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55
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 135 (24 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.
HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 68 (17 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
Localization theorems in topological Hochschild homology and topological cyclic homology
, 2008
"... We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of ..."
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Cited by 48 (8 self)
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We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of ThomasonTrobaugh in Ktheory. We also deduce versions of Thomason’s blowup formula and the projective bundle formula for THH and TC.
KTheory and Derived Equivalences
, 2003
"... 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. ..."
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Cited by 46 (7 self)
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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.
Spectral enrichments of model categories
, 2005
"... We prove that every stable, combinatorial model category can be enriched in a natural way over symmetric spectra. As a consequence of the general theory, every object in such a model category has an associated homotopy endomorphism ring spectrum. Basic properties of these invariants ..."
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Cited by 27 (5 self)
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We prove that every stable, combinatorial model category can be enriched in a natural way over symmetric spectra. As a consequence of the general theory, every object in such a model category has an associated homotopy endomorphism ring spectrum. Basic properties of these invariants
Derived Azumaya algebras and generators for twisted derived categories
, 2011
"... We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of [Gr]. We prove that any such algebra B on a scheme X provides a class φ(B) in H 1 et(X, Z) × H 2 et(X, Gm). We prove that for X a quasicompact and quasiseparated scheme φ defines ..."
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Cited by 26 (0 self)
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We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of [Gr]. We prove that any such algebra B on a scheme X provides a class φ(B) in H 1 et(X, Z) × H 2 et(X, Gm). We prove that for X a quasicompact and quasiseparated scheme φ defines a bijective correspondence, and in particular that any class in H 2 et(X, Gm), torsion or not, can be represented by a derived Azumaya algebra on X. Our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories, with coefficients in any local system of reasonable dgcategories, generalizing the well known existence of compact generators in derived categories of quasicoherent sheaves of [BoVa]. A huge part of this paper concerns the treatment of twisted derived categories, as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator. We present explicit examples of derived Azumaya algebras that are not represented by classical Azumaya algebras, as well as applications of our main result to the localization for twisted algebraic Ktheory and to the stability of saturated dgcategories by direct pushforwards along smooth and proper maps. Contents 1 The local theory 5
Homotopy Theory of Higher Categories
"... portant la référence ANR09BLAN015102 (HODAG). This is draft material from a forthcoming book to be published by Cambridge University Press in the New Mathematical Monographs series. This publication is in copyright. c○Carlos T. Simpson 2010. v This is the first draft of a book about higher categ ..."
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Cited by 23 (0 self)
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portant la référence ANR09BLAN015102 (HODAG). This is draft material from a forthcoming book to be published by Cambridge University Press in the New Mathematical Monographs series. This publication is in copyright. c○Carlos T. Simpson 2010. v This is the first draft of a book about higher categories approached by iterating Segal’s method, as in Tamsamani’s definition of nnerve and Pelissier’s thesis. If M is a tractable left proper cartesian model
Topological equivalences for differential graded algebras
 Adv. Math
, 2006
"... Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are ..."
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Cited by 22 (8 self)
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Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents
ENRICHED MODEL CATEGORIES AND AN APPLICATION TO ADDITIVE ENDOMORPHISM SPECTRA
"... Abstract. We define the notion of an additive model category and prove that ..."
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Cited by 17 (4 self)
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Abstract. We define the notion of an additive model category and prove that