Donghoon Park (Brown)

1-motives with torsion and Cartier duality

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 3 more 1-motives corresponding to its cohomology with compact support, homology, Borel-Moore homology and Ramachandran showed dual relations of them. For instance, cohomological and homological 1-motives are Cartier dual to each other.

I will give the category of 1-motives with torsion and its Cartier dual functor, and show Cartier duality for this category. Some people including Barbieri-Viale already considered such a question and got some result, but this category is something 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 dual but my category is closed under Cartier dual.