The Stacks project

Lemma 65.8.2. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Let $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$. Given a set $I$ and sheaves $F_ i$ on $\mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, if $U \cong \coprod _{i\in I} F_ i$ as sheaves, then each $F_ i$ is representable by an open and closed subscheme $U_ i$ and $U \cong \coprod U_ i$ as schemes.

Proof. By Lemma 65.8.1 the map $F_ i \to U$ is representable by open and closed immersions. Hence $F_ i$ is representable by an open and closed subscheme $U_ i$ of $U$. We have $U = \coprod U_ i$ because we have $U \cong \coprod F_ i$ as sheaves and we can test the equality on points. $\square$

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