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The Stacks project

Lemma 67.41.5. 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 67.41.1,

  2. given any commutative solid diagram

    \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{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 26.20.3.

Proof. We have to show that (1) implies (2). Suppose given a commutative diagram

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y }

as in part (2). By (1) there exists a commutative diagram

\xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y }

as in Definition 67.41.1 with K \subset K' arbitrary. By Lemma 67.41.4 we can find a morphism \mathop{\mathrm{Spec}}(A) \to X fitting into the diagram, i.e., (2) holds. \square


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