On the derived category of 1-motives, I

Overview & Synopsis

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 an etale 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 Roitman type theorems (in characteristic 0).