# Ideal sheaf is quasi-coherent if and only if its generated by local sections.

My confusion is lies in Schemes Lemma 10.1 of the Stacks project.

First, Modules Definition 8.1 states that a sheaf $$\mathcal{F}$$ of $$\mathcal{O}_X$$-modules is locally generated by sections if for all $$x\in X$$, there is a neighborhood $$U$$ of $$x$$ and a surjection $$\mathcal{O}_U^{(I)}\to\mathcal{F}|_U$$ where $$\mathcal{O}_U^{(I)} = \bigoplus_{i\in I}\mathcal{O}_U$$.

Now Schemes Lemma 10.1 states "Let $$(X,\mathcal{O}_X)$$ be a scheme, $$i:Z\to X$$ be a closed immersion of locally ringed spaces:

(1) The locally ringed space $$Z$$ is a scheme.
(2) The kernel $$\mathcal{I}$$ of the map $$\mathcal{O}_X\to i_*\mathcal{O}_Z$$ is a quasi-coherent sheaf of ideals.
(3) for every affine open $$U = \operatorname{Spec}(R)$$ of $$X$$, the morphism $$i^{-1}(U)\to U$$ can be identified with $$\operatorname{Spec}(R/I)\to\operatorname{Spec}(R)$$ for some ideal $$I$$ of $$R$$, and
(4) we have $$\mathcal{I}|_U = \widetilde{I}$$.

In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of $$X$$ is a scheme."

Question: It is this remark "In particular..." at the end which I do not understand at all. I don't even see where a module that is locally generated by sections appears in the statement, how am I supposed to conclude that a module locally generated by sections is quasi-coherent?

I am quite new to scheme theory, so I would very much appreciate a reasonably detailed response.

The way that the condition "locally generated by sections" shows up in this result is via the definition of a closed immersion:

Definition (01HK): Let $$i:Z\to X$$ be a morphism of locally ringed spaces. We say that $$i$$ is a closed immersion if:

• (1) The map $$i$$ is a homeomorphism of $$Z$$ on to a closed subspace of $$X$$.
• (2) The map $$\mathcal{O}_X\to i_*\mathcal{O}_Z$$ is surjective; let $$\mathcal{I}$$ denote the kernel.
• (3) The $$\mathcal{O}_X$$-module $$\mathcal{I}$$ is locally generated by sections.

Property (3) from this definition is how locally generated by sections enters the picture.

Now to see the remark you're struggling with. Suppose $$\mathcal{I}$$ is a sheaf of ideals which is locally generated by sections. Form a closed subscheme $$V(\mathcal{I})\subset X$$ as the support of $$\mathcal{O}_X/\mathcal{I}$$ equipped with the structure sheaf $$\mathcal{O}_X/\mathcal{I}$$. (All you have to do to check this is a closed subscheme is to verify the support is closed, which follows directly from looking at stalks.) Now you can apply conclusion (2) from your lemma: $$\mathcal{I}$$ is quasi-coherent.

The vice-versa comes from observing that if $$\mathcal{I}$$ is quasi-coherent, then over any affine open $$\operatorname{Spec} R\subset X$$, we have that $$\mathcal{I}\cong\widetilde{I}$$ for $$I\subset R$$ an ideal. Such a sheaf is generated by sections over $$\operatorname{Spec} R$$ by applying the $$\widetilde{-}$$ functor to a choice of generators $$R^{\oplus A}\to I$$.