Let $\mathcal J$ be a sheaf of ideals on a scheme $(X,\mathcal O_X)$ . Is it true that $\mathcal J$ is quasi-coherent if and only if for every affine open subset $U=Spec A$ of $X$, there is an ideal $J$ of $A$ such that $\mathcal J|_U \cong \tilde{J}$ ?
(Here by $\tilde{J}$ I mean , as usual, the sheaf on $U$ which takes every basic open set $D(f)$ of $U=Spec A$ to $J_f$ )