Hochschild constructions for Green functors

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 paper provides a possible generalization to any Green functor for G over R of the construction of the Hochschild cohomology ring HH^{∗}(G, R) of the group algebra RG from the ordinary cohomology functor H^{∗}(−, R). This construction also involves as a special case 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 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 article presents some properties of those Green functors A_{Γ}, and the functorial relations between the corresponding categories of modules. Instead of exposing first the general framework, and since the Green functor setting is rather abstract, the first sections only deal with the two above special cases.

In particular, they provide a new proof for a result of S. Siegel and S. Witherspoon ([6]), which was conjectured by C. Cibils ([3]) and C. Cibils and A. Solotar ([4]). This result describes the multiplicative structure of the Hochschild cohomology ring of a finite group in terms of cup products, transfers and restrictions on the ordinary cohomology.

The case of the crossed Burnside ring can be viewed as a conceptual reason for the existence of a ring homomorphism from the crossed Burnside ring to the center of the Mackey algebra, which was used in [1] to determine the p-blocks of the Mackey algebra.

Date : February 28, 2002

The next sections present the general theoretic setting : first the definition of crossed G-monoids, and then the construction of the Green functor structure on A_{Γ}, for a Green functor A and a crossed G-monoid Γ. A general formula for the product in the ring A_{Γ}(G) is stated. Here one more example is built : it is possible to attach to any normal subgroup N of a finite group G a kind of “relative” Hochschild cohomology ring HH^{∗}(G, N, R), which is the ordinary cohomology ring for N = 1, and the usual Hochschild cohomology ring for N = G.

The next section is an exposition of the functorial relations between categories of A-modules and A_{Γ}-modules. These relations explain in particular the existence of a natural ring homomorphism from the crossed Burnside ring to the center of the Mackey algebra.

2. Green functors and G-sets

When dealing with Mackey and Green functors, two different (but equivalent) points of view are possible : in the first one, a Mackey functor for G over R is a collection of R-modules M(H), indexed by the subgroups H of G, together with transfer maps t_{H}^{K} : M(H) → M(K), with restrictions maps r_{H}^{K} : M(K) → M(H), and conjugation maps c_{x,H} : M(H) → M(^{x}H), whenever H ⊆ K are subgroups of G, and x ∈ G. These maps are subject to a list of conditions : transitivity, commutation, triviality, and Mackey formula.

In the second point of view, 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).

Similarly, a Green functor A is a Mackey functor “with a compatible ring structure” : each evaluation A(H) has a ring structure (with unit), for a product (a, b) ↦ a.b, restrictions maps and conjugation maps are homomorphisms of rings (with unit), and the Frobenius relations on products and transfers hold. There is also a definition of Green functors in terms of G-sets ([2] 2.2) :

Definition 2.1 : Let R be a commutative ring. A Green functor A (over R) for the group G is a Mackey functor (over R) endowed for any G-sets X and Y with bilinear maps

A(X) × A(Y) → A(X × Y)

denoted by (a, b) ↦ a × b which are bifunctorial, associative, and unitary, in the following sense:

• (Bifunctoriality) If f : X → X′ and g : Y → Y′ are morphisms of G-sets, then the squares

A(X) × A(Y) \underrightarrow{×} A(X × Y)
A_{∗}(f) × A_{∗}(g) ↓  ↓ A_{∗}(f × g)
A(X′) × A(Y′) \overrightarrow{×} A(X′ × Y′)
A(X) × A(Y) \underrightarrow{×} A(X × Y)
A^{∗}(f) × A^{∗}(g) ↑  ↑ A^{∗}(f × g)
A(X′) × A(Y′) \overrightarrow{×} A(X′ × Y′)

are commutative.

• (Associativity) If X, Y and Z are G-sets, then the square

A(X) × A(Y) × A(Z) \underrightarrow{Id_{A(X)} × (×)} A(X) × A(Y × Z)
(×) × Id_{A(Z)} ↓  ↓ ×
A(X × Y) × A(Z)  \overrightarrow{×}  A(X × Y × Z)

is commutative, up to identifications (X × Y) × Z ≃ X × Y × Z ≃ X × (Y × Z).

• (Unitarity) If • denotes the G-set with one element, there exists an element ε_{A} ∈ A(•) such that for any G-set X and for any a ∈ A(X)

A_{∗}(p_{X})(a × ε_{A}) = a = A_{∗}(q_{X})(ε_{A} × a)

denoting by p_{X} (resp. q_{X}) the (bijective) projection from X × • (resp. from • × X) to X.

If A and B are Green functors for the group G, a morphism f (of Green functors) from A to B is a morphism of Mackey functors such that for any G-sets X and Y, the square

A(X) × A(Y) \underrightarrow{×} A(X × Y)
f_{X} × f_{Y} ↓  ↓ f_{X×Y}
B(X) × B(Y) \overrightarrow{×} B(X × Y)

is commutative. The composition of morphisms of Green functors is defined in the obvious way. The category of Green functors for G over R is denoted by Green_{R}(G).

Remark 2.2 : One can pass from the second definition of Green functors to the first one by the following procedure : suppose that A is a Green functor in the second sense. Then setting A(H) = A(G/H), for a subgroup H of G, one can define the maps t_{H}^{K} and r_{H}^{K}, for H ⊆ K by

t_{H}^{K} = A_{∗}(p_{H}^{K}) r_{H}^{K} = A^{∗}(p_{H}^{K})

where p_{H}^{K} : G/H → G/K is the natural projection. If x ∈ G, then the conjugation map c_{x,H} is defined by

c_{x,H} = A_{∗}(γ_{x,H})

where γ_{x,H} : G/H → G/^{x}H maps gH to gx^{−1} ^{x}H. One can also define a product (a, b) ↦ a.b on A(H) by

a.b = A^{∗}(δ_{G/H})(a × b)

where δ_{G/H} is the diagonal inclusion from G/H to (G/H) × (G/H).

Remark 2.3 : One can pass from the first definition to the second by the following construction ([2] 2.3) : suppose that A is a Green functor in the first sense. Then setting A(G/H) = A(H) leads by linearity to a definition of A(X) for any finite G-set X : one can set

A(X) = (⨁_{x∈X} A(G_{x}))^{G}

where the exponent denotes fixed points under the natural action of G on ⨁_{x∈X} A(G_{x}) by permutation of the components, and G_{x} is the stabilizer of x

in G. If f : X → Y is a map of G-sets, then the maps A_{∗}(f) and A^{∗}(f) are defined by

A_{∗}(f)(u)_{y} = ∑_{x∈[G_{y}\f^{−1}(y)]} t_{G_{x}}^{G_{y}}(u_{x})  for u ∈ A(X) and y ∈ Y
A^{∗}(f)(v)_{x} = r_{G_{x}}^{G_{f(x)}}v_{f(x)}  for v ∈ A(Y) and x ∈ X

where G_{y} is the stabilizer of y in G, and [G_{y}\f^{−1}(y)] is a set of representatives of the orbits of G_{y} on f^{−1}(y). The summation in the formula does not depend on the choice of such a set of representatives.

