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 $\cal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the $(n+1)$-cube induced by the cyclotomic trace

$$K(\cal

is A) \to TC(\cal A)$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.