THH(R) ≅ R ⊗ S1 for E∞ ring spectra

McClure, James E.; Schwänzl, R.; Vogt, R. M.

doi:10.1016/s0022-4049(97)00118-7

Cited by 54 publications

(50 citation statements)

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“…Consequently R n E lim F is a subquotient as claimed, and vanishes for all n ą 2. (One could also compute the maps in (35) to get an explicit presentation of R 1 E lim F and R 2 E lim F, but while interesting, this is unnecessary for the present paper. )…”

Section: A1 Filtered Coefficientsmentioning

confidence: 99%

A May-type spectral sequence for higher topological Hochschild homology

Angelini-Knoll

Salch

2018

Algebr. Geom. Topol.

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Given a filtration of a commutative monoid A in a symmetric monoidal stable model category C, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of A, and whose output is the higher order topological Hochschild homology of A. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring R, we get an upper bound on the size of the THH-groups of E 8 -ring spectra A such that π˚pAq -R. 55P42; 55T05

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Section: A1 Filtered Coefficientsmentioning

confidence: 99%

A May-type spectral sequence for higher topological Hochschild homology

Angelini-Knoll

Salch

2018

Algebr. Geom. Topol.

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“…This can be made most precise in the case when R is a commutative S -algebra, in which case there is a formula THH (R) ∼ = R ⊗ S 1 in terms of the simplicial tensor structure on the category of commutative S -algebras. A corresponding formula THH (R) ≃ R ⊗ |S 1 | in terms of the topological tensor structure was discussed by McClure, Schwänzl and Vogt in [MSV97]. We shall stick to the simplicial context, since we will make use of the resulting skeletal filtrations to form spectral sequences.…”

Section: Hochschild and Topological Hochschild Homologymentioning

confidence: 99%

Hopf algebra structure on topological Hochschild homology

Angeltveit

Rognes

2005

Algebr. Geom. Topol.

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The topological Hochschild homology THH (R) of a commutative S -algebra (E ∞ ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that 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 the study of some interesting examples, including the commutative S -algebras 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 K -theory of S -algebras, by means of the cyclotomic trace map to topological cyclic homology.

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“…When R is commutative, McClure, Schwanzl, and Vogt [29] made precise an insight of Bokstedt's that there should be a homeomorphism…”

Section: Introductionmentioning

confidence: 99%

“…Recall that the category of commutative S-algebras is enriched and tensored over unbased spaces, and more generally has all indexed colimits [12, 7.2.9]. When R is commutative, McClure, Schwanzl, and Vogt [29] made precise an insight of Bokstedt's that there should be a homeomorphism |N cyc R| ∼ = R ⊗ S 1 .…”

Section: Introductionmentioning

confidence: 99%

Topological Hochschild homology of Thom spectra which are E _∞ -ring spectra

Blumberg¹

2010

Journal of Topology

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Abstract. We identify the topological Hochschild homology (T HH) of the Thom spectrum associated to an E∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U , O, and F ). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that T HH of a cofibrant commutative S-algebra (E∞ ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result T HH(M f ) ≃ M f ∧BX + which yields a convenient description of T HH(M U ). This splitting holds even when the classifying map f : X → BG is only a homotopy commutative A∞ map, provided that the induced multiplication on M f extends to an E∞ ring structure; this permits us to recover Bokstedt's calculation of T HH(HZ).

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THH(R) ≅ R ⊗ S1 for E∞ ring spectra

Cited by 54 publications

References 4 publications

A May-type spectral sequence for higher topological Hochschild homology

A May-type spectral sequence for higher topological Hochschild homology

Hopf algebra structure on topological Hochschild homology

Topological Hochschild homology of Thom spectra which are E _∞ -ring spectra

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THH(R) ≅ R ⊗ S1 for E∞ ring spectra

Cited by 54 publications

References 4 publications

A May-type spectral sequence for higher topological Hochschild homology

A May-type spectral sequence for higher topological Hochschild homology

Hopf algebra structure on topological Hochschild homology

Topological Hochschild homology of Thom spectra which are E ∞ -ring spectra

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Product

Resources

About

Topological Hochschild homology of Thom spectra which are E _∞ -ring spectra