Green functors, crossed G-monoids, and Hochschild constructions

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 Gc 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 Gc. Then the Mackey functor AΓ obtained from A by the Dress construction has a natural structure of Green functor. In particular AΓ(G) is a ring. In the case where Γ is the crossed G-monoid Gc, and A is the cohomology functor (with trivial coefficients R), the ring AΓ(G) is the Hochschild cohomology ring of G over R. If A is the Burnside functor for G over R, then the ring AΓ(G) is the crossed Burnside ring of G over R. This note presents some properties of those Green functors AΓ, and the functorial relations between the corresponding categories of modules. In particular, it states a general formula for the product in the ring AΓ(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]).