Green functors, crossed G-monoids, and Hochschild constructions

Serge Bouc

Institut de Mathématiques de Jussieu Université Paris 7-Denis Diderot, 75251, Paris Cedex 05, France email: bouc@math.jussieu.fr

1. Introduction

Let G be a finite group, and R be a commutative ring. This note proposes a generalization to any Green functor for G over R of the construction of the Hochschild cohomology ring HH^{∗}(G, R) from the ordinary cohomology functor H^{∗}(−, R). Another special case is the construction of the crossed Burnside ring of G from the ordinary Burnside functor.

The general abstract setting is the following : let A be a Green functor for the group G. Let G^{c} denote the group G, on which G acts by conjugation. Suppose Γ is a crossed G-monoid, i.e. that Γ is a G-monoid over the G-group G^{c}. Then the Mackey functor A_{Γ} obtained from A by the Dress construction has a natural structure of Green functor. In particular A_{Γ}(G) is a ring.

In the case where Γ is the crossed G-monoid G^{c}, and A is the cohomology functor (with trivial coefficients R), the ring A_{Γ}(G) is the Hochschild cohomology ring of G over R. If A is the Burnside functor for G over R, then the ring A_{Γ}(G) is the crossed Burnside ring of G over R.

This note presents some properties of those Green functors A_{Γ}, and the functorial relations between the corresponding categories of modules. In particular, it states a general formula for the product in the ring A_{Γ}(G), shedding a new light on a result of S. Siegel and S. Witherspoon ([6]), which was conjectured by C. Cibils ([3]) and C. Cibils and A. Solotar ([4]).

2. Green functors and G-sets

For the various definitions of Mackey and Green functors for a finite group G over a commutative ring R, the reader is referred to [2]. The definition in use here is the one in terms of G-sets : a Mackey functor for G over R is a bivariant functor from the category of finite G-sets to the category of R-modules, which transforms disjoint unions into direct sums, and has some compatibility property with cartesian squares (see [2] 1.1.2 for details).

A Green functor A for G over R is a Mackey functor for G over R, together with product maps A(X) ⊗_{R} A(Y) → A(X × Y), for any finite G-sets X and Y, which are denoted by (a, b) ↦ a × b. Those maps have to be bivariant, associative, and unital in some suitable sense (see [2] 2.2 for details).

Mackey and Green functors for G over R are naturally the objects of categories, denoted respectively by Mack_{R}(G) and Green_{R}(G). The category Mack_{R}(G) is an abelian category, whereas Green_{R}(G) should be viewed as a generalization of the category of R-algebras.

Date : January 28, 2002

Remark 2.1 : One can recover the usual definition of Mackey and Green functors by setting A(H) = A(G/H), for a subgroup H of G. The ordinary product on A(H) can be recovered by setting

a.b = A^{∗}(δ_{G/H})(a × b)

for a, b ∈ A(H), where δ_{G/H} is the diagonal inclusion from G/H to (G/H) × (G/H), and A^{∗}(δ_{G/H}) is the image of δ_{G/H} by the contravariant part of the bivariant functor A.

Example 2.2 : Let X and Y be finite G-sets, and set

H^{⊕}(G, RX) = ⨁_{n=0}^{∞} H^{n}(G, RX) .

Then the cup-product on cohomology give maps

H^{⊕}(G, RX) × H^{⊕}(G, RY) → H^{⊕}(G, RX ⊗_{R} RY)

and identifying RX ⊗_{R} RY with R(X × Y), this gives cross product maps

H^{⊕}(G, RX) × H^{⊕}(G, RY) → H^{⊕}(G, R(X × Y)) .

This gives H^{⊕}(G, −) a Green functor structure, and if K is a subgroup of G, the induced ring structure on

H^{⊕}(G, R(G/K)) ≅ H^{⊕}(K, R)

coincides with the ordinary ring structure of H^{⊕}(K, R) for cup-products.

Example 2.3 : Let B denote the Burnside functor. If X is a finite G-set, then B(X) is the Grothendieck group of the category of G-sets over X. The obvious product

( \begin{matrix}Z \\ ↓ \\ X \end{matrix} , \begin{matrix}T \\ ↓ \\ Y \end{matrix} ) ↦ \begin{matrix}Z × T \\ ↓ \\ X × Y \end{matrix}

extends linearly to a cross product B(X) × B(Y) → B(X × Y), which gives B its structure of Green functor ([2] 2.4).

