Étale fibration for $K[[X_1,...,X_n]]$

Question

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak P}) < n$.

We shall choose an irreducible element $f \in {\frak P}$ such that up to a variables transformation and by abuse of notations we have

\begin{equation*} f = a_e + a_{e-1}X_1 + \ldots + a_{1}(X_1)^{e-1} + X_1^e, \end{equation*} where $a_1,\ldots,a_e \in K[[X_2,\ldots,X_n]]$.

The ring $R_n \colon= A_n/f$ is finite over the ring $A_{n-1} \colon= K[[X_2,\ldots,X_n]]$. Let us denote by ${\frak Q}$ the unique prime of $A_{n-1}$ lying below ${\frak P}$.

Q. Can I always find $f$ such that ${\frak P}$ is étale over ${\frak Q}$? Or equivalently for the discriminant $\Delta_f$ of $f$, can I have $\Delta_f \notin {\frak Q}$ ?

Answer 1

A general ideal $\mathfrak P$ need not contain an element $f$ that is a polynomial in one of the variables. The existence of such an element $f$ is a strong restriction on the ideal $\mathfrak P$. An even stronger restriction is requiring, in addition, that the polynomial $f$ be monic.

If we take an ideal $\mathfrak P$ containing a monic polynomial $f$ in $X_1$ then the answer to your question is positive if $K$ is of characteristic 0 and negative if $K$ is of characteristic $p>0$. Namely, if $char\ K=0$ and $\mathfrak P$ is not étale over $\mathfrak Q$, replace $f$ by $\frac{\partial f}{\partial X_1}$ and keep repeating this process as many times as necessary.

If $char\ K=p>0$, take $n=3$ (we need $n\ge3$ in order the inequalities on $ht\ \mathfrak P$ to be respected). Let $\mathfrak P$ be an ideal generated by $X_1^p-g$ and $h$, where $g$ and $h$ are sufficiently general power series in $X_2$ and $X_3$ without constant or linear terms. Then $\mathfrak P$ contains a monic polynomial, but none that make $\mathfrak P$ étale over $\mathfrak Q$.