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On residue complexes, dualizing sheaves and local cohomology modules. (English) Zbl 0834.14003
For any variety \(X\) over a perfect field the authors prove the existence of a natural isomorphism between the Grothendieck residue complex [R. Hartshorne, “Residues and duality”, Lect. Notes Math. 20 (1966; Zbl 0212.261)] and a residue complex described by 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 J. Lipman [“Dualizing sheaves, differentials and residues on algebraic varieties”, Astérisque 117 (1984; Zbl 0562.14003)] and the trace \(\vartheta_X\) defined by 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 A. Yekutieli (loc. cit.)] is established. It should be remarked that for Cohen-Macaulay varieties similar results were obtained by R. Hübl [Math. Ann. 300, No. 4, 605-628 (1994; Zbl 0814.14022)].

14B15 Local cohomology and algebraic geometry
32A27 Residues for several complex variables
13D25 Complexes (MSC2000)
13D45 Local cohomology and commutative rings
14F20 Étale and other Grothendieck topologies and (co)homologies
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Full Text: DOI
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