Bass–Quillen conjecture

In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring ${\displaystyle A[t_{1},\dots ,t_{n}]}$. 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 ${\displaystyle \operatorname {Vect} _{r}(A)}$.

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

${\displaystyle M\mapsto M\otimes _{A}A[t_{1},\dots ,t_{n}]}$

yields a bijection

${\displaystyle \operatorname {Vect} _{r}(A){\stackrel {\sim }{\to }}\operatorname {Vect} _{r}(A[t_{1},\dots ,t_{n}]).}$

Known cases

If A = k is a field, the Bass–Quillen conjecture asserts that any projective module over ${\displaystyle k[t_{1},\dots ,t_{n}]}$ 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

${\displaystyle H_{Nis}^{1}(Spec(A),GL_{r}).}$

Positive results about the homotopy invariance of

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

of isotropic reductive groups G have been obtained by Asok, Hoyois & Wendt (2018) by means of A¹ homotopy theory.

References

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

• 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, arXiv:1507.08020, Zbl 1400.14061
• Lindel, H. (1981), "On the Bass–Quillen conjecture concerning projective modules over polynomial rings", Invent. Math., 65 (2): 319–323, Bibcode:1981InMat..65..319L, doi:10.1007/bf01389017
• Lam, T. Y. (2006), Serre’s problem on projective modules, Berlin: Springer, ISBN 3-540-23317-2, Zbl 1101.13001