The \(\ell\)-adic Galois symbol and the Suslin-Voevodsky theorem (Q928473)

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 \(H^q(F, \mathbb{Q}_l/\mathbb{Z}_l(q)\) are divisible, for every field \(F\) over \(k\); (ii) The map \(H^q(F,\mathbb{Q}_l/\mathbb{Z}_l(q) @>\delta>> H^{q+ 1}(F,\mathbb{Q}_l/\mathbb{Z}_l(q))\), associated to the exact sequence of étale sheaves \[ 0\to \mu^{\otimes q}_l\to \mu^{\otimes q}_l\to \mu^{\otimes q}_l\to 0 \] is trivial for all \(F\); (iii) \(l\)-adic Hilbert 90 holds in degree \(q\); (iv) for all extensions \(F\) of \(k\) the Galois symbol \[ h^q(F_{l,F}: K^M_q(F)/lK^M_q(k)\to H(F,\mu^{\otimes q}_l) \] is bijective.