A remark on Rickard complexes

TL;DR

This paper characterizes a special class of Rickard complexes that induce Morita equivalences between block algebras, focusing on their homological properties and relation to defect groups.

Contribution

It provides a new characterization of Rickard complexes with specific homological and vertex properties, linking them to Morita equivalences between blocks.

Findings

Homology of the Rickard complex vanishes outside degree q.

Homology at degree q induces a Morita equivalence.

Vertices of the complex match the order of defect groups.

Abstract

In this paper, we characterize a Rickard complex, which induces a Rickard equivalence between the block algebras of a block $b$ and its Brauer correspondent and whose vertices have the same order as defect groups of the block $b$. The homology of such a Rickard complex vanishes at all degree but degree $q$, and the homology at degree $q$ induces a basic Morita equivalence between the block algebras in the sense of Puig.