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 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 ...
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 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.
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