Laura algebras and quasi-directed components

Marcelo Lanzilotta and David Smith

Abstract. We introduce a notion of distance between indecomposable modules and deduce from this new characterizations of laura algebras and quasi-directed Auslander-Reiten components. Finally, we show that a convex component is quasi-directed if and only if it is almost directed.

1. Introduction

The aim of the representation theory of Artin algebras is to study an algebra A by means of the category modA of finitely generated right Amodules, which often turns out to be easier to handle. A key example arises in [11] where Happel, Reiten and Smalø show that an algebra A is quasitilted, that is A is the endomorphism ring of a tilting object in a locally finite hereditary abelian category if and only if A has global dimension at most two and any indecomposable A-module either lies in L_{A} or in R_{A}. Recall that L_{A} is the full subcategory of modA consisting of all indecomposable A-modules whose predecessors have projective dimension at most one and R_{A} is defined dually.

Following this example, the module category, and particularly L_{A} and R_{A}, gained in importance and new classes of algebras, defined by the homological properties of their indecomposable modules, appeared : the shod algebras [7], the weakly shod algebras [8] and the laura algebras [1, 19]; each of them generalizing the previous ones. We refer the reader to [2] for a complete review on these classes of algebras.

Laura algebras have been introduced independently by Assem and Coelho [1] and Reiten and Skowroński [19] as a generalization of the representation-finite algebras and the weakly shod algebras. Their nice properties have made them rather interesting and hugely investigated, see, for instance [1, 19, 28, 2, 3, 10]. In particular, it is shown in [28] that an algebra A is laura if and only if it is quasitilted or its Auslander-Reiten quiver contains a faithful quasi-directed component, which is, in this case, convex. In [29], the second author presented equivalent conditions for an Auslander-Reiten component to be quasi-directed and convex.

2000 Mathematics Subject Classification: 16G70, 16G30, 18G05, 16E10.

Key words: laura algebras, quasi-directed Auslander-Reiten components, projectives and injectives

In this paper, we introduce a notion of distance between indecomposable modules which allows us to unify the approaches used in [1] and [29]. We deduce from this new characterizations of laura algebras (see Sec. 3) and convex quasi-directed components (see Sec. 4.1). Afterwards, we show that any morphism in the infinite radical implies the existence of infinitely many modules lying on a cycle and obtain, as a consequence, that any convex component is quasi-directed if and only if it is almost directed (see Sec. 4.2).

2. Preliminaries

2.1. Usual notations. Throughout this paper, all algebras are connected basic Artin algebras. For an algebra A, we denote by modA its category of finitely generated right modules and by indA a full subcategory of modA generated by one representative from each isomorphism class of indecomposable modules. We denote by rad(modA), the radical of modA, that is the ideal generated by all non-isomorphisms between indecomposable modules. The infinite radical rad^{∞}(modA) of modA is the intersection of all powers rad^{i}(modA), with i ≥ 1, of rad(modA). For an A-module M, we denote by pd_{A}M its projective dimension and by id_{A}M its injective dimension.

We denote by Γ(modA) its Auslander-Reiten quiver and by τ_{A} = DTr the usual Auslander-Reiten translation. By a component of Γ(modA), we mean a connected component of Γ(modA). A component Γ of Γ(modA) is semiregular if it does not contain a projective module and an injective module, and non-semiregular otherwise. Finally, Γ is faithful if ann(Γ), that is the intersection of the annihilators of all modules X in Γ, vanishes.

For further details on representation theory of Artin algebras, see [4, 21].

2.2. Walks, paths and hooks. Given X, Y in indA, a walk of length t between X and Y is a sequence :

w : X = X_{0} ^{f_{1}} X_{1} ^{f_{2}} ⋯ ^{f_{t}} X_{t} = Y , (t ≥ 0)

where, for each i, X_{i} ∈ indA and f_{i} is a non-zero morphism either from X_{i−1} to X_{i} or from X_{i} to X_{i−1}. In this case, we denote by l(w) the length of w. We also denote by M^{w} the set of modules in w, that is { X_{0}, X_{1}, …, X_{t} }. If each f_{i} is irreducible, w is a walk of irreducible morphisms and we denote by W (X, Y) the set of walks of irreducible morphisms between X and Y. A path from X to Y is a walk between X and Y such that each f_{i} is a morphism from X_{i−1} to X_{i} :

δ : \xymatrix{X = X_{0} \ar[r]^{f_{1}} & X_{1} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{t}} & X_{t} = Y} , (t ≥ 0)

In this case, we write δ : X ⇝ Y and we say that X is a predecessor of Y and Y is a successor of X. We denote by M_{X,Y}^{δ} the set of modules in δ. More generally, we denote by M_{X,Y} the set of modules M in indA lying on a path from X to Y. If each f_{i} is irreducible, δ is a path of irreducible morphisms and, in this case, δ is sectional if it contains no hook, that is a triple (X_{i−1}, X_{i}, X_{i+1}) such that τ_{A}X_{i+1} = X_{i−1}. We also denote by H_{X,Y}^{δ} the set of all hooks in δ and by H_{X,Y} the set of all hooks lying on a path from

