The Stacks project

Lemma 69.5.8. Notation and assumptions as in Situation 69.5.5. Let $f_0 : Y_0 \to Z_0$ be a morphism of algebraic spaces over $X_0$. Assume (a) $Y_0 \to X_0$ and $Z_0 \to X_0$ are representable, (b) $Y_0$, $Z_0$ quasi-compact and quasi-separated, (c) $f_0$ locally of finite presentation, and (d) $Y_0 \times _{X_0} X \to Z_0 \times _{X_0} X$ an isomorphism. Then there exists an $i \geq 0$ such that $Y_0 \times _{X_0} X_ i \to Z_0 \times _{X_0} X_ i$ is an isomorphism.

Proof. Choose an affine scheme $U_0$ and a surjective ├ętale morphism $U_0 \to X_0$. Set $U_ i = U_0 \times _{X_0} X_ i$ and $U = U_0 \times _{X_0} X$. Apply Limits, Lemma 32.8.11 to see that $Y_0 \times _{X_0} U_ i \to Z_0 \times _{X_0} U_ i$ is an isomorphism of schemes for some $i \geq 0$ (details omitted). As $U_ i \to X_ i$ is surjective ├ętale, it follows that $Y_0 \times _{X_0} X_ i \to Z_0 \times _{X_0} X_ i$ is an isomorphism (details omitted). $\square$


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