The Brauer group of an affine cone
type: Item
instance of: scholarly article
title: The Brauer group of an affine cone
zbMATH Open document ID: 0974.14013
DOI: 10.1016/S0022-4049(99)00074-2
published in: Journal of Pure and Applied Algebra
publication date: 2001-11-11T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic zero. Fixing integers \(d\) and \(\nu\) both \(\geq 1\), let \(f\) be a square-free homogeneous polynomial of degree \(d\) in \(k[x_0,\cdots, x_{\nu}]\). Let \(V=V(f)\) be the variety in \({\mathbb{P}}^{\nu}=\text{Proj} k[x_0,\cdots,x_{\nu}]\) defined by \(f\). Put \(X={\mathbb{A}}^{\nu+1}=\text{Spec} k[x_0,\cdots, x_{\nu}]\) and let \(C=Z(f)\) in \({\mathbb{A}}^{\nu+1}\) denote the affine cone over \(V\) in \(X\). Further let \(P=(0,\cdots, 0)\) denote the vertex of the cone \(C\) in \({\mathbb{A}}^{\nu+1}\) and \(\overline{C}\) the projective closure of \(C\) in \(\overline X={\mathbb{P}}^{\nu+1}\). View \(V\) as the hypersurface section of \(\overline C\) at infinity. Then \(\overline C=C\cup V\). In the paper under review, the Brauer groups of the varieties \(C\), \(C-P\), \(X-C\) and \(\overline X-\overline C\) are computed using cohomological methods (where all cohomology and sheaves are for the étale topology). First fix the necessary notations: \(B(Y)\) denotes the Brauer group of a variety \(Y\), and \(B'(Y)\) the cohomological Brauer group of \(Y\), which is \(H^2(Y,{\mathbb{G}}_m)\). For an integer \(n\geq 1\), \(\mu_n\) denotes the group of \(n\)-th roots of unity (which is non-canonically isomorphic to the group \({\mathbb{Z}}/n\)), and \(\mu_n(-1)\) stands for \(\text{Hom}_{{\mathbb{Z}}}(\mu_n,{\mathbb{Z}}/n)\). The main results are summarized as follows. Proposition 1: Let \(V\) be a reduced variety in \({\mathbb{P}}^{\nu}\) and \(C\) the cone over \(V\) in \(X={\mathbb{A}}^{\nu+1}\). Then \(B(C)=0\). Theorem 2: Let \(V\) be a nonsingular hypersurface in \({\mathbb{P}}^{\nu}\) of degree \(d\) and \(C\) the affine cone over \(V\) in \(X={\mathbb{A}}^{\nu+1}\). Let \(\overline C\) denote the projective closure of \(C\) in \({\mathbb{P}}^{\nu+1}\), and let \(P\) be the vertex of \(C\). Then the following assertions hold: (a) \(B(X-C)\simeq H^1(C-P, {\mathbb{Q}}/{\mathbb{Z}})\). In particular, i…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 1563000
zbMATH Keywords: affine cone
zbMATH Keywords: Brauer group
author: Timothy J. Ford
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563000