The Auslander–Reiten conjecture on stable equivalences (ARC) predicts that whenever two Artin algebras have equivalent stable module categories, they have the same number of isomorphism classes of simple modules that are not projective. Despite fifty years of work, only special classes of algebras are covered (hereditary, representation-finite, special biserial, Frobenius-finite, centralizer matrix algebras), and the conjecture is open in general. It is stated by Xi and Zhang in 'New invariants of stable equivalences and Auslander–Reiten conjecture' (arXiv:2207.10848; Math. Ann. 2026). The same paper's companion conjecture, that finiteness of the delooping level is invariant under derived equivalences, was refuted in March 2026 by an explicit finite-dimensional counterexample of Liang Chen.

Background

An Artin algebra is a ring AA that is finitely generated as a module over a commutative artinian ring kk; the central example is a finite-dimensional algebra over a field. One studies the category A-modA\text{-}\mathrm{mod} of finitely generated left AA-modules, its simple modules (modules with no nonzero submodules) and its projective modules (direct summands of free modules). The stable module category A-mod‾\underline{A\text{-}\mathrm{mod}} has the same objects as A-modA\text{-}\mathrm{mod} but its morphisms are those of A-modA\text{-}\mathrm{mod} modulo the ideal of maps that factor through a projective module. Two Artin algebras AA and BB are called stably equivalent when A-mod‾\underline{A\text{-}\mathrm{mod}} and B-mod‾\underline{B\text{-}\mathrm{mod}} are equivalent as kk-categories. Stable equivalences arise throughout the representation theory of algebras and groups — for example, between blocks of group algebras in modular representation theory — and they identify the entire non-projective representation theory of one algebra with that of the other.

Since the stable category discards projective modules, any invariant formulated purely for non-projective modules should be shared by stably equivalent algebras. The most elementary numerical invariant of this kind is the number of isomorphism classes of simple modules that are not projective. In the group-algebra setting this question was also pressed by Alperin, and the statement is sometimes called the Auslander–Alperin conjecture; Rouquier's 2006 International Congress of Mathematicians survey restates it as Conjecture 2.5. There is, however, no formal mechanism producing the equality: simple modules are not objects of the stable category, so the stable equivalence itself provides no correspondence between them, and the conjectured equality is a genuine and long-standing open problem.

The known partial results proceed by reducing or restricting the class of algebras. Auslander and Reiten (1975) proved the conjecture when one algebra is hereditary, i.e. of finite global dimension with hereditary module category. Martínez-Villa proved it for representation-finite algebras (algebras with only finitely many indecomposable finitely generated modules, up to isomorphism) over algebraically closed fields, and in 1990 reduced the general conjecture to self-injective algebras (algebras for which projective and injective modules coincide) having no nodes — a node being a simple module SS that is neither projective nor injective and whose almost split sequence 0→S→P→Tr D(S)→00\to S\to P\to \mathrm{Tr}\,D(S)\to 0 — the distinguished non-split short exact sequence, unique up to isomorphism, ending at Tr D(S)\mathrm{Tr}\,D(S), where Tr D\mathrm{Tr}\,D is the inverse Auslander–Reiten translation — has projective middle term PP. Further cases came from special biserial algebras (algebras presented by a quiver with relations in which each vertex has at most two arrows entering and two leaving, and each arrow starts at most two paths that the relations do not kill; Antipov–Zvonareva 2019) and Frobenius-finite algebras under stable equivalences of Morita type (Hu–Xi 2018); a Frobenius part of an algebra is the endomorphism algebra of its subcategory of ν\nu-stably projective modules (those projectives whose images under all positive powers of the Nakayama functor ν=D HomA(−,A)\nu=D\,\mathrm{Hom}_A(-,A), with D=Homk(−,k)D=\mathrm{Hom}_k(-,k) the standard duality, remain projective), unique up to Morita equivalence, and the algebra is Frobenius-finite when this part is representation-finite. A stable equivalence of Morita type is one induced by tensoring with a bimodule that is projective on each side and has a two-sided inverse up to stable isomorphism — the strongest and best-behaved source of stable equivalences. A principal centralizer matrix algebra is the centralizer of a single n×nn\times n matrix in the full matrix algebra over a field.

The immediate source of the present record, Xi and Zhang's paper, contributes new stable invariants: delooping levels (the least number of syzygies — iterated kernels of projective covers — after which a module becomes a direct summand of the direct sum of a projective module and a later syzygy of some module) and Igusa–Todorov ϕ\phi- and ψ\psi-dimensions (homological measures introduced to bound the finitistic dimension, the supremum of the projective dimensions of modules of finite projective dimension). The paper verifies the conjecture for principal centralizer matrix algebras over arbitrary fields and for Frobenius-finite algebras over algebraically closed fields, with no restriction on the type of stable equivalence. The paper explicitly records that the conjecture remains open in general.

