Serre’s Conjecture on Projective Modules

Paul, Arani (2021) Serre’s Conjecture on Projective Modules. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Arani Paul (16MS161)) 16MS161_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (347kB)

Abstract

Jean-Pierre Serre proposed a question in his 1955 paper ’ Faisceaux algébriques cohérents’, proposed a question that if there exists a projective finitely generated R-module which is not free, where R is the polynomial ring A[X1,X2, . . . ,Xn] over some field A. THis problem quickly became known as the "Serre’s Conjecture" which was an open problem till 1976. Serre made some progress by proving that such modules will be definitely Stably Free, but later Andrei Suslin and Daniel Quillen independently settled the problem in 1976. They proved a more general result in the sense that "Any Finitely generated projective A[X1, . . . ,Xn]- module over a principal ideal domain A is free." In this project I have tried to cover the proof of Serre’s Conjecture by Quillen. In chapter 1 and 2 we go through the basic preliminaries and in chapter 4 we proved the conjecture by the passage from local to global. We first gave proof of the local version i.e. Horrock’s Theorem, then by proving Quillen’s theorem which is a global version of Horrock’s Theorem we gradually worked our way up to Serre’s Conjecture. Also to get the full significance of the conjecture in chapter 3 I tried to establish the geometric significance of this conjecture. the relations between vector bundles and projective modules are discussed in this section. This chapter is important in the sense that it completes the picture and gives much more insight to Serre’s conjecture.

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