The Stacks project

Lemma 28.22.4. 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 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

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

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 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

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

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01PI. Beware of the difference between the letter 'O' and the digit '0'.