AUSLANDER-REITEN THEORY FOR MODULES OF FINITE COMPLEXITY OVER SELFINJECTIVE ALGEBRAS.

OTTO KERNER AND DAN ZACHARIA

Abstract. In this paper we describe the shapes of the stable components containing modules with finite complexity over a selfinjective finite dimensional algebra over an algebraically closed field. We prove that the associated orbit graph of each such component, is either a Dynkin diagram (finite or infinite), or an extended Dynkin diagram.

Peter Webb has described in [16] the Auslander-Reiten quiver of a group algebra 𝕜G of a finite group over a field. Specifically, if C_{s} is a stable component of the Auslander-Reiten quiver, then its tree class is a Dynkin diagram (finite or infinite), or an extended Dynkin diagram. The purpose of this paper is to extend Webb’s theorem to a more general setting, namely to components of selfinjective algebras containing modules of finite complexity.

The complexity of a finitely generated module measures the growth of the terms in a minimal projective resolution of the module. To be more precise, let Λ be a finite dimensional algebra over a field 𝕜 and let

P^{•}: ⋯ ⟶ P^{2} ⟶^{δ^{2}} P^{1} ⟶^{δ^{1}} P^{0} ⟶^{δ^{0}} M → 0

be a minimal projective resolution of a finitely generated Λ-module M. The i-th Betti number of M, β_{i}(M), is defined as the number of indecomposable summands of P^{i}. The complexity of M is defined as

cx M = inf{n ∈ ℕ|β_{i}(M) ≤ ci^{n−1} for some positive c ∈ ℚ and all i ≥ 0}

If no such n exists, then we say that the complexity of M is infinite. It is well-known that, if Λ is a selfinjective algebra and C is a component of the Auslander-Reiten quiver of Λ, then all nonprojective modules in C have the same complexity [7, 2.2]. Note that over the group algebra 𝕜G of a finite group, every module has finite complexity by the Alperin-Venkov theorem.

Let Λ be a selfinjective algebra, and let τ denote the Auslander-Reiten translate, see [2]. An indecomposable non projective Λ-module M is called τ-periodic, if τ^{m}M ≅ M for some m > 0. If an Auslander-Reiten component C contains a τ-periodic module, then its stable part C_{s}, is a τ-periodic component, that is, every module in C_{s} is τ-periodic.The tree class of C_{s} is either a finite Dynkin diagram, or is of type A_{∞}, which means that C_{s} is a tube [9]. The complexity of nonprojective modules in periodic components clearly is 1. Therefore, for our classification purpose, it is enough to study stable components, which are not τ-periodic. The main result of the paper is the following:

2000 Mathematics Subject Classification. Primary 16G70. Secondary 16D50, 16E05.

Most of the results of this paper were obtained while both authors were visiting the University of Bielefeld in 2009 as part of the SFB’s “Topologische und spektrale Strukturen in der Darstellungstheorie” program. We both thank Claus Michael Ringel, the Faculty of Mathematics in Bielefeld and the SFB for inviting us and for making our stay possible. The second author is supported by a grant from NSA.

Main Theorem. Let C be a non τ-periodic component of the Auslander-Reiten quiver of a selfinjective algebra Λ over an algebraically closed field, containing a module of finite complexity. Then the stable component C_{s} is of type ℤΔ, where Δ is either an extended Dynkin diagram of type Ã_{n}, D̃_{n}, Ẽ_{i} (i = 6, 7, 8), or of the type ℤA_{∞}, ℤA_{∞}^{∞} or ℤD_{∞}. If the component is a regular component, then only the infinite Dynkin trees can occur.

The paper is organized as follows. In the first section, we study stable components of type ℤT, where T is a finite graph. We prove that if C_{s} is a non τ-periodic stable component of the Auslander-Reiten quiver, whose orbit graph is a finite tree of wild type, then C_{s} must contain only modules of infinite complexity. This is done by analyzing the behavior of the Coxeter transformation on wild trees.

In the second section, we prove the main theorem. We do this by working with a class of very well-behaved modules called Ω-perfect modules, introduced in [7, 8]. Specifically, we obtain explicit descriptions of the tree class of a component, by analyzing Auslander-Reiten sequences ending at certain Ω-perfect modules belonging to that component. This is achieved by using results from [8] describing the types of Auslander-Reiten sequences ending at Ω-perfect modules of finite complexity.

We should also mention Zhang’s paper [17] in this introduction. Using her results, the classification would follow immediately if one would also assume that finite growth for minimal projective resolutions is equivalent to the finite growth for minimal injective resolution as well. This is the case for group algebras, but does not always happen, not even in the local commutative selfinjective case, see [10] for instance.

Throughout this paper Λ will denote a selfinjective algebra over an algebraically closed field . If P^{•} = (P^{i}, δ^{i}) is a minimal projective resolution of the module M, then Ω^{i}M, denotes the i-th syzygy of M; that is, Imδ^{i} = Ω^{i}M. For a finitely generated Λ-module M, we denote by ℓ(M) its composition length, and by τM = νΩ^{2}M its Auslander-Reiten translate where ν is the Nakayama equivalence. For basic results on Auslander-Reiten sequences and quivers, on Coxeter transformations, and other representation theory concepts, we refer the reader to [2, 3, 11].

1. Finite orbit graphs and finite complexity

In this section we study the case when a stable component C_{s} containing modules of finite complexity is of the type ℤT where T is a finite oriented tree with k vertices. We show first how one can use Coxeter matrices to compute dimensions, and hence estimate the growth of the the Betti numbers, of the modules in a “stable” part of C from the dimensions of the modules located on an “initial” slice S, parallel to T. Therefore, we may write S = (M_{1}, …, M_{k}), where the M_{i} are the indecomposable modules in C_{s} corresponding to the vertices of {m} × T in ℤT, for some integer m. We choose S such that neither indecomposable projective nor simple modules are predecessors of S in C. Let S^{≥0} = ⋃_{i≥0} τ^{i}S. By our choice of S, the sets S^{≥0}, respectively ΩS^{≥0} are the vertices of convex subquivers of C, respectively ΩC. The dimension function d, defined by d(M) = dimM is additive on short exact sequences, therefore, its restriction d: S^{≥0} → ℤ is additive on Auslander-Reiten sequences. This means that if 0 → τM → ⊕E_{i} → M → 0 is an Auslander-Reiten sequence with M ∈ S^{≥0}, then we have

d(M) + d(τM) = ∑_{i} d(E_{i})

Denoting by Φ the Coxeter matrix corresponding to the directed tree T, we have the following formula ([16], Proposition 2.3.):

Φ(d(M_{1}), …, d(M_{k}))^{t} = (d(τM_{1}), …, d(τM_{k}))^{t}

where (⋯)^{t} denotes the matrix transpose.

