Lemma 17.10.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $x \in X$ be a point. Assume that $x$ has a fundamental system of quasi-compact neighbourhoods. Consider any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Then there exists an open neighbourhood $U$ of $x$ such that $\mathcal{F}|_ U$ is isomorphic to the sheaf of modules $\mathcal{F}_ M$ on $(U, \mathcal{O}_ U)$ associated to some $\Gamma (U, \mathcal{O}_ U)$-module $M$.

Proof. First we may replace $X$ by an open neighbourhood of $x$ and assume that $\mathcal{F}$ is isomorphic to the cokernel of a map

$\Psi : \bigoplus \nolimits _{j \in J} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ X.$

The problem is that this map may not be given by a “matrix”, because the module of global sections of a direct sum is in general different from the direct sum of the modules of global sections.

Let $x \in E \subset X$ be a quasi-compact neighbourhood of $x$ (note: $E$ may not be open). Let $x \in U \subset E$ be an open neighbourhood of $x$ contained in $E$. Next, we proceed as in the proof of Lemma 17.3.5. For each $j \in J$ denote $s_ j \in \Gamma (X, \bigoplus \nolimits _{i \in I} \mathcal{O}_ X)$ the image of the section $1$ in the summand $\mathcal{O}_ X$ corresponding to $j$. There exists a finite collection of opens $U_{jk}$, $k \in K_ j$ such that $E \subset \bigcup _{k \in K_ j} U_{jk}$ and such that each restriction $s_ j|_{U_{jk}}$ is a finite sum $\sum _{i \in I_{jk}} f_{jki}$ with $I_{jk} \subset I$, and $f_{jki}$ in the summand $\mathcal{O}_ X$ corresponding to $i \in I$. Set $I_ j = \bigcup _{k \in K_ j} I_{jk}$. This is a finite set. Since $U \subset E \subset \bigcup _{k \in K_ j} U_{jk}$ the section $s_ j|_ U$ is a section of the finite direct sum $\bigoplus _{i \in I_ j} \mathcal{O}_ X$. By Lemma 17.3.2 we see that actually $s_ j|_ U$ is a sum $\sum _{i \in I_ j} f_{ij}$ and $f_{ij} \in \mathcal{O}_ X(U) = \Gamma (U, \mathcal{O}_ U)$.

At this point we can define a module $M$ as the cokernel of the map

$\bigoplus \nolimits _{j \in J} \Gamma (U, \mathcal{O}_ U) \longrightarrow \bigoplus \nolimits _{i \in I} \Gamma (U, \mathcal{O}_ U)$

with matrix given by the $(f_{ij})$. By construction (2) of Lemma 17.10.5 we see that $\mathcal{F}_ M$ has the same presentation as $\mathcal{F}|_ U$ and therefore $\mathcal{F}_ M \cong \mathcal{F}|_ U$. $\square$

Comment #1781 by Keenan Kidwell on

This is sort of a sanity check for my own understanding. It seems that this proof (and the proof of 01AI) both implicitly rely on the following simple principle: if $\mathscr{F}^\prime$ is a subsheaf of a sheaf $\mathscr{F}$ of modules on a ringed space $X$, $U$ an open subset of $X$, $U=\bigcup_j U_j$ an open cover, and $s\in\mathscr{F}(U)$ is a section such that $s\vert_{U_j}\in\mathscr{F}^\prime(U_j)$ for all $j$, then $s\in\mathscr{F}^\prime(U)$. I'm not suggesting this be made more explicit. I just want to confirm that this does in fact is playing a role in both arguments.

Comment #1820 by on

Yes, this is a lemma that could be formulated and then used in the proof where we find the $f_{ijk}$.

Comment #7707 by Ryo Suzuki on

This lemma can be deduced from Lemma 17.3.5.

First, it can be shown that: Let $(X,O_X)$, $I$, $\{F_i\}_{i\in I}$ as lemma 17.3.5. Let $E\subset X$ be a quasi-compact subset. Let $U,V\subset X$ are open subset, and assume $U\subset E\subset V$. Then, the map $(\bigoplus F_i)(V)\to (\bigoplus F_i)(U)$ factors through $\bigoplus (F_i(U))$.

Proof. Let $i\colon E\to X$ be a inclusion map. Then $(\bigoplus F_i)(V)\to (\bigoplus F_i)(U)$ factors through $(\bigoplus F_i)|_E(E)$. By lemma 17.3.5, $(\bigoplus F_i)|_E(E)\cong \bigoplus (F_i|_E(E))$. Immediately $\bigoplus (F_i|_E(E))\to (\bigoplus F_i)(U)$ factors through $\bigoplus (F_i(U))$. Hence $(\bigoplus F_i)(V)\to (\bigoplus F_i)(U)$ factors through $\bigoplus (F_i(U))$. q.e.d.

Using it, lemma 17.10.8 can be shown as follows: $F$ is locally isomorphic to the cokernel of $\Psi\colon \bigoplus_{j\in J} O_X\to \bigoplus_{i\in I} O_X$. By assumption, there exists $U\subset E\subset X$ such that U is open and E is quasi-compact. Then $(\bigoplus_{j\in J} O_X)(X)\to (\bigoplus_{i\in I} O_X)(X)$ factors through $\bigoplus_{i\in I} (O_X(U))$. Hence $\Psi|_U$ is given by a matrix. The rest of proof is same as present proof.

Comment #7967 by on

Sorry, but I think this isn't super helpful. Still, if others disagree I will reconsider.

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