A note on simple-minded systems and weakly simple-minded systems over self-injective algebras

Abstract: Let A be a self-injective algebra over an algebraically closed field. We study the relationship between simple-minded systems and weakly simple-minded systems in A-stmod. We present a necessary and sufficient condition for an orthogonal system to be a simple-minded system over domestic Brauer graph algebras. As a byproduct, we construct a class of simple-minded systems over 2-domestic Brauer graph algebras.

• The paper proves that weakly simple-minded systems become simple-minded when syzygy closure and a small-to-large perpendicularity condition hold, recovering the representation-finite case.
• It characterizes simple-minded systems over domestic Brauer graph algebras as orthogonal systems containing a non-periodic module and satisfying Ω(𝒮) ⊆ 𝔽(𝒮).
• It gives an explicit construction for balanced 2-domestic Brauer graph algebras, producing finite systems whose size equals the number of non-projective simple modules.

Context and motivation

Simple-minded systems, introduced by Koenig and Liu, are families of stable bricks in the stable module category $\underline{A\text{-mod}}$ of an artin algebra $A$ that are mutually orthogonal and whose extension closure $\mathcal{F}(\mathcal{S})$ equals the whole stable category. Dugas extended the notion to arbitrary Hom-finite Krull–Schmidt triangulated categories and showed that subfamilies of simple-minded systems induce torsion pairs. Weakly simple-minded systems relax the generating condition to the requirement that every non-zero object receives a non-zero morphism from some member of the family. Koenig and Liu proved that the two notions coincide over representation-finite self-injective algebras, while a prior example due to Zhang exhibits a weakly simple-minded system that is not simple-minded in the stable module category of a 2-domestic Brauer graph algebra. The paper under review addresses precisely this gap: it identifies conditions under which weakly simple-minded systems become genuine simple-minded systems, and gives a complete characterization of orthogonal systems that are simple-minded systems over domestic Brauer graph algebras.

The key technical observation is that for any simple-minded system $\mathcal{S}$, the syzygy class $\Omega(\mathcal{S})$ lies in $\mathcal{F}(\mathcal{S})$, and in fact $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$. The paper investigates when this necessary condition becomes sufficient.

From weakly simple-minded systems to simple-minded systems

The first main result (Theorem 3.x) states: let $A$ be self-injective and let $\mathcal{S}$ be a finite orthogonal system with $$\Sigma(\mathcal{S}) \subseteq \mathcal{F}(\mathcal{S})$$. If $A$0 is a weakly simple-minded system — i.e., $A$1 in $A$2 — and the "small-to-large" condition holds, namely

$A$3

then $A$4 is a simple-minded system. The proof combines two ingredients: first, the closure $A$5 propagates to coproduct closures via Lemma on coproduct extensions; second, a recent theorem of Canonaco–Neeman–Stellari on t-structures in weakly approximable triangulated categories shows that $A$6 is a t-structure on the big stable category $A$7, where $A$8 denotes closure under small coproducts and extensions. Since the right perpendicular category vanishes by hypothesis, one obtains $A$9, and intersecting back with the finitely presented stable category yields $\mathcal{F}(\mathcal{S})$0.

This recovers the Koenig–Liu coincidence result for representation-finite self-injective algebras as a corollary, since finite type forces all modules to be direct sums of finitely presented indecomposables, so the small-to-large condition holds automatically.

Several auxiliary results sharpen the picture for Nakayama-stable orthogonal systems:

• $\mathcal{F}(\mathcal{S})$1 if and only if $\mathcal{F}(\mathcal{S})$2, equivalently $\mathcal{F}(\mathcal{S})$3 satisfies the two-out-of-three property on triangles.
• Under Nakayama-stability, $\mathcal{F}(\mathcal{S})$4 implies $\mathcal{F}(\mathcal{S})$5, hence $\mathcal{F}(\mathcal{S})$6 is a triangulated subcategory of $\mathcal{F}(\mathcal{S})$7.
• For representation-finite self-injective $\mathcal{F}(\mathcal{S})$8, the characterization of Guo–Liu–Ye–Zhang (orthogonal + Nakayama-stable + cardinality equal to the number of simples) admits a variant in which the cardinality condition is replaced by $\mathcal{F}(\mathcal{S})$9: the proof propagates membership in $\mathcal{S}$0 along irreducible maps using almost split sequences and the connectedness of the finite AR-quiver.

Characterization over domestic Brauer graph algebras

For a domestic Brauer graph algebra $\mathcal{S}$1 (1-domestic or 2-domestic), the stable AR-quiver consists of at most two Euclidean components of type $\mathcal{S}$2, up to four quasi-tubes of ranks $\mathcal{S}$3 and $\mathcal{S}$4, and infinitely many homogeneous tubes of band modules. The second main result characterizes simple-minded systems among orthogonal systems: $\mathcal{S}$5 is a simple-minded system if and only if (1) $\mathcal{S}$6 contains at least one non-periodic module and (2) $\mathcal{S}$7.

