THE HOCHSCHILD COHOMOLOGY RING OF A GROUP ALGEBRA

STEPHEN F. SIEGEL AND SARAH J. WITHERSPOON

Abstract. There is a standard additive decomposition of the Hochschild cohomology ring of the group algebra of a finite group G as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of G. A special case of our main result describes the cup product in terms of this decomposition. As applications, we determine presentations for the Hochschild cohomology rings of (1) the mod-3 group algebra of the symmetric group S_{3}, (2) the mod-2 group algebra of the alternating group A_{4}, and (3) the mod-2 group algebras of the dihedral 2-groups.

1. Introduction

Historical background. The theory of the cohomology of associative algebras was initiated by G. Hochschild in 1945 [16] and evolved in tandem with group cohomology. To any associative algebra Λ, and any Λ-bimodule M, Hochschild assigned Abelian groups H^{n}(Λ, M) (n ≥ 0). In modern language these may be defined by

(1.1) H^{n}(Λ, M) = Ext_{Λ⊗_{R}Λ^{op}}^{n}(Λ, M).

Hochschild assumed the ground ring R was a field, but there is no problem with taking R to be an arbitrary commutative ring as long as Λ is R-projective.

Around this time, Eilenberg and Mac Lane gave a purely algebraic definition of the cohomology groups H^{n}(G, M) for a group G and left RG-module M [9]. They also showed that the Hochschild and ordinary theories agree, in the following sense. If M is an RG-bimodule then it may be considered a left RG-module via conjugation. Eilenberg and Mac Lane observed that the Hochschild cohomology of RG with coefficients in M is isomorphic to the ordinary cohomology of G with coefficients in that left module. (They used R = ℤ but the same arguments go through for any commutative ring.) This shows that for group algebras, nothing new is obtained by considering bimodules instead of just left modules. Moreover, since any left module arises in this way (just let RG act trivially on the right), this also shows that Hochschild cohomology is a true generalization of group cohomology.

For an arbitrary associative algebra Λ, there is a natural choice for the bimodule M, namely the bimodule Λ itself. In fact, the groups H^{∗}(Λ, Λ) have been investigated for many different classes of algebras. In low degrees, these groups are related to “algebraic deformations,” and have found applications in the study of quantum groups (see [13] and references cited).

Eilenberg and Mac Lane defined cup products for group cohomology, and in essence for Hochschild cohomology as well ([9, §§4,5]). The cup product gives H^{∗}(G, R) or H^{∗}(Λ, Λ) the structure of an associative graded algebra. Much progress has been made in describing the ring structure of the former. In particular, the landmark results of Quillen [23, 24] give a complete description, up to homeomorphism, of the spectrum of the ordinary cohomology ring of G with coefficients in a field of prime characteristic. Much less is known about Hochschild cohomology rings. One of the few general results, due to Gerstenhaber [12], is that H^{∗}(Λ, Λ) is graded-commutative, generalizing another fact from group cohomology.

Date: October 4, 1997.

1991 Mathematics Subject Classification. Primary 16E40, 20J06.

It follows from Eilenberg and Mac Lane’s observation that H^{∗}(RG, RG) is isomorphic to the cohomology of G with coefficients in RG, where RG is considered a module by conjugation. From this and the Eckmann-Shapiro Lemma, one sees that H^{∗}(RG, RG) is isomorphic to the direct sum of the cohomology of the centralizers of representatives of the conjugacy classes of G. The isomorphism, however, is not in general multiplicative. In fact, very little is known at present about the multiplicative structure of the Hochschild cohomology of group algebras. The case where G is Abelian was completely explained by Holm [18] and Cibils and Solotar [8]—in this case the Hochschild ring is just the tensor product of RG and the ordinary cohomology ring—but almost nothing else is known.

One of the intriguing aspects of Hochschild cohomology for representation theorists is that it may be applied to a block of a group algebra. With suitable restrictions on R, one may write RG as a direct sum of ideals, each of which is indecomposable as an algebra. These ideals turn out to be unique, and are called the blocks of RG ([3, §1.8]). One of the central themes in the representation theory of finite groups involves how each aspect of the theory “breaks up” into blocks. Since these algebras are not, in general, augmented, they do not come with trivial modules, and therefore the ordinary definition of cohomology does not apply.

But Hochschild cohomology does. Moreover, it can be shown that the Hochschild cohomology of the group algebra is isomorphic, as an algebra, to the direct sum of the Hochschild cohomology rings of the blocks. J. Rickard showed that under appropriate hypotheses, two blocks of RG which are derived equivalent have isomorphic Hochschild cohomology algebras ([26, Prop. 2.5]). Using this, Holm was able to calculate the even part of the Hochschild cohomology ring of a block with cyclic defect, and show that this ring distinguished between derived equivalence classes of those blocks. But almost no other cases appear in the literature. One of the reasons for this is undoubtedly the complexity of the calculations required to compute products.

Overview. Our main result, Theorem 5.1, gives a formula for the product in H^{∗}(RG, RG) in terms of the additive decomposition, proving a conjecture of Cibils [7] and Cibils and Solotar [8]. It reduces the computation of Hochschild products to standard operations within the ordinary cohomology rings of certain subgroups of G. As applications, we use this formula to determine presentations for the Hochschild cohomology rings of (1) the mod-3 group algebra of the symmetric group S_{3}, (2) the mod-2 group algebra of the alternating group A_{4}, and (3) the mod-2 group algebras of the dihedral 2-groups D_{4m}.

The material is organized as follows. In §2 we review the basic definitions of Hochschild cohomology and cup product, using arbitrary projective resolutions in place of the bar resolutions used by Hochschild and Eilenberg-Mac Lane. In §3 we specialize to the group algebra case, making explicit the isomorphism observed by Eilenberg-Mac Lane, and also showing that that isomorphism preserves the cup product. In this section we also introduce a generalization of the Hochschild cohomology ring of a group algebra: if a second group H acts on G then we may consider the rings H^{∗}(H, RG); the Hochschild cohomology ring is the special case where H = G acts on G by conjugation. Working on this level of generality, in §4 we prove an explicit version of the additive decomposition described above, and in §5 we prove our main result, the Product Formula.

Some authors use the term Hochschild cohomology of Λ to refer to H^{∗}(Λ, Λ^{∗}), where Λ^{∗} is the dual bimodule Hom_{R}(Λ, R) (e.g., Loday [20, §1.5.5] and Benson [4, §2.11]). This definition has certain advantages (e.g., it is a functor in Λ) though in general it does not seem to have an obvious multiplicative structure. In the group algebra case, however, a natural product can be defined, and in this case we show that the additive isomorphism not only holds, but is in fact multiplicative. This is covered in §6.

The next three sections contain the specific applications. The calculations are relatively straightforward and, we hope, demonstrate the usefulness of the Product Formula. In §10, we try to obtain general structure theorems for the Hochschild cohomology ring of a group algebra, and meet with some limited successes. If G is a p-group, we show that the mod-p Hochschild and ordinary cohomology rings are isomorphic modulo radicals. We then show that our rings H^{∗}(H, RG) have the structure of a Green functor, and exploit this theory to obtain a description of H^{∗}(H, RG) similar to the description of Hochschild cohomology for H = G Abelian, but modulo a certain ideal. These results were inspired by analogy with the Grothendieck ring of the category of modules for the quantum double of G (see [30]), an analogy that was essentially noted by Cibils [7]. We conclude in §11 with some intriguing questions which are suggested by the examples and these results.

Notation. Throughout this paper, R denotes an arbitrary commutative ring. G will always denote a finite group, and in §§7–9 a specific finite group. All rings and algebras are assumed to possess a unit, and all modules are assumed to be left modules unless stated otherwise. We will abbreviate H^{∗}(G, R) by H^{∗}(G) when the coefficient ring is clear. If H is a subgroup of G then restriction and corestriction (transfer) are denoted res_{H}^{G} and cor_{H}^{G}, respectively. If M is an RG-module then M↓_{H}^{G} denotes M considered as an RH-module; this is the restriction of M from G to H. If U is an RH-module then U↑_{H}^{G} denotes the RG-module RG ⊗_{RH} U; this is the module induced from H to G. If g, x ∈ G then ^{g}x = gxg^{−1}. More generally, if G is acting on a set X then we will often write ^{g}x to denote the action of g on an element x of X. We let δ_{g} denote the R-linear function from RG to R which takes the value 1 on g and 0 on any other element of G.

2. Definitions of Hochschild cohomology and products

Let R be a commutative ring and Λ an R-algebra which is finitely generated and projective as an R-module. Let Λ^{e} = Λ ⊗_{R} Λ^{op}. If M is a Λ-bimodule (i.e., a Λ^{e}-module) then the Hochschild cohomology of Λ with coefficients in M is defined to be

H^{∗}(Λ, M) = Ext_{Λ^{e}}^{∗}(Λ, M), (2.1)

where Λ is considered to be a Λ-bimodule in the usual way. These were originally defined by Hochschild [16] when R is a field; see [6, IX§4] for the more general definition and basic properties. Clearly H^{∗}(Λ, −) is a covariant functor from the category of Λ-bimodules to the category of graded R-modules.

If Λ happens to be an augmented R-algebra, then any left Λ-module M may be considered a Λ-bimodule with trivial right action. Then (2.1) agrees with the usual definition of H^{∗}(Λ, M) ([4, §2.11]). The advantage of Hochschild cohomology is that it may be applied to algebras which are not necessarily augmented, such as blocks of group algebras.

If N is another Λ-bimodule then there is a Hochschild cup product

H^{∗}(Λ, N) ⊗_{R} H^{∗}(Λ, M) ⟶^{⌣} H^{∗}(Λ, N ⊗_{Λ} M),

which can be defined as follows. Let X →^{ϵ} Λ be a Λ^{e}-projective resolution of Λ, and let D: X → X ⊗_{Λ} X be a Hochschild diagonal approximation map, i.e., a Λ^{e}-chain map such that (ϵ ⊗ ϵ) ∘ D = ϵ. In the literature it is common to take X to be the standard complex [6, IX§6], in which case there is an explicit formula for D [28, 1.2]. However, we will show below through standard homological arguments that a map D not only exists but is unique up to chain homotopy. Assuming this, let α and β be cohomology classes represented (resp.) by cocycles f ∈ Hom_{Λ^{e}}(X, N) and g ∈ Hom_{Λ^{e}}(X, M). Then α ⌣ β is represented by the cocycle (f ⊗ g) ∘ D. It is easily verified that the cup product is independent of the choices of X and D, and of the cocycles f and g representing α and β.

