## Tag `0CLD`

Chapter 86: Morphisms of Algebraic Stacks > Section 86.34: Valuative criteria

Lemma 86.34.3. In Definition 86.34.1 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 $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \mathop{\rm Spec}(A) \ar[r]^-y \ar[ru]_a & \mathcal{Y} } $$ 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 $\mathop{\rm Spec}(K)$. Hence if $\mathit{Isom}_\mathcal{Y}(y, y) \to \mathop{\rm Spec}(A)$ is proper, then by the valuative criterion of properness (Morphisms of Spaces, Lemma 55.43.1) we can find $\delta : y \to y$ whose restriction to $\mathop{\rm Spec}(K)$ 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. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 7676–7694 (see updates for more information).

```
\begin{lemma}
\label{lemma-cat-dotted-arrows-independent}
In Definition \ref{definition-fill-in-diagram}
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
$$
\xymatrix{
\Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\
\Spec(A) \ar[r]^-y \ar[ru]_a & \mathcal{Y}
}
$$
with $2$-commutative triangles
(without checking the $2$-morphisms compose correctly).
\end{lemma}
\begin{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 $\Spec(K)$.
Hence if $\mathit{Isom}_\mathcal{Y}(y, y) \to \Spec(A)$
is proper, then by the valuative criterion of properness
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-characterize-proper})
we can find $\delta : y \to y$ whose restriction to
$\Spec(K)$ 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.
\end{proof}
```

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