Necessity follows from the earlier criterion of Zhang requiring one object per Euclidean component plus containment of all quasi-simples in $\mathcal{S}$8. Sufficiency uses the fact that $\mathcal{S}$9 swaps the two Euclidean components ($\Omega(\mathcal{S})$0): condition (2), together with closure of $\Omega(\mathcal{S})$1 under all powers of $\Omega(\mathcal{S})$2, forces both Euclidean components into $\Omega(\mathcal{S})$3 — one component suffices when $\Omega(\mathcal{S})$4 is 1-domestic, while the non-periodic module in condition (1) guarantees coverage of both components when $\Omega(\mathcal{S})$5 is 2-domestic. A structural lemma then shows that the third term of any triangle induced by an irreducible map between non-projective indecomposables in a Euclidean component is a coray point, which yields all quasi-simples outside homogeneous tubes; Chan–Liu–Zhang's theorem that no member of a simple-minded system lies in a homogeneous tube ensures the simple modules are then contained in $\Omega(\mathcal{S})$6, completing the generation argument.

A notable corollary of this characterization is constructive rather than merely classificatory: it converts the abstract generating condition into two checkable homological conditions, enabling explicit constructions.

Explicit construction over 2-domestic Brauer graph algebras

The final section exploits the characterization to build simple-minded systems over 2-domestic Brauer graph algebras whose Brauer graph has a unique even cycle with equally many additional edges inside and outside the cycle (equivalently, the two Euclidean components have equal rank, $\Omega(\mathcal{S})$7). Starting from an arbitrary non-zero non-periodic indecomposable module $\Omega(\mathcal{S})$8 in $\Omega(\mathcal{S})$9, Serre duality produces a non-zero morphism $\mathcal{F}(\mathcal{S})$0; the triangle extending $\mathcal{F}(\mathcal{S})$1, rotated twice, is induced by the almost split sequence ending at $\mathcal{F}(\mathcal{S})$2 (via Auslander–Reiten–Smalø's criterion applied to the stable brick $\mathcal{F}(\mathcal{S})$3). Its middle term contributes two non-projective summands $\mathcal{F}(\mathcal{S})$4 in $\mathcal{F}(\mathcal{S})$5. Iterating the construction from $\mathcal{F}(\mathcal{S})$6 returns $\mathcal{F}(\mathcal{S})$7 as a summand, and orthogonality of each successive triple follows from stability, generalized standardness of the Euclidean components, and Serre duality. Because $\mathcal{F}(\mathcal{S})$8, the process closes after finitely many steps, producing a finite orthogonal system

$\mathcal{F}(\mathcal{S})$9

whose cardinality equals the number of non-projective simple $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$0-modules, with $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$1 and hence $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$2. By the characterization theorem, $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$3 is a simple-minded system. Two worked examples illustrate the procedure: a six-module system over a 2-domestic Brauer graph algebra on six vertices, and, over a 1-domestic example, the construction degenerates to the set of all three simple modules — itself a simple-minded system, as expected for symmetric special biserial algebras of finite type.

Limitations and open questions

The sufficient condition linking weakly and genuinely simple-minded systems depends on the small-to-large condition relating perpendicular categories in $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$4 and $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$5; the paper verifies this condition only for representation-finite algebras and does not provide criteria for checking it in the representation-infinite case beyond the Brauer graph setting. The explicit construction requires the equal-rank hypothesis $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$6 (equal numbers of additional edges inside and outside the unique even cycle); whether analogous finite constructions exist for 2-domestic Brauer graph algebras with $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$7 is not addressed. More broadly, the paper does not determine whether $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$8 together with orthogonality characterizes simple-minded systems over self-injective algebras outside the domestic Brauer graph class, nor whether the reliance on the Canonaco–Neeman–Stellari t-structure theorem can be replaced by arguments internal to $$\Omega(\mathcal{F}(\mathcal{S})) \subseteq \mathcal{F}(\mathcal{S})$$9.

Conclusion

The paper establishes that the syzygy-closure condition $A$0, long known to be necessary for simple-minded systems, becomes sufficient in two settings: for finite weakly simple-minded systems satisfying a coproduct-level perpendicularity condition, and for orthogonal systems over domestic Brauer graph algebras containing a non-periodic module. The latter yields an explicit, algorithmic construction of simple-minded systems over 2-domestic Brauer graph algebras with balanced Brauer graphs, parameterized by an arbitrary choice of non-periodic starting module. These results narrow the gap between weakly and genuinely simple-minded systems in the representation-infinite case and reduce their recognition to verifiable homological data.