Cardinality of simple-minded systems in stable module categories (Koenig–Liu Question 6.1): does every simple-minded system over an artin algebra have exactly as many objects as the non-projective simples?

A simple-minded system (s.m.s.) in the stable module category of an artin algebra $A$ is a finite set of pairwise Hom-orthogonal stable bricks from which every module without projective summands can be rebuilt by iterated extensions; the non-isomorphic non-projective simple modules themselves form one such system, and simple-minded systems are carried to simple-minded systems by stable equivalences. Koenig and Liu asked (Question 6.1, Q. J. Math. 63 (2012)) whether every s.m.s. over $A$ has cardinality exactly equal to the number of non-isomorphic non-projective simple $A$ -modules. Because s.m.s. are invariants of the stable module category, a positive answer would imply the Auslander–Reiten conjecture on stable equivalences — that stably equivalent artin algebras have the same number of non-isomorphic non-projective simple modules — in full generality. The question is open: it is proven for triangular algebras and Nakayama algebras (Koenig–Liu), representation-finite self-injective algebras (Chan–Koenig–Liu, Guo–Liu–Ye–Zhang), and domestic Brauer graph algebras (Zhang, 2023–2026, via explicit construction of all s.m.s.), with general structural constraints (tube-intersection bounds) available for all self-injective algebras; no counterexample and no general proof is known.

Background

For a finite-dimensional algebra over a field — more generally, an artin algebra, i.e. an algebra that is finitely generated as a module over a commutative artin ring — the category $\mathrm{mod}\,A$ of finitely generated modules is abelian, and its atomic building blocks are the simple modules. The projective modules are the direct summands of $A$ as a left module; equivalently, every surjection onto a projective module splits. Two algebras are Morita equivalent when their module categories are equivalent. A much older, coarser notion is a stable equivalence: an additive equivalence $\underline{\mathrm{mod}}\,A \simeq \underline{\mathrm{mod}}\,B$ between the stable module categories, where $\underline{\mathrm{mod}}\,A$ has the same objects as $\mathrm{mod}\,A$ but morphism spaces quotiented by the maps that factor through a projective module. Stable equivalences arise naturally, for instance between certain blocks of finite group algebras, but unlike Morita equivalences they need not preserve simple modules, and no general Morita-type theory for them is known.

Koenig and Liu proposed candidates for the simple objects of $\underline{\mathrm{mod}}\,A$ . A simple-minded system (s.m.s.) is a set $\mathcal{S}$ of objects of $\underline{\mathrm{mod}}\,A$ satisfying two conditions: (i) orthogonality — $\underline{\mathrm{Hom}}_A(X,Y)=0$ for distinct $X,Y \in \mathcal{S}$ and each stable endomorphism ring $\underline{\mathrm{End}}_A(X)$ is a division ring (stable bricks); and (ii) a generating condition — every object of $\mathrm{mod}\,A$ without nonzero projective direct summands lies in $\langle \mathcal{S} \rangle$ , the class built from $\mathcal{S}$ by finitely many iterated extensions computed in $\mathrm{mod}\,A$ while allowing projective summands. A complete set of non-isomorphic non-projective simple modules is an s.m.s.; every s.m.s. is finite; and the key invariance theorem of the paper (Theorem 3.2) says that any stable equivalence sends s.m.s. to s.m.s. The definition axiomatizes a 1990s precursor, Pogorzaly's maximal systems of orthogonal stable bricks, with which the conjecture discussed below was proved for self-injective special biserial algebras (an algebra is self-injective if projective and injective modules coincide; special biserial algebras are those describable by strings and bands on a quiver with relations).

A name-collision warning is essential: the Auslander–Reiten conjecture meant here is not the better-known homological conjecture on Ext-vanishing and projectivity. It is the conjecture, open since the 1970s, that any two stably equivalent artin algebras have the same number of isomorphism classes of non-projective simple modules. Since the non-projective simples form an s.m.s. and s.m.s. are stable invariants, a positive answer to the cardinality question below would settle this conjecture for all artin algebras — this is precisely the implication stated by Koenig and Liu after their Question 6.1, and it is the reason the question is studied.

Koenig and Liu confirmed the positive answer for triangular algebras (algebras given by a quiver whose underlying graph has no oriented cycles, i.e. finite global dimension) and for Nakayama algebras (those whose indecomposable modules are all uniserial, i.e. each has a unique composition series). Dugas subsequently extended the notion of s.m.s. to arbitrary Hom-finite Krull–Schmidt triangulated categories — recall $\underline{\mathrm{mod}}\,A$ is triangulated when $A$ is self-injective, with shift given by the inverse syzygy $\Omega^{-1}$ , where $\Omega X$ is the kernel of a projective cover of $X$ — and connected s.m.s. to torsion pairs; a large adjacent literature studies s.m.s. in negative Calabi–Yau and cluster-theoretic settings, but the cardinality question over general artin algebras has been advanced only through the class-by-class program summarized in the research objective above.

Problem Statement

For an artin algebra $A$ , is the cardinality $|\mathcal{S}|$ of every simple-minded system $\mathcal{S}$ in the stable module category $\underline{\mathrm{mod}}\,A$ equal to the number of isomorphism classes of non-projective simple $A$ -modules? This is Question 6.1 of Koenig–Liu (Q. J. Math. 63 (2012)), stated for arbitrary artin algebras with no restriction on representation type or geometry; a positive answer would imply the Auslander–Reiten conjecture that any two stably equivalent artin algebras have the same number of isomorphism classes of non-projective simple modules. A negative resolution is witnessed by an artin algebra together with a simple-minded system whose cardinality differs from the number of non-isomorphic non-projective simples.

References

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