Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent?
By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that $\mathcal{F}|_{U_i}$ is isomorphic to the sheaf $\widetilde{\mathcal{O}(U_i)^n}$. But we don't know each $U_i$ affine or not!
So, it that true or not?
How about $X$ being locally noetherian? It $X$ is, we can find $V_{ij} \subset U_i$ s.t. $V_{ji} = Spec(A_{ji})$. And $\mathcal{F}|_{V_{ij}} = \mathcal{F}|_{U_i}|_{V_{ij}}$...?
For example, $X(\Delta)$ is a toric varity with $\Delta$ consists of strongly convex polyhedral cones.
Thank you very much!!