X to Y. A refinement of δ is a path \xymatrix{X = X′_{0} \ar[r]^{f′_{1}} & X′_{1} \ar[r]^{f′_{2}} & ⋯ \ar[r]^{f′_{s}} & X′_{s} = Y,}

with s ≥ t, together with an order preserving function σ : {1, …, t − 1} → {1, …, s − 1} such that X_{i} = X′_{σ(i)} when 1 ≤ i ≤ t − 1. Finally, a path δ is a cycle if X = Y and at least one f_{i} is not an isomorphism. An A-module X is directing if it does not lie on any cycle and a component Γ of Γ(modA) is directed if it contains only directing modules.

2.3. Laura algebras. We recall from [1] that an Artin algebra A is called laura whenever indA\(L_{A} ∪R_{A}) is a finite set, where, as mentionned in the introduction, L_{A} and R_{A} are defined as follows :

L_{A} = { X ∈ indA | pd_{A}Y ≤ 1 for each predecessor Y of X },

R_{A} = { X ∈ indA | id_{A}Y ≤ 1 for each successor Y of X }.

We recall the following key result:

THEOREM 2.3.1. [1] The following statements are equivalent for an Artin algebra A:

(a) A is laura;

(b) There are only finitely many modules lying on a path from an injective module to a projective module;

(c) There are only finitely many modules lying on a path from a module not in L_{A} to a module not in R_{A}. □

2.4. Quasi-directed components. We recall that a component Γ of Γ(modA) is quasi-directed [1, 2, 29] if it is generalized standard [25], that is rad^{∞}(X, Y) = 0 for all X, Y in Γ, and almost directed, that is it contains only finitely many non-directing modules. Moreover, Γ is convex if any path from X to Y, with X, Y in Γ, contains only modules from Γ.

The following characterizations will play in important role in Sec. 4.

THEOREM 2.4.1. [29] Let A be an Artin algebra and assume that Γ is a semiregular component of Γ(modA) without projective modules. The following conditions are equivalent :

(a) Γ is quasi-directed;

(b) Γ is directed;

(c) For each X in Γ, there exists an integer n_{X} such that any cycle from X to X contains at most n_{X} modules;

(d) For each X in Γ, there exists an integer m_{X} such that any cycle from X to X contains at most m_{X} distinct hooks;

(e) Given X, Y in Γ, there exists an integer n_{X,Y} such that any path from X to Y contains at most n_{X,Y} modules;

(f) Given X, Y in Γ, there exists an integer m_{X,Y} such that any path

from X to Y contains at most m_{X,Y} distinct hooks. Further, if Γ contains injective modules, these conditions are equivalent to :

(g) For each X in Γ, there exists an integer i_{X} such that any path from an injective in Γ to X contains at most i_{X} modules;

(h) For each X in Γ, there exists a integer j_{X} such that any path from

an injective in Γ to X contains at most j_{X} distinct hooks. Furthermore, Γ is convex, B = A/ann(Γ) is a tilted algebra and Γ is a connecting component of Γ(modB). □

THEOREM 2.4.2. [29] Let A be an Artin algebra and assume that Γ is a non-semiregular component of Γ(modA). The following conditions are equivalent :

(a) Γ is quasi-directed and convex;

(b) Given X, Y in Γ, there exists an integer n_{X,Y} such that any path from X to Y contains at most n_{X,Y} modules;

(c) Given X, Y in Γ, there exists an integer m_{X,Y} such that any path from X to Y contains at most m_{X,Y} distinct hooks;

(d) There exists an integer n_{0} such that any path from an injective in Γ to a projective in Γ contains at most n_{0} modules;

(e) There exists an integer m_{0} such that any path from an injective in Γ

to a projective in Γ contains at most m_{0} distinct hooks. Furthermore, B = A/ann(Γ) is a laura algebra and Γ is the unique nonsemiregular and faithful component of Γ(modB). □

3. The main result

This section is devoted to our main Theorem which gives new characterizations of laura algebras by unifying the results stated in subsections 2.3 and 2.4, that is by completing (2.3.1) with the notions of hooks and bounds. The Theorem is the following :

THEOREM 3.0.3. The following statements are equivalent for an Artin algebra A:

(a) A is laura; (b) There are only finitely many modules lying on a path from an injective

module to a projective module; (b’) There are only finitely many modules lying on a path from a module

not in L_{A} to a module not in R_{A};

(c) There are only finitely many distinct hooks lying on a path from an injective module to a projective module;

(c’) There are only finitely many distinct hooks lying on a path from a

module not in L_{A} to a module not in R_{A};

(d) There exists an integer n such that any path from an injective module to a projective module contains at most n modules;

(d’) There exists an integer n′ such that any path from a module not in

L_{A} to a module not in R_{A} contains at most n′ modules;

(e) There exists an integer m such that any path from an injective module to a projective module contains at most m distinct hooks;

(e’) There exists an integer m′ such that any path from a module not in L_{A} to a module not in R_{A} contains at most m′ distinct hooks.

