My question is about the first part of the proof.
Let $X$ be a scheme. For any closed subscheme $Y$ of $X$, the corresponding ideal sheaf $I_Y$ is a quasi-coherent sheaf of ideals.
Proof: Since why is a closed subscheme of X, $ i : Y \rightarrow X$ is quasi compact and separated and by Proposition 5.8 in Hartshorne, $i_*\mathcal{O}_Y$ is quasi coherent. Since $I_Y$ is the kernel of a morphism of quasi-coherent sheaves, $I_Y$ is quasi coherent.
My questions are the following:
a) Proposition 5.8 states:
Let $f: Y \rightarrow X$ be a morphism of schemes.
If $\mathcal{G}$ is a quasi coherent sheaf of $\mathcal{O}_X$ modules then $f^*\mathcal{G}$ is a quasi coherent sheaf of $\mathcal{O}_Y$ modules.
If f is quasi compact and separated, then if $\mathcal{F}$ is a quasi coherent sheaf of $\mathcal{O}_Y$ modules, $f_*\mathcal{F}$ is a quasi coherent sheaf of $\mathcal{O}_X$ modules
There's another statement about when X and Y are noetharian but those (I think) don't apply since neither $X$ nor $Y$ is assumed to be Noetherian.
But to use this proposition, it seems like we need that either $\mathcal{O}_Y$ or $\mathcal{O}_X$ is quasi coherent, which a-priori doesn't seem to be true.
b) This might be a trivial question but I really don't understand why the kernel of a morphism of quasi coherent sheaves should be quasi coherent itself.
c) I wanted to ask if someone could show me a direct proof for finding and explicitly showing the sheaf $\tilde M$ such that $I_Y|_U \cong \tilde M$ where $U = Spec A$ is some affine neighborhood in Y and for some $A$-module M, $\tilde M(D(f)) = M_f$
Or if this is directly evident from the proof in Hartshone, then I don't see what $\tilde M$ and the isomorphism should look like.
Any help would be appreciated. Thanks!