The Stacks project

[II Theorem 7.3.8, EGA]

Lemma 29.42.1 (Valuative criterion for properness). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

  1. $f$ is proper,

  2. $f$ satisfies the valuative criterion (Schemes, Definition 26.20.3),

  3. given any commutative solid diagram

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

    where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute.

Proof. Part (3) is a reformulation of (2). Thus the lemma is a formal consequence of Schemes, Proposition 26.20.6 and Lemma 26.22.2 and the definitions. $\square$

Comments (3)

Comment #2709 by Ariyan Javanpeykar on

A reference for the valuative criterion of properness: EGA II, Theorem 7.3.8

Comment #3815 by Kestutis Cesnavicius on

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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BX5. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 29.42.1 as pdf