Remote Access Journal of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0894-0347-99-00294-5
MathSciNet review: 1647069
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • 1. M. André, Cinq exposés sur la désingularization, handwritten manuscript, École Polytechnique Fédérale de Lausanne, 1992.
  • 2. M. André, Homologie des Algèbres Commutatives, Springer-Verlag, Berlin Heidelberg New York, 1974. MR 50:4707
  • 3. M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. IHES 36 (1969), 23-58. MR 42:3087
  • 4. M. Artin, Algebraic structure of power series rings, Contemp. Math. 13 (1982), 223-227. MR 84b:13014
  • 5. M. Artin and J. Denef, Smoothing a ring homomorphism along a section, Arithmetic and Geometry, Vol II, Birkhäuser, Boston, 1983, pp. 5-29. MR 84m:14007
  • 6. M. Artin and C. Rotthaus, A structure theorem for power series rings, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (1987), 35-44. MR 90b:14006
  • 7. R. Elkik, Solutions d'équations à coefficients dans un anneau hensélién, Ann. Sci. Éc. Norm. Super. $4^{e}$ sér. 6 (1973), 533-604. MR 49:10692
  • 8. A. Grothendieck and J. Dieudonne, Élements de géometrie algébrique, Publ. IHES 20, 1964. MR 30:3885
  • 9. S. Lang, Algebra, Addison-Wesley, Reading, Massachusetts, 1967. MR 33:5416
  • 10. H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. MR 42:1813
  • 11. M. Nagata, Local Rings, Krieger Publishing Co., Huntington, N.Y., 1975. MR 57:301
  • 12. V. Nica and D. Popescu, A structure theorem on formally smooth morphisms in positive characteristic, J. of Algebra 100 (1986), 436-455. MR 87j:14007
  • 13. T. Ogoma, General Néron desingularization based on the idea of Popescu, J. of Algebra 167 (1994), 57-84. MR 96a:13008
  • 14. D. Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97-126. MR 87f:13019
  • 15. D. Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85-115. MR 91e:14003
  • 16. D. Popescu, General Néron desingularization and approximation, Nagoya Math. J. 104 (1986), 85-115. MR 88a:14007
  • 17. C. Rotthaus, On the approximation property of excellent rings, Invent. Math. 88 (1987), 39-63. MR 88c:14005
  • 18. R. Swan, Néron-Popescu desingularization, to appear, Proceedings of the International Conference on Algebra and Geometry, Taipei, Taiwan (1995).
  • 19. B. Teissier, Résultats récents sur l'approximation des morphismes en algèbre commutative, [d'après Artin, Popescu, André et Spivakovsky], Séminaire Bourbaki 784 (1994), 1-15. MR 96c:13023

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 13B40, 13C10, 14B05, 14B12, 14E40

Retrieve articles in all journals with MSC (1991): 13B40, 13C10, 14B05, 14B12, 14E40


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: https://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

American Mathematical Society