF} implies that same is true for H^{∗}(C(P)) ⊗ H^{e(P)}(P)_{LF}. Thus we just need to show that the trivial Out(P)–module occurs as a composition factor in H^{∗}(C(P))⊗H^{e(P)}(P)_{LF} = H^{∗}(C(P)) ⊗ ω(P), or, equivalently, that ω(P)^{−1} occurs as an Out(P)-composition factor H^{∗}(C(P)). We are done with the following lemma^{15}.

Lemma 7.8. If P is a p–central p–group with p an odd prime, then every irreducible 𝔽_{p}[Out(P)]–module occurs as a composition factor of H^{∗}(C(P)).

The lemma, in a stronger form than stated, follows by combining [K3, Prop.5.7 and Cor.6.8]. The key point is that, since C(P) = Ω_{1}(P), the kernel of Aut(P) → Aut(C(P)) will be a p–group if p is odd [Gor, Thm.5.3.10].

15This lemma is false if p = 2, as the example P = Q_{8} illustrates.

Example 7.9. Let p = 3 and G be the semidirect product ℤ/9 ⋊ ℤ/2. Then d_{0}(G) = d_{0}(ℤ/9) = 1, but H̃^{∗}(G)_{LF} = 0.

7.5. d_{1}(G) when G is p–central. In this subsection, let G be p–central. We show that d_{1}(G) = e(G) + h(G).

We get control of d_{1}(G) by working directly with the desuspended composition factors R_{d}H^{∗}(G) of H^{∗}(G), rather than their N il_{1}–localizations R̄_{d}H^{∗}(G), as was done in our calculation of d_{0}(G).

To simplify notation, write R_{d} for R_{d}H^{∗}(G), R̄_{d} for R̄_{d}H^{∗}(G), and C for C(G). We have that R̄_{0} = H^{∗}(C), and R_{0} = im(i^{∗}), where i : C ↪ G is the inclusion.

In the nilpotent filtration of H^{∗}(G), the last nonzero submodule, nil_{e(G)}H^{∗}(G) = Σ^{e(G)}R_{e(G)}, has been shown to be isomorphic to Σ^{e(G)}R_{0} as an unstable module. Thus d_{1} of this submodule of H^{∗}(G) equals e(G) + d_{1}(R_{0}). By Lemma 3.13, the next lemma implies that if d < e(G), d_{1}(Σ^{d}R_{d}) is strictly smaller than this.

Lemma 7.10. Each R_{d} with d < e(G) admits a filtration by unstable modules with subquotients all of the form Σ^{k}R_{0} with d + k < e(G).

Again appealing to Lemma 3.13, we then have the next corollary.

Corollary 7.11. d_{1}(G) = d_{1}(Σ^{e(G)}R_{e(G)}) = e(G) + d_{1}(R_{0}).

Thus we will have proved that, when G is p–central, d_{1}(G) = e(G) + h(G), once we have proved Lemma 7.10, and calculated that d_{1}(R_{0}) = h(G). We begin the proof of Lemma 7.10 here, and then both finish it, and calculate d_{1}(R_{0}), in the two subsections that follow, which correspond to the cases p = 2 and p odd.

Proof of Lemma 7.10. For all d, we have inclusions

R_{0} ⊗ P_{C}H^{d}(G) ⊆ R_{d} ⊆ R̄_{0} ⊗ P_{C}H^{d}(G),

where P_{C}H^{d}(G) is regarded as an unstable module concentrated in degree 0. These are inclusions of unstable modules, enriched with compatible R_{0}–module structures and R̄_{0}–comodule structures. Call the category of such objects R_{0} − R̄_{0} − U.

Say that M ∈ R_{0} − R̄_{0} − U admits a nice filtration if it admits a filtration in R_{0} − R̄_{0} − U with subquotients all of the form Σ^{k}R_{0}. We will show that each R_{d} admits a nice filtration.

(That the composition factors will then also satisfy d + k < e(G) follows immediately from the fact that Q_{R_{0}}(Σ^{d}R_{∗}) is a graded object associated to Q_{A}H^{∗}(G), which we know is one dimensional in degree e(G) and zero above that.)

We claim that, if N admits a nice filtration, and M ⊆ N, then M also admits a nice filtration. To see this, suppose F_{0}N ⊆ F_{1}N ⊆ … is a filtration of N with F_{j}N/F_{j−1}N = Σ^{k_{j}}R_{0}. Let F_{j}M = M ∩F_{j}N. Then F_{j}M/F_{j−1}M ⊆ F_{j}N/F_{j−1}N = Σ^{k_{j}}R_{0} will be an inclusion of objects in R_{0}−R̄_{0}−U that will be split as R_{0}–modules, thanks to Corollary 5.4. We conclude that F_{j}M/F_{j−1}M is either 0 or Σ^{k_{j}}R_{0}.

Thus to prove R_{d} has a nice filtration, it suffices to prove that R̄_{0} ⊗ P_{C}H^{d}(G) has a nice filtration, or just that R̄_{0} has a nice filtration. We show this in the next two subsections, which separately deal with the cases p = 2 and p is odd. □

7.6. A calculation of d_{1}(R_{0}), and a nice filtration of R̄_{0}, when p = 2. Suppose that p = 2, and that

R_{0} = 𝔽_{2}[x_{1}^{2^{j_{1}}} , …, x_{1}^{2^{jc}}] ⊆ 𝔽_{2}[x_{1}, …, x_{c}] = R̄_{0},

with j_{1} ≥ ⋯ ≥ j_{c}. We show the following.

Lemma 7.12. R̄_{0} has a good filtration as an object in R_{0} − R̄_{0} − U.

Lemma 7.13. If j_{1} > 0, d_{1}(R_{0}) = 2^{j_{1}−1}.

Proof of Lemma 7.12. For all 1 ≤ b ≤ c and 1 ≤ i_{b} ≤ j_{b}, the module

R(i_{1}, …, i_{c}) = 𝔽_{2}[x_{1}^{2^{i_{1}}} , …, x_{c}^{2^{ic}}]

will be an object in R_{0} − R̄_{0} − U in the evident way.

Clearly R(j_{1}, …, j_{c}) = R_{0} admits a nice filtration. The short exact sequences

0 → R(i_{1}, …, i_{c}) → R(i_{1}, …, i_{b} − 1, …, i_{c}) → Σ^{2^{i_{b}}} R(i_{1}, …, i_{c}) → 0

then shows that if R(i_{1}, …, i_{c}) admits a nice filtration, so does R(i_{1}, …, i_{b} − 1, …, i_{c}). By downward induction, we conclude that R(0, …, 0) = R̄_{0} admits a nice filtration. □

Proof of Lemma 7.13. By Proposition 3.12(c), it suffices to prove that, if j > 0,

d_{1}(𝔽_{2}[x^{2^{j}}]) = 2^{j−1}.

In the short exact sequence

0 → 𝔽_{2}[x^{2^{j}}] → 𝔽_{2}[x^{2^{j−1}}] → Σ^{2^{j−1}}𝔽_{2}[x^{2^{j}}] → 0,

d_{1} of the middle term is strictly less than d_{0}(Σ^{2^{j−1}}𝔽_{2}[x^{2^{j}}]) = 2^{j−1}: this is clear when j = 1, and for larger j this follows by an inductive hypothesis. Thus Proposition 3.13(b) applies to say that d_{1}(𝔽_{2}[x^{2^{j}}]) = 2^{j−1}. □

7.7. A calculation of d_{1}(R_{0}), and a nice filtration of R̄_{0}, when p is odd. Suppose that p is odd. We can assume that

R_{0} = 𝔽_{p}[y_{1}^{p^{j_{1}}} , …, y_{1}^{p^{jc}}] ⊆ Λ^{∗}(x_{1}, …, x_{c}) ⊗ 𝔽_{p}[y_{1}, …, y_{c}] = R̄_{0},

with j_{1} ≥ ⋯ ≥ j_{c}, and we show the following.

Lemma 7.14. R̄_{0} has a good filtration as an object in R_{0} − R̄_{0} − U.

