Topological Hochschild homology of ring functors and exact categories

Dundas, Bjørn Ian; McCarthy, Randy

doi:10.1016/0022-4049(95)00089-5

Cited by 40 publications

(62 citation statements)

References 10 publications

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“…As indicated above, this is the appropriate setting for studying THH , TR and TC, and is a stable homotopy theory generalization of the setting of DG-categories. In Section 3, we review the definition of THH of small spectral categories due to Bökstedt [3] and Dundas and McCarthy [13]. Because of the work of Shipley [39], the technical hypotheses of connectivity and convergence on the input symmetric spectra that seemed necessary for the last 20 years may now be omitted.…”

Section: Theorem 12mentioning

confidence: 99%

“…We begin with a review of the cyclic bar construction for small spectral categories and the variant defined by Bökstedt [3] and Dundas and McCarthy [13] necessary for the construction of TC. We finish with a brief review of the definition of cyclotomic spectra and the construction of TR and TC.…”

Section: Review Of Thh Tr and Tcmentioning

confidence: 99%

“…The correct definition, due to Bökstedt [3] for symmetric ring spectra and generalized by Dundas and McCarthy [13] to small spectral categories, does not suffer from either of these deficiencies.…”

Section: Review Of Thh Tr and Tcmentioning

confidence: 99%

“…Dundas and McCarthy [13] generalized Bökstedt's construction of THH to small spectral categories. We build on the foundations there and study more general invariance properties; see in particular Theorems 5.9 and 5.12 below.…”

mentioning

confidence: 99%

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Localization theorems in topological Hochschild homology and topological cyclic homology

Blumberg

Mandell

2012

Geom. Topol.

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We construct localization cofibration sequences for the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of small spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofibration sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K -theory. We also deduce versions of Thomason's blow-up formula and the projective bundle formula for THH and TC .

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Section: Theorem 12mentioning

confidence: 99%

Section: Review Of Thh Tr and Tcmentioning

confidence: 99%

Section: Review Of Thh Tr and Tcmentioning

confidence: 99%

mentioning

confidence: 99%

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Localization theorems in topological Hochschild homology and topological cyclic homology

Blumberg

Mandell

2012

Geom. Topol.

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“…We next recall the cyclotomic trace map of Bökstedt-Hsiang-Madsen [6], but use the technically better construction given by Dundas-McCarthy [9]. The T-fixed point spectrum T (C ) T is a symmetric orthogonal spectrum.…”

Section: The Cyclotomic Trace Mapmentioning

confidence: 99%

Topological Hochschild homology and the Bass trace conjecture

Berrick

Hesselholt

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We use the methods of topological Hochschild homology to shed new light on groups satisfying the Bass trace conjecture. Factorization of the Hattori-Stallings rank map through the Bökstedt-Hsiang-Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G wherein every subgroup isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G.

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