The bilinear form of the Brauer group of a surface
type: Item
instance of: scholarly article
title: The bilinear form of the Brauer group of a surface
zbMATH Open document ID: 0871.13001
published in: Inventiones Mathematicae
publication date: 1996-11-25T00:00:00Z
review text: Let \(X\) be a proper smooth surface over a finite field \({\mathbb{F}}\) of characteristic \(p\). Let \({\text{Br}}(X)\) denote the Brauer group of \(X\). If \({\text{Br}}(X)\) is a torsion group, one has the direct sum decomposition \({\text{Br}}(X)=\sum_{\ell}{\text{Br}}(X)(\ell)\) where \(\ell\) runs through all primes and \({\text{Br}}(X)(\ell)\) stands for the \(\ell\)-torsion subgroup. Let \({\text{Br}}(X)(\ell)_{\text{{div}}}\) denote the maximal divisible subgroup of \({\text{Br}}(X)(\ell)\). \textit{J. Tate} [in: Sém. Bourbaki 1965/66, Exposé 306 (1966; Zbl 0199.55604)] proved that if \(\ell\neq p\), the quotient \({\text{Br}}(X)(\ell)_{\text{{nd}}}:={\text{Br}}(X)(\ell)/{\text{Br}} (X)(\ell)_{\text{{div}}}\) is finite and it is equipped with a non-degenerate skew-symmetric bilinear form with values in \({\mathbb{Q}}/{\mathbb{Z}}\). Further, Tate conjectured that the bilinear form on \({\text{Br}}(X)(\ell)_{\text{{nd}}}\) is alternating. It is easily seen that if \(\ell\neq 2\) and \(\ell\neq p\), Tate's form is alternating on \({\text{Br}}(X)(\ell)_{\text{{nd}}}\) and the order \(\#{\text{Br}}(X)(\ell)_{\text{{nd}}}\) is a square. The problem remains open when \(\ell=p\neq 2\), and indeed, the main result of this paper addresses this case. Let \(\overline X =X\times_{\text{{Spec}}{\mathbb{F}}} {\text{Spec}}(\overline{\mathbb{F}})\) be the lift of \(X\) to the algebraic closure \(\overline{\mathbb{F}}\) of \({\mathbb{F}}\). \textbf{ {Theorem}}. Assume that \(p\neq 2\). Then, on the subgroup \({\text{Ker}}({\text{Br}}(X)(2)_{\text{{nd}}}\to {\text{Br}}(\overline X)(2)_{\text{{nd}}})\) Tate's skew-symmetric bilinear form is alternating, and \(\#{\text{Br}}(X)(2)_{\text{{nd}}}\) is a square. (Here the morphism is induced by the covering morphism \(\overline X\to X\).) \textbf{ {Corollary}}. Let \(\Gamma=\overline{\mathbb{F}}/{\mathbb{F}}\) be the Galois group of \({\mathbb{F}}\). If the group \(H^3(\overline X_{\text{et}},{\mathbb{Z}}_2(1))\) has no non-trivi…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14J99
zbMATH DE Number: 931981
zbMATH Keywords: smooth surfaces
zbMATH Keywords: Brauer groups
zbMATH Keywords: characteristic \(p\)
author: Tohsuke Urabe
MaRDI profile type: Publication
full work available at URL: s002220050086
OpenAlex ID: W2041684125
DOI: 10.1007/S002220050086
description: scientific article; zbMATH DE number 931981