The Stacks project

Maybe it would be good to add, in addition to the reference to 01KF, a reference (in the same place) to 01IQ, since the valuative criterion shows (in particular) that the diagonal is closed, while 01IQ says that this closedness is enough to conclude that the diagonal is a closed immersion.

THanks, fixed here.

A reference for the valuative criterion of separatedness: EGA II, Proposition 7.2.3

Added reference; thanks!

Same comment as for https://stacks.math.columbia.edu/tag/01KE

It should be the uniqueness part of valuative criterion in the proof.

In last part of proof, it seems need to show that morphism $a,b$ satisfy commutative diagram. (non trivial part)

Dear Zhaodong and minsom, I think what it says is correct. We have to show that $f$ is separated. This is equivalent to showing the diagonal $\Delta$ is closed. This is equivalent to the diagonal being closed as a map of topological spaces by Lemma 26.10.4 and the fact that the diagonal is always an immersion by Lemma \ref{01KJ. By Proposition 26.20.6 it is enough to show the existence part of the valuative criterion for $\Delta$. But by the functorial nature of the diagonal, the existence part for the diagonal $\Delta$ is equivalent to the uniquenes part for $X \to S$!!! Please try to read the proof again and see if you still disagree.

If this doesn't help, then please exactly say how to improve the wording.

Meanwhile I have added a reference to Lemma \ref{01KJ in this commit.