The Stacks project

Lemma 65.8.1. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Let $F$ and $G$ be sheaves on $(\mathit{Sch}/S)_{fppf}^{opp}$ and denote $F \amalg G$ the coproduct in the category of sheaves. The map $F \to F \amalg G$ is representable by open and closed immersions.

Proof. Let $U$ be a scheme and let $\xi \in (F \amalg G)(U)$. Recall the coproduct in the category of sheaves is the sheafification of the coproduct presheaf (Sites, Lemma 7.10.13). Thus there exists an fppf covering $\{ g_ i : U_ i \to U\} _{i \in I}$ and a disjoint union decomposition $I = I' \amalg I''$ such that $U_ i \to U \to F \amalg G$ factors through $F$, resp. $G$ if and only if $i \in I'$, resp. $i \in I''$. Since $F$ and $G$ have empty intersection in $F \amalg G$ we conclude that $U_ i \times _ U U_ j$ is empty if $i \in I'$ and $j \in I''$. Hence $U' = \bigcup _{i \in I'} g_ i(U_ i)$ and $U'' = \bigcup _{i \in I''} g_ i(U_ i)$ are disjoint open (Morphisms, Lemma 29.25.10) subschemes of $U$ with $U = U' \amalg U''$. We omit the verification that $U' = U \times _{F \amalg G} F$. $\square$

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