Serre’s problem on projective modules over polynomial rings

Determine whether every finitely generated projective module over the polynomial ring K[X1,…,Xr] is free; equivalently, ascertain whether every algebraic vector bundle over affine r-space Ar over a field K is trivial.

Background

In his 1955 paper Faisceaux algébriques cohérents, Serre identified algebraic vector bundles with finitely generated projective modules over coordinate rings. Specializing to polynomial rings raised a basic structural question: are all such projective modules free? Equivalently, are all algebraic vector bundles over affine space trivial?

This question became known as Serre’s problem and motivated substantial work connecting algebraic geometry, K-theory, and module theory.

References

Signalons que, lorsque V = K' (auquel cas A = K[X_1,\ldots, X_r]), on ignore s'il existe des A-modules projectifs de type fini qui ne soient pas libres, ou, ce qui revient au même, s'il existe des espaces fibrés algébriques à fibres vectorielles, de base Kr, et non triviaux.

— Constructing projective modules  (2412.05250 - Asok, 2024) in Section Fiber bundles and projective modules, subsection From analogies to concrete problems (quoting Serre, FAC)