Smooth profinite groups, III: the Smoothness Theorem

TL;DR

The paper proves the Smoothness Theorem for cyclotomic pairs, offering a new proof of the Norm Residue Isomorphism Theorem without motivic cohomology and extending it to algebraic fundamental groups of curves.

Contribution

It establishes the Smoothness Theorem for cyclotomic pairs and provides a novel proof of the Norm Residue Isomorphism Theorem independent of motivic cohomology.

Findings

Proves the Smoothness Theorem for all n ≥ 1.

Provides a new proof of the Norm Residue Isomorphism Theorem.

Extends results to algebraic fundamental groups of curves.

Abstract

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod $p^2$ Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.