Modules as Bundles

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


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)


Vector Bundle Projective Module Free Module Hermitian Structure Smooth Section 
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© Springer-Verlag Berlin Heidelberg 2002

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