The Stacks project

Lemma 42.46.9. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object whose Chern classes are defined. Then $c_ i(E^\vee ) = (-1)^ i c_ i(E)$, $P_ i(E^\vee ) = (-1)^ iP_ i(E)$, and $ch_ i(E^\vee ) = (-1)^ ich_ i(E)$ in $A^ i(X)$.

Proof. First proof: argue as in the proof of Lemma 42.46.6 to reduce to the case where $E$ is represented by a bounded complex of finite locally free modules of fixed rank and apply Lemma 42.43.3. Second proof: use the splitting principle discussed in Remark 42.46.8 and use that the chern roots of $E^\vee $ are the negatives of the chern roots of $E$. $\square$

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