The Stacks project

Lemma 99.39.3. In Definition 99.39.1 assume $\mathcal{I}_\mathcal {Y} \to \mathcal{Y}$ is proper (for example if $\mathcal{Y}$ is separated or if $\mathcal{Y}$ is separated over an algebraic space). Then the category of dotted arrows is independent (up to noncanonical equivalence) of the choice of $\gamma $ and the existence of a dotted arrow (for some and hence equivalently all $\gamma $) is equivalent to the existence of a diagram

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

with $2$-commutative triangles (without checking the $2$-morphisms compose correctly).

Proof. Let $\gamma , \gamma ' : y \circ j \longrightarrow f \circ x$ be two $2$-morphisms. Then $\gamma ^{-1} \circ \gamma '$ is an automorphism of $y$ over $\mathop{\mathrm{Spec}}(K)$. Hence if $\mathit{Isom}_\mathcal {Y}(y, y) \to \mathop{\mathrm{Spec}}(A)$ is proper, then by the valuative criterion of properness (Morphisms of Spaces, Lemma 65.44.1) we can find $\delta : y \to y$ whose restriction to $\mathop{\mathrm{Spec}}(K)$ is $\gamma ^{-1} \circ \gamma '$. Then we can use $\delta $ to define an equivalence between the category of dotted arrows for $\gamma $ to the category of dotted arrows for $\gamma '$ by sending $(a, \alpha , \beta )$ to $(a, \alpha , \beta \circ \delta )$. The final statement is clear. $\square$


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