\(K\)-theory of semi-local rings with finite coefficients and étale cohomology
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.
scientific article; zbMATH DE number 1786887