Crystalline realizations of 1-motives

Abstract

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 via de Jong's alterations, coincides with the crystalline realization of the Picard 1-motive, over perfect fields of cahracteristic > 2.

Universal extension crystals of 1-motives and applications

“…do not work in general but only work under the assumption that S 0 is of characteristic 0 or that we work in a p-adic context as in Section 6( * ). This second case is sufficient for the applications in [1]. See Remarks 3.4 and 5.5 for explanations of where the problem lies and Proposition 6.2 for a proof that no problem appears if we have the stronger assumptions imposed above.…”

“…[14, 10.1.7]). (For the existence of universal extensions see also [1,19,5]). Note that E(M) is endowed with a weight filtration where W 0 …”

“…This theorem answers a question raised by Buium; see Proposition 4.7. The crystalline nature of the universal extension of a 1-motive was already established in [1,Section 3]. Unfortunately, despite the fact that in loc.…”

Deligne's duality for de Rham realizations of 1‐motives

“…Two extensions (E i , v i ), i = 1, 2, are isomorphic if there exists an isomorphism ϕ : E 1 → E 2 (as extension of G by H) such that v 2 = ϕ • v 1 . As usual, Ext 1 (M, H) denotes the group of isomorphism classes of extensions of M by H. In the following, we will simply speak of 1-motives meaning S-1-motives.…”

On Deformations of 1-motives

“…Note that V(M) is killed by p s since the multiplication by p s morphism is the 0 map on G a,R . We recall that M ♮ is denoted by [1,2].…”