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

Part of: Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  25 January 2012

WILSON ONG
WILSON ONG*
Affiliation:
Mathematical Sciences Institute, Australian National University, ACT 0200, Australia (email: [email protected])
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Abstract

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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 {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.

Keywords

Galois cohomologyWitt vectorsalgebraic integersHesselholt’s conjecture

MSC classification

Secondary: 11S25: Galois cohomology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 456 - 460
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Hesselholt, L., ‘Galois cohomology of Witt vectors of algebraic integers’, Math. Proc. Cambridge Philos. Soc. 137(3) (2004), 551–557.CrossRefGoogle Scholar
[2]Hogadi, A. and Pisolkar, S., ‘On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt’, J. Number Theory 131(10) (2011), 1797–1807.Google Scholar
[3]Serre, J.-P., Local Fields, Graduate Texts in Mathematics, 67 (Springer, New York, 1979).CrossRefGoogle Scholar