The Stacks project

Lemma 66.41.7. The base change of a morphism of algebraic spaces which satisfies the existence part of (resp. uniqueness part of) the valuative criterion by any morphism of algebraic spaces satisfies the existence part of (resp. uniqueness part of) the valuative criterion.

Proof. Let $f : X \to Y$ be a morphism of algebraic spaces over the scheme $S$. Let $Z \to Y$ be any morphism of algebraic spaces over $S$. Consider a solid commutative diagram of the following shape

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

Then the set of north-west dotted arrows making the diagram commute is in 1-1 correspondence with the set of west-north-west dotted arrows making the diagram commute. This proves the lemma in the case of “uniqueness”. For the existence part, assume $f$ satisfies the existence part of the valuative criterion. If we are given a solid commutative diagram as above, then by assumption there exists an extension $K'/K$ of fields and a valuation ring $A' \subset K'$ dominating $A$ and a morphism $\mathop{\mathrm{Spec}}(A') \to X$ fitting into the following commutative diagram

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

And by the remarks above the skew arrow corresponds to an arrow $\mathop{\mathrm{Spec}}(A') \to Z \times _ Y X$ as desired. $\square$


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