Foncteur de Picard d'un champ alg\'ebrique

TL;DR

This paper investigates the properties of the Picard functor and stack of algebraic stacks, providing new proofs of representability and analyzing their geometric features, including properness and quasi-separation.

Contribution

It offers a new, direct proof of the representability of the Picard stack and establishes its quasi-separatedness and properness under certain conditions.

Findings

Picard stack is representable and quasi-separated.

Connected component of the identity is proper for geometrically normal fibers.

Includes examples of Picard functors of classical stacks.

Abstract

In this article we study the Picard functor and the Picard stack of an algebraic stack. We give a new and direct proof of the representability of the Picard stack. We prove that it is quasi-separated, and that the connected component of the identity is proper when the fibers of the stack are geometrically normal. We study some examples of Picard functors of classical stacks. In an appendix, we review the lisse-etale cohomology of abelian sheaves on an algebraic stack.