Morphisms between quasi-coherent sheaves

Question

Let $X$ be a noetherian scheme and let $F,G$ be coherent sheaves. Let $Z$ be a closed subscheme of $X$ not containing any associated point of $X$. Let $F|_U\to G|_U$ be a morphism of sheaves of modules, where $U=X\setminus Z$. Is it true that this extends at most uniquely to a morphism $F\to G$?

Answer 1

Let $R$ be DVR, $K$ its quotient field and $M,N$ torsion modules, i.e. $M \otimes_R K = 0= N \otimes_R K$. Then any morphism $M \to N$ is the zero morphism, after restricting to the generic point (which is an open set consisting of the only associated point of $R$). But you can easily choose $M$ and $N$, such that $Hom_R(M,N) \neq 0$

I think restricting $F$ and $G$ to be torsion free should make the statement true. It is then common known to be true if $X$ is integral and you should be able to reduce to this case, since $U$ does not miss an irreducible component. Haven't thought about non-reduced things though.