Vortrag
Axiomatizing Kummer's theory
Vortrag von Dr. Mathieu Florence
Sprecher eingeladen von: Prof. Dr. Joseph Ayoub
Datum: 30.10.17 Zeit: 13.15 - 14.45 Raum: Y27H25
Let F be a field. Let p be a prime number invertible in F. Kummer theory states that the natural map (between Galois cohomology groups) H^1(F,mu_{p^n}) ---> H^1(F,mu_p) is surjective, for any n>1. It is a formal consequence of Hilbert's Theorem 90 for G_m. Our main motivation here is to axiomatize Kummer theory, in order to achieve an explicit proof of the Bloch-Kato conjecture, proved by Voevodsky. We will give an overview of our work, as follows. Let k be a perfect field of characteristic p. Denote by W(k) the ring of Witt vectors over k. Let n,m be two positive integers. Using divided powers, we first define how to canonically lift (finite) p^m-torsion W(k)-modules to (finite) p^(m+n)-torsion W(k)-modules, in a way that commutes to Pontryagin duality. This is a rich functorial process, especially if k is finite. We will discuss a nice consequence for p-adic deformations. We then define the notion of a smooth profinite group, and of Kummer-type extensions of G-modules. We will finally prove lifting theorems for the cohomology of smooth profinite groups (with values in (k,G)-modules)- notably the 'stable lifting' theorem. We shall explain the connection with lifting arbitrary Galois representations, and with the norm-residue isomorphism theorem. Note that the stable lifting theorem we present here is, for the time being, not enough to achieve a proof of the latter. This is joint work with Charles De Clercq (a preprint should be available on the Arxiv at the time of the talk).