A simplified proof of Hesselholt's conjecture on Galois cohomology of Witt vectors of algebraic integers

Abstract

Let K be a complete discrete valuation field of characteristic zero with residue field kK of characteristic p > 0. Let L=K be a finite Galois extension with Galois group G = Gal(L=K) and suppose that the induced extension of residue fields kL=kK is separable. Let Wn ( ) denote the ring of p-typical Witt vectors of length n. Hesselholt ['Galois cohomology of Witt vectors of algebraic integers', Math. Proc. Cambridge Philos. Soc. 137(3) (2004), 551-557] conjectured that the pro-abelian group fH1 (G;Wn (OL))gn≥1 is isomorphic to zero. Hogadi and Pisolkar ['On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt', J. Number Theory 131(10) (2011), 1797-1807] have recently provided a proof of this conjecture. In this paper, we provide a simplified version of the original proof which avoids many of the calculations present in that version.