The \(\ell\)-adic Galois symbol and the Suslin-Voevodsky theorem
type: Item
instance of: scholarly article
title: The \(\ell\)-adic Galois symbol and the Suslin-Voevodsky theorem
zbMATH Open document ID: 1157.19002
author: P. Gille
published in: Journal of Mathematics of Kyoto University
publication date: 2008-06-18T00:00:00Z
review text: The paper deals with the famous Bloch-Kato (BK) conjecture, recently proved by Voevodsky and Rost: B-K Conjecture. Let \(k\) be a field and let \(m\geq 1\) be invertible in \(k\). Then for every integer \(q\geq 0\) the Galois symbol \[ h^q_{m,k}: K^M_q(k)/mK^M_q(k)\to H^q(k, \mu^{\otimes q}_m) \] is an isomorphism. Here \(K^M_*(k)\) are the Milnor \(K\)-groups and \(H^*(k, \mu^{\otimes q}_m)\) are the Galois cohomology groups with coefficients in the \(m\)th roots of unity, twisted \(q\) times. If \(l\) is a prime, different from the characteristic of the field \(k\) (which may assumed to be infinity), then \textit{A. Suslin} and \textit{V. Voevodsky} proved in [Bloch-Kato conjecture and motivic cohomology with finite coefficients, NATO ASI Ser., Ser. C, Math. Phys. Sci. 548, 117--189 (2000; Zbl 1005.19001)] that the Galois symbol \(K^M_q(F)/lK^M_q(F)\to H(F,\mu^{\otimes q}_l)\) is an isomorphism, for every extension \(F\) of \(k\), iff the étale cohomology groups \(H^q(F,\mathbb{Q}_l/\mathbb{Z}_l(q))\) are divisible. In this paper the author gives an alternative proof of the result above, completed by new equivalent formulations of the BK conjecture. Recall that for every cyclic extension \(L/k\) of degree \(l\) one has a commutative diagram \[ \begin{tikzcd} K^M_q(L) \rar["1-\sigma"]\dar["h^q_L" '] & K^M_q(L) \rar["N_{L/K}"]\dar["h^q_L" '] & K^M_q(k)\dar["h^q_k"]\\ H^q(L,\mathbb{Z}_l(q)) \rar["1-\sigma" '] & H^q(L,\mathbb{Z}_l(q)) \rar["\mathrm{Cor}^L_k" '] & H^q(k,\mathbb{Z}_l(q))\rlap{\,,}\end{tikzcd} \] where \(\mathbb{Z}/l\mathbb{Z}= \langle\sigma\rangle\), \(N_{L/k}\) is the norm in Milnor \(K\)-theory and \(\text{Cor}^L_k\) is the corestriction. In the case \(q= 2\) the exactness of the first row is a consequence of Hilbert 90 for \(K_2\) as proved by Merkurjev and Suslin, while exactness of the bottom row is the so-called \(l\)-adic Hilbert 90. Theorem. The following statements are equivalent: (i) The étale cohomology groups \(H^q(F, \mathbb{Q}_l/\mathbb{Z}…
reviewed by: Claudio Pedrini
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14F42
zbMATH DE Number: 5290070
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250692284
OpenAlex ID: W2003337975
DOI: 10.1215/KJM/1250692284
description: scientific article; zbMATH DE number 5290070