Generalized Bass–Quillen conjecture: extension of Zariski-locally trivial torsors under strictly isotropic reductive groups over regular rings

The generalized Bass–Quillen conjecture predicts that for a regular ring $A$ of finite Krull dimension and a strictly isotropic reductive group scheme $G$ over $A$, every Zariski-locally trivial principal $G$-bundle over the affine space $\mathbb{A}^n_A$ is extended from $A$; equivalently, the evaluation-at-zero map on non-abelian cohomology $H^1_{\mathrm{Zar}}(A[x_1,\ldots,x_n],G)\to H^1_{\mathrm{Zar}}(A,G)$ is bijective for every $n\ge 1$. For $G=\mathrm{GL}_m$ this recovers the classical Bass–Quillen conjecture on finitely generated projective modules over polynomial rings. The conjecture is known for equicharacteristic regular bases of arbitrary dimension, for bases ind-smooth over a discrete valuation ring when the group descends to the base, and — by a December 2025 preprint of Stavrova — for bases of Krull dimension $\le 2$ when the derived group of $G$ is split. What remains open is the mixed-characteristic regime: ramified regular bases of dimension $\ge 3$ (already open for $\mathrm{GL}_m$, the classical case) and dimension $\le 2$ for strictly isotropic groups with non-split derived group.

Background

The starting point is Serre's problem, posed in 1955 and solved independently by Quillen and Suslin in 1976: every finitely generated projective module over a polynomial ring $k[x_1,\ldots,x_n]$ over a field $k$ is free. Bass and Quillen then asked whether the same homotopy invariance holds over a regular base ring. A commutative noetherian ring $A$ is called regular if its localizations at prime ideals have finite global dimension (for instance, fields, the ring $\mathbb{Z}$ of integers, and polynomial rings over a field); its Krull dimension is the supremal length of chains of prime ideals. The classical Bass–Quillen conjecture asserts that for such $A$, every finitely generated projective $A[x_1,\ldots,x_n]$-module is extended from $A$, meaning of the form $P\otimes_A A[x_1,\ldots,x_n]$ for a projective $A$-module $P$; equivalently, every vector bundle (algebraic vector bundle, i.e. locally free sheaf) on the affine space $\mathbb{A}^n_A$ is the pullback of one on $\operatorname{Spec}A$.

The known classical cases shape the problem: Quillen and Suslin settled dimension $\le 1$; Murthy settled dimension $\le 2$ for a single variable; Lindel settled the case where $A$ is essentially of finite type over a field (meaning a localization of a finitely generated algebra over that field); and combining Quillen, Suslin, Lindel, and Popescu's approximation theorem — which expresses a regular ring containing a field as a filtered colimit of smooth algebras over that field — settles all unramified cases, including all equicharacteristic ones (equicharacteristic meaning the ring contains a field, as opposed to mixed characteristic). According to Česnavičius's 2022 survey of problems on torsors over regular rings, the main case in which the classical conjecture remains widely open is that of ramified regular local rings of mixed characteristic — rings whose residue field has characteristic $p>0$ while the fraction field has characteristic $0$ and $p$ lies in the square of the maximal ideal, such as complete ramified regular local rings of dimension $\ge 3$.

The generalized conjecture replaces vector bundles, which are torsors under the groups $\mathrm{GL}_m$, by torsors under reductive group schemes. A reductive group scheme is a smooth affine group scheme $G$ over $A$ whose fibers are (connected) reductive linear algebraic groups; its derived group $G^{\mathrm{der}}$ is the smallest normal subgroup scheme with a torus quotient. A principal $G$-bundle (or $G$-torsor) over a scheme $X$ is a scheme $E\to X$ with a $G$-action that is locally isomorphic, in a specified topology, to the trivial torsor $G\times X\to X$; here Zariski-locally trivial means locally trivial for the Zariski topology, and $H^1_{\mathrm{Zar}}(X,G)$ denotes the set of isomorphism classes of such torsors. A semisimple group is called split (or Chevalley–Demazure) if it is constructed from a based root system over $\mathbb{Z}$. A parabolic subgroup of $G$ is a smooth subgroup $P$ such that the quotient $G/P$ is a proper scheme over the base; a reductive $A$-group is strictly isotropic (also called totally isotropic) if it contains a parabolic $A$-subgroup intersecting properly every semisimple normal subgroup — for example, every reductive group with split derived group. The isotropicity restriction is essential: Fedorov (2016) constructed semisimple simply-connected groups over regular local rings carrying torsors on the affine line that do not descend to the base, so the naive statement for arbitrary reductive groups is false.

