# Regularity of certain schemes

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?

• Maybe I'm mistaken, but this has dimension $n$ and $T_1, \ldots, T_n$ is a regular sequence. After dividing by the ideal $(T_1, \ldots, T_n)$ you obtain $\mathcal{O}_S/(\pi)$, which is a field. The underlying criterion is that if $f$ is a nonzerodivisor in a local ring $R$ such that $R/(f)$ is regular, then $R$ is regular. Jun 4, 2019 at 7:03
• For intuition, let's consider $\mathcal{O}_S = k[\pi]$, then your ring is $k[\pi][T_1, \ldots, T_n]/(T_1^{e_1} \ldots T_n^{e_n} - \pi) = k[T_1, \ldots, T_n]$ i.e. just a polynomial ring. (And of course it is irrelevant for regularity that we have a monomial in the $T_i$.) Jun 4, 2019 at 7:16
• @Piotr Achinger: the ring is not local.
– abx
Jun 4, 2019 at 7:33
• @abx Oops, of course you are right! Part of my brain was thinking of $\mathcal{O}_S[[T_1, \ldots, T_n]]/(\ldots)$... Thinking of an easy fix that doesn't need $\mathcal{O}_S$ quasi-excellent. Jun 4, 2019 at 7:41
• That was exactly my first thought as well, but indeed the ring doesn't seem to be local as far as I could see. Jun 4, 2019 at 10:07

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 $$R$$.
There is an effective action of the torus $$G=\mathbb G_{m,S}^{n-1}=\operatorname{Spec} O_S[s_1,...,s_n]/(s_1...s_n-1)$$ on $$V$$ (the co-action is given by $$T_i\mapsto s_i\otimes T_i$$) such that the closed point $$0$$ defined by the ideal $$(T_1,...,T_n,\pi)$$ lies in the closure of every $$G$$-orbit in the closed fiber of $$V\to S$$. Since the non-regular locus of $$V$$ is closed, it is enough to verify that $$V$$ is regular at $$0$$, for which Piotr Achinger's first comment suffices.