The product of elements a ∈ A(X) and b ∈ A(Y) is defined as the element of A(X × Y) whose component in A(G_{(x,y)}), for (x, y) ∈ X × Y, is equal to

(a × b)_{x,y} = r_{G_{(x,y)}}^{G_{x}}(a_{x}).r_{G_{(x,y)}}^{G_{y}}(b_{y}) .

One can also express the previous formulae after choosing sets of representatives of orbits of G on X, Y, and X×Y. For instance, there is an isomorphism

A(X) ≅ ⨁_{x∈[G\X]} A(G_{x})

where [G\X] is a set of representatives of orbits of G on X. Now if H and K are subgroups of G, then there is an isomorphism of G-sets

(G/H) × (G/K) ≅ ⨆_{x∈[H\G/K]} G/(H ∩^{x}K)

hence an isomorphism

A((G/H) × (G/K)) ≅ ⨁_{x∈[H\G/K]} A(H ∩^{x}K) .

and the product a × b of a ∈ A(G/H) and b ∈ A(G/K) is the element

⨁_{x∈[H\G/K]} r_{H∩^{x}K}^{H}a . r_{H∩^{x}K}^{^{x}K}^{x}b ∈ ⨁_{x∈[H\G/K]} A(H ∩^{x}K)

where ^{x}b = c_{x,K}(b).

Remark 2.4 : There is an alternative procedure to pass from the first definition of Green functors to the second one, which is often easier to handle for computations. It uses co-invariants rather than invariants : if A is a Green functor for G in the first sense, and X is a finite G-set, then one can set

A(X) = (⨁_{x∈X} A(G_{x}))_{G}

where the subscript denotes co-invariants (i.e. the largest quotient with trivial G-action). If x ∈ X and u ∈ A(G_{x}), denote by [x, u]_{G} the image in A(X) of the element u of A(G_{x}). Then if f : X → Y is a map of G-sets

A_{∗}(f)([x, u]_{G}) = [f(x), t_{G_{x}}^{G_{f(x)}}u]_{G}  for x ∈ X and u ∈ A(G_{x})
A^{∗}(f)([y, v]_{G}) = ∑_{x∈G_{y}\f^{−1}(y)}[x, r_{G_{x}}^{G_{y}}v]_{G} for y ∈ Y and v ∈ A(G_{y}) .

The product of the elements [x, u]_{G} and [y, v]_{G} is given by

[x, u]_{G} × [y, v]_{G} = ∑_{w∈[G_{x}\G/G_{y}]}[(x, wy), r_{G_{x}∩G_{wy}}^{G_{x}}u.r_{G_{x}∩G_{wy}}^{G_{wy}}^{w}v]_{G} .

Example 2.5 : Let X and Y be finite G-sets, and set

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

This is usually denoted by H^{∗}(G, RX), but this notation could be confusing here with the Mackey functor formalism. Then the cup-product on cohomology gives 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)) .

It is easy to check that this gives H^{⊕}(G, −) a Green functor structure, and that 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 2.6 : 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

The Dress construction is a fundamental endofunctor of the category of Mackey functors for G over R, 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}) (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 the definitions that the evaluation of M_{Γ} at the trivial G-set • = G/G is equal to

M_{Γ}(•) = M(Γ) .

Suppose now that Γ is a finite G-monoid, i.e. that Γ is a finite monoid on which G acts by monoid automorphisms. Let A be a Green functor for G over R. Apart from the product on A(Γ) defined by the Green functor structure, defined by

(a, b) ∈ A(Γ) ↦ a.b = A^{∗} (\begin{matrix}γ \\ ↓ \\ γ,γ \end{matrix}) (a × b) ,

where (\begin{matrix}γ \\ ↓ \\ γ,γ \end{matrix}) denotes the diagonal inclusion of Γ into Γ × Γ, there is another natural product on A(Γ), defined by

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

where A_{∗} (\begin{matrix}γ_{1},γ_{2} \\ ↓ \\ γ_{1}γ_{2} \end{matrix}) denotes the image by A_{∗} of the multiplication Γ × Γ → Γ.

Let (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) denote the map from • to Γ sending the unique element of • to the

unit of Γ. Set ε_{A_{Γ}} = A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) (ε_{A}).

Lemma 3.1 : The above product gives A(Γ) a ring structure, with unit ε_{A_{Γ}}.

Proof : This will follow from theorem 5.1 below, since A(Γ) will be the evaluation at the trivial G-set of the Green functor A_{Γ^{u}}.

Notation 3.2 : In the sequel, for a Green functor A for G over R, and a G-monoid Γ, the ring A(Γ) will always be understood as the R-module A(Γ), equipped with the product (a, b) ↦ a ×_{Γ} b.

Example 3.3 : Let A denote the cohomology functor H^{⊕}(G, −), and Γ = G^{c} denote the group G, on which G acts by conjugation. Then

A(Γ) = A(G^{c}) = H^{⊕}(G, RG^{c})

is isomorphic to the Hochschild cohomology HH^{⊕}(G, R), as an R-module. But Proposition 3.1 of [6] shows that this isomorphism is actually a ring isomorphism.

Example 3.4 : Let B denote the Burnside functor for G, and Γ = G^{c} denote the group G, on which G acts by conjugation. Then

A(Γ) = B(G^{c})

is isomorphic to the crossed Burnside ring B^{c}(G) of G, as a ℤ-module. It is clear moreover from the definitions of the ring structure on B^{c}(G) ([1] 2.1) that this is actually a ring isomorphism.

Example 3.5 : Let k be a commutative ring, and let M_{k}(G) denote the Grothendieck group of the category of finitely generated kG-modules, for relations given by direct sum decompositions. The usual operations of induction and restriction of modules endow M_{k} with a structure of Mackey functor, and the tensor product of modules (over k) gives a structure of Green functor on M_{k} (for G, over ℤ). The ring M_{k}(G) is usually called the Green ring of kG-modules.

Let Γ = G^{c} denote the group G, on which G acts by conjugation, as in the previous example. Then the ring M_{k}(Γ) is isomorphic to the Grothendieck ring of Hopf bimodules for the Hopf algebra kG. This was originally a question of Cibils, and follows easily from his theorems 2.1 and 3.1 in [3], using the product formula of theorem 6.1.

4. Crossed G-monoids

Definition 4.1 : 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 4.2 : Generally the map φ : Γ → G^{c} will be clear from context, and will be understood in the notation.

