The Stacks project

Remark 50.23.7. Let $X \to S$, $i : Z \to X$, and $c \geq 0$ be as in Lemma 50.23.3. Let $p \geq 0$ and assume that $\mathcal{H}^ i_ Z(\Omega ^{p + c}_{X/S}) = 0$ for $i = 0, \ldots , c - 1$. This vanishing holds if $X \to S$ is smooth and $Z \to X$ is a Koszul regular immersion, see Derived Categories of Schemes, Lemma 36.6.9. Then we obtain a map

\[ \gamma ^{p, q} : H^ q(Z, \Omega ^ p_{Z/S}) \longrightarrow H^{q + c}(X, \Omega ^{p + c}_{X/S}) \]

by first using $\gamma ^ p : \Omega ^ p_{Z/S} \to \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$ to map into

\[ H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})) = H^ q(Z, R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S})[c]) = H^ q(X, i_*R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S})[c]) \]

and then using the adjunction map $i_*R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S}) \to \Omega ^{p + c}_{X/S}$ to continue on to the desired Hodge cohomology module.

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