Quasi-coherent sheaf

Remarks

On $X = \operatorname{Spec} A$, the sheaf $\widetilde{I}$ of an ideal $I \subseteq A$ is quasi-coherent, with the cover by $X$ itself. Let $R$ be a discrete valuation ring with fraction field $K$, and $X = \operatorname{Spec} R = \set{\eta, s}$, with $\eta$ the generic point. The sheaf $\mathcal{F}$ with $\mathcal{F}(X) = 0$ and $\mathcal{F}(\set{\eta}) = K$ is an $\mathcal{O}_X$-module but is not quasi-coherent. The only open set containing $s$ is $X$, so $\mathcal{F} \cong \widetilde{M}$ would force $M = \mathcal{F}(X) = 0$ and then $\mathcal{F}(\set{\eta}) = 0$. If $\mathcal{F}$ is quasi-coherent, then $\mathcal{F}|_U \cong \widetilde{\mathcal{F}(U)}$ for every affine open $U$, not only for those of the chosen cover. On a general ringed space, $\mathcal{F}$ is called quasi-coherent if it is locally the cokernel of a map between free modules; on schemes the two definitions agree.