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15
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
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]).
Cohomological quotients and smashing localizations
 Amer. J. Math
"... Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the coh ..."
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Cited by 14 (2 self)
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Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic Ktheory and demonstrates the relevance of the telescope
Relative singularity categories and Gorensteinprojective modules, submitted. Also see: arXiv:0709.1762
"... and under some reasonable conditions, we show that the ..."
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Cited by 6 (4 self)
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and under some reasonable conditions, we show that the
ALGEBRAIC KTHEORY AND ABSTRACT HOMOTOPY THEORY
"... terms of its DwyerKan simplicial localization. This leads to a criterion for functors to induce equivalences of Ktheory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of ..."
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Cited by 2 (1 self)
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terms of its DwyerKan simplicial localization. This leads to a criterion for functors to induce equivalences of Ktheory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of Ktheory spectra. 1.
On the Construction of Triangle Equivalences
, 1994
"... F11.52> DA ! DB ; RHom B (X; ?) : DB ! DA are defined and form an adjoint pair without any hypotheses on the rings A, B, or the complex X of ABbimodules. These functors may or may not give rise to functors between the bounded categories, but in any case, the question about the existence of functo ..."
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Cited by 1 (0 self)
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F11.52> DA ! DB ; RHom B (X; ?) : DB ! DA are defined and form an adjoint pair without any hypotheses on the rings A, B, or the complex X of ABbimodules. These functors may or may not give rise to functors between the bounded categories, but in any case, the question about the existence of functors between the bounded categories becomes much simpler to study once the phenomena at the unbounded level are understood. The situation is similar to that encountered in studying a variety over a field. It is natural to pass to the algebraic closure in a first step, and to study the problem of descent in a second step. 1.1 Unbounded resolutions. Let A be a ring (associative, with one) and denote by HA the homotopy category of (unbounded
THE PORDER OF TOPOLOGICAL TRIANGULATED CATEGORIES
"... p annihilates objects of the form Y/p. In this paper we show that the porder of a topological ..."
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Cited by 1 (1 self)
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p annihilates objects of the form Y/p. In this paper we show that the porder of a topological
CLUSTER EQUIVALENCE AND GRADED DERIVED EQUIVALENCE
"... Abstract. In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce an invariant of these algebras called cluster equivalence, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how ..."
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Abstract. In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce an invariant of these algebras called cluster equivalence, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster tilting objects in the generalized cluster categories: We first observe that cluster tilting objects in generalized cluster categories are in natural bijection to cluster tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that if two cluster equivalent algebras are not derived equivalent, then
Heller triangulated categories
, 2007
"... complexes with entries in E. Shifting a complex by 3 positions yields an outer shift ..."
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complexes with entries in E. Shifting a complex by 3 positions yields an outer shift