Lemma 50.23.2. The gysin map (50.23.1.1) is compatible with the de Rham differentials on $\Omega ^\bullet _{X/S}$ and $\Omega ^\bullet _{Z/S}$.

**Proof.**
This follows from an almost trivial calculation once we correctly interpret this. First, we recall that the functor $\mathcal{H}^ c_ Z$ computed on the category of $\mathcal{O}_ X$-modules agrees with the similarly defined functor on the category of abelian sheaves on $X$, see Cohomology, Lemma 20.34.8. Hence, the differential $\text{d} : \Omega ^ p_{X/S} \to \Omega ^{p + 1}_{X/S}$ induces a map $\mathcal{H}^ c_ Z(\Omega ^ p_{X/S}) \to \mathcal{H}^ c_ Z(\Omega ^{p + 1}_{X/S})$. Moreover, the formation of the extended alternating Čech complex in Derived Categories of Schemes, Remark 36.6.4 works on the category of abelian sheaves. The map

used in the construction of $c_{f_1, \ldots , f_ c}$ in Derived Categories of Schemes, Remark 36.6.10 is well defined and functorial on the category of all abelian sheaves on $X$. Hence we see that the lemma follows from the equality

which is clear. $\square$

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