  We show that in the case of primary field extensions, the extension of
scalars of Deligne $1$-motives admits a left adjoint, called Chow image, and a
right adjoint, called Chow trace. This generalizes Chow's results on abelian
varieties. Then we study the Chow trace in the framework of Voevodsky's
triangulated categories of (\'etale) motives. With respect to the $1$-motivic
$t$-structure on the category of Voevodsky's homological $1$-motives, the
zero-th direct image of an abelian variety is given by the Chow trace, and the
first direct image is the $0$-motive defined by the (geometric) Lang-N\'eron
group.
