Editing Serre–Swan theorem (section)

==Algebraic geometry==
The analogous result in [[algebraic geometry]], due to {{harvtxt|Serre|1955|loc=§50}} applies to vector bundles in the category of [[affine variety|affine varieties]].  Let ''X'' be an affine variety with structure sheaf <math>\mathcal{O}_X,</math> and <math>\mathcal{F}</math> a [[coherent sheaf]] of <math>\mathcal{O}_X</math> -modules on ''X''.  Then <math>\mathcal{F}</math> is the sheaf of germs of a finite-dimensional vector bundle if and only if <math>\Gamma(\mathcal{F}, X),</math> the space of sections of <math>\mathcal{F},</math> is a projective module over the commutative ring <math>A = \Gamma(\mathcal{O}_X, X).</math>