Conjecture 5.18 (Zhang, 2025): every finite weakly simple-minded system over a self-injective algebra is a simple-minded system

In the stable module category of a self-injective algebra, a simple-minded system is an orthogonal family of stable bricks whose extension closure is the whole category, while a weakly simple-minded system only needs to generate in the weaker sense that every nonzero object receives a nonzero morphism from some family member. Every simple-minded system is weakly simple-minded, and over representation-finite self-injective algebras the converse holds (Koenig–Liu, 2012); over representation-infinite algebras the converse fails, but all known counterexamples have infinite cardinality and contain objects in homogeneous tubes (Zhang, 2024). Zhang's 2025 classification of simple-minded systems over 2-domestic Brauer graph algebras proved that, there, every finite weakly simple-minded system is a simple-minded system, and posed Conjecture 5.18: the same implication holds over an arbitrary self-injective algebra. As of mid-2026 the conjecture is confirmed for representation-finite self-injective algebras and for all domestic Brauer graph algebras, and a sufficient condition is known in general, but the conjecture itself remains open for self-injective algebras beyond these classes, and its resolution bears directly on the program linking simple-minded systems to the Auslander–Reiten conjecture on stable equivalences.

Background

Let $k$ be a field and $A$ a finite-dimensional $k$ -algebra. $A$ is called self-injective if every projective $A$ -module is injective (equivalently, the left regular module ${}_A A$ is injective). For a self-injective algebra one forms the stable module category $\underline{A\text{-mod}}$ : objects are the finite-dimensional left $A$ -modules, and morphism spaces are the usual $\mathrm{Hom}$ -groups modulo those homomorphisms that factor through a projective module. This category is triangulated; its suspension (shift) is $\Sigma=\Omega^{-1}$ , the inverse of the syzygy functor $\Omega$ , which sends a module $M$ to the kernel of its projective cover. The Auslander–Reiten translate $\tau=\Omega^2\circ\nu$ (with $\nu$ the Nakayama functor $D\mathrm{Hom}_A(-,A)$ ) organizes the indecomposable modules into the Auslander–Reiten quiver (AR-quiver), whose connected components for tame self-injective algebras are of three kinds: Euclidean components of the form $\mathbb{Z}\tilde{\mathbb{A}}_{p,q}$ , quasi-tubes (tubes $\mathbb{Z}A_\infty/\langle\tau^r\rangle$ of finite rank $r$ ), and homogeneous tubes (tubes of rank $1$ ).

Simple-minded systems were introduced by Koenig and Liu in the stable module category of an artin algebra and extended by Dugas to any Hom-finite Krull–Schmidt triangulated category $\mathcal{T}$ . A family $\mathcal{S}$ of objects of $\mathcal{T}$ is a simple-minded system (SMS) if: (1) (orthogonality) $\mathcal{S}$ is an orthogonal system of stable bricks, i.e. $\mathcal{T}(S,S)\cong k$ for each $S\in\mathcal{S}$ and $\mathcal{T}(S,S')=0=\mathcal{T}(S',S)$ for distinct $S,S'\in\mathcal{S}$ ; and (2) (generation) the extension closure $\mathcal{F}(\mathcal{S})$ , i.e. the smallest subcategory containing $\mathcal{S}$ and closed under extensions, equals $\mathcal{T}$ . Koenig and Liu also introduced the weaker notion: a family $\mathcal{S}$ is a weakly simple-minded system if it is an orthogonal system satisfying only the weak generating condition that for every nonzero object $X$ of $\mathcal{T}$ there is some $S\in\mathcal{S}$ with $\mathcal{T}(S,X)\neq 0$ (equivalently, $\mathcal{S}^{\perp}=0$ ). Every SMS is weakly simple-minded; the converse is the subtle point. Koenig and Liu proved that the two notions coincide over representation-finite self-injective algebras, and Dugas showed that an SMS yields torsion pairs $(^{\perp}\mathcal{X},\mathcal{F}(\mathcal{X}))$ and $(\mathcal{F}(\mathcal{X}),\mathcal{X}^{\perp})$ for any subset $\mathcal{X}$ of an SMS. Chan, Liu and Zhang proved that no member of an SMS over a self-injective algebra lies in a homogeneous tube and that a rank- $n$ quasi-tube contains fewer than $n$ members of an SMS, and Guo, Liu, Ye and Zhang characterized SMS over representation-finite self-injective algebras as Nakayama-stable (i.e. $\nu(\mathcal{S})=\mathcal{S}$ ) orthogonal systems of the correct cardinality.

The main testing ground beyond representation-finite type is the class of Brauer graph algebras. A Brauer graph is a finite connected graph equipped with a positive-integer multiplicity at each vertex and a cyclic ordering of the edges at each vertex; the associated Brauer graph algebra is the basic symmetric algebra determined by these data, and by Roggenkamp's theorem this class coincides with the symmetric special biserial algebras, whose indecomposable modules are explicitly described as string and band modules. Bocian and Skowroński classified the domestic Brauer graph algebras — those of polynomial growth with minimal growth number — into 1-domestic and 2-domestic algebras (there are no $n$ -domestic ones for $n\geq 3$ ): roughly, the Brauer graph is a tree with two vertices of multiplicity $2$ or has a unique cycle of odd length (1-domestic), or has a unique cycle of even length with all multiplicities $1$ (2-domestic).

