Lemma 73.8.10. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $P$ be a property of objects in $(\textit{Spaces}/X)_{fppf}$ such that whenever $\{ U_ i \to U\} $ is a covering in $(\textit{Spaces}/X)_{fppf}$, then
If $P(U)$ for all $U$ affine and flat, locally of finite presentation over $X$, then $P(X)$.
Proof. Let $U$ be a separated algebraic space locally of finite presentation over $X$. Then we can choose an étale covering $\{ U_ i \to U\} _{i \in I}$ with $V_ i$ affine. Since $U$ is separated, we conclude that $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ is always affine. Whence $P(U_{i_0} \times _ U \ldots \times _ U U_{i_ p})$ always. Hence $P(U)$ holds. Choose a scheme $U$ which is a disjoint union of affines and a surjective étale morphism $U \to X$. Then $U \times _ X \ldots \times _ X U$ (with $p + 1$ factors) is a separated algebraic space étale over $X$. Hence $P(U \times _ X \ldots \times _ X U)$ by the above. We conclude that $P(X)$ is true. $\square$
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