nLab Quillen-Suslin theorem

Statement

The most memorable formulation of the Quillen-Suslin theorem states that for a field $k$ (or more generally a principal ideal domain), finitely generated projective modules over a finitary polynomial algebra $A = k[x_1, \ldots, x_r]$ are free.

In Serre’s FAC appears the sentence “It is not known if there exist projective A-modules of finite type which are not free.” This question became known as Serre’s problem or Serre’s conjecture (over repeated objections from Serre). Serre had made partial progress by proving that f.g. projective $A$-modules are stably free?, but the question remained unresolved until 1976 when an affirmative solution was produced by Daniel Quillen and independently by Andrei Suslin.

A later simplified proof was given by Leonid Vaserstein; this is recounted in Lang’s Algebra.

References

• Jean-Pierre Serre, Faisceaux algébriques cohérents, Annals of Mathematics, Second Series, 61 (2), 197–278 (1955). (doi)

• Daniel Quillen, Projective modules over polynomial rings, Inventiones Mathematicae 36 (1) (1976), 167–171. (doi)

• Andrei Suslin, Проективные модули над кольцами многочленов свободны, Doklady Akademii Nauk SSSR 229 (5) (1976), 1063-1066. Translated as Projective modules over polynomial rings are free, Soviet Mathematics 17 (4) (1976), 1160–1164.

• Serge Lang, Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), Springer-Verlag New York, 2002.