Translation-Invariant Line Bundles On Linear Algebraic Groups

TL;DR

This paper investigates the structure of translation-invariant line bundles on linear algebraic groups, proving finiteness over global function fields and constructing examples of unusual cohomological behavior.

Contribution

It provides a detailed analysis of the Picard groups of linear algebraic groups and introduces new examples of pathological cohomology phenomena.

Findings

Translation-invariant line bundles form a finite subgroup over global function fields.

Constructs examples of pathological cohomology behavior in algebraic groups.

Provides insights into the structure of Picard groups for algebraic groups.

Abstract

We study the Picard groups of connected linear algebraic groups, and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.