Integral excision for $K$-theory

Abstract

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace $K(\mathcal{A}) \to TC(\mathcal{A})$ 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 $\mathbf{T}$-Tate spectrum of topological Hochschild homology, where $\mathbf{T}$ is the circle group.