Proposition 4.3 : 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 .

This action of Γ has the following properties :

1. If X is a G-set, if g ∈ G, if γ ∈ Γ and x ∈ X, then

g(γ.x) = gγ.gx .

2. If f : X → Y is a morphism of G-sets, then for all γ ∈ Γ and all

x ∈ X

f(γ.x) = γ.f(x) .

Moreover the square

Γ × X \underrightarrow{Id_{Γ} × f} Γ × Y
↓  ↓
X  \overrightarrow{f} Y

is cartesian, where the vertical arrows are given by the actions of Γ on X and Y.

3. If X is a G-set and if θ : (Γ, φ) → (Γ′, φ′) is a morphism of crossed G-monoids, then for all γ ∈ Γ and all x ∈ X

θ(γ).x = γ.x .

Proof : The only non-obvious point concerns the cartesian square in 2). Suppose x ∈ X and (γ, y) ∈ Γ × Y are such that f(x) = γ.y. Then setting z = φ(γ)^{−1}x gives an element (γ, z) ∈ Γ×X such that (Id_{Γ}×f)(γ, z) = (γ, y) and γ.z = x. And if (γ′, z′) is another pair in Γ × X satisfying these two conditions, then γ = γ′ and γ.z = γ′.z, thus φ(γ)z = φ(γ)z′, and z = z′.

Remark 4.4 : Property 1) is equivalent to saying that the action (γ, x) ↦ γ.x of Γ on X is a map of G-sets from Γ × X to X. Property 2) implies in particular that if X and Y are two G-sets, if (x, y) ∈ X × Y and γ ∈ Γ, then γ.(x, y) = (γ.x, γ.y).

Example 4.5 : 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.

Example 4.6 : 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.

Example 4.7 : 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}.

Example 4.8 : 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 unit 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.

5. 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} defined on the G-set X by

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

by

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 mapping (x, γ_{1}, y, γ_{2}) to (x, γ_{1}.y, γ_{1}γ_{2}).

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}(•).

Theorem 5.1 : 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 : First A_{Γ} is a Mackey functor for G over R. Moreover, the product ×_{Γ} is bifunctorial : suppose that f : X → X′ and g : Y → Y′ are maps of G-sets. I must check that for any a ∈ A_{Γ}(X) and b ∈ A_{Γ}(Y)

(A_{Γ})_{∗}(f)(a) ×_{Γ} (A_{Γ})_{∗}(g)(b) = (A_{Γ})_{∗}(f × g)(a ×_{Γ} b) .

This is equivalent to

A_{∗}(f × Id_{Γ})(a) ×_{Γ} A_{∗}(g × Id_{Γ})(b) = A_{∗}(f × g × Id_{Γ})(a ×_{Γ} b)  (5.2)

The right hand side of this equation is equal to

A_{∗}(f × g × Id_{Γ})A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (a × b) = A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ f(x),g(γ_{1}.y),γ_{1}γ_{2} \end{matrix}) (a × b) .

The left hand side of equation (5.2) is equal to

A_{∗} (\begin{matrix}x′,γ_{1},y′,γ_{2} \\ ↓ \\ x′,γ_{1}.y′,γ_{1}γ_{2} \end{matrix}) (A_{∗}(f × Id_{Γ})(a) × A_{∗}(g × Id_{Y})(b)) .

Since A is a Green functor, this is equal to

A_{∗} (\begin{matrix}x′,γ_{1},y′,γ_{2} \\ ↓ \\ x′,γ_{1}.y′,γ_{1}γ_{2} \end{matrix}) A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ f(x),γ_{1},g(y),γ_{2} \end{matrix}) (a × b) = …

… = A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ f(x),γ_{1}.g(y),γ_{1}γ_{2} \end{matrix}) (a × b) .

It follows that equation (5.2) holds, since γ_{1}.g(y) = g(γ_{1}.y) for any γ_{1} ∈ Γ and y ∈ Y.

Similarly, I must check that for any a′ ∈ A_{Γ}(X′) and b′ ∈ A_{Γ}(Y′)

A^{∗}(f × Id_{Γ})(a′) ×_{Γ} A^{∗}(g × Id_{Γ})(b′) = A^{∗}(f × g × Id_{Γ})(a′ ×_{Γ} b′) (5.3)

The left hand side of this equation is equal to

A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (A^{∗}(f × Id_{Γ})(a′) × A^{∗}(g × Id_{Γ})(b′)) .

Since A is a Green functor, this is equal to

A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) A^{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ f(x),γ_{1},g(y),γ_{2} \end{matrix}) (a′ × b′)  (5.4)

The right hand side of equation (5.3) is equal to

A^{∗}(f × g × Id_{Γ})A_{∗} (\begin{matrix}x′,γ_{1},y′,γ_{2} \\ ↓ \\ x′,γ_{1}.y′,h′_{1}h′_{2} \end{matrix}) (a′ × b′)  (5.5)

Now the square

XΓ \underrightarrow{(\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ f(x),γ_{1},g(y),γ_{2} \end{matrix})} X′ΓY′Γ
(\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) ↓  ↓ (\begin{matrix}x′,γ_{1},y′,γ_{2} \\ ↓ \\ x′,γ_{1}.y′,γ_{1}γ_{2} \end{matrix})
XY Γ \overrightarrow{f × g × Id_{Γ}} X′Y′Γ

is cartesian if Γ is a crossed G-monoid. It follows that expression (5.5) is equal to expression (5.4), and equation (5.3) holds.

Next I have to check that the product on A_{Γ} is associative. In other words, if X, Y, and Z are finite G-sets, if a ∈ A_{Γ}(X), if b ∈ A_{Γ}(Y) and c ∈ A_{Γ}(Z), I must show that

a ×_{Γ} (b ×_{Γ} c) = (a ×_{Γ} b) ×_{Γ} c  (5.6)

in A_{Γ}(X × Y × Z). The left hand side is equal to

A_{∗} (\begin{matrix}x,γ_{1},y,z,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}.z,γ_{1}γ_{2} \end{matrix}) (a × A_{∗} (\begin{matrix}y,γ_{1},z,γ_{2} \\ ↓ \\ y,γ_{1}.z,γ_{1}γ_{2} \end{matrix}) (b × c)) .

Since A is a Green functor, this is also equal to

A_{∗} (\begin{matrix}x,γ_{1},y,z,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}.z,γ_{1}γ_{2} \end{matrix}) A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z,γ_{3} \\ ↓ \\ x,γ_{1},y,γ_{2}.z,γ_{2}γ_{3} \end{matrix}) (a × b × c)

which is also equal to

A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z,γ_{3} \\ ↓ \\ x,γ_{1}.y,γ_{1}.(γ_{2}.z),γ_{1}(γ_{2}γ_{3}) \end{matrix}) (a × b × c) .  (5.7)

The right hand side of equation (5.6) is equal to

A_{∗} (\begin{matrix}x,y,γ_{1},z,γ_{2} \\ ↓ \\ x,y,γ_{1}.z,γ_{1}γ_{2} \end{matrix}) (A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (a × b) × c) .