3. The Dress construction and crossed G-monoids

The Dress construction is a fundamental endo-functor of the category Mack_{R}(G), defined as follows. Let Γ be a fixed finite G-set. If M is a Mackey functor for G over R, then the Mackey functor M_{Γ} is the bivariant functor defined on the finite G-set Y by M_{Γ}(Y) = M(Y × Γ). If f : Y → Z is a map of G-sets, then (M_{Γ})_{∗}(f) = M_{∗}(f × Id_{Γ}) and (M_{Γ})^{∗}(f) = M^{∗}(f × Id_{Γ}). One checks easily ([2] 1.2) that M_{Γ} is a Mackey functor for G over R.

It follows from this definition that the evaluation of M_{Γ} at the trivial G-set • = G/G is equal to M_{Γ}(G) = M_{Γ}(•) ≅ M(Γ).

When A is a Green functor for G over R, and when the G-set Γ has some additional structure (see below), then A_{Γ} is another Green functor for G over R.

Definition 3.1 : Let G be a finite group. A crossed G-monoid (Γ, φ) is a pair consisting of a finite monoid Γ with a left action of G by monoid automorphisms (denoted by (g, γ) ↦ gγ or (g, γ) ↦ ^{g}γ, for g ∈ G and γ ∈ Γ), and a map of G-monoids φ from Γ to G^{c} (i.e. a map φ which is both a map of monoids and a map of G-sets). A morphism of crossed G-monoids from (Γ, φ) to (Γ′, φ′) is a map of G-monoids θ : Γ → Γ′ such that φ′ ∘ θ = φ.

A crossed G-group (Γ, φ) is a crossed G-monoid for which Γ is a group.

Remark 3.2 : Generally the map φ : Γ → G^{c} will be clear from context, and will be understood in the notation.

Example 3.3 :

1. Let H be a normal subgroup of G, and φ be the inclusion homomorphism from H to G. Then H^{c} = (H, φ) is a crossed G-group.

2. Let Γ be any G-monoid (i.e. any monoid with a left action of G by monoid automorphisms). Let u be the trivial monoid homomorphism from Γ to G. Then Γ^{u} = (Γ, u) is a crossed G-monoid.

3. Let (Γ, φ) be a crossed G-monoid. Then φ(Γ) is a normal subgroup of G, and φ^{−1}(1) is a G-submonoid of Γ. There is a natural inclusion of crossed G-monoids from φ^{−1}(1)^{u} to (Γ, φ), and a natural surjection from (Γ, φ) to φ(Γ)^{c}.

4. Let 𝔼 be a group of cardinality 1, with trivial G-action. Let u : 𝔼 → G^{c} be the map sending the unique element of 𝔼 to the identity of G. Then (𝔼, u) is an initial object in the category of crossed G-monoids. On the other hand the crossed G-monoid G^{c} = (G, Id_{G}) is a final object in the category of crossed G-monoids.

Notation 3.4 : Let (Γ, φ) be a crossed G-monoid. If X is any G-set, there is a natural monoid action of Γ on X, denoted by (γ, x) ∈ Γ × X ↦ γ.x ∈ X and defined by γ.x = φ(γ)x .

4. The Green functor structure on A_{Γ}

Let R be a commutative ring, and Γ be a crossed G-monoid. If A is a Green functor for G over R, then the Dress construction gives a Mackey functor A_{Γ}, whose evaluation at the G-set X is A_{Γ}(X) = A(X × Γ). If X and Y are finite G-sets, define maps

A_{Γ}(X) ⊗_{R} A_{Γ}(Y) → A_{Γ}(X × Y) : a ⊗ b ↦ a ×_{Γ} b = A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (a × b) .

The notation A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) means A_{∗}(f), where f = (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) is the map from X × Γ × Y × Γ

to X × Y × Γ sending (x, γ_{1}, y, γ_{2}) to (x, γ_{1}.y, γ_{1}γ_{2}), and A_{∗} is the covariant part of A.

This definition makes sense, since the map f is a map of G-sets if Γ is a crossed G-monoid. Moreover if a ∈ A(X × Γ) and b ∈ A(Y × Γ), then a × b ∈ A(X × Γ × Y × Γ), hence a ×_{Γ} b ∈ A(X × Y × Γ) = A_{Γ}(X × Y).

Let moreover ε_{A_{Γ}} denote the element A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) (ε_{A}) of A(Γ) = A_{Γ}(•), where (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) is the map sending the unique element of • to the identity of Γ, and ε_{A} ∈ A(•) is the unit element of A.

Theorem 4.1 : The functor A_{Γ} is a Green functor for G over R, with unit ε_{A_{Γ}}. Moreover the correspondence A ↦ A_{Γ} is an endo-functor of the category Green_{R}(G).

