Hom functor of quasi-coherent sheaf maybe not quasi-coherent

Question

I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give me a counterexample?

Answer 1

Consider an affine scheme the spectrum of a DVR $M$, then $N=\oplus_{\mathbb{N}}M$ and $M$ are q.c. and via 4.21, Algebraic_Geometry:_Sheaves_and_cohomology $Hom_{O_{spec M}} (N,M)$ is q.c. iff isomorphic to its associated sheaf. Localizing at a non-unit should preserve this ismorphism via Hartshorne II.5.1.c, but it doesn't.