The Stacks project

Lemma 94.7.2. The functor $p : \mathcal{S}\! \mathit{paces}\to (\mathit{Sch}/S)_{fppf}$ satisfies conditions (1) and (2) of Stacks, Definition 8.4.1.

Proof. It is follows from Lemma 94.7.1 that $\mathcal{S}\! \mathit{paces}$ is a fibred category over $(\mathit{Sch}/S)_{fppf}$ which proves (1). Suppose that $\{ U_ i \to U\} _{i \in I}$ is a covering of $(\mathit{Sch}/S)_{fppf}$. Suppose that $X, Y$ are algebraic spaces over $U$. Finally, suppose that $\varphi _ i : X_{U_ i} \to Y_{U_ i}$ are morphisms of $\textit{Spaces}/U_ i$ such that $\varphi _ i$ and $\varphi _ j$ restrict to the same morphisms $X_{U_ i \times _ U U_ j} \to Y_{U_ i \times _ U U_ j}$ of algebraic spaces over $U_ i \times _ U U_ j$. To prove (2) we have to show that there exists a unique morphism $\varphi : X \to Y$ over $U$ whose base change to $U_ i$ is equal to $\varphi _ i$. As a morphism from $X$ to $Y$ is the same thing as a map of sheaves this follows directly from Sites, Lemma 7.26.1. $\square$


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