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On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
PERVERSE BUNDLES AND CALOGEROMOSER SPACES
, 2007
"... Abstract. We present a simple description of moduli spaces of torsionfree Dmodules (Dbundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with CalogeroMoser quiver varieties. Namely, we show that the moduli of Dbundles form twiste ..."
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Cited by 4 (2 self)
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Abstract. We present a simple description of moduli spaces of torsionfree Dmodules (Dbundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with CalogeroMoser quiver varieties. Namely, we show that the moduli of Dbundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T ∗ X, which contain as open subsets the moduli of framed torsionfree sheaves (the Hilbert schemes T ∗ X [n] in the rank one case). The proof is based on the description of the derived category of Dmodules on X by a noncommutative version of the Beilinson transform on P 1. 1.
unknown title
"... The following thesis plays a central role in deformation theory: (∗) If X is a moduli space over a field k of characteristic zero, then a formal neighborhood of any point x ∈ X is controlled by a differential graded Lie algebra. This idea was developed in unpublished work of Deligne, Drinfeld, and F ..."
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The following thesis plays a central role in deformation theory: (∗) If X is a moduli space over a field k of characteristic zero, then a formal neighborhood of any point x ∈ X is controlled by a differential graded Lie algebra. This idea was developed in unpublished work of Deligne, Drinfeld, and Feigin, and has powerfully influenced
RESEARCH STATEMENT
"... 2.1. Compatible autoequivalences 2 2.2. Application of Theorem 2.3 to structure of Mσn ..."
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2.1. Compatible autoequivalences 2 2.2. Application of Theorem 2.3 to structure of Mσn
Deformed Calabi–Yau completions
"... With an appendix by Michel Van den Bergh at Diepenbeek Abstract. We define and investigate deformed nCalabi–Yau completions of homologically smooth di¤erential graded ( dg) categories. Important examples are: deformed preprojective algebras of connected nonDynkin quivers, Ginzburg dg algebras ass ..."
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With an appendix by Michel Van den Bergh at Diepenbeek Abstract. We define and investigate deformed nCalabi–Yau completions of homologically smooth di¤erential graded ( dg) categories. Important examples are: deformed preprojective algebras of connected nonDynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed Calabi–Yau completions do have the Calabi–Yau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the Calabi–Yau property. We show that deformed 3Calabi–Yau completions of algebras of global dimension at most 2 are quasiisomorphic to Ginzburg dg algebras and apply this to the study of clustertilted algebras and to the construction of derived equivalences associated to mutations of quivers with potentials. In the appendix, Michel Van den Bergh uses noncommutative di¤erential geometry to give an alternative proof of the fact that Ginzburg dg algebras have the Calabi–Yau property.
Abstract
, 2008
"... In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of Omodules on schemes, as well as its quasicoherent and p ..."
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In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of Omodules on schemes, as well as its quasicoherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere. Contents 1 Motivations
Author manuscript, published in "Topics in Algebraic and Topological KTheory (2011) 243301" DOI: 10.1007/9783642157080 Lectures on DGcategories
, 2013
"... The purpose of these four lectures is to provide an introduction to the theory of dgcategories. There are several possible point of views to present the subject, and my choice ..."
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The purpose of these four lectures is to provide an introduction to the theory of dgcategories. There are several possible point of views to present the subject, and my choice
DOI: 10.1007/9783642012006_11 Note of Chern character, loop spaces and derived algebraic geometry
, 2008
"... In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of Omodules on schemes, as well as its quasicoherent and p ..."
Abstract
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In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of Omodules on schemes, as well as its quasicoherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere. Contents 1 Motivations
THE FUNDAMENTAL ISOMORPHISM CONJECTURE VIA NONCOMMUTATIVE MOTIVES
"... Abstract. Given a group, we construct a fundamental additive functor on ..."