The Stacks project

Lemma 69.18.1. Let $S = U \cup W$ be an open covering of a scheme. Then the functor

\[ FP_ S \longrightarrow FP_ U \times _{FP_{U \cap W}} FP_ W \]

given by base change is an equivalence where $FP_ T$ is the category of algebraic spaces of finite presentation over the scheme $T$.

Proof. First, since $S = U \cup W$ is a Zariski covering, we see that the category of sheaves on $(\mathit{Sch}/S)_{fppf}$ is equivalent to the category of triples $(\mathcal{F}_ U, \mathcal{F}_ W, \varphi )$ where $\mathcal{F}_ U$ is a sheaf on $(\mathit{Sch}/U)_{fppf}$, $\mathcal{F}_ W$ is a sheaf on $(\mathit{Sch}/W)_{fppf}$, and

\[ \varphi : \mathcal{F}_ U|_{(\mathit{Sch}/U \cap W)_{fppf}} \longrightarrow \mathcal{F}_ W|_{(\mathit{Sch}/U \cap W)_{fppf}} \]

is an isomorphism. See Sites, Lemma 7.26.5 (note that no other gluing data are necessary because $U \times _ S U = U$, $W \times _ S W = W$ and that the cocycle condition is automatic for the same reason). Now, if the sheaf $\mathcal{F}$ on $(\mathit{Sch}/S)_{fppf}$ maps to $(\mathcal{F}_ U, \mathcal{F}_ W, \varphi )$ via this equivalence, then $\mathcal{F}$ is an algebraic space if and only if $\mathcal{F}_ U$ and $\mathcal{F}_ W$ are algebraic spaces. This follows immediately from Algebraic Spaces, Lemma 64.8.5 as $\mathcal{F}_ U \to \mathcal{F}$ and $\mathcal{F}_ W \to \mathcal{F}$ are representable by open immersions and cover $\mathcal{F}$. Finally, in this case the algebraic space $\mathcal{F}$ is of finite presentation over $S$ if and only if $\mathcal{F}_ U$ is of finite presentation over $U$ and $\mathcal{F}_ W$ is of finite presentation over $W$ by Morphisms of Spaces, Lemmas 66.8.8, 66.4.12, and 66.28.4. $\square$

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