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66
DG quotients of DG categories
 J. Algebra
"... 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. ..."
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Cited by 77 (0 self)
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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.
Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 66 (4 self)
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These are slightly expanded notes of four introductory talks on
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
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 projective m ..."
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Cited by 43 (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.
The stable derived category of a Noetherian scheme
 Compos. Math
"... Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday. Abstract. 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 s ..."
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Cited by 34 (5 self)
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Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday. Abstract. 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.
DUALIZING COMPLEXES, MORITA EQUIVALENCE AND THE DERIVED PICARD GROUP OF A RING
, 1998
"... Two rings A and B are said to be derived Morita equivalent if the derived categories Db (Mod A) and Db (Mod B) are equivalent. By results of Rickard in [Ri1] and [Ri2], if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T s.t. the functor T ⊗L A − ..."
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Cited by 29 (10 self)
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Two rings A and B are said to be derived Morita equivalent if the derived categories Db (Mod A) and Db (Mod B) are equivalent. By results of Rickard in [Ri1] and [Ri2], if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T s.t. the functor T ⊗L A −: Db (Mod A) → Db (Mod B) is an equivalence. The complex T is called a tilting complex. When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group K0(A). We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables us to compute DPic(A) in these cases. Assume A is noetherian. Dualizing complexes over A were defined in [Ye]. These are complexes of bimodules which generalize the commutative definition of [RD]. We prove that the group DPic(A) classifies the set of isomorphism
Quantum groups, the loop Grassmannian, and the Springer resolution
 J. Amer. Math. Soc
"... We establish equivalences of the following three triangulated categories: Dquantum(g) ← → D G coherent (Ñ) ← → Dperverse(Gr). Here, Dquantum(g) is the derived category of the principal block of finite dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a comp ..."
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Cited by 23 (8 self)
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We establish equivalences of the following three triangulated categories: Dquantum(g) ← → D G coherent (Ñ) ← → Dperverse(Gr). Here, Dquantum(g) is the derived category of the principal block of finite dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g; the category DG coherent (Ñ) is defined in terms of coherent sheaves on the cotangent bundle on the (finite dimensional) flag manifold for G ( = semisimple group with Lie algebra g), and the category Dperverse(Gr) is the derived category of perverse sheaves on the Grassmannian Gr associated with the loop group LG ∨ , where G ∨ is the Langlands dual group, smooth along the Schubert stratification. The equivalence between Dquantum(g) and DG coherent (Ñ) is an ‘enhancement ’ of the known expression (due to GinzburgKumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between Dperverse(Gr) and DG coherent (Ñ) can be viewed as a ‘categorification ’ of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the DeligneLanglandsLusztig conjecture. One realization is in terms of locally constant functions on the flag
Derived categories, resolutions, and Brown representability
, 2004
"... These notes are based on a series of five lectures given during the ..."
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Cited by 20 (2 self)
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These notes are based on a series of five lectures given during the