Lifting Theorems and Smooth Profinite Groups

Abstract

This work is motivated by the search for an "explicit" proof of the Bloch-Kato conjecture in Galois cohomology, proved by Voevodsky. Our concern here is to lay the foundation for a theory that, we believe, will lead to such a proof- and to further applications. Let p be a prime number. Let k be a perfect field of characteristic p. Let m be a positive integer. Our first goal is to provide a canonical process for "lifting" a module M, over the ring of Witt vectors $W_m(k)$ (of length m), to a $W_{m+1}(k)$-module, in a way that deeply respects Pontryagin duality. These are our big, medium and small Omega powers, each of which naturally occurs as a direct factor of the previous one. In the case where M is a k-vector space, they come equipped with Verschiebung and Frobenius operations. If moreover the field k is finite, Omega powers are endowed with a striking extra operation: the Transfer, to shifted Omega powers of finite-codimensional linear subspaces. To show how this formalism fits into Galois theory, we first offer an axiomatized approach to Hilbert's Theorem 90 (or more precisely, to its consequence for cohomology with finite coefficients: Kummer theory). In the context of profinite group cohomology, we thus define the notions a cyclotomic G-module, and of a smooth profinite group. We bear in mind that the fundamental example is that of an absolute Galois group, together with the Tate module of roots of unity. We then define the notion of exact sequences of G-modules of Kummer type. To finish, we give applications of this formalism. The first ones are the Stable Lifting Theorems, enabling the lifting to higher torsion in the cohomology of smooth profinite groups, with p-primary coefficients. We finish by an application to p-adic deformations. We state and prove a general descent statement, for the quotient map $\mathbb{Z}/p^2\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$.