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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