Lemma 7.15. If j_{1} = 0, d_{1}(R_{0}) = 1. If j_{1} > 0, d_{1}(R_{0}) = 2p^{j_{1}−1}.

Proof of Lemma 7.14. As a first step, we note that the filtration of R̄_{0} given by letting F_{k}R̄_{0} = Λ^{≤k}(x_{1}, …, x_{c})⊗𝔽_{p}[y_{1}, …, y_{c}] is a filtration in the category R_{0}−R̄_{0}− U, and the associated subquotients are direct sums of suspensions of 𝔽_{p}[y_{1}, …, y_{c}]. It follows that it suffices to prove the lemma with R̄_{0} replaced by 𝔽_{p}[y_{1}, …, y_{c}].

Our next reduction will allow us to reduce to the case when c = 1.

If K is a subHopf algebra of a Hopf algebra H, and both objects and all structure maps are in U, one has a category K − H − U, analogous to R_{0} − R̄_{0} − U. One can then say that M ∈ K − H − U has a good filtration if it has a filtration with subquotients that are all suspensions of K. It is easy to see that if M_{1} ∈ K_{1}−H_{1}−U and M_{2} ∈ K_{2} − H_{2} − U has a good filtration, then so does M_{1} ⊗ M_{2}, viewed as an object in K_{1} ⊗ K_{2} − H_{1} ⊗ H_{2} − U.

Applying this observation to the evident tensor decompositions of K = R_{0} and H = 𝔽_{p}[y_{1}, …, y_{c}], we are left just needing to show that 𝔽_{p}[y] has a nice filtration, when viewed as an object in 𝔽_{p}[y^{p^{j}}] − 𝔽_{p}[y] − U.

By downwards induction on i, we show that, for 0 ≤ i ≤ j, 𝔽_{p}[y^{p^{i}}] has a nice filtration, when viewed as an object in 𝔽_{p}[y^{p^{j}}] − 𝔽_{p}[y] − U. The case i = j is clear.

For the inductive step, we filter 𝔽_{p}[y^{p^{i}}]. For 0 ≤ r ≤ p − 1, define M(r) to be the span of {y^{p^{i}m} | m ≡ s mod p, for some 0 ≤ s ≤ r}. Using the formulae P^{k}y^{n} = \binom{n}{k}y^{n+k(p−1)} and Δ(y^{n}) = ∑_{k} \binom{n}{k}y^{k} ⊗ y^{n−k}, one easily checks that each M(r) is an object in 𝔽_{p}[y^{p^{j}}] − 𝔽_{p}[y] − U: if \binom{n}{k} ≢ 0 mod p, and n has the form p^{i}(pa + s) with 0 ≤ s ≤ r ≤ p − 1, then both n + k(p − 1) and k also have this form.

Thus we have a filtration in 𝔽_{p}[y^{p^{j}}] − 𝔽_{p}[y] − U:

𝔽_{p}[y^{p^{i+1}}] = M(0) ⊆ M(1) ⊆ ⋯ ⊆ M(p − 1) = 𝔽_{p}[y^{p^{i}}],

and we are assuming by induction that 𝔽_{p}[y^{p^{i+1}}] has a good filtration. Now one checks that M(r)/M(r − 1) ≃ Σ^{2p^{i}r}M(0) as objects in 𝔽_{p}[y^{p^{j}}] − 𝔽_{p}[y] − U, so by upwards induction on r we conclude that each M(r) has a good filtration. □

Proof of Lemma 7.15. By Proposition 3.12(c), it suffices to prove that d_{1}(𝔽_{p}[y]) = 1, and, if j > 0, d_{1}(𝔽_{p}[y^{p^{j}}]) = 2p^{j−1}.

Corollary 3.14 (or Proposition 3.13(b)), applied to the short exact sequence

0 → 𝔽_{p}[y] → Λ^{∗}(x) ⊗ 𝔽_{p}[y] → Σ𝔽_{p}[y] → 0,

shows that d_{1}(𝔽_{p}[y]) = d_{0}(Σ𝔽_{p}[y]) = 1.

If j > 1, we consider the short exact sequence:

0 → 𝔽_{p}[y^{p^{j}}] → 𝔽_{p}[y^{p^{j−1}}] → 𝔽_{p}[y^{p^{j−1}}]/𝔽_{p}[y^{p^{j}}] → 0.

We claim that d_{0} of the last term is 2p^{j−1}, which, by induction, will be strictly more than d_{1} of the middle term. Thus Proposition 3.13(b) applies to say that d_{1}(𝔽_{p}[y^{p^{j}}]) = 2p^{j−1}.

To verify the claim, one checks that the map 𝔽_{p}[y^{p^{j−1}}] → Σ^{2p^{j−1}}𝔽_{p}[y^{p^{j−1}}] sending y^{p^{j−1}n} to the 2p^{j−1}th suspension of ny^{p^{j−1}(n−1)} is a map of unstable A–modules, and thus induces an embedding 𝔽_{p}[y^{p^{j−1}}]/𝔽_{p}[y^{p^{j}}] ↪ Σ^{2p^{j−1}}𝔽_{p}[y^{p^{j−1}}] in U. Since the range of this embedding is the 2p^{j−1}th suspension of a reduced module, the same is true of the domain, which thus has d_{0} = 2p^{j−1}. □

7.8. d_{1}(G) when G has a p–central p–Sylow subgroup. Now suppose that G is not necessarily p–central, but has a p–central p–Sylow subgroup P. Here we show that then d_{1}(G) = d_{1}(P). As d_{1}(G) ≤ d_{1}(P) is always true, the point is to show that d_{1}(G) is as big as it could be.

Let e_{ω} ∈ 𝔽_{p}[Out(P)] be an idempotent chosen so that 𝔽_{p}[Out(P)]e_{ω} is the projective cover of the one dimensional module ω(P)^{−1}.

Lemma 7.16. d_{1}(G) = d_{1}(P) if and only if d_{1}(R_{e(P)}H^{∗}(G)) = h(P), and either of these equalities are implied by d_{1}(e_{ω}R_{0}H^{∗}(P)) = h(P).

Proof. As we proved that d_{1}(P) = e(P)+h(P), we showed that d_{1}(nil_{e(P)}H^{∗}(P)) = e(P) + h(P) and d_{1}(H^{∗}(P)/nil_{e(P)}H^{∗}(P)) < e(P) + h(P). This second fact implies that d_{1}(H^{∗}(G)/nil_{e(P)}H^{∗}(G)) < e(P) + h(P) also holds, as H^{∗}(G) is a direct summand of H^{∗}(P) in U. We conclude that d_{1}(G) = d_{1}(P) if and only if d_{1}(nil_{e(P)}H^{∗}(G)) = e(P) + h(P). As d_{1}(nil_{e(P)}H^{∗}(G)) = e(P) + d_{1}(R_{e(P)}H^{∗}(G)), we deduce that d_{1}(G) = d_{1}(P) if and only if d_{1}(R_{e(P)}H^{∗}(G)) = h(P).

Now reasoning as in §7.4, this last equality would follow if one could show that d_{1}(e_{ω}R_{0}H^{∗}(P)) = h(P). □

We now sketch a proof that d_{1}(e_{ω}R_{0}H^{∗}(P)) = h(P). This involves redoing the calculation that d_{1}(R_{0}H^{∗}(P)) = h(P) in a way that allows one to keep track of the Out(P)–action. Let C = C(P).

The 0–line of the spectral sequence associated to C → P → P/C is natural with respect to the action of Out(P). Thus the filtration studied in §6,

H^{∗}(C) = E_{2}^{0,∗} ⊃ E_{3}^{0,∗} ⊃ E_{2p+1}^{0,∗} ⊃ ⋯ ⊃ E_{2p^{k−1}+1}^{0,∗} ⊃ E_{2p^{k}+1}^{0,∗} = E_{∞}^{0,∗} = R_{0}H^{∗}(P),

is a filtration by unstable modules with an Out(P)–action.