For a finite dimensional module M we have d(M)/d(Λ) ≤ β_{0}(M) ≤ d(M) and, since Λ is selfinjective

d(τ^{m}M)/d(Λ) ≤ β_{2m}(M) ≤ d(τ^{m}M),

for all m ≥ 0. For r ≥ 0 we hence get β_{r+1}(M) ≤ d(Ω^{r+1}M) < d(P^{r}) ≤ d(Λ)β_{r}(M), Therefore, we always have the double inequality

β_{r+2}(M)/d(Λ) < β_{r+1}(M) < β_{r}(M)d(Λ).

We start by showing that underlying graphs of wild trees cannot occur as orbit graphs of stable components having modules of finite complexity.

We need first to recall a few facts about properties of Coxeter transformations for wild trees. By [1], (see also [14]), the Coxeter transformation of a wild tree, has a dominant eigenvalue ρ > 1 whose geometric multiplicity equals 1, and all the eigenvalues with magnitude different from 1, are positive real numbers. It is also known that in this case, the Jordan canonical form of Φ consists entirely of Jordan blocks of size 1, with the possible exception of some blocks of size 2, that may appear if 1 is also an eigenvalue of Φ ([15]).

The case when λ = 1 is an eigenvalue of Φ, will require some discussion in the proof to come, so assume that λ = 1 is an eigenvalue of Φ, and let

W = Ker(Φ − I)^{2}

be the corresponding generalized eigenspace. Then W has a decomposition

W = W_{1} ⊕ W_{2},

where W_{1} = Ker(Φ − I) is the eigenspace of Φ, corresponding to the eigenvalue 1, and (Φ − I)(W_{2}) ⊂ W_{1}. Thus every element w ∈ W can be written in the form w = z + z′ with z ∈ W_{1}, and z′ ∈ W_{2}. Set (Φ − I)z′ = u ∈ W_{1}. Then we have for each m ≥ 1 that Φ^{m}w = w + mu, where u ≠ 0 if and only if z′ ≠ 0, or equivalently, if and only if w is not an eigenvector of Φ. We may also consider the Coxeter transformation as a linear automorphism of ℝ^{k}. Denote again by W_{1} the real eigenspace of Φ corresponding to the eigenvalue 1. Let T̄ be the underlying graph of the quiver T and let q = qT̄: ℝ^{k} → ℝ be the Tits form of T̄, again considered as a real quadratic form:

qT̄(r_{1}, …, r_{k}) = ∑_{i} r_{i}^{2} − ∑_{i<j} a(i, j)r_{i}r_{j},

where a(i, j) denotes the number of edges between the vertices i and j of the graph T̄. If rad q denotes the radical of the real Tits form, then it is well known and easy to check, the rad q = W_{1}. Moreover it is well known that rad q contains a nonzero positive vector x = (x_{1}, …, x_{k}) only if q = qT̄ is positive semidefinite, that is if T̄ is an extended Dynkin diagram. This normally is shown for integral vectors, but the same arguments hold for real vectors.

Proposition 1.1. Let Λ be a selfinjective algebra and let C be a non τ-periodic component of the Auslander-Reiten quiver of Λ such that the orbit graph of its stable part C_{s} is a finite tree, but neither a Dynkin diagram, nor an extended Dynkin diagram. Then every module in C_{s} has infinite complexity.

Proof. Let S = (M_{1}, …, M_{k}) be a slice in C_{s}, where we assume that none of the modules M_{i} has a predecessor in C, which is indecomposable projective or simple. Let x = (d(M_{1}), …, d(M_{k}))^{t} considered as a vector in ℂ^{k}. Using the remarks at the beginning of this section, we may write for each m > 0

Φ^{m}x = (d(τ^{m}M_{1}), …, d(τ^{m}M_{k}))^{t}

and all the entries of Φ^{m}x are positive. Since the Coxeter polynomial is palindromic, λ is an eigenvalue of Φ if and only if λ^{−1} is an eigenvalue of Φ. Using this fact, write all the distinct eigenvalues of the Coxeter transformation in an order of decreasing magnitude:

ρ = λ_{0} > λ_{1} > ⋯ > λ_{r} > 1 = |λ_{r+1}| = ⋯ = |λ_{r+s}| > λ_{r+s+1} > ⋯ > λ_{2r+2+s}

and decompose ℂ^{k} into a direct sum of generalized eigenspaces corresponding to these eigenvalues:

ℂ^{k} = V_{λ_{0}} ⊕ V_{λ_{1}} ⊕ ⋯ ⊕ V_{λ_{2r+2+s}}.

Moreover, by ([15]) for each λ_{i} ≠ 1, the spaces V_{λ_{i}} consist entirely of eigenvectors of Φ corresponding to λ_{i}. If 1 is an eigenvalue, say λ_{r+1} = 1, then V_{r+1} = Ker(Φ−I)^{2}, and we get Φ^{m}(x_{r+1}) = x_{r+1} + mu_{r+1}, where u_{r+1} is nonzero if and only if x_{r+1} ∉ Ker(Φ − I). Write x as

x = x_{0} + x_{1} + x_{2} + ⋯ + x_{2r+2+s},

where for each i ≥ 0, x_{i} ∈ V_{λ_{i}}. We look now at the growth of the dimensions of the modules τ^{m}M_{i} by analyzing positive powers of Φ applied to the vector x. For each m > 0 we have

(a) Φ^{m}x = Φ^{m}x_{0} + Φ^{m}x_{1} + ⋯ + Φ^{m}x_{2r+2+s}
= λ_{0}^{m}x_{0} + λ_{1}^{m}x_{1} + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s}

if Φ is diagonalizable, and

(b) Φ^{m}x = Φ^{m}x_{0} + Φ^{m}x_{1} + ⋯ + Φ^{m}x_{2r+2+s}
= λ_{0}^{m}x_{0} + λ_{1}^{m}x_{1} + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s} + mu_{r+1}

if V_{r+1} is not an eigenspace, that is, Φ is not diagonalizable.