Our claims about the existence and uniqueness of D will follow from the Comparison Theorem [3, Thm. 2.4.2] if we can show that

X ⊗_{Λ} X ⟶^{ϵ⊗ϵ} Λ

is also a Λ^{e}-projective resolution. To see that X ⊗_{Λ} X is projective in each degree, it suffices to show that Λ^{e} ⊗_{Λ} Λ^{e} is a projective Λ^{e}-module. But as Λ^{e}-modules, Λ^{e} ⊗_{Λ} Λ^{e} ≅ Λ^{e} ⊗_{R} Λ. By hypothesis, Λ is a summand of a direct sum of copies of R; hence Λ^{e} ⊗_{Λ} Λ^{e} is isomorphic to a summand of a direct sum of copies of Λ^{e}, i.e., it is projective. To see that the complex is acyclic, we argue as follows. Since Λ ⊗_{R} Λ is projective as a right Λ-module (again because the left copy of Λ is R-projective), we have that X → Λ is a projective resolution of the free right Λ-module Λ. Hence the differentials in X are all split as right Λ-homomorphisms, and the cycles, boundaries, and homology in X are all projective right Λ-modules. In particular they are Λ-flat, so we may apply the Künneth Theorem [3, Thm. 2.7.1] to the right Λ-complex X and the left Λ-complex X. The Tor-term vanishes since the homology of X is Λ-projective, and so

H_{∗}(X ⊗_{Λ} X) ≅ H_{∗}(X) ⊗_{Λ} H_{∗}(X) ≅ Λ ⊗_{Λ} Λ ≅ Λ,

as required.

One important case is where N = M = Λ. In this case, the cup product gives H^{∗}(Λ, Λ) the structure of an associative graded algebra with unit over R, called the Hochschild cohomology ring of Λ. Gerstenhaber [12, §7, Cor. 1] showed that in fact H^{∗}(Λ, Λ) is graded-commutative (see also [28, Prop. 1.2] for a simplified proof). Another important case is where only N is replaced by Λ; this gives H^{∗}(Λ, M) the structure of a graded H^{∗}(Λ, Λ)-module.

3. The group algebra case

Reduction to ordinary cohomology. We are interested in the case where G is a finite group and Λ = RG, where as above R is any commutative ring. In this case the definitions above can all be reduced to ordinary group cohomology with no mention of bimodules. For in this case, Λ^{e} ≅ R[G × G], and if M is any RG-bimodule then by the Eckmann-Shapiro Lemma [3, Cor. 2.8.4],

H^{∗}(RG, M) ≅ Ext_{R[G×G]}^{∗}(RG, M) ≅ Ext_{R[G×G]}^{∗}(R↑_{ΔG}^{G×G}, M)
≅ Ext_{RG}^{∗}(R, M↓_{ΔG}^{G×G}) ≅ H^{∗}(G, M),

(3.1)

where ΔG = {(g, g) | g ∈ G}, and M is considered a left RG-module through the diagonal action g.x = gxg^{−1} (g ∈ G, x ∈ M). Hence the Hochschild cohomology of G with coefficients in the bimodule M is just the ordinary cohomology of G with coefficients in M under the diagonal action (cf. [21, Ch. X, Thm. 5.5], [9, §5]). In particular,

H^{∗}(RG, RG) ≅ H^{∗}(G, RG), (3.2)

where RG is considered a left RG-module by conjugation: g.a = gag^{−1} (g ∈ G, a ∈ RG).

Conversely, given any left RG-module M, we may consider it a bimodule by having RG act trivially on the right; and when we restrict this to the diagonal action we end up with the original left RG-module M. So we may as well forget all about bimodules and just consider left modules M, keeping in mind that M may be considered a bimodule in this way if the context demands.

The cup products are the same. Let M and N be RG-modules. Recall that there is already the (ordinary) cup product

H^{∗}(G, N) ⊗_{R} H^{∗}(G, M) → H^{∗}(G, N ⊗_{R} M).

The definition is entirely analogous to that of the Hochschild cup product; see [3, §3.2] or [10, Ch. 3] for details (but note that the restrictions placed on the coefficient ring in [10] are not needed for the proofs of the results which we will use). Furthermore, given an RG-homomorphism μ: N ⊗_{R} M → L, we may follow the cup product with the map μ^{∗} from H^{∗}(G, N ⊗_{R} M) to H^{∗}(G, L). We call the resulting product the cup product with respect to the pairing μ (cf. [10, §3.1]). One important case is where L = M = N is an R-algebra on which G acts as automorphisms and μ is the structure map. In this case the product gives H^{∗}(G, L) the structure of an associative graded algebra.

We will be interested here in the case N = RG, where RG is again considered a module via conjugation. In this case the structure map μ: RG ⊗_{R} M → M is an RG-homomorphism. We may therefore form the cup product with respect to μ:

H^{∗}(G, RG) ⊗_{R} H^{∗}(G, M) → H^{∗}(G, RG ⊗_{R} M) ⟶^{μ^{∗}} H^{∗}(G, M). (3.3)

Our claim is that under the isomorphism described above, this cup product corresponds to the Hochschild cup product. Specifically,

Proposition 3.1. The following diagram commutes:

[figure: H^{∗}(RG, RG) ⊗_{R} H^{∗}(RG, M); ⌣; ≅; H^{∗}(RG, M); ≅; H^{∗}(G, RG) ⊗_{R} H^{∗}(G, M); ⌣; H^{∗}(G, M).]

Proof. Let P → R be a projective resolution of the trivial RG-module. We will use this to construct an R[G × G]-projective resolution X of RG as follows. First, we may consider any RG-module as an RΔG-module via the diagonal map; we will abuse notation slightly by using the same symbol to denote either module, letting the context determine which one is meant. The same applies to complexes of RGmodules. With this in mind, set X = P↑_{ΔG}^{G×G}. Since induction is exact and takes projectives to projectives, and R↑_{ΔG}^{G×G} ≅ RG, this yields a projective resolution of RG.

We can make the isomorphism (3.1) explicit as follows. There is an RΔG-chain map ι: P ↪ X defined by ι(x) = (1, 1) ⊗ x (x ∈ P). Now given a cohomology class in H^{∗}(RG, M) represented by the cocycle f: X → M, the corresponding class in H^{∗}(G, M) is represented by the cocycle f ∘ ι.

Now let D: P → P⊗_{R} P be an RG-diagonal map. We may use this to construct a Hochschild diagonal map D′: X → X⊗_{RG} X as follows. There is an isomorphism of R[G × G]-complexes

(P ⊗_{R} P)↑_{ΔG}^{G×G} ⟶^{θ} X ⊗_{RG} X
(g, h) ⊗ x ⊗ y ⟼ ((g, 1) ⊗ x) ⊗ ((1, h) ⊗ y) g, h ∈ G, x, y ∈ P.

Set D′ = θ ∘ D↑_{ΔG}^{G×G}; this is the desired diagonal map.

Given cocycles f: X → RG and g: X → M, it follows from these definitions that we have a commutative diagram

[figure: X; D′; X ⊗_{RG} X; f⊗g; RG ⊗_{RG} M; ≅; M; P; ι; D; P ⊗_{R} P; (fι)⊗(gι); RG ⊗_{R} M; μ; M]

The bottom row represents the cup product in ordinary cohomology, the top row the product in Hochschild cohomology.

An alternative proof is to apply the results of Sanada [28, Thm. 1.1], which show that there is only one product in Hochschild cohomology satisfying a handful of basic axioms, and then verify those axioms for the ordinary cup product.

The upshot is that we may just take (3.2) and (3.3) to be our definitions of Hochschild cohomology and product for a group algebra, and interpret the rest of this section as a proof that our definitions correspond to the ones usually found in the literature.

A generalization. We now describe a generalization of the Hochschild cohomology ring of a group algebra. If H and G are finite groups, with H acting as automorphisms on G, then RG becomes an RH-module. The multiplication map RG ⊗_{R} RG → RG is an RH-homomorphism, and so we obtain a ring H^{∗}(H, RG) via cup product. This is also an associative graded R-algebra with unit, and in the case H = G, with action given by conjugation, it is the Hochschild cohomology ring. In general however, it is not necessarily graded-commutative. (For example, H^{∗}(1, RG) = RG.) The calculations in this paper are all concerned with the case H = G and action given by conjugation, but in §10 we will indicate that H ↦ H^{∗}(H, RG), for subgroups H of G, is a Green functor and exploit this to obtain some structure theorems.

Note. If R is Noetherian, then H^{∗}(H, RG) is a finitely generated R-algebra. For there is an algebra homomorphism (which is in fact injective) from H^{∗}(H) to H^{∗}(H, RG) induced by the map R → RG. By Evens’ Theorem, H^{∗}(H, RG) is finitely generated as an H^{∗}(H)-module, and H^{∗}(H) is a finitely generated R-algebra [10, Thm. 7.4.1, Cor. 7.4.6]. So the union of the images of a set of algebra generators for H^{∗}(H) with a set of module generators for H^{∗}(H, RG) will generate H^{∗}(H, RG) as an algebra.

While in general H^{∗}(H, RG) can be a very complicated ring, there is a special case in which we can say exactly what it is:

Proposition 3.2. If H acts trivially on G, then H^{∗}(H, RG) ≅ RG ⊗_{R} H^{∗}(H) as graded R-algebras. In particular, if G is Abelian then

H^{∗}(RG, RG) ≅ RG ⊗_{R} H^{∗}(G).

Proof. Let P → R be an RH-projective resolution which is finitely generated over RH in each degree. As an RH-module, RG is trivial, and moreover it is free as an R-module. So we may apply the Universal Coefficient Theorem [6, VI,Thm. 3.3] to the complex Hom_{RH}(P, R) and the RH-module RG to obtain an isomorphism of graded R-modules

RG ⊗_{R} H^{∗}(H) →^{θ} H^{∗}(H, RG)

(see [10, §3.4]). This map may be defined as follows: let a ∈ RG, ζ ∈ H^{∗}(H). If ζ is represented by the cocycle f: P → R then θ(a ⊗ ζ) is represented by the cocycle x ↦ af(x) (x ∈ P). It is immediate from the definition of cup product that θ is a ring homomorphism.

The case where G is Abelian in the Proposition was proved using very different methods by Cibils and Solotar [8, Thm. 2.1] (see also Holm [18] for the case where R is a field). In §10 we obtain a similar but weaker result for a group H acting not necessarily trivially on G.

4. Additive decomposition

We continue with the notation of the previous section: R is a commutative ring, and H and G are finite groups with H acting as automorphisms on G. The additive decomposition which we are about to describe is well-known (at least when H = G; cf. Burghelea [5] or [4, Thm. 2.11.2]), but for us it will be important to know the isomorphism explicitly. To do this we will need the following explicit version of the Eckmann-Shapiro Lemma (cf. [3, Cor. 2.8.4]):

Lemma 4.1. Let K ≤ H and U an RK-module. Let ι: H^{∗}(K, U) → H^{∗}(K, U↑_{K}^{H}) be the map induced by the RK-homomorphism U → U↑_{K}^{H} given by x ↦ 1 ⊗ x. Let π: H^{∗}(K, U↑_{K}^{H}) → H^{∗}(K, U) be the map induced by the RK-homomorphism U↑_{K}^{H} → U which sends h ⊗ x to hx (if h ∈ K) and 0 (otherwise). Then

cor_{K}^{H} ∘ι: H^{∗}(K, U) → H^{∗}(H, U↑_{K}^{H})

is an isomorphism and its inverse is π ∘ res_{K}^{H}.

Proof. Compose in either order and obtain the identity.

