Generalized double tilted algebras

By Idun REITEN and Andrzej SKOWRONSKI ^{∗} ^{1}

Abstract

We introduce the class of generalized double tilted artin algebras and prove that it coincides with the class of artin algebras whose ARquiver admits a faithful generalized standard almost directed component. A homological characterization of faithful generalized standard almost directed components is also established.

12000 Mathematics Subject Classification. Primary 16G70, 18G20; Secondary 16G10.

Key Words and Phrases. Tilted algebra, AR-quiver, artin algebra.

∗ Partially Supported by Polish Scientific KBN Grant 5 P03A 008 21.

1 Introduction

Tilted algebras have for a long time played a central role in the representation theory of artin algebras. Being a more general class than the class of hereditary algebras, they occur more frequently, and at the same time the module theory for tilted algebras is closely related to the module theory for hereditary algebras. The more general class of quasitilted algebras was introduced in [5]. This class contains also the canonical algebras, and has been the focus of much attention during the last years. A useful feature of the class of quasitilted algebras is the nice homological characterization as the algebras where each indecomposable module has projective or injective dimension at most one, and where in addition the global dimension is at most two. Dropping the last property we have the class of shod algebras, introduced and investigated in [2]. A shod algebra which is not quasitilted is said to be a strict shod algebra. Both for quasitilted algebras and for shod algebras there are several interesting descriptions , for example in terms of paths from indecomposable injective to indecomposable projective modules ( [5] [2]).

As shown in [11] the strict shod algebras are closely related to tilted algebras. A tilted algebra is characterized in terms of the existence of a faithful section such that Hom(M, DT rN) = 0 for M and N on Δ. The more general concepts of double section and double tilted algebras were introduced in [11], and the strict shod algebras were characterized in this terms. In particular, there are (left and right) tilted algebras naturally associated with a strict shod algebra, and the module theory for the strict shod algebras is closely related to the module theory for the associated tilted algebras.

In this paper we generalize the concepts of section and double section to what we call a multisection Δ, along with its left and right parts Δ_{l} and Δ_{r}. Modelled on defining double tilted algebras on the basis of double sections in [11] , we define generalized double tilted algebras on the basis of a multisection. We also characterize the generalized double tilted algebras as those haveing a component with a faithful multisection Δ such that Hom(M, DT rN) = 0 for M in Δ_{r} and N in Δ_{l}. On the other hand this connects up with the work in [11], since we show that an algebra Λ is generalized double tilted if and only if there is an almost directed generalized standard faithful component. We also describe the shape of (most) components for such algebras. For tilted algebras we have global dimension at most two, and for shod algebras it is at most three. For multisections we have the concept of n−section , where n may be infinite, and where a section is a 1−section and a double section a 2−section. Correspondingly we have n−double tilted algebras and we show that they have global dimension at most n + 1.

We now describe the content of this paper section by section. In section one we recall some background material, including essential definitions. In section two we introduce the concept of multisection in a component of the AR-quiver of an artin algebra along with its core, and the concept of generalized tilted algebras is introduced and investigated in section 3. The results on global dimension are given in section 4, and in section 5 we give a homological characterization of faithful generalized standard almost directed components.

We refer to the recent papers [][] for generalizations of (strict) shod algebras in other, but related, directions.

2 Preliminaries

In this section we recall some definitions which will be used in this paper. We also refer to [11] for relevant background material.

By an algebra we mean a basic artin algebra Λ over a fixed commutative artin ring R, and by a Λ−module is meant a finitely generated left Λ−module. We denote the category of these modules by mod Λ, and ind Λ denotes the full subcategory formed by the indecomposable modules. Then rad(mod Λ) denotes the radical of mod Λ, that is, the ideal in mod Λ generated by all noninvertible morphisms beteween modules in ind Λ. The infinite radical rad^{∞}(mod Λ) of mod Λ is the intersection of all powers rad^{n}(mod Λ), n ≥ 1, of rad(mod Λ). A component C of AR-quiver of Λ is said to be generalized standard if rad^{∞}(X, Y) = 0 for all X and Y in C.

