The Stacks project

Lemma 87.21.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces which is locally of finite type. Let $\{ g_ i : Y_ i \to Y\} $ be a family of morphisms of formal algebraic spaces which are representable by algebraic spaces and ├ętale such that $\coprod g_ i$ is surjective. Then $f$ is rig-surjective if and only if each $f_ i : X \times _ Y Y_ i \to Y_ i$ is rig-surjective.

Proof. Namely, if $f$ is rig-surjective, so is any base change (Lemma 87.21.4). Conversely, if all $f_ i$ are rig-surjective, so is $\coprod f_ i : \coprod X \times _ Y Y_ i \to \coprod Y_ i$. By Lemma 87.21.7 the morphism $\coprod g_ i : \coprod Y_ i \to Y$ is rig-surjective. Hence $\coprod X \times _ Y Y_ i \to Y$ is rig-surjective (Lemma 87.21.3). Since this morphism factors through $X \to Y$ we see that $X \to Y$ is rig-surjective by Lemma 87.21.6. $\square$

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