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

Question

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.

Answer 1

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 $R^{\oplus A}\to I$.