# Regarding a sheaf of $\mathcal O_X$-modules as a sheaf of $\mathcal O_Z$-modules, where $Z$ is a closed subscheme

Let $(X,\mathcal O_X)$ be a scheme, and $(Z,\mathcal O_Z)$ a closed subscheme, with $i: Z \rightarrow X$ the defining closed immersion. The underlying map of topological spaces is the inclusion. Since $i^{\#}: \mathcal O_X \rightarrow i_{\ast} \mathcal O_Z$ is a surjective morphism of sheaves of rings, the kernel $\mathscr I$ of $i^{\#}$ induces an isomorphism of sheaves of rings $\mathcal O_X/\mathscr I \cong i_{\ast} \mathcal O_Z$. Since we are dealing with a sheaf of rings $\mathcal O_Z$ on a closed subset, we recover $\mathcal O_Z$ by taking the inverse image sheaf:

$$\mathcal O_Z \cong i^{-1} (\mathcal O_X/\mathscr I)$$

Now, let $F$ be a sheaf of $\mathcal O_X$-modules. Suppose that for each open $U$, $\mathscr I(U)$ annihilates $F(U)$. Then $F$ is well defined as a sheaf of $\mathcal O_X/\mathscr I$-modules.

This implies $i^{-1}F$ is a sheaf of $\mathcal O_Z = i^{-1}(\mathcal O_X/\mathscr I)$-modules.

Suppose furthermore that for each $x \in X - Z$, the stalk $F_x = 0$. Then $F$ can be recovered as $i_{\ast} i^{-1}F$.

In this case, does it make sense to say that $F$ is a sheaf of $\mathcal O_Z$-modules? (even though $F$ and $\mathcal O_Z$ are sheaves on different topological spaces) If so, what is the meaning of "$F$ is a sheaf of $\mathcal O_Z$-modules?"

I have encountered this language here (Mumford, "Red Book of Varieties and Schemes"):