2
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Suppose $(V, \mathcal{O}_V)$ be a quasi compact algebraic variety over $k$. Is $ \Gamma (V, \mathcal{O}(V))$ always finitely generated as an $k$ algebra ?

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  • $\begingroup$ Is an algebraic variety necessarily of finite type over $k$ for you? $\endgroup$ – Watson Jun 29 '18 at 13:56
  • $\begingroup$ No it is just a quasicompact space with a sheaf of regular functions that is locally isomorphic to an affine variety $\endgroup$ – Ignorant Mathematician Jun 29 '18 at 14:07
  • $\begingroup$ What is an affine variety for you? Just an separated affine scheme? It should be of finite type over $k$, I guess. $\endgroup$ – Watson Jun 29 '18 at 14:09
  • $\begingroup$ Yes. We haven’t been introduced to schemes. But an affine variety is just zero set of polynomials in A_n with the usual structure sheaf. $\endgroup$ – Ignorant Mathematician Jun 29 '18 at 14:16
3
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This is not true. The following note by Vakil gives a 3 dimensional counterexample:

http://math.stanford.edu/~vakil/files/nonfg.pdf

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