Let n be smallest such that the component x_{n} of x is non zero.

Assume first that 0 ≤ n ≤ r, so λ_{n} > 1. Since all the entries of Φ^{m}x are positive and x_{n} ≠ 0, we may assume without loss of generality that the first entry of Φ^{m}x_{n} is nonzero, and that we have:

d(τ^{m}M_{1}) = λ_{n}^{m}x_{n}^{(1)} + λ_{n+1}^{m}x_{n+1}^{(1)} + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s}^{(1)}

= λ_{n}^{m}(x_{n}^{(1)} + \frac{λ_{n+1}^{m}}{λ_{n}^{m}} x_{n+1}^{(1)} + ⋯ + \frac{λ_{2r+2+s}^{m}}{λ_{n}^{m}} x_{2r+2+s}^{(1)})

in case (a) and

d(τ^{m}M_{1}) = λ_{n}^{m}x_{n}^{(1)} + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s}^{(1)}

= λ_{n}^{m}(x_{n}^{(1)} + ⋯ + \frac{1}{λ_{n}^{m}} x_{r+1}^{(1)} + \frac{m}{λ_{n}^{m}} u_{r+1}^{(1)} + ⋯ + \frac{λ_{2r+2+s}^{m}}{λ_{n}^{m}} x_{2r+2+s}^{(1)})

in case (b). Since λ_{n} > 1, it follows that the dimension of τ^{m}M_{1} and hence the Betti numbers β_{2m}(M_{1}) grow exponentially yielding a contradiction.

If n > r + s, then we obtain that for each 1 ≤ i ≤ k,

d(τ^{m}M_{i}) = λ_{n}^{m}x_{n}^{(i)} + λ_{n+1}^{m}x_{n+1}^{(i)} + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s}^{(i)}

and the eigenvalues λ_{n}, λ_{n+1}, …, λ_{2r+s+2} are all less than 1, hence

lim_{m→∞} β_{2m}(M_{i}) = lim_{m→∞} d(τ^{m}M_{i}) = 0

for each module M_{i} on the slice S, yielding also a contradiction, since none of the modules on the slice is projective.

Assume now that r + 1 ≤ n ≤ r + s, so that we have |λ_{n}| = 1. In case (a) we have

Φ^{m}x = Φ^{m}x_{n} + ⋯ + Φ^{m}x_{r+s} + ⋯ + Φ^{m}x_{r+2+s}
= λ_{n}^{m}x_{n} + ⋯ + λ_{r+s}^{m}x_{r+s} + ⋯ + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s}

and the right hand side is bounded in magnitude, hence the even, and consequently all the Betti numbers are also bounded. This yields that the complexity of all M_{i} is 1, but C_{s} is neither a stable tube, nor a ℤA_{∞} component, nor its orbit graph is of Dynkin type. So we have a contradiction, see [7].

In case (b) and n = r + 1 we get

Φ^{m}(x) = x_{r+1} + mu_{r+1} + ⋯ + λ_{2r+2+s}^{m}x_{2r+2+s}

If u_{r+1} is zero, then we are in the situation just discussed, so assume that u_{r+1} ≠ 0 The left hand side is an integral vector, and has only positive entries, for every m > 0, so a quick analysis shows that the entries u_{r+1}^{(i)} are all real and positive. This means that u_{r+1} is a positive eigenvector of Φ corresponding to the eigenvalue 1. This only can happen, it T is of Euclidean type, hence we get again a contradiction. The proof is now complete. □

If T is a Dynkin diagram, there are no stable components C_{s} of type ℤT. This follows directly from [4]. Indeed, if T is a Dynkin diagram, there is no positive integral vector x, with Φ^{m}x positive, for all m ≥ 0, where again Φ denotes the Coxeter transformation. The “tame situation” was studied in Webb’s paper, and his proof works in the more general context too with a small adaptation. We present the result and its proof for completeness.

Proposition 1.2. Let Λ be a selfinjective algebra and let C be a component of the Auslander-Reiten quiver of Λ. Assume that the stable part of C is of type ℤΔ where Δ is an extended Dynkin diagram. Then C ≠ C_{s}, that is, C is not a regular component. Moreover, every non projective module in such a component has complexity 2.

Proof. Assume that the component is a regular component, and assume that Δ is a an extended Dynkin diagram with k vertices. Let S = (M_{1}, …, M_{k}) be a slice in C. Consider again d: C → .ℤ, given by d(M) = dimM. By the remarks at the beginning of this section we have that

Φ^{n}(d(M_{1}), …, d(M_{k}))^{t} = (d(τ^{n}M_{1}), …, d(τ^{n}M_{k}))^{t}

for all m ∈ ℤ, where Φ is the appropriate Coxeter transformation for the slice S. Setting x = (dim M_{1}, …, dim M_{k})^{t}, we have that Φ^{m}(x) = x + αn for a positive integer m, and some n ∈ ℕ^{k}, with Φ(n) = n and integral scalar α ([13]). If α = 0, then the dimension vectors in each τ-orbit are bounded, hence each module in C has complexity 1. By [7], C must be either a stable tube, or a ℤA_{∞}-component. In either case we obtain a contradiction to the number of τ-orbits being finite. Assume now that α ≠ 0. Then for a large enough integer t, we have that the vector Φ^{−mt}x = x − tαn for α > 0, respectively Φ^{mt}x = x + tαn, for negative α has negative entries, and this contradicts the fact that C is a regular component.

