We know a morphism between sheaf is isomorphic iff the induced morphism of stalks is isomorphic at every point.
- My question is: if $\alpha :\mathcal{F}\rightarrow \mathcal{F}$, if there exists a point $p$ such that the induced morphism of stalks at $p$ is isomorphic (identity, respectively), is there exists a neighborhood ${U}$ of $p$, such that $\alpha \mid_{U}:\mathcal{F}\mid_{U}\rightarrow \mathcal{F}\mid_{U} $ is isomorphic (identity)?
- If not, what additional condition do we need to have this property?