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Broué’s abelian defect group conjecture

Establish an equivalence of triangulated categories between the bounded derived category of finitely generated modules over a block OG.e of the group algebra OG of a finite group G with abelian defect group D and the bounded derived category over its Brauer correspondent ON_G(D).f; that is, construct an equivalence D^b(mod_{OG.e}) ≃ D^b(mod_{ON_G(D).f}).

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Background

In modular representation theory, blocks of group algebras encapsulate deep structural information. The Brauer construction produces a block ON_G(D).f in the normalizer of the defect group D, and Broué conjectured that, for abelian defect groups, the bounded derived categories of the original block and its Brauer correspondent are equivalent.

This conjecture provides a far-reaching ‘global-to-local’ bridge in modular representation theory, predicting that representation-theoretic complexity is reflected and preserved at the level of derived categories.

References

Brou\ e put forward the idea that a perfect isometry should be a shadow of a structural correspondence at the level of the corresponding derived categories, giving rise to Brou\ e's abelian defect group conjecture : Let $G$ be a finite group and let $ \operatorname{O} G. e$ be a block of $ \operatorname{O} G$ with abelian defect group given by the $p$-group $D$. Then, there is an equivalence of triangulated categories $$\operatorname{D}{b}(\operatorname{mod}_{ \operatorname{O} G.e}) \simeq \operatorname{D}{b}(\operatorname{mod}_{ \operatorname{O} N_G(D).f}),$$ where $ \operatorname{O} N_G(D).f$ is the Brauer correspondent of $ \operatorname{O} G.e$.

— Rickard's Derived Morita Theory: Review and Outlook (2509.06369 - Jasso et al., 8 Sep 2025) in Conjecture, Subsection 1.4 (Modular representation theory and splendid equivalences)