Cohomology of the Weil group of a \(p\)-adic field
type: Item
instance of: scholarly article
title: Cohomology of the Weil group of a \(p\)-adic field
zbMATH Open document ID: 1304.11133
author: David A. Karpuk
published in: Journal of Number Theory
publication date: 2013-02-21T00:00:00Z
review text: Let \(K\) be a finite extension of the \(p\)-adic field \({\mathbb Q}_p\) and let \(\overline{K}\) be an algebraic closure of \(K\). The Weil group \(W\) of \(K\) is the subgroup of \(G=\mathrm {Gal}(\overline{K}/K)\) consisting of those automorphisms which induce an integer power of Frobenius on residue fields. The topology of \(W\) is determined by the requirement that the inertia subgroup \(I\) of \(G\) (with the Krull topology) is an open subgroup of \(W\). This paper proves several theorems about Weil groups cohomology. For instance, it is shown that if \(M\) is a \(W\)-module which is finitely generated as an abelian group then \(H^3(W,M)=0\). In the case where \(M\) is a discrete group a comparison theorem is proved relating the Weil cohomology groups \(H^n(W,M)\) to the Galois cohomology groups \(H^n(G,M)\). In the case where \(A\) is a connected commutative algebraic group a similar theorem is proved relating \(H^n(W,A(\overline{L}))\) to \(H^n(G,A(\overline{K}))\), where \(L\) is the completion of the maximal unramified extension of \(K\). In both settings, the comparison theorem allows one to determine the Galois cohomology groups in terms of the Weil cohomology groups.
reviewed by: Kevin Keating
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 55U30
Mathematics Subject Classification ID: 11E72
zbMATH DE Number: 6138269
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Weil group
zbMATH Keywords: Weil-étale cohomology
zbMATH Keywords: duality
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.08.028
OpenAlex ID: W2058341139
cites work: Néron Models
cites work: Q4542308
cites work: Cohomology of topological groups with applications to the Weil group
cites work: On Suslin's singular homology and cohomology
cites work: Affine Deligne–Lusztig varieties in affine flag varieties
cites work: Schemata over local rings. II
cites work: Duality theorems for curves over p-adic fields
cites work: The Weil-étale topology on schemes over finite fields
cites work: The Weil-étale topology for number rings
cites work: Q3871006
cites work: Q3413444
cites work: Group Extensions and Cohomology for Locally Compact Groups. III
cites work: Q3445917
cites work: On the vanishing of the measurable Schur cohomology groups of Weil groups
cites work: Q3856855
cites work: Q2756583
cites work: Q3266818
cites work: Q4296949
DOI: 10.1016/J.JNT.2012.08.028
description: scientific article; zbMATH DE number 6138269