# Questions about Hartshorne Proposition II.5.9

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!

• (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. – user10354138 May 14 at 10:13
• $\mathcal O$ is always quasi-coherent, but not necessarily coherent in the non-noetherian case. – asdq May 14 at 10:29
• 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 $\mathcal O_X(U)\to\mathcal O_Y(U\cap Y)$. – asdq May 14 at 10:34