Vector Bundles from an Algebraic Viewpoint
Published in Indian Institute of Science Education and Research, Kolkata, 2019
Abstract: In this Master’s thesis, we will give a brief introduction to projective modules and vector bundles and discuss the correspondence between modules over the ring of continuous functions (C(X)) and vector bundles over X. We will also introduce covering space theory to get a better understanding of the correspondence between algebraic and topological objects. A detailed discussion about the compactifications is also presented. We will also establish the notion of localization of vector bundles in terms of localization of C(X)-modules at a point using Swan’s theorem. We will discuss some examples of Swan’s theorem. Finally, we motivate the statement of the Quillen-Suslin theorem.