Integral Excision for K-Theory

TL;DR

This paper proves that under certain conditions, the cyclotomic trace from algebraic K-theory to topological cyclic homology satisfies excision, leading to new insights and proofs in related homological theories.

Contribution

It establishes integral excision for the cyclotomic trace in algebraic K-theory, providing new proofs and results in cyclic homology and topological Hochschild homology.

Findings

Homotopy cartesian cube induced by the cyclotomic trace satisfies excision.

New proofs for classical results in periodic cyclic homology.

Insights into the T-Tate spectrum of topological Hochschild homology.

Abstract

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace from K(A) to TC(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision. The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the T-Tate spectrum of topological Hochschild homology, where T is the circle group