Reconstructing a vector bundle from a projective module

Definition

Let $M$ be a compact smooth manifold, let $\mathbb F=\mathbb R$ or $\mathbb C$ , 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.

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 Swan, §§1–3.

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. On compact $M$ , this gives the Serre–Swan equivalence between finite-rank smooth vector bundles and finitely generated projective $C^\infty(M)$ -modules.

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

Compactness is the standard hypothesis guaranteeing the stated equivalence without conditions at infinity. Noncompact versions require care about the function algebra, finite type, support, or bounded geometry.

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.