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Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.

Q) Are there examples of ideals $J$ such that $l(J_p)\leq l(J)-1$ for all $p\supset J$ such that $ht p=ht J+1$ and $J^n\neq J^{(n)}$ for some $n$ (where $J^{(n)}$ is the nth symbolic power)?

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    $\begingroup$ For $p=m$, clearly $l(J_p)=l(J)$. $\endgroup$
    – Mohan
    Apr 10, 2017 at 17:34
  • $\begingroup$ @Mohan Thanks for pointing out the mistake. I have edited the question. $\endgroup$
    – Cusp
    Apr 10, 2017 at 18:42
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    $\begingroup$ I believe if $R$ is a regular local ring of dimension 3 and $P$ is any prime ideal of height 2 which is not a complete intersection, then $l(P)=3, l(P_P)=2$. $\endgroup$
    – Mohan
    Apr 10, 2017 at 18:48

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Let $(R, \mathfrak{m}): = \mathbb{C}[[x,y,z]]$ a formal power series. Let $J = (x^2, xy, xz) = \mathfrak{m}(x)$. We can check that $\ell(J) = 3$, $J^{(n)} = (x^n)$. This ideal satisfies the requirements.

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  • $\begingroup$ Thanks for your answer. In this case the ideal J is integrally closed. Is it always true that ideals which satisfies these conditions (other than equimultiple ideals) are always integrally closed? $\endgroup$
    – Cusp
    Apr 18, 2017 at 8:59
  • $\begingroup$ In the construction $J$, we replace $\mathfrak{m}$ by any $\mathfrak{m}$-primary ideal. Now we choose a non-integrally closed ideal (for example $(x^2, y^2, z^2)$) we have a non-integrally closed ideal that satisfies the requirements. $\endgroup$ Apr 18, 2017 at 15:21

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