It remains to prove that the projective resolutions grow linearly for the modules in C_{s}. As we have seen above, we have that Φ^{m}(x) = x + αn for some positive integer m, where the scalar α > 0. Thus, for each t > 0 we have

Φ^{mt}x = x + tαn

and this shows that the Betti numbers of the modules on the slice S increase linearly. □

2. Proof of the Theorem

This section is devoted to proving our main theorem. For the proof, we have to distinguish between two cases. First we consider the case, when every module in C is eventually Ω-perfect, a concept introduced in [7]. Then we will consider the case when C contains a non projective module that is not eventually Ω-perfect. We recall the following definitions. An irreducible map g: B → C is called Ω-perfect if the induced irreducible maps Ω^{n}g: Ω^{n}B → Ω^{n}C are either all monomorphisms, for every n ≥ 0, or are all epimorphisms, for every n ≥ 0. An indecomposable non projective module M is called Ω-perfect, if and only if all irreducible homomorphisms g: E → M are Ω-perfect. Note that if M is Ω-perfect, then automatically all irreducible maps f: τM → E are also Ω-perfect. Finally, M is called eventually Ω-perfect, if Ω^{m}M is perfect, for some m ≥ 0. Since the Nakayama functor ν is an exact equivalence, and τ = νΩ^{2} for a selfinjective algebra, then M being Ω-perfect implies that τM is also Ω-perfect. We also recall that if there are no simple Ω-periodic Λ-modules, then every indecomposable non projective Λ-module is eventually Ω-perfect [8].

We will prove the following:

Theorem 2.1. Let C be a non τ-periodic component of the Auslander-Reiten quiver of some selfinjective algebra Λ over an algebraically closed field, containing a module of finite complexity. If every module in C is eventually Ω-perfect, then the stable component C_{s} is of type ℤΔ, where Δ is either an extended Dynkin diagram of type Ã_{n}, D̃_{n}, Ẽ_{i} (i = 6, 7, 8), or of the the type ℤA_{∞}, ℤA_{∞}^{∞} or ℤD_{∞}. If the component is a regular component, then only the infinite Dynkin trees can occur.

The proof of this theorem is organized as follows: We first show that all the indecomposable middle terms in the Auslander-Reiten sequences, are pairwise nonisomorphic, or else C_{s} is of type ℤÃ_{1}, where Ã_{1} denotes the Kronecker quiver.

When there are no multiple arrows in C_{s} or equivalently, if the indecomposable middle terms in all Auslander-Reiten sequences in C are non-isomorphic, we can apply Riedtmann’s Structure Theorem, [12, 1.4] which states that there exists a directed tree T, such that C_{s} is isomorphic to ℤT/G, where G is an admissible group of automorphisms of ℤT. We will start by considering an Auslander-Reiten sequence 0 → τM → E → M → 0, such that M is Ω-perfect, without projective predecessors in C and such that α(M) is maximal in C. We construct the tree T by “knitting” together its directed edges. The possible shapes of those Auslander-Reiten sequences are listed below. It turns out that T is either a finite tree with 3 or 4 arms, or is an infinite tree of type A_{∞}, A_{∞}^{∞} or D_{∞}.

It is easy to see that ⟨τ⟩ is a normal subgroup of Aut(ℤT), where τ corresponds to the Auslander-Reiten translation. Since C_{s} contains no τ-periodic modules,

G ∩ ⟨τ⟩ = 1.

The cokernel of the embedding ⟨τ⟩ ↪ Aut(ℤT) is p: Aut(ℤT) ↠ Aut(T̄) where T̄ denotes the underlying tree, [12, 3].

If T is a finite tree, or A_{∞} or D_{∞}, then Aut(T̄) is finite, possibly trivial. If a ∈ G and p(a) has finite order m, then a^{m} ∈ ker p = ⟨τ⟩. Hence a^{m} = τ^{i} for some integer i, but this implies that a^{m} = 1, hence a has finite order. By [12, 4.1,Corollary] the identity e is the only element of finite order in G, hence a = e, which means G = {e}. This shows that in all the cases where T ≠ A_{∞}^{∞}, the stable component C_{s} = ℤT. Observe also, that, by the results of the previous section, the tree T can be neither wild nor of finite Dynkin type.

If T = A_{∞}^{∞}, then Aut(T̄) contains many elements of infinite order. Hence G can be nontrivial in this case.

Throughout this section C_{s} will be the stable part of some infinite, non τ-periodic, Auslander-Reiten component C. We will also assume that C contains a module of finite complexity, and also, unless explicitly stated, that every module in C_{s} is eventually Ω-perfect.

For an indecomposable nonprojective module M, the number of indecomposable nonprojective direct summands in the middle term of the Auslander-Reiten sequence ending in M, is denoted by α(M). Denote by

α(C) = sup{α(M)|M indecomposable Ω-perfect in C}

By [8, section 3], α(C) ≤ 4, and the possible shapes of the Auslander-Reiten sequences ending at an Ω-perfect module M belong to the following list:

Proposition 2.2. (a) If α(M) = 1 or α(M) = 2, then the Auslander-Reiten sequences ending in M have the following shapes:

\xymatrix{ & X \ar@{->>}[dr] &  &  & X \ar@{->>}[dr] &  &  & X \ar@{->>}[dr] \\ τM \ar@{^{(}->}[ur] &  & M & τM \ar@{^{(}->}[ur] \ar@{->>}[dr] &  & M & τM \ar@{->>}[dr] \ar@{->>}[ur] &  & M \\  &  &  &  & Y \ar@{^{(}->}[ur] &  &  & Y \ar@{->>}[ur] \\  & (1) &  &  & (2a) &  &  & (2b)}

(b) If α(M) = 3 and α(M) = 4, the possible shapes are:

\xymatrix{ &  &  &  &  &  &  & W \ar@{^{(}->}[ddr] \\  & X \ar@{^{(}->}[dr] &  &  & X \ar@{->>}[dr] &  &  & X \ar@{^{(}->}[dr] \\ τM \ar@{->>}[r] \ar@{->>}[ur] \ar@{->>}[dr] & Y \ar@{^{(}->}[r] & M & τM \ar@{->>}[r] \ar@{->>}[ur] \ar@{->>}[dr] & Y \ar@{^{(}->}[r] & M & τM \ar@{->>}[uur] \ar@{->>}[ur] \ar@{->>}[dr] \ar@{->>}[r] & Y \ar@{^{(}->}[r] & M \\  & Z \ar@{^{(}->}[ur] &  &  & Z \ar@{^{(}->}[ur] &  &  & Z \ar@{^{(}->}[ur] \\  & (3a) &  &  & (3b) &  &  & (4)}

We have the following:

Lemma 2.3. Let 0 → τM ⟶^{[ f_{1},f_{2} ]^{T}} U_{1} ⊕ V ⟶^{[ g_{1},g_{2} ]} M → 0 be the Auslander-Reiten sequence ending at M, and assume that the module U_{1} is indecomposable.

