The Stacks project

Lemma 50.23.5. Let $X \to S$ and $i : Z \to X$ be as in Lemma 50.23.3. Given $\alpha \in H^ q(X, \Omega ^ p_{X/S})$ we have $\gamma ^ p(\alpha |_ Z) = i^{-1}\alpha \wedge \gamma ^0(1)$ in $H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}))$. Please see proof for notation.

Proof. The restriction $\alpha |_ Z$ is the element of $H^ q(Z, \Omega ^ p_{Z/S})$ given by functoriality for Hodge cohomology. Applying functoriality for cohomology using $\gamma ^ p : \Omega ^ p_{Z/S} \to \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$ we get get $\gamma ^ p(\alpha |_ Z)$ in $H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}))$. This explains the left hand side of the formula.

To explain the right hand side, we first pullback by the map of ringed spaces $i : (Z, i^{-1}\mathcal{O}_ X) \to (X, \mathcal{O}_ X)$ to get the element $i^{-1}\alpha \in H^ q(Z, i^{-1}\Omega ^ p_{X/S})$. Let $\gamma ^0(1) \in H^0(Z, \mathcal{H}_ Z^ c(\Omega ^ c_{X/S}))$ be the image of $1 \in H^0(Z, \mathcal{O}_ Z) = H^0(Z, \Omega ^0_{Z/S})$ by $\gamma ^0$. Using cup product we obtain an element

\[ i^{-1}\alpha \cup \gamma ^0(1) \in H^{q + c}(Z, i^{-1}\Omega ^ p_{X/S} \otimes _{i^{-1}\mathcal{O}_ X} \mathcal{H}^ c_ Z(\Omega ^ c_{X/S})) \]

Using Cohomology, Remark 20.34.9 and wedge product there are canonical maps

\[ i^{-1}\Omega ^ p_{X/S} \otimes _{i^{-1}\mathcal{O}_ X}^\mathbf {L} R\mathcal{H}_ Z(\Omega ^ c_{X/S}) \to R\mathcal{H}_ Z(\Omega ^ p_{X/S} \otimes _{\mathcal{O}_ X}^\mathbf {L} \Omega ^ c_{X/S}) \to R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S}) \]

By Derived Categories of Schemes, Lemma 36.6.8 the objects $R\mathcal{H}_ Z(\Omega ^ j_{X/S})$ have vanishing cohomology sheaves in degrees $> c$. Hence on cohomology sheaves in degree $c$ we obtain a map

\[ i^{-1}\Omega ^ p_{X/S} \otimes _{i^{-1}\mathcal{O}_ X} \mathcal{H}^ c_ Z(\Omega ^ c_{X/S}) \longrightarrow \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}) \]

The expression $i^{-1}\alpha \wedge \gamma ^0(1)$ is the image of the cup product $i^{-1}\alpha \cup \gamma ^0(1)$ by the functoriality of cohomology.

Having explained the content of the formula in this manner, by general properties of cup products (Cohomology, Section 20.31), it now suffices to prove that the diagram

\[ \xymatrix{ i^{-1}\Omega ^ p_ X \otimes \Omega ^0_ Z \ar[rr]_{\text{id} \otimes \gamma ^0} \ar[d] & & i^{-1}\Omega ^ p_ X \otimes \mathcal{H}^ c_ Z(\Omega ^ c_ X) \ar[d]^\wedge \\ \Omega ^ p_ Z \otimes \Omega ^0_ Z \ar[r]^\wedge & \Omega ^ p_ Z \ar[r]^{\gamma ^ p} & \mathcal{H}^ c_ Z(\Omega ^{p + c}_ X) } \]

is commutative in the category of sheaves on $Z$ (with obvious abuse of notation). This boils down to a simple computation for the maps $\gamma ^ j_{f_1, \ldots , f_ c}$ which we omit; in fact these maps are chosen exactly such that this works and such that $1$ maps to $\frac{\text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c}$. $\square$

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