The Stacks project

Proof. The beginning of the proof is a repeat of the beginning of the proof of Lemma 28.22.2. We write it out carefully anyway.

Let $n$ be the minimal number of affine opens $U_ i \subset X$, $i = 1, \ldots , n$ such that $X = U \cup \bigcup U_ i$. (Here we use that $X$ is quasi-compact.) Suppose we can prove the lemma for the case $n = 1$. Then we can successively extend the pair $(\mathcal{G}, \varphi )$ to a pair $(\mathcal{G}_1, \varphi _1)$ over $U \cup U_1$ to a pair $(\mathcal{G}_2, \varphi _2)$ over $U \cup U_1 \cup U_2$ to a pair $(\mathcal{G}_3, \varphi _3)$ over $U \cup U_1 \cup U_2 \cup U_3$, and so on. Thus we reduce to the case $n = 1$.

Thus we may assume that $X = U \cup V$ with $V$ affine. Since $X$ is quasi-separated and $U$ quasi-compact, we see that $U \cap V \subset V$ is quasi-compact. Suppose we prove the lemma for the system $(V, U \cap V, \mathcal{F}|_ V, \mathcal{G}|_{U \cap V}, \varphi |_{U \cap V})$ thereby producing $(\mathcal{G}', \varphi ')$ over $V$. Then we can glue $\mathcal{G}'$ over $V$ to the given sheaf $\mathcal{G}$ over $U$ along the common value over $U \cap V$, and similarly we can glue the map $\varphi '$ to the map $\varphi$ along the common value over $U \cap V$. Thus we reduce to the case where $X$ is affine.

Assume $X = \mathop{\mathrm{Spec}}(R)$. By Lemma 28.22.1 above we may find a quasi-coherent sheaf $\mathcal{H}$ with a map $\psi : \mathcal{H} \to \mathcal{F}$ over $X$ which restricts to $\mathcal{G}$ and $\varphi$ over $U$. By Lemma 28.22.2 we can find a finite type quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{H}' \subset \mathcal{H}$ such that $\mathcal{H}'|_ U = \mathcal{G}$. Thus after replacing $\mathcal{H}$ by $\mathcal{H}'$ and $\psi$ by the restriction of $\psi$ to $\mathcal{H}'$ we may assume that $\mathcal{H}$ is of finite type. By Lemma 28.16.2 we conclude that $\mathcal{H} = \widetilde{N}$ with $N$ a finitely generated $R$-module. Hence there exists a surjection as in the following short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules

where $\mathcal{K}$ is defined as the kernel. Since $\mathcal{G}$ is of finite presentation and $\mathcal{H}|_ U = \mathcal{G}$ by Modules, Lemma 17.11.3 the restriction $\mathcal{K}|_ U$ is an $\mathcal{O}_ U$-module of finite type. Hence by Lemma 28.22.2 again we see that there exists a finite type quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{K}' \subset \mathcal{K}$ such that $\mathcal{K}'|_ U = \mathcal{K}|_ U$. The solution to the problem posed in the lemma is to set

which is clearly of finite presentation and restricts to give $\mathcal{G}$ on $U$ with $\varphi '$ equal to the composition

This finishes the proof of the lemma. $\square$