On Morita and derived equivalences for cohomological Mackey algebras

TL;DR

This paper establishes that equivalences between categories of cohomological Mackey functors imply Morita or derived equivalences between the original block algebras of finite groups, extending known results.

Contribution

It proves a partial converse showing that categorical equivalences induce algebraic Morita or derived equivalences between blocks.

Findings

Equivalence of cohomological Mackey functor categories implies Morita equivalence.

Rickard equivalence of categories induces derived equivalence of blocks.

Extends the understanding of relationships between algebraic and categorical equivalences.

Abstract

By results of the second author, a source algebra equivalence between two $p$-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras.