On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt

Abstract

Let K be a complete discrete valued field of characteristic zero with residue field k K of characteristic p > 0 . Let L / K be a finite Galois extension with Galois group G such that the induced extension of residue fields k L / k K is separable. Hesselholt (2004) [2] conjectured that the pro-abelian group { H 1 ( G , W n ( O L ) ) } n ∈ N is zero, where O L is the ring of integers of L and W ( O L ) is the ring of Witt vectors in O L w.r.t. the prime p. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholtʼs conjecture for all Galois extensions.