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