MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

It is remarked in Shafarevich's Basic Algebraic Geometry 1 that Rees and Nagata constructed examples of quasiprojective varieties such that the ring of regular functions is not finitely generated, but I cannot find the source he is referring to. Can anyone give such examples here? Does that mean we can't really say anything about the ring of regular functions of a quasi-projective variety?

share|cite|improve this question
up vote 7 down vote accepted

It's a theorem that a quasi-projective variety is affine if and only if it is Stein (we're working over C, say) and its ring of functions is finitely generated. So find a Stein manifold that isn't affine, and that will do it.

And, after a bit of looking, it appears that Vakil may have rediscovered the Rees and Nagata example, here.

share|cite|improve this answer
I don't think that a Stein manifold that isn't affine will do the trick. For instance, if we take Serre's example ( P^1 bundle over an elliptic curve obtained as the projectivization of the unique non-trivial extension 0-> O -> V -> O -> 0 minus the section determined by O -> V) is Stein, and every regular function on it is constant since the section has zero self-intersection. – Jorge Vitório Pereira Oct 23 '09 at 11:49
I was certain that I'd read that Stein + f.g. => Affine for varieties, but that seems like a counterexample. I must be missing a hypothesis. Asking a question to try to find out exactly what is true. Must be some nontriviality hypothesis for the ring of regular functions, I'm guessing (maybe separates points?) – Charles Siegel Oct 23 '09 at 12:52
Ok, so according to Tony Pantev on my related question, what we need is a quasi-affine variety that is Stein. Then we have affine if and only if finitely generated. So it's a matter of looking for non-affine Stein manifolds which are quasi-affine. – Charles Siegel Oct 23 '09 at 21:26

"Does that mean we can't really say anything about the ring of regular functions of a quasi-projective variety?"

Since every variety contains an open affine, the ring of regular functions is always a subring of a finitely generated ring. (I assume that you consider varieties to be integral.) This is a nontrivial restriction. Also, the ring of regular functions will be noetherian, since any infinite ascending chain of ideals would give an infinite descending chain of subschemes.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.