Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is there an example where the induced morphism $X \to$ Spec $\Gamma(O_X)$ is not proper?
As the contributer a-fortiori notes in the comments to this question http://mathoverflow.net/questions/89473/is-hix-f-finitely-generated-over-gammao-x-if-f-is-coherent, there is no such example if all the groups $H^i(X,\mathcal{F})$ are known to be finitely generated over $k$. Not being strong in algebraic geometry, I can't off-hand tell whether his argument can be generalized.
