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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