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21
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 45 (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.
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 19 (7 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 19 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
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 16 (1 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 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 13 (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.
Geometry of chain complexes and outer automorphisms under derived equivalence
 Transactions of the American Mathematical Society
"... The authors wish to dedicate this paper to Idun Reiten on the occasion of her sixtieth birthday. Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms o ..."
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Cited by 11 (0 self)
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The authors wish to dedicate this paper to Idun Reiten on the occasion of her sixtieth birthday. Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of A is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization CompA d of the family of finite Amodule complexes with fixed sequence d of dimensions and an “almost projective ” complex X ∈ CompA d, there exists a canonical vector space embedding TX(Comp A d)/TX(G.X) −→
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 11 (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.
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 5 (0 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
Morita theory in stable homotopy theory
, 2004
"... We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ fr ..."
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Cited by 3 (2 self)
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We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.
ON SERRE DUALITY FOR COMPACT HOMOLOGICALLY SMOOTH DG ALGEBRAS
, 2007
"... Let X be a smooth projective variety over a perfect field k. It is a classical fact that the ..."
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Cited by 3 (1 self)
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Let X be a smooth projective variety over a perfect field k. It is a classical fact that the