1-motives with torsion and Cartier duality

We define a category of smooth 1-motives with torsion over a locally noetherian base scheme and prove its Cartier duality. More precisely, we prove that the category of smooth 1-motives with torsion is equivalent to the category trivializations of particular Gm -biextensions, and this implies the Cartier duality for smooth 1-motives with torsion. We also show that this category has realization functors when the base scheme is a spectrum of a field. Cartier duality theorem was already proved in the case of 1-motives over a field by Deligne or Ramachandran. We will extend this result to any locally noetherian base scheme and moreover to 1-motives with torsion. The category of smooth 1-motives with torsion is not an abelian category, but there are many realization functors as the category of 1-motives.