Bass–Quillen conjecture

In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring $$A[t_1, \dots, t_n]$$. The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture.

Statement of the conjecture
The conjecture is a statement about finitely generated projective modules. Such modules are also referred to as vector bundles. For a ring A, the set of isomorphism classes of vector bundles over A of rank r is denoted by $$\operatorname{Vect}_r(A)$$.

The conjecture asserts that for a regular Noetherian ring A the assignment
 * $$M \mapsto M \otimes_A A [t_1, \dots, t_n]$$

yields a bijection
 * $$\operatorname{Vect}_r(A) \stackrel \sim \to \operatorname{Vect}_r(A[t_1, \dots, t_n]).$$

Known cases
If A = k is a field, the Bass–Quillen conjecture asserts that any projective module over $$k[t_1, \dots, t_n]$$ is free. 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 in the case that A is a smooth algebra over a field k. Further known cases are reviewed in.

Extensions
The set of isomorphism classes of vector bundles of rank r over A can also be identified with the nonabelian cohomology group
 * $$H^1_{Nis}(Spec (A), GL_r).$$

Positive results about the homotopy invariance of
 * $$H^1_{Nis}(U, G)$$

of isotropic reductive groups G have been obtained by by means of A1 homotopy theory.