Lemma 34.10.13. Let $X \to Y$ be a quasi-compact morphism of schemes. The following are equivalent

1. $\{ X \to Y\}$ is a V covering,

2. for any valuation ring $V$ and morphism $g : \mathop{\mathrm{Spec}}(V) \to Y$ there exists an extension of valuation rings $V \subset W$ and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & Y }$
3. for any morphism $Z \to Y$ and specialization $z' \leadsto z$ of points in $Z$, there is a specialization $w' \leadsto w$ of points in $Z \times _ Y X$ mapping to $z' \leadsto z$.

Proof. Assume (1) and let $g : \mathop{\mathrm{Spec}}(V) \to Y$ be as in (2). Since $V$ is a local ring there is an affine open $U \subset Y$ such that $g$ factors through $U$. By Definition 34.10.7 we can find a standard V covering $\{ U_ j \to U\}$ refining $\{ X \times _ Y U \to U\}$. By Definition 34.10.1 we can find a $j$, an extension of valuation rings $V \subset W$ and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & U_ j \ar[d] \ar@{..>}[r] & X \ar[ld] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & Y }$

We have the dotted arrow making the diagram commute by the refinement property of the covering and we see that (2) holds.

Assume (2) and let $Z \to Y$ and $z' \leadsto z$ be as in (3). By Schemes, Lemma 26.20.4 we can find a valuation ring $V$ and a morphism $\mathop{\mathrm{Spec}}(V) \to Z$ such that the closed point of $\mathop{\mathrm{Spec}}(V)$ maps to $z$ and the generic point of $\mathop{\mathrm{Spec}}(V)$ maps to $z'$. By (2) we can find an extension of valuation rings $V \subset W$ and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[rr] \ar[d] & & X \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & Z \ar[r] & Y }$

The generic and closed points of $\mathop{\mathrm{Spec}}(W)$ map to points $w' \leadsto w$ in $Z \times _ Y X$ via the induced morphism $\mathop{\mathrm{Spec}}(W) \to Z \times _ Y X$. This shows that (3) holds.

Assume (3) holds and let $U \subset Y$ be an affine open. Choose a finite affine open covering $U \times _ Y X = \bigcup _{j = 1, \ldots , m} U_ j$. This is possible as $X \to Y$ is quasi-compact. We claim that $\{ U_ j \to U\}$ is a standard V covering. The claim implies (1) is true and finishes the proof of the lemma. In order to prove the claim, let $V$ be a valuation ring and let $g : \mathop{\mathrm{Spec}}(V) \to U$ be a morphism. By (3) we find a specialization $w' \leadsto w$ of points of

$T = \mathop{\mathrm{Spec}}(V) \times _ X Y = \mathop{\mathrm{Spec}}(V) \times _ U (U \times _ X Y)$

such that $w'$ maps to the generic point of $\mathop{\mathrm{Spec}}(V)$ and $w$ maps to the closed point of $\mathop{\mathrm{Spec}}(V)$. By Schemes, Lemma 26.20.4 we can find a valuation ring $W$ and a morphism $\mathop{\mathrm{Spec}}(W) \to T$ such that the generic point of $\mathop{\mathrm{Spec}}(W)$ maps to $w'$ and the closed point of $\mathop{\mathrm{Spec}}(W)$ maps to $w$. The composition $\mathop{\mathrm{Spec}}(W) \to T \to \mathop{\mathrm{Spec}}(V)$ corresponds to an inclusion $V \subset W$ which presents $W$ as an extension of the valuation ring $V$. Since $T = \bigcup \mathop{\mathrm{Spec}}(V) \times _ U U_ j$ is an open covering, we see that $\mathop{\mathrm{Spec}}(W) \to T$ factors through $\mathop{\mathrm{Spec}}(V) \times _ U U_ j$ for some $j$. Thus we obtain a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[d] \ar[r] & U_ j \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & U }$

and the proof of the claim is complete. $\square$

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