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88
Braid group actions on derived categories of coherent sheaves
 DUKE MATH. J
, 2001
"... This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is ..."
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Cited by 264 (8 self)
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This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim X ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.
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 154 (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
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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Cited by 53 (1 self)
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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
Localization theorems in topological Hochschild homology and topological cyclic homology
, 2008
"... We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of ..."
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Cited by 48 (8 self)
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We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of ThomasonTrobaugh in Ktheory. We also deduce versions of Thomason’s blowup formula and the projective bundle formula for THH and TC.
CONSTRUCTIBLE SHEAVES AND THE FUKAYA CATEGORY
, 2006
"... Abstract. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on a manifold X. Let TwFuk(T ∗ X) be the triangulated A∞category of twisted complexes in the Fukaya category of the cotangent bundle T ∗ X. We prove that Sh(X) embeds as an A∞subcategory of TwFuk(T ∗ ..."
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Cited by 44 (11 self)
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Abstract. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on a manifold X. Let TwFuk(T ∗ X) be the triangulated A∞category of twisted complexes in the Fukaya category of the cotangent bundle T ∗ X. We prove that Sh(X) embeds as an A∞subcategory of TwFuk(T ∗ X). Taking cohomology gives an embedding of the corresponding derived categories.
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 44 (16 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
A GerstenWitt spectral sequence for regular schemes
 Ann. Sci. ENS
"... Abstract. A spectral sequence is constructed whose nonzero E1terms are the Witt groups of the residue fields of a regular scheme X, arranged in GerstenWitt complexes, and whose limit is the four global Witt groups of X. There are several immediate consequences concerning purity for Witt groups of ..."
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Cited by 40 (7 self)
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Abstract. A spectral sequence is constructed whose nonzero E1terms are the Witt groups of the residue fields of a regular scheme X, arranged in GerstenWitt complexes, and whose limit is the four global Witt groups of X. There are several immediate consequences concerning purity for Witt groups of lowdimensional schemes. The Witt groups of punctured spectra of regular local rings are also computed. Let X be a regular integral separated noetherian scheme in which 2 is everywhere invertible. (We will maintain these hypotheses throughout the introduction.) It is now known to the experts that the Witt groups of the residue fields of X form a nonexact cochain complex
Cyclic homology, cdhcohomology and negative Ktheory
, 2005
"... We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheor ..."
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Cited by 39 (11 self)
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We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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Cited by 27 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
Algebraic deformations arising from orbifolds with discrete torsion
"... Abstract. We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This ..."
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Cited by 26 (17 self)
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Abstract. We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This connects with earlier calculations by Vafa and Witten of chiral numbers and deformations of a similar orbifold. 1.