\(1\)-smooth pro-\(p\) groups and Bloch-Kato pro-\(p\) groups (Q2107532)

For a prime number \(p\) let \(\mathbb Z_p^*\) denote the group of units of the ring of \(p\)-adic integers \(\mathbb Z_p\). A pair \((G, \theta)\) consisting of a pro-\(p\) group \(G\) endowed with a continuous homomorphism \(\theta : G \longrightarrow \mathbb{Z}^*_p\) is called a cyclotomic pro-\(p\) pair. Then there is a continuous action of \(G\) on \(\mathbb{Z}_p\) given by \(gz=\theta(g)z\) and this \(G\)-module is denoted by \(\mathbb{Z}_p(1)\). A cyclotomic pair \((G,\theta)\) is called Kummerian if for every \(n\geq 1\) , the map \(H^1(G, \mathbb{Z}_p(1)/p^n)\longrightarrow H^1(G,\mathbb{Z}_p(1)/p)\) induced by the natural homomorphism \(\mathbb{Z}_p(1)/p^n\longrightarrow \mathbb{Z}_p(1)/p\) is surjective. The pair \((G,\theta)\) is said to be 1-smooth if for every open subgroup \(U\) of \(G\) the pair \((U, \theta_{|U})\) is Kummerian. A pro-\(p\) group \(G\) is called Kummerian (resp. 1-smooth) if there exists a continuous homomorphism \(\theta : G \longrightarrow \mathbb{Z}_p\) such that the pair \((G,\theta)\) is Kummerian (resp. 1-smooth). The first result of the paper is Theorem 1.1 If a \(p\)-adic analytic pro-\(p\) group is 1-smooth, then its Galois \(\mathbb{Z}/p\)-cohomology is a quadratic algebra. The author also introduces a subgroup \(\mathcal{K}(G) = \{h^{-\theta(g)}ghg-1\mid g \in G, h \in \mathrm{Ker}(\theta)\}\) and proves the following Theorem 1.2 For a pair \((G,\theta)\) such that \(\theta(G)\) is torsion free the following two conditions are equivalent: \begin{itemize} \item[(i)] \((G,\theta)\) is Kummerian; \item[(ii)] the canonical quotient \(G/\mathcal K(G)\) of \(G\) is a torsion free pro-\(p\) group; \item[(iii)] for a closed normal subgroup \(N\) of \(G\), \(N \leq \mathcal K(G)\), the quotient \(G/N\), endowed with the module induced by \(\mathbb{Z}_p(1)\), is Kummerian. \end{itemize} The next theorem shows why the property of 1-smoothness is important. Theorem 4.1 Let \(K\) be a field containing a root of unity of order \(p\), and let \(L/K\) be a Galois \(p\)-extension of \(K\) such that \(L\supset K\) containing all \(p\)-power roots of elements of \(K\). Then the Galois group \(G_{L/K}\) is Kummerian. In particular, the \(G_K\) is 1-smooth.