# On residue complexes, dualizing sheaves and local cohomology modules Academic Article

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### 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.

### publication date

• January 1, 1995