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

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.