The Stacks project

Proposition 15.48.7. The following types of rings are J-2:

  1. fields,

  2. Noetherian complete local rings,

  3. $\mathbf{Z}$,

  4. Noetherian local rings of dimension $1$,

  5. Nagata rings of dimension $1$,

  6. Dedekind domains with fraction field of characteristic zero,

  7. finite type ring extensions of any of the above.

Proof. For cases (1), (3), (5), and (6) this is proved by checking condition (4) of Lemma 15.47.6. We will only do this in case $R$ is a Nagata ring of dimension $1$. Let $\mathfrak p \subset R$ be a prime ideal and let $L/\kappa (\mathfrak p)$ be a finite purely inseparable extension. If $\mathfrak p \subset R$ is a maximal ideal, then $R \to L$ is finite and $L$ is a regular ring and we've checked the condition. If $\mathfrak p \subset R$ is a minimal prime, then the Nagata condition insures that the integral closure $R' \subset L$ of $R$ in $L$ is finite over $R$. Then $R'$ is a normal domain of dimension $1$ (Algebra, Lemma 10.112.3) hence regular (Algebra, Lemma 10.157.4) and we've checked the condition in this case as well.

For case (2), we will use condition (3) of Lemma 15.47.6. Let $R$ be a Noetherian complete local ring. Note that if $R \to R'$ is finite, then $R'$ is a product of Noetherian complete local rings, see Algebra, Lemma 10.160.2. Hence it suffices to prove that a Noetherian complete local ring which is a domain is J-0, which is Lemma 15.48.6.

For case (4), we also use condition (3) of Lemma 15.47.6. Namely, if $R$ is a local Noetherian ring of dimension $1$ and $R \to R'$ is finite, then $\mathop{\mathrm{Spec}}(R')$ is finite. Since the regular locus is stable under generalization, we see that $R'$ is J-1. $\square$

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