Smooth profinite groups, I: geometrizing Kummer theory

TL;DR

This paper introduces a geometric framework for Kummer theory using smooth profinite groups and cyclotomic pairs, leading to new proofs of classical theorems without relying on motivic cohomology.

Contribution

It develops the theory of cyclotomic pairs and smooth profinite groups, establishing a lifting theorem and a smoothness criterion that provide new insights into Galois cohomology and Kummer theory.

Findings

Proves a lifting theorem for G-linearized torsors under line bundles.

Establishes a smoothness criterion for profinite groups based on cohomological surjectivity.

Provides a new proof of the Norm Residue Isomorphism Theorem.

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.