Editing Quillen–Suslin theorem

{{Short description|Commutative algebra theorem}}
{{redirect|Serre's problem|other uses|Serre's conjecture (disambiguation)}}
{{Infobox mathematical statement
| name = Quillen–Suslin theorem
| image = 
| caption = 
| field = [[Commutative algebra]]
| conjectured by = [[Jean-Pierre Serre]]
| conjecture date = 1955
| first proof by = [[Daniel Quillen]]<br>[[Andrei Suslin]]
| first proof date = 1976
}}

The '''Quillen–Suslin theorem''', also known as '''Serre's problem''' or '''Serre's conjecture''', is a [[theorem]] in [[commutative algebra]] concerning the relationship between [[free module]]s and [[projective module]]s over [[polynomial ring]]s. In the geometric setting it is a statement about the triviality of [[vector bundle]]s on [[affine space]].

The theorem states that every [[finitely generated module|finitely generated]] [[projective module]] over a [[polynomial ring]] is [[free module|free]].

==History==
===Background===
Geometrically, finitely generated projective modules over the ring <math>R[x_1,\dots,x_n]</math> correspond to [[vector bundle]]s over [[affine space]] <math>\mathbb{A}^n_R</math>, where free modules correspond to trivial vector bundles. This correspondence (from modules to (algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending <math> M\to \widetilde{M}</math> (Hartshorne II.5, page 110).  [[Affine space]] is topologically [[contractible space|contractible]], so it admits no non-trivial topological vector bundles.  A simple argument using the [[exponential sheaf sequence|exponential exact sequence]] and the [[d-bar Poincaré lemma]] shows that it also admits no non-trivial [[holomorphic vector bundle]]s.

[[Jean-Pierre Serre]], in his 1955 paper [[List of important publications in mathematics#Faisceaux Algébriques Cohérents|''Faisceaux algébriques cohérents'']], remarked that the corresponding question was not known for algebraic vector bundles: "It is not known whether there exist projective ''A''-modules of finite type which are not free."<ref>"On ignore s'il existe des ''A''-modules projectifs de type fini qui ne soient pas libres."  Serre, ''FAC'', p. 243.</ref>  Here <math>A</math> is a [[polynomial ring]] over a [[field (mathematics)|field]], that is, <math>A</math> = <math>k[x_1,\dots,x_n]</math>.

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]."<ref>Lam, p. 1</ref>)  The statement does not immediately follow from the proofs given in the topological or holomorphic case. These cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization.

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 of modules|direct sum]] with a finitely generated free module, it became free.  The problem remained open until 1976, when [[Daniel Quillen]] and [[Andrei Suslin]] independently proved the result. 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''.

==Generalization==
A generalization relating projective modules over [[regular ring|regular]] [[Noetherian ring]]s ''A'' and their polynomial rings is known as the [[Bass–Quillen conjecture]].

Note that although <math>GL_n</math>-bundles on affine space are all trivial, this is not true for ''G''-bundles where ''G'' is a general [[reductive algebraic group]].

==Notes==
{{reflist}}

==References==
*{{Citation
 | last = Serre
 | first = Jean-Pierre
 | authorlink = Jean-Pierre Serre
 |date=March 1955
 | title = Faisceaux algébriques cohérents
 | journal = [[Annals of Mathematics]] |series = Second Series
 | volume = 61
 | pages = 197–278
 | doi = 10.2307/1969915
 | jstor = 1969915
 | issue = 2 
 | mr = 0068874
}}
*{{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
 | mr = 0177011
}}
*{{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
 | mr = 0427303
 | issue = 1
| bibcode = 1976InMat..36..167Q
 }}
*{{Citation
 | last = Suslin
 | first = Andrei A.
 | authorlink = Andrei Suslin
 | year = 1976
 | script-title=ru:Проективные модули над кольцами многочленов свободны
 | trans-title = Projective modules over polynomial rings are free
 | language = Russian
 | journal = [[Doklady Akademii Nauk SSSR]]
 | volume = 229
 | pages = 1063–1066
 | issue = 5
 | mr = 0469905
 | postscript = .
}}  Translated in {{Citation
 | year = 1976
 | title = Projective modules over polynomial rings are free
 | journal = [[Soviet Mathematics]]
 | volume = 17
 | pages = 1160–1164
 | issue = 4
 | postscript = .
}}
*{{Lang Algebra}}

An account of this topic is provided by:

*{{Citation
 | last = Lam
 | first = Tsit Yuen
|authorlink = Tsit Yuen Lam
 | year = 2006
 | title = Serre's problem on projective modules
 | publisher = [[Springer Science+Business Media]]
 | location = Berlin; New York
 | pages = 300pp
 | series = Springer Monographs in Mathematics
 | isbn = 978-3-540-23317-6
 | mr = 2235330
}}

{{DEFAULTSORT:Quillen-Suslin theorem}}
[[Category:Commutative algebra]]
[[Category:Theorems in abstract algebra]]