Localization theorems in topological Hochschild homology and topological cyclic homology

Blumberg, Andrew J¹ ; Mandell, Michael A²

¹ Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin TX 78712, USA
² Department of Mathematics, Indiana University, Rawles Hall, 831 E 3rd St, Bloomington IN 47405, USA

Geometry & topology, Tome 16 (2012) no. 2, pp. 1053-1120.

Résumé

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 $TC$.

Détail

DOI : 10.2140/gt.2012.16.1053

Classification : 19D55, 14F43
Keywords: topological Hochschild homology, topological cyclic homology, localization sequence, Mayer–Vietoris sequence, projective bundle theorem, blow-up formula

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