The Stacks project

Lemma 50.23.6. Let $c \geq 0$ be a integer. Let

\[ \xymatrix{ Z' \ar[d]_ h \ar[r] & X' \ar[d]_ g \ar[r] & S' \ar[d] \\ Z \ar[r] & X \ar[r] & S } \]

be a commutative diagram of schemes. Assume

  1. $Z \to X$ and $Z' \to X'$ satisfy the assumptions of Lemma 50.23.3,

  2. the left square in the diagram is cartesian, and

  3. $h^*\mathcal{C}_{Z/X} \to \mathcal{C}_{Z'/X'}$ (Morphisms, Lemma 29.31.3) is an isomorphism.

Then the diagram

\[ \xymatrix{ h^*\Omega ^ p_{Z/S} \ar[rr]_-{h^{-1}\gamma ^ p} \ar[d] & & \mathcal{O}_{X'}|_{Z'} \otimes _{h^{-1}\mathcal{O}_ X|_ Z} h^{-1}\mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}) \ar[d] \\ \Omega ^ p_{Z'/S'} \ar[rr]^{\gamma ^ p} & & \mathcal{H}^ c_{Z'}(\Omega ^{p + c}_{X'/S'}) } \]

is commutative. The left vertical arrow is functoriality of modules of differentials and the right vertical arrow uses Cohomology, Remark 20.34.12.

Proof. More precisely, consider the composition

\begin{align*} \mathcal{O}_{X'}|_{Z'} \otimes _{h^{-1}\mathcal{O}_ X|_ Z}^\mathbf {L} h^{-1}R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S}) & \to R\mathcal{H}_{Z'}(Lg^*\Omega ^{p + c}_{X/S}) \\ & \to R\mathcal{H}_{Z'}(g^*\Omega ^{p + c}_{X/S}) \\ & \to R\mathcal{H}_{Z'}(\Omega ^{p + c}_{X'/S'}) \end{align*}

where the first arrow is given by Cohomology, Remark 20.34.12 and the last one by functoriality of differentials. Since we have the vanishing of cohomology sheaves in degrees $> c$ by Derived Categories of Schemes, Lemma 36.6.8 this induces the right vertical arrow. We can check the commutativity locally. Thus we may assume $Z$ is cut out by $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Then $Z'$ is cut out by $f'_ i = g^\sharp (f_ i)$. The maps $c_{f_1, \ldots , f_ c}$ and $c_{f'_1, \ldots , f'_ c}$ fit into the commutative diagram

\[ \xymatrix{ h^*i^*\Omega ^ p_{X/S} \ar[rr]_-{h^{-1}c_{f_1, \ldots , f_ c}} \ar[d] & & \mathcal{O}_{X'}|_{Z'} \otimes _{h^{-1}\mathcal{O}_ X|_ Z} h^{-1}\mathcal{H}^ c_ Z(\Omega ^ p_{X/S}) \ar[d] \\ (i')^*\Omega ^ p_{X'/S'} \ar[rr]^{c_{f'_1, \ldots , f'_ c}} & & \mathcal{H}^ c_{Z'}(\Omega ^ p_{X'/S'}) } \]

See Derived Categories of Schemes, Remark 36.6.14. Recall given a $p$-form $\omega $ on $Z$ we define $\gamma ^ p(\omega )$ by choosing (locally on $X$ and $Z$) a $p$-form $\tilde\omega $ on $X$ lifting $\omega $ and taking $\gamma ^ p(\omega ) = c_{f_1, \ldots , f_ c}(\tilde\omega ) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$. Since the form $\text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$ pulls back to $\text{d}f'_1 \wedge \ldots \wedge \text{d}f'_ c$ we conclude. $\square$

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