Serre and Swan's theorem

In mathematics , Serre and Swan's theorem establishes a connection between vector bundles and projective modules or, in K-theoretical formulation, between the K-theory of a space and its functional algebra.

Vector bundles and projective modules

For a vector bundle over a topological space, let the vector space of its intersections be . This is a module above the ring of continuous functions. ${\displaystyle E\to X}$ ${\displaystyle X}$ ${\displaystyle \Gamma (E)}$ ${\displaystyle C(X)}$

One can show that there is a finitely generated , projective module. ${\displaystyle \Gamma (E)}$ ${\displaystyle C(X)}$

Let be the semigroup of the isomorphism classes of the vector bundles with the Whitney sum as a link and the semigroup of the isomorphism classes of finitely generated projective modules. The assignment defined on representatives ${\displaystyle \mathrm {Vect} (X)}$ ${\displaystyle X}$ ${\displaystyle \oplus }$ ${\displaystyle \mathrm {Proj} (C(X))}$ ${\displaystyle C(X)}$

${\displaystyle {\begin{aligned}\Gamma \colon \mathrm {Vect} (X)&\to \mathrm {Proj} (C(X))\\E&\mapsto \Gamma (E)\end{aligned}}}$

is well defined and a homomorphism of monoids , that is, it holds . This formula does not differentiate between isomorphism classes and their representatives, which is possible because of the well-defined nature. ${\displaystyle \Gamma (E_{1}\oplus E_{2})=\Gamma (E_{1})\oplus \Gamma (E_{2})}$

Serre and Swan's theorem says that for a compact Hausdorff space this assignment is a bijection . ${\displaystyle X}$ ${\displaystyle \mathrm {Vect} (X)\to \mathrm {Proj} (C(X))}$

K-theoretical formulation

Since the topological K-theory of a space is the Grothendieck group of the semigroup and the topological K-theory of the Banach algebra is the Grothendieck group of the semigroup , the isomorphism follows directly from Serre and Swan's theorem ${\displaystyle X}$ ${\displaystyle \mathrm {Vect} (X)}$ ${\displaystyle C(X)}$ ${\displaystyle \mathrm {Proj} (C(X))}$

${\displaystyle K^{0}(X)\cong K_{0}(C(X))}$

for every compact Hausdorff room . ${\displaystyle X}$

literature

• Jean-Pierre Serre: Faisceaux algébriques cohérents. In: Annals of Mathematics. 61 (2): 197-278 (1955).
• Richard Swan: Vector bundles and projective modules. In: Transactions of the American Mathematical Society. 105 (2): 264-277 (1962).