Smooth profinite groups, III: the Smoothness Theorem

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 $n\ge 1$. 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.

Same theorem as in the previous version, using the new corrected version of Smooth Profinite groups, II. Comments are welcome