On the completeness of factor rings

Authors:
S. Loepp and C. Rotthaus

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2189-2195

MSC (2000):
Primary 13J05, 13J10

DOI:
https://doi.org/10.1090/S0002-9939-02-06334-7

Published electronically:
January 17, 2002

MathSciNet review:
1896398

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a complete local domain containing the integers with maximal ideal such that is at least the cardinality of the real numbers. Let be a nonmaximal prime ideal of such that is a regular local ring. We construct an excellent local ring such that the completion of is , the generic formal fiber of is local with maximal ideal and if is a nonzero ideal of , then is complete.

**1.**S. Abhyankar, W. Heinzer, and S. Wiegand,*On the Compositum of Two Power Series Rings,*Proc. Amer. Math. Soc.**112**(1991), 629-636.MR**91j:13015****2.**A. Grothendieck and J. Dieudonné,*Éléments de Géométrie Algébrique IV, Partie 2*, Publ. Math. I.H.E.S.**24**(1965).MR**33:7330****3.**W. Heinzer, C. Rotthaus, and J. D. Sally,*Formal fibers and birational extensions,*Nagoya Math. J.,**131**(1993), 1-38. MR**95a:13008****4.**R. Heitmann,*Characterization of completions of unique factorization domains,*Trans. Amer. Math. Soc,**337**(1993), 379-387. MR**93g:13006****5.**R. Heitmann,*Completions of Local Rings with an Isolated Singularity,*J. Algebra**163**(1994), 538-567. MR**95f:13032****6.**S. Loepp,*Excellent Rings with Local Generic Formal Fibers,*J. Algebra**201**(1998), 573-583. MR**99a:13013****7.**H. Matsumura,*Commutative Ring Theory,*Cambridge University Press, 1986.MR**88h:13001****8.**H. Matsumura,*On the dimension of formal fibres of a local ring,*Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (Kinokuniya, Tokyo, 1987), 261-266.MR**90a:13027****9.**C. Rotthaus,*On rings with low dimensional formal fibres,*J. Pure and Applied Algebra,**71**(1991), 287-296.MR**92j:13006**

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Additional Information

**S. Loepp**

Affiliation:
Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267

Email:
sloepp@williams.edu

**C. Rotthaus**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Email:
rotthaus@math.msu.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06334-7

Keywords:
Local rings,
completions,
factor rings,
excellent rings

Received by editor(s):
June 28, 2000

Received by editor(s) in revised form:
February 22, 2001

Published electronically:
January 17, 2002

Additional Notes:
The first author appreciates the hospitality of Michigan State University, where this project was conducted, and is grateful for the support of the National Science Foundation via DMS #9973069

The second author thanks the National Science Foundation for their support via DMS #980122

Communicated by:
Wolmer V. Vasconselos

Article copyright:
© Copyright 2002
American Mathematical Society