# Ring of functions is finitely generated in a quasi compact variety.

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 ?

• Is an algebraic variety necessarily of finite type over $k$ for you? – Watson Jun 29 '18 at 13:56
• No it is just a quasicompact space with a sheaf of regular functions that is locally isomorphic to an affine variety – Ignorant Mathematician Jun 29 '18 at 14:07
• What is an affine variety for you? Just an separated affine scheme? It should be of finite type over $k$, I guess. – Watson Jun 29 '18 at 14:09
• 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. – Ignorant Mathematician Jun 29 '18 at 14:16