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One can define the notion of a locally free sheaf (of finite rank) on any locally ringed space.

If you restrict to the category of (noetherian?) schemes, this category is equivalent to the category of vector bundles. (This is an exercise in Hartshorne.)

If you restrict to the category of complex manifolds, this category is equivalent to the category of complex vector bundles.

Can one unify these two observations? That is, can one describe the category of locally free sheaves on a locally ringed space $(X,\mathcal{O}_X)$ as a category of "vector bundles"?

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    $\begingroup$ The category of locally free sheaves on a complex manifold is equivalent to the category of complex vector bundles. I'm not sure what the anti-equivalence you have in mind is (the usual functor followed by duality?). $\endgroup$ – Akhil Mathew Oct 1 '11 at 19:07
  • $\begingroup$ You're completely right. I was confused. $\endgroup$ – Gooz Oct 1 '11 at 19:17
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    $\begingroup$ It seems to me that locally free sheaves are the only reasonable generalisation of vector bundles to (locally) ringed spaces. Is there another definition you have in mind? $\endgroup$ – Zhen Lin Oct 1 '11 at 22:23
  • $\begingroup$ @Gooz This is a long shot since it's 6 years later, but do you have a reference for which exercise this is? I originally though you were referring to exercise 5.18, but there doesn't seem to be any mention of restricting to the case of a noetherian scheme there, so I'm confused about where that is necessary. $\endgroup$ – Luke Oct 20 '17 at 16:06

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