Problem Statement

Decide the Auslander–Reiten conjecture on stable equivalences: if AA and BB are Artin algebras whose stable module categories A-mod‾\underline{A\text{-}\mathrm{mod}} and B-mod‾\underline{B\text{-}\mathrm{mod}} are equivalent (as kk-categories), must AA and BB have the same number of isomorphism classes of simple modules that are not projective? The question concerns all Artin algebras, with no restriction on the field or coefficient ring, on the class of algebras, or on the type of stable equivalence.

References

  1. Changchang Xi, Jinbi Zhang, 'New invariants of stable equivalences and Auslander–Reiten conjecture', Mathematische Annalen 394(4) (2026; online 12 March 2026), DOI 10.1007/s00208-026-03422-8, https://doi.org/10.1007/s00208-026-03422-8 (preprint arXiv:2207.10848, posted under the different title 'New invariants of stable equivalences of algebras', v5 of 12 Oct 2025, https://arxiv.org/abs/2207.10848)
  2. M. Auslander, I. Reiten, 'Stable equivalence of dualizing R-varieties V: Artin algebras stably equivalent to hereditary algebras', Advances in Mathematics 17(2) (1975) 167–195, DOI 10.1016/0001-8708(75)90091-2, https://doi.org/10.1016/0001-8708(75)90091-2
  3. R. Rouquier, 'Derived equivalences and finite dimensional algebras', Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. II, Eur. Math. Soc., pp. 191–221 (published 2007), DOI 10.4171/022-2/9, https://doi.org/10.4171/022-2/9
  4. R. Martínez-Villa, 'The stable equivalence for algebras of finite representation type', Communications in Algebra 13(5) (1985) 991–1018, DOI 10.1080/00927878508823203, https://doi.org/10.1080/00927878508823203
  5. R. Martínez-Villa, 'Properties that are left invariant under stable equivalence', Communications in Algebra 18(12) (1990) 4141–4169, DOI 10.1080/00927872.1990.12098256, https://doi.org/10.1080/00927872.1990.12098256
  6. M. A. Antipov, A. O. Zvonareva, 'On stably biserial algebras and the Auslander–Reiten conjecture for special biserial algebras', Journal of Mathematical Sciences 240(4) (2019) 375–394, DOI 10.1007/s10958-019-04358-y, https://doi.org/10.1007/s10958-019-04358-y
  7. W. Hu, C. C. Xi, 'Derived equivalences and stable equivalences of Morita type, II', Revista Matemática Iberoamericana 34 (2018) 59–110, DOI 10.4171/rmi/981, https://doi.org/10.4171/rmi/981
  8. V. Gélinas, 'The depth, the delooping level and the finitistic dimension', Advances in Mathematics 394 (2022) 108052, DOI 10.1016/j.aim.2021.108052, https://doi.org/10.1016/j.aim.2021.108052
  9. L. Kershaw, J. Rickard, 'A finite dimensional algebra with infinite delooping level', Annals of Representation Theory 1(1) (2024) 61–65, DOI 10.5802/art.3, https://doi.org/10.5802/art.3
  10. C. M. Ringel, P. Zhang, 'Gorenstein-projective and semi-Gorenstein-projective modules', Algebra & Number Theory 14(1) (2020) 1–36, DOI 10.2140/ant.2020.14.1, https://doi.org/10.2140/ant.2020.14.1
  11. R. Guo, K. Igusa, 'Derived delooping levels and finitistic dimension', Advances in Mathematics 464 (2025) 110152, DOI 10.1016/j.aim.2025.110152, https://doi.org/10.1016/j.aim.2025.110152
  12. Liang Chen, 'Derived equivalences and delooping levels', arXiv:2603.12974 (2026), https://arxiv.org/abs/2603.12974
  13. Xiaogang Li, Changchang Xi, 'Stable equivalences and homological dimensions', arXiv:2511.19051 (2025, v2 of 4 Mar 2026), https://arxiv.org/abs/2511.19051
  14. Xiaogang Li, Changchang Xi, 'Derived equivalences, new matrix equivalences, and homological conjectures', arXiv:2509.26353 (2025, v2 of 12 Nov 2025), https://arxiv.org/abs/2509.26353
  15. Zhenxian Chen, Changchang Xi, 'Singular equivalences and homological conjectures', arXiv:2603.20643 (2026, v2 of 29 Jun 2026), https://arxiv.org/abs/2603.20643