Relative algebraic K-theory and topological cyclic homology

On relative and bi-relative algebraic 𝐾-theory of rings of finite characteristic

“…B] that the mapping fiber of the cyclotomic trace map K(X) → {TC n (X; p)} satisfies descent for the cdh-topology on the category of schemes essentially of finite type over an infinite perfect field k of positive characteristic p, provided that the resolution of singularities holds over k. The main advantage of the functor {TC n q (−; p)} that appears in the statement of Theorems B and D in comparison to the functor TC q (−; p) that appears in McCarthy's theorem [23] is that the former preserves filtered colimits while the latter, in general, does not. Therefore, replacing the latter functor by the former, the methods of sheaf cohomology become available.…”

Algebraic Topology: Applications and New Directions

“…Moreover, as we explain in Theorem 4.1 below, results of McCarthy [28] and of Geisser and the author [10] implies that the righthand square, too, is homotopy cartesian. Hence, the mapping fiber K(A, a) of the map of K-theory spectra induced by h is canonically weakly equivalent to the mapping fiber of the map of topological cyclic homology spectra induced by f .…”

“…We first use the comparison theorems between K-theory and topological cyclic homology proved by McCarthy [28] and by Geisser and the author [10,11] to prove the following general result. Theorem 4.1.…”

Excision for $K$-theory of connective ring spectra

“…It follows by the theorems of McCarthy [16], the first author [4], and Geisser and Hesselholt [7] in addition to an elementary observation about homotopy cartesian diagrams of ring spectra.…”