Definition 26.20.3. Let $f : X \to S$ be a morphism of schemes. We say $f$ satisfies the existence part of the valuative criterion if given any commutative solid diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & S }$

where $A$ is a valuation ring with field of fractions $K$, the dotted arrow exists. We say $f$ satisfies the uniqueness part of the valuative criterion if there is at most one dotted arrow given any diagram as above (without requiring existence of course).

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