The Stacks project

Lemma 101.41.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion.

Proof. Since $f$ is separated we see that all categories of dotted arrows are setoids by Lemma 101.39.2. Consider a diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y} } \]

and a $2$-morphism $\gamma : y \circ j \to f \circ x$ as in Definition 101.39.1. Consider two objects $(a, \alpha , \beta )$ and $(a', \beta ', \alpha ')$ of the category of dotted arrows. To finish the proof we have to show these objects are isomorphic. The isomorphism

\[ f \circ a \xrightarrow {\beta ^{-1}} y \xrightarrow {\beta '} f \circ a' \]

means that $(a, a', \beta ' \circ \beta ^{-1})$ is a morphism $\mathop{\mathrm{Spec}}(A) \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. On the other hand, $\alpha $ and $\alpha '$ define a $2$-arrow

\[ (a, a', \beta ' \circ \beta ^{-1}) \circ j = (a \circ j, a' \circ j, (\beta ' \star \text{id}_ j) \circ (\beta \star \text{id}_ j)^{-1}) \xrightarrow {(\alpha , \alpha ')} (x, x, \text{id}) = \Delta _ f \circ x \]

Here we use that both $(a, \alpha , \beta )$ and $(a', \alpha ', \beta ')$ are dotted arrows with respect to $\gamma $. We obtain a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[d]_ j \ar[rr]_ x & & \mathcal{X} \ar[d]^{\Delta _ f} \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^{(a, a', \beta ' \circ \beta ^{-1})} & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]

with $2$-commutativity witnessed by $(\alpha , \alpha ')$. Now $\Delta _ f$ is representable by algebraic spaces (Lemma 101.3.3) and proper as $f$ is separated. Hence by Lemma 101.39.13 and the valuative criterion for properness for algebraic spaces (Morphisms of Spaces, Lemma 67.44.1) we see that there exists a dotted arrow. Unwinding the construction, we see that this means $(a, \alpha , \beta )$ and $(a', \alpha ', \beta ')$ are isomorphic in the category of dotted arrows as desired. $\square$

Comments (0)

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Your email address will not be published.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