Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-26T15:06:15.013Z Has data issue: false hasContentIssue false
  • English
  • Français

On a property of ideals of differentiable functions

Published online by Cambridge University Press:  17 April 2009

Oscar P. Bruno
Oscar P. Bruno
Affiliation:
Cuidad Universitaria, Pab. no. 1, Depto de Mathematics, 1428 – BUENOS AIRES, Republica Argentina.
Rights & Permissions [Opens in a new window]

Abstract

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 J ⊆ C∞ (Rn) be any ideal. Since a function of the variables = (t1,…,tn) is a function of the variables which does not depend on , we have J ⊆ C∞ (IRn+P). Of course, J is not an ideal of C∞ (IRn+P), but it generates an ideal that we call . Consider the following statement (1) on J: “Given any if and only if for every fixed .

In this paper we show that statement (1) holds for a large class of finitely generated ideals although not for all of them. We say that ideals satisfying statement (1) have line determined extensions. We characterize these ideals to be closed ideals J() (in the sense of Whitney) such that for all p ∈ ℕ, the ideal is also closed. Finally, some non-trivial examples are developed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 2 , April 1986 , pp. 293 - 305
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Dubuc, E., “C∞-Schemes”, Amer. J. Math. 103 (1981) 683–690.CrossRefGoogle Scholar
[2]Malgrange, B., Ideals of Differentiable Functions, (OUP, 1966).Google Scholar
[3]Reyes, G. E. and Van Quê, N., “Smooth functors and synthetic calculus”, The L.E.J. Brouwer Centenary Symposium, Troelstra, A. S. and Van Dalen, D. (eds.), (North Holland 1982), 377–395.Google Scholar
[4]Weil, A., “Theorie des points proches sur les variétés différetiables”, Géométrie différentielle. (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourt, 1953. Centre National de la Recherche Scientifique, Paris, 1953), 111–117.Google Scholar
[5]Whitney, H., “On ideals of differentiable functions”, Amer. J. Math. 70 (1948), 635–658.CrossRefGoogle Scholar