Let $(X,O_X)$ be a scheme, let $F$ be a locally free sheaf of $O_X$-modules on $X$.
According to this MSE answer, it is not necessarily true that for any affine open $U \subset X$ we have $F|_U$ is (isomorphic to) a direct sum of $O_U$'s.
However, is it possible to find an affine open base $\{U_i\}_{i\in I}$ of $X$,
such that $F|_{U_i}$ is isomorphic to $\bigoplus_{j\in J_i}(O_{U_i})$ for some index set $J_i$, for each $i \in I$?
My thinking so far is, I would like to show if $F$ is "free over" some open set $U$, then it is also "free over" any open subset of $U$ (or at least, for each $x \in U$ there is an open $V$ with $x \in V \subset U$ such that $F|_U$ is free over $V$).
Since a localization of a free module is free, I see why the above is true when $U$ is affine, but I'm not sure what to do for the non-affine case. Could anyone explain how to approach this? Or, is my thinking incorrect here?
Best Answer
a few remarks that are maybe helpful:
So without loss of generality you can choose your neighbourhood $U$ around $x$ to be affine, but you don't need that: