# Valuative criterion over non-locally Noetherian base

Let $$X$$ be an irreducible scheme and $$f:X\rightarrow S$$ be a morphism of finite type. Let $$\eta$$ be the generic point of $$X$$. Assume that for any (not necessarily discrete) valuation ring $$A \subset K=k(\eta)$$ with $$\mathrm{Frac}(A)=K$$ and any diagram such that $$\mathrm{Spec}\,K\rightarrow X$$ is the inclusion of generic point there exists a unique dotted arrow. Is $$f$$ necessarily proper?

• No. Any separated universally closed morphism $f$ satisfies your condition, see EGA I, Proposition 5.5.8. If $f$ is not of finite type it is not proper. – abx Mar 4 at 15:59
• Welcome new contributor. Under the finite type hypothesis, this follows from EGA IV_3, Section 8. To prove that $f$ is proper, it suffices to work locally on the target. For every open subset of $S$ that does not contain $f(\eta)$, the image of $f$ is contained in the closed complement. Every open affine $U$ that does contain $f(\eta)$ is a projective limit of Noetherian affine schemes. Over one of these, there exists a morphism whose base change to $U$ is $f^{-1}(U)\to U$. Now use: mathoverflow.net/questions/493/… – Jason Starr Mar 4 at 18:06