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Tag 0CLF

Chapter 91: Morphisms of Algebraic Stacks > Section 91.38: Valuative criteria

Lemma 91.38.5. Assume given a $2$-commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-x \ar[dd]_j & \mathcal{X} \ar[d]^f \\ & \mathcal{Y} \ar[d]^g \\ \mathop{\rm Spec}(A) \ar[r]^-z & \mathcal{Z} } $$ Choose a $2$-arrow $\gamma : z \circ j \to g \circ f \circ x$. Let $\mathcal{C}$ be the category of dotted arrows for the outer rectangle and $\gamma$. Let $\mathcal{C}'$ be the category of dotted arrows for the square $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-{f \circ x} \ar[d]_j & \mathcal{Y} \ar[d]^g \\ \mathop{\rm Spec}(A) \ar[r]^-z & \mathcal{Z} } $$ and $\gamma$. There is a canonical functor $\mathcal{C} \to \mathcal{C}'$ which turns $\mathcal{C}$ into a category fibred in groupoids over $\mathcal{C}'$ and whose fibre categories are categories of dotted arrows for certain squares of the form $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \mathop{\rm Spec}(A) \ar[r]^-y & \mathcal{Y} } $$ and some choice of $y \circ j \to f \circ x$.

Proof. Omitted. Hint: If $(a, \alpha, \beta)$ is an object of $\mathcal{C}$, then $(f \circ a, \text{id}_f \star \alpha, \beta)$ is an object of $\mathcal{C}'$. Conversely, if $(y, \delta, \epsilon)$ is an object of $\mathcal{C}'$ and $(a, \alpha, \beta)$ is an object of the category of dotted arrows of the last displayed diagram with $y \circ j \to f \circ x$ given by $\delta$, then $(a, \alpha, (\text{id}_g \star \beta) \circ \epsilon)$ is an object of $\mathcal{C}$. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 8752–8783 (see updates for more information).

    \begin{lemma}
    \label{lemma-cat-dotted-arrows-composition}
    Assume given a $2$-commutative diagram
    $$
    \xymatrix{
    \Spec(K) \ar[r]_-x \ar[dd]_j & \mathcal{X} \ar[d]^f \\
    & \mathcal{Y} \ar[d]^g \\
    \Spec(A) \ar[r]^-z & \mathcal{Z}
    }
    $$
    Choose a $2$-arrow $\gamma : z \circ j \to g \circ f \circ x$.
    Let $\mathcal{C}$ be the category of dotted arrows for
    the outer rectangle and $\gamma$. Let $\mathcal{C}'$ be the
    category of dotted arrows for the square
    $$
    \xymatrix{
    \Spec(K) \ar[r]_-{f \circ x} \ar[d]_j & \mathcal{Y} \ar[d]^g \\
    \Spec(A) \ar[r]^-z & \mathcal{Z}
    }
    $$
    and $\gamma$. There is a canonical functor $\mathcal{C} \to \mathcal{C}'$
    which turns $\mathcal{C}$ into a category fibred in groupoids over
    $\mathcal{C}'$ and whose fibre categories are categories of dotted arrows
    for certain squares of the form
    $$
    \xymatrix{
    \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\
    \Spec(A) \ar[r]^-y & \mathcal{Y}
    }
    $$
    and some choice of $y \circ j \to f \circ x$.
    \end{lemma}
    
    \begin{proof}
    Omitted. Hint: If $(a, \alpha, \beta)$ is an object of $\mathcal{C}$,
    then $(f \circ a, \text{id}_f \star \alpha, \beta)$ is an object
    of $\mathcal{C}'$. Conversely, if $(y, \delta, \epsilon)$ is an
    object of $\mathcal{C}'$ and $(a, \alpha, \beta)$ is an object
    of the category of dotted arrows of the last displayed diagram
    with $y \circ j \to f \circ x$ given by $\delta$, then
    $(a, \alpha, (\text{id}_g \star \beta) \circ \epsilon)$ is an
    object of $\mathcal{C}$.
    \end{proof}

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