The set of points at which a morphism of affine schemes is not finite (Q2773357)

Let \(f:X\to Y\) be a morphism of affine varieties over an algebraically closed field \(k\). Let \(y\in Y\). We say that \(f\) is not finite at \(y\) if there exists no open affine neighbourhood \(U\) of \(y\) such that \(F|_{f^{-1}(U)}:f^{-1}(U)\to U\) is finite (in the case \(k=\mathbb C\) it means that \(f\) is not proper at \(y\)). \textit{Z. Jelonek} proved [Bull. Pol, Acad. Sci. Math. 49, 279-283 (2001; Zbl 1065.14074) and Math. Ann. 315, 1-35 (1999; Zbl 0946.14039)] that if \(f\) is generically finite and dominant then the set \(S_f\) of points at which \(f\) is not finite is either empty or a hypersurface. The main theorem of the paper is a generalization of this theorem to general affine schemes. Namely, if \(X,Y\) are affine integral schemes and \(f:X\to Y\) a dominant, generically finite morphism of finite type and \(Y\) is noetherian then \(S_f\) is either empty or a hypersurface in \(Y\). An example is also given to show that this is no longer true in the non-noetherian case.