The classical Bass–Quillen conjecture predicts that for a regular Noetherian ring RR, every finitely generated projective module over a polynomial ring R[X1,…,Xn]R[X_1,\dots,X_n] is extended from RR; it is settled in low dimension and for rings essentially of finite type over a field or Dedekind ring, but remains open in general. This record concerns the analogous question when the Noetherian hypothesis is dropped: for which coherent commutative rings of finite weak dimension (regular coherent rings in the non-Noetherian sense) does every finitely generated projective R[X1,…,Xn]R[X_1,\dots,X_n]-module extend from RR, and is every such ring stably coherent, i.e. are all its polynomial rings still coherent? The Prüfer-domain case (including all valuation domains, any rank) was settled affirmatively by Lequain and Simis in 1980 and recently generalized to torsors under reductive groups by Guo and Liu, and special classes of local domains of weak dimension two (Wang's U² rings, generalized umbrella rings) are known; beyond these classes both questions are open, and no counterexample within regular coherent rings of finite weak dimension is known.

Background

Serre's problem (1955) asked whether every finitely generated projective module over a polynomial ring k[X1,…,Xn]k[X_1,\dots,X_n] over a field kk is free; Quillen and Suslin proved this affirmatively in 1976. A module PP over R[X1,…,Xn]R[X_1,\dots,X_n] is called extended from RR if P≅P0⊗RR[X1,…,Xn]P \cong P_0 \otimes_R R[X_1,\dots,X_n] for some RR-module P0P_0. Quillen's proof introduced patching: a finitely generated projective R[X1,…,Xn]R[X_1,\dots,X_n]-module is extended if and only if it is extended locally on Spec R\mathrm{Spec}\,R, and this reduction works over arbitrary commutative rings. Roitman (1979) moreover showed, for arbitrary commutative RR, that if every finitely generated projective R[X]R[X]-module is extended then the same holds for any number of variables, so the question is genuinely one-variable in nature. The Bass–Quillen conjecture asserts that every regular Noetherian ring RR (regular meaning every finitely generated module has finite projective dimension) has this extension property. It is known in dimension at most two and when RR is (essentially of finite type, i.e. a localization of a finitely generated algebra) regular over a field or a Dedekind ring (a Noetherian integrally closed domain of dimension at most one) (Quillen 1976, Lindel 1981, Popescu 1989), and as stated in print by Guo–Liu in 2025 it remains open in general, e.g. for ramified regular local rings of dimension at least three. Outside the Noetherian world the natural hypotheses change: a commutative ring is coherent if every finitely generated ideal is finitely presented, and its weak (Tor-)dimension is the supremum of flat dimensions of its modules; a coherent ring of finite weak dimension behaves like a regular ring in the sense that every finitely generated ideal then has finite projective dimension, and such rings are called regular coherent. For domains of weak dimension at most one this recovers Prüfer domains (domains in which every finitely generated ideal is invertible), which include all valuation domains (domains whose ideals are totally ordered, of arbitrary rank and typically non-Noetherian). Lequain and Simis proved in 1980 that every Prüfer domain has the Bass–Quillen property for all numbers of variables, settling the non-Noetherian weak-dimension-one domain case completely; Guo and Liu (2025) recently generalized this far beyond projective modules, showing that over a ring ind-smooth over a Prüfer ring (i.e. expressible as a filtered colimit of smooth algebras over the Prüfer base), torsors under suitable isotropic reductive group schemes (including GLn\mathrm{GL}_n, i.e. vector bundles) on affine spaces descend. Two obstructions make the higher-dimensional non-Noetherian question delicate. First, coherence is not stable under polynomial extension: Soublin (1970) constructed coherent rings whose polynomial rings are not coherent, so even the arena of the question can collapse; a ring is called stably coherent if RR and all R[X1,…,Xn]R[X_1,\dots,X_n] are coherent. Greenberg and Vasconcelos (1976) proved that a coherent ring of weak dimension at most one, or of global dimension two (every module has projective dimension at most two), is stably coherent, and Wang proved coherence of RR and R[X]R[X] for his U² rings (local GCD domains admitting an element uu with R/(u)R/(u) a valuation domain and RuR_u Bézout, i.e. every finitely generated ideal of the localization RuR_u principal); for coherent rings of finite weak dimension at least two the question is open in general. Second, no structure theory analogous to regular local rings or Prüfer domains is available for regular coherent domains of weak dimension two or more. Partial progress on the extension property in weak dimension two is limited to special local classes: Wang's U² rings (where finitely generated projectives over R[X]R[X] are free when dim⁡R/(u)=1\dim R/(u)=1) and the companion line of Tang–Huang–Wang on generalized umbrella rings. For stably coherent regular rings, K-theoretic results (Waldhausen, Swan, and recent work of Dahlhausen on semi-valuation rings) give only stable extension of projectives from R[X]R[X], which is strictly weaker than the Bass–Quillen property.

