Swan's theorem relates vector bundles to projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces". In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

Differential geometry

Suppose M is a compact C^∞-manifold, and a smooth vector bundle V is given on M. The space of smooth sections of V is then a module over C^∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is finitely generated and projective over C^∞(M). Several specialized usages of the terms compact and compactness exist. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ...

Even more: every finitely generated projective module over C^∞(M) arises in this way from some smooth vector bundle on M, in essentially only one way. More precisely: the category of smooth vector bundles on M is equivalent to the category of finitely generated projective modules over C^∞(M). Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Topology

Suppose X is a compact Hausdorff space, and C(X) is the ring of continuous real-valued functions on X. Analogous to the result above, the category of real vector bundles on X is equivalent to the category of finitely generated projective modules over C(X). In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...