Brauer pairs for splendid Rickard equivalences

TL;DR

The paper extends the Brauer-pair framework from $p$-permutation equivalences to splendid Rickard equivalences, proving that Brauer pairs for a splendid Rickard complex $C$ coincide with those for its Lefschetz Lef invariant $\\Lambda(C)$. This yields a robust transfer of local-structure results: splendid Rickard equivalences induce local equivalences on normalizer and centralizer blocks and align with fusion-system data. The work also develops extended tensor product machinery and Brauer-construction techniques to establish a coherent, block-wise picture across sums of blocks. Consequently, foundational invariants and local-global correspondences known for $p$-permutation equivalences extend to the realm of derived-equivalence contexts, strengthening Broué-type correspondences in the setting of block algebras. These results provide a unified view of local and global behavior for splendid Rickard equivalences, with clear structural consequences for normalizer and centralizer block algebras.

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.

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Theorems & Definitions (75)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Definition 3.1
• Theorem 3.2
• Definition 3.3