On reduction and gluing technique of simple-minded systems over self-injective algebras

Abstract: Let A be a self-injective algebra over an algebraically closed field. We study reduction of simple-minded systems over stable module category A-stmod. We present a recollement and study gluing technique of simple-minded systems through a subset of a simple-minded system in A-stmod. As a byproduct, we also study the extendible property of simple-minded systems.

• The paper identifies the stable bi-perpendicular category of a Nakayama-stable set of simples with the stable module category of a self-injective corner algebra, enabling reduction and inductive analysis.
• The paper constructs simple-minded systems by gluing systems across recollements under syzygy and functorial-finiteness conditions, while showing such nontrivial recollements cannot occur for symmetric or representation-finite self-injective algebras.
• The paper characterizes extendibility by proving that an orthogonal system extends precisely when its stable bi-perpendicular category contains a simple-minded system, establishing a deletion–union bijection and applications to domestic Brauer graph algebras.

This paper develops reduction and gluing techniques for simple-minded systems in the stable module category $A\text{-}\underline{\mathrm{mod}}$ of a self-injective algebra $A$ over an algebraically closed field. Simple-minded systems, introduced by Koenig–Liu for stable module categories and generalized by Dugas to arbitrary Hom-finite Krull-Schmidt triangulated categories, are families of stable bricks satisfying orthogonality and a generating condition. The paper's contributions are threefold: an identification of the stable bi-perpendicular category of a Nakayama-stable set of simples with the stable module category of a corner algebra; a recollement-based gluing construction of simple-minded systems; and a characterization of when a Nakayama-stable orthogonal system extends to a simple-minded system.

Reduction via Nakayama-stable subsets of simples

The first main result concerns the stable bi-perpendicular category $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$ attached to a Nakayama-stable subset $\mathcal{S}'$ of the set of simple $A$-modules. The author proves that $\mathcal{D}$ carries a triangulated structure — not as a triangulated subcategory of $A\text{-}\underline{\mathrm{mod}}$, but via the new shift and standard distinguished triangles constructed by Coelho Simões–Pauksztello in their reduction theory for negative Calabi-Yau categories — and that $\mathcal{D}$ is triangulated equivalent to the stable module category of the corner algebra $eAe$, where $e$ is the sum of primitive idempotents complementary to those corresponding to $A$0. A key preliminary observation is that Nakayama-stability of the idempotent set forces $A$1 itself to be self-injective, proved by tracking projective covers under the exact restriction functor $A$2. For symmetric algebras, every subset of simples is Nakayama-stable, so the hypothesis disappears. An explicit example over a six-vertex Brauer graph algebra illustrates both a finite case ($A$3) and an infinite one ($A$4), where the objects of $A$5 do not visibly match modules of $A$6 yet become isomorphic after transport of the triangulated structure.

The proof constructs a functor $A$7 by applying $A$8 to non-projective parts, and verifies in three steps that it is triangulated, dense, and fully faithful. Density uses the two torsion pairs $A$9 and $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$0 arising from Dugas' theorem on subsets of simple-minded systems, together with the idempotent embedding functor. Faithfulness relies on a claim specific to objects of $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$1: if $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$2 for $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$3 then $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$4 in the stable category — a statement the author notes fails without the membership condition in $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$5.

As an application, the paper gives a new proof that for a representation-finite symmetric algebra, every simple-minded system has cardinality equal to the number of non-isomorphic non-projective simples. The argument combines Hu–Xi's configuration-theoretic result — that under a stable equivalence of Morita type some Auslander-Reiten translate and syzygy iterate of a simple-minded system object becomes simple — with induction on the number of simples, using the reduction equivalence to peel off one simple at a time. This cardinality statement bears directly on the Auslander–Reiten conjecture: if such cardinalities are fixed for all finite-dimensional algebras, the conjecture follows, so reduction techniques of this kind are a plausible route toward it.

Recollements and gluing

The second main result establishes that, for a Nakayama-stable orthogonal system $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$6 with $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$7 and $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$8 functorially finite, the stable category $\mathcal{D} = {}^{\perp}(\mathcal{S}')^{\perp}$9 admits a recollement of $\mathcal{S}'$0 and $\mathcal{S}'$1. The mechanism is a TTF triple: the hypotheses imply that $\mathcal{S}'$2 is a triangulated subcategory (via the symmetry $\mathcal{S}'$3 for Nakayama-stable systems), hence $\mathcal{S}'$4 and $\mathcal{S}'$5 are triangulated as well, and the two torsion pairs become stable t-structures, which by Mitchell-type results yield the recollement. Under these hypotheses the torsion pairs are moreover splitting, and $\mathcal{S}'$6.

