Lemma 42.55.3. 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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