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17
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.
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 60 (7 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
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 29 (18 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
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 20 (4 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,
Derived Hall algebras
"... The purpose of this work is to define a derived Hall algebra DH(T), associated to any dgcategory T (under some finiteness conditions). Our main theorem states that DH(T) is associative and unital. This provides an alternative to the nonexistence of Hall algebras for triangulated categories. Conten ..."
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Cited by 12 (1 self)
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The purpose of this work is to define a derived Hall algebra DH(T), associated to any dgcategory T (under some finiteness conditions). Our main theorem states that DH(T) is associative and unital. This provides an alternative to the nonexistence of Hall algebras for triangulated categories. Contents
Moduli of objects in dgcategories
, 2006
"... To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness ..."
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Cited by 10 (2 self)
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To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −stack MT is locally geometric (i.e. union of open and geometric substacks). As a consequence we prove the algebraicity of the group of autoequivalences of saturated dgcategories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as
The basic geometry of Witt vectors
"... Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “ptypical” Witt vectors of padic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two di ..."
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Cited by 8 (2 self)
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Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “ptypical” Witt vectors of padic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two different ways. We allow not just ptypical Witt vectors but also, for example, those taken with respect to any set of primes in any ring of integers in any global field. We also allow not just padic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of the Greenberg transform. We investigate whether many standard geometric properties of spaces and maps are preserved by Witt vector functors.