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.
Studies moduli spaces for two families of Enriques categories over K3 surfaces from specific threefolds, recovering classical constructions modularly and providing a criterion for Enriques categories in the appendix.
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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Homological Aspects of Separable Extensions of Triangulated Categories
Separable extensions preserve finiteness of global dimension, Gorensteinness and regularity in compactly generated triangulated categories while relating their singularity categories up to retracts.
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Group actions on relative cluster categories and Higgs categories
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.
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On two families of Enriques categories over K3 surfaecs
Studies moduli spaces for two families of Enriques categories over K3 surfaces from specific threefolds, recovering classical constructions modularly and providing a criterion for Enriques categories in the appendix.
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Twisted Kazhdan-Lusztig conjecture for $p$-adic general linear group
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.
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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.