nLab
smooth Serre-Swan theorem

Contents

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

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

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          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 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)

          Last revised on September 8, 2018 at 03:17:45. See the history of this page for a list of all contributions to it.