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 ≥ 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 p2 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.
0