Let g_{1}, …, g_{r} be representatives of the orbits of the action of H on G, and let H_{i} = Stab_{H}(g_{i}) be the stabilizer of g_{i}. Next, fix g ∈ G. Then there are two R(Stab_{H}(g))-homomorphisms

\begin{matrix}θ_{g}: R & ⟶ & RG & π_{g}: RG & ⟶ & R \\ λ & ⟼ & λg & ∑_{a∈G} λ_{a}a & ⟼ & λ_{g}. \end{matrix}

If W is any subgroup of Stab_{H}(g) then these maps induce maps on cohomology

θ_{g}^{∗}: H^{∗}(W) → H^{∗}(W, RG), π_{g}^{∗}: H^{∗}(W, RG) → H^{∗}(W).

Now define maps γ_{i}: H^{∗}(H_{i}) → H^{∗}(H, RG) by

γ_{i}(α) = cor_{H_{i}}^{H} θ_{g_{i}}^{∗}(α), α ∈ H^{∗}(H_{i})

Lemma 4.2. The map

H^{∗}(H, RG) ⟶ ⨁_{i} H^{∗}(H_{i})
ζ ⟼ (π_{g_{i}}^{∗} res_{H_{i}}^{H}(ζ))_{i}

is an isomorphism of graded R-modules, and its inverse sends α ∈ H^{∗}(H_{i}) to γ_{i}(α).

Proof. (cf. [27, Exer. 6.1.45].) We have RG = ⨁_{i} M_{i}, where M_{i} is the free Rmodule generated by the elements of the orbit containing g_{i}. Hence H^{∗}(H, RG) ≅ ⨁_{i} H^{∗}(H, M_{i}). Now there is an isomorphism M_{i} ≅ R↑_{H_{i}}^{H} = RH ⊗_{RH_{i}} R given by λ^{h}g_{i} ↔ h ⊗ λ, and this induces an isomorphism in cohomology. Composing that with the isomorphism of Lemma 4.1 (with K = H_{i}), one obtains the desired isomorphism.

5. The Product Formula

We continue with the notation above; in particular H is a finite group acting on a finite group G. We seek a way to describe the cup product in H^{∗}(H, RG) in terms of the direct sum decomposition described in Lemma 4.2. This is accomplished as follows. Fix i, j ∈ {1, …, r}. Let D be a set of double coset representatives for H_{i}\H/H_{j}. For each x ∈ D, there is a unique k = k(x) such that

g_{k} = ^{y}g_{i}^{yx}g_{j} (5.1)

for some y ∈ H. Moreover, k is independent of the choice of representative x of the double coset H_{i}xH_{j}. The set of all y satisfying (5.1) is also a double coset. To see this, let us fix one such y = y(x) for which (5.1) holds. Then

{y′ ∈ H | g_{k} = ^{y′}(g_{i}^{x}g_{j})} = H_{k}y = H_{k}y(^{x}H_{j} ∩ H_{i}) ∈ H_{k}\H/^{x}H_{j} ∩ H_{i},

since by (5.1), ^{yx}H_{j} ∩^{y}H_{i} ⊆ H_{k}. We may now state our main result:

Theorem 5.1 (Product Formula). Let α ∈ H^{∗}(H_{i}), β ∈ H^{∗}(H_{j}). Then

γ_{i}(α) ⌣ γ_{j}(β) = ∑_{x∈D} γ_{k}(cor_{W}^{H_{k}}(res_{W}^{^{y}H_{i}} y^{∗}α ⌣ res_{W}^{^{yx}H_{j}}(yx)^{∗}β))

where D is a set of double coset representatives for H_{i}\H/H_{j}, k = k(x) and y = y(x) are chosen to satisfy (5.1), and W = W (x) = ^{yx}H_{j} ∩^{y}H_{i}.

The statement requires a bit of explanation. First, recall that given an RHmodule U, a subgroup K ≤ H, and h ∈ H, there is a map

h^{∗}: H^{∗}(K, U) → H^{∗}(^{h}K_{,}U),

which can be defined on the cochain level by fixing an RH-projective resolution P → R and setting

(h^{∗}f)(v) = hf(h^{−1}v), f ∈ Hom_{RK}(P, U), v ∈ P

(see [10, §4.1]). Now it follows from the theorem, and the fact that the cup product is well defined, that the sum in the statement of the theorem is independent of the choices for x and y. However, this can also be seen directly. For y is unique up to left multiplication by an element z of H_{k}. Since z^{∗} respects the cup product and commutes with restriction and corestriction, and since H_{k} acts trivially on its own cohomology, any term of the sum is unchanged by replacing y with zy. If x is multiplied on the right by an element of H_{j} the term is unchanged for similar reasons. If x is replaced by wx, where w ∈ H_{i}, then one must replace y with yw^{−1} so that (5.1) holds, and again the term is unchanged.

We now turn to the proof.

Lemma 5.2. Let h ∈ H and a, b ∈ G.

(i) Suppose W ≤ Stab_{H}(a). Then h^{∗}θ_{a}^{∗} = θ_{^{h}a}^{∗}h^{∗} as maps from H^{∗}(W) to H^{∗}(^{h}W, RG).

(ii) Suppose W ≤ Stab_{H}(a) ∩ Stab_{H}(b), and α, β ∈ H^{∗}(W). Then

θ_{a}^{∗}(α) ⌣ θ_{b}^{∗}(β) = θ_{ab}^{∗}(α ⌣ β).

(iii) Suppose W′ ≤ W ≤ Stab_{H}(a). Then θ_{a}^{∗} and π_{a}^{∗} commute with res_{W′}^{W} and

cor_{W′}^{W}. (iv) Suppose W ≤ Stab_{H}(a) ∩ Stab_{H}(b). Then π_{a}^{∗}θ_{b}^{∗} = δ_{a,b} id.

Proof. (i) Let f ∈ Hom_{W} (P, R). Then h^{∗}(θ_{a}f)(x) = f(h^{−1}x)^{h}a = θ_{^{h}a}(h^{∗}(f))(x). (ii) Let μ: RG ⊗_{R} RG → RG be the multiplication map. Let D: P → P ⊗_{R} P be a diagonal approximation map. For f, g ∈ Hom_{W} (P, R),

μ ∘ ((θ_{a} ∘ f) ⊗ (θ_{b} ∘ g)) ∘ D = μ ∘ (θ_{a} ⊗ θ_{b}) ∘ (f ⊗ g) ∘ D = θ_{ab} ∘ (f ⊗ g) ∘ D.

(iii) We will show cor_{W′}^{W} commutes with π_{a}^{∗}; the other three cases are similar. For f ∈ Hom_{W′}(P, RG),

π_{a}( ∑_{w∈W/W′} ^{w}f(w^{−1}x)) = ∑_{w} π_{^{w^{−1}}a}(f(w^{−1}x)) = ∑_{w}(π_{a} ∘ f)(w^{−1}x),

as ^{w^{−1}}a = a. If f is a cocycle representing α then the left side above represents π_{a}^{∗}(cor_{W′}^{W}(α)) and the right side represents cor_{W′}^{W}(π_{a}^{∗}(α)). (iv) Clearly π_{a}θ_{b} = δ_{a,b} id; apply the cohomology functor.

Proof of Theorem 5.1. By Lemma 4.2,

γ_{i}(α) ⌣ γ_{j}(β) = cor_{H_{i}}^{H}(θ_{g_{i}}^{∗}α) ⌣ cor_{H_{j}}^{H}(θ_{g_{j}}^{∗}β)

= cor_{H_{i}}^{H}(θ_{g_{i}}^{∗}α ⌣ res_{H_{i}}^{H} cor_{H_{j}}^{H} θ_{g_{j}}^{∗}β)

= ∑_{x∈D} cor_{H_{i}}^{H}(θ_{g_{i}}^{∗}α ⌣ cor_{^{x}H_{j}∩H_{i}}^{H_{i}} res_{^{x}H_{j}∩H_{i}}^{^{x}H_{j}} x^{∗}θ_{g_{j}}^{∗}β)

= ∑_{x} cor_{^{x}H_{j}∩H_{i}}^{H}(res_{^{x}H_{j}∩H_{i}}^{H_{i}} θ_{g_{i}}^{∗}α ⌣ res_{^{x}H_{j}∩H_{i}}^{^{x}H_{j}} x^{∗}θ_{g_{j}}^{∗}β)

= ∑_{x} cor_{^{x}H_{j}∩H_{i}}^{H} θ_{g_{i}^{x}g_{j}}^{∗}(res_{^{x}H_{j}∩H_{i}}^{H_{i}} α ⌣ res_{^{x}H_{j}∩H_{i}}^{^{x}H_{j}} x^{∗}β),

by Lemma 5.2 (i)–(iii), as well as the double coset formula [10, Thm. 4.2.6], [10, Prop. 4.2.4], and [10, Prop. 4.2.1]. So again by Lemma 4.2,

γ_{i}(α) ⌣ γ_{j}(β)

= ∑_{k} ∑_{x} γ_{k}(π_{g_{k}}^{∗} res_{H_{k}}^{H} cor_{^{x}H_{j}∩H_{i}}^{H} θ_{g_{i}^{x}g_{j}}^{∗}(res_{^{x}H_{j}∩H_{i}}^{H_{i}} α ⌣ res_{^{x}H_{j}∩H_{i}}^{^{x}H_{j}} x^{∗}β))

= ∑_{k} ∑_{x,y} γ_{k}(π_{g_{k}}^{∗} cor_{W′}^{H_{k}} res_{W′}^{^{yx}H_{j}∩^{y}H_{i}} y^{∗}θ_{g_{i}^{x}g_{j}}^{∗}(res_{^{x}H_{j}∩H_{i}}^{H_{i}} α ⌣ res_{^{x}H_{j}∩H_{i}}^{^{x}H_{j}} x^{∗}β))

= ∑_{k} ∑_{x,y} γ_{k}(cor_{W′}^{H_{k}} π_{g_{k}}^{∗}θ_{^{y}g_{i}^{yx}g_{j}}^{∗}(res_{W′}^{^{y}H_{i}} y^{∗}α ⌣ res_{W′}^{^{yx}H_{j}}(yx)^{∗}β)),

where y runs over a set of representatives for H_{k}\H/^{x}H_{j} ∩ H_{i}, and

W′ = W′(k, x, y) = ^{yx}H_{j} ∩^{y}H_{i} ∩ H_{k}.

However, Lemma 5.2 (iv) implies the only terms that can be non-zero are those for which g_{k} = ^{y}g_{i}^{yx}g_{j}. But we have seen that each x determines a unique k and double coset H_{k}y(^{x}H_{j} ∩ H_{i}) for which this holds. So we may take y = y(x), and then ^{yx}H_{j} ∩^{y}H_{i} ⊆ H_{k}, so W′ = W.