For M and N in ind Λ a path from M to N in ind Λ is given by sequence of nonzero morphisms

M = X_{o} →^{f_{1}} X_{1} → … →^{f_{t}} X_{t} = N

where, for each i, X_{i} is in ind Λ and each f_{i} is a nonzero nonisomorphism, and t ≥ 1. Then M is said to be a predecessor of N and N a successor of M. If the morphisms f_{i} are in addition irreducible, then the path is a path of irreducible morphisms. If M is isomorphc to N, then the path is an oriented cycle. A module M in ind Λ is directing if it does not lie on any oriented cycle, and a component C of the AR-quiver is directing if all modules in C are directing. The component C is said to be almost directing if all but a finite number of modules in C are directing.

Let Z_{0} → Z_{1} → ⋯ → Z_{n−1} → Z_{n} be a path of irreducible morphisms in ind Λ. If Z_{i−1} ≃ τZ_{i+1} (where τ = DT r) with D the ordinary duality and T r the transpose ) for some i with 1 ≤ i ≤ n − 1, we say that Z_{i} is a hook of the path. The path is said to be sectional if it has no hook and almost sectional if it has exactly one hook.

3 Almost directed components

In this section we introduce some concepts playing a fundamental role in our further investigations.

Let Λ be an artin algebra and let C be a component of Γ_{Λ} . A full connected valued subquiver Δ of C is said to be a multisection in C if the following conditions are satisfied:

(1) Δ is almost directed.

(2) Δ is convex in C.

(3) For each τ_{Λ} − orbit τO in C we have 1 ≤| Δ ∩ O |< ∞.

(4) | Δ ∩ O |= 1 for all but finitely many τ_{Λ} − orbits O in C.

(5) No proper full convex valued subquiver of Δ satisfies the conditions (1) − (4).

For a multisection Δ in a component C of Γ_{Λ}, we consider also the following full valued subquivers of C:

Δl^{′} = {X ∈ Δ; there is an almost sectional path X → ⋯ → P with P projective }, Δr^{′} = {X ∈ Δ; there is an almost sectional path I → ⋯ → X with I

injective },

Δl^{′′} = {X ∈ Δl^{′}; τ_{Λ}^{−}X ∉ Δl^{′}},

Δr^{′′} = {X ∈ Δr^{′}; τ_{Λ}X ∉ Δr^{′}},

Δ_{l} = (Δ\Δr^{′}) ∪ τ_{Λ}Δr^{′′},

Δ_{r} = (Δ\Δl^{′}) ∪ τ_{Λ}^{−}Δl^{′′},

Δ_{c} = Δ′_{l} ∩ Δ′′_{r} (the core of Δ).

In the above notation, we have the following

Proposition 2.1. Let C be a component of Γ_{Λ} with a multisection Δ. Then we have the following.

(i) Δ_{c} is finite. (ii) Every cycle in C lies in Δ_{c}.

(iii) Every indecomposable module M in C is in Δ_{c} or is a predecessor of Δ_{l} or a successor of Δ_{r} in C.

Proof: For (i), observe that every module X from Δ_{c} lies on a path I → ⋯ → X → ⋯ → P in C with I injective and P projective. Moreover, both I and P belong to Δ, because Δ is convex in C and contains modules from the τ_{Λ} − orbits of I and P. Finally, Δ contains at most finitely many modules lying on oriented cycles in Δ (equivalently in C). Therefore, Δ_{c} is finite.

