Results 1  10
of
30
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 ..."
Abstract

Cited by 15 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
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.
UNIQUENESS OF E ∞ STRUCTURES FOR CONNECTIVE COVERS
, 2006
"... Abstract. We refine our earlier work on the existence and uniqueness of E ∞ structures on Ktheoretic spectra to show that at each prime p, the connective Adams summand ℓ has a unique structure as a commutative Salgebra. For the pcompletion ℓp we show that the McClureStaffeldt model for ℓp is equ ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. We refine our earlier work on the existence and uniqueness of E ∞ structures on Ktheoretic spectra to show that at each prime p, the connective Adams summand ℓ has a unique structure as a commutative Salgebra. For the pcompletion ℓp we show that the McClureStaffeldt model for ℓp is equivalent as an E ∞ ring spectrum to the connective cover of the periodic Adams summand Lp. We establish a Bousfield equivalence between the connective cover of the LubinTate spectrum En and BP〈n〉.
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 ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
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.
Rings, modules, and algebras in infinite loop space theory
 ADV. MATH
, 2004
"... 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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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. Our method