Abstract
Let G be a finite group, and R be a commutative ring. If A is a Green functor for G over R, and Γ is a crossed G-monoid, then the Mackey functor AΓ obtained by the Dress construction has a natural structure of Green functor, and its evaluation AΓ(G) is an R-algebra. This framework involves as special cases the construction of the Hochschild cohomology algebra of the group algebra from the ordinary cohomology functor, and the construction of the crossed Burnside ring from the ordinary Burnside functor. This article presents some properties of those Green functors A Γ, and the functorial relations between the corresponding categories of modules. As a consequence, a general product formula for the algebra A Γ(G) is stated.