Noethérianité de certaines algèbres de fonctions analytiques et applications

Abdelhafed Elkhadiri; Mouttaki Hlal

Annales Polonici Mathematici (2000)

  • Volume: 75, Issue: 3, page 247-256
  • ISSN: 0066-2216

Abstract

top
Let M ⊂ ℝ n be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider S K = f ∈ H ( M ) | f ( x ) ≠ 0 f o r a l l x ∈ K ; S K is a multiplicative subset of H ( M ) . Let S K - 1 H ( M ) be the localization of H(M) with respect to S K . In this paper we prove, first, that S K - 1 H ( M ) is a regular ring (hence noetherian) and use this result in two situations:    1) For each open subset Ω ⊂ ℝ n , we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, f x , is algebraic on H ( ℝ n ) . We prove that if Ω is a bounded subanalytic subset, then O(Ω) is a regular ring (hence noetherian).    2) Let M ⊂ ℝ n be a Nash submanifold and N(M) the ring of Nash functions on M; we have an injection N(M) → H(M). In [2] it was proved that every prime ideal p of N(M) generates a prime ideal of analytic functions pH(M) if M or V(p) is compact. We use our Theorem 1 to give another proof in the situation where V(p) is compact. Finally we show that this result holds in some particular situation where M and V(p) are not assumed to be compact.

How to cite

top

Elkhadiri, Abdelhafed, and Hlal, Mouttaki. "Noethérianité de certaines algèbres de fonctions analytiques et applications." Annales Polonici Mathematici 75.3 (2000): 247-256. <http://eudml.org/doc/208398>.

@article{Elkhadiri2000,
author = {Elkhadiri, Abdelhafed, Hlal, Mouttaki},
journal = {Annales Polonici Mathematici},
keywords = {Nash functions; regular rings; analytic algebra; subanalytic sets},
language = {fre},
number = {3},
pages = {247-256},
title = {Noethérianité de certaines algèbres de fonctions analytiques et applications},
url = {http://eudml.org/doc/208398},
volume = {75},
year = {2000},
}

TY - JOUR
AU - Elkhadiri, Abdelhafed
AU - Hlal, Mouttaki
TI - Noethérianité de certaines algèbres de fonctions analytiques et applications
JO - Annales Polonici Mathematici
PY - 2000
VL - 75
IS - 3
SP - 247
EP - 256
LA - fre
KW - Nash functions; regular rings; analytic algebra; subanalytic sets
UR - http://eudml.org/doc/208398
ER -

References

top
  1. [1] J. Bochnak, M. Coste et M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. 12, Springer, New York, 1987. Zbl0633.14016
  2. [2] M. Coste, J. M. Ruiz and M. Shiota, Approximation in compact Nash manifolds, Amer. J. Math. 117 (1995), 905-927. Zbl0873.32007
  3. [3] A. Elkhadiri et J.-Cl. Tougeron, Familles noethériennes de modules sur k ̲ [ [ x ] ] et applications, Bull. Sci. Math. 120 (1996), 253-292. 
  4. [4] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. 
  5. [5] D. Popescu, General Neron desingularization, Nagoya Math. J. 100 (1985), 97-126. Zbl0561.14008

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.