Excision for $K$-theory of connective ring spectra

Abstract

We extend Geisser and Hesselholt’s result on “bi-relative $K$-theory” from discrete rings to connective ring spectra. That is, if $\mathcal{A}$ is a homotopy cartesian n-cube of ring spectra (satisfying connectivity hypotheses), then the $(n + 1)$-cube induced by the cyclotomic trace $$K(\mathcal{A}) \to TC(\mathcal{A})$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.