In mathematics, the Bass–Quillen conjecture relates

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...

s over a

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...

''A'' and over the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...

A_1, \dots, t_n /math>. The conjecture is named for

Hyman Bass Hyman Bass (; born October 5, 1932)

Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 19 ...

, who formulated the conjecture.

Statement of the conjecture

The conjecture is a statement about finitely generated

projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...

s. Such modules are also referred to as vector bundles. For a ring ''A'', the set of

isomorphism class In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the st ...

es 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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...

and was later proved by Quillen and Suslin, see

Quillen–Suslin theorem The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it i ...

. 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 group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

s ''G'' have been obtained by by means of A¹ homotopy theory.

References

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