Does there exist an ideal sheaf $\mathcal F$ on some affine scheme $X$ such that $\mathcal F$ is not quasi-coherent?

Question

Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $\mathcal F$ on $X$ such that $\mathcal F$ is not quasi-coherent.

Answer 1

It often helps to translate the problem into commutative algebra. I will define ideal sheaves on a scheme $X$ to be an $\mathcal{O}_X$ module $\mathcal{I}$ such that for all open sets $U\subset X$, $\mathcal{I}(U)$ is an ideal of $\mathcal{O}_X(U)$. So let $\mathcal{I}$ be an ideal sheaf on $X=\operatorname{Spec}A$. Recall that quasicoherence can be checked by showing that the natural map $\Gamma(\operatorname{Spec}A,\mathcal{I})_f\rightarrow \Gamma(\operatorname{Spec}A_f,\mathcal{I})$ is an isomorphism for all $f$. We clearly have $\Gamma(\operatorname{Spec}A,\mathcal{I})_f=I_f$ where $I$ is some ideal of $A$. Also $\Gamma(\operatorname{Spec}A_f,\mathcal{I})=J$ where $J$ is some ideal of $A_f$. Can you find an ideal $J$ of $A_f$ that doesn't come from localization of an ideal $I$ of $A$?

Answer 2

Now that I understood the problem with my (deleted) answer, I try to write an easy example of an ideal sheaf that it is not quasicoherent:

Let $R$ be a discrete valuation ring, with field of fractions $K$, and let $X=\operatorname {Spec}(R)=\{x,\eta\}$ be the associated affine scheme with generic point $\eta$ and closed point $x$. Consider now the ideal sheaf $\mathcal I\subset \mathcal O_X$ defined by $\mathcal I(X)=0$ and $\mathcal I(\{\eta\})=K$.

This ideal sheaf is not quasicoherent.

For example, you can see that the "associated closed subscheme", in case it were quasicoherent, would correspond to the set $\{\eta\}$, which it is not closed.

Or you can follow Samir Canning suggestion and localize at the uniformizer $\pi$ of $R$. Then $R_{\pi}=K$, $\Gamma(\operatorname{Spec}(R_{\pi}),\mathcal I)=K$, while $\Gamma(\operatorname{Spec}(R),\mathcal I)_{\pi}=0$.