Notice that γ_{1} is an algebra monomorphism. This is because it is the map in cohomology induced by the algebra homomorphism R → RG sending λ to λ1. (Alternatively, one may see this as the special case of Theorem 5.1 where i = j = 1.) So we may view H^{∗}(H, RG) as an H^{∗}(H)-module. Now, each H^{∗}(H_{i}) may also be considered as an H^{∗}(H)-module via restriction. As an immediate corollary of Theorem 5.1, we obtain

Corollary 5.3. The additive isomorphism of Lemma 4.2

H^{∗}(H, RG) ⟶^{≈} ⨁_{i} H^{∗}(H_{i})

is an isomorphism of graded H^{∗}(H)-modules.

Proof. For i = 1, the Theorem reduces to γ_{1}(α) ⌣ γ_{j}(β) = γ_{j}(res_{H_{j}}^{H}(α) ⌣ β).

Note. In the case where H = G and the action is conjugation, Theorem 5.1 verifies a formula conjectured by Cibils [7] and Cibils and Solotar [8]. Our sum over k corresponds to their sum over C ∈ C. Our sum over x corresponds to their sum over E_{A,B}^{C}/Z_{c}. Their pair (K, L) is our (y^{−1}, x^{−1}y^{−1}).

6. Hochschild cohomology with dual coefficients

There is an isomorphism of RH-bimodules RG → RG^{∗}, sending g to δ_{g^{−1}} (g ∈ G). It follows from functorality that H^{∗}(RH, RG) ≅ H^{∗}(RH, RG^{∗}) as graded Rmodules. Arguing as above, we have H^{∗}(RH, RG^{∗}) ≅ H^{∗}(H, RG^{∗}), where RG^{∗} is now considered a left RH-module as follows: (h · θ)(x) = θ(h^{−1} · x) (h ∈ H, θ ∈ RG^{∗}, x ∈ RG).

There is a natural product on H^{∗}(H, RG^{∗}) which we now describe. Since RG is a bialgebra, RG^{∗} is an R-algebra with multiplication

RG^{∗} ⊗_{R} RG^{∗} →^{ν} RG^{∗}
θ ⊗ ϕ ↦ θϕ

where (θϕ)(g) = θ(g)ϕ(g) (g ∈ G). One can check that ν is an RH-homomorphism, and therefore there is a cup product on H^{∗}(H, RG^{∗}) with respect to the pairing ν.

Using the familiar identification of H^{0}(H, M) with the invariants M^{H}, we see that H^{0}(H, RG^{∗}) may be identified with the algebra of functions from G to R constant on each H-orbit of G, under pointwise multiplication; this is isomorphic as an algebra to the direct product of r copies of R, where r is the number of H-orbits on G. However, H^{0}(H, RG) ≅ (RG)^{H}. As an R-module this is also isomorphic to R^{r}, but in general it has a completely different ring structure. For example, if R = 𝔽 is a field of characteristic dividing |G| then (RG)^{G} = Z(𝔽G) has nilpotent elements (such as ∑_{g∈G} g).

We will show next that the additive decomposition of Lemma 4.2 (using RG^{∗} ≅ RG) is also multiplicative, by contrast to the case for H^{∗}(G, RG).

Proposition 6.1. There is an isomorphism of graded R-algebras H^{∗}(H, RG^{∗}) ≅ ⨁_{i} H^{∗}(H_{i}). In particular, H^{∗}(G, RG^{∗}) and H^{∗}(LBG; R) are isomorphic as algebras, where LBG is the space of free loops on the classifying space BG of G.

Proof. There is an ideal direct sum decomposition RG^{∗} = ⨁_{i} M_{i} where M_{i} is the free R-module generated by the dual functions δ_{g} for all g in the H-orbit of g_{i}. Therefore H^{∗}(H, RG^{∗}) ≅ ⨁_{i} H^{∗}(H, M_{i}) as algebras, where H^{∗}(H, M_{i}) is endowed with the cup product with respect to the multiplication map M_{i} ⊗ M_{i} → M_{i}. We claim that H^{∗}(H, M_{i}) ≅ H^{∗}(H_{i}) as algebras. Indeed, we may identify M_{i} with R↑_{H_{i}}^{H} via the RH-isomorphism δ_{^{h}g_{i}} ↔ h ⊗ 1. Lemma 4.1 says that there is an additive isomorphism π ∘ res_{H_{i}}^{H}: H^{∗}(H, M_{i}) → H^{∗}(H_{i}). But restriction is multiplicative, and π is induced by the projection M_{i} → Rδ_{g_{i}} ≅ R, which is a ring homomorphism (and an RH_{i}-homomorphism). Hence π is multiplicative as well, and our claim is proved. Finally, H^{∗}(LBG; R) ≅ ⨁_{i} H^{∗}(H_{i}) as algebras because LBG is homotopy equivalent to the disjoint union of the BH_{i} ([4, §2.12]).

In contrast, H^{∗}(G, 𝔽G) cannot in general be the cohomology ring of any topological space X. For H^{0}(X; 𝔽) is isomorphic, as an algebra, to a product of copies of 𝔽, one for each connected component of X. But as we have seen, H^{0}(G, 𝔽G) may have nilpotent elements.

7. The symmetric group of degree 3

Generators and relations. Before beginning the calculations it is helpful to make some general remarks about generators and relations. Let 𝔽 be a field and A a f.g. graded-commutative 𝔽-algebra which is finite dimensional in each degree. Suppose A = B ⊕ U, where B is a graded subalgebra of A and U is a graded B-submodule. Such a situation arises, for example, if A = H^{∗}(G, 𝔽G), B = γ_{1}(H^{∗}(G, 𝔽)), and U = ∑_{i≥2} γ_{i}(H^{∗}(H_{i}, 𝔽)).

Suppose B is generated, as an algebra, by x_{1}, …, x_{N}, subject to the relations r_{i} = 0 (i ∈ I), in addition to the graded-commutative relations. Suppose in addition that U is generated, as a B-module, by y_{1}, …, y_{M}, subject to the relations s_{j} = 0 (j ∈ J). (Hence U is isomorphic to the graded-free B-module on the y_{i} modulo the B-submodule generated by the s_{j}). It is assumed that all of these generators and relations are homogeneous. By hypothesis, for each 1 ≤ l ≤ m ≤ M, we may choose f_{lm}^{n} ∈ B (0 ≤ n ≤ M) such that

y_{l}y_{m} = f_{lm}^{0} + ∑_{n=1}^{M} f_{lm}^{n}y_{n}. (7.1)

Now it is clear that A is generated as an 𝔽-algebra by x_{1}, …, x_{N}, y_{1}, …, y_{M}. We claim one obtains a presentation for A by taking those generators, together with the relations of type 1 r_{i} = 0, the relations of type 2 s_{j} = 0, and the relations of type 3 given by equation (7.1), in addition to the graded-commutative relations. Indeed, let A′ be the algebra presented abstractly with these generators and relations, and let B′ be the subalgebra of A′ generated by the x_{i}. Clearly there is a homomorphism θ from A′ onto A, and the relations of type 1 guarantee θ maps B′ isomorphically onto B. The relations of type 3 show that every element of A′ can be expressed as the sum of an element of B′ and a B′-linear combination of the y_{i}. Together with the relations of type 2, this implies that the dimension, in each degree, of A′ is less than or equal to that of A, so θ is an isomorphism, and we have established the claim.

Hochschild cohomology of 𝔽_{3}S_{3}. Let G = S_{3} = ⟨a, b | a^{3} = 1 = b^{2}, bab = a^{−1}⟩. In this section we will show how to find the Hochschild cohomology ring of 𝔽_{3}G using the Product Formula. To do this, choose conjugacy class representatives g_{1} = 1, g_{2} = a, and g_{3} = b. The centralizers of these elements are, resp., H_{1} = G, H_{2} = ⟨a⟩, and H_{3} = ⟨b⟩. From now on we will assume all coefficients are in 𝔽_{3} unless stated otherwise. Now H^{∗}(G) is generated by elements u and v, of degrees (resp.) 3 and 4, subject only to the graded-commutative relations. Similarly, H^{∗}(⟨a⟩) is generated by elements w_{1}, w_{2}, of degrees (resp.) 1, 2, subject only to the graded-commutative relations. Of course H^{∗}(⟨b⟩) is just 𝔽_{3}, concentrated in degree 0.

Now define the following elements in the Hochschild cohomology ring. Since γ_{1} is an algebra monomorphism, by a slight abuse of notation we may identify any element of H^{∗}(G) with its image under γ_{1}. Let E_{i} = γ_{i}(1) (1 ≤ i ≤ 3), let C_{1} = 1 + E_{2}, C_{2} = E_{3}, and X_{j} = γ_{2}(w_{j}) (j = 1, 2).

Theorem 7.1. H^{∗}(S_{3}, 𝔽_{3}S_{3}) is generated as an algebra by elements u, v, C_{1}, C_{2}, X_{1}, and X_{2}, of degrees (resp.) 3, 4, 0, 0, 1, and 2, subject to the relations

uX_{1} = 0, vX_{1} = uX_{2}, uC_{2} = 0 = vC_{2}
C_{i}X_{j} = 0 = C_{i}C_{j} (i, j ∈ {1, 2}), X_{1}X_{2} = uC_{1}, X_{2}^{2} = vC_{1}

in addition to the graded-commutative relations. In particular, the algebra monomorphism γ_{1} : H^{∗}(S_{3}) → H^{∗}(S_{3}, 𝔽_{3}S_{3}) induces an isomorphism modulo radicals.

Proof. Relations in degree 0 are most easily handled by the identification of degree-0 Hochschild cohomology with the center of the group algebra. Under this identification, E_{i} corresponds to the sum of the group elements conjugate to g_{i}. So one obtains, for example,

E_{2}^{2} = (a + a^{−1})^{2} = a + a^{−1} − 1 = E_{2} − 1,

which implies C_{1}^{2} = 0. The other degree-0 relations are handled in a similar way.

Restriction from G to ⟨a⟩ is injective and takes u to w_{1}w_{2} and v to w_{2}^{2}. It follows that H^{∗}(⟨a⟩) is generated as an H^{∗}(G)-module by 1, w_{1}, and w_{2}, subject to the relations u.w_{1} = 0 and v.w_{1} = u.w_{2}. Of course H^{∗}(⟨b⟩) is generated as an H^{∗}(G)- module by 1, subject to the relations u.1 = 0 = v.1. This implies H^{∗}(G, 𝔽_{3}G) is generated by u, v, E_{2}, E_{3}, X_{1}, X_{2}; we have replaced E_{2} with 1 + E_{2} because it simplifies the relations somewhat. It also implies the first line of relations hold, which are the “relations of type 2.”

We now turn to the “relations of type 3.” According to the Product Formula,

γ_{2}(α)γ_{2}(β) = γ_{2}(b^{∗}(αβ)) + γ_{1}(cor_{⟨a⟩}^{G}(αb^{∗}(β))).

for α, β ∈ H^{∗}(⟨a⟩). It is easily checked that b^{∗}(w_{i}) = −w_{i}, and

cor_{⟨a⟩}^{G}(w_{1}w_{2}^{n}) = \begin{cases}−uv^{(n−1)/2} & n odd \\ 0 & n even \end{cases} cor_{⟨a⟩}^{G}(w_{2}^{n}) = \begin{cases}0 & n odd \\ −v^{n/2} & n even. \end{cases}

Letting α = 1 and β = w_{1}, this yields E_{2}X_{1} = −X_{1}, i.e., C_{1}X_{1} = 0. Letting α = β = w_{2}, one obtains

X_{2}^{2} = v + γ_{2}(w_{2}^{2}) = v + vE_{2} = vC_{1}.

