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Tag 0A3W

Chapter 55: Morphisms of Algebraic Spaces > Section 55.40: Valuative criteria

Lemma 55.40.5. Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of algebraic spaces over $S$. The following are equivalent

  1. $f$ satisfies the existence part of the valuative criterion as in Definition 55.40.1,
  2. 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 dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition 25.20.3.

Proof. We have to show that (1) implies (2). Suppose given a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\rm Spec}(A) \ar[r] & Y } $$ as in part (2). By (1) there exists a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(K') \ar[r] \ar[d] & \mathop{\rm Spec}(K) \ar[r] & X \ar[d] \\ \mathop{\rm Spec}(A') \ar[r] \ar[rru] & \mathop{\rm Spec}(A) \ar[r] & Y } $$ as in Definition 55.40.1 with $K \subset K'$ arbitrary. By Lemma 55.40.4 we can find a morphism $\mathop{\rm Spec}(A) \to X$ fitting into the diagram, i.e., (2) holds. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-morphisms.tex and is located in lines 8263–8282 (see updates for more information).

    \begin{lemma}
    \label{lemma-usual-enough}
    Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of
    algebraic spaces over $S$. The following are equivalent
    \begin{enumerate}
    \item $f$ satisfies the existence part of the valuative criterion
    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 dotted arrow, i.e., $f$ satisfies the existence part of the valuative
    criterion as in
    Schemes, Definition \ref{schemes-definition-valuative-criterion}.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    We have to show that (1) implies (2). Suppose given a commutative diagram
    $$
    \xymatrix{
    \Spec(K) \ar[r] \ar[d] & X \ar[d] \\
    \Spec(A) \ar[r] & Y
    }
    $$
    as in part (2). By (1) there exists a commutative diagram
    $$
    \xymatrix{
    \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\
    \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y
    }
    $$
    as in Definition \ref{definition-valuative-criterion} with $K \subset K'$
    arbitrary. By Lemma \ref{lemma-push-down-solution} we can find a morphism
    $\Spec(A) \to X$ fitting into the diagram, i.e., (2) holds.
    \end{proof}

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