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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
Classification spaces of maps in model categories
"... Abstract. We correct a mistake in [DK2] and use this to identify homotopy function complexes in a model category with the nerves of certain categories of zigzags. ..."
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Cited by 5 (3 self)
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Abstract. We correct a mistake in [DK2] and use this to identify homotopy function complexes in a model category with the nerves of certain categories of zigzags.
Universal Toda brackets of ring spectra
, 2006
"... Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines ..."
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Cited by 4 (1 self)
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Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an Rmodule spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex Ktheory spectra serve as our main examples. 1.
HOMOTOPIC HOPFGALOIS EXTENSIONS: FOUNDATIONS AND EXAMPLES
, 902
"... Abstract. HopfGalois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the HopfGalois extensions with respect to the dual of the group algebra of G. Rognes recently extended the not ..."
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Cited by 2 (1 self)
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Abstract. HopfGalois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the HopfGalois extensions with respect to the dual of the group algebra of G. Rognes recently extended the notion of
The plus construction, Bousfield localization, and derived completion
, 2009
"... We define a plusconstruction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plusconstruction is related to both Bousfield localization and Carlsson’s derived completion. 1 ..."
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We define a plusconstruction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plusconstruction is related to both Bousfield localization and Carlsson’s derived completion. 1
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
"... We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and p ..."
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We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.
DG ALGEBRAS WITH EXTERIOR HOMOLOGY
"... Abstract. We study differential graded algebras (DGAs) whose homology is an exterior algebra over a commutative ring R on a generator of degree n, and also certain types of differential modules over these DGAs. We obtain a complete classification with R = Z or R = Fp and n ≥ −1. The examples are une ..."
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Abstract. We study differential graded algebras (DGAs) whose homology is an exterior algebra over a commutative ring R on a generator of degree n, and also certain types of differential modules over these DGAs. We obtain a complete classification with R = Z or R = Fp and n ≥ −1. The examples are unexpectedly interesting. 1.