The relations involving X_{1}X_{2} and C_{1}X_{2} are handled similarly. Finally, the Product Formula implies that the product of an element in the image of γ_{2} with one in the image of γ_{3} lies in the image of γ_{3}. Since this is 0 in positive degrees, we obtain the remaining relations.

It follows from our comments above that we have found a complete set of generators and relations. The remark on isomorphism modulo radicals follows from the observation that X_{2}^{3} = vC_{1}X_{2} = 0, so C_{1}, C_{2}, X_{1}, and X_{2} all lie in the radical of the Hochschild cohomology ring.

Hochschild cohomology of 𝔽_{2}S_{3}. The mod-2 Hochschild cohomology of S_{3} can be found by more elementary measures. For 𝔽_{2}S_{3} has two blocks: the principal block, which is isomorphic to 𝔽_{2}(ℤ/2), and a block which is isomorphic to the algebra of 2 by 2 matrices over 𝔽_{2}. The Hochschild cohomology of the first is determined by Prop. 3.2, the second is a simple algebra and so has Hochschild cohomology isomorphic to 𝔽_{2} concentrated in degree 0. Hence

H^{∗}(S_{3}, 𝔽_{2}S_{3}) ≅ 𝔽_{2}(ℤ/2) ⊗_{𝔽_{2}} H^{∗}(ℤ/2, 𝔽_{2}) ⊕ 𝔽_{2}
≅ 𝔽_{2}[u, v, w | v^{2} = uw = vw = 0, w^{2} = w],

where u and v have degree 1 and w has degree 0.

8. The Alternating group A_{4}

Let G = A_{4} = ⟨a, b, c | a^{2} = b^{2} = c^{3} = 1, ab = ba, c^{−1}ac = b, c^{−1}bc = ab⟩.

Proposition 8.1. H^{∗}(A_{4}, 𝔽_{2}A_{4}) is generated as a commutative algebra by elements u, v, w, C_{1}, C_{2}, C_{3}, X_{1}, Y_{1}, X_{2}, and Y_{2}, of degrees (resp.) 2, 3, 3, 0, 0, 0, 1, 1, 2, and 2, subject to the relation of type 1

u^{3} + v^{2} + vw + w^{2} = 0,

the 10 relations of type 2

\begin{aligned}uX_{2} + vX_{1} + vY_{1} + wY_{1} & = 0 & uY_{2} + vX_{1} + wX_{1} + wY_{1} & = 0 \\ u^{2}Y_{1} + vX_{2} + wX_{2} + vY_{2} & = 0 & u^{2}X_{1} + u^{2}Y_{1} + vX_{2} + wY_{2} & = 0 \end{aligned}

uC_{2} = vC_{2} = wC_{2} = uC_{3} = vC_{3} = wC_{3} = 0

and the 28 relations of type 3

X_{1}^{2} = 0, X_{1}Y_{1} = uC_{1}, X_{1}X_{2} = (v + w)C_{1}, X_{1}Y_{2} = wC_{1}, Y_{1}^{2} = 0,
Y_{1}X_{2} = vC_{1}, Y_{1}Y_{2} = (v + w)C_{1}, X_{2}^{2} = 0, X_{2}Y_{2} = u^{2}C_{1}, Y_{2}^{2} = 0,
X_{1}C_{i} = Y_{1}C_{i} = X_{2}C_{i} = Y_{2}C_{i} = 0 (1 ≤ i ≤ 3),
C_{i}C_{j} = 0 (1 ≤ i ≤ j ≤ 3).

In particular, the monomorphism H^{∗}(A_{4}, 𝔽_{2}) ↪^{γ_{1}} H^{∗}(A_{4}, 𝔽_{2}A_{4}) induces an isomorphism modulo radicals.

The remainder of this section is devoted to the proof of the proposition. To begin, the conjugacy classes of G with chosen representatives are

{g_{1} = 1}, {g_{2} = a, b, ab}, {g_{3} = c, ac, bc, abc}, {g_{4} = c^{2}, ac^{2}, bc^{2}, abc^{2}}.

The centralizers are H_{1} = G, H_{2} = ⟨a, b⟩ ≅ ℤ/2 × ℤ/2, H_{3} = H_{4} = ⟨c⟩ ≅ ℤ/3.

The cohomology rings are as follows (coefficients are assumed to be in 𝔽_{2} unless stated otherwise):

H^{∗}(G) = 𝔽_{2}[u, v, w | deg(u) = 2, deg(v) = deg(w) = 3, u^{3} + v^{2} + vw + w^{2} = 0],

(see [2, Appendix]), H^{∗}(H_{2}) = 𝔽_{2}[x, y | deg(x) = deg(y) = 1], and H^{∗}(H_{3}) = H^{∗}(H_{4}) = 𝔽_{2}. We may take x, y to be the dual basis to a, b; restriction from G to H_{2} is injective, and is defined by

u ↦ x^{2} + xy + y^{2}, v ↦ x^{3} + x^{2}y + y^{3}, w ↦ x^{3} + xy^{2} + y^{3}.

G acts on H^{∗}(H_{2}) as follows: a and b act trivially, while c sends x to y and y to x + y, and the image of restriction from G is the invariant subring. If we identify H^{∗}(G) with this subring, then corestriction from H_{2} to G is just the trace map ∑_{i=0}^{2}(c^{i})^{∗}.

As above, we will identify α with γ_{1}(α) (α ∈ H^{∗}(G)), and let E_{i} = γ_{i}(1) (1 ≤ i ≤ 4). Let C_{1} = E_{2} + 1, C_{2} = E_{3}, C_{3} = E_{4}, X_{1} = γ_{2}(x), Y_{1} = γ_{2}(y), X_{2} = γ_{2}(x^{2}), and Y_{2} = γ_{2}(y^{2}).

Let A = H^{∗}(H_{2}), B = H^{∗}(G); we may consider B as a subalgebra of A via restriction. We first show that A is generated as a B-module by 1, x, y, x^{2}, y^{2}. This is clearly the case for degrees 0 and 1. In degree 2, we have xy = u1 + x^{2} + y^{2}. In degree 3,

\begin{aligned}x^{3} & = ux + (v + w)1, & x^{2}y & = v1 + ux + uy, \\ xy^{2} & = w1 + ux + uy, & y^{3} & = (v + w)1 + uy. \end{aligned} (8.1)

Now suppose α is a monomial in x and y of degree n (n ≥ 4). Let α = βγ, with β of degree 3. By induction on n, γ may be written as a B-linear combination of 1, x, y, x^{2}, y^{2}, which have degree at most 2. By (8.1), β is a B-linear combination of 1, x, y, which have degree at most 1. Hence α is a B-linear combination of monomials of degree at most 3. Again using (8.1), this means α is a B-linear combination of 1, x, y. Hence A is generated by 1, x, y, x^{2}, y^{2}, as claimed.

We next show that the first four relations of type 2 define A as a B-module. It is straightforward to check that these relations hold. Hence there is a complex of graded B-modules

P_{1} →^{δ_{1}} P_{0} →^{δ_{0}} A → 0

in which P_{0} = B ⊕ B[1]^{⊕2} ⊕ B[2]^{⊕2}, P_{1} = B[4]^{⊕2} ⊕ B[5]^{⊕2} (B[n] is B as an 𝔽_{2}- module but with the grading shifted so that B[n]_{i} = B_{i−n}), and the differentials are defined by

δ_{0}(λ_{1}, …, λ_{5}) = λ_{1}1 + λ_{2}x + λ_{3}y + λ_{4}x^{2} + λ_{5}y^{2}  λ_{1}, …, λ_{5} ∈ B
δ_{1}(α_{1}, …, α_{4}) = α_{1}r_{1} + ⋯ + α_{4}r_{4}  α_{1}, …, α_{4} ∈ B,

where

r_{1} = (0, v, v + w, u, 0)  r_{2} = (0, v + w, w, 0, u)
r_{3} = (0, 0, u^{2}, v + w, v)  r_{4} = (0, u^{2}, u^{2}, v, w).

To show that the relations suffice is equivalent to showing that this complex is exact in degree 0.

We show the complex is exact by extending it slightly to a complex

0 → P_{2} →^{δ_{2}} P_{1} →^{δ_{1}} P_{0} →^{δ_{0}} A → 0, (8.2)

and showing that this bigger complex is exact. Here, P_{2} = S[7]^{⊕2} ⊕S[8]^{⊕2} ⊕S[9]^{⊕2}, where S = 𝔽_{2}[v, w]. Note that B = S ⊕ Su ⊕ Su^{2}. In particular, B is a graded-free S-module, so what we are showing is that (8.2) is actually a graded-free resolution of A as an S-module. We may define δ_{2} by

\begin{matrix}δ_{2}(s_{1}, …, s_{6}) = ( & (v + w)s_{1} + vs_{2} + ((v + w)s_{5} + vs_{6})u + s_{3}u^{2}, \\ vs_{1} + ws_{2} + (vs_{5} + ws_{6})u + s_{4}u^{2}, \\ ws_{3} + vs_{4} + s_{1}u + s_{5}u^{2}, \\ vs_{3} + (v + w)s_{4} + s_{2}u + s_{6}u^{2}), \end{matrix} (8.3)

where s_{1}, …, s_{6} ∈ S.

It is tedious but straightforward to check that δ_{1}δ_{2} = 0, so this is actually a complex. Also, it follows easily from (8.3) that δ_{2} is injective. We will now show that Ker(δ_{1}) = Im(δ_{2}). Suppose ∑_{i=1}^{4} α_{i}r_{i} = 0. Let α_{i} = μ_{i} + ν_{i}u + ξ_{i}u^{2} (μ_{i}, ν_{i}, ξ_{i} ∈ S). Setting each coordinate of ∑_{i} α_{i}r_{i} equal to 0 and using the relation of type 1, we have

0 = μ_{1} + (v + w)ν_{3} + vν_{4}  0 = ν_{1} + (v + w)ξ_{3} + vξ_{4}
0 = μ_{2} + vν_{3} + wν_{4}  0 = vξ_{1} + (v + w)ξ_{2} + μ_{4}
0 = ν_{2} + vξ_{3} + wξ_{4}  0 = wξ_{1} + vξ_{2} + μ_{3}.

Hence (α_{1}, …, α_{4}) = δ_{2}(ν_{3}, ν_{4}, ξ_{1}, …, ξ_{4}).

We now know that the complex (8.2) is exact everywhere except possibly in degree 0. However, if p_{i}(t) is the Poincaré series of P_{i}, then p_{2}(t) − p_{1}(t) + p_{0}(t) is

\frac{2(t^{7} + t^{8} + t^{9})}{(1 − t^{3})^{2}} − \frac{2(t^{4} + t^{5})(1 − t + t^{2})}{(1 − t)(1 − t^{3})} + \frac{(1 + 2t + 2t^{2})(1 − t + t^{2})}{(1 − t)(1 − t^{3})} = \frac{1}{(1 − t)^{2}} ,

which is the Poincaré series for A. Hence the complex is exact everywhere.

