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 Broué's abelian defect group conjecture for blocks of $p$-nilpotent groups with abelian Sylow $p$-subgroup.

Fixed an assumption error in Theorem 6 and added a necessary lemma to prove Corollary 8. 7 pages

References in corpus (1)

• Descent of equivalences for blocks with Klein four defect groups

Cited by in corpus (1)

• On endosplit $p$-permutation resolutions and Broué's conjecture for $p$-solvable groups