In the answer to  [this](https://mathoverflow.net/questions/2071/non-finitely-generated-ring-of-regular-functions/)  question we saw that  [there exists](http://math.stanford.edu/~vakil/files/nonfg.pdf)  a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.


I was talking about this previous question today and the following question came up - given any countably generated noetherian k-algebra R which is an integral domain and whose field of fractions has finite transcendence degree over k, where k is a field does there exist some quasi-projective variety X (by variety I mean an integral separated scheme of finite type over k) such that the ring of global sections of X is R?


It is possible one needs more hypotheses to make this work - if this is false I think it would be interesting to know the class of algebras which can occur.



Can any countably generated k-algebra occur as the ring of global sections of some variety?
No. Take k[t] and invert countably many relatively prime polynomials. This obeys all of your adjectives (localization preserves noetherianness, the others are obvious.)


However, a ring of global sections must be a subring of some finitely generated k-algebra. (I pointed this out in the last discussion.) Hence, its unit group must be a subgroup of the group of units of a finitely generated k-algebra. If A is any finitely generated k-algebra, then Units(A)/Units(k) is a finitely generated abelian group, and thus can't contain the countably generated group of the above example.


I can't find a reference for the fact about Units(A)/Units(k) at the moment, Tevelev describes this as well known in


* *Compactifications of subvarieties of tori*, American Journal of Mathematics **129** no. 4 (2007) pp. 1087–1104, doi:[10.1353/ajm.2007.0029](https://doi.org/10.1353/ajm.2007.0029), arXiv:[math/0412329](https://arxiv.org/abs/math/0412329)


