Galois descent of splendid Rickard equivalences for blocks of $p$-nilpotent groups

TL;DR

This paper advances the understanding of Galois descent in block theory of p-nilpotent groups, proving new criteria and confirming conjectures for blocks with abelian Sylow p-subgroups.

Contribution

It generalizes Galois descent criteria for chain complexes and verifies a refined Broué's conjecture for specific p-nilpotent group blocks.

Findings

Strengthens Galois descent results for p-nilpotent groups.

Provides a general descent criterion for chain complexes.

Verifies Broué's abelian defect group conjecture in new cases.

Abstract

We strengthen the results of Boltje and Yilmaz regarding the Galois descent of equivalences of blocks of $p$-nilpotent groups and a result of Kessar and Linckelmann regarding Galois descent of splendid Rickard equivalences for blocks with compatible Galois stabilizers. A more general descent criteria for chain complexes is proven along the way, which requires the adaptation of a theorem of Reiner for chain complexes. This verifies Kessar and Linckelmann's refinement of Brou\'{e}'s abelian defect group conjecture for blocks of $p$-nilpotent groups with abelian Sylow $p$-subgroup.