Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Stably free modules over $ \mathbf{R}{[X]}$ of rank $ > \dim \mathbf{R}$ are free


Author: Ihsen Yengui
Journal: Math. Comp. 80 (2011), 1093-1098
MSC (2010): Primary 13C10, 19A13, 14Q20, 03F65
DOI: https://doi.org/10.1090/S0025-5718-2010-02427-5
Published electronically: September 27, 2010
MathSciNet review: 2772113
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for any finite-dimensional ring $ \mathbf{R}$ and $ n\geq \dim \mathbf{R} +2$, the group $ {\rm E}_{n}(\R[X])$ acts transitively on $ {\rm Um}_{n}(\mathbf{R}[X])$. In particular, we obtain that for any finite-dimensional ring $ \mathbf{R}$, all finitely generated stably free modules over $ \mathbf{R}[X]$ of rank $ > \dim \mathbf{R}$ are free. This result was only known for Noetherian rings. The proof we give is short, simple, and constructive.


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

  • 1. H. Bass, Libération des modules projectifs sur certains anneaux de polynômes, Sém. Bourbaki 1973/74, exp. 448, Lecture Notes in Math., vol. 431, Springer-Verlag, Berlin and New York, 1975, 228-254. MR 0472826 (57:12516)
  • 2. T. Coquand, H. Lombardi, C. Quitté, Generating non-Noetherian modules constructively, Manuscripta Mathematica 115 (2004) 513-520. MR 2103665 (2005h:13014)
  • 3. T. Coquand, H. Lombardi, M.-F. Roy, An elementary characterization of Krull dimension, From sets and types to analysis and topology: towards practicable foundations for constructive mathematics (L. Crosilla, P. Schuster, eds.), Oxford University Press, 2005. MR 2188647 (2007a:54023)
  • 4. A. Ellouz, H. Lombardi, I. Yengui, A constructive comparison between the rings $ \mathbf{R}(X)$ and $ \mathbf{R} \langle X \rangle$ and application to the Lequain-Simis induction theorem, J. Algebra 320 (2008) 521-533. MR 2422305 (2009e:13022)
  • 5. T.Y. Lam, Serre's conjecture, Lecture Notes in Mathematics, vol. 635, Springer-Verlag, Berlin-New York, 1978. MR 0485842 (58:5644)
  • 6. T.Y. Lam, Serre's Problem on Projective Modules, Springer Monographs in Mathematics, 2006. MR 2235330 (2007b:13014)
  • 7. Y. Lequain, A. Simis, Projective modules over $ \mathbf{R}[X_1,... ,X_n]$, $ \mathbf{R}$ a Prüfer domain, J. Pure Appl. Algebra 18 (2) (1980) 165-171. MR 585221 (82e:13010)
  • 8. H. Lombardi, C. Quitté, Constructions cachées en algèbre abstraite (2) Le principe local-global, dans: Commutative ring theory and applications. Eds: Fontana M., Kabbaj S.-E., Wiegand S. Lecture notes in pure and applied mathematics, vol. 131, M. Dekker, 2002, 461-476. MR 2029844 (2005b:13015)
  • 9. H. Lombardi, C. Quitté, Algèbre commutative, Méthodes Constructives, to appear, available at http://hlombardi.free.fr/publis/A--PTFCours.html. An english version is to be published by Springer.
  • 10. R. Mines, F. Richman, W. Ruitenburg, A Course in Constructive Algebra, Universitext. Springer-Verlag, 1988. MR 919949 (89d:03066)
  • 11. A. Mnif, I. Yengui, An algorithm for unimodular completion over Noetherian rings, J. Algebra 316 (2007) 483-498. MR 2356840 (2008h:13016)
  • 12. D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976) 167-171. MR 0427303 (55:337)
  • 13. R.A. Rao, The Bass-Quillen conjecture in dimension three but characteristic $ \neq 2, 3$ via a question of A. Suslin, Invent. Math. 93 (1988) 609-618. MR 952284 (89d:13011)
  • 14. M. Roitman, On stably extended projective modules over polynomial rings, Proc. Amer. Math. Soc. 97 (1986) 585-589. MR 845969 (87f:13007)
  • 15. J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955) 191-278. MR 0068874 (16:953c)
  • 16. J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Sém. Dubreil-Pisot, no. 23, Paris (1957/58). MR 0177011 (31:1277)
  • 17. A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR-Izv. 11 (1977) 221-238. MR 0472792 (57:12482)
  • 18. I. Yengui, Making the use of maximal ideals constructive, Theoretical Computer Science 392 (2008) 174-178. MR 2394992 (2009a:13012)
  • 19. I. Yengui, The Hermite ring conjecture in dimension one, J. Algebra 320 (2008) 437-441. MR 2417998 (2009g:16019)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 13C10, 19A13, 14Q20, 03F65

Retrieve articles in all journals with MSC (2010): 13C10, 19A13, 14Q20, 03F65


Additional Information

Ihsen Yengui
Affiliation: Department of Mathematics, Faculty of Sciences of Sfax, 3000 Sfax, Tunisia
Email: ihsen.yengui@fss.rnu.tn

DOI: https://doi.org/10.1090/S0025-5718-2010-02427-5
Keywords: Stably free modules, unimodular vectors, Quillen-Suslin theorem, Hermite rings, Hermite ring conjecture, constructive mathematics
Received by editor(s): June 13, 2009
Published electronically: September 27, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society