Localization theorems in topological Hochschild homology and topological cyclic homology

TL;DR

This paper develops localization cofiber sequences for topological Hochschild and cyclic homology of spectral categories, connecting them with algebraic K-theory localization sequences and geometric formulas.

Contribution

It introduces a global construction of THH and TC for schemes using spectral categories, deriving localization, blow-up, and projective bundle formulas.

Findings

Constructed localization cofiber sequences for THH and TC.

Connected THH and TC localization sequences to Thomason-Trobaugh K-theory.

Extended formulas like blow-up and projective bundle to THH and TC.

Abstract

We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of 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 cofiber 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.