Closed subschemes and quasi-coherent sheaves of ideals

Question

Let me quote two results from Qing Liu's Algebraic Geometry and Arithmetic Curves:

Lemma 2.2.23: Let $X$ be a ringed space, $\mathcal{J}$ be a sheaf of ideals on $X$, $V(\mathcal{J}) = \{x\in X\;:\;\mathcal{J}_x \neq \mathcal{O}_{X,x}\}$, and $j:V(\mathcal{J})\to X$ be the inclusion. Then $V(\mathcal{J})$ is a closed subset of $X$, $(V(\mathcal{J}),j^{-1}(\mathcal{O}_X/\mathcal{J}))$ is a ringed space, and $(j,j^\#)$ is a closed immersion where $$j^\#:\mathcal{O}_X\to j_*j^{-1}(\mathcal{O}_X/\mathcal{J}) = \mathcal{O}_X/\mathcal{J}$$ is the canonical projection.

The very next proposition states the converse, that is a closed immersion $Y\to X$ gives rise to a sheaf of ideals (namely the kernel) whose closed subspace is isomorphic to $Y$. Explicitly,

Proposition 2.2.24: Let $f:Y\to X$ be a closed immersion of ringed spaces, $\mathcal{J}:=\operatorname{ker}f^\#$, and $Z = V(\mathcal{J})$. Then $f$ factors into an isomorphism $Y\simeq Z$ followed by the canonical closed immersion $Z\to X$.

Together, these imply that there is a one-to-one correspondence between sheaves of ideals $\mathcal{J}$ of $\mathcal{O}_X$ and closed subschemes $Z$ of $X$ via $\mathcal{J}\mapsto V(\mathcal{J})\mapsto \ker(\mathcal{O}_X\to\mathcal{O}_X/\mathcal{J}) = \mathcal{J}$. Am I interpreting this correctly?

I believe this is all good so far. What confuses me is what happens one introduces quasi-coherence.

Proposition 5.1.15: Let $X$ be a scheme, $Z$ be a closed subscheme, and $j:Z\to X$ denote the canonical closed immersion. Then $\operatorname{ker}j^\#$ is a quasi-coherent sheaf of ideals on $X$, and the map $Z\mapsto\operatorname{ker}j^\#$ establishes a bijection between closed subschemes of $X$ and quasi-coherent sheaves of modules on $X$.

Here are my two questions:

1. When working over a scheme $(X,\mathcal{O}_X)$, are all sheaves of ideals of $\mathcal{O}_X$ quasi-coherent?

Proof? If $\mathcal{J}$ is any sheaf of ideals of $\mathcal{O}_X$, then the canonical map $(j,j^\#):(V(\mathcal{J}),j^{-1}\mathcal{O}_X/\mathcal{J})\to(X,\mathcal{O}_X)$ is a closed immersion, i.e. $(V(\mathcal{J}),j^{-1}\mathcal{O}_X/\mathcal{J})$ is a closed subscheme of $X$, hence by Proposition 5.1.15, $\ker j^\# = \mathcal{J}$ is a quasi-coherent sheaf of ideals.

2. Could someone kindly provide an example of a locally ringed space $(X,\mathcal{O}_X)$ and a sheaf of ideals $\mathcal{J}$ that is not quasi-coherent?

Assuming 1. is actually correct, it ought to be the case that $X$ cannot be a scheme, right?

Answer 1

The answer to question 1 is no, and the counterexample we produce is also valid for question 2. Let $X=\Bbb A^1_k$, let $U$ be the complement of the origin, and consider $j_!\mathcal{O}_U$, the sheaf which takes the value $\mathcal{O}_U(W)$ for $W$ an open subset contained in $U$ and $0$ else. This is a sheaf of ideals, but it's not quasi-coherent: it has no nonzero global sections, despite not being the zero sheaf.

The correct statement is that there is a one-to-one correspondence between quasi-coherent sheaves of ideals on $X$ and closed subschemes of $X$.