(a). If g_{1} is a monomorphism, and f_{1} is an epimorphism, then either α(U_{1}) = 1, or α(U_{1}) = 2 and the Auslander-Reiten sequence ending at U_{1} is of type (2a).

(b). If f_{1} and g_{1} are both epimorphisms, and U_{1} is Ω-perfect, then we either have that α(U_{1}) = 2 and the Auslander-Reiten sequence ending at U_{1} is of type (2b), or α(U_{1}) = 3, and the Auslander-Reiten sequence ending at U_{1} is of type (3b).

Proof. (a). Since the module M is perfect, the induced irreducible map τU_{1} → τM is again a monomorphism. Since g_{1} is an irreducible monomorphism, the U_{1} is also Ω-perfect, by [8], hence the Auslander-Reiten sequence ending at U_{1} has the shape (1) or (2a) from our list, and the first part follows.

(b) Since M is perfect, the map τg_{1} : τU_{1} → τM is again an epimorphism, and since U_{1} is also Ω-perfect, the statement follows from an examination of our list. □

A repeated application of part (a) of this lemma yields immediately the following consequence:

Corollary 2.4. If the Auslander-Reiten sequence ending at the Ω-perfect module M has the form

0 → τM ⟶^{[ f_{1},f_{2} ]^{T}} U_{1} ⊕ V ⟶^{[ g_{1},g_{2} ]} M → 0

with U_{1} indecomposable, and where f_{1} is an epimorphism and g_{1} is a monomorphism, then there exists a finite chain of irreducible monomorphisms

U_{r} ↪ U_{r−1} ↪ ⋯ ↪ U_{2} ↪ U_{1}

with r ≥ 1, α(U_{r}) = 1, and α(U_{i}) = 2 for all 1 ≤ i < r. □

We show now that the possibility of having multiple arrows in the Auslander-Reiten components containing modules of finite complexity is rather slim. We have the following:

Proposition 2.5. Let M ∈ C_{s} be a module of finite complexity, and let 0 → τM → ⨁_{1≤i≤k} X_{i} → M → 0 be an Auslander-Reiten sequence ending at M. Then, for all i ≠ j , X_{i} ≇ X_{j}, unless k = 2 and the component C_{s} is of type ℤÃ_{1}.

Proof. We prove the proposition using a case by case analysis on the allowable possibilities from our list. There is nothing to prove if k = 1. Assume that k = 2. The result is clear if we are in the situation (2a) from our list since the two summands in the middle have distinct lengths, so assume that we are in situation (2b). This means that there exists an Auslander-Reiten sequence of the form 0 → τM → X ⊕ X → M → 0. Without loss of generality, we may assume that X is also Ω-perfect, and, using the list, we have an Auslander-Reiten sequence 0 → τX → τM ⊕ τM → X → 0, and, continuing in this way we obtain a “flat” component

\xymatrix{ &  & Y_{−1} \ar@{-->}[dr] \ar@{-->}[dr] &  & Y_{1} \ar@{-->}[dr] \ar@{-->}[dr] \\ … & Y_{−2} \ar@{-->}[ur] \ar@{-->}[ur] &  & Y_{0} \ar@{-->}[ur] \ar@{-->}[ur] &  & Y_{2} & …}

This means that the entire component C_{s} is of type ℤÃ_{1}.

Consider now the situation when we have an Auslander-Reiten sequence of the type 0 → τM → X ⊕ X ⊕ X → M → 0. This means that we must be in situation (3a), and since X is Ω-perfect. by [8], α(X) ≥ 3. But each of the three irreducible maps τM → X is an epimorphism, and therefore the type of the Auslander-Reiten sequence ending at X cannot belong to our list, obtaining a contradiction.

Let us also consider, the possibility that the Auslander-Reiten sequence ending at M is of the form 0 → τM → X ⊕ X ⊕ Y → M → 0 where X ≇ Y. We are now either in situation (3a), or (3b). In either case, α(X) ≥ 2, and there are two irreducible epimorphisms τM → X. Using our list, it follows that we must have α(X) = 2, case (2b).. But then the irreducible maps τX → τM have to be epimorphisms, whereas the irreducible maps X → M are monomorphisms, so we obtain a contradiction.

It remains to examine the situation when α(M) = 4. We cannot have a summand X appearing with multiplicity 3 or 4, in the middle of the Auslander-Reiten sequence ending at M, since then we would obtain that α(X) ≥ 3 and three or more irreducible epimorphisms τM → X which would contradict the restrictions imposed by our list. The only case remaining is when the Auslander-Reiten sequence ending at M has the form 0 → τM → X ⊕ X ⊕ Y ⊕ Z → M → 0 where Y ≇ X ≇ Z, but as above, this case cannot exist either. □

The proof of the following lemma is immediate.

Lemma 2.6. If α(C) = 2 and M is an Ω-perfect module in C having no simple or projective predecessors in C. Then all the predecessors of M are Ω-perfect. □

For the remaining part of the proof of theorem 2.1 we assume that the indecomposable middle terms of the Auslander-Reiten sequences in C are pairwise nonisomorphic.

First we characterize the shapes of the stable components C with α(C) = 2.

