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Brauer pairs for splendid Rickard equivalences
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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 $\Lambda(C)$ 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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Permutation twisted cohomology, remixed
For every finite p-group, the remixed twisted cohomology ring gives an injective comparison map from the Balmer spectrum of permutation modules, an open immersion when the ring is Noetherian.
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