Since A is a Green functor, this is also

A_{∗} (\begin{matrix}x,y,γ_{1},z,γ_{2} \\ ↓ \\ x,y,γ_{1}.z,γ_{1}γ_{2} \end{matrix}) A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z,γ_{3} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2},z,γ_{3} \end{matrix}) (a × b × c)

which is also equal to

A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z,γ_{3} \\ ↓ \\ x,γ_{1}.y,(γ_{1}γ_{2}).z,(γ_{1}γ_{2})γ_{3} \end{matrix}) (a × b × c) .

This is equal to (5.7). Hence A_{Γ} is associative.

Finally I have to check that ε_{A_{Γ}} is a unit for A_{Γ}. Let X be a finite G-set, and let a ∈ A_{Γ}(X). Then

ε_{A_{Γ}} ×_{Γ} a = A_{∗} (\begin{matrix}γ_{1},x,γ_{2} \\ ↓ \\ γ_{1}.x,γ_{1}γ_{2} \end{matrix}) (A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) (ε_{A}) × a)
= A_{∗} (\begin{matrix}γ_{1},x,γ_{2} \\ ↓ \\ γ_{1}.x,γ_{1}γ_{2} \end{matrix}) A_{∗} (\begin{matrix}x,γ \\ ↓ \\ 1,x,γ \end{matrix}) (a)
= A_{∗} (\begin{matrix}x,γ \\ ↓ \\ 1_{Γ}.x,1_{Γ}γ \end{matrix}) (a) = a .

One checks similarly that a ×_{Γ} ε_{A_{Γ}} = a.

To complete the proof of theorem 5.1, it remains to check that the construction A ↦ A_{Γ} is an endo-functor of Green_{R}(G). So let θ : A → A′ be a morphism of Green functors. Then θ is in particular a morphism of Mackey functors, and it induces a morphism of Mackey functors θ_{Γ} from A_{Γ} → A′_{Γ}, whose evaluation at the finite G-set X is the map

(θ_{Γ})_{X} = θ_{X×Γ} : A_{Γ}(X) = A(X × Γ) → A′(X × Γ) = A_{Γ′}(X) .

It is clear moreover that with obvious notation (θ ∘ θ′)_{Γ} = θ_{Γ} ∘ θ′_{Γ}, and that if θ is the identity endomorphism of A, then θ_{Γ} is the identity of A_{Γ}.

It remains to check that θ_{Γ} is a morphism of Green functors, i.e. that it is compatible with the product. So let X and Y be finite G-sets, let a ∈ A_{Γ}(X) and b ∈ A_{Γ}(Y). Then

(θ_{Γ})_{X×Y} (a ×_{Γ} b) = θ_{X×Y ×Γ} (A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (a × b))
= A′_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (θ_{X×Γ×Y ×Γ}(a × b))

(because θ:A →A′ is a morphism of Mackey functors)

= A′_{∗} (\begin{matrix}x,γ_{1},y,Γ_{2} \\ ↓ \\ x,γ_{1}.y,Γ_{1}γ_{2} \end{matrix}) (θ_{X×Γ}(a) × θ_{Y ×Γ}(b))

(because θ is a morphism of Green functors)

= θ_{X×Γ}(a) ×_{Γ} θ_{Y ×Γ}(b)
= (θ_{Γ})_{X}(a) ×_{Γ} (θ_{Γ})_{Y} (b) .

This shows that θ_{Γ} is a morphism of Green functors from A_{Γ} to A′_{Γ}, and completes the proof of theorem 5.1.

Remark 5.8 : The evaluation at the trivial G-set of the Green functor A_{Γ} is

A_{Γ}(•) = 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) .

This product coincides with the product defined in section 3, and justifies lemma 3.1.

Proposition 5.9 : Let f : (Γ, φ) → (Γ′, φ′) be a morphism of crossed Gmonoids. 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′}. Moreover, if f is injective, then A_{f} is a split injection of Mackey functors.

Proof : The maps A_{f,X} define a morphism of Mackey functors from A_{Γ} to A_{Γ′} ([2] 1.2). Moreover clearly

A_{f,•}(ε_{A_{Γ}}) = A_{∗}(f)A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) (ε_{A}) = A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ′} \end{matrix}) (ε_{A}) = ε_{A_{Γ′}} .

Moreover if X and Y are finite G-sets, if a ∈ A_{Γ}(X) and b ∈ A_{Γ}(Y), then

A_{∗} (\begin{matrix}x,y,γ \\ ↓ \\ x,y,f(γ) \end{matrix}) (a ×_{Γ} b) = A_{∗} (\begin{matrix}x,γ \\ ↓ \\ x,f(γ) \end{matrix}) (a) ×_{Γ′} A_{∗} (\begin{matrix}y,γ \\ ↓ \\ y,f(γ) \end{matrix}) (b) .

Indeed, the left hand side is equal to

A_{∗} (\begin{matrix}x,y,γ \\ ↓ \\ x,y,f(γ) \end{matrix}) A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (a × b) = A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,f(γ_{1}γ_{2}) \end{matrix}) (a × b)

whereas the right hand side is equal to

A_{∗} (\begin{matrix}x,γ′_{1},y,γ′_{2} \\ ↓ \\ x,γ′_{1}.y,γ′_{1}γ′_{2} \end{matrix}) (A_{∗} (\begin{matrix}x,γ \\ ↓ \\ x,f(γ) \end{matrix}) (a) × A_{∗} (\begin{matrix}y,γ \\ ↓ \\ y,f(γ) \end{matrix}) (b))

or

A_{∗} (\begin{matrix}x,γ′_{1},y,γ′_{2} \\ ↓ \\ x,γ′_{1}.y,γ′_{1}γ′_{2} \end{matrix}) A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,f(γ_{1}),y,f(γ_{2}) \end{matrix}) (a × b) = …

… = A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,f(γ_{1}).y,f(γ_{1})f(γ_{2}) \end{matrix}) (a × b) .

Now f(γ_{1}).y = γ_{1}.y and f(γ_{1})f(γ_{2}) = f(γ_{1}γ_{2}) for all y ∈ Y and γ_{1}, γ_{2} ∈ Γ, since f is a morphism of crossed G-monoids.

Finally 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}, i.e. such that A^{f} ∘ A_{f} = Id_{A_{Γ}}. The proposition follows.

