How to define product of subsheaf

Question

Let X be a scheme. If $\mathcal F$ be a sheaf of $\mathcal O_ X$ module. IF $\mathcal G$ and $\mathcal H$ are two $\mathcal O_X$ submodules of $\mathcal F$. Then what is $\mathcal GH$

Answer 1

I'm assuming you mean to ask about $\mathscr{G}\mathscr{H}$; this doesn't make sense in general. For a general sheaf of $\mathscr{O}_X$-modules there is no way to multiply sections.

There are some contexts where it does make sense, and it is defined as the sheaf image of some morphism. For example, if $\mathscr{F}=\mathscr{O}_X$, so $\mathscr{G}$ and $\mathscr{H}$ are ideal sheaves, then $\mathscr{G}\mathscr{H}$ is (by definition) the sheaf image of the natural ``multiplication morphism" $\mathscr{G}\otimes_{\mathscr{O}_X}\mathscr{G}\rightarrow\mathscr{O}_X$. More generally, if $\mathscr{G}$ is an ideal sheaf and $\mathscr{H}$ a sheaf of $\mathscr{O}_X$-modules, then $\mathscr{G}\mathscr{H}$ is the image of the morphism $\mathscr{G}\otimes_{\mathscr{O}_X}\mathscr{H}\rightarrow\mathscr{H}$ given by (restricting) the $\mathscr{O}_X$-module structure on $\mathscr{G}$.