Lemma 27.6.3 (062N)—The Stacks project

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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 July 21, 2020 at 20:38

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

Comment #5636 by Johan on November 14, 2020 at 13:50

Thanks and fixed here.

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