Bass–Quillen conjecture

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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 [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 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, doi:10.1007/bf01390008, Bibcode1976InMat..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, Bibcode1981InMat..65..319L 
  • Lam, T. Y. (2006), Serre’s problem on projective modules, Berlin: Springer, ISBN 3-540-23317-2