The Stacks project

Remark 50.9.2. Here is a reformulation of the calculations above in more abstract terms. Let $p : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If we view $\text{d}\log $ as a map

\[ \mathcal{O}_ X^*[-1] \to \sigma _{\geq 1}\Omega ^\bullet _{X/S} \]

then using $\mathop{\mathrm{Pic}}\nolimits (X) = H^1(X, \mathcal{O}_ X^*)$ as above we find a cohomology class

\[ \gamma _1(\mathcal{L}) \in H^2(X, \sigma _{\geq 1}\Omega ^\bullet _{X/S}) \]

The image of $\gamma _1(\mathcal{L})$ under the map $\sigma _{\geq 1}\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}$ recovers $c_1^{dR}(\mathcal{L})$. In particular we see that $c_1^{dR}(\mathcal{L}) \in F^1H^2_{dR}(X/S)$, see Section 50.7. The image of $\gamma _1(\mathcal{L})$ under the map $\sigma _{\geq 1}\Omega ^\bullet _{X/S} \to \Omega ^1_{X/S}[-1]$ recovers $c_1^{Hodge}(\mathcal{L})$. Taking the cup product (see Section 50.7) we obtain

\[ \xi = \gamma _1(\mathcal{L}_1) \cup \ldots \cup \gamma _1(\mathcal{L}_ a) \in H^{2a}(X, \sigma _{\geq a}\Omega ^\bullet _{X/S}) \]

The commutative diagrams in Section 50.7 show that $\xi $ is mapped to $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_ a)$ in $H^{2a}_{dR}(X/S)$ by the map $\sigma _{\geq a}\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}$. Also, it follows $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_ a)$ is contained in $F^ a H^{2a}_{dR}(X/S)$. Similarly, the map $\sigma _{\geq a}\Omega ^\bullet _{X/S} \to \Omega ^ a_{X/S}[-a]$ sends $\xi $ to $c_1^{Hodge}(\mathcal{L}_1) \cup \ldots \cup c_1^{Hodge}(\mathcal{L}_ a)$ in $H^ a(X, \Omega ^ a_{X/S})$.

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