Corollary 5.10 : Let Γ be a crossed G-monoid. For any finite G-set X, denote by ι_{X} the map A_{∗} (\begin{matrix}x \\ ↓ \\ x,1_{Γ} \end{matrix}) from A(X) to A(X × Γ) = A_{Γ}(X). Then ι_{X} is a split injective map of R-modules, and the maps ι_{X} define an injective morphism of Green functors from A to A_{Γ}.

Proof : This is the special case of the proposition, where f is the (unique) morphism of crossed G-monoids from (𝔼, u) to (Γ, φ) (see example 4.8), sending the unique element of 𝔼 to 1_{Γ}.

6. The product formula

This section states a product formula for the ring A_{Γ}(G), which generalizes theorem 5.1 of [6].

Theorem 6.1 : 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}) .

Corollary 6.2 : Taking first sets of orbit representatives, there is an isomorphism of R-modules

A(Γ) ≅ ⨁_{γ∈[G\Γ]} A(G_{γ})

where [G\Γ] is a set of representatives of the orbits of G in Γ. With this notation, the product of a ∈ A(G_{γ}) and b ∈ A(G_{δ}) is equal to

⨁_{ε∈[G\Γ]} ⨁_{w∈[G_{γ}\G/G_{δ}]} t_{G_{g(w,ε)γ}∩G_{g(w,ε)wδ}}^{G_{ε}}^{g(w,ε)}(r_{G_{γ}∩G_{wδ}}^{G_{γ}}a.r_{G_{γ}∩G_{wδ}}^{G_{wδ}}^{w}b)

where g(w, ε) is an element of the unique class G_{ε}g in G_{ε}\G such that g(γ wδ) = ε.

Proof : This follows from the formulae given in remark 2.3 : the product a × b is the element of A(Γ × Γ) whose component (α, β) is equal to

r_{G_{(α,β)}}^{G_{α}}a_{α}.r_{G_{(α,β)}}^{G_{β}}b_{β}

and a ×_{Γ} b is the image of a × b by A_{∗}(μ), where μ : Γ → Γ is the product in Γ. The formula of the theorem follows.

The formula in the corollary is a translation using sets of representatives, and is a generalization of the product formula of [6] theorem 5.1.

Remark 6.3 : There is a similar formula for the product in A(Γ), if one uses co-invariants rather than invariants, as in remark 2.4. There is an isomorphism of R-modules

A(Γ) ≅ (⨁_{γ∈Γ} A(G_{γ}))_{G}

and denoting by [γ, a]_{G} the image in A(G_{Γ}) of the element a of A(γ), for γ ∈ Γ, the product of [γ, a]_{G} and [δ, b]_{G} (for δ ∈ Γ and b ∈ A(G_{δ})) is equal to

[γ, a]_{G} ×_{Γ} [δ, b]_{G} = ∑_{w∈[G_{γ}\G/G_{δ}]}[γ^{w}δ, t_{G_{γ}∩G_{^{w}δ}}^{G_{γ ^{w}δ}} (r_{G_{γ}∩G_{^{w}δ}}^{G_{γ}}a.r_{G_{γ}∩G_{^{w}δ}}^{G_{^{w}δ}}^{w}b)]_{G} .

This kind of formula was used in [1].

Corollary 6.4 : 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.

Proof : Let γ be an element of H, and consider the G_{γ}-set P_{γ} of pairs (α, β) of elements of H such that αβ = γ. The correspondence

σ : (α, β) ↦ (β, α^{β})

is a permutation of P, which commutes with the action of G_{γ}. Moreover

G_{α} ∩ G_{β} = G_{β} ∩ G_{α^{β}} .

Let a and b be elements of A(H^{c}). If A is graded, suppose moreover that a and b are homogeneous. The γ-component of the product of a and b is equal to

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

Since A is (graded) commutative, this is also equal (up to sign if A is graded) to

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

Changing the order of summation via the permutation σ gives

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

Now for (β, α) ∈ P_{γ}, set K = G_{β,α} = C_{G}(<β, α>). Then β ∈ H ∩ C_{G}(K), thus β acts trivially on A(K). Hence ^{β}r_{G_{(α,β)}}^{G_{α}}a_{α} = r_{G_{(α,β)}}^{G_{α}}a_{α} and

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

and A(H^{c}) is (graded) commutative.

Remark 6.5 : Corollary 6.4 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]).

Remark 6.6 : Let N be a normal subgroup of G, viewed as a crossed Gmonoid via the inclusion morphism from N to G. Then taking for A the cohomology functor H^{⊕}(−, R) leads to a ring A_{N}(G). This ring is isomorphic to the cohomology ring H^{⊕}(G, R) if N is trivial, and to the Hochschild cohomology ring HH^{⊕}(G, R) if N = G. It is a kind of “relative” Hochschild cohomology, and could be denoted by HH^{⊕}(G, N, R). As an R-module, it is isomorphic to

HH^{⊕}(G, N, R) ≅ ⨁_{\begin{matrix}n∈N \\ mod. G \end{matrix}} H^{⊕}(C_{G}(n), R)

In other words HH^{⊕}(G, N, R) ≅ H^{⊕}(G, RN), where the action of G on RN comes from the conjugation action of G on N. Corollary 6.4 shows that HH^{⊕}(G, N, R) is graded commutative.

7. Semi-direct products of crossed G-monoids

Theorem 5.1 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.

Proposition 7.1 : 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 unit (1_{Γ′}, 1_{Γ}).

Proof : This is a series of straightforward verifications.

Definition 7.2 : The crossed G-monoid (Γ”, φ”) of proposition 7.1 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 7.3 : 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}.

Proof : Let X be a finite G-set. Then

(A_{Γ})_{Γ′}(X) = A_{Γ}(X × Γ′) = A(X × Γ′ × Γ)

and this induces clearly an isomorphism of Mackey functors

(A_{Γ})_{Γ′} ≅ A_{Γ′×Γ}

where Γ′ × Γ is the direct product of Γ′ and Γ, with diagonal G-action, i.e. the underlying G-set of Γ′ ⋊ Γ.

Now let X and Y be finite G-sets, let a ∈ A(X × Γ′ × Γ), viewed as (A_{Γ})_{Γ′}(X), and b ∈ A(Y × Γ′ × Γ) , viewed as (A_{Γ})_{Γ′}(Y). Then the product of a and b in (A_{Γ})_{Γ′}(X × Y) is equal to

a ×_{Γ,Γ′} b = A_{Γ}_{∗} (\begin{matrix}x,γ′_{1},y,γ′_{2} \\ ↓ \\ x,γ′_{1}.y,γ′_{1}γ′_{2} \end{matrix}) (a ×_{Γ} b)
= A_{∗} (\begin{matrix}x,γ′_{1},y,γ′_{2},γ \\ ↓ \\ x,γ′_{1}.y,γ′_{1}γ′_{2},γ \end{matrix}) A_{∗} (\begin{matrix}x,γ′_{1},γ_{1},y,γ′_{2},γ_{2} \\ ↓ \\ x,γ′_{1},γ_{1}.y,γ_{1}.γ′_{2},γ_{1}γ_{2} \end{matrix}) (a × b)
= A_{∗} (\begin{matrix}x,γ′_{1},γ_{1},y,γ′_{2},γ_{2} \\ ↓ \\ x,γ′_{1}.(γ_{1}.y),γ′_{1}(γ_{1}.γ′_{2}),γ_{1}γ_{2} \end{matrix}) (a × b)
= a ×_{Γ′⋊Γ} b

since moreover γ′_{1}.(γ_{1}.y) = φ′(γ′)φ(γ)y = φ”(γ′, γ)y = (γ′, γ).y.

