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199
Stability structures, motivic DonaldsonThomas invariants and cluster transformations
, 2008
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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 187 (4 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.
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
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 94 (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.
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
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Cited by 86 (17 self)
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Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,
Higher and derived stacks: a global overview
, 2005
"... These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the sub ..."
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Cited by 58 (9 self)
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These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.
Higher Ktheory via universal invariants
 Duke Math. J
"... Abstract. Using the formalism of Grothendieck’s derivators, we construct ‘the universal localizing invariant of dg categories’. By this, we mean a morphism Ul from the pointed derivator HO(dgcat) associated with the Morita homotopy theory of dg categories to a triangulated strong derivator Mloc dg s ..."
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Cited by 42 (21 self)
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Abstract. Using the formalism of Grothendieck’s derivators, we construct ‘the universal localizing invariant of dg categories’. By this, we mean a morphism Ul from the pointed derivator HO(dgcat) associated with the Morita homotopy theory of dg categories to a triangulated strong derivator Mloc dg such that Ul commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle and is universal for these properties. Similary, we construct the ‘universal additive invariant of dg categories’, i.e. the universal morphism of derivators Ua from HO(dgcat) to a strong triangulated derivator Madd dg which satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen Ktheory appears