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 Picard 1-motive, over perfect fields.
This paper is in part a fitting sequel to the following:
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L. Barbieri Viale, A. Rosenschon & M. Saito: Deligne's conjecture on
1-motives, to appear on Annals of Math (Princeton, USA).
http://arXiv.org/abs/math/0102150
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L. Barbieri Viale: On algebraic 1-motives related to Hodge cycles, in
Algebraic Geometry -- A Volume in Memory of P. Francia -- Walter de Gruyter,
Berlin/New York, 2002, p. 25-60.
http://arXiv.org/abs/math/0103179
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L. Barbieri Viale & V. Srinivas: Albanese and Picard 1-motives, Memoire SMF
87, vi+104 pages, Paris, 2001.
http://smf.emath.fr/Publications/Memoires/2001/87/html/smf_mem-ns_87.html