For strictly isotropic groups the generalized conjecture is known when the base ring contains a field, in any Krull dimension (Stavrova 2022, via $\mathbb{A}^1$-invariance of non-stable $K_1$-functors, i.e. of the quotients $G(R)/E(R)$ of group-valued points by the elementary subgroup; alternative proof by Česnavičius 2024), and when the base is ind-smooth over a discrete valuation ring with the group defined over that ring (Guo–Liu 2025, building on their Bass–Quillen-type theorem for valuation rings); ind-smooth means a filtered colimit of smooth algebras, and a discrete valuation ring is a principal ideal domain with exactly one nonzero prime ideal, for example the ring $\mathbb{Z}_p$ of $p$-adic integers. A December 2025 preprint of Stavrova adds the case of bases of Krull dimension $\le 2$ with $G^{\mathrm{der}}$ split, with no restriction on the characteristic, generalizing Murthy's classical dimension-2 theorem. After this work, the conjecture is unresolved precisely in mixed characteristic: dimension $\ge 3$ bases outside the valuation-ring setting, and dimension $\le 2$ bases for strictly isotropic groups with non-split derived group.

Problem Statement

Determine the truth of the generalized Bass–Quillen conjecture: for every regular ring $A$ of finite Krull dimension, every strictly isotropic reductive group scheme $G$ over $A$, and every integer $n\ge 1$, is the evaluation-at-zero map

$$H^1_{\mathrm{Zar}}\bigl(A[x_1,\ldots,x_n],G\bigr)\longrightarrow H^1_{\mathrm{Zar}}(A,G)$$

bijective — equivalently, is every Zariski-locally trivial principal $G$-bundle over $\mathbb{A}^n_A$ isomorphic to the pullback of a $G$-bundle over $\operatorname{Spec}A$? In the present state of knowledge (August 2026) this is open precisely in mixed characteristic: (i) regular bases of Krull dimension $\ge 3$ that are not ind-smooth over a discrete valuation ring on which $G$ is defined — already open for $G=\mathrm{GL}_m$, where it is the classical Bass–Quillen conjecture in its ramified mixed-characteristic case, e.g. over complete ramified regular local rings of dimension $\ge 3$; and (ii) mixed-characteristic bases of dimension $\le 2$ for strictly isotropic $G$ whose derived group is not split and which do not fall under the valuation-ring theorem of Guo–Liu. A complete proof in the stated generality, or an explicit counterexample within the strictly isotropic class, settles the problem.

References

1. Anastasia Stavrova, On the generalized Bass–Quillen conjecture in dimension 2, arXiv:2512.18868 (2025), https://arxiv.org/abs/2512.18868
2. Hyman Bass, Some problems in "classical" algebraic K-theory, in Algebraic K-Theory II (Proc. Conf., Battelle Memorial Inst., 1972), Lecture Notes in Mathematics 342, Springer (1973), pp. 3–73, DOI 10.1007/BFB0073718, https://doi.org/10.1007/bfb0073718
3. Daniel Quillen, Projective modules over polynomial rings, Inventiones Mathematicae 36 (1976), 167–171, DOI 10.1007/BF01390008, https://doi.org/10.1007/bf01390008
4. A. A. Suslin, Projective modules over a polynomial ring are free, Soviet Mathematics Doklady 17 (1976), 1160–1164; translation of Doklady Akademii Nauk SSSR 229 (1976), 1063–1066, MR0469905, Zbl 0354.13010, https://zbmath.org/?q=an%3A0354.13010
5. M. Pavaman Murthy, Projective $A[X]$-modules, Journal of the London Mathematical Society 41 (1966), 453–456, DOI 10.1112/jlms/s1-41.1.453, https://doi.org/10.1112/jlms/s1-41.1.453
6. 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
7. Roman Fedorov, Affine Grassmannians of group schemes and exotic principal bundles over $\mathbb{A}^1$, American Journal of Mathematics 138 (2016), no. 4, 879–906, DOI 10.1353/ajm.2016.0036, https://doi.org/10.1353/ajm.2016.0036
8. Anastasia Stavrova, $\mathbb{A}^1$-invariance of non-stable $K_1$-functors in the equicharacteristic case, Indagationes Mathematicae 33 (2022), no. 2, 322–333, DOI 10.1016/j.indag.2021.08.002, arXiv:1912.05424, https://arxiv.org/abs/1912.05424
9. Kęstutis Česnavičius, Problems about torsors over regular rings, Acta Mathematica Vietnamica 47 (2022), no. 1, 39–107, DOI 10.1007/s40306-022-00477-y, arXiv:2201.06424, https://arxiv.org/abs/2201.06424
10. Kęstutis Česnavičius, Torsors on the complement of a smooth divisor, Cambridge Journal of Mathematics 12 (2024), no. 4, 903–936, DOI 10.4310/cjm.241128010204, arXiv:2204.08233, https://arxiv.org/abs/2204.08233
11. Ning Guo and Fei Liu, The Bass–Quillen conjecture for torsors over valuation rings, International Mathematics Research Notices 2025 (2025), no. 11, article rnaf142, DOI 10.1093/imrn/rnaf142, arXiv:2503.10163, https://arxiv.org/abs/2503.10163
12. M. S. Raghunathan and A. Ramanathan, Principal bundles on the affine line, Proceedings of the Indian Academy of Sciences (Mathematical Sciences) 93 (1984), 137–145, DOI 10.1007/BF02840656, https://doi.org/10.1007/BF02840656