The Bass–Quillen conjecture for torsors under totally isotropic reductive group schemes in mixed characteristic

For a Noetherian regular ring $A$ and a totally isotropic reductive $A$ -group scheme $G$ , the torsor form of the Bass–Quillen conjecture (Conjecture 1.5 of Guo–Liu, 2025) predicts that every Nisnevich-locally trivial $G$ -torsor on the affine space $\mathbb{A}^N_A$ descends to $A$ , that is, pullback gives a bijection $H^1_{\mathrm{Nis}}(A,G)\simeq H^1_{\mathrm{Nis}}(\mathbb{A}^N_A,G)$ . The equi-characteristic case is fully settled, and Guo–Liu settled the case where $A$ is ind-smooth over a Prüfer ring $R$ and $G$ descends to a totally isotropic reductive $R$ -group scheme. The problem posed here is the remaining mixed-characteristic core: the conjecture for regular rings not covered by the ind-smooth-over-Prüfer hypothesis (for instance ramified regular rings), and the extension in which $G$ is an arbitrary totally isotropic reductive group scheme over a Prüfer-smooth $A$ without descending to the base. Since $G=\mathrm{GL}_n$ is totally isotropic, this contains the classical Bass–Quillen conjecture on projective modules over polynomial rings in mixed characteristic. Both regimes remain open as of August 2026; the only later partial result (Stavrova, 2025) covers dimension at most 2 with split derived group.

Background

The classical Bass–Quillen conjecture grew out of Serre's problem on whether finitely generated projective modules over a polynomial ring $k[x_1,\dots,x_n]$ over a field are free, settled affirmatively in 1976. Its general form asserts that for a Noetherian regular ring $A$ (a commutative noetherian ring whose localizations at prime ideals are regular local rings), every finitely generated projective module over $A[x_1,\dots,x_N]$ is extended from $A$ , equivalently every vector bundle on the affine space $\mathbb{A}^N_A$ is pulled back from $\operatorname{Spec} A$ . This is known when $A$ is regular over a Dedekind ring (in particular for fields, semilocal Dedekind domains, and unramified rings), and remains open in general, especially in mixed characteristic (rings not containing a field, such as ramified regular local rings).

A reductive group scheme is a smooth affine group scheme $G\to\operatorname{Spec} A$ whose geometric fibres are connected reductive algebraic groups (built from a torus and a semisimple part, such as $\mathrm{GL}_n$ , $\mathrm{SL}_n$ , $\mathrm{PGL}_n$ , symplectic and orthogonal groups). A $G$ -torsor (principal $G$ -bundle) is a scheme $P\to X$ with a $G$ -action that is, locally on $X$ in a specified topology, isomorphic to the trivial torsor $G$ ; isomorphism classes of torsors form a pointed cohomology set $H^1(X,G)$ , and for $G=\mathrm{GL}_n$ these are exactly rank- $n$ vector bundles. The Nisnevich topology is a Grothendieck topology finer than the Zariski topology but coarser than the étale topology: a Nisnevich cover is a family of étale maps such that every point admits a preimage inducing an isomorphism of residue fields; Nisnevich-local triviality is intermediate between Zariski- and étale-local triviality.

The torsor analogue of Bass–Quillen replaces $\mathrm{GL}_n$ by a general reductive $A$ -group scheme $G$ and vector bundles by Nisnevich-locally trivial $G$ -torsors. Here an isotropicity hypothesis is forced: a reductive group scheme is totally isotropic if, at every point, every simple factor of its adjoint part contains a copy of the multiplicative group $\mathbb{G}_m$ over a finite étale extension (equivalently, admits a fibrewise proper parabolic subgroup); tori and quasi-split groups (groups containing a Borel subgroup over the base, in particular split groups such as $\mathrm{GL}_n$ ) are totally isotropic. Without this condition the analogue fails: anisotropic groups (for instance the real orthogonal group $\mathrm{SO}(x_1^2+\cdots+x_4^2)$ over $\mathbf{R}$ ) carry Zariski-locally trivial torsors on affine spaces that do not descend.

Conjecture 1.5 of Guo–Liu (2025) — Bass–Quillen for torsors — asserts that for every Noetherian regular ring $A$ and every totally isotropic reductive $A$ -group scheme $G$ , pullback induces a bijection $H^1_{\mathrm{Nis}}(A,G)\xrightarrow{\sim}H^1_{\mathrm{Nis}}(\mathbb{A}^N_A,G)$ for all $N\geq 1$ ; the case $G=\mathrm{GL}_n$ recovers the classical conjecture. Prior state: the case where $A$ contains a field was fully settled around 2019–2022 (notably by Asok–Hoyois–Wendt for $A$ smooth over a field, and by Stavrova and by Česnavičius for general $A$ containing a field). A key input in the equi-characteristic arguments is a relative form of the Grothendieck–Serre conjecture proved by Fedorov: the classical Grothendieck–Serre conjecture predicts that a torsor under a reductive group scheme over a regular semilocal ring that becomes trivial over the fraction field is already trivial, and Fedorov's relative version states that for a regular local ring $R$ containing a field $k$ with fraction field $K$ , a totally isotropic reductive $R$ -group $G$ , and an affine $k$ -scheme $W$ , no nontrivial $G$ -torsor over $W\times_k R$ trivializes over $W\times_k K$ .

