DG-modules over de Rham DG-algebra

Rybakov, Sergey

doi:10.1007/s40879-014-0014-4

Cited by 3 publications

(3 citation statements)

References 10 publications

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“…Furthermore, both versions of Koszul duality are reinterpreted in this framework in op. cit., and likewise functoriality in [Ry15].…”

Section: It Contains the Category Of Coherent Sheaves Cohp Pmentioning

confidence: 94%

“…Remark 3.2.14. There is a third sheaf of algebras and corresponding category from [BG17,Ry15] that one can consider, which corresponds to the category CohpT X r´1sq grTate . The de Rham algebra is the dg sheaf of dg-commutative k-algebras Ω ‚ X,d " pΩ ‚ X , dq underlying the de Rham complex.…”

Section: 24mentioning

confidence: 99%

“…D X Rees alg. r D X " ReespD X , F ord q classical limit O T X D ´Ω duality "filtered" Koszul duality linear Koszul duality [BD91, Ry15,Pr15] [Ka91, Po11, BN12, TV20a, TV20b] [MR10, MR15, MR16, AO20]…”

Section: Introductionmentioning

confidence: 99%

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Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Chen¹

2023

Preprint

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In this paper we establish an equivalence between the category of S 1 -equivariant coherent sheaves on the formal loop space p LX of a smooth stack X and the category of coherent filtered D-modules on X, generalizing results from the case where X is a smooth variety. This equivalence interpolates between categories of coherent sheaves on stacks appearing as categorical traces in algebro-geometric settings, and categories of equivariant constructible sheaves in topological settings. Unlike in the case of varieties, the subcategory of coherent Dmodules on a smooth stack X is larger than the subcategory of compact objects; we identify the corresponding Koszul dual subcategory as the category of S 1 -equivariant ind-continuous coherent sheaves on p LX.

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“…Furthermore, both versions of Koszul duality are reinterpreted in this framework in op. cit., and likewise functoriality in [Ry15].…”

Section: It Contains the Category Of Coherent Sheaves Cohp Pmentioning

confidence: 94%

Section: 24mentioning

confidence: 99%

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Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Chen¹

2023

Preprint

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Positselski

2021

Relative Nonhomogeneous Koszul Duality

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On some categories of structured sets

Chiaselotti,

Gentile,

Infusino

2024

European Journal of Mathematics

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Given an arbitrary set $$\Omega $$ Ω , we consider the collections $$\textrm{SS}\hspace{0.55542pt}(\Omega )$$ SS ( Ω ) , $$\textrm{SR}\hspace{0.55542pt}(\Omega )$$ SR ( Ω ) and $$\textrm{SO}\hspace{0.55542pt}(\Omega )$$ SO ( Ω ) of all the set systems, the binary set relations and the set operators on $$\Omega $$ Ω . We introduce the notion of linking map on $$\Omega $$ Ω as any map whose domain and codomain may be chosen between the above collections. After providing a descriptive overview useful for framing the notion of linking map in a broad non-specialized context, we explain how linking maps occur in a very natural way in two specific results. The first of these results concerns the classic identification between the subfamily $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) of all the equivalence relations on $$\Omega $$ Ω and the subfamily $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) of all the set partitions on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) of closure operators on $$\Omega $$ Ω and four linking maps whose restrictions to the subfamilies $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) , $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) and $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) are bijections. The second result concerns the identification between the subfamily $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) of all closure set operators on $$\Omega $$ Ω and the subfamily $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) of all closure set systems on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) of binary set relations and four linking maps whose restrictions to the subfamilies $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) , $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) and $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) are again bijections. In an attempt to extend in a natural way the above linking maps to categorical isomorphisms, after fixing a nonnegative integer k, we introduce three categories $$\mathbf{SS^k}$$ SS k , $$\mathbf{SR^k}$$ SR k and $$\mathbf{SO^k}$$ SO k , whose detailed study mainly occupies the first part of the present work. Objects and arrows of these three categories are obtained by means of k-iterations of the powerset functor "Equation missing", and they generalize the notions of set systems, set relations and set operators, respectively. In the second part of the paper, we extend the linking maps previously described at a categorical level in terms of isomorphisms between specific categories of set systems, binary set relations and set operators generalizing the occurring collections introduced before, and also prove numerous other results concerning the main properties of all these categories, such as completeness, cocompleteness and Cartesian closedness.

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DG-modules over de Rham DG-algebra

Cited by 3 publications

References 10 publications

Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks I: Koszul duality

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On some categories of structured sets

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