The rest of this section will be occupied by the proof of this statement. We refer to [12] for other characterizations of laura algebras using the Gabriel-Roiter measure.

3.1. A metric for indA. Among other things, we need in the proof of this Theorem to gather the modules lying on a path into a finite neighborhood of a certain module. This is done with the introduction of a distance in indA : given two modules X and Y in indA, set

d(X, Y) = \{ \begin{matrix}min\{ |M^{w}| − 1 | w ∈ W (X, Y) \}, if W (X, Y) is not empty; \\ ∞, otherwise. \end{matrix}

LEMMA 3.1.1. Let X, Y be modules in indA such that d(X, Y) = n < ∞. Then, there exists a walk of irreducible morphisms X = X_{0} − X_{1} − ⋯ − X_{n} = Y, with X_{i} ≠ X_{j} when i ≠ j.

Proof. If d(X, Y) = 0, then X = Y and there is nothing to show.

Otherwise, let ϵ : X = Y_{0} −^{f_{1}} Y_{1} −^{f_{2}} ⋯ −^{f_{t}} Y_{t} = Y be a walk in W (X, Y) such that |M^{ϵ}| = n + 1 = d(X, Y) + 1, with n ≥ 1. Clearly n ≤ t. If n < t, there exist i, j, with 0 ≤ i < j ≤ t, such that Y_{i} = Y_{j}. This yields a walk of

irreducible morphisms ϵ′ : X = Y_{0} −^{f_{1}} Y_{1} − ⋯ −^{f_{i}} Y_{i} = Y_{j} −^{f_{j+1}} ⋯ −^{f_{t}} Y_{t} = Y,

such that n ≤ l(ϵ′) = t − (j − i) < t. Inductively, one gets a walk X = X_{0} − X_{1} − ⋯ − X_{n} = Y, in W (X, Y), with X_{i} ≠ X_{j}, for i ≠ j (because d(X, Y) = n). □

PROPOSITION 3.1.2. (a) The map d : indA × indA → IR ∪ {∞} is a metric; (b) Let d̂ : indA × indA → IR ∪ {∞} given by

d̂(X, Y) = \{ \begin{matrix}min\{ l(w) | w ∈ W (X, Y) \}, if W (X, Y) is not empty; \\ ∞, otherwise. \end{matrix}

The maps d, d̂ are equal.

Proof. (a). The proof is easy and omitted. (b). Let X, Y ∈ indA. Since it is obvious that d(X, Y) = d̂(X, Y) = ∞ if and only if W (X, Y) is empty, assume that W (X, Y) is not empty. First, if d̂(X, Y) = n < ∞, then there exists a walk of minimal length w : X = X_{0} − X_{1} − ⋯ − X_{n} = Y in W (X, Y). The minimality implies that X_{i} ≠ X_{j} when i ≠ j. Therefore d(X, Y) ≤ n = d̂(X, Y). Conversely, if d(X, Y) = n < ∞, then it follows from (3.1.1), that there exists a walk of irreducible morphisms of length n in W (X, Y), and hence d̂(X, Y) ≤ n = d(X, Y). □

DEFINITION 3.1.3. Let G ⊆ indA and n ≥ 0. The ball of center G and radius n is the set:

B(G, n) = { X ∈ indA | there exists M ∈ G such that d(M, X) ≤ n }.

REMARK 3.1.4. Since the Auslander-Reiten quiver of an Artin algebra A is locally finite, then G ⊆ indA is finite if and only if B(G, n) is finite for all n ≥ 0.

3.2. The proof of the Theorem. We now start proving Theorem 3.0.3. We first recall the following result obtained in [27], in which we denote (as in the sequel) by r the rank of the Grothendieck group of A.

LEMMA 3.2.1. Let A be an Artin algebra and X_{1}, X_{2}, …, X_{t} be distinct modules in indA, with t > r. Then Hom_{A}(X_{i}, τX_{j}) ≠ 0, for some i, j, with 1 ≤ i, j ≤ t. □

Our first lemma connects the number of distinct hooks on paths with the number of distinct modules.

LEMMA 3.2.2. Let X, Y be modules in indA and assume that |H_{X,Y}^{δ} | ≤ b for each path δ : X ⇝ Y, for some b ≥ 0. Then |M_{X,Y}^{δ} | ≤ (b+1)(r+1)−1 for each path δ : X ⇝ Y.

