[Theorem 23.1.3, EGA]

Lemma 10.161.16 (Tate). Let $R$ be a ring. Let $x \in R$. Assume

1. $R$ is a normal Noetherian domain,

2. $R/xR$ is a domain and N-2,

3. $R \cong \mathop{\mathrm{lim}}\nolimits _ n R/x^ nR$ is complete with respect to $x$.

Then $R$ is N-2.

Proof. We may assume $x \not= 0$ since otherwise the lemma is trivial. Let $K$ be the fraction field of $R$. If the characteristic of $K$ is zero the lemma follows from (1), see Lemma 10.161.11. Hence we may assume that the characteristic of $K$ is $p > 0$, and we may apply Lemma 10.161.12. Thus given $L/K$ a finite purely inseparable field extension we have to show that the integral closure $S$ of $R$ in $L$ is finite over $R$.

Let $q$ be a power of $p$ such that $L^ q \subset K$. By enlarging $L$ if necessary we may assume there exists an element $y \in L$ such that $y^ q = x$. Since $R \to S$ induces a homeomorphism of spectra (see Lemma 10.46.7) there is a unique prime ideal $\mathfrak q \subset S$ lying over the prime ideal $\mathfrak p = xR$. It is clear that

$\mathfrak q = \{ f \in S \mid f^ q \in \mathfrak p\} = yS$

since $y^ q = x$. Observe that $R_{\mathfrak p}$ is a discrete valuation ring by Lemma 10.119.7. Then $S_{\mathfrak q}$ is Noetherian by Krull-Akizuki (Lemma 10.119.12). Whereupon we conclude $S_{\mathfrak q}$ is a discrete valuation ring by Lemma 10.119.7 once again. By Lemma 10.119.10 we see that $\kappa (\mathfrak q)/\kappa (\mathfrak p)$ is a finite field extension. Hence the integral closure $S' \subset \kappa (\mathfrak q)$ of $R/xR$ is finite over $R/xR$ by assumption (2). Since $S/yS \subset S'$ this implies that $S/yS$ is finite over $R$. Note that $S/y^ nS$ has a finite filtration whose subquotients are the modules $y^ iS/y^{i + 1}S \cong S/yS$. Hence we see that each $S/y^ nS$ is finite over $R$. In particular $S/xS$ is finite over $R$. Also, it is clear that $\bigcap x^ nS = (0)$ since an element in the intersection has $q$th power contained in $\bigcap x^ nR = (0)$ (Lemma 10.51.4). Thus we may apply Lemma 10.96.12 to conclude that $S$ is finite over $R$, and we win. $\square$

Comment #2749 by Ariyan Javanpeykar on

A reference: EGA IV_0, Theorem 23.1.3

Comment #4300 by bogdan on

The proof implicitly uses the fact that $S_{\mathfrak q}$ is noetherian (when it cites 00PD). This follows from Tag 00PG but this result is not explicitly cited now (the proof can be significantly simplified in this situation).

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