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?
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1$\begingroup$ 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. $\endgroup$– abxCommented Mar 4, 2019 at 15:59
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$\begingroup$ 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/… $\endgroup$– Jason StarrCommented Mar 4, 2019 at 18:06
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