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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