Our work above shows that

d_{1}(E_{r}^{0,∗}) = \begin{cases}2p^{j−1} & if r = 2p^{j} + 1 with j ≥ 1 \\ 1 & if r = 3 \\ 0 & if r = 2. \end{cases}

We now suppose that p > 2 and k ≥ 1: the cases when p = 2 or when E_{∞}^{0,∗} equals E_{2}^{0,∗} or E_{3}^{0,∗} are similar and easier. Recall that then h(P) = 2p^{k−1}. Using Proposition 3.13 in the usual way, we conclude that d_{1}(e_{ω}R_{0}H^{∗}(P)) = h(P) if and only if d_{0}(e_{ω}B) = 2p^{k−1}, where B = E_{2p^{k−1}+1}^{0,∗}/E_{2p^{k}+1}^{0,∗}.

From §6, we see that

(7.1) B = Λ^{∗}(C_{0}^{#}) ⊗ S^{∗}(β(C_{1}^{#}) + Φβ(C_{2}^{#}) + ⋯ + Φ^{k−1}β(C_{k}^{#}))

⊗ S^{∗}(Φ^{k−1}β(C^{#}/C_{k}^{#}))/S^{∗}(Φ^{k}β(C^{#}/C_{k}^{#})),

where C_{0}^{#} ⊆ C_{1}^{#} ⊆ ⋯ ⊆ C_{k}^{#} ⊆ C^{#} is a filtration of C^{#} as an Out(P)–module. As an unstable module, B thus has the form

M ⊗ (S^{∗}(Φ^{k−1}β(V))/S^{∗}(Φ^{k}β(V))),

where M is reduced. Now one observes that S^{∗}(β(V))/S^{∗}(Φβ(V)) = Σ^{2}N where N is reduced. Thus

S^{∗}(Φ^{k−1}β(V))/S^{∗}(Φ^{k}β(V)) = Φ^{k−1}(S^{∗}(β(V))/S^{∗}(Φβ(V)))
= Φ^{k−1}(Σ^{2}N)
= Σ^{2p^{k−1}}(Φ^{k−1}N),

which is the 2p^{k−1}st suspension of a reduced module.

We conclude that d_{0}(e_{ω}B) = 2p^{k−1} if and only if e_{ω}B is nonzero.

The image of Out(P) → GL(C) lands in the parabolic subgroup GL(C, P) respecting the filtration of C, and the idempotent e_{ω} will project to a nonzero idempotent in 𝔽_{p}[GL(C, P)].

We claim that if e ∈ 𝔽_{p}[GL(C, P)] is any nonzero idempotent, then e acts nontrivially on B as described in (7.1). Equivalently, we claim that all irreducible 𝔽_{p}[GL(C, P)]–modules occur as composition factors in B.

To prove the claim, we note that all irreducible GL(C, P) modules will be pullbacks from the associated Levi factor (i.e. the product of ‘block diagonal’ GL(V_{j})’s), as the projection from the one to the other has kernel which is a p–group. This reduces us quickly to verifying the following lemma.

Lemma 7.17. Every irreducible 𝔽_{p}[GL(V)]–module occurs as a composition factor in S^{∗}(β(V))/S^{∗}(Φβ(V)).

Proof. It is well known that every such irreducible S occurs in S^{∗}(β(V)). Choosing an occurrence of lowest polynomial degree, it is clear that it will remain nonzero in the quotient S^{∗}(β(V))/S^{∗}(Φβ(V)). □

8. Central essential cohomology

Recall that Cess^{∗}(G) is defined to be the kernel of the restriction map

H^{∗}(G) → ∏_{\begin{matrix}C(G)<U \\ C(G)≠U \end{matrix}} H^{∗}(C_{G}(U)).

The invariants e′(G) and e′′(G) are then defined by letting

e′(G) = max {d | Q_{A}Cess^{d}(G) ≠ 0} ∪ {−1},

where A is a Duflot subalgebra of H^{∗}(G), and

e′′(G) = max {d | P_{C(G)}Cess^{d}(G) ≠ 0} ∪ {−1}.

In this section we study Cess^{∗}(G), e′(G), and e′′(G), and connect them to the invariant d_{0}(G).

8.1. The structure of Cess^{∗}(G). We begin by proving Theorem 2.12. Most of this theorem is restated in the following. We let C = C(G), as usual.

Proposition 8.1. If A is a Duflot subalgebra of H^{∗}(G), then the following hold.

(a) Cess^{∗}(G) is a free A–module.

(b) The composite P_{C}Cess^{∗}(G) ↪ Cess^{∗}(G) ↠ Q_{A}Cess^{∗}(G) is monic.

(c) The sequence 0 → Q_{A}Cess^{∗}(G) → Q_{A}H^{∗}(G) → ∏_{C(G)<U} Q_{A}H^{∗}(C_{G}(U)) is

exact.

Proof. It is convenient to let M_{1} = Cess^{∗}(G), M_{2} = H^{∗}(G), M_{3} = M_{2}/M_{1}, M_{4} = ∏_{C(G)<U} H^{∗}(C_{G}(U)), and M_{5} = M_{4}/M_{3}. We have short exact sequences of A–

modules

0 → M_{1} → M_{2} → M_{3} → 0,

and

0 → M_{3} → M_{4} → M_{5} → 0.

Statements (a) and (c) of the proposition follow if we verify that all the M_{j} are free A–modules. This we do by arguing as in [CTVZ, proof of Thm.12.3.3].

The homomorphisms C×C_{G}(V) → C_{G}(V) make each M_{j} into a H^{∗}(C) comodule and each M_{j} ⊗ H^{∗}(C) into an A–module such that the comodule structure map M_{j} → M_{j}⊗H^{∗}(C) is a map of A–modules, and the composite M_{j} → M_{j}⊗H^{∗}(C) → M_{j} is the identity. Thus M_{j} is a direct summand of M_{j} ⊗ H^{∗}(C) as an A–module, and we conclude that M_{j} is free if M_{j} ⊗ H^{∗}(C) is.

To show that M_{j} ⊗ H^{∗}(C) is a free A–module, we give it a decreasing A–module filtration by letting F^{n} = M_{j}^{≥n} ⊗ H^{∗}(C). Then each F^{n}/F^{n+1} = M_{j}^{n} ⊗ H^{∗}(C) is a direct sum of copies of H^{∗}(C), and so is a free A–module, and the freeness of M_{j} easily follows.

Finally, statement (b) follows from consideration of the commutative square

\xymatrix{P_{C}Cess^{∗}(G) \ar[d] \ar[r] & Q_{A}Cess^{∗}(G) \ar[d] \\ P_{C}H^{∗}(G) \ar[r] & Q_{A}H^{∗}(G).}

The left map is clearly monic, and Theorem 2.1 says that the bottom map is also. Thus so is the top map. □

To finish the proof of Theorem 2.12, it remains to show that Cess^{∗}(G) is finitely generated as an A–module (with the corollary that e′(G) and e′′(G) are well defined finite numbers). As A has Krull dimension equal to the rank of C, and Cess^{∗}(G) is a free A–module, it is equivalent to prove the next result.

Proposition 8.2. The Krull dimension of Cess^{∗}(G) is at most the rank of C.

Proof. The proposition follows from a result of Carlson. It is convenient to let R = H^{∗}(G), and I = Cess^{∗}(G). By definition, the Krull dimension of the R– module I is the Krull dimension of the algebra R/Ann(I).

Let J be the image of

∑_{\begin{matrix}C<U \\ C≠U \end{matrix}} Ind_{C_{G}(U)}^{G} : ⨁_{\begin{matrix}C<U \\ C≠U \end{matrix}} H^{∗}(C_{G}(U)) → H^{∗}(G).

By standard arguments^{16}, J ⊂ Ann(I). Thus R/J → R/Ann(I) is a surjection, and so the Krull dimension of R/Ann(I) is at most the Krull dimension of R/J.

In the notation of [Ca1], J = J_{c+1}, where c is the rank of C. Then [Ca1, Cor.2.2] says that the Krull dimension of R/J is at most c. □

The next proposition is easily verified.