Proof. Assume this is not the case. Then, there exists a path of the form δ : X = X_{0} ⇝ X_{1} ⇝ ⋯ ⇝ X_{(b+1)(r+1)−1} ⇝ Y, with X_{i} ≠ X_{j} when i ≠ j. This yields a family of subpaths {δ_{k}}_{k=1,...,b+1}, where δ_{k} is given by δ_{k} : X_{(k−1)(r+1)} ⇝ ⋯ ⇝ X_{kr+(k−1)}. Since each subpath δ_{k} contains r + 1 distinct X_{i}’s, then (3.2.1) gives, for each k, a path of the form X_{i_{k}} → τX_{j_{k}} → V_{k} → X_{j_{k}}. Gluing these paths together gives a path:

\begin{matrix}X = X_{0} ⇝ & X_{i_{1}} → τX_{j_{1}} → V_{1} → X_{j_{1}} ⇝ ⋯ \\ ⋯ ⇝ X_{i_{b+1}} → τX_{j_{b+1}} → V_{b+1} → X_{j_{b+1}} ⇝ Y, \end{matrix}

containing at least b + 1 distinct hooks, a contradiction to the hypothesis. □

LEMMA 3.2.3. Let X, Y be modules in indA and assume that |M_{X,Y}^{δ} | ≤ b for each path δ : X ⇝ Y, for some b ≥ 0. Then M_{X,Y} is a finite set.

Proof. Assume δ : X = X_{0} →^{f_{1}} X_{1} →^{f_{2}} ⋯ →^{f_{t}} X_{t} = Y is a path. By [29, (1.1)], f_{i} ∉ rad^{∞}(modA) for each i. Therefore δ can be refined to a path of irreducible morphisms δ′ : X = X′_{0} → X′_{1} → ⋯ → X′_{s} = Y. Since |M_{X,Y}^{δ′} | ≤ b by hypothesis, we have |M_{X,X′_{i}}^{δ′_{i}}| ≤ b for each subpath δ′_{i} : X = X′_{0} ⇝ X′_{i} of δ′, with 1 ≤ i ≤ s. This entails that d(X, X_{i}) ≤ b for each i with 1 ≤ i ≤ t. Hence, M_{X,Y} ⊆ B({X}, b) and M_{X,Y} is a finite set by (3.1.4). □

As a consequence we obtain the following result. Observe that the equivalence of (c) and (d) has been obtained in [29] under the assumption that the given modules X and Y belong to the same component of Γ(modA); this assumption turns out to be unnecessary.

COROLLARY 3.2.4. Let A be an Artin algebra and X, Y be modules in indA. The following conditions are equivalent:

(a) M_{X,Y} is finite;

(b) H_{X,Y} is finite;

(c) There exists an n such that |M_{X,Y}^{δ} | ≤ n, for each path δ : X ⇝ Y;

(d) There exists an m such that |H_{X,Y}^{δ} | ≤ m, for each path δ : X ⇝ Y.

Proof. Clearly, (a) implies (b) and (b) implies (d). On the other hand,

(d) implies (c) by (3.2.2) and (c) implies (a) by (3.2.3). □ As we see, we easily deduce from this corollary the equivalence of the

statements (b), (c), (d) and (e) of our main theorem. However, since indA generally contains infinitely many modules neither in L_{A}, nor in R_{A}, the above corollary is useless for proving the equivalence of the other statements since the bounds imposed by (b’), (c’), (d’) and (e’) may thus tend to infinity. The idea is to impose a finiteness condition on these modules and use the distance introduced in Sec. 3.1. This leads to the following result (compare with [1, (1.5)]).

PROPOSITION 3.2.5. (a) Let P be an indecomposable projective A-module and M be a predecessor of P. If M ∈ R_{A}, then M ∈ B({P }, r);

(b) Let I be an indecomposable injective A-module and N be a successor of I. If N ∈ L_{A}, then N ∈ B({I}, r).

Proof. We only prove (a) since the proof of (b) is dual. (a). We first show that any path of irreducible morphisms ϵ : N = X_{s} → X_{s−1} → ⋯ → X_{0} = P, with N a successor of M, is sectional. Indeed, if this is not the case there exists a minimal j such that (X_{j+1}, X_{j}, X_{j−1}) is a hook in ϵ and the subpath X_{j} → X_{j−1} → ⋯ → X_{0} = P is sectional. In particular Hom_{A}(X_{j−1}, P) ≠ 0 and so id_{A}X_{j+1} ≥ 2 by [21], contradicting N ∈ R_{A}. Therefore, any such path is sectional and, in particular, any such path has length at most r by (3.2.1) and the non-sectionality of cycles [5, 6].

Now, let δ : M = M_{s} →^{f_{s}} M_{s−1} →^{f_{s−1}} ⋯ →^{f_{1}} M_{0} = P be an arbitrary path from M to P. We claim that δ contains no morphism in rad^{∞}(modA), and hence that δ can be refined to a (sectional) path of irreducible morphisms. Indeed, if f_{1} ∈ rad^{∞}(modA), then it follows from [29, (1.1)] that there

exists a path M ⇝ M_{1} → Nr →^{gr} N_{r−1} →^{g_{r−1}} ⋯ →^{g_{1}} N_{0} →^{g_{0}} M_{0} = P, where

g := g_{0}g_{1} ⋯ gr ≠ 0 and N_{i} ≅ N̸_{j} for any i ≠ j. By (3.2.1), there exist i, j such that Hom_{A}(N_{i}, τN_{j}) ≠ 0. This yields a path M ⇝ M_{1} ⇝ N_{i} → τN_{j} → V_{j} → N_{j} ⟶^{g_{0}g_{1}···g_{j}} M_{0} = P. By [21], we have, as before, id_{A}(τN_{j}) ≥ 2, a contradiction since M ∈ R_{A}. Therefore, f_{1} ∉ rad^{∞}(modA), and hence can be refined to a (sectional) path of irreducible morphisms.

