# Power series rings and the formal generic fibre

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 $$S$$?

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).