Equivalences between blocks of cohomological Mackey algebras

Abstract

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system "large enough". Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra $RG$ and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra $co\mu_{R}(G)$. Here, we prove that a so-called permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Brou\'e's abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.