Lemma 29.20.2. Let $X$ be a scheme. The following are equivalent
$X$ is quasi-excellent, and
$X$ is a G-scheme and J-2.
Proof. The implication (1) $\Rightarrow $ (2) follows on combining the definition of quasi-excellent schemes (Definition 29.20.1), the definition of quasi-excellent rings (More on Algebra, Definition 15.53.1), the definition of G-schemes (Properties, Definition 28.14.1), and Lemma 29.19.2. Conversely, if $X$ is a G-scheme and J-2, then for any $x \in X$ we can find an affine open neighbourhood $x \in U \subset X$ such that $\mathcal{O}_ X(U)$ is a G-ring. By Lemma 29.19.2 the ring $\mathcal{O}_ X(U)$ is also J-2, whence quasi-excellent. $\square$
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