Lemma 29.32.10. Let $X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map

of Lemma 29.32.8 is an isomorphism.

Lemma 29.32.10. Let $X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map

\[ (g')^*\Omega _{X/S} \to \Omega _{X'/S'} \]

of Lemma 29.32.8 is an isomorphism.

**Proof.**
This is the sheafified version of Algebra, Lemma 10.131.12.
$\square$

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