Lemma 49.2.1. Let

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

be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms $f^*\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$.

Lemma 49.2.1. Let

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

be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms $f^*\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$.

**Proof.**
Combine Morphisms, Lemma 28.31.10 with the fact that formation of exterior power commutes with base change.
$\square$

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