Homology, Homotopy and Applications

Volume 10 (2008)

Number 1

Excision for $K$-theory of connective ring spectra

Pages: 29 – 39

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n1.a2

Authors

Bjørn Ian Dundas (Department of Mathematics, University of Bergen, Norway)

Harald Øyen Kittang (Faculty of Engineering, Oslo University College, Oslo, Norway)

Abstract

We extend Geisser and Hesselholt’s result on “bi-relative $K$-theory” from discrete rings to connective ring spectra. That is, if $A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the $(n + 1)$-cube induced by the cyclotomic trace

$\mathcal{K(A) \to TC(A)}$

is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.

Keywords

algebraic $K$-theory, excision, ring spectrum

2010 Mathematics Subject Classification

18G30, 19C40, 19D55

Published 1 January 2008