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Rigid Dualizing Complexes Over Commutative Rings

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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.

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References

  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. Conrad, B.: Grothendieck duality and base change. Lecture Notes in Math. Springer 1750 (2000)

  3. Etingof, P., Ginzburg, V.: Symplectic reflection algebras, Calogero–Moser space, and deformed Harish–Chandra homomorphism. Invent. Math. 147(2), 243–348 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hakim, M.: Topos Annelés et Schémas Relatifs. Springer (1972)

  6. Hübl, R., Kunz, E.: Regular differential forms and duality for projective morphisms. J. Reine Angew. Math. 410, 84–108 (1990)

    MATH  MathSciNet  Google Scholar 

  7. Hübl, R.: Traces of differential forms and hochschild homology. Lecture Notes in Math. Springer-Verlag, Berlin 1368 (1989)

  8. Kapustin, A., Kuznetsov, A., Orlov, D.: Noncommutative instantons and twistor transform. Comm. Math. Phys. 221, 385–432 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kleiman, S.L.: Relative duality for quasi-coherent sheaves. Compositio Math. 41(1), 39–60 (1980)

    MATH  MathSciNet  Google Scholar 

  10. Kunz, E.: Kähler Differentials. Vieweg, Braunschweig/Wiesbaden (1986)

    MATH  Google Scholar 

  11. Lipman, J.: Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque 117 (1984)

  12. Lipman, J.: Residues and traces of differential forms via Hochschild homology. Contemp. Math. AMS, Providence 61, (1987)

  13. Lipman, J., Nayak, S., Sastry, P.: Pseudofunctorial behavior of Cousin complexes on formal schemes. Contemp. Math. AMS 375, 3–133 (2005)

    MathSciNet  Google Scholar 

  14. Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9(1), 205–236 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hartshorne, R.: Residues and duality. Lecture Notes in Math. Springer-Verlag, Berlin 20, (1966)

  16. Sastry, P.: Residues and duality on algebraic schemes. Compositio Math. 101, 133–178 (1996)

    MATH  MathSciNet  Google Scholar 

  17. Sancho de Salas, F.: Residue: a geometric construction. Canad. Math. Bull. 45(2), 284–293 (2002)

    MATH  MathSciNet  Google Scholar 

  18. Van den Bergh, M.: Existence theorems for dualizing complexes over non-commutative graded and filtered ring. J. Algebra 195(2), 662–679 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Verdier, J.L.: Base change for twisted inverse image of coherent sheaves. Algebraic Geometry. Oxford Univ. Press, pp. 393–408 (1969)

  20. Yekutieli, A.: Dualizing complexes over noncommutative graded algebras. J. Algebra 153(1), 41–84 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yekutieli, A.: An explicit construction of the Grothendieck residue complex, (with an appendix by Sastry, P.) Astérisque 208 (1992)

  22. Yekutieli, A.: Smooth formal embeddings and the residue complex. Canad. J. Math. 50, 863–896 (1998)

    MATH  MathSciNet  Google Scholar 

  23. Yekutieli, A.: Rigid dualizing complexes and perverse aoherent sheaves on schemes (in preparation)

  24. Yekutieli, A., Zhang, J.J.: Rings with Auslander dualizing complexes. J. Algebra 213(1), 1–51 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Yekutieli, A., Zhang, J.J.: Residue complexes over noncommutative rings. J. Algebra 259(2), 451–493 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yekutieli, A., Zhang, J.J.: Dualizing complexes and perverse modules over differential algebras. Compositio Math. 141, 620–654 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Yekutieli, A., Zhang, J.J.: Dualizing complexes and perverse sheaves on noncommutative ringed schemes. Selecta Math. 12, 137–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Yekutieli, A., Zhang, J.J.: Rigid complexes via DG algebras. Trans. AMS 360, 3211–3248 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Ben Gurion University, Be’er Sheva, 84105, Israel

    Amnon Yekutieli

  2. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    James J. Zhang

Authors
  1. Amnon Yekutieli
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  2. James J. Zhang
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Corresponding author

Correspondence to Amnon Yekutieli.

Additional information

This research was supported by the US–Israel Binational Science Foundation. The second author was partially supported by the US National Science Foundation.

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Yekutieli, A., Zhang, J.J. Rigid Dualizing Complexes Over Commutative Rings. Algebr Represent Theor 12, 19–52 (2009). https://doi.org/10.1007/s10468-008-9102-9

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  • DOI: https://doi.org/10.1007/s10468-008-9102-9

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