Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\mathcal{X}$. Is $H^i(X,\mathcal{F})$ finitely generated over $\Gamma(O_X)$ if $\mathcal{F}$ is coherent ? This statement is simple enough that I probably would have heard it if it were true.

If it makes a difference, the statement that I really would like to understand is whether $\Gamma(O_X) \to Ext^i(\mathcal{G},\mathcal{G})$ is module finite for any coherent sheaf.

Most of the schemes that I am "friendly with" are either projective or affine. The statements are correct in those particular cases, so probably I just need to learn more examples... Thanks!