Bass–Quillen conjecture

Short description: Would relate vector bundles over a regular Noetherian ring and over a polynomial ring

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 (Lindel 1981) in the case that A is a smooth algebra over a field k. Further known cases are reviewed in (Lam 2006).

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 (Asok Hoyois) by means of A¹ homotopy theory.

References

• Asok, Aravind; Hoyois, Marc; Wendt, Matthias (2018), "Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces", Geom. Topol. 22 (2): 1181–1225, doi:10.2140/gt.2018.22.1181 
• Lindel, H. (1981), "On the Bass–Quillen conjecture concerning projective modules over polynomial rings", Invent. Math. 65 (2): 319–323, doi:10.1007/bf01389017, Bibcode: 1981InMat..65..319L 
• Lam, T. Y. (2006), Serre's problem on projective modules, Berlin: Springer, ISBN 3-540-23317-2