The Stacks project

Lemma 28.13.6. Let $X$ be a scheme. The following are equivalent:

1. The scheme $X$ is Nagata.

2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is Nagata.

3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is Nagata.

4. There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is Nagata.

Moreover, if $X$ is Nagata then every open subscheme is Nagata.