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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 project ..."
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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.
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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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
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 37 (12 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.
Fukaya categories and deformations
 Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 351–360, Higher Ed
, 2002
"... Soon after their first appearance [7], Fukaya categories were brought to the attention of a wider audience through the homological mirror conjecture [14]. Since then Fukaya and his collaborators have undertaken the vast project of laying down the foundations, and as a result a fully general definiti ..."
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Cited by 37 (6 self)
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Soon after their first appearance [7], Fukaya categories were brought to the attention of a wider audience through the homological mirror conjecture [14]. Since then Fukaya and his collaborators have undertaken the vast project of laying down the foundations, and as a result a fully general definition is available [9, 6]. The task that symplectic geometers are now facing is to make these categories into an effective tool, which in particular means developing more ways of doing computations in and with them. For concreteness, the discussion here is limited to projective varieties which are CalabiYau (most of it could be carried out in much greater generality, in particular the integrability assumption on the complex structure plays no real role). The first step will be to remove a hyperplane section from the variety. This makes the symplectic form exact, which simplifies the pseudoholomorphic map theory considerably. Moreover, as far as Fukaya categories are concerned, the affine piece can be considered as a first approximation to the projective variety. This is a fairly obvious idea, even though its proper formulation requires some algebraic formalism of deformation theory. A basic question is the finitedimensionality of the relevant deformation spaces. As Conjecture 4 shows, we hope for a favourable answer in many cases. It remains to be seen whether this is really a viable strategy for understanding Fukaya categories in interesting examples. Lack of space and ignorance keeps us from trying to survey related developments, but we want to give at least a few indications. The idea of working relative to a divisor is very common in symplectic geometry; some papers whose viewpoint is close to ours are [12, 16, 3, 17]. There is also at least one entirely different approach to Fukaya categories, using Lagrangian fibrations and Morse theory [8, 15, 4]. Finally, the example of the twotorus has been studied extensively [18]. Acknowledgements. Obviously, the ideas outlined here owe greatly to Fukaya
Hochschild homology and semiorthogonal decompositions
"... Abstract. We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the co ..."
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Cited by 29 (2 self)
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Abstract. We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hochschild homology is additive with respect to semiorthogonal decompositions and construct some long exact sequences relating the Hochschild cohomology of a category and its semiorthogonal components. We also compute Hochschild homology and cohomology of some interesting admissible subcategories, in particular of the nontrivial components of derived categories of some Fano threefolds and of the nontrivial components of the derived categories of conic bundles. 1.
Hochschild cohomology and derived Picard groups
, 2003
"... We interpret Hochschild cohomology as the Lie algebra of the derived Picard group and deduce that it is preserved under derived equivalences. ..."
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Cited by 28 (1 self)
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We interpret Hochschild cohomology as the Lie algebra of the derived Picard group and deduce that it is preserved under derived equivalences.
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
On the Cyclic Homology of Ringed Spaces and Schemes
 DOC. MATH. J. DMV
, 1998
"... We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space. ..."
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Cited by 25 (0 self)
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We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.