Hochschild constructions for Green functors

Serge Bouc

Abstract:

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 $HH^*(G,R)$ from the ordinary cohomology ring $H^*(G,R)$. This construction also involves as a special case the construction of the crossed Burnside ring of $G$ from the ordinary Burnside ring.\par
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 $\Gamma$ is a crossed $G$-monoid, i.e. that $\Gamma$ is a $G$-monoid over the $G$-group $G^c$. Then the Mackey functor $A_\Gamma$ obtained from $A$ by Dress construction has a natural structure of Green functor. In particular $A_\Gamma(G)$ is a ring. \par
In the case where $\Gamma$ is the crossed $G$-monoid $G^c$, and $A$ is the cohomology functor (with trivial coefficients $R$), the ring $A_\Gamma(G)$ is the Hochschild cohomology ring of $G$ over $R$. If $A$ is the Burnside functor for $G$ over $R$, then the ring $A_\Gamma(G)$ is the crossed Burnside ring of $G$ over $R$.\par
Some general properties of the functors $A_\Gamma$ are stated (product formula, relations between the categories of modules, composition of these constructions).