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The Stacks project

Lemma 28.13.5. Let X be a scheme. The following are equivalent:

  1. The scheme X is universally Japanese.

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

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

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

Moreover, if X is universally Japanese then every open subscheme is universally Japanese.


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