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Inventiones mathematicae

, Volume 161, Issue 2, pp 427–452 | Cite as

The minimal model program for orders over surfaces

  • Daniel Chan
  • Colin Ingalls
Article

Abstract

We develop the minimal model program for orders over surfaces and so establish a noncommutative generalisation of the existence and uniqueness of minimal algebraic surfaces. We define terminal orders and show that they have unique étale local structures. This shows that they are determined up to Morita equivalence by their centre and algebra of quotients. This reduces our problem to the study of pairs (Z,α) consisting of a surface Z and an element α of the Brauer group Brk(Z). We then extend the minimal model program for surfaces to such pairs. Combining these results yields a noncommutative version of resolution of singularities and allows us to show that any order has either a unique minimal model up to Morita equivalence or is ruled or del Pezzo.

Keywords

Model Program Local Structure Minimal Model Algebraic Surface Minimal Model Program 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Artin, M.: Local structure of maximal orders on surfaces. Brauer groups in ring theory and algebraic geometry (Wilrijk 1981). Lect. Notes Math., vol. 917, pp. 146–181. Berlin, New York: Springer 1982Google Scholar
  2. 2.
    Artin, M.: Geometry of quantum planes. Azumaya algebras, actions, and modules (Bloomington, IN 1990). Contemp. Math., vol. 124, pp. 1–15. Providence, RI: Am. Math. Soc. 1992Google Scholar
  3. 3.
    Artin, M.: Two-dimensional orders of finite representation type. Manuscr. Math. 58, 445–471 (1987)CrossRefGoogle Scholar
  4. 4.
    Artin, M.: Some problems on three-dimensional graded domains. Representation theory and algebraic geometry (Waltham, MA 1995), pp. 1–19. Lond. Math. Soc. Lect. Note Ser., vol. 238. Cambridge: Cambridge Univ. Press 1997Google Scholar
  5. 5.
    Artin, M., deJong, A.J.: Stable Orders over Surfaces. PreprintGoogle Scholar
  6. 6.
    Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc., III. Ser. 25, 75–95 (1972). Prog. Math., vol. 86. Boston, MA: Birkhäuser Boston 1990Google Scholar
  7. 7.
    Artin, M., Zhang, J.J.: Noncommutative projective schemes. Adv. Math. 109, 228–287 (1994)CrossRefGoogle Scholar
  8. 8.
    Auslander, M., Goldman, O.: Maximal Orders. Trans. Am. Math. Soc. 97, 1–24 (1960)Google Scholar
  9. 9.
    Hijikata, H., Nishida, K.: When is Λ1⊗Λ2 hereditary? Osaka J. Math. 35, 493–500 (1998)Google Scholar
  10. 10.
    Iskovskih, V.A.: Minimal Models of rational surfaces over arbitrary fields (russian). Izv. Akad. Nauk SSSR, Ser. Mat. 43, 19–43, 237 (1979)Google Scholar
  11. 11.
    Chan, D., Kulkarni, R.: Del Pezzo Orders on Projective Surfaces. Adv. Math. 173, 144–177 (2003)CrossRefGoogle Scholar
  12. 12.
    Kollár, J., Mori, S.: Birational geometry of algebraic varieties. With the collaboration of C.H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134. Cambridge: Cambridge University Press 1998Google Scholar
  13. 13.
    Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116, 133–176 (1982)Google Scholar
  14. 14.
    Ramras, M.: Orders with finite global dimension. Pac. J. Math. 50, 583–587 (1974)Google Scholar
  15. 15.
    Reiner, I.: Maximal orders. London Mathematical Society Monographs, no. 5. A subsidiary of Harcourt Brace Jovanovich, Publishers. London, New York: Academic Press 1975Google Scholar
  16. 16.
    Serre, J.-P.: Local class field theory. Algebraic Number Theory (Proc. Instructional Conf., Brighton 1965) (1967)Google Scholar
  17. 17.
    Stafford, J.T., Zhang, J.J.: Homological properties of (graded) Noetherian PI rings. J. Algebra 168, 988–1026 (1994)CrossRefGoogle Scholar
  18. 18.
    Van den Bergh, M.: Blowing up of non-commutative smooth surfaces. Mem. Am. Math. Soc. 154 (2001)Google Scholar
  19. 19.
    Yekutieli, A., Zhang, J.J.: Serre duality for noncommutative projective schemes. Proc. Am. Math. Soc. 125, 697–707 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.School of MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.Department of MathematicsUniversity of New BrunswickFrederictonCanada

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