Lemma 50.23.8. Let $X \to S$ and $i : Z \to X$ be as in Lemma 50.23.3. Assume $X \to S$ is smooth and $Z \to X$ Koszul regular. The gysin maps $\gamma ^{p, q}$ are compatible with the de Rham differentials on $\Omega ^\bullet _{X/S}$ and $\Omega ^\bullet _{Z/S}$.

**Proof.**
This follows immediately from Lemma 50.23.4.
$\square$

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