In order to characterize profinite groups which are realizable as absolute Galois groups \(G_F\) of fields \(F\), the authors look here for cohomological constrains derived from the Milnor-Bloch-Kato conjecture (recently proved by Voevodsky and others). Recall that this theorem asserts that for all \(r \geq 0\) and all \(m\) prime to \(\text{char}\,F\), there is a canonical isomorphism \(K^M_r (F)/m \displaystyle\buildrel\sim \over \to H^r (G_F, \mu^{\otimes r}_m).\) Using this, the authors reveal a surprisingly close connection between the “small” quotient \(G^{[3]}_F = G_F/G^{(3)}_F\) in the descending \(q\)-central sequence \(G^{(i)}_F\) and the cohomology algebra \(H^\ast (G_F) = H^\ast (G_F, {\mathbb Z}/q),\) where \(q\) is a prime power and \(F\) is assumed to contain \(\mu_q.\) The connection consists in four main properties:
A. The inflation map gives an isomorphism \(H^\ast (G^{[3]}_F) \displaystyle\buildrel \sim \over \to H^\ast(G_F)\).
B. \(G^{[3]}_F\) is uniquely determined by \(H^r (G_F)\) for \(r = 1,2\), the cup product \[ H^1 (G_F) \times H^1(G_F) \to H^2 (G_F) \] and the Bockstein morphism \(H^1 (G_F) \to H^2 (G_F)\).
C. Let \(F_1, F_2\) be fields and \(\pi: G_{F_1}\to G_{F_2}\) a continuous homomorphism. The following conditions are equivalent :
(i) the induced map \(\pi^\ast\;:\;H^\ast (G_{F_2}) \to H^\ast (G_{F_1})\) is an isomorphism
(ii) the induced map \(\pi^{[3]}: G^{[3]}_{F_1} \to G^{[3]}_{F_2}\) is an isomorphism.
An interesting consequence is:
D. Let \(F_1, F_2\) be fields containing \(\mu_p\) and let \(\pi: G_1(F) (p) \to G_{F_2} (p)\) be a continuous homomorphisms between the maximal pro-\(p\) quotients of the \(G_{F_i}\)’s. Then \(\pi\) is an isomorphism if and only if the induced map \(\pi^{[3]} : G^{[3]}_{F_1} \to G^{[3]}_{F_2}\) is an isomorphism.
Using these results, the authors can give new examples of pro \(p\)-groups which cannot be realized as maximal pro-\(p\) Galois groups of fields.
The approach in this paper is group-theoretic, and the main results above are actually proved for arbitrary profinite groups \(G\) which satisfy certain cohomological conditions which are implied (when \(G = G_F)\) by the Milnor-Bloch-Kato conjecture. This “axiomatization” requires, as a counterpart, an ingenuous layout of subtle cohomological properties. Note that when restricted to absolute Galois groups, the arguments can be made more direct by using techniques from the embedding problem of fields ; see e.g. the report by the reviewer [“Étude kummerienne de la \(q\)-suite centrale descendante d’un groupe de Galois”, Algèbre et Théorie des Nombres, Publ. Math. Besançon, 2, 123–139 (2012; Zbl 1309.12007)].