Is $\mathcal F$ isomorphic to $\widetilde M$ for some ideal $M$ of $A$?

Question

Let $A$ be a commutative ring with unity, and let $\mathcal F$ be a sub-$\mathcal O_{\operatorname{Spec}A}$-module of $\mathcal O_{\operatorname{Spec}A}$.

Is $\mathcal F$ isomorphic to $\widetilde M$ for some ideal $M$ of $A$?

Answer 1

What you are looking for are ideal sheaves which are not quasi-coherent.

Here's an example: Let $A$ be a DVR. Consider the sheaf $\mathcal{F}$ on $X=\operatorname{Spec}A$ defined by $\mathcal{F}(X) = 0 $ and $\mathcal{F}(\eta) = K$, where $\eta$ is the generic point of $A$ and $K$ is the fraction field of $A$. Then this is not isomorphic to $\tilde{M}$ for any $A-$module $M$.