On the Derived Category Of 1-Motives

The authors embed the derived category of Deligne 1-motives over a perfect field into the etale version of Voevodsky's triangulated category of geometric motives after inverting the exponential characteristic. They then show that this full embedding ``almost'' has a left adjoint LAlb. Applying LAlb to the motive of a variety, the authors get a bounded complex of 1-motives that they compute fully for smooth varieties and partly for singular varieties. Among applications, the authors...
More Description

The authors embed the derived category of Deligne 1-motives over a perfect field into the etale version of Voevodsky's triangulated category of geometric motives after inverting the exponential characteristic. They then show that this full embedding ``almost'' has a left adjoint LAlb. Applying LAlb to the motive of a variety, the authors get a bounded complex of 1-motives that they compute fully for smooth varieties and partly for singular varieties. Among applications, the authors give motivic proofs of Roitman type theorems and new cases of Deligne's conjectures on 1-motives.