Bass's finiteness conjecture for $K_0$ of regular rings finitely generated over $\mathbb{Z}$

For a commutative ring $R$ that is regular and finitely generated as a $\mathbb{Z}$ -algebra (a ring of finite type over $\mathbb{Z}$ , such as $\mathbb{Z}[t]$ , coordinate rings of affine varieties over finite fields, or affine pieces of arithmetic schemes), Bass conjectured in 1973 that the algebraic K-groups $K_n(R)$ , $n>0$ , are finitely generated abelian groups; the $n=0$ instance says that the Grothendieck group $K_0(R)$ of finitely generated projective $R$ -modules is finitely generated, i.e. that projective modules over arithmetic rings admit an essentially finite classification. The conjecture is classical in Krull dimension one but remains open in general — already for smooth affine surfaces over finite fields it subsumes conjectural finite-generation statements for Chow groups of zero-cycles tied to the Tate conjecture. The conjecture resurfaced as a bottleneck in the theory of the genus of division algebras: Chernousov–Rapinchuk–Rapinchuk (Israel J. Math. 2020) derived, conditionally on it, triviality of Čech $H^1$ for $\mathrm{GL}_n$ after localization and Condition (T) for adele class sets of $\mathrm{GL}_n$ and $\mathrm{GL}_{1,C}$ over fraction fields of such rings (the Condition (T) statements in Krull dimension at most two, with order hypotheses in the $\mathrm{GL}_{1,C}$ case), and no later work has removed this dependence except in the toric case.

Background

For a unital commutative ring $R$ , the Grothendieck group $K_0(R)$ is the abelian group generated by isomorphism classes of finitely generated projective $R$ -modules, i.e. direct summands of finite free modules, subject to the relation $[P]=[P']+[P'']$ for every exact sequence $0\to P'\to P\to P''\to 0$ ; it is the degree-zero part of algebraic K-theory and the home of the classification problem for vector bundles over affine schemes. A ring is called regular if every finitely generated module has finite projective dimension, and a finitely generated $\mathbb{Z}$ -algebra (ring of finite type over $\mathbb{Z}$ ) is one obtained from $\mathbb{Z}$ by adjoining finitely many elements and passing to quotients of the result, i.e. a quotient of a polynomial ring $\mathbb{Z}[x_1,\dots,x_m]$ ; standard examples are $\mathbb{Z}[t_1,\dots,t_m]$ , coordinate rings of affine varieties over finite fields, and affine open subsets of schemes of finite type over $\operatorname{Spec}\mathbb{Z}$ . Krull dimension is the combinatorial length of chains of prime ideals, the basic dimension notion of commutative algebra.

In his 1973 problem list 'Some problems in "classical" algebraic K-theory', H. Bass conjectured that for a regular ring $R$ of finite type over $\mathbb{Z}$ all the groups $K_n(R)$ , $n>0$ , are finitely generated abelian groups. The case relevant here, finite generation of $K_0(R)$ , is the natural higher-dimensional continuation of classical ideal theory: in Krull dimension one, $K_0(R)$ splits as $\mathbb{Z}$ plus the ideal class group (the group of invertible fractional ideals modulo principal ones), which is finite by the classical finiteness theorems for one-dimensional arithmetic rings. The arithmetic base is essential for plausibility: over algebraically closed base fields the analogous statement already fails in dimension one, since for a smooth complex curve the Picard group of line bundles contains a complex torus as a divisible, hence non-finitely-generated, subgroup. Beyond dimension one the conjecture subsumes major open problems on algebraic cycles: for a regular ring that is smooth over a finite field, standard comparisons between K-theory and Chow groups (groups of algebraic cycles modulo rational equivalence) show that finite generation of $K_0$ entails the conjectural finite generation of the Chow group of zero-cycles, already an open and difficult question for smooth affine surfaces over finite fields, closely tied to the Tate conjecture on cycles over finite fields (the prediction that Galois-invariant cycle classes on a variety over a finite field are spanned by classes defined over finite extensions of that field).

The conjecture recently reappeared as an explicit bottleneck in a seemingly distant problem on division algebras. The genus $\operatorname{gen}(D)$ of a finite-dimensional division algebra $D$ with center a field $K$ is the set of $K$ -isomorphism classes of division $K$ -algebras having exactly the same maximal subfields as $D$ , where a maximal subfield is a field extension of $K$ contained in $D$ of the maximal possible degree $(\dim_K D)^{1/2}$ . Chernousov, A. Rapinchuk and I. Rapinchuk proved that $\operatorname{gen}(D)$ is finite whenever the center is a finitely generated field of any characteristic. Their method routes through adelic double cosets: for a set $V$ of discrete valuations of $K$ , one forms the ring of adeles $\mathbf{A}(K,V)$ restricted to $V$ (a restricted product of the completions $K_v$ ), and for an algebraic $K$ -group $G$ the class set $\operatorname{cl}(G,K,V)$ is the double-coset space $G(\mathbf{A}^\infty(K,V))\backslash G(\mathbf{A}(K,V))/G(K)$ , where $\mathbf{A}^\infty$ denotes the integral adeles; $G$ satisfies Condition (T) with respect to $V$ if after deleting finitely many valuations from $V$ this class set reduces to one element. For $K$ the fraction field of a regular finitely generated $\mathbb{Z}$ -algebra $R$ of Krull dimension $d$ , the authors derive — but only assuming Bass's conjecture — that some principal open subset $U\subset\operatorname{Spec}R$ has trivial Čech cohomology $\check H^1(U,\mathrm{GL}_n)$ for all $n>d$ (Čech $H^1$ with coefficients in a group scheme classifies algebraic bundles, i.e. torsors, over $U$ ; the paper calls its triviality Conjecture 5.2 for $G=\mathrm{GL}_n$ ; Corollary 6.15), that $\mathrm{GL}_n$ for $n>3$ satisfies Condition (T) with respect to the divisorial valuations of $R$ (those attached to height-one prime ideals) when $R$ is a regular domain of Krull dimension at most $2$ (Corollary 6.16), and, in the descent section, Condition (T) for $\mathrm{GL}_{1,C}$ for central simple algebras $C$ of degree $>3$ admitting suitable orders over an integrally closed finitely generated $\mathbb{Z}$ -algebra of Krull dimension at most $2$ (Theorem 7.5); here a central simple algebra over a field is a finite-dimensional algebra whose center is exactly that field and which has no nontrivial two-sided ideals, and its degree is the square root of its dimension. The paper labels Bass's conjecture 'old and notoriously difficult'; the main genus-finiteness theorem itself is unconditional, and it is these structural generalizations that remain conditional. The conjecture is also revisited in the motivic context by Geisser, who relates Bass-type finite generation of motivic cohomology (the cohomology theory of algebraic cycles that underlies algebraic K-theory) over finite fields to finite generation of Weil-étale cohomology (a cohomology theory for varieties over finite fields built from the Weil group), again without resolving it.

