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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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