The Stacks project

Lemma 67.41.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable. The following are equivalent

  1. $f$ satisfies the existence part of the valuative criterion as in Definition 67.41.1,

  2. 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 dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition 26.20.3.

Proof. It suffices to show that given a commutative diagram of the form

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar[rru]^\varphi & \mathop{\mathrm{Spec}}(A) \ar[r] & Y } \]

as in Definition 67.41.1, then we can find a morphism $\mathop{\mathrm{Spec}}(A) \to X$ fitting into the diagram too. Set $X_ A = \mathop{\mathrm{Spec}}(A) \times _ Y Y$. As $f$ is representable we see that $X_ A$ is a scheme. The morphism $\varphi $ gives a morphism $\varphi ' : \mathop{\mathrm{Spec}}(A') \to X_ A$. Let $x \in X_ A$ be the image of the closed point of $\varphi ' : \mathop{\mathrm{Spec}}(A') \to X_ A$. Then we have the following commutative diagram of rings

\[ \xymatrix{ K' & K \ar[l] & \mathcal{O}_{X_ A, x} \ar[l] \ar[lld] \\ A' \ar[u] & A \ar[l] & A \ar[l] \ar[u] } \]

Since $A$ is a valuation ring, and since $A'$ dominates $A$, we see that $K \cap A' = A$. Hence the ring map $\mathcal{O}_{X_ A, x} \to K$ has image contained in $A$. Whence a morphism $\mathop{\mathrm{Spec}}(A) \to X_ A$ (see Schemes, Section 26.13) as desired. $\square$


Comments (3)

Comment #2101 by Kestutis Cesnavicius on

In (1), "valuation criterion" ---> "valuative criterion", and similarly later in the section.


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Your email address will not be published.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 03K8. Beware of the difference between the letter 'O' and the digit '0'.