Lemma 100.41.3. 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 100.39.2. 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 100.39.1. 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 100.3.3) and proper as $f$ is separated. Hence by Lemma 100.39.13 and the valuative criterion for properness for algebraic spaces (Morphisms of Spaces, Lemma 66.44.1) 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. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)