Proposition 8.3. Cess^{∗}(G×H) is naturally isomorphic to Cess^{∗}(G)⊗Cess^{∗}(H). Thus e′(G × H) = e′(G) + e′(H) and e′′(G × H) = e′′(G) + e′′(H).

We end this subsection by proving Proposition 2.16, which we recall here.

Proposition 8.4. Assuming that V = C(C_{G}(V)), Cess^{∗}(C_{G}(V)) = 0 unless the rank of V is at least equal to the depth of H^{∗}(G).

Proof. Let r(U) denote the p–rank of an elementary abelian p–group U, and let d be the depth of H^{∗}(G). Suppose that V = C(C_{G}(V)) and r(V) < d. We wish to show that Cess ∗ (C_{G}(V)) = 0. Note that, if V < U, then C_{C_{G}(V)}(U) = C_{G}(U). Thus Cess^{∗}(C_{G}(V)) is the kernel of the restriction map

H^{∗}(C_{G}(V)) → ∏_{\begin{matrix}V <U<C_{G}(V) \\ V ≠U \end{matrix}} H^{∗}(C_{G}(U)).

The kernel of this is contained in the kernel of

f : H^{∗}(C_{G}(V)) → ∏_{\begin{matrix}V <U<C_{G}(V) \\ r(U)=d \end{matrix}} H^{∗}(C_{G}(U)).

16This follows immediately from the fact that Ind_{C_{G}(U)}^{G} : H^{∗}(C_{G}(U)) → H^{∗}(G) is a map of H^{∗}(G)–modules.

Thus it suffices to show that f is monic.

Meanwhile, Carlson’s theorem [Ca1] implies that the product of restriction maps

g : H^{∗}(G) → ∏_{\begin{matrix}U<G \\ r(U)=d \end{matrix}} H^{∗}(C_{G}(U))

is monic.

There is a commutative diagram

\xymatrix{∐_{V <U<C_{G}(V), r(U)=d} V × C_{G}(U) \ar[d] \ar[r] & V × C_{G}(V) \ar[d] \\ ∐_{U<G, r(U)=d} C_{G}(U) \ar[r] & G.}

This induces a commutative diagram in U

\xymatrix{H^{∗}(V) ⊗ H^{∗}(C_{G}(V)) \ar[d] \ar[r]^{1⊗f} & ∏_{V <U<C_{G}(V), r(U)=d} H^{∗}(V) ⊗ H^{∗}(C_{G}(U)) \ar[d] \\ H^{∗}(G) \ar[r]^{g} & ∏_{U<G, r(U)=d} H^{∗}(C_{G}(U)).}

Adjointing, we get a commutative diagram

\xymatrix{H^{∗}(C_{G}(V)) \ar[d] \ar[r]^{f} & ∏_{V <U<C_{G}(V), r(U)=d} H^{∗}(C_{G}(U)) \ar[d] \\ T_{V} H^{∗}(G) \ar[r]^{T_{V} g} & ∏_{U<G, r(U)=d} T_{V} H^{∗}(C_{G}(U)).}

The left vertical map here is an inclusion, and the exactness of T_{V} shows that T_{V} g is monic. We conclude that f is also monic. □

8.2. Proof of Theorem 2.14. Theorem 2.14 says that

d_{0}(G) = max{e′′(C_{G}(V)) | V < G}.

In the right hand side of this equation, one can restrict to subgroups V such that V = C(C_{G}(V)), because if V < U and U is central in C_{G}(V), then C_{G}(V) = C_{G}(U), so that e′′(C_{G}(V)) = e′′(C_{G}(U)). Thus Theorem 2.14 will follow from the next theorem.

Theorem 8.5. R̄_{d}H^{∗}(G) ≠ 0 if and only if P_{V} Cess^{d}(C_{G}(V)) ≠ 0 for some V < G satisfying V = C(C_{G}(V)).

As we begin the proof of this, it is convenient to let, for V < G,

Ess^{∗}(V) = ker{P_{V} H^{∗}(C_{G}(V)) → ∏_{V <U} P_{V} H^{∗}(C_{G}(U))},

with the product over all U that are strictly bigger than V. One easily verifies that

Ess^{∗}(V) = \begin{cases}P_{V} Cess^{d}(C_{G}(V)) & if V = C(C_{G}(V)) \\ 0 & otherwise. \end{cases}

Thus we wish to show that R̄_{d}H^{∗}(G) = 0 if and only if Ess^{d}(V) = 0 for all V < G.

We prove the ‘only if’ implication first. Let c_{V} ∈ 𝔽_{p}[GL(V)] be the top Dickson invariant, and let W_{G}(V) = N_{G}(V)/V. Then [K3, Lemma 7.8] (proved by using the formula in this paper’s Proposition 4.6) says that, for all V, there is an embedding

(c_{V} H^{∗}(V) ⊗ Ess^{d}(V))^{W_{G}(V)} ⊆ R̄_{d}H^{∗}(G).

Thus R̄_{d}H^{∗}(G) = 0 implies that (c_{V} H^{∗}(V) ⊗ Ess^{d}(V))^{W_{G}(V)} = 0 for all V. But then Ess^{d}(V) = 0, by the next lemma.

Lemma 8.6. If W < GL(V) and M is an 𝔽_{p}[W]–module, then [c_{V} H^{∗}(V)⊗M]^{W} = 0 implies that M = 0.

Proof. It is well known (see, e.g. [Alp, p.45]) that 𝔽_{p}[W] embeds in H^{∗}(V) as 𝔽_{p}[W]–modules, and thus in c_{V} H^{∗}(V), as multiplication by c_{V} is a monic GL(V)– module map. Thus (𝔽_{p}[W]⊗M)^{W} embeds in (c_{V} H^{∗}(V)⊗M)^{W}. But 𝔽_{p}[W]⊗M ≃ 𝔽_{p}[W] ⊗ M_{triv} as 𝔽_{p}[W]–modules, where M_{triv} denotes M with trivial W –action. Finally, (𝔽_{p}[W] ⊗ M_{triv})^{W} ≃ M as 𝔽_{p}–vector spaces. Putting this all together, we have shown that M embeds in (c_{V} H^{∗}(V) ⊗ M)^{W}, and the lemma follows. □

Returning to the proof of the theorem, we now assume that Ess^{d}(V) = 0 for all V, and deduce that R̄_{d}H^{∗}(G) = 0. Using the formula in Proposition 4.6, given

x = (x_{V}) ∈ R̄_{d}H^{∗}(G) ⊆ ∏_{V} H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(V)),

we show that each component x_{V} of x is zero by downwards induction on V.

So assume that x_{U} = 0 for all V < U. Proposition 4.6 tells us that, under the restriction map

H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(V)) → ∏_{V <U} H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(U)),

x_{V} will have the same image as (x_{U}) under the map

∏_{V <U} H^{∗}(U) ⊗ P_{U}H^{d}(C_{G}(U)) → ∏_{V <U} H^{∗}(V) ⊗ P_{V} H^{d}(C_{G}(U)).

Since the latter is zero by inductive assumption, we conclude that x_{V} ∈ H^{∗}(V) ⊗ Ess^{d}(V), and is thus zero also.

Remark 8.7. Note that Theorem 8.5 includes a second proof that e′′(G) is a well defined finite number.

8.3. The Depth Conjecture, the Regularity Conjecture, and a bound on e′(G). By Theorem 2.12, e′′(G) ≤ e′(G). Thus we get the bound

d_{0}(G) ≤ max{e′(C_{G}(V)) | V < G}.

If G is p–central, we know that e′′(G) = e′(G) = e(G). Here we discuss work towards Conjecture 2.18 which said that, if G is not p–central, then e′(G) < e(G). In particular, we show that this is true if the p–rank of G is no more than 2 more than the rank of C(G), and link the general conjecture to Benson’s Regularity Conjecture. Enroute, a similar argument will also prove Theorem 2.13, the special case of Carlson’s Depth Conjecture in which the depth of H^{∗}(G) is as small as possible.

