Lemma 30.26.4. Consider a cartesian diagram of schemes
with f locally of finite type. If Z is a closed subset of X proper over S, then (g')^{-1}(Z) is a closed subset of X' proper over S'.
Lemma 30.26.4. Consider a cartesian diagram of schemes
with f locally of finite type. If Z is a closed subset of X proper over S, then (g')^{-1}(Z) is a closed subset of X' proper over S'.
Proof. Observe that the statement makes sense as f' is locally of finite type by Morphisms, Lemma 29.15.4. Endow Z with the reduced induced closed subscheme structure. Denote Z' = (g')^{-1}(Z) the scheme theoretic inverse image (Schemes, Definition 26.17.7). Then Z' = X' \times _ X Z = (S' \times _ S X) \times _ X Z = S' \times _ S Z is proper over S' as a base change of Z over S (Morphisms, Lemma 29.41.5). \square
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