Projective Modules and the Quillen Suslin Theorem

Abstract

The notion of a module over a ring is a generalization of that of a vector space over a field k. The axioms are identical. But whereas every vector space possesses a basis, a module need not always have one. Modules possessing a basis are called free. So a finitely generated free /2-module is of the form for some n € N, equipped with the usual operations. A module is called projective, iff it is a direct summand of a free module. Especially a finitely generated i?-module P is projective iff there is an i?-module Q with P ^ Q = R^iov some n. Remarkably enough there do exist non free projective modules. Even there are non free P such that P 0 for some m and n. Modules P having the latter property are called stably free. On the other hand there are many rings, all of whose projective modules are free, e.g. local rings and principal ideal domains. (A commutative ring is called local iff it has exactly one maximal ideal.) In 1955, J.R Serre posed the question whether any finitely generated projective module over the polynomial ring k[Xi,X2i * • * ,X„] in several variables over a field is actually free. Later this was called Serre’s conjecture. For two decades it remained a challenging problem. It was known from the beginning that such a module had to be stably free. Serre^s conjecture was proved independently by D. Quillen and A. Suslin in 1976. The aim of this thesis is to present these independent proofe in a lucid and thorough manner. The layout of the thesis is as following. In first chapter we note down some essential commutative algebra results needed for our purpose. The second chapter will be a study of projective modules with several examples. In our third and final chapter we develop the Quillen Suslin theory that leads to the proof of Serre^s conjecture