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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.