On the $K$-theory of pullbacks

Abstract

To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic $K$-theory. The construction of this new ring spectrum is categorical and hence allows to determine the failure of excision for any localizing invariant in place of $K$-theory.

As immediate consequences we obtain an improved version of Suslin’s excision result in $K$-theory, generalizations of results of Geisser and Hesselholt on torsion in (bi)relative $K$-groups, and a generalized version of pro-excision for $K$-theory. Furthermore, we show that any truncating invariant satisfies excision, nilinvariance, and cdh-descent. Examples of truncating invariants include the fibre of the cyclotomic trace, the fibre of the rational Goodwillie–Jones Chern character, periodic cyclic homology in characteristic zero, and homotopy $K$-theory.

Various of the results we obtain have been known previously, though most of them in weaker forms and with less direct proofs.

Authors

Markus Land

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Georg Tamme

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany