## Geometry & Topology

### Localization theorems in topological Hochschild homology and topological cyclic homology

#### Abstract

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

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

Dates
Revised: 7 February 2012
Accepted: 7 March 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732414

Digital Object Identifier
doi:10.2140/gt.2012.16.1053

Mathematical Reviews number (MathSciNet)
MR2928988

Zentralblatt MATH identifier
1282.19004

#### Citation

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. https://projecteuclid.org/euclid.gt/1513732414

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