The approximation property for some $5$-dimensional Henselian rings
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- by Joseph Becker, J. Denef and L. Lipshitz PDF
- Trans. Amer. Math. Soc. 276 (1983), 301-309 Request permission
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.References
- M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 232018, DOI 10.1007/BF01389777
- M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188
- M. Artin, Construction techniques for algebraic spaces, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 419–423. MR 0427316 —, Lectures on deformations of singularities, Tata Institute Notes 54, Bombay, 1976.
- Joseph Becker, A counterexample to Artin approximation with respect to subrings, Math. Ann. 230 (1977), no. 2, 195–196. MR 480508, DOI 10.1007/BF01370664
- Joseph Becker, J. Denef, L. Lipshitz, and L. van den Dries, Ultraproducts and approximations in local rings. I, Invent. Math. 51 (1979), no. 2, 189–203. MR 528023, DOI 10.1007/BF01390228 C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
- J. Denef and L. Lipshitz, Ultraproducts and approximation in local rings. II, Math. Ann. 253 (1980), no. 1, 1–28. MR 594530, DOI 10.1007/BF01457817
- Renée Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553–603 (1974) (French). MR 345966 A. Grothendieck, Eléments de géométrie algébrique. IV, Inst. Hautes Études Sci. Publ. Math. 24 (1965); 32 (1967).
- Marvin J. Greenberg, Rational points in Henselian discrete valuation rings, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 59–64. MR 207700
- A. M. Gabrièlov, The formal relations between analytic functions, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 64–65 (Russian). MR 0302930
- André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI 10.1007/bf02684271
- Dorin Popescu, A remark on two-dimensional local rings with the property of approximation, Math. Z. 173 (1980), no. 3, 235–240. MR 592372, DOI 10.1007/BF01159662
- Gerhard Pfister and Dorin Popescu, On three-dimensional local rings with the property of approximation, Rev. Roumaine Math. Pures Appl. 26 (1981), no. 2, 301–307. MR 616044
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249
Additional Information
- © Copyright 1983 American Mathematical Society
- 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