Excision for $K$-theory of connective ring spectra

Abstract

We extend Geisser and Hesselholt's result on "bi-relative Ktheory" from discrete rings to connective ring spectra. That is, if A is a homotopy cartesian n-cube of ring spectra (satisfying connectivity hypotheses), then the (n + 1)-cube induced by the cyclotomic traceis homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.

Integral excision for $K$-theory

“…Just as we did in [8], this extends to the case where A is a homotopy cartesian square of connective ring spectra with f 1 0-connected (though there are also other and even simpler alternatives since we are only concerned with rational results).…”

“…We will recall the necessary details when we need them in Section 3. If we take the profinite completion, then Theorem 1.1 is a special case of [8], which itself is an extension of the discrete case established by Geisser and Hesselholt [10]. If we replace topological cyclic homology with negative cyclic homology and work rationally, then it is closely related to Cortiñas' result [4].…”

“…Theorem 1.1 claims that ifib hofib trc (A) * , and so it clearly suffices to show that ifib hofib trc (A) (0) ifib hofib trc (A) * . The profinite completion part, namely that ifib hofib trc (A) is contractible, is the main result of [8], which relied heavily on the work of Geisser and Hesselholt [10] in the discrete ring case, which again used ideas from Cortiñas' rational paper [4].…”

“…It would be desirable to have a statement where just one of the maps, say f 1 , were 0-connected (as was the case in [4,8] and [10]). With the present line of proof this is not obtainable, essentially because of a technicality (Ext-completion of infinite sums of torsion modules need not be torsion), which vanishes under certain finiteness conditions.…”

Excision in algebraic -theory revisited

“…Both use pro versions of the results of Suslin and Wodzicki. Building on these results, Dundas and Kittang [DK08, DK13] prove that the fibre of the cyclotomic trace satisfies excision also for connective ring spectra, and with integral coefficients (under the technical assumption that both, and are surjective).…”