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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 Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ 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.

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  • $\begingroup$ I would suggest to consider the relative compactification $\bar X$ of $X$ over $Spec \Gamma(O_X)$. If $\bar X \ne X$ then most probably it would be easy to produce a coherent sheaf with non finitely generated cohomology group. $\endgroup$
    – Sasha
    Commented Mar 3, 2012 at 4:25

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See this question. Basically, if your variety is not proper, then valuative criterion gives you non-complete curves and you can deal with them directly.

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