Geometry & Topology

Localization theorems in topological Hochschild homology and topological cyclic homology

Andrew J Blumberg and Michael A Mandell

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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.

Article information

Geom. Topol., Volume 16, Number 2 (2012), 1053-1120.

Received: 18 November 2010
Revised: 7 February 2012
Accepted: 7 March 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]
Secondary: 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

topological Hochschild homology topological cyclic homology localization sequence Mayer–Vietoris sequence projective bundle theorem blow-up formula


Blumberg, Andrew J; Mandell, Michael A. Localization theorems in topological Hochschild homology and topological cyclic homology. Geom. Topol. 16 (2012), no. 2, 1053--1120. doi:10.2140/gt.2012.16.1053.

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