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

Alexei N. Skorobogatov

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.

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

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.
Paper Structure (5 sections, 17 theorems, 66 equations)

This paper contains 5 sections, 17 theorems, 66 equations.

Table of Contents

  1. Cohomology of the product
  2. Brauer group of the product
  3. Abelian varieties
  4. Kummer surfaces
  5. Appendix, by Alexander Petrov

Key Result

Proposition 1.1

Let $X$ and $Y$ be pointed projective, geometrically reduced and geometrically connected varieties over a field $k$. Then there is a natural isomorphism

Theorems & Definitions (19)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 3.1
  • ...and 9 more