Kyoto Journal of Mathematics

A few examples of local rings, I

Jun-ichi Nishimura

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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

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

First available in Project Euclid: 19 February 2012

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Zentralblatt MATH identifier

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


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

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