Chow trace of 1-motives and the Lang-Néron groups

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 (étale) 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-Néron group.