In a book I am reading, "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents" by Luc Illusie, Yves Laszlo, Fabrice Orgogozo (<https://arxiv.org/abs/1207.3648>), there is the following assertion whose proof I would like to understand.


Let $e\_1, ..., e\_n$ be natural numbers, not all zero. For any trait $S$ with uniformizer $\pi$, define $$ V(S,\pi,e\_1,...,e\_n) = Spec(O\_S[T\_1,\ldots,T\_n]/(T\_1^{e\_1}...T\_n^{e\_n}-\pi)). $$ It is claimed that this scheme is regular in Exposé XVI, Lemma 3.5.13. However the proof is only briefly outlined. If someone can provide details, I would appreciate it.


The first case is if at least one of $e\_i$ are "invertible in $\eta$" (generic point of $S$), which, I don't even see why that would be relevant, but this case is supposed to be an easy exercise. Is there a regularity criterion which would be easy to apply here?



Regularity of certain schemes
Here is another argument to prove the desired regularity, along the lines of Piotr Achinger's comment.


As explained in the linked seminar notes, the essential case to consider is $V(S, \pi, e\_1, \dots, e\_n)$ (for which we write $V$), where for some $i$, the exponent $e\_i$ is invertible in $\eta$.


From the hypothesis that $e\_i$ is invertible we may conclude that $V\_\eta$ is smooth over $\eta$. Therefore, all points of $V$ in the fiber over $\eta$ are regular. To show that $V$ is regular, it thus remains to show that the points in the special fiber are regular, i.e. it remains to show that any prime ideal of $\mathcal{O}\_S[T\_1, \dots, T\_n]/(T\_1^{e\_1} \dots T\_n^{e\_n} - \pi)$ (for which we write $R$) containing $\pi$ is generated by a regular sequence.


Consider a prime ideal $P$ of $R$ containing $\pi$. Since we have $\pi = T\_1^{e\_1} \dots T\_n^{e\_n}$ in $R$, we have $T\_j$ in $P$ for some $j$ such that $e\_j$ is non-zero. The quotient ring $R/T\_j R$ is isomorphic to $(\mathcal{O}\_S/\pi \mathcal{O}\_S)[T\_1, \dots, T\_{j-1}, T\_{j+1}, \dots, T\_n]$. (Here we use the hypothesis that $e\_j$ is non-zero: the ideal of $\mathcal{O}\_S[T\_1, \dots, T\_n]$ generated by $T\_1^{e\_1} \dots T\_n^{e\_n} - \pi$ and $T\_j$ is the same as the ideal generated by $\pi$ and $T\_j$.) Since $R/T\_j R$ is a polynomial ring over a field, any prime ideal of $R/ T\_j R$ is generated by a regular sequence. In particular, the prime ideal $P(R/T\_j R)$ is generated by a regular sequence. Let $g\_1, \dots, g\_k$ be elements of $R$ mapping to such a regular sequence. We then have that the sequence $T\_j, g\_1, \dots, g\_k$ generates $P$. Furthermore, $T\_j$ is regular in $R$, since $R/T\_j R$ is a domain, and so, by the definition of ``regular sequence'', we have that $T\_j, g\_1, \dots, g\_k$ is a regular sequence in $R$. 


