Descent theory for open varieties

TL;DR

This paper generalizes descent theory for smooth algebraic varieties, connecting Brauer--Manin obstructions for integral points to torsors under multiplicative groups, thus broadening the scope of previous results.

Contribution

It extends descent theory to all smooth varieties without the constant invertible functions condition, linking obstructions for integral points to torsors under multiplicative groups.

Findings

Extended descent theory to arbitrary smooth varieties.

Linked Brauer--Manin obstruction to torsors under groups of multiplicative type.

Broadened understanding of obstructions for integral points.

Abstract

We extend the descent theory of Colliot-Th\'el\`ene and Sansuc to arbitrary smooth algebraic varieties by removing the condition that every invertible regular function is constant. This links the Brauer--Manin obstruction for integral points on arithmetic schemes to the obstructions defined by torsors under groups of multiplicative type.