In the chapter Schemes of the Stacks project, I am confused about Lemma 4.5, which I state here.
"Let $X$, $Y$ be locally ringed spaces, $\mathcal{I}\subset\mathcal{O}_X$ be a sheaf of ideals locally generated by sections. Let $i:Z\to X$ be the associated closed subspace. A morphism $f:Y\to X$ factors through $Z$ iff the map $f^*\mathcal{I}\to f^*\mathcal{O}_X = \mathcal{O}_Y$ is zero. If this is the case, the morphism $g:Y\to Z$ satisfying $f = i\circ g$ is unique."
I do not want to reproduce the proof here, it is available at the link above. I have parsed the proof many times over and for the life of me I cannot identify where the "locally generated by sections" hypothesis is used. Is it necessary at all? If so, could someone kindly provide a counter example?
The assumption is necessary to have $i:Z\to X$ is the associated closed subspace. The simplest example for a sheaf of ideals that not locally generated by sections (and hence, doesn't define a closed subscheme) is the following:
Let $R=k[x]_{(x)}$ so this scheme has only two points: $(0)$ and $(x)$ and only one open set $U=\{ (0) \}$ (beside $\emptyset $ and $X=\text{Spec}(R)$). We can define the sheaf $\mathcal{I}$ by $\mathcal{I}(U)=\mathcal{O}_X(U)=k(x)$ and $\mathcal{I}(X)=0$. This sheaf is not locally generated by sections (since $0$ can not generate the fiber $\mathcal{I}(X)_{(0)}=\mathcal{I}(U)=k(x)$). The support of this sheaf is precisely $\{ (0) \}$, which is not closed.