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Rigid Dualizing Complexes Over Commutative Rings

  • Amnon YekutieliEmail author
  • James J. Zhang
Article
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Abstract

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes.

Keywords

Commutative rings Derived categories Dualizing complexes Rigid complexes 

Mathematics Subject Classifications (2000)

Primary: 14F05 Secondary: 14B25 14F10 13D07 18G10 16E45 
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References

  1. 1.
    Alonso, L., Jeremías, A., Lipman, J.: Duality and flat base change on formal schemes. In: Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes. Contemp. Math. 244, 3–90 (1999)Google Scholar
  2. 2.
    Conrad, B.: Grothendieck duality and base change. Lecture Notes in Math. Springer 1750 (2000)Google Scholar
  3. 3.
    Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero–Moser space, and deformed Harish–Chandra homomorphism. Invent. Math. 147(2), 243–348 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Frankild, A., Iyengar, S., P. Jørgensen: Dualizing differential graded modules and gorenstein differential graded algebras. J. London Math. Soc. 68(2), 288–306 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hakim, M.: Topos Annelés et Schémas Relatifs. Springer (1972)Google Scholar
  6. 6.
    Hübl, R., Kunz, E.: Regular differential forms and duality for projective morphisms. J. Reine Angew. Math. 410, 84–108 (1990)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hübl, R.: Traces of differential forms and hochschild homology. Lecture Notes in Math. Springer-Verlag, Berlin 1368 (1989)Google Scholar
  8. 8.
    Kapustin, A., Kuznetsov, A., Orlov, D.: Noncommutative instantons and twistor transform. Comm. Math. Phys. 221, 385–432 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kleiman, S.L.: Relative duality for quasi-coherent sheaves. Compositio Math. 41(1), 39–60 (1980)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Kunz, E.: Kähler Differentials. Vieweg, Braunschweig/Wiesbaden (1986)zbMATHGoogle Scholar
  11. 11.
    Lipman, J.: Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque 117 (1984)Google Scholar
  12. 12.
    Lipman, J.: Residues and traces of differential forms via Hochschild homology. Contemp. Math. AMS, Providence 61, (1987)Google Scholar
  13. 13.
    Lipman, J., Nayak, S., Sastry, P.: Pseudofunctorial behavior of Cousin complexes on formal schemes. Contemp. Math. AMS 375, 3–133 (2005)MathSciNetGoogle Scholar
  14. 14.
    Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9(1), 205–236 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hartshorne, R.: Residues and duality. Lecture Notes in Math. Springer-Verlag, Berlin 20, (1966)Google Scholar
  16. 16.
    Sastry, P.: Residues and duality on algebraic schemes. Compositio Math. 101, 133–178 (1996)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Sancho de Salas, F.: Residue: a geometric construction. Canad. Math. Bull. 45(2), 284–293 (2002)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Van den Bergh, M.: Existence theorems for dualizing complexes over non-commutative graded and filtered ring. J. Algebra 195(2), 662–679 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Verdier, J.L.: Base change for twisted inverse image of coherent sheaves. Algebraic Geometry. Oxford Univ. Press, pp. 393–408 (1969)Google Scholar
  20. 20.
    Yekutieli, A.: Dualizing complexes over noncommutative graded algebras. J. Algebra 153(1), 41–84 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Yekutieli, A.: An explicit construction of the Grothendieck residue complex, (with an appendix by Sastry, P.) Astérisque 208 (1992)Google Scholar
  22. 22.
    Yekutieli, A.: Smooth formal embeddings and the residue complex. Canad. J. Math. 50, 863–896 (1998)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Yekutieli, A.: Rigid dualizing complexes and perverse aoherent sheaves on schemes (in preparation)Google Scholar
  24. 24.
    Yekutieli, A., Zhang, J.J.: Rings with Auslander dualizing complexes. J. Algebra 213(1), 1–51 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Yekutieli, A., Zhang, J.J.: Residue complexes over noncommutative rings. J. Algebra 259(2), 451–493 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Yekutieli, A., Zhang, J.J.: Dualizing complexes and perverse modules over differential algebras. Compositio Math. 141, 620–654 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Yekutieli, A., Zhang, J.J.: Dualizing complexes and perverse sheaves on noncommutative ringed schemes. Selecta Math. 12, 137–177 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Yekutieli, A., Zhang, J.J.: Rigid complexes via DG algebras. Trans. AMS 360, 3211–3248 (2008)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion UniversityBe’er ShevaIsrael
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

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