Vector Bundles and Projective Modules

Abstract

Vector bundles—the most important particular case of fiber bundle in the differential calculus—are studied in this chapter, both in the classical and in the general algebraic situation. The notion of section of a vector bundle and its relationship with projective modules is described. The equivalence of the category of vector bundles over a manifold and the category of projective finite-type modules over the correspondence algebra of smooth functions is established. The direct sum of vector bundles, their tensor product, subbundles, pullback vector bundles, sections of vector bundles, pseudo-bundles are considered.