A SIMPLIFIED PROOF OF HESSELHOLT’S CONJECTURE ON GALOIS COHOMOLOGY OF WITT VECTORS OF ALGEBRAIC INTEGERS

Abstract

AbstractLet 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 {H1 (G,𝕎n (𝒪L))}n≥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 Theory131(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.