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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 1

### Excision for $K$-theory of connective ring spectra

Pages: 29 – 39

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

#### Authors

#### 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