Definition 28.13.1. Let $X$ be a scheme.

Assume $X$ integral. We say $X$ is

*Japanese*if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_ X(U)$ is Japanese (see Algebra, Definition 10.161.1).We say $X$ is

*universally Japanese*if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_ X(U)$ is universally Japanese (see Algebra, Definition 10.162.1).We say $X$ is

*Nagata*if for every $x \in X$ there exists an affine open neighbourhood $x \in U \subset X$ such that the ring $\mathcal{O}_ X(U)$ is Nagata (see Algebra, Definition 10.162.1).

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