Inductively, assume that there exists an integer l, with l ≥ 1, such that the subpath M_{l} →^{f_{l}} M_{l−1} →^{f_{l−1}} ⋯ →^{f_{1}} M_{0} = P of δ can be refined to a (sectional) path of irreducible morphisms. We show that f_{l+1} ∉ rad^{∞}(modA). Indeed, if f_{l+1} ∈ rad^{∞}(modA), then it follows from (the proof of) [29, (1.1)] that there exists a path of irreducible morphisms Y_{t} → ⋯ → Y_{1} → Y_{0} = M_{l} containing at least r + 1 distinct modules Y′_{0}, Y′_{1}, ⋯, Yr^{′} and such that Hom_{A}(M_{l+1}, Y_{t}) ≠ 0. Applying (3.2.1) to these modules entails the existence of a path of irreducible morphisms τY′_{j} → V′_{j} → Y′_{j} → ⋯ → Y_{0} = M_{l}, where Hom_{A}(Y′_{i}, τY′_{j}) ≠ 0 for some i, j. By the induction hypothesis, we get a path of irreducible morphisms τY′_{j} → V′_{j} → Y′_{j} → ⋯ → Y_{0} = M_{l} → ⋯ → M_{0} = P containing at least one hook. By the first part of the proof, τ_{A}Y′_{j} ∉ R_{A}, which, again, contradicts the fact that M ∈ R_{A}. This shows that f_{l+1} ∉ rad^{∞}(modA). Consequently, the path δ can be refined to a path of irreducible morphisms. Moreover, by the first part of the proof, δ is sectional and, by (3.2.1) and the nonsectionality of cycles, l(δ) ≤ r. Therefore d(M, P) ≤ r and the proof is complete. □

This yields a new proof of the following corollary, first stated in [1], which proof now directly follows from (3.2.5) and (3.1.4).

COROLLARY 3.2.6. (a) There are only finitely many indecompos-

able modules in R_{A} which are predecessors of a projective module;

(b) There are only finitely many indecomposable modules in L_{A} which are successors of an injective module. □

As a consequence, we get the proof of the Theorem 3.0.3.

The proof. The equivalence of (a), (b) and (b’) follows from (2.3.1). On the other hand, the equivalence of (b), (c), (d) and (e) follows from (3.2.4) and the fact that there are only finitely many injective modules and projective modules in indA. For the other statements, we only prove the equivalence of the conditions (c’) and (c), the equivalence of the other statements with their corresponding ones are similar. We first prove that (c’) implies (c). Indeed, if this is not the case, there exists an infinite family (τM_{λ}, V_{λ}, M_{λ})_{λ∈Λ} of distinct hooks lying on paths from an injective I to a projective P. By (3.2.6), there exists an infinite subset Λ′ of Λ such that τM_{λ} ∉ L_{A} and M_{λ} ∉ R_{A} for each λ ∈ Λ′. This yields an infinity of hooks between a module not in L_{A} and a module not in R_{A}, a contradiction to the hypothesis.

Conversely, assume that (c) holds true and that (τM, V, M) is a hook lying on a path L ⇝ τM → V → M ⇝ N, with L ∉ L_{A} and N ∉ R_{A}. Since L ∉ L_{A}, there exists a predecessor L′ of L such that pd_{A}L′ ≥ 2, and hence an indecomposable injective module I such that Hom_{A}(I, τL′) ≠ 0. This yields a path δ : I → τL′ → V′ → L′ ⇝ L. Dually, there exists a path ε : N ⇝ N′ → W′ → τ^{−1}N′ ⇝ P, for some indecomposable projective module P. Gluing these paths together yields a path I ⇝^{δ} L ⇝ τM → V → M ⇝ N ⇝^{ε} P, showing that the hook (τM, N, M) belongs to a finite set, and hence (c’) holds true. □

4. Quasi-directed components

In this section, we first deduce new characterizations of quasi-directed and convex components from the results obtained in Sec. 3. Finally, we study the infinite radical of modA and show that any convex and almost directed component is generalized standard, and hence quasi-directed (4.2.3).

4.1. The immediate consequences. Our first result, which directly follows from (2.4.1) and (3.2.4), characterizes the semiregular quasi-directed components.

