A characterization of representation-finite algebras

Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ (A) the Auslander–Reiten quiver of A. We show that A is representation-finite if and only if Γ (A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander–Reiten translation. Let K denote a fixed algebraically closed field and A a finite-dimensional K-algebra (associative, with an identity) which we shall assume to be basic and connected. By an A-module is meant a finite-dimensional right A-module. Throughout the paper we shall freely use the terminology and notation introduced in [7]. In particular, we denote by modA the category of A-modules, by rad(modA) the Jacobson radical of modA, and by rad∞(modA) the intersection of all powers rad(modA), i ≥ 0, of rad(modA). From the existence of the Auslander-Reiten sequences in modA we know that rad(modA) is generated by irreducible maps as a left and as a right ideal. By Γ (A) we denote the Auslander–Reiten quiver of A whose vertices are the isoclasses of indecomposable objects in modA and arrows correspond to irreducible maps, and by τ and τ−1 the Auslander–Reiten translations DTr and TrD, respectively. For the sake of simplicity we identify an A-module with its isomorphism class. The τ -orbit of an indecomposable A-module X is the family of non-zero modules of the form τX, n ∈ Z, where Z is the set of all integers. An A-module X is called periodic if τX ' X for some n 6= 0. By a path from M to N in Γ (A) we mean a sequence of vertices and arrows M → M1 → . . . → Mn → N in Γ (A). In this case, M is called a predecessor of N and N a successor of M . An oriented cycle is a non-trivial path from a point to itself. Recall that an algebra A is called representation-finite if Γ (A) is finite. In [4] the following results were proved: (a) An algebra A is representation-finite if and only if rad∞(modA) = 0. 32 A. Skowroński and M. Wenderlich (b) If rad∞(modA) is nilpotent, then A is tame (in the sense of [1] and [8]). (c) If A is either a tilted algebra or a standard selfinjective algebra then rad∞(modA) is nilpotent if and only if A is domestic (in the sense of [8]). Representation-finite algebras are domestic and domestic algebras are tame (see [8]). We shall prove here the following characterization of representation-finite algebras. Theorem. Let A be an algebra. The following conditions are equivalent. (i) A is representation-finite. (ii) Γ (A) admits at most finitely many vertices lying on an oriented cycle and the number of τ -orbits in Γ (A) is finite. (iii) rad∞(modA) is nilpotent and the number of τ -orbits in Γ (A) is finite. P r o o f. Obviously (i) implies (ii). Moreover, (iii) implies (i). Indeed, suppose that A is representation-infinite and satisfies (iii). Then the nilpotency of rad∞(modA) implies that A is tame and then, by the validity of the Brauer–Thrall II conjecture (for a proof, see for example [2]) and [1, 6.7], there are infinitely many pairwise non-isomorphic indecomposable A-modules X with X ' τX, impossible by the second part of (iii). Therefore, in order to prove the theorem, it is enough to show that (ii) implies the nilpotency of rad∞(modA). Observe first that, if M0 → M1 → . . . → Mn → M0 is an oriented cycle in Γ (A), then either all modules Mi are periodic or τMi = 0 (resp. τMi = 0) for some m > 0 and some i. Indeed, if one of the modules Mi, say M0, is not periodic and τMi 6= 0 for all m ≥ 0 (resp. m ≤ 0) and all i, 0 ≤ i ≤ n, then the modules τM0, m ≥ 0 (resp. m ≤ 0) are pairwise non-isomorphic and lie on oriented cycles τM0 → τM1 → . . .