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$\colon$

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

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

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 $S$?