Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-19T12:01:05.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomial rings and their projective modules

Published online by Cambridge University Press:  22 January 2016

Dorin Popescu
Dorin Popescu*
Affiliation:
Mathematics Department INCREST, Bucharest 79622, 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 R be a regular noetherian ring. A central question concerning projective modules over polynomial R-algebras is the following.

(1.1) BASS-QUILLEN CONJECTURE ([2] Problem IX, [10]). Every finitely generated projective module P over a polynomial R-algebra R[T], T = (T1,…, Tn) is extended from R, i.e.

P≊R[T]⊗R P/(T)P.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 121 - 128
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Artin, M. and Denef, J., Smoothing of a ring homomorphism along a section, in: Arithmetic and Geometry II, Birkhäuser, Boston, 1983, 5–32.Google Scholar
[2] Bass, H., Some problems in “classical” algebraic k-theory, in: Algebraic K-theory II, Lecture Notes in Math., Vol. 342, Berlin, 1973, 1–70.Google Scholar
[3] Bhatwadekar, S. M. and Rao, R. A., On a question of Quillen, Trans. Amer. Math. Soc., 279 (1983), 801–810.CrossRefGoogle Scholar
[4] Grothendieck, A. and Dieudonné, J., Eléments de géometrie algébrique IV, Part 1, Publ. Math. IHES, Paris, 1964.Google Scholar
[5] Lindel, H., On the Bass—Quillen Conjecture concerning projective modules over polynomial rings, Invent. Math., 65 (1981), 319–323.CrossRefGoogle Scholar
[6] Matsumura, H., Commutative algebra, Benjamin, New York, 1980.Google Scholar
[7] Murthy, M. P., A letter containing Swan’s notes on Lindel’s results.Google Scholar
[8] Popescu, D., General Néron desingularization and approximation, Nagoya Math. J., 104 (1986), 85–115.Google Scholar
[9] Popescu, D., On a question of Quillen, to appear.Google Scholar
[10] Quillen, D., Projective modules over polynomial rings, Invent. Math., 36 (1976), 167–171.CrossRefGoogle Scholar
[11] Raynaud, M., Anneaux locaux henséliens, Lecture Notes in Math., Vol. 169, Berlin, 1970.Google Scholar
[12] Roitman, M., On projective modules over polynomial rings, J. Algebra, 58 (1979), 51–63.CrossRefGoogle Scholar
[13] Suslin, A. A., Projective modules over a polynomial ring are free, Soviet. Math. Dokl., 17 (1976), 1160–1164.Google Scholar