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Derived equivalences of K3 surfaces and orientation
"... Abstract. Every Fourier–Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive ..."
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Abstract. Every Fourier–Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms. 1.
DIMENSIONS OF TRIANGULATED CATEGORIES VIA KOSZUL OBJECTS
"... Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin alge ..."
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Cited by 20 (10 self)
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Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras. 1.
THE GLOBAL DECOMPOSITION THEOREM FOR HOCHSCHILD (CO)HOMOLOGY OF SINGULAR SPACES VIA THE ATIYAH–CHERN CHARACTER
, 2006
"... Abstract. We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X → Y, the Hochschild complex HX/Y, as ..."
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Cited by 13 (1 self)
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Abstract. We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X → Y, the Hochschild complex HX/Y, as introduced in [BFl2], decomposes naturally in the derived category D(X) into ⊕ p≥0 Sp (LX/Y [1]), the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case. Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah–Chern character, the exponential of the universal Atiyah class. We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah–Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.
Loop spaces and Langlands parameters
, 2007
"... Abstract. We apply the technique of S 1equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free l ..."
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Cited by 3 (1 self)
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Abstract. We apply the technique of S 1equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free loop spaces, cyclic homology and de Rham cohomology to
Homotopy coherent centers versus centers of homotopy categories
 IN 46556 USA DWYER.1@ND.EDU MARKUS SZYMIK DEPARTMENT OF MATHEMATICAL SCIENCES NTNU NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY 7491 TRONDHEIM NORWAY MARKUS.SZYMIK@MATH.NTNU.NO
, 2013
"... Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are no ..."
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Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a BousfieldKan type spectral sequence that computes the homotopy groups. Nontrivial classes of examples are given as illustration throughout.
COHOMOLOGICAL SYMMETRY IN TRIANGULATED CATEGORIES
"... Abstract. We give a criterion for cohomological symmetry in a triangulated ..."
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Abstract. We give a criterion for cohomological symmetry in a triangulated
ON GROWTH IN TOTALLY ACYCLIC MINIMAL COMPLEXES
, 904
"... Abstract. Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth. 1. ..."
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Abstract. Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth. 1.