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 ?
2026-03-25 03:02:58.1774407778
Ring of functions is finitely generated in a quasi compact variety.
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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