Galois descent on the Brauer group
type: Item
instance of: scholarly article
zbMATH Open document ID: 1317.14042
author: Jean-Louis Colliot-Thélène
author: Alexei N. Skorobogatov
publication date: 2013-10-30T00:00:00Z
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 6221007
zbMATH Keywords: Brauer group
zbMATH Keywords: Galois descent
MaRDI profile type: Publication
full work available at URL: crelle-2012-0039
OpenAlex ID: W2807056217
DOI: 10.1515/CRELLE-2012-0039
title: Galois descent on the Brauer group
published in: Journal für die Reine und Angewandte Mathematik
review text: The main object of the paper under review is the map NEWLINE\[NEWLINE \alpha: \mathrm{Br}(X)\to \mathrm{Br}(\overline X)^\Gamma, NEWLINE\]NEWLINE where \(X\) is a smooth, geometrically integral, projective variety \(X\) over a field \(k\) of characteristic zero, \(\overline X=X\times_k\bar k\), \(\Gamma=\mathrm{Gal}(\bar k/k)\). The main result states that the cokernel of \(\alpha\) is finite. In the case where \(H^1(X,O_X)=0\) or \(k\) is a number field, the authors provide explicit estimates for the order and exponent of this finite group. Particular applications include \(K3\) surfaces and products of two curves. Finally, the main finiteness theorem is extended to quasi-projective varieties, under the assumption that \(k\) is finitely generated over \(\mathbb Q\).
reviewed by: Boris Kunyavskiĭ
description: scientific article; zbMATH DE number 6221007