In this setting Zhang first gave a practical criterion for orthogonal systems over domestic Brauer graph algebras to be SMS, and, crucially, produced the first examples of weakly simple-minded systems that are not simple-minded systems; all of these examples have infinite cardinality and their members lie in homogeneous tubes, so they evade the Chan–Liu–Zhang obstruction only by being infinite. Zhang then constructed all simple-minded systems over any 2-domestic Brauer graph algebra (giving a new proof of the Auslander–Reiten conjecture for these algebras) and proved as Corollary 5.17 that over a 2-domestic Brauer graph algebra every weakly simple-minded system of finite cardinality is an SMS, by showing that a finite weakly simple-minded system must meet a Euclidean component and then applying the torsion-pair generation criterion. Immediately after that corollary he posed the general statement, Conjecture 5.18 of arXiv:2509.24184. The question is tied to the Auslander–Reiten conjecture (AR-conjecture), which asserts that a stable equivalence between self-injective algebras preserves the number of isomorphism classes of non-projective simple modules; since SMS are invariant under stable equivalences and play the role of simples in the stable category, rigidity statements such as the conjecture are a route to cardinality results that would imply the AR-conjecture (Pogorzaly reduced the general AR-conjecture to the self-injective case).

In June 2026 the same author released two further preprints that shape the current status: part II of the construction program extends the classification to 1-domestic Brauer graph algebras by covering theory, yielding the unified statement that over any domestic Brauer graph algebra the SMS are exactly the maximal orthogonal systems meeting every Euclidean component, and as Corollary 3.5 that every finite weakly simple-minded system over a domestic Brauer graph algebra is an SMS; and a separate note proves that over an arbitrary self-injective algebra a finite weakly simple-minded system satisfying two auxiliary hypotheses — the cosyzygy condition $\Sigma(\mathcal{S})\subseteq\mathcal{F}(\mathcal{S})$ and a lifting condition stating that vanishing of $\mathcal{S}^{\perp}$ among finite-dimensional modules implies vanishing among all modules — is an SMS, recovering the Koenig–Liu theorem in the representation-finite case. Neither work claims the general conjecture, and no counterexample to the finite statement is known.

Problem Statement

Conjecture 5.18 of arXiv:2509.24184, stated for a finite-dimensional self-injective algebra $A$ over an arbitrary field $k$ : if $\mathcal{S}$ is a weakly simple-minded system in the stable module category $\underline{A\text{-mod}}$ — an orthogonal system of stable bricks such that every nonzero object $X$ of $\underline{A\text{-mod}}$ admits a nonzero stable morphism from some $S\in\mathcal{S}$ — and $\mathcal{S}$ has finite cardinality, must the extension closure $\mathcal{F}(\mathcal{S})$ be all of $\underline{A\text{-mod}}$ , i.e. must $\mathcal{S}$ be a simple-minded system? The implication is known for representation-finite self-injective algebras (Koenig–Liu) and for all domestic Brauer graph algebras (Zhang, Corollaries 5.17 and 3.5); the problem is to decide it, in the source's full generality and with no restriction on the representation type, for an arbitrary self-injective algebra — or to exhibit a finite weakly simple-minded system that is not a simple-minded system.

References

S. Koenig and Y. Liu, 'Simple-minded systems in stable module categories', Quarterly Journal of Mathematics 63 (2012), 653–674. DOI: 10.1093/qmath/har009, https://doi.org/10.1093/qmath/har009 (arXiv:1009.1427, https://arxiv.org/abs/1009.1427)
A. Dugas, 'Torsion pairs and simple-minded systems in triangulated categories', Applied Categorical Structures 23 (2015), 507–526. DOI: 10.1007/s10485-014-9365-8, https://doi.org/10.1007/s10485-014-9365-8 (preprint arXiv:1207.7338, 2012, https://arxiv.org/abs/1207.7338)
A. Chan, S. Koenig and Y. Liu, 'Simple-minded systems, configurations and mutations for representation-finite self-injective algebras', Journal of Pure and Applied Algebra 219 (2015), 1940–1961. DOI: 10.1016/j.jpaa.2014.07.018, https://doi.org/10.1016/j.jpaa.2014.07.018
A. Chan, Y. Liu and Z. Zhang, 'On simple-minded systems and τ-periodic modules of self-injective algebras', Journal of Algebra 560 (2020), 416–441. DOI: 10.1016/j.jalgebra.2020.05.024, https://doi.org/10.1016/j.jalgebra.2020.05.024
J. Guo, Y. Liu, Y. Ye and Z. Zhang, 'On simple-minded systems over representation-finite self-injective algebras', Algebras and Representation Theory 25 (2022), 983–1002. DOI: 10.1007/s10468-021-10056-8, https://doi.org/10.1007/s10468-021-10056-8 (arXiv:2006.14289, https://arxiv.org/abs/2006.14289)
R. Bocian and A. Skowroński, 'Symmetric special biserial algebras of Euclidean type', Colloquium Mathematicum 96 (2003), 121–148. DOI: 10.4064/cm96-1-11, https://doi.org/10.4064/cm96-1-11
Z. Zhang, 'On simple-minded systems over domestic Brauer graph algebras', European Journal of Mathematics 10 (2024), article 1. DOI: 10.1007/s40879-023-00717-x, https://doi.org/10.1007/s40879-023-00717-x
Z. Zhang, 'A construction of simple-minded systems over domestic Brauer graph algebras I: the 2-domestic case', arXiv:2509.24184 (2025), https://arxiv.org/abs/2509.24184
Z. Zhang, 'A note on simple-minded systems and weakly simple-minded systems over self-injective algebras', arXiv:2606.21881 (2026), https://arxiv.org/abs/2606.21881
Z. Zhang, 'A construction of simple-minded systems over domestic Brauer graph algebras II: the 1-domestic case', arXiv:2606.16085 (2026), https://arxiv.org/abs/2606.16085
Z. Zhang, 'On reduction and gluing technique of simple-minded systems over self-injective algebras', arXiv:2606.16081 (2026; Communications in Algebra, DOI: 10.1080/00927872.2026.2704957), https://arxiv.org/abs/2606.16081