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Given a surjective morphism

$$\frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{I}\twoheadrightarrow \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{J}$$

where $I,J$ are genereated by regular sequences.

Question

Can we find a regular sequence $f_{1},\dots,f_{k},\dots,f_{l}$ such that there exist a commutative square

$$\begin{array}{ccc} \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{I} & \twoheadrightarrow & \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{J} \\ \downarrow\simeq & & \downarrow\simeq \\ \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{f_{1},\dots,f_{k}} & \twoheadrightarrow & \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{f_{1},\dots,f_{l}} , \end{array}$$

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    $\begingroup$ No. Take the case of two variables $ x,y$. Take $ I=( x^2,y^2)$ and $J=( x,y)$. $\endgroup$
    – Mohan
    Commented Feb 12 at 15:04

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