Existence of Non-Vanishing Section and Cancellation Problem of Projective Modules

Kumbhargire, Rudresh (2022) Existence of Non-Vanishing Section and Cancellation Problem of Projective Modules. Masters thesis, Indian Institute of Science, Education and Research Kolkata.

Text (MS dissertation of Rudresh Kumbhargire (17MS213)) 17MS213_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (556kB)

Abstract

The study of projective modules is inspired by classical results in topology. A classical result from topology says that if V is a vector bundle over a compact manifold X with rank greater than dimension of X, then V has a nowhere vanishing section. Now a result by Swan says that there is a one-to-one correspondence between the topological vector bundles and projective modules. The algebraic analog of this result is the Splitting Theorem by Serre, i.e. when we have a projective module P, when can we have a surjective map from P to R. So in this report we are going to study the Splitting Theorem by Serre and see that this result cannot be improved further. Also we will look at the Cancellation Theorem by Hyman Bass and again see that this result in general is the best possible result.

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