The aim of this talk is to explain how to axiomatize Hilbert's Theorem 90, in the setting of (the cohomology with finite coefficients of) profinite groups. I shall first explain the general framework. It includes, in particular, the use of divided power modules over Witt vectors; a process which appears to be of independent interest in the theory of modular representations. I shall then give several applications to Galois cohomology, notably to the problem of lifting mod p Galois representations (or more accurately: torsors under these) modulo higher powers of p. I'll also explain the connection with the Bloch-Kato conjecture in Galois cohomology, proved by Rost, Suslin and Voevodsky. This is joint work in progress with Charles De Clercq.
- Number Theory Seminar