Crystalline realizations of 1-motives

We consider the crystalline realization of Deligne's 1-motives in positive characteristics and prove a comparison theorem with the De Rham realization of (formal) liftings to zero characteristic. We then show that one dimensional crystalline cohomology of an algebraic variety, defined by forcing universal cohomological descent {\em via}\, de Jong's alterations, coincides with the crystalline realization of the Picard 1-motive, over perfect fields of cahracteristic $>2$