Brauer pairs for splendid Rickard equivalences

Abstract

We define the notion of a Brauer pair of a chain complex, extending the notion of a Brauer pair of a $p$-permutation module introduced by Boltje and Perepelitsky. In fact, the Brauer pairs of a splendid Rickard equivalence $C$ coincide with the set of Brauer pairs of the corresponding $p$-permutation equivalence $Λ(C)$ induced by $C$. As a result, we derive structural results for splendid Rickard equivalences that correspond to known structural properties for $p$-permutation equivalences. In particular, we show splendid Rickard equivalences induce local splendid Rickard equivalences between normalizer block algebras as well as centralizer block algebras.