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The Stacks project

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

The hint should read "In one direction, one uses the functor Spec_S(Sym(--))" instead of just Spec

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