Lemma 2.7. Assume that α(C) = 2 and that

0 → τM ⟶^{[ f_{1},f_{2} ]^{T}} X_{1} ⊕ Y_{1} ⟶^{[ g_{1},g_{2} ]} M → 0

is an Auslander-Reiten sequence ending at the Ω-perfect module M ∈ C, with X_{1} and Y_{1} indecomposable. Then (a) If f_{1}, g_{2} are monomorphisms and f_{2}, g_{1} are epimorphisms, then the stable

component C_{s} is of type ℤA_{∞}.

(b) If each of the irreducible maps f_{1}, f_{2}.g_{1}, g_{2} is an epimorphism, then C_{s} is of the type ℤA_{∞}^{∞}, or ℤÃ_{n} for some n > 1.

Proof. (a). Since f_{2} is an epimorphism and g_{1} is a monomorphism, we may apply lemma 2.2 and corollary 2.3 to obtain the existence of a (necessarily) finite chain of irreducible monomorphisms

Y_{r} ↪ Y_{r−1} ↪ ⋯ ↪ Y_{2} ↪ Y_{1}

where α(Y_{r}) = 1, and α(Y_{i}) = 2 for all 1 ≤ i < r. A similar analysis allows us to construct an arbitrary long chain of irreducible epimorphisms ending at M:

⋯ ↠ X_{s} ↠ X_{s−1} ↠ ⋯ ↠ X_{2} ↠ X_{1} ↠ M

where for each i ≥ 1, we have α(X_{i}) = 2. Therefore, by Riedtmann, C_{s} is of type ℤA_{∞}.

(b). Since α(C) = 2, the previous results show that every predecessor of M has an Auslander-Reiten sequence ending at it of type (2b). If C_{s} has infinitely many τ–orbits, then C_{s} = ℤA_{∞}^{∞}, so assume that C_{s} has finitely many orbits, that is C_{s} is of the form ℤA_{∞}^{∞}/G where G is an admissible group of automorphisms of ℤA_{∞}^{∞}. Assume that, by taking sufficient powers of τ, M is isomorphic to a predecessor in ℤA_{∞}^{∞}. This means that there are two different vertices x and y in different τ-orbits of ℤA_{∞}^{∞}, a finite path from x to y, an element g ∈ G such that y = x^{g}. and both vertices represent the module M. By taking sufficiently large powers of τ, we can assume that all the modules, corresponding to the vertices on this path are Ω-perfect. Since the Auslander-Reiten sequence ending at each predecessor of M, is of type (2b), it follows that there must be a proper surjection M → M. This clearly cannot happen. Therefore we may not identify M with predecessors, so we get that C_{s} = ℤA_{∞}^{∞}/G is of type ℤÃ_{n} for some n > 1. The proof of the lemma is now complete. □

We turn our attention to the components having the property that α(C) = 3.

Lemma 2.8. Assume that α(C) = 3 and let

0 → τM ⟶^{[ f_{1},f_{2},f_{3} ]^{T}} X ⊕ Y ⊕ Z ⟶^{[ g_{1},g_{2},g_{3} ]} M → 0

be an Auslander-Reiten sequence with three indecomposable non projective summands in the middle.

(a) If g_{1}, g_{2}, g_{3} are monomorphisms, the C_{s} is of type ℤẼ_{i}, i = 6, 7, 8.

(b) If g_{1} is an epimorphism, and g_{2}, g_{3} are monomorphisms, then C_{s} is either of type ℤD_{∞}, or of type ℤD̃_{n} for some n ≥ 5.

Proof. (a) It follows from our list that, in this case, f_{1}, f_{2} and f_{3} are all epimorphisms. Therefore, by 2.4., the tree type of C_{s} is T_{rst} where T_{rst} is a rooted tree with three arms of lengths r, s and t, respectively. Therefore, C_{s} ≅ ℤT_{rst}, and, since C_{s} is not τ-periodic, the tree T_{rst} cannot be of Dynkin type. It cannot be of wild type either by our results in the previous section. Therefore, it must be an extended Dynkin diagram Ẽ_{i}, i = 6, 7, 8.

(b) We are now in the case (3b) of our list. Applying 2.4 to Y_{1} = Y and Z_{1} = Z, we obtain chains of irreducible monomorphisms Y_{r} ↪ Y_{r−1} ↪ ⋯ ↪ Y_{2} ↪ Y_{1}, and Z_{s} ↪ Z_{s−1} ↪ ⋯ ↪ Z_{2} ↪ Z_{1}, with α(Y_{r}) = α(Z_{s}) = 1, and α(Y_{i}) = α(Z_{j}) = 2, for 1 ≤ i < r and 1 ≤ j < s. Note also, that by taking enough powers of the Auslander-Reiten translate τ, we may also assume that X_{1} = X is Ω-perfect. The Auslander-Reiten sequence ending at X_{1} has the form

0 → τX_{1} ⟶^{[ h_{1},τg_{1} ]^{T}} V ⊕ τM ⟶^{[ h,f_{1} ]} X_{1} → 0

where V ≠ 0 but possibly decomposable, and both h_{1} and h are epimorphisms. We show first that r = s = 1. Assume not, and say, s > 1, and let

0 → τZ_{1} ⟶ Z_{2} ⊕ τM ⟶ Z_{1} → 0

be the Auslander-Reiten sequence ending at Z_{1}. We have the following diagram representing the Auslander-Reiten sequences ending at M, and its immediate predecessors:

[figure: V; τX_{1}; X_{1}; τ^{2}M; τM; M; τY_{1}; Y_{1}; τZ_{1}; Z_{1}; Z_{2}]

We now compute lengths along these Auslander-Reiten sequences, and if we denote by ℓ(C) the length of a module C, we obtain

ℓ(τM) + ℓ(M) = ℓ(X_{1}) + ℓ(Y_{1}) + ℓ(Z_{1})
ℓ(τM) + ℓ(τ^{2}M) = ℓ(τX_{1}) + ℓ(τY_{1}) + ℓ(τZ_{1})
ℓ(X_{1}) + ℓ(τX_{1}) = ℓ(V) + ℓ(τM)
ℓ(Z_{1}) + ℓ(τZ_{1}) = ℓ(Z_{2}) + ℓ(τM)
ℓ(Y_{1}) + ℓ(τY_{1}) ≥ ℓ(τM)

Hence we have

ℓ(M) + ℓ(τ^{2}M) + 2ℓ(τM) = ℓ(X_{1}) + ℓ(Y_{1}) + ℓ(Z_{1}) + ℓ(τX_{1}) + ℓ(τY_{1}) + ℓ(τZ_{1})
≥ 3ℓ(τM) + ℓ(Z_{2}) + ℓ(V)

and consequently, we get

ℓ(M) + ℓ(τ^{2}M) ≥ ℓ(τM) + ℓ(Z_{2}) + ℓ(V).

