Serre's Problem
Serre's problem, also called Serre's conjecture, asserts that the implication "free module ==> projective module" can be reversed for every module over the polynomial ring k[X_1,...,X_n], where k 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.
See also
Cole Prize, Fields Medal, Free Module, Projective ModuleThis entry contributed by Margherita Barile
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References
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. SSSR 229, 1063-1066, 1976.Referenced on Wolfram|Alpha
Serre's ProblemCite this as:
Barile, Margherita. "Serre's Problem." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SerresProblem.html