The Stacks project

Proof. 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 $\mathcal{G}$ to a $\mathcal{G}_1$ over $U \cup U_1$ to a $\mathcal{G}_2$ over $U \cup U_1 \cup U_2$ to a $\mathcal{G}_3$ 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 $U$, $V$ are quasi-compact open, we see that $U \cap V$ is a quasi-compact open. It suffices to prove the lemma for the system $(V, U \cap V, \mathcal{F}|_ V, \mathcal{G}|_{U \cap V})$ since we can glue the resulting sheaf $\mathcal{G}'$ over $V$ to the given sheaf $\mathcal{G}$ over $U$ along the common value over $U \cap V$. Thus we reduce to the case where $X$ is affine.

Assume $X = \mathop{\mathrm{Spec}}(R)$. Write $\mathcal{F} = \widetilde M$ for some $R$-module $M$. By Lemma 28.22.1 above we may find a quasi-coherent subsheaf $\mathcal{H} \subset \mathcal{F}$ which restricts to $\mathcal{G}$ over $U$. Write $\mathcal{H} = \widetilde N$ for some $R$-module $N$. For every $u \in U$ there exists an $f \in R$ such that $u \in D(f) \subset U$ and such that $N_ f$ is finitely generated, see Lemma 28.16.1. Since $U$ is quasi-compact we can cover it by finitely many $D(f_ i)$ such that $N_{f_ i}$ is generated by finitely many elements, say $x_{i, 1}/f_ i^ N, \ldots , x_{i, r_ i}/f_ i^ N$. Let $N' \subset N$ be the submodule generated by the elements $x_{i, j}$. Then the subsheaf $\mathcal{G}' = \widetilde{N'} \subset \mathcal{H} \subset \mathcal{F}$ works. $\square$