Bass–Quillen conjecture

Template:Short description 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 ${\mathrm{Vect}}_{r}A$.

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

${\mathrm{Vect}}_{r}A\stackrel{\sim }{\to }{\mathrm{Vect}}_r(A[t_1,\dots ,t_n]).$

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 Template:Harvtxt in the case that A is a smooth algebra over a field k. Further known cases are reviewed in Template:Harvtxt.

Extensions

The set of isomorphism classes of vector bundles of rank r over A can also be identified with the nonabelian cohomology group

$H_{Nis}^1(Spec(A),G{L}_r).$

Positive results about the homotopy invariance of

$H_{Nis}^1(U,G)$

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

References

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