Lemma 42.56.3. Let X be a scheme. There is a ring map \psi ^{-1} : K_0(\textit{Vect}(X)) \to K_0(\textit{Vect}(X)) which sends [\mathcal{E}] to [\mathcal{E}^\vee ] when \mathcal{E} is finite locally free and is compatible with pullbacks.
Proof. The only thing to check is that taking duals is compatible with short exact sequences and with pullbacks. This is clear. \square
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