Bass–Quillen conjecture

In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring . The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture., Section 4.1

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 .

The conjecture asserts that for a regular Noetherian ring A the assignment

yields a bijection

Known cases If A = k is a field, the Bass–Quillen conjecture asserts that any projective module over 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

Positive results about the homotopy invariance of

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

References