The relative units-Picard complex and the Brauer group of a product

TL;DR

This paper introduces the relative units-Picard complex to analyze the Brauer group of product schemes, providing new exact sequences that help compute these groups in specific geometric contexts.

Contribution

It develops the relative units-Picard complex framework and applies it to derive exact sequences for the Brauer group of fiber products of schemes.

Findings

Derived a five-term exact sequence relating Brauer groups of products and factors.

Solved the Brauer group problem for certain ruled varieties over characteristic zero fields.

Provided a new cohomological approach to understanding the Brauer group of scheme products.

Abstract

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors. Under certain hypotheses, we obtain a five-term exact sequence involving the preceding groups which enables us to solve the indicated problem in the case of, e.g., (certain types of) ruled varieties over a field of characteristic zero.