(ii) Let X = X_{1} → X_{2} → ⋯ → X_{m} → X_{m+1} = X be an oriented cycle in C. We claim that there exist i, j ∈ {1, …, m} and natural numbers p and q such that τ_{Λ}^{p}X_{i} is projective and τ_{Λ}^{−q}X_{j} is injective. We note first that the cycle X = X_{1} → ⋯ → X_{m} = X is not sectional (see [1, V II.2.6]). Then for any natural numbers r, s such that the modules τ_{Λ}^{r}X_{i} and τ_{Λ}^{−s}X_{i} are nonzero for all i ∈ {1, 2, …, m}, all modules τ_{Λ}^{t}X_{i}, i ∈ {1, 2, …, m}, −s ≤ t ≤ r, lie on one oriented cycle in C. Suppose X_{1}, X_{2}, …, X_{m} are stable modules. Then there exists an integer n such that τ_{Λ}^{n}X = τ_{Λ}^{n}X_{1} belongs to Δ, and so the convexity of Δ in C implies that all modules τ_{Λ}^{t}X_{i}, i ∈ {1, …, m}, t ∈ Z, belong to Δ and obviously lie on oriented cycles, a contradiction because Δ is almost directed. Hence, without loss of generality, we may assume that one of the modules X_{1}, X_{2}, …, X_{m} is not left stable. Let p ≥ 0 be the minimal natural number such that τ_{Λ}^{p}X_{i} is projective for some i ∈ {1, …, m}. Assume all modules X_{1}, X_{2}, …, X_{m} are right stable. Since Δ intersects the τ_{Λ} −orbit of X_{i}, there exists an integer l ≤ p such that τ_{Λ}^{l}X_{i} ∈ Δ. But then all modules τ_{Λ}^{t}X_{j}, j ∈ {1, …, m}, t ≤ p, belong to Δ and lie on oriented cycles, again a contradiction because Δ is almost directed. Therefore there is a minimal q ≥ 0 such that τ_{Λ}^{−q}X_{j} is injective for some j ∈ {1, …, m}. Finally, observe that there are nonsectional paths τ_{Λ}^{−q}X_{j} → ⋯ → X and X → ⋯ → τ_{Λ}^{p}X_{i}, and consequently all modules X = X_{1}, X_{2}, …, X_{m} lie in Δ_{c}.

(iii) Let M be an indecomposable module in C and O the τ_{Λ} − orbit of M. Since Δ is a multisection in C, we have m =| Δ ∩ O |≥ 1. Let Δ ∩ O = {τ_{Λ}^{m−1}X, …, X}. Assume first that m ≥ 2. Then it follows from the minimality of Δ that there are sectional paths I → ⋯ → τ_{Λ}^{m−1}X and X → ⋯ → P with I injective and P projective. Hence {τ_{Λ}^{m−2}X, …, X} ⊆

Δr^{′} and {τ_{Λ}^{m−1}X, …, τ_{Λ}X} ⊆ Δl^{′} , and τ_{Λ}^{m−2}X ∈ Δr^{′′}, τ_{Λ}X ∈ Δ′′_{l}. Therefore

τ_{Λ}^{m−1}X ∈ Δ_{l}, X ∈ Δ_{r}, and {τ_{Λ}^{m−2}X, …, τ_{Λ}X} ⊆ Δ_{c}.

Assume now m = 1. Observe that if X ∉ Δl^{′} and X ∉ Δr^{′} then X ∈ Δl^{′} ∩ Δ_{r}. Suppose X ∈ Δl^{′}. Then X ∉ Δr^{′}, because Δ ∩ O = {X}. But then τ_{Λ}^{−}X ∉ Δ, and so X ∈ Δ_{l} and τ_{Λ}^{−}X ∈ Δ_{r}. Similarly, if X ∈ Δr^{′} then τ_{Λ}X ∈ Δ_{l} and X ∈ Δ_{r}. Summing up, we conclude that the module M lies in Δ_{c} or is a predecessor of a module from Δ_{l} or a successor of a module from Δ_{r}.

THEOREM 2.2. Let C be a component of Γ_{Λ}. Then C is almost directed if and only if C admits a multisection Δ.

Proof. The sufficiency part follows from Proposition 2.1(ii). We shall prove the necessity part. Assume that C is almost directed. In order to prove that C admits a multisection it is enough to show that there exists a full connected valued subquiver Σ of C satisfying the conditions (1) − (4). Indeed, then any minimal full convex valued subquiver Δ of Σ satisfying the conditions (1) − (4) is a multisection in C. If C is finite we may take Σ = C. Therefore, assume that C is infinite.

