The Stacks project

Lemma 50.23.2. The gysin map ( 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

\[ \mathop{\mathrm{Coker}}\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^ c_ Z(\mathcal{F}) \]

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

\[ \text{d}\left( \frac{\tilde\omega \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c}\right) = \frac{\text{d}(\tilde\omega ) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c} \]

which is clear. $\square$

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