Deligne 1-motives with torsion and \'etale motives

Abstract

We construct the motivic t-structure on 1-motives with integral coefficients over a scheme of characteristic zero or a Dedekind scheme. When we invert the residue characteristic exponents of the base, this t-structure induces a t-structure on the category of smooth 1-motives whose heart is the category of Deligne 1-motives with torsion which we prove to be abelian. This relies on the fact that over a normal scheme, Deligne 1-motives with torsion are determined by their fiber on the generic point and on an explicit description of good reduction Deligne 1-motives with torsion.