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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!

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  • $\begingroup$ (a) $\mathcal{O}$ is always coherent --- this is the first example immediately following the definition of quasicoherence and coherence. (b) is Proposition II.5.7. $\endgroup$ – user10354138 May 14 at 10:13
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    $\begingroup$ $\mathcal O$ is always quasi-coherent, but not necessarily coherent in the non-noetherian case. $\endgroup$ – asdq May 14 at 10:29
  • $\begingroup$ As for c), over an affine neighbourhood $U$ of $X$ (Note that $I_Y$ lives on $X$) the sheaf is just given by the kernel of the homomorphism $\mathcal O_X(U)\to\mathcal O_Y(U\cap Y)$. $\endgroup$ – asdq May 14 at 10:34

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