\(K\)-theory of semi-local rings with finite coefficients and étale cohomology
type: Item
instance of: scholarly article
title: \(K\)-theory of semi-local rings with finite coefficients and étale cohomology
zbMATH Open document ID: 1013.19001
DOI: 10.1023/A:1015608422475
published in: \(K\)-Theory
publication date: 2002-08-21T00:00:00Z
review text: In this article the author derives some consequences of Voevodsky's proof of the Milnor conjecture and the existence of the spectral sequence of \textit{E. M. Friedlander} and \textit{A. A. Suslin} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No.~6, 773-875 (2002; Zbl 1047.14011)]. Let \(A\) be a commutative semi-local ring containing \(1/2\). For most such \(A\) (called ``non-exceptional'' in the paper) the author constructs a natural isomorphism of the form \[ \oplus_{i=0}^{n} \;H_{\text{ét}}^{2i-n}(A ; (\mu_{2^{m}})^{\oplus^{i}}) @ >{\cong} >> K_{n}(A ; {\mathbb Z}/2^{m}) \] for all \(m \geq 2\). From this it is shown that for a non-exceptional scheme \(X\), quasi-projective or regular over \({\mathbb Z}[1/2]\), \( K_{n}(X ; {\mathbb Z}/2^{m}) \) and \( K_n'(X ; {\mathbb Z}/2^{m})\) is finite for \(n \geq dim(X) - 1\). Similar finiteness results are obtained when \(X\) is a variety over \({\mathbb F}_{p}\) or \({\mathbb Q}_{p}\) with \(p\) odd. Also, using Chern class with values in truncated étale cohomology, the author shows that for \(X\) over \({\mathbb Z}[1/2]\) of Krull dimension \(d\) (resp. over a field or a discrete valuation ring) \( K_{n}(X ; {\mathbb Z}/2^{m})\) is isomorphic to \( \oplus_{i \geq 1} \;H_{\text{ Zar}}^{2i-n}( X ; \tau_{ \leq i}R \alpha_{*}(\mu_{2^{m}})^{\oplus^{i}}) \) for \(n \geq 3\) (resp. \(n \geq 2\)). Here \(\alpha\) is the projection from the étale site of \(X\) to the Zariski site of \(X\) and \(\tau\) denotes truncation in the derived category.
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 19D55
zbMATH DE Number: 1786887
zbMATH Keywords: étale cohomology
zbMATH Keywords: semi-local ring
zbMATH Keywords: motivic cohomology
author: Bruno Kahn
reviewed by: Victor P. Snaith
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1786887