Serre's problem, also called Serre's conjecture, asserts that the implication "free moduleprojective module"
can be reversed for every module over the polynomial
ring,
where
is a field (Serre 1955).
The hard part of the proof, the one concerning finitely generated modules, was given simultaneously, and independently, by D. Quillen in Cambridge, Massachusetts and A. A. Suslin in Leningrad (St. Petersburg) in 1976. As a result, the statement is often referred to as the "Quillen-Suslin theorem."
The solution to this difficult problem is part of the work for which Quillen was
awarded the Fields Medal in 1978.
Quillen and Suslin received, for other contributions in algebra, the Cole
Prize in 1975 and 2000 respectively.
Eisenbud, D. "Solution du problème de Serre par Quillen-Suslin." In Séminaire d'Algèbre Paul Dubreil. Paris
1975-1976 (Ed. M. P. Malliavin). Berlin: Springer-Verlag, pp. 9-19,
1977.Ferrand, D. "Les modules projectifs de type fini sur un anneau
de polynômes sur un corps sont libres." In Séminaire
Bourbaki, Vol. 1975/76. Berlin: Springer-Verlag, pp. 202-221, 1977.Gupta,
S. K. and Murthy, M. P. Suslin's Work on Linear Groups Over Polynomial
Rings and Serre Problem. New Delhi, India: Macmillan, 1980.Lam,
T. Y. Serre's
Conjecture. Berlin: Springer-Verlag, 1978.Lam, T. Y. Serre's
Problem on Projective Modules. Berlin: Springer-Verlag, 2005.Quillen,
D. "Projective Modules over Polynomial Rings." Invent. Math.36,
167-171, 1976.Serre, J. P. "Faisceaux Algébriques Cohérents."
Ann. Math.61, 191-278, 1955.Simis, A. When
Are Projective Modules Free? Queen's Papers in Pure and Applied Mathematics,
Vol. 21. Kingston, Ontario, Canada: Queen's University, 1969.Suslin,
A. A. "Projective Modules Over Polynomial Rings Are Free." Dokl.
Akad. Nauk. SSSR229, 1063-1066, 1976.
projective module"
can be reversed for every module over the polynomial
ring ![k[X_1,...,X_n]](/images/equations/SerresProblem/Inline2.svg)
,
where 
is a field (Serre 1955).
