A few examples of local rings, I

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 $xA=\mathfrak{p}\in Spec\left(A\right)$ such that $A/\mathfrak{p}$ 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.