Lemma 50.15.6 (0FMX)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 50: de Rham Cohomology
Section 50.15: Log poles along a divisor
Lemma 50.15.6 (cite)

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Lemma 50.15.6. Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$ . Let $b \in H^ m_{dR}(X/S)$ be a de Rham cohomology class whose restriction to $Y$ is zero. Then $c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = 0$ in $H^{m + 2}_{dR}(X/S)$ .

Proof. This follows immediately from Lemma 50.15.5. Namely, we have

$c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = -c_1^{dR}(\mathcal{O}_ X(-Y)) \cup b = -\xi '(b) = -\delta (b|_ Y) = 0$

as desired. For the second equality, see Remark 50.4.3. $\square$

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