Editing Projective module (section)

== Vector bundles and locally free modules ==
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A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of [[vector bundle]]s. This can be made precise for the ring of [[continuous function (topology)|continuous]] [[real number|real]]-valued functions on a [[compact space|compact]] [[Hausdorff space]], as well as for the ring of [[smooth function]]s on a [[manifold|smooth manifold]] (see [[Serre–Swan theorem]] that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a [[smooth vector bundle]]).

Vector bundles are ''locally free''. If there is some notion of "localization" that can be carried over to modules, such as the usual [[localization of a ring]], one can define locally free modules, and the projective modules then typically coincide with the locally free modules.