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Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated by them.

Suppose that ${\frak P}$ satisfies the following three conditions$\colon$

  1. ${\frak P} \cap S = 0$.

  2. ${\mathrm{ht}}({\frak P}) = d$.

  3. Neither of $\overline{f_1},\ldots,\overline{f_d} \in K[[X_1,\ldots,X_d]]$ is zero.

(The condition 3. is equivalent to that every $f_i$ contains at least one term $c_{e_1,\ldots,e_d}X_1^{e_1} \ldots X_d^{e_d}$ such that $c_{e_1,\ldots,e_d} \notin (S_1,\ldots,S_n)$.)

Q. Is the ring $S[[X_1,\ldots,X_d]]/{\frak P}$ necessariy finite over $S$?

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No. Take $n=d=2$, $f_1=X_1-S_1$, $f_2=X_1-S_2X_2$, and check. (Observe in particular that $S[[X_1,X_2]]/(f_1,f_2)$ is isomorphic to $S[[X_2]]/(S_1-S_2X_2)$ as an $S$-algebra, and to $K[[S_2,X_2]]$ as a ring).

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