Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture (Q864962)

From the introduction: Let \(F\) be a field and \(m\geq 2\) be an integer not divisible by the characteristic of \(F\). Consider the absolute Galois group \(G_F=\text{Gal}(\overline F/F)\), where \(\overline F\) denotes the (separable) algebraic closure of \(F\). The famous Milnor-Kato conjecture claims that the natural homomorphism of graded rings \(K^M_n(F)\otimes{\mathbb Z}/m\rightarrow H^n(G_F,\mu_m^ {\otimes n})\) from the Milnor \(K\)-theory of the field \(F\) modulo \(m\) to the Galois cohomology of \(F\) with cyclotomic coefficients is an isomorphism (here, as usual, \(\mu_m\) denotes the group of \(m\)-roots of unity in \(F\)). One can see that it suffices to verify this conjecture in the case when \(m\) is a prime number. \textit{V. Voevodsky} [Publ.Math., Inst. Hautes Étud. Sci. 98, 1--57 (2003; Zbl 1057.14028)] proved this conjecture for \(m\) equal to a power of 2. In his paper he also outlined a general approach to the Milnor-Kato conjecture for arbitrary \(m\). The first step of his argument dealt with a prime number \(\ell\), a field \(F\) having no finite extensions of degree prime to \(\ell\), and an integer \(n\geq 1\) such that the group \(K^M_{n+1}(F)\) is \(\ell\)-divisible. Assuming that the conjecture holds in degree less or equal to \(n\) for \(m=\ell\) and any field containing \(F\), Voevodsky was proving that \(H^{n+1}(G_F,\mathbb Z/\ell)=0.\) The main goal of this paper is to give a simplified elementary version of Voevodsky's proof of this step. In particular, we do not need to consider fields transcendental over \(F\) and we make no use of motivic cohomology at all. Our main result is formulated as follows. Theorem 1. Let \(\ell\) be a prime number, \(n\geq 1\) be an integer, and \(F\) be a field of characteristic not equal to \(\ell\), having no finite extensions of degree prime to \(\ell\). Suppose that the homomorphism \(K^M_n(L)/\ell K^M_n(L)\rightarrow H^n(G_L,{\mathbb Z}/\ell \) is an isomorphism for any finite extension \(L\) of the field \(F\). Furthermore, assume that \(K^M_{n+1}(F)=\ell K^M_{n+1}(F).\) Then one has \(H^{n+1}(G_F, \mathbb Z/\ell)=0.\) In the second half of this paper we apply the same techniques to obtain further exact sequences of Galois cohomology for cyclic, biquadratic, and dihedral field extensions. In particular, our method proves the biquadratic exact sequences conjectured by \textit{A. S. Merkurjev} and \textit{J.-P. Tignol} [Comment. Math. Helv. 68, No. 1, 138--169 (1993; Zbl 0781.12005)] and \textit{B. Kahn} [Comment. Math. Helv. 69, No. 1, 120--136 (1994; Zbl 0798.12007)]. In addition, we introduce an extended version of the classical Bass-Tate lemma and deduce some corollaries about generators and relations of annihilator ideals in Galois cohomology rings.