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Apologies for the question which I am quite sure is an exercise in Hartshorne, but I am at a conference without my Hartshorne, and I want to claim a fact in a talk.

By definition, a scheme $X$ is finite type (over the base $k$, which can be $\mathbb{C}$ if you prefer) if it can be covered by finitely many affine schemes, each of which is Spec of an finitely-generated ring (over $k$).

If an affine scheme $X$ is finite type, then the definition tells us only that there are plenty of finitely-generated localizations of $\mathcal{O}_X$. However, I vaguely recall this was one of the properties that was 'independent of affine cover', as in, if it was true for one affine cover, then it was true for all of them. If so, then it is true for the affine cover of $X$ by itself; ie, $\mathcal{O}_X$ is finitely-generated.

I haven't yet succeeded in coming up with a proof. So, I ask, if $X$ is finite type, is $\mathcal{O}_X$ finitely-generated?

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Someone asked this at MO: mathoverflow.net/questions/52014/… –  Matt Sep 9 '11 at 14:53
Thank you, that is most helpful. –  Greg Muller Sep 9 '11 at 19:02

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