I know that if $Y \hookrightarrow X$ is a closed embedding i of schemes, then the sheaf of ideals
$I_Y(U) = $ {$f \in \mathcal{O}_X(U)\text{ } | i^*(f) = 0$}
is quasi coherent. I sort of understand the proof that Hartshorne gives for this (Proposition 5.9, Chapter 2), but I'm having trouble with one thing conceptually.
By the definition, $I_Y$ being quasi coherent means that there is an affine cover {$U_\alpha$} of X such that $I_Y|_{U_\alpha} \cong \widetilde M$ for some $\mathcal{O}_X(U_\alpha)$ module M.
What exactly is $\widetilde M$ and the isomorphism between $I_Y|_{U_\alpha}$ and $\widetilde M$? I don't see understand this structure based on the proof given. I'm not really sure intuitively what it should be either. (Still trying to get aquainted with algebraic geometry).
I'd appreciate any help.