The Stacks project

[II Proposition 7.2.3, EGA]

Proof. By assumption (1), Proposition 26.20.6, and Lemmas 26.21.2 and 26.10.4 we see that it suffices to prove the morphism $\Delta _{X/S} : X \to X \times _ S X$ satisfies the existence part of the valuative criterion. Let a solid commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & X \times _ S X } \]

be given. The lower right arrow corresponds to a pair of morphisms $a, b : \mathop{\mathrm{Spec}}(A) \to X$ over $S$. By (2) we see that $a = b$. Hence using $a$ as the dotted arrow works. $\square$

Comments (8)

Comment #1725 by Keenan Kidwell on

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.

Comment #2706 by Ariyan Javanpeykar on

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

Comment #3814 by Kestutis Cesnavicius on

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

Comment #5388 by Zhaodong Cai on

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

Comment #5504 by minsom on

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

Comment #5622 by on

Dear Zhaodong and minsom, I think what it says is correct. We have to show that is separated. This is equivalent to showing the diagonal 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 . But by the functorial nature of the diagonal, the existence part for the diagonal is equivalent to the uniquenes part for !!! 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.

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