The Stacks project

Definition 28.13.1. Let $X$ be a scheme.

1. 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).

2. 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).

3. 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).