The Stacks project

Definition 66.41.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ satisfies the uniqueness part of the valuative criterion if given any commutative solid diagram

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

where $A$ is a valuation ring with field of fractions $K$, there exists at most one dotted arrow (without requiring existence). We say $f$ satisfies the existence part of the valuative criterion if given any solid diagram as above there exists an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ and a morphism $\mathop{\mathrm{Spec}}(A') \to X$ such that the following diagram commutes

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

We say $f$ satisfies the valuative criterion if $f$ satisfies both the existence and uniqueness part.


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