Motivic cohomology with \(\mathbb Z/2\)-coefficients
type: Item
instance of: scholarly article
title: Motivic cohomology with \(\mathbb Z/2\)-coefficients
zbMATH Open document ID: 1057.14028
published in: Publications Mathématiques
publication date: 2004-10-28T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104197
review text: Let \(k\) be a field and \(\ell\) a prime number different from \(\text{char}(k)\). For a fixed separable closure \(k_{\text{sep}}\) of \(k\), the group \(\mu_\ell\) of \(\ell\)th roots of unity in \(k_{\text{sep}}\) is a \(\text{Gal}(k_{\text{sep}}/k)\)-module occuring in the short exact Kummer sequence \[ 1\to \mu_\ell\to k^*_{\text{sep}}@> z^\ell>> k^*_{\text{sep}}\to 1. \] By the classical work of J. Milnor, H. Bass and J. Tate there is then a natural homomorphism \[ K^M_*(k)\to H^*(k, \mu^{\otimes_*}_\ell) \] from the \(K\)-theoretic Milnor ring to the group cohomology ring, which factors through a map \[ K^M_*(k)/\ell\to H^*(k, \mu^{\otimes_*}_\ell) \] called the norm residue homomorphism. There is a recent general conjecture about this particular homomorphism, the so-called Bloch-Kato conjecture, stating that the norm residue homomorphism is an isomorphism of graded rings for any field \(k\) with \(\text{char}(k)\neq\ell\). This conjecture generalizes the famous Milnor conjecture [\textit{J. Milnor}, Invent. Math. 9, 318--344 (1970; Zbl 0199.55501)] predicting that this homomorphism is at least injective in degree two. The general Bloch-Kato conjecture was formulated in 1980 [\textit{K. Kato}, J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 603--683 (1980; Zbl 0463.12006)], and partial results in confirming it have been obtained by \textit{A. Merkurjev} [Sov. Math. Dokl. 24, 546--551 (1981; Zbl 0496.16020)] for \(\ell= 2\) and degree 2, by \textit{A. Merkurjev} and \textit{A. Suslin} [Math. USSR. Izv. 21, 307--340 (1983; Zbl 0525.18008)] for all \(\ell\) and degree 2, by \textit{M. Rost} [On Hilbert Satz 90 for \(K_3\) for degree-two extensions, Preprint, \url{http://www.math.uni-bielefeld.de/~rost/K3-86.html}] for \(\ell= 2\) and degree 3, and independently by \textit{A. Merkurjev} and \textit{A. Suslin} [Math. USSR Izv. 36, 541--565 (1991; Zbl 0725.19003)] for the same case of \(\ell= 2\) and degree 3. One of the main objectives of the paper under review is to pre…
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 2111677
zbMATH Keywords: norm residue homomorphism
zbMATH Keywords: Bloch-Kato conjecture
zbMATH Keywords: Milnor conjecture
zbMATH Keywords: motivic cohomology operations
zbMATH Keywords: Beilinson-Lichtenbaum conjectures
Wikidata QID: Q29395066
author: Vladimir Voevodsky
reviewed by: Werner Kleinert
MaRDI profile type: Publication
OpenAlex ID: W2071886435
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DOI: 10.1007/S10240-003-0010-6
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description: scientific article; zbMATH DE number 2111677