×

zbMATH — the first resource for mathematics

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)].

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. El Zein,Complexe Dualizant et Applications à la Classe Fondamentale d’un Cycle, Bulletin de la Société Mathématique de France58 (1978). · Zbl 0388.14002
[2] R. Hübl and E. Kunz,Integration of differential forms on schemes, Journal für die Reine und Angewandte Mathematik410 (1990), 53–83. · Zbl 0712.14006 · doi:10.1515/crll.1990.410.53
[3] R. Hübl and E. Kunz,Regular differential forms and duality for projective morphisms, Journal für die Reine und Angewandte Mathematik410 (1990), 84–108. · Zbl 0709.14014 · doi:10.1515/crll.1990.410.84
[4] R. Hübl and P. Sastry,Regular differential forms and relative duality, American Journal of Mathematics115 (1993), 749–787. · Zbl 0796.14012 · doi:10.2307/2375012
[5] R. Hübl,Traces of Differential Forms and Hochschild Homology, Lecture Notes in Mathematics1368, Springer-Verlag, Berlin, 1989. · Zbl 0675.13019
[6] R. Hübl,Residues of regular and meromorphic differential forms, to appear in Mathematische Annalen. · Zbl 0814.14022
[7] E. Kunz,Kähler Differentials, Vieweg, Braunschweig, Wiesbaden, 1986.
[8] E. Kunz,Differentialformen auf algebraischen Varietäten mit Singulatitäten II, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg47 (1978), 42–70. · Zbl 0379.14005 · doi:10.1007/BF02941351
[9] A. Grothendieck,Local Cohomology, Lecture Notes in Mathematics 41, Springer-Verlag, Berlin, 1967.
[10] J. Lipman,Dualizing sheaves, differentials and residues on algebraic varieties, Asterisque117 (1984). · Zbl 0562.14003
[11] J. Lipman and P. Sastry,Regular differentials and equidimensional scheme-maps, Journal of Algebraic Geometry1 (1992), 101–130. · Zbl 0812.14011
[12] H. Matsumura,Commutative Algebra, Benjamin/Cummings, Reading, Mass., 1980.
[13] M. Nagata,Imbedding of an abstract variety in a complete variety, Journal of Mathematics of Kyoto University2 (1962), 1–10. · Zbl 0109.39503
[14] R. Hartshorne,Residues and Duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin, 1966.
[15] P. Sastry,Errata to: Regular differentials and equidimensional scheme-maps, in preparation. · Zbl 1080.14515
[16] J.-L. Verdier,Base change for twisted inverse image of coherent sheaves, inAlgebraic Geometry (Bombay 1968), Oxford Univ. Press, 1969, pp. 393–408.
[17] A. Yekutieli,An explicit construction of the Grothendieck residue complex (with an appendix by P. Sastry), Astérisque208 (1992). · Zbl 0788.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.