It follows that the previous isomorphism of Mackey functors is compatible with the product. Moreover, the unit of (A_{Γ})_{Γ′} is by definition equal to the following element of A(Γ′ × Γ) :

ε_{(A_{Γ})_{Γ′}} = A_{Γ}_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ′} \end{matrix}) (ε_{A_{Γ}})
= A_{∗} (\begin{matrix}γ \\ ↓ \\ 1_{Γ′},γ \end{matrix}) A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) (ε_{A})
= A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ′},1_{Γ} \end{matrix}) (ε_{A})
= ε_{A_{Γ′⋊Γ}}

and the proposition follows.

8. From A-modules to A_{Γ}-modules

It follows in particular from corollary 5.10 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 8.1 : Let A be a Green functor for G over R, and M be an Amodule. 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 8.2 : Let Γ be a crossed G-monoid. If A is a Green functor for G over R, and 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). 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).

Proof : Again I have to check that this product is bifunctorial, associative, and that ε_{A_{Γ}} is a left unit. Let f : X → X′ and g : Y → Y′ be maps of G-sets. Let a ∈ A_{Γ}(X) and m ∈ M(Y). I must check that

A_{∗}(f)(a) ×_{Γ} M_{∗}(g)(m) = M_{∗}(f × g)(a ×_{Γ} m) .

From the definitions, this amounts to checking that

M_{∗} (\begin{matrix}x′,γ,y′ \\ ↓ \\ x′,γ.y′ \end{matrix}) (A_{∗}(f)(a) × M_{∗}(g)(m)) = M_{∗}(f × g)M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) (a × m)

or equivalently that

M_{∗} (\begin{matrix}x′,γ,y′ \\ ↓ \\ x′,γ.y′ \end{matrix}) M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ f(x),γ,g(y) \end{matrix}) (a × m) = …

… = M_{∗} (\begin{matrix}x,y \\ ↓ \\ f(x),g(y) \end{matrix}) M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) (a × m) .

This holds, since both sides are equal to

M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ f(x),γ.g(y) \end{matrix}) (a × m) .

I must also check that for a′ ∈ A_{Γ}(X′) and m′ ∈ M(Y′)

A^{∗}(f)(a′) ×_{Γ} M^{∗}(g)(m′) = M^{∗}(f × g)(a′ ×_{Γ} m′) .

By similar transformations, this amounts to checking that

M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix})M^{∗} (\begin{matrix}x,γ,y \\ ↓ \\ f(x),γ,g(y) \end{matrix})(a′ × m′) = …

… = M^{∗} (\begin{matrix}x,y \\ ↓ \\ f(x),g(y) \end{matrix})M_{∗} (\begin{matrix}x′,γ,y′ \\ ↓ \\ x′,γ.y′ \end{matrix})(a′ × m′) .

This holds because M is a Mackey functor, and because the square

XΓY \underrightarrow{(\begin{matrix}x,γ,y \\ ↓ \\ f(x),γ,g(y) \end{matrix})} X′ΓY′
(\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) ↓  ↓ (\begin{matrix}x′,γ,y′ \\ ↓ \\ x′,γ.y′ \end{matrix})
XY \overrightarrow{(\begin{matrix}x,y \\ ↓ \\ f(x),g(y) \end{matrix})} X′Y′

is a cartesian square of G-sets if Γ is a crossed G-monoid. This shows that the product (a, m) ↦ a ×_{Γ} m is bifunctorial.

It is moreover associative : let X, Y, Z be finite G-sets, and let a ∈ A_{Γ}(X), b ∈ A_{Γ}(Y), and m ∈ M(Z). Then I must check that

a ×_{Γ} (b ×_{Γ} m) = (a ×_{Γ} b) ×_{Γ} m  (8.3)

in M_{Γ}(X × Y × Z). The left hand side is equal to

M_{∗} (\begin{matrix}x,γ,y,z \\ ↓ \\ x,γ.y,γ.z \end{matrix}) (a × M_{∗} (\begin{matrix}y,γ,z \\ ↓ \\ y,γ.z \end{matrix}) (b × m)) .

Since M is a module for the Green functor A, this is also equal to

M_{∗} (\begin{matrix}x,γ,y,z \\ ↓ \\ x,γ.y,γ.z \end{matrix}) M_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z \\ ↓ \\ x,γ_{1},y,γ_{2}.z \end{matrix}) (a × b × m)

which is also equal to

M_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z \\ ↓ \\ x,γ_{1}.y,γ_{1}.(γ_{2}.z) \end{matrix}) (a × b × m) .  (8.4)

The right hand side of equation (8.3) is equal to

M_{∗} (\begin{matrix}x,y,γ,z \\ ↓ \\ x,y,γ.z \end{matrix}) (A_{∗} (\begin{matrix}x,γ_{1},y,γ_{2} \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2} \end{matrix}) (a × b) × m) .

Since M is a module for the Green functor A, this is also

M_{∗} (\begin{matrix}x,y,γ,z \\ ↓ \\ x,y,γ.z \end{matrix}) M_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z \\ ↓ \\ x,γ_{1}.y,γ_{1}γ_{2},z \end{matrix}) (a × b × m) .

This is equal to

M_{∗} (\begin{matrix}x,γ_{1},y,γ_{2},z \\ ↓ \\ x,γ_{1}.y,(γ_{1}γ_{2}).z \end{matrix}) (a × b × m)

which is equal to (8.4). Hence the product is associative.

Finally, if X is any G-set and m ∈ M(X), then

ε_{A_{Γ}} ×_{Γ} m = M_{∗} (\begin{matrix}γ,x \\ ↓ \\ γ.x \end{matrix}) (A_{∗} (\begin{matrix}• \\ ↓ \\ 1 \end{matrix}) (ε_{A}) × m)
= M_{∗} (\begin{matrix}γ,x \\ ↓ \\ γ.x \end{matrix}) M_{∗} (\begin{matrix}x \\ ↓ \\ 1,x \end{matrix}) (m)
= M_{∗} (\begin{matrix}x \\ ↓ \\ 1.x \end{matrix}) (m) = m .

