Lemma 34.8.2. Let $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ be a standard ph covering. Let $T' \to T$ be a morphism of affine schemes. Then $\{ U_ j \times _ T T' \to T'\} _{j = 1, \ldots , m}$ is a standard ph covering.
Proof. Let $f : U \to T$ be proper surjective and let an affine open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$ be given as in Definition 34.8.1. Then $U \times _ T T' \to T'$ is proper surjective (Morphisms, Lemmas 29.9.4 and 29.41.5). Also, $U \times _ T T' = \bigcup _{j = 1, \ldots , m} U_ j \times _ T T'$ is an affine open covering. This concludes the proof. $\square$
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