Lemma 29.4.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G} \subset \mathcal{F}$ be a $\mathcal{O}_ X$-submodule. There exists a unique quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{G}' \subset \mathcal{G}$ with the following property: For every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{H}$ the map

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{H}, \mathcal{G}') \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{H}, \mathcal{G}) \]

is bijective. In particular $\mathcal{G}'$ is the largest quasi-coherent $\mathcal{O}_ X$-submodule of $\mathcal{F}$ contained in $\mathcal{G}$.

**Proof.**
Let $\mathcal{G}_ a$, $a \in A$ be the set of quasi-coherent $\mathcal{O}_ X$-submodules contained in $\mathcal{G}$. Then the image $\mathcal{G}'$ of

\[ \bigoplus \nolimits _{a \in A} \mathcal{G}_ a \longrightarrow \mathcal{F} \]

is quasi-coherent as the image of a map of quasi-coherent sheaves on $X$ is quasi-coherent and since a direct sum of quasi-coherent sheaves is quasi-coherent, see Schemes, Section 26.24. The module $\mathcal{G}'$ is contained in $\mathcal{G}$. Hence this is the largest quasi-coherent $\mathcal{O}_ X$-module contained in $\mathcal{G}$.

To prove the formula, let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_ X$-module and let $\alpha : \mathcal{H} \to \mathcal{G}$ be an $\mathcal{O}_ X$-module map. The image of the composition $\mathcal{H} \to \mathcal{G} \to \mathcal{F}$ is quasi-coherent as the image of a map of quasi-coherent sheaves. Hence it is contained in $\mathcal{G}'$. Hence $\alpha $ factors through $\mathcal{G}'$ as desired.
$\square$

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