Question

We know that any submodule of a quotient module $\frac{M}{N}$ is of the form $\frac{K}{N}$, where $K$ is a submodule of $M$ containing $N$. Now here is a question: Let $\cal F$ and ${\cal G}$ be quasi coherent sheaves on a scheme $X$. Is any quasi coherent subsheaf of a quotient sheaf $\frac{\cal F}{{\cal G}}$ of the form $\frac{\cal H}{{\cal G}}$ where $\cal H$ contains ${\cal G}$?

Answer 1

With all due respect to abelian categories the pedestrian answer is this: Let $\alpha:\mathscr {F\to F/G}$ be the natural morphism and $\mathscr{M\subseteq F/G}$ a subsheaf. Then setting $\mathscr{H=\alpha^* M}$ gives you $\mathscr{M\simeq H/G}$.

Answer 2

This is true in every abelian category. If $M \subseteq F/G$, define $H = M \times_{F/G} F$, then $G \subseteq H \subseteq F$ and $M=H/G$. Alternatively, use the local fact and gluing.