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
Comments (2)
Comment #5406 by Leo on
Comment #5636 by Johan on
There are also: