Bass–Quillen Conjecture

In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring ''A'' and over the polynomial ring A _1, \dots, t_n/math>. The conjecture is named for Hyman Bass, 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 $\operatorname_r A$.

The conjecture asserts that for a regular Noetherian ring ''A'' the assignment
:M \mapsto M \otimes_A A _1, \dots, t_n/math>
yields a bijection
:\operatorname_r A \,\stackrel \sim \to \operatorname_r(A _1, \dots, t_n.

Known cases

If ''A'' = ''k'' is a field, the Bass–Quillen conjecture asserts that any projective module over k _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 in the case that ''A'' is a smooth algebra over a field ''k''. Further known cases are reviewed in .

Extensions

The set of isomorphism classes of vector bundles of rank ''r'' over ''A'' can also be identified with the nonabelian cohomology group
:$H^1_(Spec (A), GL_r).$
Positive results about the homotopy invariance of
:$H^1_(U, G)$
of isotropic reductive groups ''G'' have been obtained by by means of A¹ homotopy theory.

References

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{{DEFAULTSORT:Bass-Quillen conjecture
Commutative algebra
Algebraic K-theory
Algebraic geometry