Purity for the Brauer group
type: Item
instance of: scholarly article
title: Purity for the Brauer group
zbMATH Open document ID: 1430.14047
DOI: 10.1215/00127094-2018-0057
author: Kęstutis Česnavičius
published in: Duke Mathematical Journal
publication date: 2019-07-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1711.06456
full work available at URL: https://projecteuclid.org/euclid.dmj/1558145274
review text: The purity conjecture is a famous conjecture due to \textit{A. Grothendieck} [Adv. Stud. Pure Math. 3, 88--188 (1968; Zbl 0198.25901)]. It predicts that the Brauer group of a regular scheme \(X\) is not changed when removing a closed subscheme \(Z \subset X\) of codimension at least two. Combining several results due to Grothendieck and Gabber, the conjecture was proved except in some cases. In the present paper, the author settles the these remaining cases. They concern \(p\)-torsion Brauer classes in mixed characteristic \((0,p)\). This finishes the proof of the following theorem. Theorem [\textit{K. Česnavičius}, Duke Math. J. 168, No. 8, 1461--1486 (2019; Zbl 07080116)]. For each locally noetherian scheme \(X\) and a closed subscheme \(Z\subset X\), such that for every \(z \in Z\) the local ring \(\mathcal{O}_{X,z}\) of \(X\) at \(z\) is regular of dimension \(\geq 2\), we have \[ H^2_{\text{et}} \left(X, \mathbb{G}_m\right) \overset{\sim}{\longrightarrow} H^2_{\text{et}} \left(X - Z, \mathbb{G}_m\right) \] and \[ H^3_{\text{et}} \left(X, \mathbb{G}_m\right)\hookrightarrow H^3_{\text{et}} \left(X - Z, \mathbb{G}_m\right). \]
reviewed by: Charlotte Ure
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 16K50
zbMATH DE Number: 7080116
zbMATH Keywords: Brauer group
zbMATH Keywords: étale cohomology
zbMATH Keywords: perfectoid ring
zbMATH Keywords: punctured spectrum
zbMATH Keywords: purity
MaRDI profile type: Publication
OpenAlex ID: W3104984180
arXiv ID: 1711.06456
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description: scientific article