The Stacks project

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. $\square$