DG quotients of DG categories

By V. Drinfeld. 2002

471 citations recorded by the source corpus.

Abstract Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/816fa650c339c87e06810b9749f5ab26849310a3

How researchers see this paper

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In Algebra and Number Theory

1. B. Keller
2. Gonçalo Tabuada
3. Osamu Iyama
4. Alexander Kuznetsov
5. Dong Ping Yang
6. Michel van den Bergh
7. Dmitri Orlov
8. P. Seidel

In Geometry and Topology

1. B. Keller
2. M. Abouzaid
3. I. Smith
4. P. Seidel
5. Gonçalo Tabuada
6. D. Auroux
7. L. Katzarkov
8. Dmitri Orlov

In Mathematical Physics

1. M. Abouzaid
2. I. Smith
3. B. Keller
4. P. Seidel
5. D. Auroux
6. Gonçalo Tabuada
7. Nick Sheridan
8. L. Katzarkov

Citations in the corpus

• Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (2008, unknown): “The internal hom assigns to presentable ∞categories C and D the ∞-category of colimit-preserving functors FunL(C, D), which is presentable by [L2, Proposition 5.5.3.8].”
• Higher K-theory via universal invariants (2007, background): “Stabilization 26 10.”
• Equivalence of the Derived Category of a Variety with a Singularity Category (2010, methodology): “erfect A-modules in Db gr(A) by PerfA. Definition 1.3. The singularity category of (X,A) is defined to be the Verdier quotient: Dgr sg(A) = D b gr(A)/PerfA. At the dg level, we can take the dg quotient [Dri04], [Kel99], [To¨e07] of the dg derived category Db gr(A) by the full dg subcategory of perfect”
• Augmentations are sheaves (2015, background): “ory, we write Fun naive(C;A) for the dg category of functors from Cto the category whose objects are cochain complexes in A, and whose maps are the cochain maps. We write Fun(C;A) for the dg quotient [Dri04] of Fun naive(C;A) by the thick subcategory of functors taking values in acyclic complexes. F”
• Loop spaces and connections (2010, background): “(Among the many excellent references for dg and A∞-categories, we recommend [10] and the survey [19].”
• Grothendieck ring of pretriangulated categories (2004, background): “quivalence of triangulated categories Ho(SF(A)) ≃D(A). A DG functor G: A→Binduces DG functors Res G: B 0-mod →A-mod, Ind G: A0-mod →B0-mod, where Res G is the obvious restriction functor. We refer to [Dr] for the definition of Ind G; let us only list some of the properties. 1). The functor Ind G is l”
• Non-connective K-theory via universal invariants (2009, methodology): “where B is a pre-triangulated dg category, A is a thick dg subcategory of B, and B/A is the Drinfeld dg quotient [14] of B by A.”
• Categorical resolutions of irrational singularities (2012, background): “In [10], Drinfeld defined a new DG-category D/D0, which is known as Drinfeld quotient.”
• The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks (2009, background): “If X is a topological space we let Sh(X) denote the dg category of bounded chain complexes of sheaves of C-vector spaces on X , localized with respect to acyclic complexes (see [16] for localizations of dg categories).”
• Compact generators in categories of matrix factorizations (2009, unknown): “This theory of dg quotients was extended and made more explicit in [Dri04].”
• A Guided Tour Through the Garden of Noncommutative Motives (2011, background): “Theorem 3.1 allowed us to complete this aspect of Drinfeld’s work by characterizing the DG quotient as a homotopy cofiber construction; see [44].”