nLab smooth Serre-Swan theorem

Idea

The smooth Serre-Swan theorem (Nestruev 03, 11.33) states that over a smooth manifold $X$,

1. the section functor

   $$\Gamma_X(-) \;\colon\; SmoothVectorBundles_{/X} \hookrightarrow C^\infty(X) Mod$$

   that sends smooth vector bundles (with fibers of uniformly bounded dimension) over $X$ of finite rank to their spaces of smooth sections, regarded as modules over the algebra of smooth functions on $X$, is a fully faithful functor

   (Nestruev 03, theorem 11.29);

2. its essential image consists precisely of the finitely generated projective modules

   (Nestruev 03, 11.32).

This is the variant for differential geometry of what the Serre-Swan theorem asserts in algebraic geometry and in topology.

Remark

(base smooth manifold not required to be compact)

Contrary to the original theorem of Swan 62 for topological vector bundles, here in the smooth case the base smooth manifold $X$ of the smooth vector bundle is not required to be compact. While for topological spaces compactness is needed to deduce that every topological vector bundle is a direct summand of a trivial vector bundle (this prop.) for smooth vector bundles this conclusion follows without assuming compactness, by embedding of smooth manifolds into Cartesian spaces (this prop.).

Remark

(other algebraic apects of differential geometry)

Together with the embedding of smooth manifolds into formal duals of R-algebras, the smooth Serre-Swan theorem states that that differential geometry is “more algebraic” than it may superficially seem. A third fact in this vein is that derivations of smooth functions are vector fields.

Related theorems

• the Hadamard lemma

• the Tietze extension theorem

• the Whitney extension theorem

• the Steenrod-Wockel approximation theorem

• embedding of smooth manifolds into formal duals of R-algebras

• derivations of smooth functions are vector fields

References

• Jet Nestruev, Smooth manifolds and observables, Graduate texts in mathematics, 220, Springer-Verlag, ISBN 0-387-95543-7 (2003)