Serre–Swan theorem

In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".

The two precise formulations of the theorems differ somewhat. The original theorem, as stated by

over C^∞(M). In other words, every vector bundle is a direct summand of some trivial bundle: ${\displaystyle M\times \mathbb {R} ^{k}}$ for some k. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle ${\displaystyle M\times \mathbb {R} ^{k}\to E.}$ This can be done by, for instance, exhibiting sections s₁...s_k with the property that for each point p, {s_i(p)} span the fiber over p.

When M is

finitely generated projective module

over C^∞(M) arises in this way from some smooth vector bundle on M. Such a module can be viewed as a smooth function f on M with values in the n × n idempotent matrices for some n. The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dimensional 2-point manifold, the module ${\displaystyle \mathbb {R} \oplus 0}$ is finitely-generated and projective over ${\displaystyle C^{\infty }(M)\cong \mathbb {R} \times \mathbb {R} }$ but is not

Another way of stating the above is that for any connected smooth manifold M, the section

. If ${\displaystyle \tau :(E_{1},\pi _{1})\to (E_{2},\pi _{2})}$ is a

${\displaystyle \pi _{2}\circ \tau =\pi _{1}}$

${\displaystyle \forall s\in \Gamma (X,E_{1})\quad \pi _{2}\circ \tau \circ s=\pi _{1}\circ s={\text{id}}_{X},}$

giving the map

${\displaystyle {\begin{cases}\Gamma \tau :\Gamma (X,E_{1})\to \Gamma (X,E_{2})\\s\mapsto \tau \circ s\end{cases}}}$

which respects the module structure (Várilly, 97). Swan's theorem asserts that the functor Γ is an equivalence of categories.

Algebraic geometry

The analogous result in algebraic geometry, due to Serre (1955, §50) applies to vector bundles in the category of affine varieties. Let X be an affine variety with structure sheaf ${\displaystyle {\mathcal {O}}_{X},}$ and ${\displaystyle {\mathcal {F}}}$ a coherent sheaf of ${\displaystyle {\mathcal {O}}_{X}}$ -modules on X. Then ${\displaystyle {\mathcal {F}}}$ is the sheaf of germs of a finite-dimensional vector bundle if and only if ${\displaystyle \Gamma ({\mathcal {F}},X),}$ the space of sections of ${\displaystyle {\mathcal {F}},}$ is a projective module over the commutative ring ${\displaystyle A=\Gamma ({\mathcal {O}}_{X},X).}$

References

• ISBN 978-0-387-08090-1
• Manoharan, Palanivel (1995), "Generalized Swan's theorem and its application",
  MR 1264823
  .
• MR 0068874
  .
• JSTOR 1993627
  .
• Nestruev, Jet (2003), Smooth manifolds and observables, Graduate texts in mathematics, vol. 220, Springer-Verlag,
  ISBN 0-387-95543-7
• Giachetta, G.; Mangiarotti, L.;
  ISBN 981-256-129-3
  .