Reasonable triangulated categories have filtered enhancements

Modoi, George Ciprian

doi:10.1090/proc/14474

Cited by 7 publications

(7 citation statements)

References 15 publications

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“…For the statement about weight structures, we refer to [13] for 𝑓-categories and [1,34] for ∞-categories. In fact, the derivator assumption implies the 𝑓-category assumption by [25]. □ There are different assumptions under which the above functors can be shown to be fully faithful.…”

Section: A2mentioning

confidence: 99%

K$K$‐Motives and Koszul duality

Eberhardt

2022

Bulletin of London Math Soc

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We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K$K$‐theory. For this, we introduce and study categories DKS(X)$\hbox{DK}_\mathcal {S}(X)$ of S$\mathcal {S}$‐constructible K$K$‐motivic sheaves on varieties X$X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S$\mathcal {S}$‐constructible mixed sheaves DSmix(X)$\hbox{D}^{mix}_\mathcal {S}(X)$ to DKS(X)$\hbox{DK}_\mathcal {S}(X)$. We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K$K$‐motivic sheaves on Langlands dual flag varieties.

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Section: A2mentioning

confidence: 99%

K$K$‐Motives and Koszul duality

Eberhardt

2022

Bulletin of London Math Soc

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“…3. Every triangulated category which is the underlying category of a stable derivator admits a filtered enhancement; this is the content of [19].…”

Section: Realized Triangulated Categoriesmentioning

confidence: 99%

Compatibility of t-Structures in a Semiorthogonal Decomposition

Antonio

2022

Appl Categor Struct

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“…In the appendix we formulate the filtered derived category of a DG category, which assures the existence of a realization functor for algebraic triangulated categories (Corollary A.9). We refer to [KV,Ke1,ChR,M] for different approaches.…”

Section: Introductionmentioning

confidence: 99%

Yoneda algebras and their singularity categories

Hanihara¹

2019

Preprint

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For a finite dimensional algebra Λ of finite representation type and an additive generator M for mod Λ, we investigate the properties of the Yoneda algebra Γ = i≥0 Ext i Λ (M, M ). We show that Γ is graded coherent and Gorenstein of self-injective dimension at most 1, and the graded singularity category D Z sg (Γ) of Γ is triangle equivalent to the derived category of the stable Auslander algebra of Λ. These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category Y of Λ as the additive closure of the shifts of the Λ-modules in the derived category D b (mod Λ). We show that Y is coherent and Gorenstein of self-injective dimension at most 1, and the singularity category of Y is triangle equivalent to the derived category D b (mod(mod Λ)) of the stable category mod Λ. To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an fcategory over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.

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Reasonable triangulated categories have filtered enhancements

Abstract: We prove that a triangulated category which is the underlying category of a stable derivator has a filtered enhancement, providing an affirmative answer to a conjecture in [3].

Cited by 7 publications

References 15 publications

K$K$‐Motives and Koszul duality

K$K$‐Motives and Koszul duality

Compatibility of t-Structures in a Semiorthogonal Decomposition

Yoneda algebras and their singularity categories

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