The Bass-Quillen conjecture relates vector bundles over a regular Noetherian ring A to vector bundles over the polynomial ring formed by adjoining variables to A. It is named for Hyman Bass and Daniel Quillen, who formulated it: Bass presented the idea in 1973 in a paper on problems in classical algebraic K-theory, and Quillen published related work on projective modules over polynomial rings in 1976.
Facts
StatementFor a regular Noetherian ring A, sending a module M to M tensored over A with the polynomial ring A adjoin t_1 through t_n gives a bijection between rank r vector bundles over A and rank r vector bundles over A adjoin t_1 through t_n. 1 Proposed Year Progress Toward ResolutionThe conjecture was proved by Lindel in 1981 for the case that A is a smooth algebra over a field k. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Bass-Quillen Conjecture, Wikipedia
Statement of the conjecture section
The conjecture asserts that for a regular Noetherian ring A the assignment M maps to M tensor_A A[t1, ..., tn] yields a bijection Vect_r A to Vect_r(A[t1, ..., tn]).
References section, Bass citation
Bass, H. (1973), Some problems in 'classical' algebraic K-theory. Algebraic K-Theory II
Known cases section
The conjecture was shown by Lindel (1981) in the case that A is a smooth algebra over a field k.
- Lead section
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