On residue complexes, dualizing sheaves and local cohomology modules
For any variety \(X\) over a perfect field the authors prove the existence of a natural isomorphism between the Grothendieck residue complex [\textit{R. Hartshorne}, ``Residues and duality'', Lect. Notes Math. 20 (1966; Zbl 0212.261)] and a residue complex described by \textit{A. Yekutieli} [``An explicit construction of the Grothendieck residue complex'', Astérisque 208 (1992; Zbl 0788.14011)]. As a result they get that the trace map \(\widetilde \vartheta_X\) defined by \textit{J. Lipman} [``Dualizing sheaves, differentials and residues on algebraic varieties'', Astérisque 117 (1984; Zbl 0562.14003)] and the trace \(\vartheta_X\) defined by \textit{A. Yekutieli} (loc. cit.) agree up to sign. -- Then formulas for residues of local cohomology classes of differential forms are written down explicitly. Thus, a clear relation between local cohomology residues and the Parshin residues [see \textit{A. Yekutieli} (loc. cit.)] is established. It should be remarked that for Cohen-Macaulay varieties similar results were obtained by \textit{R. Hübl} [Math. Ann. 300, No. 4, 605-628 (1994; Zbl 0814.14022)].
local cohomology classes of differential forms
scientific article; zbMATH DE number 784924
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