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123
DG quotients of DG categories
 J. ALGEBRA
, 2008
"... 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. ..."
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Cited by 152 (0 self)
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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.
The homotopy theory of dgcategories and derived Morita Theory
, 2006
"... The main purpose of this work is to study the homotopy theory of dgcategories up to quasiequivalences. Our main result is a description of the mapping spaces between two dgcategories C and D in terms of the nerve of a certain category of (C, D)bimodules. We also prove that the homotopy category ..."
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Cited by 152 (6 self)
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The main purpose of this work is to study the homotopy theory of dgcategories up to quasiequivalences. Our main result is a description of the mapping spaces between two dgcategories C and D in terms of the nerve of a certain category of (C, D)bimodules. We also prove that the homotopy category Ho(dg −Cat) possesses internal Hom’s relative to the (derived) tensor product of dgcategories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dgcategories of modules over two dgcategories C and D as the dgcategory of (C, D)bimodules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dgcategories (i.e. the nerve of the category of dgcategories and quasiequivalences between them). The second application is the existence of a good theory of localization for dgcategories, defined in terms of a natural universal property. Our last application states that the dgcategory of (continuous) morphisms between the dgcategories of quasicoherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasiequivalent
Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 117 (4 self)
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These are slightly expanded notes of four introductory talks on
The stable derived category of a Noetherian scheme
 COMPOS. MATH
, 2004
"... For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an anal ..."
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Cited by 95 (12 self)
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For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal CohenMacaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. In addition, the relevance of the stable derived category in modular representation theory is indicated.
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
"... ..."
On the Cyclic Homology of Exact Categories
 JPAA
"... The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated project ..."
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Cited by 88 (1 self)
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The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra. However, we show that McCarthy's theory cannot be both compatible with localizations and invariant under functors inducing equivalences in the derived category. This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.
Compact generators in categories of matrix factorizations
 MR2824483 (2012h:18014), Zbl 1252.18026, arXiv:0904.4713
"... Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an ex ..."
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Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this quasiequivalence we establish a derived Morita theory which identifies the functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of matrix factorization categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry modelled on dg categories. Contents
Two kinds of derived categories, Koszul duality, and comodulecontramodule correspondence
, 2009
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CLUSTER ALGEBRAS VIA CLUSTER CATEGORIES WITH INFINITEDIMENSIONAL MORPHISM SPACES
"... Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an e ..."
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Cited by 49 (3 self)
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Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the Einvariant and show that an arbitrary decorated representation with vanishing Einvariant is characterized by its gvector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid