# nLab smooth Serre-Swan theorem

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

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

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

that sends smooth vector bundles 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

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

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.

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