Title : 1-motives with torsion

The category of 1-motives given by Deligne has realization functors and
Cartier duality. He also proved any semi-normal complex algebraic curve has
a 1-motive over C whose realizations are isomorphic to the first
singular, l-adic, and De Rham cohomology groups of this curve. For such a
curve, Lichtenbaum gave three more 1-motives corresponding to its cohomology
with compact support, homology, Borel-Moore homology and Ramachandran showed
duality relations among them. For instance, cohomological and homological
1-motives are dual to each other.

I will give the category of 1-motives with torsion and its Cartier duality
functor, and show Cartier duality for this category. Some people including
Barbieri-Viale already considered such a question and got some result, but
their category is different. For example, it is an abelian category
but my category is not, and they need one more (in fact, dual) category to
define Cartier duality but my category is closed under Cartier duality.