Lemma 67.43.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion.

Proof. Let a diagram as in Definition 67.41.1 be given. Suppose there are two distinct morphisms $a, b : \mathop{\mathrm{Spec}}(A) \to X$ fitting into the diagram. Let $Z \subset \mathop{\mathrm{Spec}}(A)$ be the equalizer of $a$ and $b$. Then $Z = \mathop{\mathrm{Spec}}(A) \times _{(a, b), X \times _ Y X, \Delta } X$. If $f$ is separated, then $\Delta$ is a closed immersion, and this is a closed subscheme of $\mathop{\mathrm{Spec}}(A)$. By assumption it contains the generic point of $\mathop{\mathrm{Spec}}(A)$. Since $A$ is a domain this implies $Z = \mathop{\mathrm{Spec}}(A)$. Hence $a = b$ as desired. $\square$

Comment #970 by Kestutis Cesnavicius on

Perhaps it would be worthwhile to have a reference to http://stacks.math.columbia.edu/tag/03IX at the end of the statement of the lemma?

Comment #1004 by on

This kind of thing happens all over the place. The solution would be to add more structure to our underlying LaTeX. For example, any reference to a mathematical property would have a reference back to the definition and we could then decorate the webpages to show this. But this would make it essentially harder to write latex for the Stacks project than it is to write LaTeX for the Stacks project which we do not want.

Now in this case in particular, there is a reference to the definition of "uniqueness part of the valuative criterion" immediately in the text of the proof (on the first line). Generally speaking when this happens we do not also put in a reference in the text of the lemma, because we expect the reader to start reading the proof in order to find out what the exact definitions of things are.

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