The remaining relations of type 2 follow trivially from degree considerations. To get the relations of type 3 we use the Product Formula, which yields

γ_{2}(α)γ_{2}(β) = γ_{1}(cor_{H_{2}}^{G}(αβ)) + γ_{2}(c^{∗}(α)(c^{2})^{∗}(β) + (c^{2})^{∗}(α)c^{∗}(β)). (8.4)

This yields the products of any pair of generators from γ_{2}(H^{∗}(H_{2})); for example

X_{2}Y_{2} = γ_{1}(cor_{H_{2}}^{G}(x^{2}y^{2})) + γ_{2}(c^{∗}(x^{2})(c^{∗})^{2}(y^{2}) + (c^{∗})^{2}(x^{2})c^{∗}(y^{2}))
= γ_{1}(u^{2}) + γ_{2}(x^{4} + x^{2}y^{2} + y^{4}) = u^{2}C_{1}.

Now for i ∈ {1, 2}, j ∈ {3, 4}, the Product Formula tells us that the product of any element of γ_{i}(H^{∗}(H_{i})) with one of γ_{j}(H^{∗}(H_{j})) will lie in γ_{j}(H^{∗}(H_{j})), and will hence be 0 unless possibly both elements have degree 0. Finally, the relations C_{i}C_{j} = 0 are easily verified using the identification of 0-degree Hochschild cohomology with the center of 𝔽_{2}G.

It follows from our comments on generators and relations that we have given a presentation of H^{∗}(A_{4}, 𝔽_{2}A_{4}) as an 𝔽_{2}-algebra. It remains only to prove the remark about the isomorphism modulo radicals. This is immediate since the X_{i}, Y_{i} (i = 1, 2), and C_{j} (j = 1, 2, 3) all square to 0.

9. The dihedral 2-groups

Let m ≥ 2 be a power of 2 and G = D_{4m} = ⟨a, b | a^{2m} = b^{2} = 1, bab = a^{−1}⟩.

Proposition 9.1. H^{∗}(D_{4m}, 𝔽_{2}D_{4m}) is generated as a commutative algebra by elements u, v, w, C_{1}, C_{2}, C_{3}, C_{4}, T_{1}, T_{2}, of degrees (resp.) 1, 1, 2, 0, 0, 0, 0, 1, 1, subject to the 10 degree-0 relations

C_{1}^{2} = XC_{2} = C_{1}C_{3} = C_{1}C_{4} = C_{2}^{m} = C_{2}C_{3} = C_{2}C_{4} = C_{3}^{2} = C_{3}C_{4} = C_{4}^{2} = 0,

where

X = C_{1} + ∑_{i=0}^{m/2−2} \binom{m/2 + i}{2i + 2}C_{2}^{2i+2},

the 11 degree-1 relations

vC_{3} = C_{1}T_{1}, vC_{4} = C_{1}T_{2}, vC_{2} = uC_{3} = (u + v)C_{4} = C_{2}T_{1} = C_{2}T_{2} = 0,
C_{3}T_{1} = uX + vC_{1}, C_{3}T_{2} = uC_{2}^{m−1} = C_{4}T_{1}, C_{4}T_{2} = uX,

and the 6 degree-2 relations

u^{2} = uv, uT_{1} = (u + v)T_{2} = T_{1}T_{2} = 0, T_{1}^{2} = v(u + v), T_{2}^{2} = u^{2}.

Note. It is also true in this case that γ_{1} induces an isomorphism modulo radicals, but this is a special case of the more general Theorem 10.1.

In the above and in what follows, we use the convention that the binomial coefficient \binom{n}{k} is 0 if k > n. The rest of this section is devoted to a proof of the proposition.

Generators. The m + 3 conjugacy classes of G are

{1}, {a^{m}}, {a^{r}, a^{−r}}(1 ≤ r ≤ m − 1), {a^{s}b | s even}, {a^{s}b | s odd}.

and from these we choose representatives g_{1} = 1, g_{2} = a^{m}, g_{r+2} = a^{r}, g_{m+2} = b, g_{m+3} = ab. The centralizers are H_{1} = H_{2} = G, H_{r+2} = ⟨a⟩, and the Klein 4-groups H_{m+2} = ⟨a^{m}, b⟩ and H_{m+3} = ⟨a^{m}, ab⟩. We have

H^{∗}(G) = 𝔽_{2}[u, v, w | deg(u) = deg(v) = 1, deg(w) = 2, u^{2} = uv]
H^{∗}(⟨a⟩) = 𝔽_{2}[u_{1}, w_{1} | deg(u_{1}) = 1, deg(w_{1}) = 2, u_{1}^{2} = 0]
H^{∗}(H_{m+j+1}) = 𝔽_{2}[x_{j}, y_{j} | deg(x_{j}) = deg(y_{j}) = 1] (j = 1, 2)

(all coefficients are assumed to be in 𝔽_{2}; see [22] for H^{∗}(G)). We may take u, v to be the basis dual to the basis ā, b̄ of G/⟨a^{2}⟩, and w to be the degree-2 Stiefel-Whitney class of the natural representation of G on the plane. There is only one choice for u_{1} (resp. w_{1}), namely, the unique non-zero element of H^{∗}(⟨a⟩) in degree 1 (resp. 2). Take x_{1}, y_{1} (resp. x_{2}, y_{2}) to be dual to a^{m}, b (resp. a^{m}, ab). Also let H^{∗}(⟨a^{m}⟩) = 𝔽_{2}[z]. As before, we will identify α with γ_{1}(α) (α ∈ H^{∗}(G)). Let E_{i} = γ_{i}(1) (1 ≤ i ≤ m + 3), C_{1} = 1 + E_{2}, C_{2} = E_{3}, C_{3} = E_{m+2}, C_{4} = E_{m+3}, and T_{j} = γ_{m+j+1}(x_{j}) (j = 1, 2).

Lemma 9.2. The restrictions of u to the subgroups ⟨a⟩, H_{m+2}, H_{m+3}, ⟨a^{m}⟩ are (resp.) u_{1}, 0, y_{2}, 0. The restrictions of v are 0, y_{1}, y_{2}, 0. The restrictions of w are w_{1}, x_{1}(x_{1} + y_{1}), x_{2}(x_{2} + y_{2}), z^{2}. Moreover, restriction from ⟨a⟩ to ⟨a^{m}⟩ maps u_{1} to 0 and w_{1} to z^{2}.

Proof. Restrictions from ⟨a⟩ to ⟨a^{m}⟩ follow from [1, Cor. II.5.7]. Of the others, only w presents some difficulties. Its restrictions to H_{m+2} and H_{m+3} follow from [22, Prop. 2.1]. It follows that

res_{⟨a^{m}⟩}^{G}(w) = res_{⟨a^{m}⟩}^{H_{m+2}} res_{H_{m+2}}^{G}(w) = res_{⟨a^{m}⟩}^{H_{m+2}}(x_{1}^{2} + x_{1}y_{1}) = z^{2},

whence res_{⟨a⟩}^{G}(w) = w_{1}.

We claim that H^{∗}(G, 𝔽_{2}G) is generated as an H^{∗}(G)-module by the E_{i} (1 ≤ i ≤ m+3), T_{1}, and T_{2}. For 1 ≤ i ≤ m+1, the restriction from G to H_{i} is surjective (by Lemma 9.2, if i ≥ 3), so E_{i} generates the image of γ_{i}. By Lemma 9.2, the image of restriction to H_{m+2+j} is the subring generated by y_{j} and x_{j}(x_{j} + y_{j}) (j = 1, 2). Hence the proof of our claim is completed by

Lemma 9.3. The polynomial ring 𝔽_{2}[x, y] is generated as a module over the subring 𝔽_{2}[y, x(x + y)] by 1 and x.

Proof. We show that x^{i}y^{j} is in the submodule generated by 1 and x by induction on i + j. This is clear if i + j < 2. If j > 0 then x^{i}y^{j} = y(x^{i}y^{j−1}). If i ≥ 2 then x^{i} = (x^{2} + xy)x^{i−2} + y(x^{i−1}).

Using the identification of degree-0 Hochschild cohomology with the center of 𝔽_{2}G, one can see that E_{4} = C_{2}^{2} and E_{r+2} = E_{r+1}C_{2} + E_{r} (3 ≤ r ≤ m − 1). By

Figure 1. Product Formula data for G = D_{4m}

i  j  x  g_{i}^{x}g_{j} k  y  ^{y}H_{i}  ^{yx}H_{j}  W
1  j  1  g_{j}  j  1  G  H_{j}  H_{j}
2  2  1  1  1  1  G  G  G
r+2 1  a^{r−m} m − r + 2  b  G  ⟨a⟩  ⟨a⟩
m+2 1  a^{m}b m + 2  a^{m/2} G  H_{m+2}  H_{m+2}
m+3 1  a^{m+1}b m + 3  a^{m/2} G  H_{m+3}  H_{m+3}
r+2 s+2 1  a^{r+s} r + s + 2 ^{(0)} 1  ⟨a⟩  ⟨a⟩  ⟨a⟩
2  ^{(1)} 1  ⟨a⟩  ⟨a⟩  ⟨a⟩
2m−r−s+2 ^{(2)} b  ⟨a⟩  ⟨a⟩  ⟨a⟩
b  a^{r−s} r − s + 2 ^{(3)} 1  ⟨a⟩  ⟨a⟩  ⟨a⟩
1  ^{(4)} 1  ⟨a⟩  ⟨a⟩  ⟨a⟩
s − r + 2 ^{(5)} b  ⟨a⟩  ⟨a⟩  ⟨a⟩
m+2 1  a^{r}b m + 2  ^{(6)} a^{−r/2} ⟨a⟩  ⟨a^{m}, a^{−r}b⟩ ⟨a^{m}⟩
m + 3  ^{(7)} a^{\frac{1−r}{2}} ⟨a⟩  ⟨a^{m}, a^{1−r}b⟩ ⟨a^{m}⟩
m+3 1  a^{r+1}b m + 3  ^{(6)} a^{−r/2} ⟨a⟩  ⟨a^{m}, a^{1−r}b⟩ ⟨a^{m}⟩
m + 2  ^{(7)} a^{\frac{−1−r}{2}} ⟨a⟩  ⟨a^{m}, a^{−r}b⟩ ⟨a^{m}⟩
m+2 m+2 1  1  1  1  H_{m+2}  H_{m+2}  H_{m+2}
a^{l}  ^{(8)} a^{−2l} 2 + 2l  b  H_{m+2}  ⟨a^{m}, a^{−2l}b⟩ ⟨a^{m}⟩
a^{m/2}  a^{m}  2  1  H_{m+2}  H_{m+2}  H_{m+2}
m+3 a^{l}  ^{(9)} a^{−1−2l} 3 + 2l  b  H_{m+2}  ⟨a^{m}, a^{−1−2l}b⟩ ⟨a^{m}⟩
m+3 m+3 1  1  1  1  H_{m+3}  H_{m+3}  H_{m+3}
a^{l}  ^{(8)} a^{−2l} 2 + 2l  b  ⟨a^{m}, a^{−1}b⟩ ⟨a^{m}, a^{−1−2l}b⟩ ⟨a^{m}⟩
a^{m/2}  a^{m}  2  1  H_{m+3}  H_{m+3}  H_{m+3}
(0) 1 ≤ r + s < m ^{(1)} r + s = m ^{(2)} r + s > m ^{(3)} r > s ^{(4)} r = s
(5) r < s ^{(6)} r even ^{(7)} r odd ^{(8)} 1 ≤ l < m/2 ^{(9)} 0 ≤ l < m/2

iterating this formula one obtains

(9.1) E_{k} = \begin{cases}∑_{i=0}^{k/2−2} \binom{k/2 − 1 + i}{1 + 2i}C_{2}^{2+2i} & k even and 4 ≤ k ≤ m \\ ∑_{i=0}^{(k−3)/2} \binom{(k − 3)/2 + i}{2i}C_{2}^{1+2i} & k odd and 3 ≤ k ≤ m + 1. \end{cases}

So in fact the powers of C_{2} generate the images of γ_{i} (3 ≤ i ≤ m + 1). From this we obtain the algebra generators of H^{∗}(G, 𝔽_{2}G) listed in the proposition.

