Let $X$ be a scheme. You may assume that it is nice enough, perhaps of finite type over a field $k$ and smooth. In my application $X$ is not separated, a priori.
Assume that every coherent sheaf on $X$ is generated by global sections. Does it follow that $X$ is quasi-affine?
You can take any scheme $X$ of finite type over a noetherian affine scheme $S$. The condition on the generation by global sections says exactly that $O_X$ is ample (EGA II, 4.5.5(d)). The existence of an invertible ample sheaf implies $X$ is separated.
By EGA, II.5.1.2, $X$ is quasi-affine. In fact, as $O_X$ is relatively ample for $X\to S$ (EGA, loc.cit.), $X$ is even open in an affine scheme of finite type over $S$.
