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

In this paper we refine the Voevodsky/Orgogozo relationship between the
derived category of Deligne's 1-motives over a perfect field and
Voevodsky's triangulated category of geometric motives. We get an integral
full embedding, provided we replace the latter by its étale counterpart.
The main tool of this paper is that this full embedding "almost" has a
left adjoint, that we call LAlb. Composing with duality, we obtain a
related functor RPic. Applied to the motive of a variety X, we thus get a
bounded complex of 1-motives LAlb(X) that we compute completely when X is
smooth and to a large extent for general X (in characteristic 0).
Actually, we get six complexes of 1-motives associated to an algebraic
variety X, namely: LAlb(X), LAlb^c (X), LAlb^*(X), RPic(X), RPic^c(X) and
RPic^*(X). These complexes provide natural candidates for realizing
Deligne's conjectures on 1-motives.

In particular, we prove that degree 1 homology L_1Alb(X) is canonically
isomorphic to the 1-motive Alb^-(X) of Barbieri-Viale--Srinivas, at least
when X is proper. One application is a new proof of Roitman's torsion
theorem and its generalisation by Spiess-Szamuely when X is smooth, and an
extension of this theorem to any singular X in characteristic 0.

This is a preliminary version. In the final version, we hope to also capture
the 1-motive Alb^+(X), hence get a motivic proof of Roitman's theorem for the
Levine-Weibel cohomological Chow group of X, and on the other hand to approach
Deligne's conjecture through realization functors.

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

Bruno Kahn <kahn@math.jussieu.fr>