Lemma 34.10.4. Let $\{ T_ j \to T\} _{j = 1, \ldots , m}$ be a standard V covering. Let $T' \to T$ be a morphism of affine schemes. Then $\{ T_ j \times _ T T' \to T'\} _{j = 1, \ldots , m}$ is a standard V covering.

Proof. Let $\mathop{\mathrm{Spec}}(V) \to T'$ be a morphism where $V$ is a valuation ring. By assumption we can find an extension of valuation rings $V \subset W$, an $i$, and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & T_ i \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & T }$

By the universal property of fibre products we obtain a morphism $\mathop{\mathrm{Spec}}(W) \to T' \times _ T T_ i$ as desired. $\square$

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