# Bass–Quillen conjecture

__: Would relate vector bundles over a regular Noetherian ring and over a polynomial ring__

**Short description**In mathematics, the **Bass–Quillen conjecture** relates vector bundles over a regular Noetherian ring *A* and over the polynomial ring [math]\displaystyle{ A[t_1, \dots, t_n] }[/math]. The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture.^{[1]}^{[2]}

## 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 [math]\displaystyle{ \operatorname{Vect}_r(A) }[/math].

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

- [math]\displaystyle{ M \mapsto M \otimes_A A [t_1, \dots, t_n] }[/math]

yields a bijection

- [math]\displaystyle{ \operatorname{Vect}_r(A) \stackrel \sim \to \operatorname{Vect}_r(A[t_1, \dots, t_n]). }[/math]

## Known cases

If *A* = *k* is a field, the Bass–Quillen conjecture asserts that any projective module over [math]\displaystyle{ k[t_1, \dots, t_n] }[/math] 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

- [math]\displaystyle{ H^1_{Nis}(Spec (A), GL_r). }[/math]

Positive results about the homotopy invariance of

- [math]\displaystyle{ H^1_{Nis}(U, G) }[/math]

of isotropic reductive groups *G* have been obtained by (Asok Hoyois) by means of **A**^{1} homotopy theory.

## References

- ↑ Bass, H. (1973),
*Some problems in ‘classical’ algebraic K-theory. Algebraic K-Theory II*, Berlin-Heidelberg-New York: Springer-Verlag, Section 4.1 - ↑ Quillen, D. (1976), "Projective modules over polynomial rings",
*Invent. Math.***36**: 167–171, doi:10.1007/bf01390008, Bibcode: 1976InMat..36..167Q

- 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 - 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

Original source: https://en.wikipedia.org/wiki/Bass–Quillen conjecture.
Read more |