Advertisement

Modules as Bundles

Chapter
Part of the Lecture Notes in Physics book series (LNPMGR, volume 51)

Abstract

The algebraic analogue of vector bundles has its origin in the fact that a vector bundle EM over a manifold M is completely characterized by the space ε = Γ(E, M) of its smooth sections. In this context, the space of sections is thought of as a (right) module over the algebra C (M) of smooth functions over M. Indeed, by the Serre-Swan theorem [143], locally trivial, finite-dimensional complex vector bundles over a compact Hausdorff space M correspond canonically to finite projective modules over the algebra A = C (M). To the vector bundle E one associates the C (M)-module ε = Γ(M, E) of smooth sections of E. Conversely, if E is a finite projective module over C (M), the fiber E m of the associated bundle E over the point mM is E m = ε/ε\( E_m = \mathcal{E}/\mathcal{E}\mathcal{I}_m \), (4.1) where the ideal \( \mathcal{I}_m \subset \mathcal{C}\left( M \right) \) , corresponding to the point mM, is given by % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaatuuDJX % wAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab-brijnaaBaaa \( \begin{gathered} \mathcal{I}_m = \ker \left\{ {\chi _m :C^\infty \left( M \right) \to \mathbb{C}|\chi _m \left( f \right) = f\left( m \right)} \right\} \hfill \\ {\mathbf{ }} = \left\{ {f \in C^\infty \left( M \right)|f\left( m \right) = 0} \right\} \hfill \\ \end{gathered} \). (4.2)

Keywords

Vector Bundle Projective Module Free Module Hermitian Structure Smooth Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Personalised recommendations