The Picard Functor

Following Grothendieck’s treatment [FGA], we introduce the relative Picard functor Pic_X/S and treat the notion of the rigidified relative Picard functor. The main purpose of this chapter is the presentation of various results on the representability of Pic_X/S. We explain Grothendieck’s theorem on the representability of Pic_X/S by a scheme and point out improvements due to Mumford [2] as well as those due to Altman and Kleiman [1]. In Section 8.3, we discuss the main steps of M. Artin’s approach [5] to the representability of Pic_X/S by an algebraic space; for details, the reader is referred to his paper. At the end of the chapter, there is a collection of some results on smoothness as well as on finiteness properties of Pic_X/S, as can be found in [SGA 6].