The Stacks project

Lemma 27.22.4. (Variant of Lemma 27.22.2 dealing with modules of finite presentation.) Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{G}$ be an $\mathcal{O}_ U$-module which is of finite presentation. Let $\varphi : \mathcal{G} \to \mathcal{F}|_ U$ be a morphism of $\mathcal{O}_ U$-modules. Then there exists an $\mathcal{O}_ X$-module $\mathcal{G}'$ of finite presentation, and a morphism of $\mathcal{O}_ X$-modules $\varphi ' : \mathcal{G}' \to \mathcal{F}$ such that $\mathcal{G}'|_ U = \mathcal{G}$ and such that $\varphi '|_ U = \varphi $.

Proof. The beginning of the proof is a repeat of the beginning of the proof of Lemma 27.22.2. We write it out carefuly 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 27.22.1 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 27.22.2 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 27.16.2 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

\[ 0 \to \mathcal{K} \to \mathcal{O}_ X^{\oplus n} \to \mathcal{H} \to 0 \]

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 27.22.2 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

\[ \mathcal{G}' = \mathcal{O}_ X^{\oplus n}/\mathcal{K}' \]

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

\[ \mathcal{G}' = \mathcal{O}_ X^{\oplus n}/\mathcal{K}' \to \mathcal{O}_ X^{\oplus n}/\mathcal{K} = \mathcal{H} \xrightarrow {\psi } \mathcal{F}. \]

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


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