### On the derived category of 1-motives, I, by Luca Barbieri-Viale and Bruno Kahn

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).

This is a stabilised version of preprint #800.

Luca Barbieri-Viale <barbieri@math.unipd.it>

Bruno Kahn <kahn@math.jussieu.fr>