Note on the completion of a local domain with geometrically normal formal fibers

Charef Beddani

doi:10.1016/j.crma.2013.03.004

Algebra/Algebraic Geometry

Note on the completion of a local domain with geometrically normal formal fibers

Présenté par : Michel Raynaud

Charef Beddani ¹

¹ Taibah University, Faculty of Science, Department of Mathematics, Madinah, Saudi Arabia

Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 173-175.

Résumé
Abstract

Dans cette note, nous allons montrer que, si $(R, m)$ est un anneau local intègre tel que ses fibres formelles soient géométriquement normales, alors le nombre des idéaux premiers minimaux dans la complétion $m$ -adique $\hat{R}$ égale exactement le nombre dʼidéaux maximaux dans la clôture algébrique $\bar{R}$ de R dans son corps de fractions.

In this note we will prove that if $(R, m)$ is a local domain such that its formal fibers are geometrically normal, then the number of minimal prime ideals in the $m$ -adic completion $\hat{R}$ equals exactly the number of maximal prime ideals in the integral closure $\bar{R}$ of R in its field of quotients.

Reçu le : 2012-12-07
Accepté le : 2013-03-08
Publié le : 2013-03-01

DOI : 10.1016/j.crma.2013.03.004

Affiliations des auteurs :

Charef Beddani ¹

¹ Taibah University, Faculty of Science, Department of Mathematics, Madinah, Saudi Arabia

@article{CRMATH_2013__351_5-6_173_0,
     author = {Charef Beddani},
     title = {Note on the completion of a local domain with geometrically normal formal fibers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {173--175},
     publisher = {Elsevier},
     volume = {351},
     number = {5-6},
     year = {2013},
     doi = {10.1016/j.crma.2013.03.004},
     language = {en},
}

TY  - JOUR
AU  - Charef Beddani
TI  - Note on the completion of a local domain with geometrically normal formal fibers
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 173
EP  - 175
VL  - 351
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2013.03.004
LA  - en
ID  - CRMATH_2013__351_5-6_173_0
ER  -

%0 Journal Article
%A Charef Beddani
%T Note on the completion of a local domain with geometrically normal formal fibers
%J Comptes Rendus. Mathématique
%D 2013
%P 173-175
%V 351
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2013.03.004
%G en
%F CRMATH_2013__351_5-6_173_0

Charef Beddani. Note on the completion of a local domain with geometrically normal formal fibers. Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 173-175. doi : 10.1016/j.crma.2013.03.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.03.004/

Bibliographie
Cité par

[1] J. Dieudonné, M. Borelli, Topics in local algebra: Lectures delivered at University of Notre-Dame, 1967.

[2] W. Heinzer; C. Rotthaus; S. Wiegand Catenary local rings with geometrically normal formal fibers, Algebra, Arithmetic and Geometry with Applications, Springer-Verlag, Berlin, Heidelberg, New York, 2004, pp. 497-510 (papers from Shreeram S. Abhyankarʼs 70th Birthday Conference)

[3] D. Katz On the number of minimal prime ideals in the completion of a local ring, Rocky Mt. J. Math., Volume 16 (1986), pp. 575-578

[4] K. Kiyek; J.L. Vicente Resolution of Curve and Surface Singularities in Characteristic Zero, Kluwer, Dordrecht, 2004

[5] H. Matsumura Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, 1986

[6] M. Nagata Local Rings, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, 1962

Cité par Sources :

Commentaires - Politique