The Stacks project

Definition 101.39.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ satisfies the existence part of the valuative criterion if for every diagram (101.39.1.1) and $\gamma $ as in Definition 101.39.1 there exists an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the outer rectangle of the diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar@/^2em/[rr]_{x'} \ar[d]_{j'} & \mathop{\mathrm{Spec}}(K) \ar[d]_ j \ar[r]_-x & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar@/_2em/[rr]^{y'} & \mathop{\mathrm{Spec}}(A) \ar[r]^-y & \mathcal{Y} } \]

with induced $2$-arrow $\gamma ' : y' \circ j' \to f \circ x'$ is nonempty.


Comments (0)

There are also:

  • 2 comment(s) on Section 101.39: Valuative criteria

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Your email address will not be published.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 0CLK. Beware of the difference between the letter 'O' and the digit '0'.