Separable extensions preserve finiteness of global dimension, Gorensteinness and regularity in compactly generated triangulated categories while relating their singularity categories up to retracts.
On equivariant triangulated categories
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abstract
Consider a finite group $G$ acting on a triangulated category $\mathcal T$. In this paper we investigate triangulated structure on the category $\mathcal T^G$ of $G$-equivariant objects in $\mathcal T$. We prove (under some technical conditions) that such structure exists. Supposed that an action on $\mathcal T$ is induced by a DG-action on some DG-enhancement of $\mathcal T$, we construct a DG-enhancement of $\mathcal T^G$. Also, we show that the relation "to be an equivariant category with respect to a finite abelian group action" is symmetric on idempotent complete additive categories.
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Constructs G-equivariant relative cluster and Higgs categories from group actions on ice quivers with potential and links them via orbit mutations to skew-symmetrizable cluster algebras, yielding an additive categorification for non-simply-laced principal coefficients.
Irreducible representations of twisted p-adic GL groups in unramified principal series are classified using enhanced Langlands parameters, with the twisted Kazhdan-Lusztig conjecture proved for Grothendieck group multiplicities via graded Hecke algebras.
Lecture notes summarizing recent progress on hyper-Kähler varieties via Lagrangian fibrations, atomic sheaves, and derived categories.
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Hyper-K\"ahler varieties: Lagrangian fibrations, atomic sheaves, and categories
Lecture notes summarizing recent progress on hyper-Kähler varieties via Lagrangian fibrations, atomic sheaves, and derived categories.