On equivalences for cohomological Mackey functors

TL;DR

This paper explores the relationships between various types of equivalences of blocks in finite groups and the induced equivalences of their associated categories of cohomological Mackey functors, providing explicit constructions and extending known results.

Contribution

It offers an intrinsic description of cohomological Mackey functors in terms of source algebras and constructs explicit tilting complexes to realize derived equivalences.

Findings

Source algebra equivalences induce category equivalences of cohomological Mackey functors.

Splendid derived equivalences lead to derived equivalences of Mackey functor categories.

Extension of Tambara's result on finitistic dimension to blocks.

Abstract

By results of Rognerud, 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. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.