We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into anétale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding "almost" has a left adjoint, that we call LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Roǐtman type theorems (in characteristic 0).

We define, in a purely algebraic way, 1-motives Alb + (X), Alb − (X), Pic + (X) and Pic − (X) associated with any algebraic scheme X over an algebraically closed field of characteristic zero. For X over C of dimension n the Hodge realizations are, respectively , H 2n−1 (X, Z(n))/(torsion), H 1 (X, Z)/(torsion), H 1 (X, Z(1)) and H 2n−1 (X, Z(1− n))/(torsion).

We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soulé on cohomology, and prove it. This implies the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1-motives with torsion.

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.

Let X be a complex projective surface with arbitrary singularities. We construct a generalized Abel-Jacobi map A 0 (X) → J 2 (X) and show that it is an isomorphism on torsion subgroups. Here A 0 (X) is the appropriate Chow group of smooth 0-cycles of degree 0 on X, and J 2 (X) is the intermediate Jacobian associated with the mixed Hodge structure on H 3 (X). Our result generalizes a theorem of Roitman for smooth surfaces: if X is smooth then the torsion in the usual Chow group A 0 (X) is isomorphic to the torsion in the usual Albanese variety J 2 (X) ∼ = Alb(X) by the classical Abel-Jacobi map.