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Tag 0CLN

Chapter 90: Morphisms of Algebraic Stacks > Section 90.38: Valuative criteria

Lemma 90.38.13. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent

  1. $f$ satisfies the existence part of the valuative criterion,
  2. for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism $\mathcal{X} \times_\mathcal{Y} T \to T$ satisfies the existence part of the valuative criterion as a morphism of algebraic spaces.

Proof. Omitted. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 8933–8944 (see updates for more information).

    \begin{lemma}
    \label{lemma-existence-representable}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks
    which is representable by algebraic spaces. Then the following are equivalent
    \begin{enumerate}
    \item $f$ satisfies the existence part of the valuative criterion,
    \item for every scheme $T$ and morphism $T \to \mathcal{Y}$
    the morphism $\mathcal{X} \times_\mathcal{Y} T \to T$ satisfies
    the existence part of the valuative criterion as a morphism
    of algebraic spaces.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

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