Lemma 67.14.2. Let S be a scheme. Let X be an algebraic space over S. 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 Properties of Spaces, Lemma 66.29.7. 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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