Serre–Swan theorem - Imported Knowlpedia Content

Statement

Let $M$ be a compact smooth manifold and $\mathbb F=\mathbb R$ or $\mathbb C$ . The smooth Serre–Swan theorem states that the functor

E\longmapsto\Gamma^\infty(M,E)

from finite-rank smooth $\mathbb F$ -vector bundles with bundle morphisms covering $\operatorname{id}_M$ to finitely generated projective $C^\infty(M,\mathbb F)$ -modules with module homomorphisms is an equivalence of categories. Thus every section module is finitely generated projective, every such module is the module of smooth sections of a vector bundle, and module homomorphisms arise uniquely from vector-bundle maps over $M$ . The compactness hypothesis is part of this formulation.

The two constructions

A vector bundle $E$ embeds as a direct summand of a trivial bundle. Taking sections then expresses $\Gamma^\infty(M,E)$ as a direct summand of a finite-rank free $C^\infty(M)$ -module, hence as a finitely generated projective module.

Conversely, a finitely generated projective module is represented by an idempotent matrix $p\in M_N(C^\infty(M))$ . Evaluating $p$ pointwise gives a smoothly varying family of projections, whose images form a vector subbundle of $M\times\mathbb F^N$ . This idempotent reconstruction is inverse to taking sections up to natural isomorphism Swan, §§1–3.

Categorical content

The theorem is stronger than a bijection of isomorphism classes. It identifies morphisms covering $\operatorname{id}_M$ and their compositions, so direct sums, direct summands, and isomorphisms agree on the geometric and algebraic sides. In particular, bundle automorphisms over $M$ correspond exactly to invertible $C^\infty(M)$ -linear endomorphisms of the section module.

This equivalence explains why vector bundles define classes in topological $K$ -theory and why finitely generated projective modules play the role of vector bundles in noncommutative geometry.

Variants and scope

“Serre–Swan duality” is common terminology, but the result is a covariant equivalence produced by the section functor, not a contravariant duality.

References

Richard G. Swan, “Vector Bundles and Projective Modules,” Transactions of the American Mathematical Society 105 (1962), 264–277. DOI record. Relevant: §§1–3, finite-type bundles, projective section modules, and reconstruction.
José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: chapter 2, projective modules as noncommutative vector bundles and the Serre–Swan correspondence.