Proof : The proof is a series of straightforward verifications.

Remark 4.2 : The evaluation at the trivial G-set of the Green functor A_{Γ} is A_{Γ}(•) ≅ A(Γ), and with this identification the product on A(Γ) is given by

(a, b) ∈ A(Γ) × A(Γ) ↦ A_{∗} (\begin{matrix}γ_{1},γ_{2} \\ ↓ \\ γ_{1}γ_{2} \end{matrix}) (a × b) .

An explicit version of this product formula will be given in theorem 5.1.

Proposition 4.3 : Let f : (Γ′, φ′) → (Γ, φ) be a morphism of crossed G-monoids.

1. For any G-set X, denote by A_{f,X} the map A_{∗}(Id_{X} × f) from A_{Γ′}(X) to A_{Γ}(X). Then these maps A_{f,X} define a morphism of Green functors A_{f} from A_{Γ′} to A_{Γ}.

2. Moreover, if f in injective, then A_{f} is a split injection of Mackey functors.

3. In particular, the inclusion (𝔼, u) → (Γ, φ) induces a morphism of Green functors ι : A → A_{Γ}, which is a split injection of Mackey functors.

Proof : Here again, the proof is straightforward, except maybe for the last assertion : if f is injective, then the maps A_{X}^{f} = A^{∗}(Id_{X} × f) define a morphism of Mackey functors A^{f} from A_{Γ} to A_{Γ′} ([2] 1.2), which is a section to A_{f}.

5. The product formula

The following product formula for the ring A_{Γ}(G) is is just a reformulation of the definition, using the translation between the different definitions of Green functors.

Theorem 5.1 : Let A be a Green functor for G over R, and Γ be a crossed G-monoid. Then

A_{Γ}(G) = A(Γ) = (⨁_{γ∈Γ} A(G_{γ}))^{G}

and for γ ∈ Γ, the γ-component of the product of the elements a and b of A(Γ) is given by

(a ×_{Γ} b)_{γ} = ∑_{{(α,β)∈G_{γ}\(Γ×Γ) \atop αβ=γ}} t_{G_{(α,β)}}^{G_{γ}} (r_{G_{(α,β)}}^{G_{α}}a_{α}.r_{G_{(α,β)}}^{G_{β}}b_{β}) .

Remark 5.2 : One can write this formula after taking sets of representatives for the action of G on Γ. In this form, when A is the ordinary cohomology functor, and Γ = G^{c}, it was the conjecture of Cibils and Solotar mentioned in the introduction. Theorem 5.1 shows that in the proof of this conjecture by Siegel and Witherspoon ([6] Theorem 5.1), the essential point is that cup products for Hochschild cohomology and for ordinary cohomology are the same. The rest of the proof appears as a formal consequence of the underlying Green functor structure.

An easy corollary of the product formula is the following :

Corollary 5.3 : Let H be a normal subgroup of G. Suppose that A is a (graded) commutative Green functor. If for any subgroup K of G, the group H ∩ C_{G}(K) acts trivially on A(K), then the ring A(H^{c}) is (graded) commutative.

Remark 5.4 : Corollary 5.3 shows in particular that the crossed Burnside ring of G is commutative. Similarly, the Hochschild cohomology ring of G is graded commutative. This was first proved by Gerstenhaber ([5]).

6. Semi-direct products of crossed G-monoids

Theorem 4.1 shows that the correspondence A ↦ A_{Γ} is an endo-functor of Green_{R}(G). It is natural to compose those endo-functors, and this leads to the notion of semi-direct product of crossed G-monoids. All the following results are straightforward :

Proposition 6.1 : Let (Γ, φ) and (Γ′, φ′) be crossed G-monoids. Let Γ” denote the direct product Γ′ × Γ, with diagonal G-action. Define the following multiplication on Γ” :

(γ′_{1}, γ_{1})(γ′_{2}, γ_{2}) = (γ′_{1}(γ_{1}.γ′_{2}), γ_{1}γ_{2}) ∀γ_{1}, γ_{2} ∈ Γ, ∀γ′_{1}, γ′_{2} ∈ Γ′ .

Define φ” : Γ” → G^{c} by φ”(γ′, γ) = φ′(γ′)φ(γ) for all γ ∈ Γ and γ′ ∈ Γ′. Then (Γ”, φ”) is a crossed G-monoid, with identity (1_{Γ′}, 1_{Γ}).

Definition 6.2 : The crossed G-monoid (Γ”, φ”) of proposition 6.1 is called the semi-direct product of the crossed G-monoids (Γ′, φ′) and (Γ, φ), and it is denoted by (Γ′, φ′) ⋊ (Γ, φ), or Γ′ ⋊ Γ for short.

