Poincaré 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 Poincaré duality with respect to De Rham homology as defined by Hartshorne [H.75], 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 [HL], 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 [H.75, IV.1.2] (resp. [BB, 1]) going in the opposite direction, are shown to be compatible with our algebraic Poincaré pairing and the analogous complex-analytic (resp. rigid) one (resp. [B.97, 3.2]).