A note on the Brauer group and the Brauer-Manin set of a product

TL;DR

This paper extends previous results on the Brauer group and Brauer-Manin set behavior under product formation to include non-projective varieties, broadening the scope of these algebraic geometry tools.

Contribution

It generalizes earlier work by relaxing the projectivity condition, showing the commutativity of Brauer groups and Brauer-Manin sets for a wider class of varieties.

Findings

Brauer groups commute with product for non-projective varieties

Brauer-Manin sets also commute under these conditions

Results expand applicability of previous theorems

Abstract

We generalize the results of Skorobogatov and Zarhin considering the commutativity of Brauer groups (and Brauer-Manin sets) with taking product of two varieties, by relaxing the condition that varieties are projective.