The Stacks project

Lemma 50.2.2. Consider a commutative diagram of schemes

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

If $X' \to X$ and $S' \to S$ are étale, then the maps discussed above induce isomorphisms $f^*\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$.

Proof. We have $\Omega _{S'/S} = 0$ and $\Omega _{X'/X} = 0$, see for example Morphisms, Lemma 29.36.15. Then by the short exact sequences of Morphisms, Lemmas 29.32.9 and 29.34.16 we see that $\Omega _{X'/S'} = \Omega _{X'/S} = f^*\Omega _{X/S}$. Taking exterior powers we conclude. $\square$

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