An Alternative Proof of Hesselholt's Conjecture on Galois Cohomology of Witt Vectors of Algebraic Integers

Details An Alternative Proof of Hesselholt's Conjecture on Galois Cohomology of Witt Vectors of Algebraic Integers An Alternative Proof of Hesselholt's Conjecture on Galois Cohomology of Witt Vectors of Algebraic Integers

2011-08-08

arxiv.org/abs/1108.1744v2

Mathematics

Number Theory

3 pages; added references, changed Remark 2.1 to a lemma and proof, updated abstract

Scientific paper

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. Let $\mathbb{W}_n(\cdot)$ denote the ring of $p$-typical Witt vectors of length $n$. Hesselholt conjectured that the pro-abelian group $\{H^1(G,\mathbb{W}_n(\mathcal{O}_L))\}_{n\geq 1}$ is isomorphic to zero. Hogadi and Pisolkar have recently provided a proof of this conjecture. In this paper, we provide an alternative proof of Hesselholt's conjecture which is simpler in several respects.

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