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

\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

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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