Homology, Homotopy and Applications

Volume 15 (2013)

Number 1

Integral excision for $K$-theory

Pages: 1 – 25

DOI: http://dx.doi.org/10.4310/HHA.2013.v15.n1.a1

Authors

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

Harald Øyen Kittang (Cappelen Damm AS, Oslo, Norway)

Abstract

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

Keywords

excision in algebraic $K$-theory, derived algebraic geometry, ring spectrum, cyclotomic trace

2010 Mathematics Subject Classification

13D15, 14A20, 19D55, 55P43

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