Stable-equivalence classification, scalar problems, and the Auslander–Reiten conjecture for self-injective algebras of polynomial growth

The source paper (Zhou–Zimmermann, Beiträge Algebra Geom. 53 (2012) 349–364; arXiv:1005.1150) proved that for indecomposable weakly symmetric algebras of polynomial growth over an algebraically closed field, the derived equivalence classification coincides with the classification up to stable equivalence of Morita type, up to unresolved 'scalar problems' in the one-parameter families $\Lambda'_3(\lambda)$ and $A_1(\lambda)$ (and $\Lambda_3(\lambda)$ in characteristic 2), and deduced the Auslander–Reiten conjecture for stable equivalences of Morita type between such algebras. The paper explicitly leaves open three linked questions: (1) whether different parameter values in those families give derived-equivalent (hence stably equivalent of Morita type) algebras; (2) the stable-equivalence classification of general self-injective algebras of polynomial growth — beyond Morita-type equivalences and beyond weakly symmetric algebras — and the validity of the Auslander–Reiten conjecture for arbitrary stable equivalences in this class; (3) whether two indecomposable tame symmetric algebras are stably equivalent of Morita type if and only if they are derived equivalent. All three items remain open; no counterexample to (3) is known, and every invariant computed so far is constant in the scalar parameters.

Background

Let $K$ be an algebraically closed field and $A$ a finite-dimensional $K$-algebra; write $\mathrm{mod}\,A$ for its category of finite-dimensional right modules. $A$ is self-injective if projective and injective modules coincide, symmetric if $A \cong \mathrm{Hom}_K(A,K)$ as $A$-$A$-bimodules, and weakly symmetric if the projective cover of each simple module has socle isomorphic to its top; symmetric implies weakly symmetric implies self-injective. By Drozd's tame–wild trichotomy every finite-dimensional algebra is of finite, tame, or wild representation type; within the tame ones, $A$ is of polynomial growth if the number of one-parameter families of $d$-dimensional indecomposable modules is bounded by a polynomial in $d$, and domestic if it is bounded linearly. Self-injective algebras of polynomial growth form one of the central tractable classes in the representation theory of associative algebras: it contains all tame blocks (indecomposable two-sided direct summands) of modular group algebras and the algebras of dihedral, semidihedral and quaternion type — the tame self-injective classes covering the blocks with those defect groups. Two equivalence relations on algebras are compared throughout. A derived equivalence is a triangle equivalence $D^b(\mathrm{mod}\,A) \simeq D^b(\mathrm{mod}\,B)$; by Rickard's theorem it is controlled by tilting complexes (bounded complexes of projective modules that generate the homotopy category and whose endomorphism complexes have homology concentrated in one degree). The stable module category $\underline{\mathrm{mod}}\,A$ is the quotient of $\mathrm{mod}\,A$ by the ideal of morphisms factoring through projectives; a stable equivalence is an additive equivalence $\underline{\mathrm{mod}}\,A \simeq \underline{\mathrm{mod}}\,B$. A stronger notion due to Broué's school: a pair of bimodules $(M,N)$ defines a stable equivalence of Morita type if $M$ and $N$ are projective as one-sided modules on both sides and $M \otimes_B N \cong A \oplus P$ and $N \otimes_A M \cong B \oplus Q$ for projective bimodules $P$ and $Q$. A theorem of Rickard (1989) says that derived-equivalent self-injective algebras are stably equivalent of Morita type; however, not every stable equivalence is of Morita type, and the relationship between the two classification problems is delicate. In 1975 Auslander and Reiten conjectured that stably equivalent Artin algebras have the same number of isomorphism classes of non-projective simple modules — for self-injective algebras, the same number of simple modules. This Auslander–Reiten conjecture (ARC) remains open in general; it is known for algebras stably equivalent to self-injective special biserial algebras (special biserial algebras are bound-quiver algebras with at most two arrows starting and ending at each vertex and at most one arrow continuing after each arrow — the standard quiver-based tame class; Antipov–Zvonareva) and for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth (the source paper), but for little else. The classification input is a programme of Bocian–Holm–Skowroński, Białkowski–Holm–Skowroński and Holm–Skowroński: the indecomposable weakly symmetric algebras of domestic type, and the standard and non-standard weakly symmetric non-domestic algebras of polynomial growth, are classified up to derived equivalence by explicit finite lists of algebras given by quivers and relations, some members of which depend on a parameter $\lambda \in K \setminus \{0,1\}$ (the families $\Lambda'_3(\lambda)$, $A_1(\lambda)$, and $\Lambda_3(\lambda)$ in characteristic 2). Zhou and Zimmermann (2012) proved that for these classes the derived equivalence classification coincides with the classification up to stable equivalence of Morita type, modulo the 'scalar problems' — it is not known whether two distinct parameter values of one family give derived-equivalent algebras — and deduced ARC for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth, as well as ARC for arbitrary stable equivalences between weakly symmetric algebras of domestic type.

Problem Statement

Over an algebraically closed field $K$, resolve the questions explicitly left open by Zhou–Zimmermann (2012), in their original generality. (1) Scalar problems: for the derived-equivalence normal forms of indecomposable weakly symmetric non-domestic algebras of polynomial growth — the standard families $\Lambda'_3(\lambda)$ and $A_1(\lambda)$, $\lambda \in K \setminus \{0,1\}$, and, for the non-standard algebras in characteristic 2, the family $\Lambda_3(\lambda)$ — determine, for $\lambda \ne \lambda'$ in $K \setminus \{0,1\}$, whether $\Lambda(\lambda)$ and $\Lambda(\lambda')$ are derived equivalent (equivalently for the stable classification: stably equivalent of Morita type), thereby deciding whether the derived equivalence classes, and the stable-equivalence-of-Morita-type classes, of these families are genuinely parametrized by $\lambda$. (2) Beyond Morita type and beyond weakly symmetric algebras: classify the indecomposable self-injective algebras of polynomial growth up to stable equivalence in general — the source paper identifies as the crucial first step the classification of the algebras stably equivalent to $\Lambda'_9$ (in characteristic $\ne 2$) or to $\Lambda_{10}$, the self-injective polynomial-growth algebras that are not symmetric, and as the second step the classification of weakly symmetric algebras of polynomial growth up to stable equivalence — and prove or disprove the Auslander–Reiten conjecture for arbitrary stable equivalences between self-injective algebras of polynomial growth. (3) Dichotomy: decide whether two indecomposable tame symmetric algebras are stably equivalent of Morita type if and only if they are derived equivalent; the 'if' direction is Rickard's theorem, so the question is whether every stable equivalence of Morita type between indecomposable tame symmetric algebras comes from a derived equivalence. Items (1), (2) and (3) are independently standing open questions, each posed in its original generality; a resolution of any one of them is a partial result, and the full objective is to resolve all three, namely to determine the relationship between stable equivalence, stable equivalence of Morita type, and derived equivalence for the tame self-injective world and to settle the Auslander–Reiten conjecture there; within (2), both the stable-equivalence classification and the Auslander–Reiten statement are required.

References

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