9
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • $\begingroup$ 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
    Mar 3, 2012 at 4:25

1 Answer 1

Reset to default
1
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.