The Auslander–Reiten conjecture for stable equivalences: preservation of non-projective simple modules and the gluing-of-idempotents characterization

When two Artin algebras have equivalent stable module categories (categories of modules modulo morphisms factoring through projectives), must they have the same number of isomorphism classes of non-projective simple modules? This is the Auslander–Reiten conjecture (ARC) for stable equivalences, a foundational open problem in the representation theory of algebras. The source paper (Koenig–Liu, J. Algebra 319, 2008) studies this question through radical embeddings $B\subset A$ obtained by gluing two idempotents of $A$ , and characterizes — but only modulo ARC — when such an embedding induces a stable equivalence modulo projectives: precisely when a sink idempotent (simple projective) and a source idempotent (simple injective) are glued, equivalently when the pair is stably equivalent modulo semisimples. Whether this characterization, and the equivalence of all its characterizing conditions, holds unconditionally remains open: a single radical embedding that is stably equivalent modulo projectives but not modulo semisimples would refute ARC. As of 2026 ARC is confirmed only for special classes (hereditary, representation-finite, special biserial, Frobenius-finite, principal centralizer matrix algebras, group algebras of $p$ -groups), and no counterexample of any kind is known; most recently Chen–Li–Liu–Xing (arXiv:2608.14253, 2026) closed the class of Brauer graph algebras up to semisimple summands under stable equivalence of Morita type.

Background

An Artin algebra is an algebra that is finitely generated as a module over a commutative artinian center; the central example is a finite-dimensional algebra $A=kQ/I$ given by a quiver (directed graph) $Q$ with relations $I$ . Modules are always finitely generated right modules. A module is simple if it has no nontrivial submodules and projective if it is a direct summand of a free module; over $kQ/I$ the indecomposable projective $P_v=e_vA$ at a vertex $v$ is simple exactly when no arrow starts at $v$ (a sink), and the corresponding injective (a module into which the simple embeds) is simple exactly when no arrow ends at $v$ (a source).

The stable module category $\underline{\mathrm{mod}}\,A$ has the same objects as the module category but its morphism sets are quotients of the usual ones by the maps that factor through a projective module. Two Artin algebras are stably equivalent when their stable categories are equivalent as $k$ -categories. Unlike module categories (Morita theory) or derived categories (Rickard's theory), no analogue of Morita theory is known for stable categories — it is not even clear what should play the role of generators — and this is regarded as a main obstruction to the following conjecture of Auslander and Reiten (recorded as Conjecture (5), p. 409, of the 1995 Cambridge book by Auslander, Reiten and Smalø): stably equivalent Artin algebras have the same number of isomorphism classes of non-projective simple modules (ARC). The conjecture is verified in few cases: for algebras stably equivalent to hereditary algebras (Auslander–Reiten 1975), for representation-finite algebras over an algebraically closed field (Martínez-Villa 1985), for group algebras of finite $p$ -groups (Linckelmann 1996), for self-injective special biserial algebras (Pogorzaly 1994, with a complete proof by Antipov–Zvonareva 2017/2019), for Frobenius-finite algebras over an algebraically closed field and for principal centralizer matrix algebras (Xi–Zhang 2022/2026). Martínez-Villa (1990) reduced the conjecture to self-injective algebras without nodes (a node is a certain indecomposable projective-injective non-simple module singled out by him as the obstruction to lifting stable equivalences to module categories). Equivalent reformulations exist: under a stable equivalence of Morita type (a stable equivalence induced by tensoring with bimodules that are projective on both sides), ARC is equivalent to invariance of the $p$ -rank of the Cartan matrix (the matrix recording the multiplicities of simples in projectives) and of degree-zero stable Hochschild homology (Liu–Zhou–Zimmermann 2012); and ARC follows if every simple-minded system of $A$ — a set of mutually orthogonal stable bricks that generates the stable category and whose image in the stable category behaves like the simples — has the same cardinality as the set of non-projective simple modules (Koenig–Liu 2012, Question 6.1).

The source paper approaches ARC through an explicit construction. Gluing two idempotents $e_1,e_2$ of $A=kQ/I$ means forming the subalgebra $B$ generated by $f=e_1+e_2$ and all other idempotents and arrows; the quiver of $B$ is that of $A$ with the two vertices identified. The inclusion $B\subset A$ is a radical embedding, i.e. an algebra monomorphism sending the Jacobson radical of $B$ onto that of $A$ . Koenig and Liu study the case where $A$ has a simple projective and a simple injective module and $B$ is obtained by gluing the corresponding idempotents, extending constructions of Martínez-Villa. They prove that gluing a sink (simple projective) and a source (simple injective) yields a stable equivalence modulo projectives, realized by tensoring with an explicitly constructed bimodule that is projective on one side only — so the equivalence is not of Morita type and not induced by an adjoint pair. They also work with a second stable category, modulo semisimple modules (morphisms factoring through semisimple modules are killed), for which the stable equivalences arising from gluings all come from induction and restriction, hence from an adjoint pair. In this framework they characterize when a radical embedding induces a stable equivalence of each kind; however, the strongest equivalences between the two notions — including the converse direction that a radical embedding inducing a stable equivalence modulo projectives must come from gluing a sink with a source, and hence the equivalence of all the characterizing conditions in the paper's summary theorem — are proved only under ARC. The later literature (Liu–Rubio y Degrassi–Wen 2023/2024) still quotes this characterization in exactly that conditional form: the embedding induces a stable equivalence modulo projectives if and only if a sink and a source were glued, 'at least when we modulo the Auslander–Reiten conjecture for stable equivalences'.

