Cohomology of groups, Abelian Sylow subgroups and splendid equivalences (Q2701751)

This paper is part of a bigger programme which aims at studying group cohomology from a new point of view. The new structure comes from certain derived equivalences acting on group cohomology. In an earlier paper [Auto-equivalences of derived categories acting on group cohomology (to appear)] the author has found an action of the group of derived auto-equivalences fixing the trivial module on the cohomology of a given finite group. In the present paper he restricts attention to the subgroup of splendid auto-equivalences which are associated with splendid tilting complexes in the sense of Rickard. This action is shown to be compatible with the Mackey functor properties of group cohomology. In subsequent work the author also proved compatibility with the structure of an unstable module over the Steenrod algebra. If the Sylow subgroup \(P\) is Abelian then the action is in fact by outer automorphisms of \(P\).