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Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.

If $a$ that is not a zero divisor of $R/I$ we have that $R/(I+\langle a\rangle)$ has finite length over R.

Can we say that for a "generic" element $a$, this length is constant? More formally, can we say that there exists an ideal $J \subset R$ such that the length of $R/(I+\langle a\rangle)$ is independent of $a$, assuming that $a\notin J$?

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    $\begingroup$ You would need to allow several different possibilities for $J$. Consider the case that $I$ is the principal ideal $\langle st\rangle$ in the power series ring $k[[s,t]]$. You need to exclude both $\langle s,t^2\rangle$ and $\langle s^2,t \rangle$. $\endgroup$ Commented Dec 4, 2024 at 23:43
  • $\begingroup$ @JasonStarr Right. So what about if we allow more than one ideal, but still finitely many of them? $\endgroup$ Commented Dec 4, 2024 at 23:52

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