I have the following: a locally free sheaf $F$ on a scheme $X$, an $\mathcal{O}_X$-module $G$ and, for a fixed $x\in X$, a morphisms of $\mathcal{O}_{X,x}$-modules $F_{x}\to G_{x}$. Is it true that we can extend this morphism locally, that is, to find $V$ open neighborhood of $x$ and a morphisms of $\mathcal{O}_V$-modules $F|_V\to G|_V$ inducing the original one on stalks?
My attempt really stops when trying to find this suitable $V$, because I know that, since we're working in abelian categories, for any $s^x\in F_x$ I can find an open neighborhood $U$ of $x$ and a section $s\in F(U)$ such that $s_x=s^x$, and mapping this with the morphism on stalks to, say, $t^x$, lead to consider $W\ni x$ and a section $t\in G(W)$ such that $t_x=t^x$. But it doesn't seem to be that this takes me somewhere in order to define a morphism of sheaves, not even on a small neighborhood, beacuse any of these small neighborhood that I can find depends on the element $s^x$.
(I also think that it would be easy to find such a morphism on any open $U$ if I had maps on stalks for every $x$ in $U$, but here just one morphism in given).
Any help is appreciated! Thanks in advance
I write this down in full detail since others on this forum may have an interest in this question - it is a much asked question.
The question is local hence we may assume $X:=Spec(A)$, $E:=A\{e_1,..,e_n\}$ a free $A$-module of rank $n$ and $F$ any $A$-module. Let $p\in X$ be a point with corresponding prime ideal $\mathfrak{p}\in A$. Let $\phi_{\mathfrak{p}}: E_{\mathfrak{p}} \rightarrow F_{\mathfrak{p}}$ be an $A_{\mathfrak{p}}$-linear map. It follows $\phi(e_i)=x_i/t_i$ with $t_i\in A-\mathfrak{p}$ for all $i$. Let $t(i):=\prod_{k\neq i}t_k$ and let $t:=\prod_i t_i$. let $y_i:=t(i)x_i$. It follows $t(i),t\in \mathfrak{p}$ and there is an equality $y_i/t=x_i/t_i$ in $F_{\mathfrak{p}}$. The element $t$ is in $A$ and by construction it follows $\mathfrak{p}\in D(t)$. We may define an $A_t$-linear map $\phi_t:A_t\otimes E \rightarrow A_t \otimes F$ by $\phi_t(e_i):=y_i/t$. The map $\phi_t$ has the property that when we take the tensor product $A_{\mathfrak{p}}\otimes_{A_t}-$ we get back the map $\phi_{\mathfrak{p}}$. Let $\mathcal{E},\mathcal{F}$ be the sheaves correspoding to $E,F$.
Question: "Is it true that we can extend this morphism locally, that is, to find V open neighborhood of x and a morphisms of OV-modules F|V→G|V inducing the original one on stalks?"
Yes this is true. Above we have constructed a map of sheaves of $\mathcal{O}_{D(t)}$-modules $\phi_t:\mathcal{E}_{D(t)} \rightarrow \mathcal{F}_{D(t)}$ which gives back the map $\phi_{\mathfrak{p}}$ when we pass to the stalk at $\mathfrak{p}$. Here $\mathcal{E}_{D(t)}$ means we restrict the sheaf $\mathcal{E}$ to the open set $D(t)$.