Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

A new proof of D. Popescu's theorem
on smoothing of ring homomorphisms


Author: Mark Spivakovsky
Journal: J. Amer. Math. Soc. 12 (1999), 381-444
MSC (1991): Primary 13B40, 13C10, 14B05, 14B12, 14E40
MathSciNet review: 1647069
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a new proof of D. Popescu's theorem which says that if $\sigma :A\rightarrow B$ is a regular homomorphism of noetherian rings, then $B$ is a filtered inductive limit of smooth finite type $A$-algebras. We strengthen Popescu's theorem in two ways. First, we show that a finite type $A$-algebra $C$, mapping to $B$, has a desingularization $C\rightarrow D$ which is smooth wherever possible (roughly speaking, above the smooth locus of $C$). Secondly, we give sufficient conditions for $B$ to be a filtered inductive limit of its smooth finite type $A$-subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.


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  • 1. M. André, Cinq exposés sur la désingularization, handwritten manuscript, École Polytechnique Fédérale de Lausanne, 1992.
  • 2. Michel André, Homologie des algèbres commutatives, Springer-Verlag, Berlin-New York, 1974 (French). Die Grundlehren der mathematischen Wissenschaften, Band 206. MR 0352220
  • 3. M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 0268188
  • 4. M. Artin, Algebraic structure of power series rings, Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981) Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 223–227. MR 685955
  • 5. M. Artin and J. Denef, Smoothing of a ring homomorphism along a section, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser, Boston, Mass., 1983, pp. 5–31. MR 717604
  • 6. Michael Artin and Christel Rotthaus, A structure theorem for power series rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 35–44. MR 977751
  • 7. Renée Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553–603 (1974) (French). MR 0345966
  • 8. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 0173675
  • 9. Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
  • 10. Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
  • 11. Masayoshi Nagata, Local rings, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Corrected reprint. MR 0460307
  • 12. Vasile Nica and Dorin Popescu, A structure theorem on formally smooth morphisms in positive characteristic, J. Algebra 100 (1986), no. 2, 436–455. MR 840587, 10.1016/0021-8693(86)90087-6
  • 13. Tetsushi Ogoma, General Néron desingularization based on the idea of Popescu, J. Algebra 167 (1994), no. 1, 57–84. MR 1282816, 10.1006/jabr.1994.1175
  • 14. Dorin Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97–126. MR 818160
  • 15. Dorin Popescu, Letter to the editor: “General Néron desingularization and approximation” [Nagoya Math. J. 104 (1986), 85–115; MR0868439 (88a:14007)], Nagoya Math. J. 118 (1990), 45–53. MR 1060701
  • 16. Dorin Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85–115. MR 868439
  • 17. Christel Rotthaus, On the approximation property of excellent rings, Invent. Math. 88 (1987), no. 1, 39–63. MR 877005, 10.1007/BF01405090
  • 18. R. Swan, Néron-Popescu desingularization, to appear, Proceedings of the International Conference on Algebra and Geometry, Taipei, Taiwan (1995).
  • 19. Bernard Teissier, Résultats récents sur l’approximation des morphismes en algèbre commutative (d’après André, Artin, Popescu et Spivakovsky), Astérisque 227 (1995), Exp. No. 784, 4, 259–282 (French, with French summary). Séminaire Bourbaki, Vol. 1993/94. MR 1321650

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Additional Information

Mark Spivakovsky
Affiliation: Department of Mathematics, University of Toronto, Erindale College, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6
Email: spiva@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00294-5
Keywords: Smooth homomorphism, N\'{e}ron desingularization, Artin approx\-i\-ma\-tion
Received by editor(s): May 8, 1992
Received by editor(s) in revised form: July 24, 1998
Additional Notes: Research supported by the Harvard Society of Fellows, NSF, NSERC and the Connaught Fund
Dedicated: Dedicated to Professor H. Hironaka on the occasion of his sixtieth birthday
Article copyright: © Copyright 1999 American Mathematical Society