Quillen–Suslin theorem
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- For the outstanding conjecture on Galois representations, see Serre conjecture (number theory).
The Quillen–Suslin theorem, also known as Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings.
In the case of finitely presented modules over a commutative ring, the condition of being projective is equivalent to being locally free, in the sense that every localization of the module at a prime ideal is free. Every free module is projective, but for modules over a general commutative ring, the converse is false. Jean-Pierre Serre found evidence that a converse might hold for polynomial rings, and asked the following question in his famous 1955 paper "Faisceaux algébriques cohérents" [1]:
- Is every finitely generated projective module over a polynomial ring (in several variables) over a field free?
An equivalent, more geometric, formulation of this question is whether every algebraic vector bundle on affine space is trivial.
Serre made some progress towards a solution in 1957 when he proved in [2] that every finitely generated projective module over a polynomial ring over a field was stably free.
Serre's conjecture remained open until 1976, when Daniel Quillen in [3], and Andrei Suslin in [4], 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 proof of the theorem, which can be found in Serge Lang's Algebra [5, p.850].
[edit] References
- [1] Serre, J.P., Faisceaux algébriques cohérents, Ann. Math., 61, 1955.
- [2] Serre, J.P., Modules projectifs et espaces fibrés à fibre vectorielle, Sém. Dubreuil-Pisot, 23, 1957/58.
- [3] Quillen D., Projective modules over polynomial rings, Invent. Math., 36, 167-171, 1976.
- [4] Suslin, A.A. Suslin, Projective modules over a polynomial ring are free, Soviet Math. Dokl. 17 (1976) 1160–1164 (English translation).
- [5] Lang, S., Algebra, 3rd ed., Addison-Wesley.
An account of this topic is provided by:
- [6] Lam, T.Y., Serre's Conjecture, Springer-Verlag, 1978.
