In Hartshorne p. 109 he defines a sheaf $\mathcal{F}$ of $O_X$-modules to be locally free if there is an open cover of $X$, s.t. on each $U$, $\mathcal{F}|_U$ is a free $O_X|_U$ module of rank $I$. Then if $X$ is connected, rank $I$ is globally well-defined. Here $(X,O_X)$ is any ringed topological space (e.g. not necessarily the structure sheaf of a ring). A similar definition is here: Locally free sheaf – Encyclopedia of Mathematics.
However, it didn't seem obvious to me that if $V$ is a smaller open set included in $U$ (say $U$ connected), then the number of copies $J$ of $(O_X|_V)^J=\mathcal{F}|_V$ would remain the same, because in general the restriction map of the sheaf $O_X$ or $\mathcal{F}$ from $U$ to $V$ need be neither subjective nor injective, why would the index $J$ stay the same as $I$?
Best Answer
Actually, there are two different restriction maps:
The first one (the one you correctly say is neither surjective nor injective in general) is that on sections: for $\mathcal{F}$ a sheaf on a scheme $X$ and two open subsets $V \subseteq U$, there is a map $\mathcal{F}(U) \to \mathcal{F}(V)$.
On the other hand, the inclusion map $i_{VU}: V \to U$ induces a functor $i_{VU}^{-1}$ from the category of sheaves on $U$ to the category of sheaves on $V$. This is called restriction of sheaves. By functoriality, $W \subseteq V \subseteq U$ yields $i_{WV}^{-1}\circ i_{VU}^{-1} = i_{WU}^{-1}$, so the notation is usually shortened to just $ -|_{W}$.
The second one is the one that you want to look at: the statement is then that $\mathcal{F}|_{U} \cong \mathcal{O}_X^{\oplus I}|_{U}$ implies $\mathcal{F}|_{V} \cong \mathcal{O}_X^{\oplus I}|_{V}$ for $V \subseteq U$.