A glance back at the Quillen-Suslin theorem and the recent status o f the Bass-Quillen conjecture

A A Lemmer,G Naudé

doi:10.4102/satnt.v8i1.859

Abstract

In his famous FAC-article J.P. Serre asked whether all finitely generated projective modules over K[X1,...,Xn] (K a field) are free. This question soon became known in the mathematical society as “Serre’s Conjecture”. This conjecture was proved independently in 1976 by A. A. Suslin and D. Quillen. Important further developments followed from the ideas in Quillen’s proof. In this article we discuss the geometric motivation of the original problem as well as certain aspects of Quillen’s proof. Then we discuss further developments, in particular a proof that finitely generated projective modules over R[X1,...,Xn], R a Bezout domain, are free. We also give attention to new results on the existence of free summands in projective modules. In the case of polynomial rings, this work strengthens Serre’s famous free summand theorem: if R is a Noetherian ring and P is a finitely generated projective R-module such that rank P > dim R, then there exists a free rank one summand in P. Finally we discuss the present status of the most important open problem in this field, namely the Bass-Quillen Conjecture: if R is a regular local ring, is every finitely generated projective R[X1,...,Xn] module free?

Highlights

A glance back at the Quillen-Suslin theorem and the recent status o f the Bass-Quillen conjecture
As M ’n eindigvoortgebringde projektiewe moduul oor S is met rang M > 1, dan bevat M ’n vry, rang een sommand: M = R ® M'. (c) As T ’n gebied is met die eienskap dat enige twee elemente van T ’n grootste gemene deler het, dan is enige rang een projektiewe moduul oor T vry
Dit is bekend dat BQ’ waar is vir R ’n reguliere lokale ring van dimensie < 2.8 Die dimensie-een-geval volg natuurlik direk uit die Quillen-Suslin-stelling, aangesien ’n lokale hoofideaalgebied regulier lokaal is

Summary

Introduction

A glance back at the Quillen-Suslin theorem and the recent status o f the Bass-Quillen conjecture. Aan die hand van die Quillen-Suslin-stelling kan ons dus aflei dat enige nie-singuliere kromme in

Results

Conclusion