Let $(R,\mathfrak m)$ be a Noetherian local ring.

Definition: $I$ is called locally complete intersection ideal if $I_p$ is a complete intersection for all $p\in V(I)$.

I want an example of an ideal $I$ satisfying the following three properties:

1) grade$(I)\geq 1$,

2) $I$ locally complete intersection ideal but not an $\mathfrak m$-primary and complete intersection ideal,

3) $I$ is not integrally closed.

Any suggestion or reference will be extremely helpful. Thank you in advance.

  • $\begingroup$ What is a locally complete intersection and non-complete intersection ideal of a local ring? $\endgroup$
    – A.G
    Commented Feb 9, 2017 at 17:19
  • $\begingroup$ An ideal I is called a locally complete intersection if its localization at every prime ideal containing I is complete intersection. $\endgroup$ Commented Feb 11, 2017 at 4:28
  • $\begingroup$ An ideal I is called a complete intersection if it is generated by regular sequence. $\endgroup$ Commented Feb 11, 2017 at 4:28
  • $\begingroup$ In a noetherian local ring both definitions are obviously the same, so you cannot find an ideal satisfying one and not the other. $\endgroup$
    – A.G
    Commented Feb 11, 2017 at 22:27
  • $\begingroup$ Do you know how to construct a variety from the embedding of $\mathbb{P}^1$ in $\mathbb{P}^3$ using $\mathcal{O}(3)$? $\endgroup$
    – 54321user
    Commented Jun 9, 2017 at 20:24

1 Answer 1


Let ,$R=\mathbb{Z}/3\mathbb{Z}[x,y,z,w]/(x^4+y^3+z^4)$. Then, $y\in (x,z)^F\subseteq (x,z)^*\subseteq \overline{(x,z)}$, and thence $(x,z)$ is not integrally closed, however $x,z$ is a regular sequence because $R$ is a complete intersection and $x,z,w$ is a system of parameters for $R$.

  • $\begingroup$ I am mainly interested in the case where the ideal itself is not complete intersection. I have examples when the ideal itself is complete intersection. Sorry for not mentioning that in the question. $\endgroup$ Commented Feb 9, 2017 at 16:01

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