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[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. $\square$

Comments (4)

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 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).

There are also:

  • 7 comment(s) on Section 10.161: Japanese rings

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