Problem Statement

Let RR be a commutative coherent ring (every finitely generated ideal is finitely presented) of finite weak dimension (the weak dimension is the supremum of the flat dimensions of all RR-modules) — equivalently, a regular coherent ring in the non-Noetherian sense, i.e. every finitely generated ideal has finite projective dimension — not assumed Noetherian. A finitely generated projective R[X1,…,Xn]R[X_1,\dots,X_n]-module PP is extended from RR if P≅P0⊗RR[X1,…,Xn]P \cong P_0 \otimes_R R[X_1,\dots,X_n] for some RR-module P0P_0. Determine, in this generality, the two linked properties:

(1) For which such rings RR does the Bass–Quillen extension property hold: every finitely generated projective R[X1,…,Xn]R[X_1,\dots,X_n]-module is extended from RR for every n≥1n \geq 1 (by Roitman's theorem it suffices to decide this for R[X]R[X] alone)? The question asks for the class of such rings with the extension property, not merely whether every such ring has it.

(2) Is every such ring stably coherent, i.e. is R[X1,…,Xn]R[X_1,\dots,X_n] coherent for every nn? This is known when the weak dimension is at most one and for coherent rings of global dimension two, and is false for coherent rings without finite-dimension hypotheses (Soublin); the coherent finite-weak-dimension case is open.

Determine the class of regular coherent rings of finite weak dimension satisfying (1) and (2).

References

  1. Daniel Quillen, Projective modules over polynomial rings, Inventiones Mathematicae 36 (1976), 167–171, DOI 10.1007/BF01390008, https://doi.org/10.1007/BF01390008
  2. Hartmut Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Inventiones Mathematicae 65 (1981), 319–323, DOI 10.1007/BF01389017, https://doi.org/10.1007/BF01389017
  3. Moshe Roitman, On projective modules over polynomial rings, Journal of Algebra 58 (1979), 51–63, DOI 10.1016/0021-8693(79)90188-1, https://doi.org/10.1016/0021-8693(79)90188-1
  4. Dorin Popescu, Polynomial rings and their projective modules, Nagoya Mathematical Journal 113 (1989), 121–128, DOI 10.1017/S0027763000001288, https://doi.org/10.1017/S0027763000001288
  5. Yves Lequain and Aron Simis, Projective modules over R[X_1,...,X_n], R a Prüfer domain, Journal of Pure and Applied Algebra 18 (1980), 165–171, DOI 10.1016/0022-4049(80)90127-9, https://doi.org/10.1016/0022-4049(80)90127-9
  6. Jean-Pierre Soublin, Anneaux et modules cohérents, Journal of Algebra 15 (1970), 455–472, DOI 10.1016/0021-8693(70)90050-5, https://doi.org/10.1016/0021-8693(70)90050-5
  7. B. V. Greenberg and Wolmer V. Vasconcelos, Coherence of polynomial rings, Proceedings of the American Mathematical Society 54 (1976), 59–64, DOI 10.1090/S0002-9939-1976-0417164-2, https://doi.org/10.1090/S0002-9939-1976-0417164-2
  8. FangGui Wang, The Bass-Quillen problem on a class of local rings with weak global dimension two, Science in China Series A: Mathematics 52 (2009), 94–108, DOI 10.1007/s11425-008-0154-4, https://doi.org/10.1007/s11425-008-0154-4
  9. Gaohua Tang, Zhaoyong Huang, and Fanggui Wang, Generalized umbrella rings, Journal of Algebra 315 (2007), 555–565, DOI 10.1016/j.jalgebra.2007.01.034, https://doi.org/10.1016/j.jalgebra.2007.01.034
  10. Fanggui Wang and Hwankoo Kim, Coherent Rings with Finite Weak Global Dimension, in Foundations of Commutative Rings and Their Modules, Algebra and Applications, Springer (2017), 535–571, DOI 10.1007/978-981-10-3337-7_9, https://doi.org/10.1007/978-981-10-3337-7_9
  11. Ning Guo and Fei Liu, The Bass–Quillen conjecture for torsors over valuation rings, International Mathematics Research Notices (2025), DOI 10.1093/imrn/rnaf142, arXiv:2503.10163, https://doi.org/10.1093/imrn/rnaf142
  12. Anastasia Stavrova, On the generalized Bass–Quillen conjecture in dimension 2, arXiv:2512.18868 (2025), https://arxiv.org/abs/2512.18868
  13. Christian Dahlhausen, Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory, Comptes Rendus. Mathématique 363 (2025), 989–1001, DOI 10.5802/crmath.786 (preprint arXiv:2403.02413), https://doi.org/10.5802/crmath.786
  14. Ryoya Ando, Polynomial Extensions and Localization of Non-Noetherian Cohen–Macaulay Rings, arXiv:2606.25384 (2026), https://arxiv.org/abs/2606.25384
  15. Moritz Kerz, Shuji Saito, and Georg Tamme, Towards a non-Archimedean analytic analog of the Bass–Quillen conjecture, Journal of the Institute of Mathematics of Jussieu 19 (2020), 1931–1946, DOI 10.1017/S147474801900001X, https://doi.org/10.1017/S147474801900001X