The Descent Problem For Topological Cyclic Homology And Algebraic K-Theory

. We investigate 'etale descent properties of topological Hochschild and cyclic homology. Using these properties we deduce a general injectivity result for the descent map in algebraic K-theory, and show that algebraic K-theory has 'etale descent for rings of integers in unramified and tamely ramified p-adic fields. x1. Introduction This article investigates the relation of Quillen's algebraic K-theory groups of a ring R to corresponding 'etale cohomology groups of R with coefficients in the sheaf defined by K-theory. This relation is best formulated as a Grothendiecktype cohomological descent problem for K-theory following [Th1]. More precisely, given a functor from R-algebras to (topological) spectra, one can imitate the Cech construction in sheaf cohomology and construct a spectrum H (R ; F), which is called here the ('etale) Cech spectrum of R with coefficients in F . When F = K, the K-theory functor, then we call this spectrum the ('etale) Cech K-theory of R. There is a canon..