Relations. Straightforward calculations in Z(𝔽_{2}G), together with formula (9.1) with k = m + 1 in the case of the product C_{1}C_{2}, yield the degree-0 relations. We now gather the data necessary to verify the remaining relations. In order to apply the Product Formula, we will use the table in Figure 1. The notation for the table is exactly as in Section 4. All values of r and s in the table range between 1 and m − 1. Next, we will need the following.

Lemma 9.4. (i) If F is a proper subgroup of an elementary abelian 2-group E

then cor_{F}^{E} = 0. (ii) cor_{H_{m+2}}^{G} maps y_{1} to 0, x_{1} to u + v, and x_{1}^{2} to v(u + v). (iii) cor_{H_{m+3}}^{G} maps x_{2} to u, y_{2} to 0, and x_{2}^{2} to u^{2}.

(iv) cor_{⟨a^{m}⟩}^{⟨a⟩} maps z to u_{1} and z^{2} to 0.

Proof. Part (i) follows from [10, Lem. 6.3.4]. To prove (ii), note that y_{1} is in the image of restriction from G, so cor_{H_{m+2}}^{G}(y_{1}) = 0. Now cor_{H_{m+2}}^{G}(x_{1}) = λ_{1}u + λ_{2}v for some λ_{1}, λ_{2} ∈ 𝔽_{2}. By the double coset formula,

res_{H_{m+2}}^{G} cor_{H_{m+2}}^{G}(x_{1}) = x_{1} + ∑_{i=1}^{m/2−1} cor_{⟨a^{m}⟩}^{H_{m+2}} res_{⟨a^{m}⟩}^{⟨a^{m},a^{2i}b⟩}((a^{i})^{∗}x_{1}) + (a^{m/2})^{∗}x_{1}
= x_{1} + 0 + (x_{1} + y_{1}) = y_{1}.

The double coset formula also yields res_{H_{m+3}}^{G} cor_{H_{m+2}}^{G}(x_{1}) = 0, since each term involves corestrictions from a proper subgroup of H_{m+3}. On the other hand,

res_{H_{m+2}}^{G} cor_{H_{m+2}}^{G}(x_{1}) = res_{H_{m+2}}^{G}(λ_{1}u + λ_{2}v) = λ_{2}y_{1}
res_{H_{m+3}}^{G} cor_{H_{m+2}}^{G}(x_{1}) = res_{H_{m+3}}^{G}(λ_{1}u + λ_{2}v) = (λ_{1} + λ_{2})y_{2}.

This forces λ_{1} = λ_{2} = 1. Now, x_{1}^{2} + x_{1}y_{1} = res_{H_{m+2}}^{G}(w) and y_{1} = res_{H_{m+2}}^{G}(v), whence

cor_{H_{m+2}}^{G}(x_{1}^{2}) = cor_{H_{m+2}}^{G}(x_{1}y_{1}) = v cor_{H_{m+2}}^{G}(x_{1}) = v(u + v),

completing the proof of (ii). Part (iii) is handled in a similar way. Part (iv) follows from [1, Cor. II.5.7].

Using this information and the Product Formula, one may obtain the remaining relations. The only complications arise in the following cases. When computing C_{3}T_{1} one obtains

C_{3}T_{1} = (u + v)C_{1} + ∑_{j=1}^{m/2−1} γ_{2+2j}(u_{1}),

where the sum is empty in case m = 2. Using γ_{2+2j}(u_{1}) = uE_{2+2j} and formula (9.1) with k = 2 + 2j, we obtain

C_{3}T_{1} = (u + v)C_{1} + ∑_{j=1}^{m/2−1} ∑_{i=0}^{j−1} \binom{j + i}{1 + 2i}uC_{2}^{2+2i}

= (u + v)C_{1} + ∑_{i=0}^{m/2−2} \binom{m/2 + i}{2 + 2i}uC_{2}^{2+2i}.

Similarly, we calculate

C_{3}T_{2} = ∑_{i=1}^{m/2} uE_{1+2i} = ∑_{i=0}^{m/2−1} \binom{m/2 + i}{1 + 2i}uC_{2}^{1+2i}. (9.2)

However,

Lemma 9.5. If 0 ≤ i ≤ m/2 − 2 then \binom{m/2 + i}{1 + 2i} ≡ 0 mod 2.

Proof. Let m/2 = 2^{k}, and let i = a_{0} + a_{1}2 + ⋯ + a_{k−1}2^{k−1} be the base 2 expansion of i. We then have base 2 expansions

m/2 + i = a_{0} + a_{1}2 + a_{2}2^{2} + ⋯ + a_{k}2^{k}
1 + 2i = b_{0} + b_{1}2 + b_{2}2^{2} + ⋯ + b_{k}2^{k},

where a_{k} = 1, b_{0} = 1 and b_{j} = a_{j−1} (1 ≤ j ≤ k). By [19, Lem. 22.4], it suffices to show a_{j} < b_{j} for some j. As i ≤ 2^{k} − 2, at least one of the a_{j} must be equal to 0. Choose the smallest j with a_{j} = 0. If j = 0, then 0 = a_{0} < b_{0} = 1. If j > 0, then b_{j} = a_{j−1} = 1 as j is smallest, so that 0 = a_{j} < b_{j} = 1 in this case as well.

Hence the sum (9.2) reduces to C_{3}T_{2} = uC_{2}^{m−1}. Similar arguments yield C_{4}T_{1} = uC_{2}^{m−1} and C_{4}T_{2} = uX.

Sufficiency. We will now show that we have found a full set of relations by considering the algebra A defined abstractly by our generators and relations. Clearly there is a homomorphism from A onto H^{∗}(G, 𝔽_{2}G). Note also that w does not occur in any of our relations, so A ≅ 𝔽_{2}[w] ⊗_{𝔽_{2}} B, where B is the algebra given by the remaining generators and all the relations.

Direct examination shows that

B_{0} = ⟨1, C_{1}, C_{2}, C_{2}^{2}, …, C_{2}^{m−1}, C_{3}, C_{4}⟩
B_{1} = ⟨u, v, T_{1}, T_{2}, uC_{1}, vC_{1}, C_{1}T_{1}, C_{1}T_{2}, uC_{2}, uC_{2}^{2}, …, uC_{2}^{m−1}⟩
B_{2} = ⟨uv, v^{2}, vT_{1}, vT_{2}, uvC_{1}, v^{2}C_{1}, vC_{1}T_{1}, vC_{1}T_{2}⟩

In particular, the dimensions of the degree-0, 1, and 2 components of B are less than or equal to (resp.) m + 3, m + 7, and 8. Note also that B_{2} is the degree-2 component of Bv, the ideal generated by v. This implies that B_{k} is the degree-k component of Bv for all k ≥ 2: as the generators of B are all of degree 0 or 1, a monomial f of degree k may be factored into a degree-2 factor and a degree-(k − 2) factor. The degree-2 factor is in Bv, so the same is true of f. This implies that multiplication by v is a surjective map from B_{k} to B_{k+1} for all k ≥ 2. Therefore dim(B_{k}) ≤ 8 for all k ≥ 2.

The Poincaré series of a graded vector space having dimension m + 3 in degree 0, m + 7 in degree 1, and 8 in each degree k ≥ 2 is

\frac{(m + 3) + 4t + (1 − m)t^{2}}{1 − t} .

We conclude that the dimension of A in each degree is less than or equal to that given by the series

\frac{(m + 3) + 4t + (1 − m)t^{2}}{(1 − t)(1 − t^{2})} = \frac{(m + 3) + (1 − m)t}{(1 − t)^{2}}.

But this is the Poincaré series of H^{∗}(G, 𝔽_{2}G) obtained from the additive decomposition. Therefore the homomorphism from A onto H^{∗}(G, 𝔽_{2}G) is an isomorphism, and Proposition 9.1 is, at long last, proved.

10. Structure theorems

Our goal in this section is to obtain some general structure theorems for the rings H^{∗}(H, RG).

The p-group case. First we give a result regarding the Hochschild cohomology rings of group algebras of p-groups in characteristic p.

Theorem 10.1. Let 𝔽 be a field of characteristic p, and G a p-group. Then the map of algebras γ_{1} : H^{∗}(G, 𝔽) → H^{∗}(G, 𝔽G) induces an isomorphism modulo radicals.

Proof. Let I be the augmentation ideal of 𝔽G. Then 𝔽G, as an 𝔽G-module under conjugation, is the direct sum of its submodules 𝔽1 and I. Applying the cohomology functor to this splitting, we see that there is an algebra homomorphism π from the Hochschild ring to the ordinary ring satisfying πγ_{1} = 1, and we may identify the kernel of π with H^{∗}(G, I).

It suffices to show that H^{∗}(G, I) is a nilpotent ideal. But from the definition of the cup product, for all n > 0 the multiplication map from H^{∗}(G, I)^{⊗n} to H^{∗}(G, I) factors through the map H^{∗}(G, I^{⊗n}) → H^{∗}(G, I) induced by multiplication in I. As G is a p-group, I is a nilpotent ideal, so for n sufficiently large the map from I^{⊗n} to I is 0. Hence H^{∗}(G, I)^{n} = 0.

A Green functor. We next generalize a weaker consequence of Proposition 3.2 to a group H acting nontrivially on a group G. Combined with a general result of Thévenaz about Green functors, this leads to a structure theorem for H^{∗}(H, RG) in Corollary 10.4.