This shows that i_{Γ}(M) is an A_{Γ}-module.

Now if f : M → N is a morphism of A-modules, if X and Y are finite G-sets, if a ∈ A_{Γ}(X) and m ∈ M(Y), then

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

(because f is a morphism of Mackey functors)

= N_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) (a × f_{Y} (m))

(because f is a morphism of A-modules)

= a ×_{Γ} f_{Y} (m) .

Hence f defines a morphism of A_{Γ}-modules from i_{Γ}(M) to i_{Γ}(N), and the proof of the theorem is complete.

Proposition 8.5 : Let Γ be a crossed G-monoid, and A be a Green functor for G over R. If M is an A-module, then the composition r_{Γ}∘i_{Γ} is isomorphic to the identity functor of A-Mod. Proof : Let M be an A-module. If X is a finite G-set, then

r_{Γ} ∘ i_{Γ}(M)(X) = M(X)

thus r_{Γ} ∘ i_{Γ}(M) is isomorphic to M as a Mackey functor. Moreover if f : M → N is a morphism of A-modules, then r_{Γ} ∘ i_{Γ}(f) = f. It remains to check that r_{Γ} ∘ i_{Γ}(M) is isomorphic to M as an A-module, i.e. that if X and Y are finite G-sets, if a ∈ A(X) and m ∈ M(Y), then

a × m = ι_{X}(a) ×_{Γ} m .

This is equivalent to

a × m = M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) (A_{∗} (\begin{matrix}x \\ ↓ \\ x,1_{Γ} \end{matrix}) (a) × m)
= M_{∗} (\begin{matrix}x,γ,y \\ ↓ \\ x,γ.y \end{matrix}) M_{∗} (\begin{matrix}x,y \\ ↓ \\ x,1_{Γ},y \end{matrix}) (a × m)
= M_{∗} (\begin{matrix}x,y \\ ↓ \\ x,1_{γ}.y \end{matrix}) (a × m)
= a × m

which proves the proposition.

Corollary 8.6 : The functor i_{Γ} is a full embedding of A-Mod into A_{Γ}-Mod.

Proof : Since r_{Γ} ∘ i_{Γ} is isomorphic to the identity functor, the functor i_{Γ} is faithful. Moreover, the functor r_{Γ} is clearly faithful : if f : M → N is a morphism of A_{Γ}-modules, then for any finite G-set X, the map r_{Γ}(f)_{X} : M(X) → N(X) is equal to the map f_{X}. Now the functors i_{Γ} and r_{Γ} induce maps

Hom_{A}(M, N) →^{i_{Γ}} Hom_{A_{Γ}}(i_{Γ}(M), i_{Γ}(N)) →^{r_{Γ}} Hom_{A}(M, N)

whose composition is the identity. Since the map r_{Γ} is moreover injective, it follows that both maps are bijections, and the corollary follows.

9. Centres and centralizers

Recall the following definitions from [2] 6.5 and 12.2 :

Definition 9.1 : Let A be a Green functor for G over R. If X and Y are finite G-sets, if a ∈ A(X) and b ∈ A(Y), set

a ×^{op} b = A_{∗} (\begin{matrix}y,x \\ ↓ \\ x,y \end{matrix}) (b × a) ∈ A(X × Y) .

If M is a Mackey subfunctor of A, define the commutant of M in A by

C_{A}(M)(X) = {a ∈ A(X) | ∀Y, ∀m ∈ M(Y), a × m = a ×^{op} m}

Also define ζ_{A}(X) as the set of natural transformations from the identity functor I of A-Mod to the endofunctor I_{X} of A-Mod given by the Dress construction associated to X.

The commutant of M in A is a Green subfunctor of A ([2] 6.5.3). 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 9.2 : 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 Gsets, 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}.

Proof : Let f : Y → Z be a map of G-sets. Then for m ∈ M(Y)

z_{X}(α)_{M,Z}(M_{∗}(f)(m)) = M_{∗} (\begin{matrix}x,z \\ ↓ \\ z,x \end{matrix}) (α ×_{Γ} M_{∗}(f)(m))
= M_{∗} (\begin{matrix}x,z \\ ↓ \\ z,x \end{matrix}) M_{∗} (\begin{matrix}x,y \\ ↓ \\ x,f(y) \end{matrix}) (α ×_{Γ} m)

(because i_{Γ}(M) is an A_{Γ}-module)

= M_{∗} (\begin{matrix}y,x \\ ↓ \\ f(y),x \end{matrix}) M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) (α ×_{Γ} m)
= (M_{X})_{∗}(f)(z_{X}(α)_{M,Y} (m)) .

A similar computation, using the fact that M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) = M^{∗} (\begin{matrix}y,x \\ ↓ \\ x,y \end{matrix}), shows that

z_{X}(α)_{M,Y} ∘ M^{∗}(f) = (M_{X})^{∗}(f) ∘ z_{X}(α)_{M,Z}

hence that the maps z_{X}(α)_{M,Y} define a morphism of Mackey functors z_{X}(α)_{M} from M to M_{X}.

Now let Y and Z be finite G-sets, let a ∈ A(Z) and m ∈ M(Y). Then

a × z_{X}(α)_{M,Y} (m) = a × M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) (α ×_{Γ} m)
= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) (a × (α ×_{Γ} m))

(because M is an A-module)

= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) (ι_{Z}(a) ×_{Γ} (α ×_{Γ} m))

(by proposition 8.5)

= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) ((ι_{Z}(a) ×_{Γ} α) ×_{Γ} m)

= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) ((ι_{Z}(a) ×_{Γ}^{op} α) ×_{Γ} m)

(because α ∈ C_{A_{Γ}}(ι(A))(X))

= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) ((A_{Γ})_{∗} (\begin{matrix}x,z \\ ↓ \\ z,x \end{matrix}) (α ×_{Γ} ι_{Z}(a)) ×_{Γ} m)
= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) M_{∗} (\begin{matrix}x,z,y \\ ↓ \\ z,x,y \end{matrix}) ((α ×_{Γ} ι_{Z}(a)) ×_{Γ} m)

(because i_{Γ}(M) is an A_{Γ}-module)

= M_{∗} (\begin{matrix}z,x,y \\ ↓ \\ z,y,x \end{matrix}) M_{∗} (\begin{matrix}x,z,y \\ ↓ \\ z,x,y \end{matrix}) (α ×_{Γ} (ι_{Z}(a) ×_{Γ} m))
= M_{∗} (\begin{matrix}x,z,y \\ ↓ \\ z,y,x \end{matrix}) (α ×_{Γ} (a × m))

(by proposition 8.5)

= z_{X}(α)_{M,Z×Y} (a × m)

which proves assertion 1).