Problem Statement

Prove or refute Bass's finiteness conjecture for $K_0$ : is $K_0(R)$ , the Grothendieck group of finitely generated projective $R$ -modules, a finitely generated abelian group for every commutative regular ring $R$ that is finitely generated as a $\mathbb{Z}$ -algebra? The question concerns the full class of such rings — all characteristics and all Krull dimensions, including smooth algebras over finite fields and regular rings of mixed characteristic such as $\mathbb{Z}[t_1,\dots,t_m]$ and its localizations at finitely many elements. An affirmative answer with a complete proof settles the $K_0$ -instance of the conjecture as attributed to Bass's 1973 problem list and formulated as Conjecture 6.14 of the cited genus paper (Bass's list additionally conjectures the analogous statement for the higher K-groups $K_n$ with $n>0$ , which is not part of this problem); an explicit counterexample, namely a regular ring finitely generated over $\mathbb{Z}$ whose Grothendieck group $K_0(R)$ is not a finitely generated abelian group, refutes it. An affirmative answer automatically renders unconditional the statements the genus program derived conditionally on the conjecture — existence of a principal open subset of $\operatorname{Spec}R$ with trivial $\check H^1(\,\cdot\,,\mathrm{GL}_n)$ for $n$ above the Krull dimension, and Condition (T) for $\mathrm{GL}_n$ and $\mathrm{GL}_{1,C}$ over the fraction fields of such rings, each under the auxiliary hypotheses stated in that paper (for the Condition (T) statements, Krull dimension at most two, and for $\mathrm{GL}_{1,C}$ additionally the order hypotheses); a refutation leaves the truth of those conditional consequences undetermined, and re-proving those group-theoretic statements by independent means when the conjecture is refuted is not part of the required deliverable.

References

V. I. Chernousov, A. S. Rapinchuk, I. A. Rapinchuk, 'The finiteness of the genus of a finite-dimensional division algebra, and some generalizations', Israel Journal of Mathematics 236 (2020), 747–799. DOI: 10.1007/s11856-020-1988-x; arXiv:1802.00299; https://arxiv.org/abs/1802.00299
H. Bass, 'Some problems in "classical" algebraic K-theory', in: Lecture Notes in Mathematics 343, Springer, Berlin, 1973, pp. 1–73. DOI: 10.1007/bfb0073718; https://doi.org/10.1007/bfb0073718
A. S. Rapinchuk, I. A. Rapinchuk, 'Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields', Journal of Number Theory 233 (2022), 228–260. DOI: 10.1016/j.jnt.2021.07.001; arXiv:2002.01520; https://arxiv.org/abs/2002.01520
A. S. Rapinchuk, I. A. Rapinchuk, 'Properness of the global-to-local map for algebraic groups with toric connected component and other finiteness properties', Mathematical Research Letters 30 (2023), 913–943. DOI: 10.4310/mrl.2023.v30.n3.a11; arXiv:2104.14625; https://arxiv.org/abs/2104.14625
I. A. Rapinchuk, 'Algebraic groups with good reduction and the genus problem', Contemporary Mathematics 800 (2024), 271–281. DOI: 10.1090/conm/800/16061; arXiv:2205.00273; https://arxiv.org/abs/2205.00273
P. Abramenko, A. S. Rapinchuk, I. A. Rapinchuk, 'Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings', Journal of Algebra 656 (2024), 3–23. DOI: 10.1016/j.jalgebra.2023.05.023; arXiv:2208.09353; https://arxiv.org/abs/2208.09353
A. S. Rapinchuk, I. A. Rapinchuk, A. Roy, 'Products of trees and ${\rm PGL}_2$ -torsors over the punctured affine line', preprint (2026). arXiv:2608.18934; https://arxiv.org/abs/2608.18934
T. H. Geisser, 'Finite generation conjectures for motivic cohomology theories over finite fields', in: Regulators (Contemporary Mathematics 571), American Mathematical Society, 2012, pp. 153–165. DOI: 10.1090/conm/571/11326; preprint arXiv:1103.5544 (posted under the title 'Finite generation conjectures for cohomology over finite fields'); https://arxiv.org/abs/1103.5544