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Tag 0A40

Chapter 55: Morphisms of Algebraic Spaces > Section 55.43: Valuative criterion of properness

Lemma 55.43.1 (Valuative criterion for properness). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

  1. $f$ is proper,
  2. the valuative criterion holds as in Definition 55.40.1,
  3. given any commutative solid diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\rm Spec}(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition 25.20.3.

Proof. Formal consequence of Lemma 55.42.3 and the definitions. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-morphisms.tex and is located in lines 8930–8951 (see updates for more information).

    \begin{lemma}[Valuative criterion for properness]
    \label{lemma-characterize-proper}
    Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces
    over $S$. Assume $f$ is of finite type and quasi-separated. Then the
    following are equivalent
    \begin{enumerate}
    \item $f$ is proper,
    \item the valuative criterion holds as in Definition
    \ref{definition-valuative-criterion},
    \item given any commutative solid diagram
    $$
    \xymatrix{
    \Spec(K) \ar[r] \ar[d] & X \ar[d] \\
    \Spec(A) \ar[r] \ar@{-->}[ru] & Y
    }
    $$
    where $A$ is a valuation ring with field of fractions $K$, there exists
    a unique dotted arrow, i.e., $f$ satisfies the valuative
    criterion as in
    Schemes, Definition \ref{schemes-definition-valuative-criterion}.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Formal consequence of
    Lemma \ref{lemma-characterize-separated-and-universally-closed}
    and the definitions.
    \end{proof}

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