Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-08T09:17:34.711Z Has data issue: false hasContentIssue false
  • English
  • Français

A remark on projective modules

Published online by Cambridge University Press:  17 April 2009

Wojciech Kucharz
Wojciech Kucharz
Affiliation:
Department of Mathematics and StatisticsUniversity of New Mexico Albuquerque, New Mexico 87131, U.S.A.
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 denote the field of real numbers and let A be the ring of regular functions on Rn, that is the localization of R[T1 …, Tn] with respect to the set of all polynomials vanishing nowhere on Rn. Let X be an algebraic subset of Rn and let I(X) be the ideal of A of all functions vanishing on X. Assume that X is compact and nonsingular and k = codim X = 1, 2, 4 or 8. we prove here that if the A/I(X)-module I(X)/I(X)2 can be generated by k elements, then there exist a projective A-module P of rank k and a homomorphism from P onto I(X).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 3 , December 1987 , pp. 515 - 520
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Akbulut, S. and King, H., “The topology of real algebraic sets with isolated singularities,” Ann. of Math. 113 (1981), 425–446.CrossRefGoogle Scholar
[2]Benedetti, R. and Tognoli, A., “On real algebraic vector bundles,” Bull. Sci. Math. (2) 104 (1980), 89–112.Google Scholar
[3]Bochnak, J., Coste, M. and Roy, M.F., Géométrie Algébrique Réelle, (Ergebnisse der Math. Vol. 12, Springer 1987).Google Scholar
[4]Bochnak, J. and Kucharz, W., “Real algebraic surfaces as complete intersections,” Bull. London Math. Soc. 19 (1987), 149–153.CrossRefGoogle Scholar
[5]Hironaka, H., “Resolution of singularities of an algebraic variety over a field of characteristic zero,” Ann. of Math. 79 (1964), 109–326.CrossRefGoogle Scholar
[6]Husemoller, D., Fibre Bundles, (Graduate Texts in Mathematics 20, second edition, Springer 1975), 209–225.Google Scholar
[7]Kucharz, W., “Vector bundles over real algebraic surfaces and three-folds,” Compositio Math 60 (1986), 209–225.Google Scholar
[8]Kucharz, W., “Real algebraic curves as complete intersections,” Math. z., 194 (1987), 259–266.CrossRefGoogle Scholar
[9]Milnor, J., Topology from the Differentiable Viewpoint, (Univ. of Virginia Press, Charlottesville 1966).Google Scholar
[10]Quillen, D., “Projective modules over polynomial rings,” Invent. Math. 36 (1976), 167–171.CrossRefGoogle Scholar
[11]Silhol, R., “Géométrie algébrique sur un corps non a1gébriquement clos,” Comm. Algebra 6 (1978), 1134–1165.CrossRefGoogle Scholar
[12]Suslin, A., “Projective modules over polynomial rings,” (Russian), Dokl. Akad. Nauk. S.S.S.R. 229 (1976), 1063–1066.Google Scholar
[13]Tognoli, A., “Algebraic approximation of manifolds and spaces,” Séminair Bourbaki vol. 1979/80 (Springer Lect. Notes in Math. 842, 73–94, 1981).Google Scholar