Inventiones mathematicae

, Volume 85, Issue 2, pp 379–414 | Cite as

Arithmetic on two dimensional local rings

  • Shuji Saito


Local Ring Dimensional Local Ring 
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  1. 1.
    Abhyankar, S.: Resolution of singularities for arithmetical surfaces. In: Arithmetical Algebraic Geometry. New York: Harper and Row, 1963, pp. 111–152Google Scholar
  2. 2.
    Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S.36, 23–58 (1969)Google Scholar
  3. 3.
    Stein, M.R., Dennis, R.K.:K 2 of radical ideals and semilocal rings revisited. Lect. Notes Math.342, 281–303 (1973)Google Scholar
  4. 4.
    Gabber, O.: A lecture at I.H.E.S. on March in 1981Google Scholar
  5. 5.
    Greenberg, M.: Rational points in henselian discrete valuation rings. Publ. Math. I.H.E.S.23 (1964)Google Scholar
  6. 6.
    Grothendieck, A.: Le groupe de Brauer III. In: Dix exposé sur la cohomologie des schémas. Amsterdam: North-Holland 1968Google Scholar
  7. 7.
    Illusie, L.: Complexe de De Rham-Witt et cohomologie cristalline. Ann. Sci. Ec. Norm. Sup.12, 501–661 (1979)Google Scholar
  8. 8.
    Kato, K.: A generalization of local class field theory by usingK-groups, I. J. Fac. Sci., Univ. Tokyo, Sec. IA26, 303–376 (1979); II, ibid. Kato, K.: A generalization of local class field theory by usingK-groups, I. J. Fac. Sci., Univ. Tokyo, Sec. IA27, 602–683 (1980); III, ibid. Kato, K.: A generalization of local class field theory by usingK-groups, I. J. Fac. Sci., Univ. Tokyo, Sec. IA29, 31–43 (1982)Google Scholar
  9. 9.
    Kato, K., Saito, S.: Unramified class field theory of arithmetical surfaces. Ann. Math.118, 241–275 (1983)Google Scholar
  10. 10.
    Kato, K., Saito, S.: Global class field theory of arithmetic schemes (Preprint)Google Scholar
  11. 11.
    Lichtenbaum, S.: Duality theorems for curves overp-adic fields. Invent. Math.7, 120–136 (1969)Google Scholar
  12. 12.
    Mercurjev, A.S., Suslin, A.A.:K-cohomology of Severi-Brauer varieties and norm residue homomorphism (Preprint)Google Scholar
  13. 13.
    Saito, S.: Class field theory for two dimensional local rings. (to appear)Google Scholar
  14. 14.
    Saito, S.: Class field theory for curves over local fields. Journal Number Theory21, 44–80 (1985)Google Scholar
  15. 15.
    Serre, J.-P.: Cohomologie galoisienne. Lect. Notes Math.5, 1965Google Scholar
  16. 16.
    Serre, J.-P.: Corps locaux. Paris: Hermann 1962Google Scholar
  17. 17.
    Shafarevich, I.R.: Lectures on minimal models and birational transformations of two dimensional schemes. Tata Institute of Foundamental Research, Bombay, 1966Google Scholar
  18. 18.
    Shatz, S.S.: Cohomology of artinian group schemes over local fields. Ann. Math.79, 411–449 (1964)Google Scholar
  19. 19.
    Tate, J.: WC-groups over ϱ-adic fields. Séminaire Bourbaki; 10e année. 1957/1958Google Scholar
  20. 20.
    Hironaka, H.: Desingularization of excellent surfaces. Lectures at Advanced Science Seminar in Algebraic Geometry. Bowdoin College, Summer 1967, noted by Bruce BennettGoogle Scholar
  21. 21.
    Bourbaki: Algèbre commutative. Chapitre 7. Paris: Hermann 1965Google Scholar
  22. 22.
    Kurke, H., Pfister, G., Popescu, D., Roczen, M., Mostowski, T.: Die Approximationseigenschaft lokaler Ringe. Lect. Notes Math.634, 1978Google Scholar
  23. 23.
    Brown, M.L.: A class of 2-dimensional local rings with Artin's approximation property. The Journal of the London Mathematical Society27 Ser. 2, 29–34 (1983)Google Scholar
  24. 24.
    Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math.30, 947–977 (1979)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Shuji Saito
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceUniversity of TokyoTokyoJapan

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