Finite-dimensional differential graded algebras and their geometric realizations

Orlov, Dmitri

doi:10.1016/j.aim.2020.107096

Cited by 14 publications

(20 citation statements)

References 18 publications

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“…Remark 2.26. Another notion of a simple dg algebra was considered in a recent paper [Orl20]. Orlov's notion is much stronger than ours, in that it is quasi-isomorphic to an ordinary simple algebra.…”

Section: Proposition 223 If a Triangulated Category Admits A Morphism-faithful Prime Rank Function Then It Is Simplementioning

confidence: 72%

Rank functions on triangulated categories

Chuang

Lazarev

2021

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

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Section: Proposition 223 If a Triangulated Category Admits A Morphism-faithful Prime Rank Function Then It Is Simplementioning

confidence: 72%

Rank functions on triangulated categories

Chuang

Lazarev

2021

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…As a first example, we can consider the dg 2-category C k . In this case, #-C k and C k are dg biequivalent and a dg algebra 1-morphism in C k corresponds to a finite-dimensional dg algebra over k. Results of [Or2] imply the following corollary.…”

Section: The Dg 2-category Cmentioning

confidence: 89%

“…The first example of such dg 2-categories is given by C k , the 2-category of bounded complexes of finite-dimensional kvector spaces. Using results of Orlov [Or2], we prove in Proposition 5.3 that C k has a unique non-acyclic quotient-simple pretriangulated 2-representation. • Finally, we explain how the theory developed in this paper can be applied to categorical braid group actions of [KS,Ro1,Ro2] in Section 5.5.…”

Section: Introductionmentioning

confidence: 94%

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Pretriangulated 2-representations via dg algebra 1-morphisms

Laugwitz¹,

Miemietz²

2022

Preprint

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This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of modules over dg algebra 1morphisms internal to associated dg 2-categories of compact dg modules. Further, we investigate the Morita theory and quasi-equivalences for such dg 2representations. We relate this theory to various classes of examples of dg categorifications from the literature. Contents 1. Introduction 2.1. Dg categories 2.2. pretriangulated categories 2.3. Compact dg modules 2.4. Compactness and adjunctions 2.5. The homotopy category of compact semi-free modules 3. Dg 2-categories and 2-representations 3.1. Dg 2-categories 3.2. Dg 2-representations 3.3. Compact pretriangulated 2-representations 3.4. Dg ideals of 2-representations and dg 2-subrepresentations 3.5. Cyclic and quotient-simple dg 2-representations 3.6. Homotopy 2-representations 4. Dg algebra 1-morphisms 4.1. Compact modules over dg algebra 1-morphisms 4.2. The algebra structure on internal homs [X, X] 4.3. An equivalence of dg 2-representations 4.4. Morphisms of dg 2-representations and dg algebra 1-morphisms

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“…2.3], consider the quotient A/J + , where J + stands for the external dg Jacobson radical of A. Theorem 3.7. Assume that the quotient dg k-algebra A/J + is separable in the sense of [16,Def. 2.11] (this holds when k is perfect) and that char(k) > 0.…”

Section: Applications To Noncommutative Geometrymentioning

confidence: 99%

Noncommutative Riemann hypothesis

Tabuada¹

2021

Preprint

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In this note, making use of noncommutative l-adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first application, we prove that the generalized Riemann hypothesis is invariant under derived equivalences and homological projective duality. As a second application, we prove the noncommutative generalized Riemann hypothesis in some new cases.

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Finite-dimensional differential graded algebras and their geometric realizations

Cited by 14 publications

References 18 publications

Rank functions on triangulated categories

Rank functions on triangulated categories

Pretriangulated 2-representations via dg algebra 1-morphisms

Noncommutative Riemann hypothesis

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