We introduce the following Green functor. Assign to any subgroup K of H the ring H^{∗}(K, RG), and consider the usual maps conjugation h^{∗}: H^{∗}(K, RG) → H^{∗}(^{h}K, RG) for h ∈ H, and res_{L}^{K} and cor_{L}^{K} for L ≤ K ≤ H. The various properties required of a Green functor in this situation follow from [10, Ch. 4], and will be used in the remainder of this section. In particular, the image of corestriction from a subgroup L of K is an ideal of H^{∗}(K, RG).

Let

\overline{H}^{∗}(H, RG) = H^{∗}(H, RG)/ ∑_{K<H} cor_{K}^{H}(H^{∗}(K, RG))
\overline{H}^{∗}(H) = H^{∗}(H)/ ∑_{K<H} cor_{K}^{H}(H^{∗}(K)).

Let G^{H} denote the subgroup of G consisting of all elements fixed by H.

Theorem 10.2. Let H and G be finite groups with H acting as automorphisms on G. Then

\overline{H}^{∗}(H, RG) ≅ R(G^{H}) ⊗_{R} \overline{H}^{∗}(H)

as graded R-algebras.

In order to prove the Theorem, we will first define a map from H^{∗}(H, RG) to R(G^{H}) ⊗_{R} H^{∗}(H) and examine some of its properties. Consider the RHhomomorphism RG → R(G^{H}) defined on G by sending g to g (if g ∈ G^{H}) and 0 (otherwise). This induces a map from H^{∗}(H, RG) to H^{∗}(H, R(G^{H})). The latter ring is isomorphic to R(G^{H}) ⊗ H^{∗}(H) by Proposition 3.2, and so we obtain a map ψ: H^{∗}(H, RG) → R(G^{H}) ⊗ H^{∗}(H). This map is given explicitly by

ψ(ζ) = ∑_{g∈G^{H}} g ⊗ π_{g}^{∗}(ζ). (10.1)

In general, ψ is not an algebra homomorphism.

Let ρ: H^{∗}(H) → \overline{H}^{∗}(H) denote the quotient map, and \overline{ψ} = (id ⊗ρ) ∘ ψ. In the next lemma, we show that \overline{ψ} is an algebra homomorphism. Finally, we prove that it induces the isomorphism in the theorem.

Lemma 10.3. \overline{ψ}: H^{∗}(H, RG) → R(G^{H}) ⊗_{R} \overline{H}^{∗}(H) is an algebra homomorphism.

Proof. By Lemma 4.2, it suffices to show that

\overline{ψ}(γ_{i}(α) ⌣ γ_{j}(β)) = \overline{ψ}(γ_{i}(α))\overline{ψ}(γ_{j}(β)) (10.2)

for all 1 ≤ i, j ≤ r and α ∈ H^{∗}(H_{i}), β ∈ H^{∗}(H_{j}). By Lemma 5.2 (iii) and (iv) and the definitions of \overline{ψ} and γ_{i}, the right side is nonzero only when both g_{i}, g_{j} ∈ G^{H}, and in this case the right side of equation (10.2) is g_{i}g_{j} ⊗ ρ(α ⌣ β). On the other hand, by Theorem 5.1, the left side of equation (10.2) is equal to

∑_{x∈D}\overline{ψ}γ_{k}(cor_{W}^{H_{k}}(res_{W}^{^{y}H_{i}} y^{∗}α ⌣ res_{W}^{^{yx}H_{j}}(yx)^{∗}β))

= ∑_{g∈G^{H}} ∑_{x∈D} g ⊗ ρπ_{g}^{∗}γ_{k} cor_{W}^{H_{k}}(res_{W}^{^{y}H_{i}} y^{∗}α ⌣ res_{W}^{^{yx}H_{j}}(yx)^{∗}β),

where D is a set of double coset representatives for H_{i}\H/H_{j}, k = k(x) and y = y(x) are chosen to satisfy g_{k} = ^{y}g_{i}^{yx}g_{j}, and W = W (x) = ^{y}H_{i} ∩^{yx}H_{j}. By Lemma 5.2 (iii) and (iv), for each g ∈ G^{H}, π_{g}^{∗}γ_{k} is the identity if g = g_{k} and 0 otherwise. Therefore this sum is equal to

∑_{g∈G^{H}} ∑_{x∈D} δ_{g,g_{k}}g_{k} ⊗ ρ cor_{W}^{H} (res_{W}^{^{y}H_{i}} y^{∗}α ⌣ res_{W}^{^{yx}H_{j}}(yx)^{∗}β).

Now, if g_{i}, g_{j} ∈ G^{H}, then H_{i} = H_{j} = W = H, and this sum contains the single term g_{k} ⊗ρ(α ⌣ β) = g_{i}g_{j} ⊗ρ(α ⌣ β), so in this case we have established equation (10.2). If one of g_{i}, g_{j} is not in G^{H}, then each W appearing in the above sum is a proper subgroup of H, and so the sum is 0, as was the case for the right side of equation (10.2). Therefore \overline{ψ} is a ring homomorphism.

Proof of Theorem 10.2. We will show that the map \overline{ψ} of the Lemma induces the required isomorphism. We first show that \overline{ψ} is surjective. Let α ∈ H^{∗}(H) and h ∈ G^{H}. If we let ζ = θ_{h}^{∗}(α), then ψ(ζ) = ∑_{g∈G^{H}} g ⊗ π_{g}^{∗}θ_{h}^{∗}(α) = h ⊗ α by Lemma 5.2 (iv). Therefore \overline{ψ} is surjective.

Next we will show that ψ factors through \overline{H}^{∗}(H, RG). This will follow once we show that ψ takes the image of cor_{K}^{H} into R(G^{H}) ⊗ Im(cor_{K}^{H}) for each subgroup K of H. Let ζ = γ_{i}(α) be an element of H^{∗}(K, RG), with α ∈ H^{∗}(K_{i}). If g_{i} is not in G^{H}, then ψ(cor_{K}^{H}(ζ)) = 0 by Lemma 5.2 (iii) and (iv) and the definition (10.1) of ψ. If g_{i} ∈ G^{H}, Lemma 5.2 (iii) and (iv) show that ψ(cor_{K}^{H}(ζ)) = g_{i} ⊗ cor_{K}^{H} α, which is in R(G^{H})⊗Im(cor_{K}^{H}). Therefore ψ takes the image of cor_{K}^{H} into R(G^{H})⊗Im(cor_{K}^{H}), and so \overline{ψ} factors through \overline{H}^{∗}(H, RG).

It remains to prove that the kernel of \overline{ψ} is contained in ∑_{K<H} Im(cor_{K}^{H}). Let ζ ∈ Ker(\overline{ψ}), and write ζ = ∑_{i} γ_{i}(α_{i}) with α_{i} ∈ H^{∗}(H_{i}). As we have shown that Im(cor_{K}^{H}) is contained in the kernel of \overline{ψ} for all proper subgroups K of H, and γ_{i}(α_{i}) = cor_{H_{i}}^{H} θ_{i}^{∗}(α_{i}) by definition, we may assume that α_{i} = 0 for all i with g_{i} ∉ G^{H}. Considering the definition (10.1) of ψ, we may further assume that α_{i} is nonzero only for a single fixed value of i, that is ζ = θ_{i}^{∗}(α_{i}) with g_{i} ∈ G^{H}. By the definition of \overline{ψ} and Lemma 5.2 (iv), we have \overline{ψ}(ζ) = g_{i} ⊗ \overline{α_{i}}. As \overline{ψ}(ζ) = 0, this implies α_{i} ∈ ∑_{K<H} Im(cor_{K}^{H}). By applying Lemma 5.2 (iii) to ζ = θ_{i}^{∗}(α_{i}), we see that ζ is in ∑_{K<H} Im(cor_{K}^{H}).

Next consider the ring homomorphism

β: H^{∗}(H, RG) → ∏_{K}(\overline{H}^{∗}(K, RG))^{N_{H}(K)},

where the product is over a set of representatives K of conjugacy classes of subgroups of H, given in the K-component by the composition of res_{K}^{H} with the quotient map. The kernel of β is nilpotent by [29, Thm. 3.2]. By Theorem 10.2, we know the structure of the factors in the above product. Thus we obtain the following corollary.

Corollary 10.4. There is a ring homomorphism, with nilpotent kernel,

β: H^{∗}(H, RG) → ∏_{K}(R(G^{K}) ⊗_{R} \overline{H}^{∗}(K))^{N_{H}(K)},

the product taken over a set of representatives K of conjugacy classes of subgroups of H.

11. Questions

We conclude with a few questions indicated by the results above. The first is suggested by our calculations and the result on p-groups: suppose 𝔽 is a field and G is a finite group such that 𝔽G is indecomposable. Then we may ask

Question 1. Does γ_{1} : H^{∗}(G, 𝔽) → H^{∗}(G, 𝔽G) induce an isomorphism modulo radicals?

As we have seen, Question 1 has an affirmative answer for S_{3} mod 3, A_{4} mod 2, and for any p-group mod p. Of course, if 𝔽G has more than one block, than Question 1 has a negative answer; this is because the center of 𝔽G modulo its radical has dimension equal to the number of blocks, so the map fails to be an isomorphism in degree 0. However, we may refine the question as follows. Suppose now G is any finite group, and consider the map f which is γ_{1} followed by the projection onto H^{∗}(B_{0}, B_{0}), where B_{0} is the principal block of 𝔽G.

Question 2. Does f: H^{∗}(G, 𝔽) → H^{∗}(B_{0}, B_{0}) induce an isomorphism modulo radicals?

Question 2 has an affirmative answer for the above cases, and also for S_{3} mod 2, as is easily verified. It also has an affirmative answer in the case where G is Abelian. For in this case we may identify the Hochschild cohomology ring with 𝔽G⊗_{𝔽} H^{∗}(G), by Prop. 3.2. Under this identification, γ_{1} takes α ∈ H^{∗}(G) to 1⊗α. The principal block component of the Hochschild ring modulo its radical is

\frac{B_{0} ⊗ H^{∗}(G)}{rad(B_{0} ⊗ H^{∗}(G))} ≅ \frac{B_{0}}{rad(B_{0})} ⊗ \frac{H^{∗}(G)}{rad(H^{∗}(G))} ≅ 𝔽 ⊗ \frac{H^{∗}(G)}{rad(H^{∗}(G))} ≅ \frac{H^{∗}(G)}{rad(H^{∗}(G))}

Hence the map modulo radicals induced by f is indeed an isomorphism.

Acknowledgments

We are grateful to Dave Benson, Frank Sottile, and Burt Totaro for helpful conversations during the preparation of this manuscript. We would also like to thank the authors of Macaulay 2 [14] and Singular [15]. Both systems helped us to gain a better understanding of the Hochschild cohomology ring of the mod-2 group algebra of the dihedral group of order 8, which in turn helped us more generally in finding a presentation of the Hochschild cohomology ring of the mod-2 group algebras of all the dihedral 2-groups.

The second author would like to thank the University of Toronto, the University of California at Berkeley, and MSRI for their hospitality during work on this manuscript, as well as NSERC for providing research support.

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Department of Mathematics & Statistics, University of Massachusetts, Amherst, MA 01003-4515

   E-mail address: siegel@math.umass.edu

Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada

   E-mail address: sjw@math.toronto.edu