Problem Statement

The problem is the Auslander–Reiten conjecture for stable equivalences, together with its concrete formulation through gluing of idempotents from the source paper.

(1) General form. Let $A$ and $B$ be Artin algebras whose stable module categories (modulo morphisms factoring through projective modules) are equivalent. Must $A$ and $B$ have the same number of isomorphism classes of non-projective simple modules? A positive answer proves the conjecture; an explicit pair of stably equivalent algebras with different numbers of non-projective simple modules refutes it.

(2) Derived form from the source paper (Koenig–Liu 2008). Let $A$ be a finite-dimensional algebra and $B\subset A$ a radical embedding obtained by gluing two idempotents of $A$ . Without assuming the Auslander–Reiten conjecture: is it true that $A$ and $B$ are stably equivalent modulo projectives only when the glued idempotents correspond to a sink with a source (equivalently, only when $A$ and $B$ are stably equivalent modulo semisimples) — that is, are all implications among the characterizing conditions of the paper's summary theorem unconditional? Equivalently: does there exist a radical embedding $B\subset A$ such that $A$ and $B$ are stably equivalent modulo projectives but not stably equivalent modulo semisimples? The existence of such an example would produce a counterexample to the general conjecture in (1).

The general conjecture (1) is open: it is refuted by an explicit counterexample of any stable-equivalence type. Note that the implication established in the source runs from ARC to the unconditional gluing characterization, not conversely; a positive answer to form (2) alone therefore does not by itself settle form (1).

References

S. Koenig and Y. Liu, 'Gluing of idempotents, radical embeddings and two classes of stable equivalences', Journal of Algebra 319 (2008) 5144–5164. DOI: 10.1016/j.jalgebra.2007.12.012, https://doi.org/10.1016/j.jalgebra.2007.12.012
M. Auslander, I. Reiten and S. O. Smalø, 'Representation Theory of Artin Algebras', Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. DOI: 10.1017/CBO9780511623608, https://doi.org/10.1017/CBO9780511623608
M. Auslander and I. Reiten, 'Stable equivalence of dualizing R-varieties. V: Artin algebras stably equivalent to hereditary algebras', Advances in Mathematics 17 (1975) 167–195. DOI: 10.1016/0001-8708(75)90091-2, https://doi.org/10.1016/0001-8708(75)90091-2
R. Martínez-Villa, 'The stable equivalence for algebras of finite representation type', Communications in Algebra 13 (1985) 991–1018. DOI: 10.1080/00927878508823203, https://doi.org/10.1080/00927878508823203
R. Martínez-Villa, 'Properties that are left invariant under stable equivalence', Communications in Algebra 18 (1990) 4141–4169. DOI: 10.1080/00927872.1990.12098256, https://doi.org/10.1080/00927872.1990.12098256
M. Linckelmann, 'Stable equivalences of Morita type for self-injective algebras and p-groups', Mathematische Zeitschrift 223 (1996) 87–100. DOI: 10.1007/BF02621590, https://doi.org/10.1007/BF02621590
Z. Pogorzaly, 'Algebras stably equivalent to selfinjective special biserial algebras', Communications in Algebra 22 (1994) 1127–1160. DOI: 10.1080/00927879408824898, https://doi.org/10.1080/00927879408824898
S. Koenig and Y. Liu, 'Simple-minded systems in stable module categories', The Quarterly Journal of Mathematics 63 (2012) 653–674. DOI: 10.1093/qmath/har009, arXiv:1009.1427, https://doi.org/10.1093/qmath/har009
Y. M. Liu, G. Zhou and A. Zimmermann, 'Higman ideal, stable Hochschild homology and Auslander–Reiten conjecture', Mathematische Zeitschrift 270 (2012) 759–781. DOI: 10.1007/s00209-010-0825-z, https://doi.org/10.1007/s00209-010-0825-z
Y. Liu, G. Zhou and A. Zimmermann, 'Two questions on stable equivalences of Morita type', arXiv:1409.1821 (2014), https://arxiv.org/abs/1409.1821
Y. Liu, L. Rubio y Degrassi and C. Wen, 'The Hochschild cohomology groups under gluing idempotents', arXiv:2211.05435 (2023), https://arxiv.org/abs/2211.05435
Y. Liu, L. Rubio y Degrassi and C. Wen, 'The Hochschild cohomology groups under gluing arrows', Communications in Algebra 52 (2024) 3871–3897. DOI: 10.1080/00927872.2024.2333472, arXiv:2307.08057, https://doi.org/10.1080/00927872.2024.2333472
M. Antipov and A. Zvonareva, 'On stably biserial algebras and the Auslander–Reiten conjecture for special biserial algebras', Journal of Mathematical Sciences 240 (2019) 375–394. DOI: 10.1007/s10958-019-04358-y, arXiv:1711.05021, https://doi.org/10.1007/s10958-019-04358-y
C. C. Xi and J. Zhang, 'New invariants of stable equivalences and Auslander–Reiten conjecture', Mathematische Annalen 394 (2026). DOI: 10.1007/s00208-026-03422-8, arXiv:2207.10848, https://doi.org/10.1007/s00208-026-03422-8
P. Chen, N. Li, Y. Liu and B. Xing, 'Brauer graph algebras are closed under stable equivalence of Morita type', arXiv:2608.14253 (2026), https://arxiv.org/abs/2608.14253
H. Jonsson, 'J-equivalence for associative algebras', Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 66 (2024/2025) 539–574. DOI: 10.1007/s13366-024-00757-8, https://doi.org/10.1007/s13366-024-00757-8