Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-21T23:34:22.087Z Has data issue: false hasContentIssue false
  • English
  • Français

General Néron desingularization and approximation

Published online by Cambridge University Press:  22 January 2016

Dorin Popescu
Dorin Popescu*
Affiliation:
INCREST Department of Mathematics, Bd. Păcii 220, 79622 Bucharest, ROMANIA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be a noetherian ring (all the rings are supposed here to be commutative with identity), a ⊂ A a proper ideal and  the completion of A in the α-adic topology. We consider the following conditions

(WAP) Every finite system of polynomial equations over A has a solution in A iff it has one in Â.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 104 , December 1986 , pp. 85 - 115
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[Ani] André, M., Homologie des algebres commutatives, Springer-Verlag, Berlin 1974.CrossRefGoogle Scholar
[An2] André, M., Localisation de la lissité formelle, Manuscripta Math., 13 (1974), 297–307.CrossRefGoogle Scholar
[An3] André, M., Artin’s theorem on the solution of analytic equations in positive characteristic, Manuscripta Math., 15, (1975), 314–348.Google Scholar
[Ai] Artin, M., On the solution of analytic equations, Invent, math., 5 (1968), 277–291.Google Scholar
[A2] Artin, M., Algebraic approximation of structures over complete local rings, Publ. Math. IHES, 36 (1969), 23–58.Google Scholar
[A3] Artin, M., Construction techniques for algebraic spaces, Actes Congrès Intern. Math., t. 1 (1970), 419–423.Google Scholar
[AD] Artin, M. and Denef, J., Smoothing of a ring homomorphism along a section, Arithmetic and Geometry, vol. II, Birkhäuser, Boston (1983), 5–32.Google Scholar
[BNP] Basarab, S., Nica, V. and Popescu, D., Approximation properties and existential completeness for ring morphisms, Manuscripta Math., 33 (1981), 227–282.Google Scholar
[BDLV] Becker, J., Denef, J., Lipshitz, L. and van den Dries, L., Ultraproducts and approximation in local rings I, Invent. Math., 51 (1979), 189–203.Google Scholar
[BDL] Becker, J., Denef, J. and Lipshitz, L., The approximation property for some 5-dimensional henselian rings, Trans. Amer. Math. Soc, 276, no. 1 (1983), 301–309.CrossRefGoogle Scholar
[BR] Brezuleanu, A. and Radu, N., Sur la localisation de la lissité formelle, C.R. Acad. Sci. Paris, Sér. A, t. 276 (1973), 439–441.Google Scholar
[B] Brown, M. L., Artin’s approximation property, Thesis, Cambridge Peterhouse, August 1979.Google Scholar
[CP] Cipu, M. and Popescu, D., Some extensions of Néron’s p-desingularization and approximation, Rev. Roumaine Math. Pures Appl., t. XXIV, no. 10 (1981), 1299–1304.Google Scholar
[DL] Denef, J. and Lipshitz, L., Ultraproducts and approximation in local rings II, Math. Ann., 253 (1980), 1–28.Google Scholar
[E] Elkik, R., Solutions d’équations á coefficients dans un anneaux hensélien, Ann. Sci. École Norm. Sup. 4° serie, t. 6 (1973), 533–604.Google Scholar
[Ga] Gabrielov, A. M., The formal relations between analytic functions (Russian), Funktional Anal, i Prilozen., 5 (1971), no. 4, 64–65.Google Scholar
[Gr] Greenberg, M., Rational points in henselian discrete valuation rings, Publ. Math. IHES, 31 (1966), 59–64.Google Scholar
[EGA] Grothendieck, A. and Dieudonné, J., Eléments de géométrie algébrique, IV, Parts 1 and 4, Publ. Math. IHES, 1964.Google Scholar
[G] Grothendieck, A., Technique de descente II, Sém. Bourbaki, 195 (1959–1960).Google Scholar
[J] Jähner, U., Der Satz, von M. Artin über die Lösungen analytischer Gleichungen mit Koeffizienten in einem Körper beliebiger Charakteristik, Arch. Math., XXIX (1977), 485–490.Google Scholar
[KPR] Kurke, H., Pfister, G., and Roczen, M., Henselsche Ringe and algebraische Geometrie, VEB, Berlin, 1975.Google Scholar
[KMPPR] Kurke, H., Mostowski, T., Pfister, G., Popescu, D., and Roczen, M., Die Approximationseigenschaft lokaler Ringe, Lect. Notes in Math., 634, Springer-Verlag, Berlin, 1978.Google Scholar
[M] Matsumura, H., Commutative algebra, Benjamin, New York, 1980.Google Scholar
[PP1] Pfister, G. and Popescu, D., Die strenge Approximationseigenschaft lokaler Ringe, Invent. Math., 30 (1975), 145–174.Google Scholar
[PP2] Pfister, G. and Popescu, D., On three dimensional local rings with the property of approximation, Rev. Roumaine Math. Pures Appl. t. XXVI, no. 2 (1981), 301–307.Google Scholar
[P1] Popescu, D., Algebraically pure morphisms, Rev. Roumaine Math. Pures Appl., t. XXIV, no. 6 (1979), 947–977.Google Scholar
[P2] Popescu, D., A remark on two dimensional local rings with the property of approximation, Math. Z., 173 (1980), 235–240.Google Scholar
[P3] Popescu, D., Global form of Néron’s p-desingularization and approximation, Proceedings “Week of Algebraic Geometry” Bucharest, June 30-July 6, 1980, Teubner Texte Band 40, Leipzig, 1981.Google Scholar
[P4] Popescu, D., General Néron desingularization, Nagoya Math. J., 100 (1985), 97–126.Google Scholar
[P5] Popescu, D., On Zariski’s Uniformization Theorem, Algebraic Geometry, Bucharest 1982, Proceedings, Lect. Notes in Math., 1056, Springer-Verlag, Berlin, 1984.Google Scholar
[R] Radu, N., Sur une critère de lissité formelle, Bull. Math. Soc. Sci. Math. R.S. Roumaine t. 21, 68, no. 1–2 (1977), 133–135.Google Scholar
[Ra] Raynaud, M., Anneaux locaux henséliens, Lect. Notes in Math., 169, Springer-Verlag, Berlin, 1970.Google Scholar
[Ro] Rotthaus, C., Potenzreihenerweiterung und formale Fasern in lokalen Ringen mit Approximationseigenschaft., Manuscripta Math., 42 (1983), 53–65.CrossRefGoogle Scholar
[S] Schemmel, K. P., Eine notwendige und hinreichende Bedingung fur die Approximationseigenschaft analytischer Potenzreihenringe über einem Körper beliebiger characteristik, Rev. Roumaine Math. Pures Appl., t. XXVII, no. 8 (1982), 875–884.Google Scholar
[ZS] Zariski, O. and Samuel, P., Commutative Algebra II, D. van Nostrand Company Inc., New York 1960.CrossRefGoogle Scholar