Smooth profinite groups, I: geometrizing Kummer theory

Abstract

In this series of three papers, we introduce and study cyclotomic pairs and smooth profinite groups. They are a geometric axiomatisation of Kummer theory for fields, with coefficients $p$-primary roots of unity, for a prime $p$. These coefficients are enhanced, to $G$-linearized line bundles in Witt vectors, over $G$-schemes of characteristic $p$. In the second paper, this upgrade is pushed even further, to the scheme-theoretic setting. In this first article, we introduce cyclotomic pairs, smooth profinite groups and $(G,S)$-cohomology. We prove a first lifting theorem for $G$-linearized torsors under line bundles (Theorem A). With the help of the algebro-geometric tools developed in the second article, this formalism is applied in the third one, to prove the Smoothness Theorem, whose essence reads as follows. Let $G$ be profinite group. Assume that, for every open subgroup $H \subset G$, and for $n=1$, the natural arrow $H^n(H,\mathbb{Z}/p^2) \to H^n(H,\mathbb{Z}/p)$ is surjective. Then, it is also surjective for every such $H$, and every $n \geq 2$. Applied to absolute Galois groups, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology.