Kyoto Journal of Mathematics

A few examples of local rings, I

Jun-ichi Nishimura

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Abstract

In this paper, we first recall and apply the fundamental techniques of constructing bad Noetherian local domains, due to C. Rotthaus, T. Ogoma, R. C. Heitmann, and M. Brodmann and C. Rotthaus, to show several basic examples:

(1) a three-dimensional Nagata normal local domain, which is a complete intersection, whose regular locus is not open;

(2) a three-dimensional Henselian Nagata normal local domain, which is not catenary.

Next we present a unified version of Brodmann and Rotthaus’s and Ogoma’s methods in order to obtain a particular local domain A with a specified prime element x such that the local domain A/xA is the bad Noetherian local domain given above:

(3) a three-dimensional unmixed local domain A that has xA=pSpec(A) such that A/p is not unmixed.

Finally we follow Ogoma’s construction of factorial local domains whose completions are designated complete local domains. Then, we gather some examples of bad factorial local domains.

Article information

Source
Kyoto J. Math., Volume 52, Number 1 (2012), 51-87.

Dates
First available in Project Euclid: 19 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1329684742

Digital Object Identifier
doi:10.1215/21562261-1503754

Mathematical Reviews number (MathSciNet)
MR2892767

Zentralblatt MATH identifier
1241.13016

Subjects
Primary: 13E05: Noetherian rings and modules 13H05: Regular local rings 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]
Secondary: 13C15: Dimension theory, depth, related rings (catenary, etc.) 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13G05: Integral domains

Citation

Nishimura, Jun-ichi. A few examples of local rings, I. Kyoto J. Math. 52 (2012), no. 1, 51--87. doi:10.1215/21562261-1503754. https://projecteuclid.org/euclid.kjm/1329684742


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