On the K-theory of pullbacks

• 成果类型: Article
• 署名作者: Land, Markus; Tamme, Georg
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2019.190.3.4
• 发表日期: 2019
• 页码: 877-930
• 关键词: cyclic homology; a(1)-homotopy theory; excision; localization; descent

摘要

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 us 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 Hessel-holt 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.