Proposition 6.3 : Let A be a Green functor for G over R. If Γ and Γ′ are crossed G-monoids, then the Green functor (A_{Γ})_{Γ′} is naturally isomorphic to A_{Γ′⋊Γ}.

7. From A-modules to A_{Γ}-modules

There is a natural notion of module over a Green functor (see [2] 2.2), and it follows in particular from proposition 4.3 that there is a functor of restriction r_{Γ} along the Green functor homomorphism ι : A → A_{Γ}, from the category A_{Γ}-Mod of A_{Γ}-modules to the category A-Mod. This section describes a functor i_{Γ} from A-Mod to A_{Γ}-Mod.

Notation 7.1 : Let A be a Green functor for G over R, and M be an A-module. If X and Y are finite G-sets, if a ∈ A_{Γ}(X) and m ∈ M(Y), denote by a ×_{Γ} m the element of M(X × Y) defined by

a ×_{Γ} m = M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) (a × m) ∈ M(X × Y).

Theorem 7.2 : Let Γ be a crossed G-monoid, and let A be a Green functor for G over R.

1. If M is an A-module, then the product (a, m) ∈ A_{Γ}(X) × M(Y) ↦ a ×_{Γ} m ∈ M(X × Y) endows M with a structure of A_{Γ}-module, denoted by i_{Γ}(M).

2. If f : M → N is a morphism of A-modules, then the maps f_{X} : M(X) → N(X) define a morphism i_{Γ}(f) of A_{Γ}-modules from i_{Γ}(M) to i_{Γ}(N).

3. This defines a functor i_{Γ} from A-Mod into A_{Γ}-Mod, which is is a full embedding.

8. Centres and centralizers

Let A be a Green functor for G over R. If M is a Mackey subfunctor of A, one can define the commutant C_{A}(M) of M in A. It is is a Green subfunctor of A ([2] 6.5.3).

If X is a finite G-set, define ζ_{A}(X) as the set of natural transformations from the identity functor I of A-Mod to the endo-functor I_{X} of A-Mod given by the Dress construction associated to X. In section 12.2 of [2], it is shown that ζ_{A} has a natural structure of Green functor. Its evaluation at the trivial G-set is the center of the category A-Mod, i.e. the set of natural transformations from the identity functor of A-Mod to itself.

Theorem 8.1 : Let Γ be a crossed G-monoid, and A be a Green functor. Let C(A, Γ) denote the commutant of ι(A) in A_{Γ}. If X and Y are finite G-sets, if M is an A-module, and if α ∈

C(A, Γ)(X), define a map z_{X}(α)_{M,Y} : M(Y) → M(Y × X) by z_{X}(α)_{M,Y} (m) = M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) (α ×_{Γ} m).

Then :

1. For given X, α and M, the maps z_{X}(α)_{M,Y} define a morphism of A-modules z_{X}(α)_{M} from M to M_{X}.

2. For given X and α, these morphisms z_{X}(α)_{M} define an element z_{X}(α) of ζ_{A}(X).

3. The maps z_{X} define a morphism of Green functors z from C(A, Γ) to ζ_{A}.

Remark 8.2 : Theorem 8.1 provides in particular a natural ring homomorphism from C(A, Γ)(•) to the center of the category A-Mod. If A is the Burnside functor B, and Γ = G^{c}, then actually C(A, Γ) = A_{Γ}, and the previous ring homomorphism is the natural morphism from the crossed Burnside ring of G over R to the center of the Mackey algebra of G over R. This morphism leads in particular to a description of the block idempotents of the Mackey algebra ([1]).

References

[1] S. Bouc. The p-blocks of the Mackey algebra. Preprint (2001), to appear in “Algebra and Representation Theory”.

[2] S. Bouc. Green-functors and G-sets, volume 1671 of Lecture Notes in Mathematics. Springer, October 1997.

[3] C. Cibils. Tensor product of Hopf bimodules over a group algebra. Proc. Amer. Math. Soc., 125:1315–1321, 1997.

[4] C. Cibils and A. Solotar. Hochschild cohomology algebra of abelian groups. Arch. Math., 68:17–21, 1997.

[5] M. Gerstenhaber. The cohomology structure of an associative ring. Ann. Math., 78:267–288, 1963.

[6] S. F. Siegel and S. J. Witherspoon. The Hochschild cohomology ring of a group algebra. Proc. London Math. Soc., 79:131–157, 1999.

[AMA - Algebra Montpellier Announcements - 01-2002] [February 2002]
Received October 2001, revised January 2002.