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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.
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 32 (10 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.
algebras and modules in general model categories, preprint arXiv:math.AT/0101102
, 2001
"... Abstract. In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give Jsemi model structures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. We prove h ..."
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Cited by 22 (0 self)
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Abstract. In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give Jsemi model structures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. We prove homotopy invariance properties for the categories of algebras and modules. In a second part we develop the theory of Smodules and algebras of [EKMM] and [KM], which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.
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
Homology and cohomology of E∞ ring spectra
 MATHEMATISCHE ZEITSCHRIFT
, 2005
"... Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads. ..."
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Cited by 16 (0 self)
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Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.
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 14 (6 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
On Voevodsky’s algebraic Ktheory spectrum BGL
, 2007
"... Under a certain normalization assumption we prove that the P 1spectrum BGL of Voevodsky which represents algebraic Ktheory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1spectrum BGL with the structure of a commutative ring P 1spectrum in the motivic stable homotopy cate ..."
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Cited by 13 (6 self)
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Under a certain normalization assumption we prove that the P 1spectrum BGL of Voevodsky which represents algebraic Ktheory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1spectrum BGL with the structure of a commutative ring P 1spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec(Z). For an arbitrary Noetherian base scheme S we pull this structure back to get a distinguished monoidal structure on BGL. 1