Guo–Liu (2025) proved Conjecture 1.5 when $A$ is ind-smooth over a Prüfer ring $R$ — ind-smooth means $A$ is a filtered colimit of smooth $R$ -algebras, and a Prüfer ring is a domain all of whose local rings are valuation rings (a valuation ring is an integral domain whose ideals are totally ordered by inclusion; a discrete valuation ring, abbreviated DVR, is a Noetherian one) — under the additional hypothesis that $G$ descends to a totally isotropic reductive $R$ -group scheme. Since a Noetherian regular ring that is unramified (essentially smooth over a Dedekind ring, e.g. over $\mathbb{Z}$ or a DVR) is ind-smooth over that base, this settles the unramified mixed-characteristic case for groups coming from the base. The mixed-characteristic case beyond this, and the case of groups not descending to the base, remain open, and Guo–Liu identify a relative Grothendieck–Serre statement over a mixed-characteristic DVR base (their Remark 4.3) as the specific missing input that blocks the extension of the equi-characteristic strategy: for a DVR $D$ , an essentially smooth local $D$ -algebra $R$ with fraction field $K$ , a totally isotropic reductive $R$ -group scheme $G$ , and a flat affine $D$ -scheme $W$ , every $G$ -torsor over $W\times_D R$ that becomes trivial over $W\times_D K$ is already trivial; this statement is itself open.

Problem Statement

Settle the remaining open cases of the Bass–Quillen conjecture for torsors (Conjecture 1.5 of Guo–Liu, 2025): for a Noetherian regular ring $A$ and a totally isotropic reductive $A$ -group scheme $G$ , is the pullback of Nisnevich cohomology sets

$H^1_{\mathrm{Nis}}(A,G)\xrightarrow{\sim}H^1_{\mathrm{Nis}}(\mathbb{A}^N_A,G)$

a bijection for every $N\geq 1$ — equivalently, does every Nisnevich-locally trivial $G$ -torsor on the affine space $\mathbb{A}^N_A$ descend to a $G$ -torsor over $A$ ? The common objective is descent of torsors from affine spaces over the bases and groups not covered by known results, specifically the two regimes the source work leaves open:

(i) Mixed characteristic, general regular rings: $A$ is a Noetherian regular ring that does not contain a field and is not itself ind-smooth over a Prüfer ring — for instance a ramified regular local ring — and $G$ is any totally isotropic reductive $A$ -group scheme. Since $\mathrm{GL}_n$ is totally isotropic, an affirmative answer here in particular settles the classical Bass–Quillen conjecture for finitely generated projective modules over polynomial rings in mixed characteristic.

(ii) Groups not descending to the base: $A$ is ind-smooth over a Prüfer ring $R$ , and $G$ is an arbitrary totally isotropic reductive group scheme over $A$ , without assuming that $G$ is pulled back from a reductive group scheme over $R$ (the extension conjectured in Remark 1.10 of Guo–Liu).

A complete answer is either a proof of the conjecture in these regimes or an explicit counterexample within them (a ring $A$ of the class stipulated in the respective regime — a Noetherian regular ring in regime (i), a ring ind-smooth over a Prüfer ring in regime (ii) — a totally isotropic reductive $A$ -group $G$ , and a Nisnevich-locally trivial $G$ -torsor on some $\mathbb{A}^N_A$ that does not descend to $A$ ).

References

Ning Guo and Fei Liu, “The Bass–Quillen conjecture for torsors over valuation rings”, 2025, arXiv:2503.10163, https://arxiv.org/abs/2503.10163; published in International Mathematics Research Notices (2025), doi:10.1093/imrn/rnaf142, https://doi.org/10.1093/imrn/rnaf142 (source of the problem statement)
Anastasia Stavrova, “On the generalized Bass–Quillen conjecture in dimension 2”, 2025, arXiv:2512.18868, https://arxiv.org/abs/2512.18868
Arnab Kundu, “Isotropic Torsors on Smooth Algebras over Prüfer Rings”, 2025, arXiv:2505.04760, https://arxiv.org/abs/2505.04760
Roman Fedorov, “On the Grothendieck-Serre Conjecture about principal bundles and its generalizations”, arXiv preprint 2018, arXiv:1810.11844, https://arxiv.org/abs/1810.11844; published in Algebra & Number Theory 16 (2022), doi:10.2140/ant.2022.16.447, https://doi.org/10.2140/ant.2022.16.447 (equicharacteristic relative Grothendieck–Serre input cited in the source paper)
Ning Guo and Fei Liu, “Grothendieck--Serre for constant reductive group schemes”, 2023, arXiv:2301.12460, https://arxiv.org/abs/2301.12460
Aravind Asok, Marc Hoyois, Matthias Wendt, “Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces”, arXiv preprint 2015, arXiv:1507.08020, https://arxiv.org/abs/1507.08020; published in Geometry & Topology 22 (2018), doi:10.2140/gt.2018.22.1181, https://doi.org/10.2140/gt.2018.22.1181