Lemma 15.47.2. Let $R$ be a Noetherian ring. Let $X = \mathop{\mathrm{Spec}}(R)$. The ring $R$ is J-1 if and only if $V(\mathfrak p) \cap \text{Reg}(X)$ contains a nonempty open subset of $V(\mathfrak p)$ for all $\mathfrak p \in \text{Reg}(X)$.

Proof. This follows from Topology, Lemma 5.16.5 and the fact that $\text{Reg}(X)$ is stable under generalization by Algebra, Lemma 10.110.6. $\square$

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