COROLLARY 4.1.1. Let A be an Artin algebra and assume that Γ is a semiregular component of Γ(modA) without projective modules. The following conditions are equivalent:

(a) Γ is quasi-directed;

(b) Γ is directed;

(c) For each X in Γ, there are only finitely many modules lying on a cycle from X to X;

(d) For each X in Γ, there are only finitely many distinct hooks lying on a cycle from X to X; (e) Given X, Y in Γ, there are only finitely many modules lying on a path

from X to Y;

(f) Given X, Y in Γ, there are only finitely many distinct hooks lying on

a path from X to Y; Further, if Γ contains injective modules, these conditions are equivalent to :

(g) For each X ∈ Γ, there are only finitely many modules lying on a path from an injective module in Γ to X;

(h) For each X ∈ Γ, there are only finitely many distinct hooks lying on a path from an injective module in Γ to X.

Furthermore, Γ is convex, B = A/ann(Γ) is a tilted algebra and Γ is a connecting component of Γ(modB). □

Of course, the dual statement on semiregular components without injective modules also holds, we leave the primal-dual translation to the reader.

We deduce from this result a surprising fact, which is that the condition of being generalized standard is unnecessary in the definition of semiregular quasi-directed components.

COROLLARY 4.1.2. Let A be an Artin algebra and assume that Γ is a semiregular component of Γ(modA). Then Γ is quasi-directed if and only if Γ is almost directed. Proof. Since the necessity is obvious, assume that Γ is not quasi-directed. By (4.1.1)(c), there exists an X in Γ such that |M_{X,X}| = ∞. It then follows from [29, (1.1)] that |M_{X,X} ∩Γ| = ∞, saying that Γ contains infinitely many non-directing modules. □

Our classification is completed with the characterization of the nonsemiregular quasi-directed and convex components, a corollary which follows directly from (2.4.2) and (3.2.4).

COROLLARY 4.1.3. Let A be an algebra and assume that Γ is a nonsemiregular component of Γ(modA). The following conditions are equivalent:

(a) Γ is quasi-directed and convex;

(b) Given X, Y in Γ, there are only finitely many modules lying on a path from X to Y;

(c) Given X, Y in Γ, there are only finitely many distinct hooks lying on a path from X to Y;

(d) There are only finitely many modules lying on a path from an injective module in Γ to a projective module in Γ;

(e) There are only finitely many distinct hooks lying on a path from an injective module in Γ to a projective module in Γ.

Furthermore, B = A/ann(Γ) is a laura algebra and Γ is the unique non-semiregular, convex and faithful component of Γ(modB). □

We stress that the statement of (4.1.2) does not hold in general for non-semiregular components as shown by the following example. Moreover, this example shows that the conditions (c) and (d) of (4.1.1) cannot be added to the five equivalent conditions of (4.1.3) (while this is trivial for (b) since the Auslander-Reiten quiver of any representation-finite algebra is quasi-directed).

EXAMPLE 4.1.4. Let A be the the radical square zero algebra given by the quiver

[figure: 1; •; •; •; •; 2; •; •; •; •; 3; •; •; 4; •; •; •; •; 5; •; •; 6]

Then, if we denote by P_{i} (or I_{i}, or S_{i}) the indecomposable projective (or injective, or simple, respectively) corresponding to the point i in the quiver, the Auslander-Reiten quiver Γ(modA) of A has the shape shown in Fig. 1, where we identify the two copies of S_{2}. If Γ is the non-semiregular compo-

[figure: •; •; •; •; •; •; •; •; •; P_{2}; I_{3}; I_{6}; S_{4}; S_{6}; Γ; P_{1}; I_{2}; S_{1}; P_{3}; S_{3}; P_{4}; S_{2}; •; •; S_{2}; I_{1}; P_{5}; •; •; •; •; •; •; •; •; •; •; •; •; •; •; I_{5}; S_{5}]

Figure 1. Γ(modA)

nent indicated in Fig. 1, it is not hard to see that Γ contains only directing modules and that the sets M_{X,X} and H_{X,X} are finite for all X in Γ. However, Γ is not generalized standard since rad^{∞}(P_{2}, I_{2}) ≠ 0, nor convex.

In the previous example, Γ is not convex. Our next aim is to show that (4.1.2) holds in general when we assume that Γ is convex. To do this, we need to establish new facts on the infinite radical.

4.2. The infinite radical.

LEMMA 4.2.1. Let A be a (representation-infinite) tilted Artin algebra and M, N be two modules in indA. If rad^{∞}(M, N) ≠ 0, then there exist infinitely many non-directing modules lying on a path from M to N.

Proof. First, if M (or N) does not belong to a connecting component, a postprojective component or a preinjective component, then it follows from [14, (3.7)] that M (or N, respectively) belongs to a component obtained from a quasi-serial component (that is a stable tube or a component of the form ℤA_{∞}, [23]) by coray insertions or ray insertions, and the result follows trivially. Otherwise, by [22, p. 41], we may assume without loss of generality that M belongs to a postprojective component while N belongs to a connecting component Γ, different from the one containing M.