Thanks to Proposition 8.3, it suffices to prove either Theorem 2.13 or Conjecture 2.18 in the special case when G has no direct summands isomorphic to ℤ/p. We will assume this. As a consequence, even in the odd prime case, a Duflot subalgebra A of H^{∗}(G) will have the form

A = 𝔽_{p}[ξ_{1}, …, ξ_{c}],

where c is the rank of C = C(G) and e(G) = ∑_{j=1}^{c}(|ξ_{j}| − 1). The sequence ξ_{1}, …, ξ_{c}

is a Duflot sequence: the sequence restricts to a regular sequence in H^{∗}(C). We let r be the p–rank of G.

Lemma 8.8. Suppose c < r. Given any Duflot sequence ξ_{1}, …, ξ_{c} ∈ H^{∗}(G), there exists ξ ∈ H^{∗}(G) such that, for all proper inclusions C < V, ξ_{1}, …, ξ_{c}, ξ restricts to a regular sequence in H^{∗}(C_{G}(V)).

Proof. Let n be the rank of G/C (so n > r − c), and let ρ be the regular representation of G/C. For 1 ≤ i ≤ r − c, let κ̄_{i} ∈ H^{2(p^{n}−p^{n−i})}(G/C) be the (p^{n} − p^{n−i})th Chern class of ρ, and then let κ_{i} = Inf_{G/C}^{G}(κ̄_{i}) ∈ H^{∗}(G). It is easy to check that ξ_{1}, …, ξ_{c}, κ_{1}, …, κ_{r−c} is a polarized system of parameters in the sense of [G1, Def.2.2]. It follows that the element ξ = κ_{1} satisfies the conclusion of the lemma. □

Proposition 8.9. Suppose c < r and A = 𝔽_{p}[ξ_{1}, …, ξ_{c}] is a Duflot algebra of H^{∗}(G). The following are equivalent for a fixed integer e ≥ 0.

(a) e′(G) < e.

(b) With ξ as in the lemma, the kernel of multiplication by ξ,

ker{ξ· : H^{d}(G)/(ξ_{1}, … ξ_{c}) → H^{d+|ξ|}(G)/(ξ_{1}, … ξ_{c})},

is zero for all d ≥ e.

(c) ⋂_{ξ∈H̃^{∗}(G)} ker{ξ· : H^{d}(G)/(ξ_{1}, … ξ_{c}) → H^{d+|ξ|}(G)/(ξ_{1}, … ξ_{c})}

is zero for all d ≥ e.

Proof. For each d and ξ ∈ H^{∗}(G), we have a commutative diagram

(8.1)  \xymatrix{H^{d}(G)/(ξ_{1}, … ξ_{c}) \ar[d]^{ξ·} \ar[r]^{f(d)} & ⨁_{C<V} H^{d}(C_{G}(V))/(ξ_{1}, … ξ_{c}) \ar[d]^{ξ·} \\ H^{d+|ξ|}(G)/(ξ_{1}, … ξ_{c}) \ar[r]^{f(d+|ξ|)} & ⨁_{C<V} H^{d+|ξ|}(C_{G}(V))/(ξ_{1}, … ξ_{c}),}

where f(d) is induced by the evident restriction maps.

By Theorem 2.12, f(d) is monic for all large d, and e′(G) is the largest d such that f(d) is not monic. Thus statement (a) is equivalent to the statement that ker f(d) is zero for all d ≥ e.

We show that statement (a) implies statement (b). Thus suppose that ker f(d) is zero for all d ≥ e, and let ξ ∈ H^{∗}(G) be as in the lemma. Then, in diagram (8.1), the right map is monic for all d, and the top map is monic for all d ≥ e. Thus the left map is monic in the same range.

Statement (b) obviously implies statement (c).

Finally, we show that statement (c) implies statement (a). Assuming statement (c), we prove, by downwards induction on d, that f(d) is monic. Thus assume f(d′) is monic for all d′ > d ≥ e. Given 0 ≠ κ ∈ H^{d}(G)/(ξ_{1}, … ξ_{c}), we need to show that f(d)(κ) ≠ 0. By (c), there exists ξ ∈ H̃^{∗}(G) such that ξ · κ ≠ 0. As f(d + |ξ|) is monic by inductive assumption, f(d + |ξ|)(ξ · κ) ≠ 0. But this equals ξ · f(d)(κ), and so f(d)(κ) ≠ 0. □

Proof of Theorem 2.13. With notation as in the proposition just proved, we wish to prove that Cess^{∗}(G) = 0 if and only if the depth of H^{∗}(G) is greater than c. Thanks to Theorem 2.12, Cess^{∗}(G) = 0 if and only if statement (a) of the last proposition holds when e = 0. But then statement (b) is true with e = 0, and thus the depth of H^{∗}(G) is at least c + 1.

Conversely, if the depth of H^{∗}(G) is at least c + 1, there exists a ξ such that ξ_{1}, …, ξ_{c}, ξ is a regular sequence on H^{∗}(G), and so statement (c) certainly holds with e = 0. Thus statement (a) does as well. □

Now we study statements (b) and (c) of the last proposition, using work by Carlson and Benson.

Proposition 8.10. If r − c = 1, then Q_{A}Cess^{∗}(G) satisfies Poincaré duality with duality degree equal to e(G). In other words, the Poincaré polynomial p_{Q_{A}Cess^{∗}(G)}(t) satisfies

p_{Q_{A}Cess^{∗}(G)}(t) = t^{e(G)}p_{Q_{A}Cess^{∗}(G)}(1/t).

Proof. The conclusion of the proposition is obvious if Cess^{∗}(G) = 0, so we can assume that the depth of H^{∗}(G) is precisely c. Let ξ_{1}, …, ξ_{c} be as in Proposition 8.9, and choose ξ as in the lemma. Replacing ξ by a large power of itself, if necessary, we can assume that, in diagram (8.1), f(d + |ξ|) is monic for all d. Thus Q_{A}Cess^{∗}(G), the kernel of the top map in (8.1), identifies with the kernel of multiplication by ξ, the left map in (8.1). But a careful reading of [BC2, Lemma 3.2 and its proof] reveals that the Poincaré series of this kernel is precisely the polynomial called ‘p_{r}(t)’ there, and then [BC2, Theorem 3.9] says that the functional equation of the proposition holds. □

Corollary 8.11. If r − c = 1, then

e′(G) = e(G) − min{d | Cess^{d}(G) ≠ 0} < e(G).

To state what we know about the situation when r −c > 1, we need to introduce local cohomology. If I is a homogeneous ideal in a graded ring R, and M is a graded R–module, H_{I}^{0,∗}(M) is defined to be the I–torsion in M, i.e. the set of x ∈ M such that I^{k}x = 0 for some k. This is a left exact functor of M, and H_{I}^{d,∗}(M) is defined to be the associated dth right derived functor.

In [Be], Benson conjectured

Conjecture 8.12 (Strong Regularity Conjecture).

H_{H̃^{∗}(G)}^{i,j}(H^{∗}(G)) = 0 for \begin{cases}j ≥ −i & if c ≤ i < r \\ j > −i & if i = r. \end{cases}

Proof of Proposition 2.19. This proposition asserted that, for a fixed finite group G, Conjecture 2.18 is implied by the Strong Regularity Conjecture. q

2

1  u, v xv
0 1 x, y
0 1 2  p
Figure 1. E_{3}^{p,q} = E_{∞}^{p,q} modulo (a^{2}, b^{2}, c^{2})

In the terminology of [Be], if the Strong Regularity Conjecture holds, then, by [Be, Thm.4.5], every filter regular sequence is of type beginning with the sequence (−1, −2, …, −(c + 1)). In particular, with ξ as in statement (b) of Proposition 8.9, the sequence ξ_{1}, …, ξ_{c}, ξ is the beginning of such a sequence. From the definition of filter regular, we see that statement (b) of Proposition 8.9 thus holds with e = e(G). □

As mentioned in the introduction, in [Be], Benson shows that his conjecture is true if r − c ≤ 2. I have my own ‘heuristic’ proof of statement (c) with e = e(G) under the same condition, and the failure of the method to go beyond r − c ≤ 2 makes one wonder if a counterexample to both of our conjectures is lurking among the groups of order 128 or 256.

