Vector Bundles from an Algebraic Viewpoint

Gunjal, Milind Vishwas (2019) Vector Bundles from an Algebraic Viewpoint. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Milind Vishwas Gunjal (14MS064)) 14MS064.pdf - Submitted Version Restricted to Repository staff only Download (791kB)

Abstract

In this 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 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.

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