Bass–Quillen conjecture explained

A[t _{1,} ... , t _{n]} . The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture.^[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

\operatorname{Vect} _{r} A .

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

M \mapsto M ⊗ _{A} A [t _{1,} ... , t _{n]} yields a bijection\operatorname{Vect} _{r} A \stackrel \sim \to \operatorname{Vect} _{r(A[t} _{1,} ... , t _{n]).}

Known cases

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

k[t _{1,} ... , 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.

Notes and References

1. , Section 4.1