The Stacks project

Lemma 69.5.11. Notation and assumptions as in Situation 69.5.5. If $X$ is a scheme, then there exists an $i$ such that $X_ i$ is a scheme.

Proof. Choose a finite affine open covering $X = \bigcup W_ j$. By Lemma 69.5.7 we can find an $i \in I$ and open subspaces $W_{j, i} \subset X_ i$ whose base change to $X$ is $W_ j \to X$. By Lemma 69.5.10 we may assume that each $W_{j, i}$ is an affine scheme. This means that $X_ i$ is a scheme (see for example Properties of Spaces, Section 65.13). $\square$

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