Following [16, (1.4)], there exists an infinite path

(∗) \xymatrix{⋯ \ar[r] & N_{i} \ar[r]^{h_{i}} & N_{i−1} \ar[r]^{h_{i−i}} & ⋯ \ar[r]^{h_{2}} & N_{1} \ar[r]^{h_{1}} & N_{0} = N}

of irreducible morphisms such that, for each i, there exists k_{i} in rad^{∞}(M, N_{i}) with h_{1}h_{2} … h_{i}k_{i} ≠ 0. Since Γ is acyclic, there exists i ≥ 1 such that no projective in Γ is a predecessor (by a path of irreducible morphism) of N_{i}. Observe that N_{j} is left stable for all j ≥ i. Let Γ′ be the left stable part of Γ containing N_{j}, for all j ≥ i. By [13, (3.6)], Γ′ is isomorphic to a full subquiver of ℤΔ, where Δ is a finite and acyclic quiver. Let Δ′ be the copy of Δ containing N_{i}. By the Liu-Skowroński criterium [15, 24], Δ′ is a complete slice of the infinite-representation algebra B = A/annΔ′. Therefore, we can assume that N = N_{i} and that N belongs to a connecting component Γ^{N} of Γ(modB) without projective modules. In this case, Γ(modB) admits a unique postprojective component Γ^{M} (necessarily different from the one containing N), and this component contains M_{B}.

Now, let T be a quasi-serial component of Γ(modB) such that Hom(Γ^{M}, T ) ≠ 0 and Hom(T , Γ^{N}) ≠ 0, and let X be a module in T . Since B is connected, there exists a path P_{1} →^{f_{1}} P_{2} →^{f_{2}} ⋯ →^{f_{n−1}} P_{n} →^{f_{n}} X, where P_{i} is projective for each i and P_{1} belongs to Γ^{M}. Moreover, we can assume that P_{2} ∉ Γ^{M}. Consequently, f_{1} ∈ rad^{∞}(P_{1}, P_{2}) and it follows from [20, 26, 29], for instance, that for each t ≥ 0 there exists a path

P_{1} = X_{0} →^{h_{1}} X_{1} →^{h_{2}} ⋯ →^{h_{t−1}} X_{t} →^{h} P_{2},

where h_{i} is irreducible for each i and h belongs to rad^{∞}(X_{t}, P_{2}). Since Γ^{M} is acyclic, right stable and contains only finitely many τ-orbits, there exists a t ≥ such that X_{t} is a successor of M, inducing a path M ⇝ X_{t} ⇝ P_{2} ⇝ X. On the other hand, since Δ′ is a complete slice, rad^{∞}(X, T) ≠ 0 for a certain indecomposable module T in Δ′ and, dually, there exists a path of the form X ⇝ N. Gluing these paths together yields a path of the form M ⇝ X ⇝ N. The result then follows from the fact that T contains infinitely many non-directing modules X. □

The fact that the previous result holds true for tilted algebras implies that it holds true for any Artin algebra.

THEOREM 4.2.2. Let A be an Artin algebra and M, N be two modules in indA. If rad^{∞}(M, N) ≠ 0, then there exist infinitely many non-directing modules lying on a path from M to N.

Proof. Let f be a non-zero morphism in rad^{∞}(M, N), and let Γ be the connected component of Γ(modA) containing N. Following [16, (1.4)] and [29, (1.1)], there exists an infinite path

(∗) \xymatrix{⋯ \ar[r] & N_{i} \ar[r]^{h_{i}} & N_{i−1} \ar[r]^{h_{i−1}} & ⋯ \ar[r]^{h_{2}} & N_{1} \ar[r]^{h_{1}} & N_{0} = N}

such that (a) For each i, there exists k_{i} ∈ rad^{∞}(M, N_{i}) such that h_{1}h_{2} ⋯ h_{i}k_{i} ≠ 0;

(b) For each i, N_{i} ∈ Γ;

(c) N_{i} ≠ N_{j} when i ≠ j. If (∗) contains infinitely many non-directing modules, this ends the proof. Otherwise, we can assume that (∗) contains only directing modules. Moreover, it follows from [18, 27] that (∗) crosses only finitely many τ-orbits. Without loss of generality, we can therefore assume that (∗) intersects infinitely many times the τ-orbit of N. In particular, N is left stable and non-periodic.

Let _{l}Γ be the left stable part of Γ containing N. Then _{l}Γ contains no oriented cycle. Indeed, assume that δ : X = X_{0} → X_{1} → ⋯ → X_{n} = X is a cycle lying in _{l}Γ. Since _{l}Γ is left stable, there exists i ≥ 0 such that N_{i} belongs to the τ-orbit of N and is a predecessor of X. Conversely, there exists r ≥ 0 such that τ^{r}X is a predecessor of N_{i}. But then, since X is a predecessor of τ^{r}X by [9, (1.4)], we obtain a path X ⇝ τ^{r}X ⇝ N_{i} ⇝ X, saying that N_{i} is not directing, a contradiction.