Remark 8.13. Slightly milder than the Strong Regularity Conjecture is Benson’s Regularity Conjecture, which asserts that the Castelnuovo–Mumford regularity of H^{∗}(G) is precisely 0. In terms of local cohomology, this is the statement that H_{H̃^{∗}(G)}^{i,j}(H^{∗}(G)) = 0 for j > −i. For our purposes, this is enough to deduce that e′(G) ≤ e(G).

9. Examples

Example 9.1. Let W (2) be the universal 2–central group whose quotient by its center C is V_{2} = (Z/2)^{2}. Thus there is a central extension

H_{2}(V_{2}; 𝔽_{2}) ⟶^{i} W (2) ⟶^{q} V_{2},

where C = H_{2}(V_{2}; 𝔽_{2}) ≃ (ℤ/2)^{3}. In terms of Hall–Senior numbering, and thus also the numbering in [CTVZ], W (2) is 32#18.

In the associated spectral sequence, one has that

E_{2}^{∗,∗} = 𝔽_{2}[x, y, a, b, c],

with a, b, c ∈ E_{2}^{0,1} and x, y ∈ E_{2}^{1,0}, and d_{2}(a) = x^{2}, d_{2}(b) = xy, and d_{2}(c) = y^{2}.

As E_{3}^{∗,0} = 𝔽_{2}[x, y]/(x^{2}, xy, y^{2}), it follows that a^{2}, b^{2}, and c^{2} must be permanent cycles. We conclude that W (2) will have type [2, 2, 2] so that d_{0}(W (2)) = e(W (2)) = 3 and d_{1}(W (2)) = 4.

With a bit more work, one can show that E_{3}^{∗,∗}/(a^{2}, b^{2}, c^{2}) is six dimensional with generators as indicated in Figure 1, where u and v are respectively represented by bx + ay and cx + by. This is a Poincaré duality algebra with relations x^{2} = xy = y^{2} = u^{2} = v^{2} = uv = xu = yv = xv + yu = 0.

It follows that E_{3}^{∗,∗} = E_{∞}^{∗,∗}, and then that

H^{∗}(W (2)) ≃ 𝔽_{2}[α, β, γ, x, y, u, v]/(x^{2}, xy, y^{2}, u^{2}, uv, v^{2}, xu, yv, xv + yu), with x, y ∈ H^{1} and α, β, γ, u, v ∈ H^{2}. Here α, β, and γ are represented by a^{2}, b^{2}, and c^{2} in the spectral sequence.

The polynomial subalgebra A = 𝔽_{2}[α, β, γ] is a Duflot subalgebra. With respect to the H^{∗}(C) = 𝔽_{2}[a, b, c] comodule structure, the elements 1, x, y are in the image of the inflation map q^{∗} and so are primitive. The top class xv is not in the image of inflation, but is primitive, by our general theory. The elements u and v are not primitive, as m^{∗}(u) = 1 ⊗ u + b ⊗ x + a ⊗ y and m^{∗}(v) = 1 ⊗ v + c ⊗ x + b ⊗ y in H^{∗}(C) ⊗ E_{∞}^{∗,∗}. Thus each of the inclusions

im(q^{∗}) ↪ P_{C}H^{∗}(W (2)) ↪ Q_{A}H^{∗}(W (2))

is proper.

The nilpotent filtration works as follows.

R_{0} = H^{∗}(W (2))/(x, y, u, v) ≃ 𝔽_{2}[α, β, γ] and R̄_{0} = 𝔽_{2}[a, b, c]. The embedding R_{0} ⊂ R̄_{0} sends α to a^{2}, β to b^{2}, and γ to c^{2}.

R_{1} is the free 𝔽_{2}[α, β, γ]–module on generators x̄, ȳ, ū, v̄ of respective degrees 0,0,1,1, and R̄_{1} is the free 𝔽_{2}[a, b, c]–module on x̄, ȳ. The embedding R_{1} ⊂ R̄_{1} sends ū to bx̄ + aȳ and v̄ to cx̄ + bȳ. Thus Sq^{1}(ū) = βx̄ + αȳ and Sq^{1}(v̄) = γx̄ + βȳ.

R_{2} and R̄_{2} are both 0, as P_{C}H^{2}(W (2)) = 0.

R_{3} is the free 𝔽_{2}[α, β, γ]–module on a single generator x̄v̄ of degree 0, and R̄_{3} is the free 𝔽_{2}[a, b, c]–module on this same element.

Finally, H^{∗}(W (2))_{LF} ⊂ H^{∗}(W (2)) is the algebra spanned by 1, x, y, xv. All nontrivial products and Steenrod operations are zero.

Example 9.2. Let G be the group of order 64 with Hall–Senior number #108. Using information from [CTVZ], we analyzed H^{∗}(G) in detail for other purposes in [K3]. Here we summarize relevant bits to illustrate how one can calculate H^{∗}(G)_{LF} and R̄_{d}H^{∗}(G) by using Proposition 4.7 and Proposition 4.6.

The commutator subgroup Z = [G, G] has order 2. The center C is elementary abelian of rank 2, and C = Φ(G), so Z < C and G/C is elementary abelian of rank 4. There is a unique maximal elementary abelian group V of rank 3, and its centralizer K has order 32, so that N_{G}(V)/C_{G}(V) = G/K ≃ ℤ/2. More precisely, K is isomorphic to (ℤ/2)^{2} × Q_{8}, with Q_{8} embedded so that V ∩ Q_{8} = Z.

We have the following picture of A^{C}(G):

[figure: C; V; ℤ/2]

and from this it is already clear that R̄_{0}H^{∗}(G) = H^{∗}(V)^{ℤ/2}.

We have maps of unstable algebras equipped with Aut(G) action:

P_{V} H^{∗}(K) ↪ P_{C}H^{∗}(K) ⟵^{j^{∗}} P_{C}H^{∗}(G),

where j : K → G is the inclusion. It is easily checked that j^{∗} is onto in degree 1.

The maps of pairs (Q_{8}, Z) → ((ℤ/2)^{2} × Q_{8}, (ℤ/2)^{2} × Z) = (K, V) induces an isomorphism of algebras:

P_{V} H^{∗}(K) ≃ P_{Z}H^{∗}(Q_{8}).

The algebra P_{Z}H^{∗}(Q_{8}) is familiar: the calculation of H^{∗}(Q_{8}) using the Serre spectral sequence associated to Z → Q_{8} → Q_{8}/Z reveals that P_{Z}H^{∗}(Q_{8}) = Im {H^{∗}(Q_{8}/Z) → H^{∗}(Q_{8})} = B^{∗}, where B^{∗} is the Poincaré duality algebra 𝔽_{2}[x, w]/(x^{2} + xw + w^{2}, x^{2}w + xw^{2}), where x and w both have degree 1. B^{∗} has dimension 1,2,2,1 in degrees 0,1,2,3.

From this we learn that P_{C}H^{∗}(K) ≃ B^{∗}[y] where y is also in degree 1, and thus is generated by elements in degree 1. It follows that j^{∗} : P_{C}H^{∗}(G) → P_{C}H^{∗}(K) is onto, and then that Inn(G) acts trivially on both P_{V} H^{∗}(K) and P_{C}H^{∗}(K).

Proposition 4.7 then tells us that there is a pullback diagram of unstable algebras:

\xymatrix{H^{∗}(G)_{LF} \ar[d] \ar[r] & P_{C}H^{∗}(G) \ar[d]^{j^{∗}} \\ P_{V} H^{d}(K) \ar[r] & P_{C}H^{∗}(K).}

Similarly, Proposition 4.6 tells us that, for all d, there is a pullback diagram of unstable modules:

\xymatrix{R̄_{d}H^{∗}(G) \ar[d] \ar[r] & H^{∗}(C) ⊗ P_{C}H^{d}(G) \ar[d]^{1⊗j^{∗}} \\ H^{∗}(V)^{ℤ/2} ⊗ P_{V} H^{d}(K) \ar[r] & H^{∗}(C) ⊗ P_{C}H^{d}(K).}

Note that the kernel of j^{∗} : H^{∗}(G) → H^{∗}(K) is precisely Cess^{∗}(G), which is described in [CTVZ]. In our terminology, we learn that a Duflot subalgebra A is polynomial on classes of degree 2 and 8 (so G has type [8, 2]), and Q_{A}Cess^{∗}(G) is a graded vector space of dimension 1,3,5,6,5,3,1 in degrees 1,2,3,4,5,6,7. Note that this evident Poincaré duality is predicted by Proposition 8.10.

