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The approximation property for some $ 5$-dimensional Henselian rings


Authors: Joseph Becker, J. Denef and L. Lipshitz
Journal: Trans. Amer. Math. Soc. 276 (1983), 301-309
MSC: Primary 13J15; Secondary 13D10, 14B12, 14D15
DOI: https://doi.org/10.1090/S0002-9947-1983-0684510-2
MathSciNet review: 684510
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Abstract: Let $ k$ be a field of characteristic 0, $ k[[{X_1},{X_2}]]$ the ring of formal power series and $ R = k[[{X_1},{X_2}]]{[{X_3},{X_4},{X_5}]^ \sim}$ the algebraic closure of $ k[[{X_1},{X_2}]][{X_3},{X_4},{X_5}]$ in $ k[[{X_1},\ldots,{X_5}]]$. It is shown that $ R$ has the Approximation Property.


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  • [A] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291. MR 0232018 (38:344)
  • [A1] -, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23-58. MR 0268188 (42:3087)
  • [A2] -, Construction techniques for algebraic spaces, Actes Congrès Internat. Math. 1 (1970), 419-423. MR 0427316 (55:350)
  • [A3] -, Lectures on deformations of singularities, Tata Institute Notes 54, Bombay, 1976.
  • [B] J. Becker, A counterexample to Artin approximation with respect to subrings, Math. Ann. 230 (1977), 195-196. MR 0480508 (58:669)
  • [BDLV] J. Becker, J. Denef, L. Lipshitz and L. van den Dries, Ultraproducts and approximation in local rings. I, Invent. Math. 51 (1979), 189-203. MR 528023 (80k:14009)
  • [CK] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
  • [DL] J. Denef and L. Lipshitz, Ultraproducts and approximation in local rings. II, Math. Ann. 253 (1980), 1-28. MR 594530 (82g:13021)
  • [E] R. Elkik, Solutions d'équations a coefficients dans un anneau Hensélien, Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 553-604. MR 0345966 (49:10692)
  • [EGA] A. Grothendieck, Eléments de géométrie algébrique. IV, Inst. Hautes Études Sci. Publ. Math. 24 (1965); 32 (1967).
  • [G] M. Greenberg, Rational points in Henselian discrete valuation rings, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 59-64. MR 0207700 (34:7515)
  • [Ga] A. M. Gabriélov, Formal relations between analytic functions, Funkcional. Anal. i Priložen. 5 (1971), 64-65 = Functional Anal. Appl. 5 (1971), 318-319. MR 0302930 (46:2073)
  • [N] A. Néron, Modèles minimaux des varietés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964). MR 0179172 (31:3423)
  • [P] D. Popescu, A remark on two dimensional local rings with the property of approximation, Mat. Z. 173 (1980), 235-240. MR 592372 (82b:13012)
  • [PP] G. Pfister and D. Popescu, On three dimensional local rings with the property of approximation, Rev. Roumaine Math. Pures Appl. 26 (1981), 301-307. MR 616044 (82i:13022)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra, Vol. II, Springer-Verlag, Berlin and New York, 1960. MR 0120249 (22:11006)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0684510-2
Article copyright: © Copyright 1983 American Mathematical Society

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