Moreover, _{l}Γ contains only finitely many τ-orbits. Indeed, if this is not the case, then there exists a stable connected component C of _{l}Γ having infinitely many τ-orbits. Let X be a module in C such that the length of any walk in Γ from a non-stable module to the τ-orbit of X is at least 2r, where r is the rank of the Grothendieck group of A. But then, since X is a predecessor of τ^{s}X for all s ≥ 1 by [9, (1.5)], the above argument gives the existence of a N_{j} which is not directing, a contradiction.

Therefore, by [13, (3.6)], _{l}Γ is isomorphic to a full subquiver of ℤΔ, which is stable under predecessor, where Δ is a finite and acyclic quiver. Let \overline{Δ} be a fixed copy of Δ such that no module in \overline{Δ} is a successor (by a path of irreducible morphisms) of a projective module in Γ. Such a \overline{Δ} exists by [17, (1.2)], for instance. Let D be the full subquiver of _{l}Γ consisting of all predecessors of \overline{Δ} in _{l}Γ. Moreover, let T be the direct sum of all modules in \overline{Δ}. We claim that Hom_{A}(T, τT) = 0. Indeed, if this is not the case, then there exist indecomposable direct factors Y and Z of T and a non-zero morphism s : Y → τZ. Observe that s ∈ rad^{∞}(modA) since D is a full subquiver of ℤΔ which is acyclic and closed under predecessors. By hypothesis, there exists N_{i} in (∗) such that N_{i} is a predecessor of Y. On the other hand, since s ∈ rad^{∞}(modA), it follows from [16, (1.4)] that there exists an infinite path of irreducible morphisms

\xymatrix{⋯ \ar[r] & Y_{j} \ar[r] & Y_{j−1} \ar[r] & ⋯ \ar[r] & Y_{1} \ar[r] & Y_{0} = τZ}

in D such that Hom_{A}(Y, Y_{j}) ≠ 0 for all j ≥ 0. Since D has only finitely many τ-orbits and no oriented cycle, there exists j ≥ 0 such that Y_{j} is a predecessor of N_{i}, giving a path N_{i} ⇝ Y → Y_{j} ⇝ N_{i}, a contradiction to the fact that N_{i} is directing. Consequently, Hom_{A}(T, τT) = 0.

Now, let B = A/annT. Observe that annT = annD by [24, (Lemma 3)]. Consequently, T is a faithful B-module and D consists of B-modules. Since Hom_{B}(T, τT) = 0, B is a tilted algebra by [15, 24]. Moreover, T is a slice B-module lying in a connecting component without projective modules.

Let p ≥ 0 be such that N_{q} ∈ D for all q ≥ p. We claim that there exists q ≥ p such that Im(k_{q}) has at least one indecomposable direct factor not in D. Indeed, let l be the length of the A-module M. By [25, (2.6)], for instance, there exists N_{q} ∈ D such that any predecessor X of N_{q} is such that l(X) > l. Obviously, k_{q} : M → N_{q} is not surjective, and hence Imk_{q} ≠ N_{q}. But then, since l(Imk_{q}) ≤ l, it follows from the hypothesis made on N_{q} that there exists an indecomposable direct factor Q of Imk_{q} such that Q ∉ D. In particular, Q belongs to a connected component of Γ(modB) different from the one containing N_{q}.

Therefore the inclusion Q ↪ N_{q} belong to rad^{∞}(modA) and it follows from (4.2.1) that there exist infinitely many indecomposable non-directing B-modules (and hence A-modules) X lying on a path Q ⇝ X ⇝ N_{q}, giving infinitely many indecomposable modules X lying on a path of the form M → Q ⇝ X ⇝ N_{q} ⇝ N. This concludes the proof. □

THEOREM 4.2.3. Let A be an Artin algebra and Γ be a component of Γ(modA). If Γ is convex, then it is quasi-directed if and only if it is almost directed.

Proof. The necessity is obvious. For the sufficiency, it remains to show that Γ is generalized standard. Assume to the contrary that rad^{∞}(M, N) ≠ 0 for some indecomposable A-modules M and N. By (4.2.2), there exist infinitely many non-directing modules X with a path M ⇝ X ⇝ N, a contradiction since Γ is convex and almost directed. □

We also get the following result.

COROLLARY 4.2.4. Let A be an Artin algebra. Then A is representation-finite if and only if Γ(modA) contains only finitely many non-directing modules.

Proof. Since the necessity is obvious, assume to the contrary that A is representation-infinite. Then rad^{∞}(modA) ≠ 0 and the contradiction follows from (4.2.2). □

ACKNOWLEDGEMENTS. This paper was written while the first author visited the Universities of Sherbrooke and Bishop’s under a CRM-ISM post-doctoral fellowship, and while the second author had an NSERC graduate studies fellowship.

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M. Lanzilotta; Centro de Matemática (CMAT), Iguá 4225, Universidad de la República, CP 11400, Montevideo, Uruguay. E-mail address: marclan@cmat.edu.uy

D. Smith; Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1. E-mail address: david.smith@usherbrooke.ca