Poincare Duality for Algebraic De Rham Cohomology

We discuss in some detail the algebraic notion of De Rham cohomology with compact supports for singular schemes over a field of characteristic zero. We prove Poincare duality with respect to De Rham homology as defined by Hartshorne, so providing a generalization of some results of that paper to the non proper case. In order to do this, we work in the setting of the categories introduced by Herrera and Lieberman, and we interpret our cohomology groups as hyperext groups. We exhibit canonical morphisms of ''cospecialization'' from complex-analytic De Rham (resp. rigid) cohomology groups with compact supports to the algebraic ones. These morphisms, together with the ''specialization'' morphisms going in the opposite direction, are shown to be compatible with our algebraic Poincare pairing and the analogous complex-analytic (resp. rigid) one.