The Stacks project

Proof. Let $\mathcal{F} = \mathop{\mathrm{Coker}}(\bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i})$ as in ( It suffices to show that the cokernel of a map $\varphi : j_{W!}\mathcal{O}_ W \to \mathcal{F}$ with $W \in \mathcal{B}$ is another module of the same type. The map $\varphi $ corresponds to $s \in \mathcal{F}(W)$. Since $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \to \mathcal{F}$ is surjective, by (1) we may choose a covering $\{ W_ k \to W\} _{k \in K}$ with $W_ k \in \mathcal{B}$ such that $s|_{W_ k}$ is the image of some section $s_ k$ of $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i})$. By (2) the object $W$ is quasi-compact. By Lemma 18.30.2 there is a finite subset $K' \subset K$ such that $\bigoplus _{k \in K'} j_{W_ k!}\mathcal{O}_{W_ k} \to j_{W!}\mathcal{O}_ W$ is surjective. We conclude that $\mathop{\mathrm{Coker}}(\varphi )$ is equal to

\[ \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{k \in K'} j_{W_ k!}\mathcal{O}_{W_ k} \oplus \bigoplus j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \right) \]

where the map $\bigoplus _{k \in K'} j_{W_ k!}\mathcal{O}_{W_ k} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$ corresponds to $\sum _{k \in K'} s_ k$. This finishes the proof. $\square$

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