Reconstructing a vector bundle from a projective module

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, let $\mathbb F\in\{\mathbb R,\mathbb C\}$ , and put $A=C^\infty(M,\mathbb F)$ . For a finitely generated projective $A$ -module $Q$ , choose $N$ and an idempotent $p\in M_N(A)$ with $Q\cong pA^N$ . The vector bundle reconstructed from $Q$ is the smooth vector bundle

E_Q=\coprod_{x\in M}\operatorname{im}p(x)\longrightarrow M.

Give it the smooth structure of the image bundle of the smooth idempotent $p$ . Its fibers have locally constant dimension, and its smooth sections satisfy $\Gamma(E_Q)\cong pA^N\cong Q$ . Different choices of the finite free summand and idempotent produce canonically isomorphic bundles up to the module isomorphism chosen.

Intrinsic fiberwise description

For $x\in M$ , let $I_x=\{f\in A:f(x)=0\}$ . Evaluation identifies the reconstructed fiber with

(E_Q)_x\cong Q/I_xQ.

This formula does not mention a chosen embedding into a free module. Projectivity ensures that these quotient spaces have locally constant dimension and fit together smoothly; the idempotent presentation supplies explicit local trivializations.

The natural map to the module of smooth sections

Q\longrightarrow \Gamma(E_Q), \qquad q\longmapsto \bigl(x\mapsto q\bmod I_xQ\bigr),

is an $A$ -module isomorphism. Swan’s construction proves this reconstruction and the converse passage from bundles to projective section modules.

Functoriality and equivalence

An $A$ -linear map $Q\to Q'$ induces fiber maps $Q/I_xQ\to Q'/I_xQ'$ varying smoothly with $x$ . Consequently, reconstruction is functorial and is inverse, up to natural isomorphism, to taking smooth sections. For connected $M$ , this gives the Serre–Swan equivalence between finite-rank smooth vector bundles and finitely generated projective $C^\infty(M,\mathbb F)$ -modules without a compactness assumption.

For $Q=eA^N$ , the construction is exactly the image bundle of the idempotent $e$ . If $Q=A^r$ , it yields the trivial rank- $r$ bundle.

Conventions and scope

On a disconnected base, a finitely generated projective module reconstructs a bundle whose locally constant rank is globally bounded by $N$ . Conversely, that bounded-rank condition is needed if the vector-bundle convention permits different ranks on different components. No condition at infinity is needed because the coefficient algebra is all of $C^\infty(M,\mathbb F)$ , rather than an algebra of functions vanishing at infinity.

References

Richard G. Swan, “Vector Bundles and Projective Modules,” Transactions of the American Mathematical Society 105 (1962), 264–277. DOI record. Relevant: §§1–3, reconstruction and equivalence of vector bundles with projective modules.
Alain Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted edition. Relevant: chapter I, finitely generated projective modules as noncommutative vector bundles.