Splendid and perverse equivalences

TL;DR

This paper explores the concept of splendid perverse equivalences, combining ideas from Rickard, Chuang, and Rouquier, to understand relations between global and local perverse equivalences in finite group blocks, especially with abelian defect groups.

Contribution

It introduces the framework of splendid perverse equivalences and proves their connection to local derived perverse equivalences under certain conditions.

Findings

Establishes a link between global and local perverse equivalences.

Shows that splendid perverse equivalences induce local derived equivalences.

Applies to blocks of finite groups with abelian defect groups.

Abstract

Inspired by the works of Rickard on splendid equivalences and of Chuang and Rouquier on perverse equivalences, we are here interested in the combination of both, a splendid perverse equivalence. This is naturally the right framework to understand the relations between global and local perverse equivalences between blocks of finite groups, as a splendid equivalence induces local derived equivalences via the Brauer functor. We prove that under certain conditions, we have an equivalence between a perverse equivalence between the homotopy category of p-permutation modules and local derived perverse equivalences, in the case of abelian defect group.