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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Localization theorems in topological Hochschild homology and topological cyclic homology

Andrew J Blumberg and Michael A Mandell

Geometry & Topology 16 (2012) 1053–1120
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.

Keywords
topological Hochschild homology, topological cyclic homology, localization sequence, Mayer–Vietoris sequence, projective bundle theorem, blow-up formula
Mathematical Subject Classification 2010
Primary: 19D55
Secondary: 14F43
References
Publication
Received: 18 November 2010
Revised: 7 February 2012
Accepted: 7 March 2012
Published: 5 June 2012
Proposed: Ralph Cohen
Seconded: Haynes Miller, Jesper Grodal
Authors
Andrew J Blumberg
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin TX 78712
USA
Michael A Mandell
Department of Mathematics
Indiana University
Rawles Hall
831 E 3rd St
Bloomington IN 47405
USA