Math 2.0

K-theory K0 and projective modules Conjecture

Bass–Quillen conjecture

Builds on Noetherian ringRegular local ringPolynomial ringFinitely generated projective moduleTensor product of modules

Statement

Let AA be a commutative Noetherian ring that is regular, that is, its localization at every prime ideal is a regular local ring. Let n≥1n \ge 1 and let A[t1,…,tn]A[t_1, \ldots, t_n] be the polynomial ring over AA. Conjecture: every finitely generated projective A[t1,…,tn]A[t_1, \ldots, t_n]-module PP is extended from AA, that is, P≅A[t1,…,tn]⊗AQP \cong A[t_1, \ldots, t_n] \otimes_A Q for some finitely generated projective AA-module QQ.

Remarks

The conjecture is open in general. For a field AA, every finitely generated projective AA-module is free, and the conjecture becomes the Quillen–Suslin theorem. Quillen's proof also gives the case where AA is a Dedekind domain. Lindel proved the conjecture when AA is essentially of finite type over a field, and with Popescu's approximation theorem this extends to regular rings that contain a field. The regularity hypothesis cannot be dropped: over a non-regular ring there can be projective modules over A[t]A[t] that are not extended from AA.

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Finitely generated projective module, concept, no evidence levelFinitely generated pr…conceptNoetherian ring, concept, no evidence levelNoetherian ringconceptPolynomial ring, concept, no evidence levelPolynomial ringconceptRegular local ring, concept, no evidence levelRegular local ringconceptTensor product of modules, concept, no evidence levelTensor product of mod…conceptBass–Quillen conjecture, conjecture, no evidence levelBass–Quillen conjectu…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Finitely generated projective module concept
  2. Noetherian ring concept
  3. Polynomial ring concept
  4. Regular local ring concept
  5. Tensor product of modules concept
  6. Bass–Quillen conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Bass%E2%80%93Quillen_conjecture
About this page
Tag8XH5R0NXXQ
URIm2://8XH5R0NXXQ
Kindconjecture
WritingW0 stub
MSC19A13, 13C10
StatusThis copy of the site has no build data. Run m2 status 8XH5R0NXXQ.
AboutFinitely generated projective module, Polynomial ring
UsesNoetherian ring, Regular local ring, Polynomial ring, Finitely generated projective module, Tensor product of modules
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Changed3 November 2026

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