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56
Generators and representability of functors in commutative and noncommutative geometry
 MOSC MATH. J
, 2002
"... We give a sufficient condition for an Extfinite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived ca ..."
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Cited by 85 (2 self)
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We give a sufficient condition for an Extfinite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 78 (16 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
Introduction to Ainfinity algebras and modules
 Homology, Homotopy and Applications
"... Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞algebras, ..."
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Cited by 68 (6 self)
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Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞algebras,
The additivity of traces in triangulated categories
 Adv. Math
"... Abstract. We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relatin ..."
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Cited by 34 (7 self)
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Abstract. We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the examples and shed light on generalized homology and cohomology theories. Contents 1. Generalized trace maps 2 2. Triangulated categories 6 3. Weak pushouts and weak pullbacks 9 4. The compatibility axioms 11
Ktheory and derived equivalences
 Duke Math. J
"... Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. Contents ..."
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Cited by 29 (6 self)
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Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. Contents
Local homology and cohomology on schemes
, 1997
"... Abstract. We prove a sheaftheoretic derivedcategory generalization of GreenleesMay duality (a farreaching generalization of Grothendieck’s local duality theorem): for a quasicompact separated scheme X and a “proregular ” subscheme Z—for example, any separated noetherian scheme and any closed su ..."
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Cited by 25 (3 self)
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Abstract. We prove a sheaftheoretic derivedcategory generalization of GreenleesMay duality (a farreaching generalization of Grothendieck’s local duality theorem): for a quasicompact separated scheme X and a “proregular ” subscheme Z—for example, any separated noetherian scheme and any closed subscheme—there is a sort of sheafified adjointness between local cohomology supported in Z and leftderived completion along Z. In particular, leftderived completion can be identified with local homology, i.e., the homology of RHom • (RΓ Z OX, −). Sheafified generalizations of a number of duality theorems scattered about the literature result: the PeskineSzpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related localglobal duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality
Invariance and localization for cyclic homology of DG algebras
 J. PURE APPL. ALGEBRA
, 1998
"... We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyc ..."
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Cited by 24 (6 self)
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We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of ThomasonTrobaugh’s [51] and Yao’s [58] localization theorems in Ktheory (cf. also [55]).
Realizability Of Modules Over Tate Cohomology
, 2001
"... Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is is ..."
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Cited by 20 (1 self)
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Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kGmodule M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
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Cited by 19 (3 self)
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Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,
Cyclic Homology For Schemes
 Proc. Amer. Math. Soc
, 1996
"... Abstract. Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring k. By ‘extend ’ we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme. The purpose of this paper is to show that ther ..."
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Cited by 16 (2 self)
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Abstract. Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring k. By ‘extend ’ we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme. The purpose of this paper is to show that there is a cyclic homology theory HC ∗ of schemes over a commutative ring k, extending the usual cyclic homology HC ∗ of kalgebras. By a cyclic homology theory for schemes over k we mean a family of graded kmodules HCn(X) associated to every scheme X over k which satisfy: (0.1) they are natural and contravariant in X; (0.2) for each affine scheme X = Spec A, there are natural isomorphisms HCn(X) ∼ = HCn(A) for all n; (0.3) if X = U ∪ V, there is a MayerVietoris sequence · · · HCn(X) → HCn(U) ⊕ HCn(V) → HCn(U ∩ V) → HCn−1(X) · · ·. We discuss uniqueness of a cyclic homology theory briefly in Remark 0.5 below. We have chosen homological indexing because of axiom (0.2), and because cohomological indexing (HC n = HC−n) would concentrate the nonzero groups in negative degrees.