×

zbMATH — the first resource for mathematics

Residues and duality for algebraic schemes. (English) Zbl 0902.14012
Let \(k\) be a field. According to Grothendieck duality theory [R. Hartshorne, “Residues and duality”, Lect. Notes Math. 20 (Berlin 1966; Zbl 0212.26101)], on every \(k\)-scheme of finite type \(X\) there exists a complex of quasi-coherent \({\mathcal O}_X\)-modules called the residue complex, and denoted \({\mathcal K}_X^\bullet\). If \(\pi:X\to \text{Spec} k\) is the structural morphism, and \(\pi^!\) is the twisted inverse image functor, then \({\mathcal K}_X^\bullet\cong\pi^!k\) in the derived category of \({\mathcal O}_X\)-modules.
The purpose of the paper under review is to give an explicit construction of the Grothendieck residue complex in terms of differential forms and cohomological residues. The base field \(k\) is assumed to be perfect. Previously, this method was applied to constructions of the dualizing sheaf, or the sheaf of regular differentials [cf., e.g., J. Lipman and P. Sastry, J. Algebr. Geom. 1, No. 1, 101-130 (1992; Zbl 0812.14011) and R. Hübl and E. Kunz, J. Reine Angew. Math. 410, 84-108 (1990; Zbl 0709.14014)]. An explicit construction of the residue complex using topological local fields and Parshin residues was achieved by the reviewer [cf. A. Yekutieli, “An explicit construction of the Grothendieck residue complex”, Astérisque 208 (1992; Zbl 0788.14011)].
The construction consists of two main ingredients. The first is a study of Cousin complexes and their behaviour with respect to equidimensional scheme morphisms. – The second ingredient involves complete local rings. For every complete local residually finitely generated \(k\)-algebra \(R\) a module \({\mathcal K} (R)\) is defined, with certain functorial properties (so-called “trace structure”). This portion of the paper builds on the works of I.-C. Huang [“Pseudofunctors on modules with zero dimensional support”, Mem. Am. Math. Soc. 548 (1995; Zbl 0851.18007)]. – Finally, these two ingredients are combined. For a point \(x\in X\) one sets \({\mathcal K} (x): ={\mathcal K} (\widehat {\mathcal O}_{X,x})\), and \({\mathcal K}_X^\bullet: =\bigoplus_{x\in X} {\mathcal K} (x)\). The coboundary map \(\delta\) on \({\mathcal K}_X^\bullet\) is defined by identifying this graded sheaf with the Cousin complex of \(\Omega^n_{X/k} [n]\), when \(X\) is smooth of dimension \(n\).