Since we have proper epimorphisms V ↠ X_{1} ↠ M and τM ↠ X_{1} ↠ M, we obtain ℓ(τ^{2}M) > ℓ(M) + ℓ(Z_{2}). Since we have irreducible monomorphisms Z_{2} → Z_{1} and

Z_{1} → M, we have (see [2], chapter V, for instance) that ℓ(Z_{2}) ≥ \frac{1}{(1 + D^{2})^{2}} ℓ(M),

where D = dim_{𝕜} Λ. Letting

c = 1 + \frac{1}{(1 + D^{2})^{2}} > 1

we get as in [8] that ℓ(τ^{2}(M)) ≥ cℓ(M), hence, for all m ≥ 1, we have

ℓ(τ^{2m}(M)) ≥ c^{m}ℓ(M)

which means that the complexity of M is infinite, contradicting our assumption. Therefore, r = s = 1.

We turn now our attention to the Ω-perfect module X_{1}. Clearly α(X_{1}) > 1. Assume first that all the indecomposable modules Y with α(Y) = 3 are in the τ-

orbit of M. This means that α(X_{1}) = 2, and the Auslander-Reiten sequence ending at X_{1} is of type (2b). Letting X_{2} = V, we obtain for each l ≥ 0, epimorphisms τ^{l}X_{2} → τ^{l}X_{1}. In this way we construct inductively an infinite sectional path of irreducible epimorphisms

⋯ ↠ X_{s} ↠ X_{s−1} ↠ ⋯ ↠ X_{2} ↠ X_{1} ↠ M

where for each i, α(X_{i}) = 2. This means that in this case T = D_{∞} and therefore the stable component C_{s} is of type ℤD_{∞}.

Assume now that there are more τ-orbits in C_{s} containing modules Y such that α(Y) = 3. If α(X_{1}) = 3, we get the following picture:

\xymatrix{ & X_{2} \ar@{-->}[dr] \\ τX_{1} \ar@{-->}[dr] \ar@{-->}[ur] \ar[r] & U_{2} \ar[r] & X_{1} \ar@{-->}[dr] \\  & τM \ar@{-->}[dr] \ar@{-->}[ur] \ar[r] & Y_{1} \ar[r] & M \\  &  & Z_{1} \ar@{-->}[ur]}

where 0 → τX_{1} ⟶^{[ h_{11},h_{12},τf_{1} ]^{T}} X_{2}⊕⊕U_{2}⊕τM ⟶^{[ h,h′,g_{1} ]^{T}} X_{1} → 0 is the Auslander-

Reiten sequence ending at X_{1}. Since g_{1} and τf_{1} are epimorphisms, we are again in the case (3b) in our list. The same argument used to show that α(Y_{1}) = α(Z_{1}) = 1 can be used now to prove that α(U_{2}) = α(X_{2}) = 1, and then C_{s} is of type ℤD̃_{5}. If α(X_{1}) = 2, then we have a sectional path of epimorphisms X_{r} ↠ X_{r−1} ↠ ⋯ ↠ X_{1}, where each X_{i} is Ω-perfect, α(X_{i}) = 2 for all 1 ≤ i < r, and α(X_{r}) = 3. Thus, the Auslander-Reiten sequences ending at each X_{i} for i < r are of type (2b). Let 0 → τX_{r} ⟶ X_{r−1} ⊕ U ⊕ V ⟶ X_{r} → 0 be the Auslander-Reiten sequence ending at X_{r}. Then again, the maps V → X_{r} and U → X_{r} are monomorphisms, and we also have α(U) = α(V) = 1. This means that C_{s} is of type ℤD̃_{n} where n = r + 4, and the proof of the lemma is complete. □

Lemma 2.9. Assume that C_{s} contains a module M such that α(M) = 4. Then the component C_{s} is of type ℤD̃_{4}.

Proof. Without loss of generality we may assume that the module M is Ω-perfect. Let 0 → τM → ⨁_{1≤i≤4} X_{i} → M → 0 be the Auslander-Reiten sequence ending at M. We are in the case (4) of our list, so each map X_{i} → M is a monomorphism, and each τM → X_{i} is an epimorphism. By 2.4., we obtain for each 1 ≤ i ≤ 4, the existence of a finite sectional path of monomorphisms X_{i}^{(r_{i})} ↪ ⋯ ↪ X_{i}^{(1)} = X_{i}, with α(X_{i}^{(r_{i})}) = 1. This means that our component is of type ℤT_{r_{1}r_{2}r_{3}r_{4}} where T_{r_{1}r_{2}r_{3}r_{4}} is star with four arms of lengths r_{1}, …, r_{4}. If r_{i} > 1 for some i, then the tree is of wild type, so C_{s} cannot contain a module of finite complexity, by the results of the previous section. Therefore, we must have T_{r_{1}r_{2}r_{3}r_{4}} = D̃_{4}. □

Theorem 2.1. follows now from the preceding lemmas.

It remains to prove that our classification result holds also in the case where a stable component C_{s} contains a module M that is not eventually Ω-perfect. It turns out that this is the easy case. Moreover, the assumption that C contains an indecomposable module M with finite complexity is not needed. The proof is based on the following result from [7, 2.6] and [8, 2.3]:

Lemma 2.10. Let C_{s} be a stable component that is not τ-periodic and contains a module M that is not eventually Ω-perfect. Let g: E → τ^{m}M be an irreducible epimorphism, such that Ωg: ΩE → Ωτ^{m}M is injective. Then the kernel S of g is simple and Ω-periodic.

