Lemma 87.15.1 (0AJ1)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.15: Fibre products
Lemma 87.15.1 (cite)

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Lemma 87.15.1. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of maps of sheaves on $(\mathit{Sch}/S)_{fppf}$ . Assume (a) $X_ i$ is a formal algebraic space over $S$ , (b) $X_ i \to X$ is representable by algebraic spaces and étale, and (c) $\coprod X_ i \to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$ .

Proof. For each $i$ pick $\{ X_{ij} \to X_ i\} _{j \in J_ i}$ as in Definition 87.11.1. Then $\{ X_{ij} \to X\} _{i \in I, j \in J_ i}$ is a family as in Definition 87.11.1 for $X$ . $\square$

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