  We consider the crystalline realization of Deligne's 1-motives in positive
characteristics and prove a comparison theorem with the De Rham realization of
liftings to zero characteristic. We then show that one dimensional crystalline
cohomology of an algebraic variety, defined by universal cohomological descent
via de Jong's alterations, coincide with the crystalline realization of the
(cohomological) Picard 1-motive, over perfect fields.
