The Stacks project

101.43 Valuative criterion for properness

Here is the statement.

Lemma 101.43.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

  1. $f$ is proper, and

  2. $f$ satisfies both the uniqueness and existence parts of the valuative criterion.

Proof. A proper morphism is the same thing as a separated, finite type, and universally closed morphism. Thus this lemma follows from Lemmas 101.41.2, 101.41.3, 101.42.1, and 101.42.2. $\square$

Comments (4)

Comment #2971 by on

Which "valuative criterion" is being refereed to in (2)? The most relevant one I could find is in tag 03IX, however this is stated for morphisms of algebraic spaces, not algebraic stacks.

Moreover, in the valuative criterion stated in tags 0A40 and 03IX, there is a base scheme S which seems to play no role.

Comment #3096 by on

The valuative criteria for morphisms between algebraic stacks are discussed in Section 101.39. See Definitions 101.39.6 and 101.39.10.

The thing about base schemes is discussed here.

Comment #5853 by Antoine Chambert-Loir on

Regarding Daniel Loughran's comment, I suggest adding a hypertext link to uniqueness part/existence part of the valuative criterion. I still have not understood whether I can/should make this modification myself and try to ask a commit.

Comment #5856 by on

@#5853. Sure, feel free to make this modification yourself and send me a pull request or send me a patch or something like that. Thanks!

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