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Compositio Mathematica

, Volume 112, Issue 1, pp 93–125 | Cite as

Noncommutative schemes

  • Alexander L. Rosenberg
Article

Abstract

The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme morphisms. In particular, we prove an analogue of the canonical bijection: \({\text{Hom}}_{{\text{schemes/}}k} \)((X, O), Spec(A))⋍ Hom\({\text{Hom}}_{k - alg} \) (A,Γ (X, O)). We define a noncommutative version of the Čech cohomology of an affine cover and show that the Čech cohomology can be used to compute higher direct images. This fact is applied here to compute cohomology of invertible sheaves on skew projective spaces and in [LR3] to study D-modules on quantum flag varieties.

Quasi-scheme Zariski cover continuous morphism flat localization skew projective space quantum flag variety continuous monad resolutions of endofunctors. 

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References

  1. [A]
    Artin M.: Geometry of quantum planes, Contemporary Mathematicsv. 124 (1992).Google Scholar
  2. [AB]
    Artin, M. and Van den Bergh, M.: Twisted homogenious coordinate rings, J. Algebra133 (1990).Google Scholar
  3. [AZ]
    Artin, M. and Zhang, J.J.: Noncommutative Projective Schemes, preprint, (1994).Google Scholar
  4. [Bass]
    Bass, H.: Algebraic K-Theory, Benjamin (1968).Google Scholar
  5. [BB1]
    Beilinson, A. and Bernstein, J.: Localization de \(\mathcal{G}\)-modules, C.R. Acad. Sci.Paris 292 (1981) 15-18.Google Scholar
  6. [BB]
    Beilinson, A. and Bernstein, J.: A proof of Jantzen conjectures, Advances in Soviet mathematicsv. 16, Part I (1993).Google Scholar
  7. [BD]
    Bucur, I. and Deleanu, A.: Introduction to the theory of categories and functors, Pure and Appl. Math., V.XIX, John Wiley & Sons LTD, London New York Sydney, (1969).Google Scholar
  8. [Dl]
    Deligne, Categories Tannakiennes, Grothendieck Festshrift, v. 1, (1990).Google Scholar
  9. [D]
    Dixmier, J.: Algèbres Enveloppantes, GauthierVillars, Paris/Bruxelles/Montreal, (1974).Google Scholar
  10. [Dr]
    Drinfeld, V.G.: Quantum groups, Proc. Int. Cong. Math., Berkeley (1986) 798-820.Google Scholar
  11. [Gab]
    Gabriel, P.: Des catégories abéliennes, Bull. Soc. Math.France, 90 (1962) 323-449.Google Scholar
  12. [GZ]
    Gabriel, P. and Zisman, M.: Calculus of Fractions and Homotopy Theory, Springer Verlag, Berlin-Heidelberg-New York, (1967).Google Scholar
  13. [Go]
    Godement, R.: Théorie des faisceaux, Actual. Scient. et Industr., no. 1252, Paris: Hermann, (1958).Google Scholar
  14. [He]
    Herstein, I. N.: Noncommutative Rings, John Wiley & Sons, (1968).Google Scholar
  15. [H]
    Hodges, T.: Ring-theoretical Aspects of the Bernstein-Beilinson Theorem, LNM v. 1448 (1990) 155-163.Google Scholar
  16. [Jo]
    Joseph, A.: Faithfully flat embeddings for minimal primitive quotients of quantized enveloping algebras, in: Quantum Deformations of Algebras and Their Representations, Joseph, A. and Shnider, S. (eds), Israel Math. Conf. Proc.(1993) 79-106.Google Scholar
  17. [Jo1]
    Joseph, A.: Quantum Groups and Their Primitive Ideals, Springer-Verlag, (1995).Google Scholar
  18. [LR1]
    Lunts, V. and Rosenberg, A.L.: Differential calculus in noncommutative algebraic geometry I. D-ticalculus on noncommutative rings(1996) MPI 9653.Google Scholar
  19. [LR2]
    Lunts, V. and Rosenberg, A.L.: Differential calculus in noncommutative algebraic geometry II, D-ticalculus in the braided case. The localization of quantized enveloping algebras (1996) MPI 9676.Google Scholar
  20. [LR3]
    Lunts, V. and Rosenberg, A.L.: Localization for Quantum Groups(1995) preprint.Google Scholar
  21. [ML]
    Mac-Lane, S.: Categories for the Working Mathematicians, Springer-Verlag; New York-Heidelberg-Berlin (1971).Google Scholar
  22. [M1]
    Manin, Yu. I.: Quantum Groups and Non-commutative Geometry, Publications du C.R.M.; Univ. de Montreal, (1988).Google Scholar
  23. [M2]
    Manin, Yu. I.: Topics in Noncommutative Geometry, Princeton University Press, Princeton New Jersey (1991).Google Scholar
  24. [OV]
    Van Oystaeyen, F. and Willaert, L.: Grothendieck topology, Coherent sheaves and Serre’s theorem for schematic algebras, J. Pure and Applied algebra104 (1995) 109-122.CrossRefGoogle Scholar
  25. [R]
    Rosenberg, A.: Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its applications, v. 330, Kluwer Academic Publishers, (1995) 316.Google Scholar
  26. [R1]
    Rosenberg, A. L.: Non-commutative Local Algebra, GAFA vol. 4, no.5 (1994) 545-585.Google Scholar
  27. [R2]
    Rosenberg, A. L.: Reconstruction of Schemes, (1996) MPI.Google Scholar
  28. [S]
    Serre, J.-P.: Faisceaux algébriques cohérents, Annals of Math.62 (1955).Google Scholar
  29. [V1]
    Verevkin, A. B.: On a noncommutative analogue of the category of coherent sheaves on a projective scheme, Amer. Math. Soc. Transl.(2) v. 151 (1992).Google Scholar
  30. [V2]
    Verevkin, A. B.: Serre injective sheaves, Math. Zametki52 (1992) 35-41.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Alexander L. Rosenberg
    • 1
  1. 1.Bures-sur-YvetteFrance

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