To prove assertion 2), I must check that if f : M → N is a morphism of A-modules, then for any finite G-set Y, the square

M(Y) \underrightarrow{f_{Y}} N(Y)
z_{X}(α)_{M,Y} ↓  ↓ z_{X}(α)_{N,Y}
M(Y × X) \overrightarrow{f_{Y ×X}} N(Y × X)

is commutative. But for m ∈ M(Y)

f_{Y ×X} ∘ z_{X}(α)_{M,Y} (m) = f_{Y ×X}M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) (α ×_{Γ} m)
= N_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) f_{X×Y} (α ×_{Γ} m)

(because f is a morphism of Mackey functors)

= N_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) (α ×_{Γ} f_{Y} (m))

(because i_{Γ}(f) is a morphism of A_{Γ}-modules)

= z_{X}(α)_{N,Y} ∘ f_{Y} (m)

as was to be shown.

For assertion 3), first recall the Mackey functor structure on ζ_{A} ([2] 12.2.1) : if X is a finite G-set and θ is a natural transformation from the identity functor I of A-Mod to I_{X}, then θ is characterized by maps of Amodules θ_{M} : M → M_{X}, for each A-module M, i.e. by maps of R-modules θ_{M,Y} : M(Y) → M(Y × X), for any finite G-set Y. Now if f : X → Z is a map of finite G-sets, then (ζ_{A})_{∗}(f)(θ) is the natural transformation from I to I_{Z} characterized by the maps

(ζ_{A})_{∗}(f)(θ)_{M,Y} = M_{∗}(Id_{Y} × f) ∘ θ_{M,Y} .

Similarly, if θ′ ∈ ζ_{A}(Z), then ζ_{A}^{∗}(f)(θ′) is the element of ζ_{A}(X) defined by

ζ_{A}^{∗}(f)(θ′) = M^{∗}(Id_{Y} × f) ∘ θ′_{M,Y} .

Let C denote the Green functor C(A, Γ). To show that the maps z_{X} : C(X) → ζ_{A}(X) define a morphism of Mackey functors, I must check that for any m ∈ M(Y)

M_{∗}(Id_{Y} × f) ∘ z_{X}(α)_{M,Y} (m) = z_{Z}(C_{∗}(f)(α))_{M,Y} (m)  (9.3)

and similarly

M^{∗}(Id_{Y} × f) ∘ z_{Z}(α)_{M,Y} (m) = z_{X}(C^{∗}(f)(α))_{M,Y} (m) .  (9.4)

The left hand side of equation 9.3 is equal to

M_{∗}(Id_{Y} × f) ∘ z_{X}(α)_{M,Y} (m) = M_{∗}(Id_{Y} × f)M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) (α ×_{Γ} m)
= M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,f(x) \end{matrix}) (α ×_{Γ} m)
= M_{∗} (\begin{matrix}z,y \\ ↓ \\ y,z \end{matrix}) M_{∗} (\begin{matrix}x,y \\ ↓ \\ f(x),y \end{matrix}) (α ×_{Γ} m)
= M_{∗} (\begin{matrix}z,y \\ ↓ \\ y,z \end{matrix}) ((A_{Γ})_{∗}(f)(α) ×_{Γ} m)

(because i_{Γ}(M) is an A_{Γ}-module)

= z_{Z}(C_{∗}(f)(α))(m)

as was to be shown, since C_{∗}(f) = (A_{Γ})_{∗}(f), because C is a subfunctor of A_{Γ}. The proof of equation 9.4 is similar, using the fact that M_{∗} (\begin{matrix}x,y \\ ↓ \\ y,x \end{matrix}) = M^{∗} (\begin{matrix}y,x \\ ↓ \\ x,y \end{matrix}).

It remains to check that the maps z_{X} define a morphism of Green functors from C = C(A, Γ) to ζ_{A}. Recall from [2] 12.2.1 that if X and Y are finite G-sets, if θ ∈ ζ_{A}(X) and ψ ∈ ζ_{A}(Y), then the product θ × ψ is the element of ζ_{A}(X × Y) determined by the maps

(θ × ψ)_{M,Z} = M_{∗} (\begin{matrix}z,y,x \\ ↓ \\ z,x,y \end{matrix}) ∘ θ_{M,Z×Y} ∘ ψ_{M,Z}

for any A-module M and any finite G-set Z.

So let α ∈ C(X) and β ∈ C(Y). Then for m ∈ M(Z)

(z_{X}(α) × z_{Y} (β))_{M,Z}(m) = M_{∗} (\begin{matrix}z,y,x \\ ↓ \\ z,x,y \end{matrix}) ∘ z_{X}(α)_{M,Z×Y} ∘ z_{Y} (β)_{M,Z}(m) = …

… = M_{∗} (\begin{matrix}z,y,x \\ ↓ \\ z,x,y \end{matrix}) M_{∗} (\begin{matrix}x,z,y \\ ↓ \\ z,y,x \end{matrix}) [α ×_{Γ} M_{∗} (\begin{matrix}y,z \\ ↓ \\ z,y \end{matrix}) (β ×_{Γ} m)]
= M_{∗} (\begin{matrix}x,z,y \\ ↓ \\ z,x,y \end{matrix}) [M_{∗} (\begin{matrix}x,y,z \\ ↓ \\ x,z,y \end{matrix}) (α ×_{Γ} β ×_{Γ} m)]

(because i_{Γ}(M) is an A_{Γ}-module)

= M_{∗} (\begin{matrix}x,y,z \\ ↓ \\ z,x,y \end{matrix}) (α ×_{Γ} β ×_{Γ} m)
= z_{X×Y} (α ×_{Γ} β)_{M,Z}(m)

This shows that z_{X}(α)×z_{Y} (β) = z_{X×Y} (α×_{Γ}β). The last verification concerns units : for any m ∈ M(Z)

z_{•}(ε_{A_{Γ}})_{M,Z}(m) = M_{∗} (\begin{matrix}•,y \\ ↓ \\ y,• \end{matrix}) (ε_{A_{Γ}} ×_{Γ} m)
= m

since ε_{A_{Γ}} is the unit of A_{Γ}. Hence z_{•}(ε_{A_{Γ}}) is the identity transformation of the identity functor of A-Mod. This completes the proof of the theorem.

Remark 9.5 : Theorem 9.2 provides in particular a natural ring homomorphism from C(A, Γ)(•) to the center of the category A-Mod. One can check from the definitions that

C(A, Γ)(•) = \{α ∈ A(Γ) | ∀X, ∀a ∈ A(X), a × α = A_{∗} (\begin{matrix}γ,x \\ ↓ \\ γ.x,γ \end{matrix}) (α × x)\}

as a subring of A(Γ). 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.