Q_{A}Cess^{∗}(G) has a basis in which every element is a product of 1 dimensional classes, and thus P_{C}Cess^{∗}(G) = Q_{A}Cess^{∗}(G). In [K3, Prop.10.2], we further showed that

P_{C}Cess^{∗}(G) ≃ ΣB^{∗}[y]/(y^{4})

as unstable modules.

It follows that there are short exact sequences in U:

0 → ΣB^{∗}[y]/(y^{4}) → H^{∗}(G)_{LF} → B^{∗} → 0,

and

0 → H^{∗}(C) ⊗ [ΣB^{∗}[y]/(y^{4})]^{d} → R̄_{d}H^{∗}(G) → H^{∗}(V)^{ℤ/2} ⊗ B^{d} → 0.

Furthermore, d_{0}(G) = e′′(G) = e′(G) = 7, and e(G) = 8.

Example 9.3. One can often determine e(G) using minimal information about the extension class τ^{∗} : C^{∗} → H^{2}(G/C) (where C = C(G)), and in situations where H^{∗}(G) has yet to be calculated.

For example, suppose that p = 2 and G has no ℤ/2 direct summands (so that τ^{∗} is monic). If the image of τ^{∗} has a basis consisting of products of 1 dimensional classes, then G has type [2, …, 2] and so e(G) equals the rank of C. To see this, we note that, if d_{2}(a) = xy, then

d_{3}(a^{2}) = d_{3}(Sq^{1}a) = Sq^{1}(d_{2}(a)) = Sq^{1}(xy) = x^{2}y + xy^{2} ≡ 0 mod (xy).

This criterion holds for the important family of groups studied in [AKM]. There the authors associate a 2-central Galois group G_{𝔽} to every field 𝔽 of characteristic different from 2 that is not formally real. They call this group a W –group due to its connections to the Witt ring W 𝔽 [MiS]. Thus d_{0}(G_{𝔽}) = e(G_{𝔽}) = r and q

8  2

7

6  6 8 8 8 4 1

5

4  8 12 8 7 4

3

2  4 7 8 8 8

1

0 1 4 8 10 8 6
0 1 2 3 4 5 6 7 8  p
Figure 2. The dimension of E_{∞}^{p,q} modulo (a^{8}, b^{8})

d_{1}(G_{𝔽}) = r+1, where G_{𝔽} has rank r. Included among these groups are the universal W –groups W (n), the 2–central group with extension sequence

H_{2}((ℤ/2)^{n}; 𝔽_{2}) → W (n) → (ℤ/2)^{n}.

Thus d_{0}(W (n)) = \binom{n+1}{2} and d_{1}(W (n)) = \binom{n+1}{2} + 1.

At odd primes p, analogous criteria exist, ensuring that G is p–central of type [2, …, 2]. Interesting families of such groups were studied by Browder–Pakianathan [BP] and Adem–Pakianathan [AP]. Included among these are the universal groups W (n, p), with extension sequence

H_{2}((ℤ/p)^{n}; 𝔽_{p}) → W (n, p) → (ℤ/p)^{n}.

Thus d_{0}(W (n, p)) = \binom{n+1}{2} and d_{1}(W (n, p)) = \binom{n+1}{2} + 1.

Example 9.4. Compared to the families in the last example, at the other extreme among 2–central 2–groups is the 2–Sylow subgroup P of the simple group SU(3, 4). This group has order 64 and Hall–Senior number #187. Its center C is elementary abelian of rank 2. In [G2], Green analyzed the associated spectral sequence^{17}. In particular, P has type [8, 8], so that d_{0}(P) = e(P) = 14 and d_{1}(P) = 18, and the analogue of Figure 1 is the impressively complex Figure 2 (reproduced from [G2]).

In spite of this complexity, it is interesting to note that one can get the bound e(P) ≤ 14 quite easily, by using representation theory and characteristic classes.

We thank David Green for the following description of some complex representations of P. Let H^{∗}(C) = 𝔽_{2}[a, b], and then let ρ_{a} and ρ_{b} be the 1–dimensional complex representations of C with respective total Stiefel–Whitney classes w(ρ_{a}) = 1 + a^{2}, w(ρ_{b}) = 1 + b^{2}. These representations extend to 1-dimensional representations ρ̃_{a} and ρ̃_{b} of subgroups Q_{a} and Q_{b} of index 4 in P. Let ω_{a} and ω_{b} be the 4 dimensional representations one gets by inducing ρ̃_{a} and ρ̃_{b} up to P: these turn out to be irreducible.

By construction Res_{C}^{P}(ω_{a}) = 4ρ_{a} and Res_{C}^{P}(ω_{b}) = 4ρ_{b}. It follows that the total Stiefel–Whitney classes of ω_{a} and ω_{b} restrict to (1 + a^{2})^{4} = 1 + a^{8} and (1 + b^{2})^{4} = 1 + b^{8} in H^{∗}(C). Thus im Res_{C}^{P} contains 𝔽_{2}[a^{8}, b^{8}] and so e(P) ≤ 14 must hold.

17A key simplification comes by computing with 𝔽_{4} coefficients rather than 𝔽_{2} coefficients.

Alternatively, one can just use the single 8 dimensional representation ω_{a} ⊕ ω_{b}. This is faithful, as it is faithful when restricted to C, the subgroup of all elements of order 2. It has characteristic classes that restrict to a^{8}+b^{8} and a^{8}b^{8} in H^{∗}(C). From this, one can formally deduce that the special Hopf algebra im Res_{C}^{P} must contain 𝔽_{2}[a^{8}, b^{8}], so that d_{0}(P) ≤ 14 and d_{1}(P) ≤ 18. By contrast, the estimate of Henn, Lannes, and Schwartz in [HLS1] just lets one conclude that d_{0}(P) ≤ 64 and d_{1}(P) ≤ 120 if one knows that P has a faithful 8 dimensional complex representation. This suggests that there might be some general bounds for d_{s}(G) for an arbitrary group G, determined by the dimensions of its faithful representations, that are much better than those in [HLS1].

Appendix A. Tables of group invariants

Here are various tables of some of our invariants for 2–groups of order dividing 64. The tables were compiled by hand using the calculations in [CTVZ] and the website version [Ca2]. The type of a group G, and thus e(G) and h(G), can be deduced by inspecting the description of restriction to maximal elementary abelian subgroups; this is particularly easy when G is 2–central. If G is not 2–central, one can immediately determine if Cess^{∗}(G) ≠ 0, since both the rank of Z(G) and the depth of H^{∗}(G) are given, and then read off the number e′(G) from the description of depth essential cohomology. The website source allows one to identify centralizers of elementary abelian subgroups as needed.

We say a group is indecomposable if it cannot be written as a nontrivial direct product of two subgroups. The numbering of groups is as in [CTVZ] which follows the Hall–Senior numbering [HS].

In Tables 1 and 2, recall that, since G is 2–central, d_{0}(G) = e(G) = e′(G) = e′′(G), and d_{1}(G) = e(G) + h(G).

In Table 3, ‘2’ means ℤ/2, etc. To compute d_{0}(G), we needed to observe that, in all cases covered by this table, e′′(G) = e′(G). Except when G is 32#41, this can be checked by noticing that elements in the top degree in Q_{A}Cess^{∗}(G) are represented by classes in the image of Inf_{G/C}^{G}, and so are primitive. When G is 32#41, elements in the Q_{A}Cess^{5}(G) are represented by essential classes of lowest degree, and so are primitive.

