Rational singularities and the Brauer group
type: Item
instance of: scholarly article
title: Rational singularities and the Brauer group
zbMATH Open document ID: 0809.13001
author: Tim Ford
author: Rick Miranda
author: Frank R. DeMeyer
published in: Journal of Algebra
publication date: 1994-04-19T00:00:00Z
review text: The authors present two examples of Brauer group calculations. First, they exhibit a local normal two dimensional domain \(R\), which is the local ring of an ordinary double point on a quartic in \(\mathbb{P}^ 3 (\mathbb{C})\) which has trivial divisor class group while its completion has class group of order 2 and the kernel \(B(K/R)\) of the map from the Brauer group of \(R\) to that of its quotient field \(K\) also has order 2. Next, they exhibit a rational complex surface \(X\) with a rational singular point \(p\) such that if \(R\) is the local ring of \(X\) at \(p\) and \(K\) its quotient field, then \(R\) has class group of order 2, its completion has class group cyclic of order 4, and \(B(K/R)\) is of order 2. The existence of such examples was predicted by Grothendieck in the 1960's; the sophisticated computations that the authors use to establish their examples explain the long interval between the prediction and discovery.
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 554222
zbMATH Keywords: Brauer group
zbMATH Keywords: class group
reviewed by: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: jabr.1993.1253
OpenAlex ID: W2060612922
DOI: 10.1006/JABR.1993.1253
description: scientific article; zbMATH DE number 554222