On the K-theory of pullbacks

TL;DR

This paper introduces a categorical construction of a new ring spectrum associated with pullback squares of ring spectra, improving understanding of excision failures in algebraic K-theory and related invariants.

Contribution

It provides a new categorical method to analyze excision failures in K-theory, generalizes several known results, and shows that truncating invariants satisfy excision, nilinvariance, and cdh-descent.

Findings

Improved version of Suslin's excision in K-theory

Generalizations of torsion results in relative K-groups

Truncating invariants satisfy excision, nilinvariance, and cdh-descent

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.