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Comptes Rendus Mathématique
Volume 351, n° 5-6
pages 173-175 (mars 2013)
Doi : 10.1016/j.crma.2013.03.004
Received : 7 December 2012 ;  accepted : 8 Mars 2013
Note on the completion of a local domain with geometrically normal formal fibers
 

Charef Beddani
 Taibah University, Faculty of Science, Department of Mathematics, Madinah, Saudi Arabia 

Abstract

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

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Résumé

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

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