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Residues of regular and meromorphic differential forms. (English) Zbl 0814.14022
For varieties over perfect fields residues of differential forms can be constructed both cohomologically and using higher dimensional local fields. Related to the two types of residues are two constructions of residual complexes. We will exhibit an explicit isomorphism between the two constructions, thus obtaining also a relation between the two types of residues and the integrals derived from them. Using this we can compare various results of the different residue theories, and we can combine them to prove a generalization of a theorem of Newton on the centroids of the intersection schemes of two plane curves.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
13N10 Commutative rings of differential operators and their modules
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
32A27 Residues for several complex variables
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References:
[1] Beilinson A.A.: Residues and adeles. Func. Anal. Appl.14 (1980), 34-35 · Zbl 0509.14018
[2] Br?derle S.: Cousinkomplexe in der kommutativen Algebra. Diplomarbeit. Regensburg, 1986
[3] Grothendieck A., J. Dieudonn?: ?lements de Ge?m?trie Alg?brique, Chap. IV. Publ. Math. IHES32 (1968)
[4] Grothendieck A.: The cohomology theory of abstract algebraic varieties. Proc. ICM Edinburgh 1958, 103-118, Cambridge University Press, New York 1960
[5] Griffiths, P., J. Harris: Principles of algebraic geometry. John Wiley and Sons, New York 1978 · Zbl 0408.14001
[6] Huang I.C.: Pseudofunctors on modules with 0-dimensional support. PhD thesis, Purdue 1992
[7] H?bl, R., E. Kunz: Integration of differential forms on schemes. J. Reine Angew. Math.410 (1990), 53-83 · Zbl 0712.14006 · doi:10.1515/crll.1990.410.53
[8] H?bl R., E. Kunz: Regular differential forms and duality for projective morphisms. J. Reine Angew. Math.410 (1990), 84-108 · Zbl 0709.14014 · doi:10.1515/crll.1990.410.84
[9] H?bl R., P. Sastry: Regular differentials and relative duality. Am. J. Math.115 (1993), 749-787 · Zbl 0796.14012 · doi:10.2307/2375012
[10] H?bl R.: Traces of differential forms and Hochschild homology. Lect. Notes Math.1368. Springer, Berlin Heidelberg New York 1989 · Zbl 0675.13019
[11] H?bl, R.: On the transitivity of regular differential forms. Manuscr. Math.65 (1989), 213-224 · Zbl 0704.13004 · doi:10.1007/BF01168300
[12] H?bl R., A. Yekutieli: Adeles and differential forms. Preprint · Zbl 0847.14006
[13] Kunz E.: K?hler differentials. Vieweg Verlag, Braunschweig Wiesbaden, 1986 · Zbl 0587.13014
[14] Kersken M.: Cousmkomplexe und Nennersysteme. Math. Z.182 (1983), 389-402 · Zbl 0502.13008 · doi:10.1007/BF01179758
[15] Kersken M.: Der Residuenkomplex in der lokalen algebraischen und analytischen Geometrie. Math. Ann.265 (1983), 423-455 · Zbl 0582.32010 · doi:10.1007/BF01455946
[16] Kersken M.: Regul?re Differentialformen. Manuscr. Math.46 (1986) 1-25 · Zbl 0556.13006 · doi:10.1007/BF01185193
[17] Kunz E.: Residuen von Differentialformen auf Cohen-Macaulay Variet?ten. Math. Z.152 (1977), 165-189 · Zbl 0342.14022 · doi:10.1007/BF01214187
[18] Kunz E.: Introduction to commutative algebra and algebraic geometry. Birkh?user, Boston 1986
[19] Kunz E.: Verallgemeinerung eines Satzes von Newton. Sitzungsber. Bayer. Akad. Wissen., Math. Naturw. Kl., 3. Abh. 1989 · Zbl 0791.51016
[20] Kunz E., R. Waldi: Regular differential forms. Contemp. Math79. AMS, Providence 1988
[21] Lipman J.: Dualizing sheaves, differentials and residues on algebraic varieties. Asterisque117, 1984 · Zbl 0562.14003
[22] Lipman J.: Residues and traces of differential forms via Hochschild homology. Contemp. Math.61. AMS, Providence 1987 · Zbl 0606.14015
[23] Lipman, J., P. Sastry: Regular differentials and equidimensional scheme-maps. J. Algebraic Geometry1 (1992), 101-130 · Zbl 0812.14011
[24] Lomadze V.G.: On residues in algebraic geometry. Math. USSR, Izv.19 (1982), 495-520 · Zbl 0528.14003 · doi:10.1070/IM1982v019n03ABEH001426
[25] Lomadze V.G.: On the intersection index of divisors. Math. USSR, Izv.17 (1981), 343-352 · Zbl 0471.14001 · doi:10.1070/IM1981v017n02ABEH001362
[26] Newton I.: Curves. In: Lexicon Technicum, vol. 2. London, 1710
[27] Parshin A.N.: On the arthmetic of two-dimensional schemes 1: Distributions and residues. Math. USSR, Izv.10 (1976), 695-729 · Zbl 0366.14003 · doi:10.1070/IM1976v010n04ABEH001810
[28] Parshin A.N.: Chern classes, adeles and L-functions. J. Reine Angew. Math.341 (1983), 174-192 · Zbl 0518.14013 · doi:10.1515/crll.1983.341.174
[29] Hartshorne R.: Residues and duality. Lect. Notes Math.20, Springer, Berlin Heidelberg New York 1966 · Zbl 0212.26101
[30] Serre J.P.: Groupes Alg?briques et Corps de Class. Hermann, Paris 1959 · Zbl 0097.35604
[31] Sastry P., A. Yekutieli: On residue complexes, dualizing sheaves and local cohomology modules. To appear in: Israel Jour. of Math. · Zbl 0834.14003
[32] Yekutieli A.: An explicit construction of the Grothendieck residue complex. Ast?risque208, 1992 · Zbl 0788.14011
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