We have the following:

Theorem 2.11. Let C_{s} be a stable component that is not τ-periodic and contains a module that is not eventually Ω-perfect. Then C_{s} ≅ ℤΔ where Δ is either an extended Dynkin diagram, or one of the infinite Dynkin trees A_{∞}^{∞} or D_{∞}.

Proof. If the component contains a module of complexity 1, then it must be a ℤA_{∞} by [7], so let C be an indecomposable module in C_{s} that is not eventually Ω-perfect and assume that cx C > 1. This means that there exists an irreducible epimorphism B → C whose kernel is isomorphic to some simple Ω-periodic module S. Let V = S ⊕ ΩS ⊕ ⋯ ⊕ Ω^{n−1}S, where n is the Ω-period of S, by 2.10. Clearly ΩV ≅ V. But the Nakayama functor ν also has finite order on every simple module, so let m be such that ν^{m}S ≅ S. Set W = V ⊕ νV ⊕ ⋯ ⊕ ν^{m−1}V, so we also have νW ≅ W. Since the syzygy functor commutes with the Nakayama functor, we have τW ≅ W. Each indecomposable direct summand of W has complexity 1, and every module in C_{s} has complexity greater than 1, so no summand of W is either in C_{s} or in ΩC_{s}. Following [5, 6], let d_{W} : C_{s} → ℕ be the function

d_{W} (M) = dim_{𝕜} \underline{Hom}_{Λ}(W, M).

Since \underline{Hom}_{Λ}(S, B) ≅ Hom_{Λ}(S, B) ≠ 0, d_{W} ≠ 0. This function satisfies the conditions of Lemma 3.2. of [6], hence it is an additive function on the stable component. Moreover, since τW ≅ W, we also have

d_{W} (τM) = dim_{𝕜} \underline{Hom}_{Λ}(W, τM) = dim_{𝕜} \underline{Hom}_{Λ}(τW, τM) = d_{W} (M)

If X is indecomposable and f: X → B is an irreducible map, the additivity of d_{W} implies 2d_{W} (X) ≥ d_{W} (B) > 0, which shows that d_{W} a positive additive function, hence it induces a positive additive function on the orbit graph Δ̄ of C_{s} ≅ ℤΔ. Hence Δ is either an extended Dynkin diagram, or one of the infinite Dynkin quivers A_{∞}, A_{∞}^{∞} or D_{∞} by [9]. Since C_{s} contains a module that is not eventually Ω-perfect, the case A_{∞} can be excluded. Indeed, if C_{s} is of type ℤA_{∞}, take an indecomposable module M in C_{s}, such that no predecessor in C is projective or simple, and such that α(M) = 1. Then also α(Ω^{m}M) = 1 and α(τ^{m}M) = 1, for all m ≥ 0, hence all irreducible maps ending in Ω^{m}M are epimorphisms. Therefore M is Ω-perfect, and by lemma 2.6., all modules in C_{s} are eventually Ω-perfect. □

Theorem 2.11 has the following unexpected direct consequence

Corollary 2.12. Let Δ be a finite or infinite quiver which is not of finite Dynkin type, or of extended Dynkin type, or A_{∞}^{∞} or D_{∞}. If C is a non-periodic Auslander-Reiten component of a selfinjective algebra, such that C_{s} is of type ℤΔ, then every nonprojective module in C is eventually Ω-perfect. □

Finally, our main result follows by putting together theorems 2.1 and 2.11.

References

[1] A’Campo, N. Sur les valeurs propres de la transformation de Coxeter. Invent. Math. 33 (1976), no. 1, 61–67.

[2] Auslander, M., Reiten, I., Smalø, S. Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol 36, 1997.

[3] Benson, D.J. Representations and Cohomology I, Cambridge Studies in Advanced Mathematics, vol 30. Cambridge University Press, Cambridge, 1998.

[4] Bernstein, I. N., Gelfand, I. M. Ponomarev, V. A. Coxeter functors, and Gabriel’s theorem. (Russian) Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33. translated in Russian Math Surveys, 28, 1973, 17-32.

[5] Erdmann, K. On Auslander-Reiten components for group algebras, J. Pure Appl.Algebra 104 (1995),149-160..

[6] Erdmann, K., Skowroński, A. On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), 165-189.

[7] Green, E. L., Zacharia, D. Auslander-Reiten components containing modules with bounded Betti numbers, Trans. Amer. Math. Soc., 361 (2009), 4195-4214.

[8] Green, E. L., Zacharia, D. On modules of finite complexity over selfinjective artin algebras. To appear.

[9] Happel, D., Preiser, U., Ringel, C. M. Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules. Representation theory, II, pp. 280–294, Lecture Notes in Math., vol 832, Springer, Berlin, 1980.

[10] Jorgensen, D. A., Şega, L. M., Asymmetric complete resolutions and vanishing of Ext over Gorenstein rings. Int. Math. Res. Not. 2005, no. 56, 3459–3477.

[11] Kerner, O. Representations of Wild Quivers, CMS Conf. Proc., 19, (1996), 65–107.

[12] Riedtmann, C. Algebren, Darstellungsköcher, Ueberlagerungen und zurück. Comment. Math. Helv. 55 (1980), no. 2, 199–224.

[13] Ringel, C.M. Tame algebras and integral quadratic forms. Lecture Notes in Mathematics, 1099. Springer-Verlag, Berlin, 1984.

[14] Ringel, C.M. The spectral radius of the Coxeter transformations for a generalized Cartan matrix. Math. Ann. 300 (1994), no. 2, 331–339.

[15] Subbotin, V.F., Stekolshchik, R.B. The Jordan form of Coxeter transformations and applications to representations of finite graphs. Funkcional. Anal. i Prilozhen. 12 (1978), no. 1, 84–85.

[16] Webb, P. The Auslander-Reiten quiver of a finite group, Math. Z. 179 (1982), no. 1, 97–121.

[17] Zhang, Y. Eigenvalues of Coxeter transformations and the structure of regular components of an Auslander-Reiten quiver. Comm. Algebra 17 (1989), no. 10, 2347–2362.

Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany

E-mail address: kerner@math.uni-duesseldorf.de

Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA

E-mail address: zacharia@syr.edu