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We prove some finiteness theorems for the Picard functor of an algebraicstack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give astacky version of Raynaud's relative representability theorem, we givesufficient conditions for the existence of the torsion component of the Picardfunctor, and for the finite generation of the Neron-Severi groups or of thePicard group itself. We give some examples and applications. In an appendix, weprove the semicontinuity theorem for a (non necessarily tame) algebraic stack.

Finiteness theorems for the Picard objects of an algebraic stack