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.