The Stacks Project


Tag 0CLA

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

Definition 91.38.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram \begin{equation} \tag{91.38.1.1} \vcenter{ \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] & \mathcal{Y} } } \end{equation} where $A$ is a valuation ring with field of fractions $K$. Let $$ \gamma : y \circ j \longrightarrow f \circ x $$ be a $2$-morphism witnessing the $2$-commutativity of the diagram. (Notation as in Categories, Sections 4.27 and 4.28.) Given (91.38.1.1) and $\gamma$ a dotted arrow is a triple $(a, \alpha, \beta)$ consisting of a morphism $a : \mathop{\rm Spec}(A) \to \mathcal{X}$ and $2$-arrows $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$ such that $\gamma = (\text{id}_f \star \alpha) \circ (\beta \star \text{id}_j)$, in other words such that $$ \xymatrix{ & f \circ a \circ j \ar[rd]^{\text{id}_f \star \alpha} \\ y \circ j \ar[ru]^{\beta \star \text{id}_j} \ar[rr]^\gamma & & f \circ x } $$ is commutative. A morphism of dotted arrows $(a, \alpha, \beta) \to (a', \alpha', \beta')$ is a $2$-arrow $\theta : a \to a'$ such that $\alpha = \alpha' \circ (\theta \star \text{id}_j)$ and $\beta' = (\text{id}_f \star \theta) \circ \beta$.

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 8605–8644 (see updates for more information).

    \begin{definition}
    \label{definition-fill-in-diagram}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
    Consider a $2$-commutative solid diagram
    \begin{equation}
    \label{equation-diagram}
    \vcenter{
    \xymatrix{
    \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\
    \Spec(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y}
    }
    }
    \end{equation}
    where $A$ is a valuation ring with field of fractions $K$. Let
    $$
    \gamma : y \circ j \longrightarrow f \circ x
    $$
    be a $2$-morphism witnessing the $2$-commutativity of the diagram.
    (Notation as in Categories, Sections \ref{categories-section-formal-cat-cat}
    and \ref{categories-section-2-categories}.)
    Given (\ref{equation-diagram}) and $\gamma$
    a {\it dotted arrow} is a triple $(a, \alpha, \beta)$ consisting of a
    morphism $a : \Spec(A) \to \mathcal{X}$ and $2$-arrows
    $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$
    such that
    $\gamma = (\text{id}_f \star \alpha) \circ (\beta \star \text{id}_j)$,
    in other words such that
    $$
    \xymatrix{
    & f \circ a \circ j \ar[rd]^{\text{id}_f \star \alpha} \\
    y \circ j \ar[ru]^{\beta \star \text{id}_j} \ar[rr]^\gamma & &
    f \circ x
    }
    $$
    is commutative. A {\it morphism of dotted arrows}
    $(a, \alpha, \beta) \to (a', \alpha', \beta')$ is a
    $2$-arrow $\theta : a \to a'$ such that
    $\alpha = \alpha' \circ (\theta \star \text{id}_j)$ and
    $\beta' = (\text{id}_f \star \theta) \circ \beta$.
    \end{definition}

    Comments (0)

    There are no comments yet for this tag.

    There are also 2 comments on Section 91.38: Morphisms of Algebraic Stacks.

    Add a comment on tag 0CLA

    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 lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?