Remark 50.23.1. Let $X \to S$ be a morphism of schemes. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Let $Z \subset X$ be the closed subscheme cut out by $f_1, \ldots , f_ c$. Below we will study the gysin map

50.23.1.1
$$\label{derham-equation-gysin} \gamma ^ p_{f_1, \ldots , f_ c} : \Omega ^ p_{Z/S} \longrightarrow \mathcal{H}_ Z^ c(\Omega ^{p + c}_{X/S})$$

defined as follows. Given a local section $\omega$ of $\Omega ^ p_{Z/S}$ which is the restriction of a section $\tilde\omega$ of $\Omega ^ p_{X/S}$ we set

$\gamma ^ p_{f_1, \ldots , f_ c}(\omega ) = c_{f_1, \ldots , f_ c}(\tilde\omega |_ Z) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$

where $c_{f_1, \ldots , f_ c} : \Omega ^ p_{X/S} \otimes \mathcal{O}_ Z \to \mathcal{H}_ Z^ c(\Omega ^ p_{X/S})$ is the map constructed in Derived Categories of Schemes, Remark 36.6.10. This is well defined: given $\omega$ we can change our choice of $\tilde\omega$ by elements of the form $\sum f_ i \omega '_ i + \sum \text{d}(f_ i) \wedge \omega ''_ i$ which are mapped to zero by the construction.

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