nLab
smooth Serre-Swan theorem

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \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 smooth manifold XX,

  1. the section functor

    Γ X():SmoothVectorBundles /XC (X)Mod \Gamma_X(-) \;\colon\; SmoothVectorBundles_{/X} \hookrightarrow C^\infty(X) Mod

    that sends smooth vector bundles over XX of finite rank to their spaces of smooth sections, regarded as modules over the algebra of smooth functions on XX, 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 XX 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.

References

  • Jet Nestruev, Smooth manifolds and observables, Graduate texts in mathematics, 220, Springer-Verlag, ISBN 0-387-95543-7 (2003)
Revised on October 22, 2017 11:04:04 by Urs Schreiber (94.220.50.90)