Quillen–Suslin theorem

Quillen–Suslin theorem

The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings. Every free module over a ring is projective, but most rings admit projective modules that are not free. Serre asked whether the converse holds in certain situations.

Geometrically, finitely generated free modules correspond to trivial vector bundles and finitely generated projective modules to more general vector bundles. Affine space is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles. Jean-Pierre Serre, in his 1955 paper "Faisceaux algébriques cohérents", remarked that the equivalent question was not known for algebraic vector bundles: "It is not known if there exist projective "A"-modules of finite type which are not free." ["On ignore s'il existe des A-modules projectifs de type finie qui ne soient pas libres." Serre, "FAC", p. 243.] Here "A" is a polynomial ring over a field, that is, "A" = "k" ["x"1, ..., "x"n] .

To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could [to the name] ." [Lam, p. 1] ) The statement is not immediately obvious from the topological and holomorphic cases, because these cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization. Instead, the problem turns out to be extremely difficult. Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its direct sum with a free module, it became free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin, independently proved that the answer was affirmative. Quillen was awarded the Fields Medal in 1978 in part for his proof of the Serre conjecture. Leonid Vaseršteĭn later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's "Algebra".

References

*Citation
last = Serre
first = Jean-Pierre
authorlink = Jean-Pierre Serre
year = 1955
month = March
title = Faisceaux algébriques cohérents
journal = Annals of Mathematics. Second Series.
volume = 61
pages = 197-278
url = http://links.jstor.org/sici?sici=0003-486X%28195503%292%3A61%3A2%3C197%3AFAC%3E2.0.CO%3B2-C
*Citation
last = Serre
first = Jean-Pierre
authorlink = Jean-Pierre Serre
year = 1958
chapter = Modules projectifs et espaces fibrés à fibre vectorielle
title = Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23
language = French
*Citation
last = Quillen
first = Daniel
authorlink = Daniel Quillen
year = 1976
title = Projective modules over polynomial rings
journal = Inventiones Mathematicae
volume = 36
pages = 167-171
doi = 10.1007/BF01390008
*Citation
last = Suslin
first = Andrei A.
authorlink = Andrei Suslin
year = 1976
title = Projective modules over polynomial rings are free
language = Russian
journal = Dokl. Akad. Nauk SSSR
volume = 229
number = 5
pages = 1063-1066. Translated in Citation
last = Suslin
first = Andrei A.
authorlink = Andrei Suslin
year = 1976
title = Projective modules over polynomial rings are free
language = Russian
journal = Soviet Math. Dokl.
volume = 17
number = 4
pages = 1160-64
*Citation
last = Lang
first = Serge
authorlink = Serge Lang
year = 2002
title = Algebra
edition = Revised third
series = Graduate Texts in Mathematics, 211
publisher = Springer Science+Business Media
isbn = 0-387-95385-X

An account of this topic is provided by:

*Citation
last = Lam
first = T. Y.
year = 2006
title = Serre's problem on projective modules
publisher = Springer Science+Business Media
location = Berlin; New York
pages = 300pp.
id = ISBN 978-3540233176


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