Lemma 28.14.2. Let $X$ be a scheme. The following are equivalent:
$X$ is a G-scheme,
$X$ is locally Noetherian and for all $x \in X$ the ring map $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge $ is regular, and
$X$ is locally Noetherian and for any closed point $x \in X$ the ring map $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge $ is regular.
Proof. It is clear that (1) implies (2) and (2) implies (3). Assume (3). Let $x \in X$ be any point. By Lemma 28.5.9 there exists a specialization $x \leadsto x'$ with $x'$ closed in $X$. Then $\mathcal{O}_{X, x'}$ is a G-ring by More on Algebra, Lemma 15.51.7. Since $\mathcal{O}_{X, x}$ is the localization of $\mathcal{O}_{X, x'}$ at a prime ideal (Schemes, Lemma 26.13.2), we see that $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^\wedge $ is a regular ring map. Since $x \in X$ was arbitrary we conclude that for any affine open $U \subset X$ the Noetherian (Lemma 28.5.2) ring $\mathcal{O}_ X(U)$ is a G-ring. This proves that (1) holds. $\square$
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