Original Research

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

A. A. Lemmer, G. Naudé
Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie | Vol 8, No 1 | a859 | DOI: https://doi.org/10.4102/satnt.v8i1.859 | © 1989 A. A. Lemmer, G. Naudé | This work is licensed under CC Attribution 4.0
Submitted: 14 March 1989 | Published: 14 March 1989

About the author(s)

A. A. Lemmer,, South Africa
G. Naudé,, South Africa

Full Text:

PDF (208KB)

Share this article

Bookmark and Share


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?


No related keywords in the metadata.


Total abstract views: 1004
Total article views: 1245

Reader Comments

Before posting a comment, read our privacy policy.

Post a comment (login required)

Crossref Citations

No related citations found.