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
f satisfies the existence part of the valuative criterion as in Definition 67.41.1,
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.
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