no. 5-6
Algebra/Algebraic Geometry
Note on the completion of a local domain with geometrically normal formal fibers

Présenté par : Michel Raynaud

Beddani, Charef1
1 Taibah University, Faculty of Science, Department of Mathematics, Madinah, Saudi Arabia
Comptes Rendus. Mathématique, Tome 351 (2013) no. 5-6, pp. 173-175.

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 Rˆ égale exactement le nombre dʼidéaux maximaux dans la clôture algébrique 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 Rˆ equals exactly the number of maximal prime ideals in the integral closure R¯ of R in its field of quotients.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.03.004
Beddani, Charef 1

1 Taibah University, Faculty of Science, Department of Mathematics, Madinah, Saudi Arabia 
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Beddani, Charef. Note on the completion of a local domain with geometrically normal formal fibers. Comptes Rendus. Mathématique, Tome 351 (2013) no. 5-6, pp. 173-175. doi : 10.1016/j.crma.2013.03.004. http://www.numdam.org/articles/10.1016/j.crma.2013.03.004/

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

[2] Heinzer, W.; Rotthaus, C.; Wiegand, S. 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] Katz, D. 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] Kiyek, K.; Vicente, J.L. Resolution of Curve and Surface Singularities in Characteristic Zero, Kluwer, Dordrecht, 2004

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

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

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