The Stacks project

Proof. Let $\{ X_ i \to X\} _{i = 1, \ldots , n}$ be a standard fpqc covering (Definition 34.9.9). Let $g : \mathop{\mathrm{Spec}}(V) \to X$ be a morphism where $V$ is a valuation ring. Let $x \in X$ be the image of the closed point of $\mathop{\mathrm{Spec}}(V)$. Choose an $i$ and a point $x_ i \in X_ i$ mapping to $x$. Then $\mathop{\mathrm{Spec}}(V) \times _ X X_ i$ has a point $x'_ i$ mapping to the closed point of $\mathop{\mathrm{Spec}}(V)$. Since $\mathop{\mathrm{Spec}}(V) \times _ X X_ i \to \mathop{\mathrm{Spec}}(V)$ is flat we can find a specialization $x''_ i \leadsto x'_ i$ of points of $\mathop{\mathrm{Spec}}(V) \times _ X X_ i$ with $x''_ i$ mapping to the generic point of $\mathop{\mathrm{Spec}}(V)$, see Morphisms, Lemma 29.25.9. By Schemes, Lemma 26.20.4 we can choose a valuation ring $W$ and a morphism $h : \mathop{\mathrm{Spec}}(W) \to \mathop{\mathrm{Spec}}(V) \times _ X X_ i$ such that $h$ maps the generic point of $\mathop{\mathrm{Spec}}(W)$ to $x''_ i$ and the closed point of $\mathop{\mathrm{Spec}}(W)$ to $x'_ i$. We obtain a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & X_ i \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & X } \]

where $V \to W$ is an extension of valuation rings. This proves the lemma. $\square$

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