# Descent of equivalences and character bijections

## TL;DR

This paper investigates how categorical equivalences between block algebras of finite groups, like Morita and derived equivalences, can be realized over non-splitting fields, with results on descent and connections to character bijections.

## Contribution

It provides new results on the descent of Morita and derived equivalences, including perfect isometries and Rouquier's Rickard complex, over non-split fields.

## Key findings

- Perfect isometries induce isomorphisms of centers in non-split cases.
- Rouquier's splendid Rickard complex descends to non-split situations.
- A descent theorem for Morita equivalences with endopermutation sources.

## Abstract

Categorical equivalences between block algebras of finite groups - such as Morita and derived equivalences - are well-known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non splitting fields. This article presents various result on the theme of descent. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centers in non-split situations, and explain connections with Navarro's generalisation of the Alperin-McKay conjecture. We show that Rouquier's splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source.