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).

Comment #7178 by Zeyn Sahilliogullari on

Perhaps it is worth mentioning that the left arrow is induced by the inclusion/localization map rather than an arbitary homomorphism.

Comment #7313 by on

Going to leave this as is for now, but of course you are right that strictly speaking this should be mentioned. There are many, many instances of this all over the place.

There are also:

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