→ τMn → τM0, a contradiction to (ii). Denote by Γ(A) (resp. Γ−(A)) the full translation subquiver of Γ (A) formed by all non-periodic indecomposable modules X such that τX 6= 0 (resp. τ−nX 6= 0) for all n ≥ 0. By the above remark, Γ(A) and Γ−(A) are quivers without oriented cycles. Then there exist finite sets X and Y of indecomposable A-modules such that the following conditions are satisfied: (1) X (resp. Y) intersects every τ -orbit in Γ(A) (resp. Γ−(A)). (2) Every path in Γ (A) with source and target in X (resp. Y) has all vertices in X (resp. Y). (3) There is no oriented cycle in Γ (A) consisting of modules from X (resp. Y). Representation-finite algebras 33 (4) Every predecessor (resp. successor) of some module of X (resp. Y) belongs to Γ(A) (resp. Γ−(A)). Denote by C (resp. C−) the full translation subquiver of Γ (A) formed by all proper predecessors (resp. proper successors) of modules of X (resp. Y). We may assume that C and C− are disjoint. Observe that C (resp. C−) is a disjoint union of translation quivers of the form (−N)∆ (resp. N∆) for some quiver ∆ without oriented cycles. Let D be the family of all indecomposable A-modules which are neither in C nor in C−. Then D is (up to isomorphism) finite; denote by d the maximum of dimensions of modules from D. Now let M and N be two indecomposable A-modules and f : M → N a non-zero map in rad∞(modA). Assume that M does not belong to C−. We claim that f factors through a direct sum of modules of Y. Since f ∈ rad∞(M,N), there exists, for each t > 0, a chain M g1 →M1 g2 →M2 → . . .→Mt−1 gt →Mt of irreducible maps in mod A and a morphism ht ∈ rad(Mt, N) such that f = htgt . . . g1. Then there exists p ≥ 0 such that, for t ≥ p, Mt does not contain direct summands from C. Applying the lemma of Harada and Sai [3] (for a proof we refer to [6, 2.2]) we conclude that, for t ≥ p+2, Mt is a direct sum of modules of C−. Observe that, for s = p+2, gs . . . g1 is a linear combination of paths in Γ (A) from M to indecomposable direct summands ofMs, hence lying in C−, which must factor through modules of Y. Similarly, we show that, if N is not in C, then f factors through a direct sum of modules of X . Let nowm = 2+1. We shall show that (rad∞(modA))2m = 0. It is enough to show that for each chain of morphisms Z0 u1 →Z1 u2 →Z2 → . . .→ Z2m−1 u2m → Z2m , where all Zi are indecomposable A-modules and where the ui belong to rad∞(modA), the composition u = u2m . . . u1 is zero. Since we assume that C and C− are disjoint, it follows that either ui factors through a direct sum of modules of X or ui+1 factors through a direct sum of modules of Y. Consequently, for each j, 1 ≤ j ≤ m, we have u2ju2j−1 = βjαj , where αj ∈ rad(Z2j−2, Vj), βj ∈ rad(Vj , Z2j) and Vj is a direct sum of indecomposable modules of X ∪ Y. Let γj = αj+1βj for j = 1, . . . ,m − 1. Applying again the lemma of Harada and Sai, we conclude that γm−1 . . . γ1 = 0. Hence u = βmγm−1 . . . γ1α1 = 0, which finishes the proof of the theorem. The following corollary is an immediate consequence of the theorem. Corollary 1. Let A be an algebra such that Γ (A) has no oriented cycles. Then A is representation-finite if and only if the number of τ -orbits in Γ (A) is finite. 34 A. Skowroński and M. Wenderlich Observe that wild hereditary algebras are representation-infinite and their Auslander–Reiten quiver does not contain oriented cycles (see [5]). Recall also that an algebra A is called representation-directed provided every indecomposable A-module is directing, that is, it does not belong to an oriented cycle M0 →M1 → . . .→Mn →M0 of non-zero non-isomorphisms between indecomposable A-modules Mi. It was recently shown in [9] that a connected Auslander–Reiten component consisting of directing modules has only finitely many τ -orbits and the number of such components is finite. Hence the above corollary also implies the following characterization of representation-directed algebras due to Ringel (see [7, 2.4]). Corollary 2. An algebra A is representation-directed if and only if A is representation-finite and Γ (A) has no oriented cycles. The authors wish to thank the Polish Ministry of Education for its support under Research Project R.P.I.10.