The Stacks project

Lemma 63.16.5. Let $\mathit{Sch}_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. Let $F$ be an algebraic space over $S$. Let $T$ be a scheme over $S$ and let $f : T \to F$ be a morphism over $S$. Let $f' : T' \to F'$ be the morphism over $S'$ we get from $f$ by applying the equivalence of categories described in Lemma 63.16.3. For any property $\mathcal{P}$ as in Definition 63.5.1 we have $\mathcal{P}(f') \Leftrightarrow \mathcal{P}(f)$.

Proof. Suppose that $U$ is a scheme over $S$, and $U \to F$ is a surjective ├ętale morphism. Denote $U'$ the scheme $U$ viewed as a scheme over $S'$. In Lemma 63.16.1 we have seen that $U' \to F'$ is surjective ├ętale. Since

\[ j(T \times _{f, F} U) = T' \times _{f', F'} U' \]

the morphism of schemes $T \times _{f, F} U \to U$ is identified with the morphism of schemes $T' \times _{f', F'} U' \to U'$. It is the same morphism, just viewed over different base schemes. Hence the lemma follows from Lemma 63.11.4. $\square$


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