Lemma 27.6.3. The category of vector bundles over a scheme $S$ is anti-equivalent to the category of quasi-coherent $\mathcal{O}_ S$-modules.

**Proof.**
Omitted. Hint: In one direction one uses the functor $\underline{\mathop{\mathrm{Spec}}}_ S(\text{Sym}^*_{\mathcal{O}_ S}(-))$ and in the other the functor $(\pi : V \to S) \leadsto (\pi _*\mathcal{O}_ V)_1$ where the subscript indicates we take the degree $1$ part.
$\square$

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