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
13D15 Grothendieck groups, \(K\)-theory and commutative rings
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Beilinson, A.A. : Residues and adeles . Func. Anal. and Appl. 14 (1980) 34-35. · Zbl 0509.14018 · doi:10.1007/BF01078412
[2] Brüderle, S. : Cousinkomplexe in der kommutativen algebra . Diplomarbeit. Regensburg, 1986.
[3] Elzein, F. : Complexe dualisant et applications à la classe fondamentale d’un cycle . Bull. Soc. Math. France, Memoire 58 (1978). · Zbl 0388.14002 · numdam:MSMF_1978__58__1_0 · eudml:94780
[4] Grothendieck, A. : The cohomology theory of abstract algebraic varieties . In Proceedings of the ICM, Edinburgh, pages 103-118. Cambridge University Press, 1958. · Zbl 0119.36902
[5] Grothendieck, A. and Dieudonné, J. : Élements de Geométrie Algébrique III . Publ. Math. IHES 11 (1961) and 17 (1963). · Zbl 0122.16102 · numdam:PMIHES_1963__17__5_0
[6] Grothendieck, A. and Dieudonné, J. : Élements de Geométrie Algébrique IV . Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966), 32 (1967). · Zbl 0153.22301 · numdam:PMIHES_1967__32__5_0 · eudml:103873
[7] Hartshorne, R. : Residues and Duality , volume 20 of Lecture Notes in Mathematics. Springer-Verlag, Heidelberg, 1966. · Zbl 0212.26101 · doi:10.1007/BFb0080482 · eudml:203789
[8] Hartshorne, R. : Local Cohomology, volume 41 of Lecture Notes in Mathematics . Springer-Verlag, Heidelberg, 1967.
[9] Hartshorne, R. : Algebraic Geometry . Springer-Verlag, Heidelberg, 1977. · Zbl 0367.14001
[10] Huang, I-C. : Psuedofunctors on modules with 0-dimensional support . PhD thesis, Purdue University, 1992.
[11] Huang, I-C. : Functorial construction of cousin complexes . preprint, 1993. · Zbl 0814.13011
[12] Huang, I-C. : Psuedofunctors on modules with 0-dimensional support , volume 114, number 548, Memoirs of the AMS , Amer. Math. Society, 1995. · Zbl 0851.18007
[13] Hübl, R. : Traces of differential forms and Hochschild homology , volume 1368 of Lecture Notes in Mathematics. Springer-Verlag, Heidelberg, 1989. · Zbl 0675.13019
[14] Hübl, R. : Residues of regular and meromorphic differentials . In preparation, 1993.
[15] Hübl, R. and Kunz, E. : Integration of differential forms on schemes . J. Reine Angew. Math. 410 (1990) 53-83. · Zbl 0712.14006 · doi:10.1515/crll.1990.410.53 · crelle:GDZPPN002207850 · eudml:153258
[16] Hübl, R. and Sastry, P. : Regular differential forms and relative duality . Amer. J. Math. 115 (1993) 749-787. · Zbl 0796.14012 · doi:10.2307/2375012
[17] Kersken, M. : Cousinkomplexe und nennersysteme . Math. Z. 182 (1983) 389-402. · Zbl 0489.13006 · doi:10.1007/BF01179758 · eudml:173292
[18] Kersken, M. : Der residuenkomplex in der lokalen algebraichen und analytischen geometrie . Math. Ann. 265 (1983) 423-455. · Zbl 0582.32010 · doi:10.1007/BF01455946 · eudml:163831
[19] Kunz, E. : Residuen von differentialformen auf cohen-macaulay varietäten . Math. Z. 152 (1977) 165-189. · Zbl 0342.14022 · doi:10.1007/BF01214187 · eudml:172465
[20] Kunz, E. : Kähler Differentials . Vieweg, Braunschweig, 1986. · Zbl 0587.13014
[21] Lipman, J. : Dualizing sheaves and residues on algebraic varieties , volume 117 of Asterisque. Société Mathematique de France , 1984. Publié avec le concours du CNRS. · Zbl 0562.14003
[22] Lipman, J. : Residues and differential forms via Hochschild homology , volume 79 of Contemp. Math. Amer. Math. Soc., Providence, RI, 1987. · Zbl 0606.14015
[23] Lipman, J. and Sastry, P. : Regular differentials and equidimensional scheme maps . J. Algebraic Geometry 1 (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] Matsumura, H. : Commutative ring theory . Cambridge University Press, 1986. · Zbl 0603.13001
[26] Parshin, A.N. : On the arithmetic of two-dimensional schemes I: Distributions and residues . Math. USSR Izv. 10 (1976) 695-729. · Zbl 0366.14003 · doi:10.1070/IM1976v010n04ABEH001810
[27] 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 · crelle:GDZPPN00220049X · eudml:152535
[28] Sastry, P. : A pointwise criterion for dualizing pairs , volume 208 of Asterisque, pages 117-126. Société Mathematique de France, 1992. Appendix to book by A. Yekutieli. · Zbl 0788.14011
[29] Sastry, P. and Yekutieli, A. : On residue complexes, dualizing sheaves and local cohomology modules . Israel J. Math., volume 90 (1995), pages 325-348. · Zbl 0834.14003 · doi:10.1007/BF02783219
[30] Verdier, J.L. : Base change for twisted inverse image of coherent sheaves . In Algebraic Geometry (Bombay, 1968) , pages 393-408, London, 1969. Oxford University Press. · Zbl 0202.19902
[31] Yekutieli, A. : An explicit construction of Grothendieck’s residue complex , volume 208 of Asterisque. Société Mathematique du France, 1992. Publié avec concours CNRS. · Zbl 0788.14011
[32] Yekutieli, A. : Residues and differential operators on schemes . Preprint, 1993. · Zbl 0962.14010 · doi:10.1215/S0012-7094-98-09509-6 · arxiv:alg-geom/9602011
[33] Yekutieli, A. : Traces and differential operators over Beilinson completion algebras . To appear in Comp. Math. · Zbl 0856.13019 · numdam:CM_1995__99_1_59_0 · eudml:90408 · arxiv:alg-geom/9502024
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.