Gluing then proceeds as expected: given simple-minded systems $\mathcal{S}'$7 of $\mathcal{S}'$8 and $\mathcal{S}'$9 of $A$0, the union $A$1 is a simple-minded system of $A$2. Orthogonality across the two pieces follows from the splitting identity, and generation from the torsion triangle decomposition. The same conclusion holds for any recollement of $A$3 whose left term carries a Nakayama-stable simple-minded system.

Two structural obstructions deserve emphasis. First, for representation-finite self-injective algebras there are no non-trivial recollements of this form: any Nakayama-stable orthogonal system with $A$4 is already a simple-minded system, shown via Riedtmann's classification of stable AR-quivers of the form $A$5 and an analysis of almost split sequences forcing $A$6 to contain all $A$7-orbits. Second, since the stable module category of a symmetric algebra is $A$8-Calabi-Yau, it admits no non-trivial t-structures by Zhou–Zhu, and hence no non-trivial recollements at all by the correspondence of Nicolás–Saorín. Consequently, the recollement theorem has content only for representation-infinite self-injective algebras that are not symmetric — notably weakly symmetric or general self-injective algebras with nontrivial Nakayama permutation.

Extendibility

The third main result characterizes extendibility: a Nakayama-stable orthogonal system $A$9 with $\mathcal{D}$0 and $\mathcal{D}$1 functorially finite extends to a simple-minded system of $\mathcal{D}$2 if and only if the stable bi-perpendicular category $\mathcal{D}$3 contains a simple-minded system. Moreover, extension yields a bijection between simple-minded systems containing $\mathcal{D}$4 and simple-minded systems of $\mathcal{D}$5, given by deletion and union respectively. The proof of surjectivity of the inclusion $\mathcal{D}$6 is an induction on the filtration length $\mathcal{D}$7, using long exact sequences to show that triangles with two vertices in $\mathcal{D}$8 have their third vertex there as well. Unlike the Coelho Simões–Pauksztello reduction, here $\mathcal{D}$9 is a genuine triangulated subcategory of $A\text{-}\underline{\mathrm{mod}}$0, so the correspondence lives inside the ambient category. As corollaries, the same bijection holds for arbitrary orthogonal systems over symmetric algebras (where Nakayama-stability is automatic).

Combining these results gives a clean criterion: if no non-trivial recollement as above exists, then any Nakayama-stable orthogonal system satisfying the syzygy and finiteness conditions is already a simple-minded system. Applied to domestic Brauer graph algebras, this yields a second characterization complementing the earlier criterion involving Euclidean components: an orthogonal system $A\text{-}\underline{\mathrm{mod}}$1 with $A\text{-}\underline{\mathrm{mod}}$2 and $A\text{-}\underline{\mathrm{mod}}$3 functorially finite is automatically a simple-minded system, since either $A\text{-}\underline{\mathrm{mod}}$4 contains a non-periodic module (whose $A\text{-}\underline{\mathrm{mod}}$5-orbit generates everything) or the splitting torsion pair forces a contradiction.

Limitations and open questions

Several restrictions bound the applicability of these results. The reduction theorem requires Nakayama-stability of the chosen subset of simples; without it, the complementary corner algebra $A\text{-}\underline{\mathrm{mod}}$6 need not be self-injective, and the comparison with its stable module category breaks down. The recollement and gluing theorems require $A\text{-}\underline{\mathrm{mod}}$7 plus functorial finiteness of $A\text{-}\underline{\mathrm{mod}}$8 — conditions that, as shown, cannot hold non-trivially for symmetric or representation-finite self-injective algebras, leaving the representation-infinite non-symmetric case as the essential setting. The author explicitly notes that providing a concrete example of a non-trivial recollement satisfying the hypotheses of the gluing theorem remains open, as does extending the extendibility analysis beyond the finite-type situation already treated in prior work. It is also worth recording that not every triangulated category admits a simple-minded system, so the "if and only if" in the extendibility theorem does not guarantee existence unconditionally.

Conclusion

The paper integrates the negative Calabi-Yau reduction machinery of Coelho Simões–Pauksztello into the module-theoretic setting of self-injective algebras, identifying the reduced category concretely as a stable module category of a corner algebra, and supplies recollement-based gluing and extendibility criteria for simple-minded systems. The new proofs of the finite-type cardinality result and of extendibility reframe known statements within a uniform torsion-pair framework, and the criteria for orthogonal systems over domestic Brauer graph algebras sharpen existing characterizations. The principal unresolved issue raised by the work is whether the recollement-gluing theorem can be instantiated by explicit non-trivial examples among representation-infinite, non-symmetric self-injective algebras.