A duality theorem for Tate-Shafarevich groups of curves over algebraically closed fields
type: Item
instance of: scholarly article
title: A duality theorem for Tate-Shafarevich groups of curves over algebraically closed fields
zbMATH Open document ID: 1451.11130
author: Timo Keller
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2019-01-25T00:00:00Z
review text: In this paper, the author proves a duality theorem for Tate-Shafarevich groups over function fields. Let \(K\) be the function field of an irreducible smooth projective curve over an algebraically closed field, and let \(F\) be a finite discrete Galois module of order invertible in \(K\). Recall that the Tate-Shafarevich groups \(\Sha^i(K,F)\) are the kernel of the map from the Galois cohomology groups of \(F\) to the product of all the cohomology groups at the completions of \(K\) at the discrete valuations (Definition 3.2). The main theorem of the paper (Theorem 3.9) states that there is a perfect pairing of finite groups \[ \Sha^1(K,F)\times \Sha^1(K,Hom(F,\mathbb{Q}/\mathbb{Z}(1)))\to\mathbb{Q}/\mathbb{Z}. \] Furthermore, the group \(Sha^0(K,F)\) is finite and \(Sha^r(K,F)=0\) for \(r>1\). The proof follows the lines of [\textit{D. Harari} and \textit{T. Szamuely}, J. Algebr. Geom. 25, No. 3, 571--605 (2016; Zbl 1355.14018)] by using the duality for open subschemes of smooth proper curves (Theorem 2.2). Proposition 4.2 contains another result on the vanishing of the first Galois cohomology of certain algebraic groups.
reviewed by: Fangzhou Jin
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14L17
zbMATH DE Number: 7006744
zbMATH Keywords: Galois cohomology
zbMATH Keywords: étale and other Grothendieck topologies and cohomologies
zbMATH Keywords: affine algebraic groups
MaRDI profile type: Publication
full work available at URL: s12188-018-0196-7
OpenAlex ID: W2895293093
cites work: Néron Models
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Wikidata QID: Q129145713
DOI: 10.1007/S12188-018-0196-7
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description: scientific article; zbMATH DE number 7006744