[1][2] The conjecture is a statement about finitely generated projective modules.
For a ring A, the set of isomorphism classes of vector bundles over A of rank r is denoted by
The conjecture asserts that for a regular Noetherian ring A the assignment yields a bijection If A = k is a field, the Bass–Quillen conjecture asserts that any projective module over
This question was raised by Jean-Pierre Serre and was later proved by Quillen and Suslin; see Quillen–Suslin theorem.
More generally, the conjecture was shown by Lindel (1981) in the case that A is a smooth algebra over a field k. Further known cases are reviewed in Lam (2006).
The set of isomorphism classes of vector bundles of rank r over A can also be identified with the nonabelian cohomology group Positive results about the homotopy invariance of of isotropic reductive groups G have been obtained by Asok, Hoyois & Wendt (2018) by means of A1 homotopy theory.