Picard Groups Of Algebraic Groups And An Affineness Criterion

TL;DR

This paper establishes a criterion linking the affineness of algebraic groups over a field to the torsion property of their Picard groups, providing detailed structure results depending on the field's perfection.

Contribution

It proves that an algebraic group is affine if and only if its Picard group is torsion, and characterizes the Picard group's structure over perfect and imperfect fields.

Findings

Algebraic group is affine iff its Picard group is torsion.

Picard group is finite over perfect fields.

Structure of Picard group over imperfect fields of characteristic p.

Abstract

We prove that an algebraic group over a field $k$is affine precisely when its Picard group is torsion, and show that in this case the Picard group is finite when $k$ is perfect, and the product of a finite group of order prime to $p$ and a $p$-primary group of finite exponent when $k$ is imperfect of characteristic $p$.