Lemma 42.45.4 (0FAB)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.45: The Chern character and tensor products
Lemma 42.45.4 (cite)

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Lemma 42.45.4. In Situation 42.7.1 let $X$ be locally of finite type over $S$ . Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$ -module with dual $\mathcal{E}^\vee$ . Then $ch_ i(\mathcal{E}^\vee ) = (-1)^ i ch_ i(\mathcal{E})$ in $A^ i(X) \otimes \mathbf{Q}$ .

Proof. Follows from the corresponding result for Chern classes (Lemma 42.43.3). $\square$

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