Terminal quasi-affine varieties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:28:18Z http://mathoverflow.net/feeds/question/67097 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67097/terminal-quasi-affine-varieties Terminal quasi-affine varieties? kummelweck 2011-06-06T22:44:03Z 2011-06-07T15:42:28Z <p>Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular functions on $U$. Write $Max(A)$ for the topological space of maximal ideals of $A$ (in the Zariski topology). Let $V\subset Max(A)$ be the union of all open subsets of the form $Max(A_f)$ where is $f\in A\setminus{0}$ is such that $A_f$ is a finitely generated $k$-algebra. Is it always the case that $V$ is quasi-affine? i.e. We know from the definition that $V$ is locally Noetherian. But is it always Noetherian?</p>