(I currently possess only the Japanese Edition of Hartshorne's book. Therefore, I kindly request that any answers provided do not rely on page numbers (of the English Edition) for reference. Thank you.)
In Corollary 4.8 in Chapter 2 of the book, it is stated that only closed immersions (and not open immersions) are considered proper. However, I believe the open immersions are proper under the assumption of the corollary. The following is the proof I wrote.
This corollary assumes that all schemes are noetherian. Since $X$ is noetherian, an open immersion $f: X \to Y$ should also be quasi-compact. Thus $f$ is of finite type, by Exercise 3.13.$\require{AMScd}$
\begin{CD} \mathrm{Spec}\ K @>>> B \\ @V{\text{$i$}}VV @VV{\text{$f$}}V \\ \mathrm{Spec}\ R @>{\text{$\phi$}}>> D \end{CD}
Now we can apply valuative criterion (Theorem 4.7). $f$ is a open immersion, so $f$ is a split mono and there is $g : Y \to X$ such that $gf = 1_X$. Then $g\phi : T \to X$ is the only morphism that remains the diagram above commutative. (The commutativity is evident. The uniqueness is followed by the fact that $f$ is separated or the fact that $f$ is a mono in the meaning of category theory.) By the criterion, it follows that $f$ is proper. $\blacksquare$
Is the proof above is wrong? Or was the fact that open immersions are proper under the assumption accidentally not written in the corollary?
