### abstract

- According to Grothendieck Duality Theory [RD], on each varietyV over a fieldk, there is a canonical complex of\(\mathcal{O}_V \)-modules, theresidue complex \(\mathcal{K}_V^{RD \cdot } \cong \pi ^! k\). These complexes satisfy (and are characterized by) functorial properties in the categoryV ofk-varieties. In [Ye] a complex\(\mathcal{K}_V^ \cdot \) is constructed explicitly (when the fieldk is perfect). The main result of this paper is that the two families of complexes,\(\{ \mathcal{K}_V^{RD \cdot } \} v \in \nu \) and\(\{ \mathcal{K}_V^{ \cdot } \} v \in \nu \), which carry certain additional data (such as trace maps…), are uniquely isomorphic. As a corollary we recover Lipman’s canonical dualizing sheaf of [Li], and we obtain formulas for residues of local cohomology classes of differential forms.