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19
Γcohomology of rings of numerical polynomials and E∞ structures on
"... Abstract. We investigate Γcohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction group ..."
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Cited by 7 (6 self)
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Abstract. We investigate Γcohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E ∞ structures. As a consequence we obtain an E∞ structure for the connective Adams summand. For the JohnsonWilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its Inadic completion.
Hopf algebra structure on topological Hochschild homology
, 2005
"... The topological Hochschild homology THH(R) of a commutative Salgebra (E ∞ ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bökstedt spectral sequence converging to the mod p homology of THH(R) in ..."
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Cited by 6 (3 self)
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The topological Hochschild homology THH(R) of a commutative Salgebra (E ∞ ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bökstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to study some interesting examples, including the commutative Salgebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic Ktheory of Salgebras, using topological cyclic homology.
M A Mandell, Rings, modules, and algebras in infinite loop space theory
 Adv. Math
"... Abstract. We give a new construction of the algebraic Ktheory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structur ..."
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Cited by 3 (0 self)
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Abstract. We give a new construction of the algebraic Ktheory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory, a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction.
Morita theory in stable homotopy theory
, 2004
"... We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ fr ..."
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Cited by 3 (2 self)
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We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.
Twovector bundles define a form of elliptic cohomology, available at arxiv:math.KT/0706.0531
"... Abstract. We prove that for wellbehaved small commutative rig categories (aka. symmetric bimoidal categories) R the algebraic Ktheory space of the Ktheory spectrum, HR, of R is equivalent to K0(π0R) × BGL(R)  + where GL(R) is the monoidal category of weakly invertible matrices over R. In parti ..."
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Cited by 3 (1 self)
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Abstract. We prove that for wellbehaved small commutative rig categories (aka. symmetric bimoidal categories) R the algebraic Ktheory space of the Ktheory spectrum, HR, of R is equivalent to K0(π0R) × BGL(R)  + where GL(R) is the monoidal category of weakly invertible matrices over R. In particular, this proves the conjecture from [BDR] that K(ku) is the Ktheory of the 2category of complex 2vector spaces. Hence, the work of the fourth author and Christian Ausoni on K(ku) [A, AR] shows that the theory of virtual 2vector bundles as in [BDR, Theorem 4.10] qualifies as a form of elliptic cohomology theory.
Cohomology theories for highly structured ring spectra. arXiv: math.AT/0211275
"... Abstract. This is a survey paper on cohomology theories for A ∞ and E ∞ ring spectra. Different constructions and main properties of topological AndréQuillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to ..."
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Cited by 3 (1 self)
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Abstract. This is a survey paper on cohomology theories for A ∞ and E ∞ ring spectra. Different constructions and main properties of topological AndréQuillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of A ∞ and E ∞ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures. 1.
DIAGRAM SPACES, DIAGRAM SPECTRA, AND SPECTRA OF UNITS
, 908
"... Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for sp ..."
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Cited by 2 (0 self)
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Abstract. We compare the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM Smodules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω ∞. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Ω ∞ agree. This comparison is then used to show that two different constructions of the spectrum of units gl1R of a structured ring spectrum R agree. Contents