Table 1: Indecomposable, 2–central, 2–groups of order ≤ 32

Order # Type d_{0}(G) d_{1}(G) Notes
2  1  [1]  0  0  ℤ/2
4  2  [2]  1  2  ℤ/4
8  3  [2]  1  2  ℤ/8
5  [4]  3  5  Q_{8}
16  5  [2]  1  2  ℤ/16
14 [4]  3  5  Q_{16}
32 18 [2,2,2]  3  4
19 [2,2]  2  3
21 [2,2]  2  3
28 [4,2]  4  6
29 [2,2]  2  3
30 [2,2]  2  3
35 [4,2]  4  6
40 [4,4]  6  8
51 [4]  3  5  Q_{32}

GROUP COHOMOLOGY PRIMITIVES 47

Table 2: Indecomposable, 2–central, groups of order 64

# Type d_{0}(G) d_{1}(G)  Notes
11  [2]  1  2  ℤ/64
30 [2,2,2]  3  4
37 [2,2,2]  3  4
38 [2,2]  2  3
39 [2,2]  2  3
41 [2,2]  2  3
59 [2,2,2]  3  4
63 [4,2]  4  6
64 [2,2]  2  3
65 [2,2]  2  3
82 [2,2,2]  3  4
87 [4,2,2]  5  7
88 [2,2,2]  3  4
90 [2,2,2]  3  4
92 [4,2,2]  5  7
93 [2,2,2]  3  4
101 [4,4]  6  8
119 [4,2]  4  6
139 [4,2]  4  6
140 [2,2]  2  3
141 [2,2]  2  3
145 [4,2,2]  5  7
149 [4,2,2]  5  7
152 [4,2,2]  5  7
153 [4,4,4]  9  11  2–Sylow of Sz(8)
162 [4,4]  6  8
187 [8,8]  14  18 2–Sylow of U_{3}(𝔽_{4})
190 [4,2]  4  6
191 [4,4]  6  8
192 [4,2]  4  6
194 [4,4]  6  8
199 [4,4]  6  8
210 [4,4]  6  8
211 [4,2]  4  6
212 [4,4]  6  8
222 [4,4]  6  8
227 [4,4]  6  8
233 [4,4]  6  8
235 [4,2]  4  6
236 [4,2]  4  6
240 [4,4]  6  8
267 [4]  3  5  Q_{64}

48 KUHN

Table 3: Indecomposable, non 2–central, 2–groups of order ≤ 32

Order # Type Depth Rank e(G) e′(G) d_{0}(G)  C_{G}(V)’s  Notes
8  4 [2]  2  2  1  -1  0  2^{2}  D_{8}
16  8 [4]  1  2  3  -1  1  4 × 2  AES_{16}
9 [2,2]  2  3  2  1  1  2^{2}
11 [4]  1  2  3  2  2  4 × 2
12 [2]  1  2  1  -1  0  2^{2}  D_{16}
13 [4]  1  2  3  2  2  2^{2}  SD_{16}
32 16 [4,2]  2  3  4  3  3  4 × 2^{2}
17 [4]  2  2  3  -1  1  8 × 2
20 [2,2]  2  3  2  1  1  4 × 2^{2}
22 [4]  1  2  3  2  2  8 × 2
26 [4]  2  2  3  -1  1  8 × 2, 4 × 2
27 [2,2]  2  3  2  1  1  2^{3}
31 [4]  2  2  3  -1  2  4 × 4, 4 × 2
32 [4]  1  2  3  2  2  8 × 2
33 [2,2]  3  4  2  -1  0  2^{4}, 2^{3}
34 [2,2]  3  3  2  -1  0  2^{3}
36 [2,2]  3  3  2  -1  1  4 × 2^{2}, 2^{3}
37 [4,2]  2  3  4  3  3  4 × 2^{2}
38 [4,2]  2  3  4  2  2  4 × 2^{2}, 2^{3}
39 [4,2]  2  3  4  3  3  2^{3}
41 [4,4]  2  3  6  5  5  2^{3}
42 [4]  3  3  3  -1  0  2^{3}  D_{8} ∗ D_{8}
43 [8]  2  2  7  -1  3  Q_{8} × 2  D_{8} ∗ Q_{8}
44 [4]  2  3  3  -1  1  4 × 2, 2^{3}
45 [8]  1  2  7  4  4  Q_{8} × 2, 4 × 2
46 [4]  2  3  3  -1  3  Q_{8} × 2, 2^{3}
47 [4]  1  3  3  1  1  D_{8} × 2, 2^{3}
48 [8]  1  2  7  6  6  Q_{8} × 2
49 [2]  2  2  1  -1  0  2^{2}  D_{32}
50 [4]  1  2  3  2  2  2^{2}  SD_{32}

GROUP COHOMOLOGY PRIMITIVES 49

Table 4: Indecomposable, non 2–central, order 64, with Cess^{∗}(G) ≠ 0

# Type Rank e′(G) # Type Rank e′(G) # Type Rank e′(G)
42  [4]  2  2  173 [4,4]  4  3  193 [4,2]  3  3
67  [4]  2  2  175 [4,4]  4  3  196 [4,2]  3  2
143 [4]  2  2  183 [4,4]  4  5  197 [4,2]  3  3
182 [4]  2  2  202 [4,2]  4  2  198 [4,2]  3  3
245 [8]  2  4  200 [4,4]  3  5
246 [4]  2  2  32 [4,2]  3  3  204 [4,2]  3  3
249 [8]  2  6  33 [4,2]  3  3  206 [4,2]  3  3
255 [8]  2  6  40 [2,2]  3  1  207 [4,2]  3  3
258 [8]  2  4  54 [4,2]  3  3  208 [4,2]  3  3
266 [4]  2  2  60 [4,2]  3  3  209 [4,2]  3  3
61 [4,2]  3  3  213 [4,2]  3  2
121 [8]  3  3  62 [2,2]  3  1  214 [4,2]  3  2
130 [8]  3  3  79 [4,4]  3  5  215 [4,4]  3  5
133 [8]  3  4  80 [4,4]  3  4  216 [4,4]  3  5
180 [8]  3  3  95 [4,2]  3  3  218 [4,2]  3  3
181 [8]  3  3  97 [4,2]  3  3  219 [4,2]  3  2
247 [4]  3  1  98 [4,2]  3  3  220 [4,2]  3  3
251 [8]  3  4  99 [4,2]  3  2  221 [4,4]  3  5
253 [8]  3  3  100 [4,2]  3  3  223 [4,4]  3  5
254 [8]  3  4  102 [4,4]  3  5  224 [4,4]  3  5
257 [8]  3  3  108 [8,2]  3  7  225 [4,2]  3  2
262 [8]  3  3  115 [8,2]  3  7  226 [4,2]  3  3
116 [4,2]  3  3  228 [4,2]  3  2
81 [2,2,2] 5  1  118 [4,2]  3  3  229 [4,2]  3  3
129 [4,2]  3  3  230 [4,2]  3  3
83 [2,2,2] 4  2  132 [4,2]  3  3  231 [4,4]  3  5
85 [2,2,2] 4  2  138 [2,2]  3  1  232 [4,4]  3  5
86 [2,2,2] 4  2  161 [4,4]  3  4  234 [2,2]  3  1
89 [4,2,2] 4  4  165 [4,4]  3  4  238 [4,4]  3  5
91 [2,2,2] 4  2  166 [4,4]  3  4  239 [4,2]  3  3
146 [2,2,2] 4  2  167 [4,4]  3  4
147 [4,2,2] 4  4  168 [4,4]  3  5
148 [2,2,2] 4  2  172 [8,2]  3  5
150 [2,2,2] 4  2  174 [4,4]  3  5
151 [2,2,2] 4  2  177 [4,4]  3  3
178 [4,4]  3  4
94 [4,2]  4  2  179 [4,4]  3  5
113 [4,2]  4  2  185 [4,4]  3  3
131 [4,2]  4  2  186 [4,4]  3  4
163 [4,4]  4  3  189 [4,2]  3  3

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Department of Mathematics, University of Virginia, Charlottesville, VA 22904

   E-mail address: njk4x@virginia.edu