Boundedness of the p-primary torsion of the Brauer group of products of varieties

2024/04/29 by Alexei N. Skorobogatov, Skorobogatov, Alexei N. · 4 citations
Mathematics · #14F22 #14F30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2404.19150

openalex publication_date 2024/04/29 · openalex created_date 2024/05/03 · openalex updated_date 2026/07/28

Abstract

We prove that the quotient of the Brauer group of a product of varieties over k by the sum of the images of the Brauer groups of factors has finite exponent. The bulk of the proof concerns p-primary torsion in characteristic p. Our approach gives a more direct proof of the boundedness of the p-primary torsion of the Brauer group of an abelian variety, as recently proved by D'Addezio. We show that the transcendental Brauer group of a Kummer surface over k has finite exponent, but can be infinite when k is an infinite field of positive characteristic. This answers a question of Zarhin and the author.

Cited by

• Brauer groups of abelian varieties over fields of finite characteristic
• Adelic Mordell-Lang and the Brauer-Manin obstruction
• Higher dimensional dominoes in de Rham-Witt cohomology
• Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic

Related

• Brauer groups of abelian varieties over fields of finite characteristic
• Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces
• On p-torsions of geometric Brauer groups
• Remarks on p-primary torsion of the Brauer group
• A remark on uniform boundedness for Brauer groups