Since C is infinite, then the left stable part C_{l} of C or the right stable part C_{r} of C is infinite. Assume C_{l} is infinite and let D be an infinite component of C_{l}. We claim that D is directed. Suppose to the contrary that D contains an oriented cycle. If D is stable, then applying [18] we conclude that D is a stable tube and consequently contains infinitely many modules on oriented cycles in D, a contradiction because C is almost directed. Thus assume D contains an injective module. Then it follows from [9, (2.3)] that there exists an infinite sectional path

⋯ → τ_{Λ}^{2t}X_{1} → τ_{Λ}^{t}X_{s} → ⋯ → τ_{Λ}^{t}X_{1} → X_{s} ⋯ → X_{1}

with t > s such that X_{1}, X_{2}, …, X_{s} is a complete set of representatives of the τ_{Λ} − orbits in D. But then again there are infinitely many modules in D, and hence in C, lying on oriented cycles. Therefore, D is directed. Applying now [9, (3.4)] we obtain that there exists a connected valued quiver Ω without oriented cycles such that D is isomorphic to a full translation subquiver of ZΩ which is closed under predecessors. In particular, D admits a section isomorphic to Ω. Similarly, we prove that if E is an infinite component of C_{r} then E is directed and admits a section Θ.

Let D_{1}, …, D_{p} be the family of all infinite components of C_{l} and E_{1}, …, E_{q} be the family of all infinite components of C_{r}. It follows from our discussion above that D_{1}, …, D_{p}, E_{1}, …, E_{q} are directed translation quivers and admit respectively sections Ω_{1}, …, Ω_{p}, Θ_{1}, …, Θ_{q}. Further, let O_{1}, …, O_{m} be all finite τ_{Λ} − orbits in C. Choose a finite family X of modules in C intersecting each of the quivers Ω_{1}, …, Ω_{p}, Θ_{1}, …, Θ_{q}, O_{1}, …, O_{m} exactly once. Since C is a connected quiver, for any modules X and X′ from X, there is in C a finite walk X = X_{1} −X_{2} −⋯−X_{t−1} −X_{t} = X′ connecting X and X′. Hence there exists a finite full valued subquiver Y in C such that X ⊆ Y and any two modules Y and Y′ in Y can be connected by a walk consisiting entirely of modules from Y. Let Σ be the convex hull of the full valued subquivers Ω_{1}, …, Ω_{p}, Θ_{1}, …, Θ_{q}, O_{1}, …, O_{m}, and Y in C. Then Σ satifies the required conditions (1) − (4). This finishes the proof.

As a consequence we obtain the following

Corollary 2.3 Let C be a component of Γ_{Λ}. Then C is directed if and only if C admits a directed multisection Δ.

Our next aim is to introduce a numerical invariant w(A) of a multisection Δ in a component C of Γ_{Λ}, called the width of Δ. We write the path p as an ordered sequence of modules associated with the vertices. A subpath (M, Z^{(1)}, τ_{Λ}^{−1}M, Z^{(2)}, τ_{Λ}^{−2}M, …, Z^{(n)}, τ_{Λ}^{−n}M) is called a hook path of length n (if n ≥ 1), and it is a maximal hook path if it is not contained in any hook path of larger length. Associated with the path p is the following sequence of maximal hook paths (if there are any hook paths). Start with a maximal hook path (M, Z^{(1)}, τ_{Λ}^{−1}M, Z^{(2)}, τ_{Λ}^{−2}M, …, Z^{(n)}, τ_{Λ}^{−n}M), where M is the first module on p which is the start of a hook subpath of p. Then take the maximal hook subpath of p with the start at the first possible successor of τ_{Λ}^{−n}M on p, etc. Denote by i(p) the sum of the length of these hook subpaths of p. In particular we have i(p) = 0 if and only if p is sectional. We define the width w(Δ) of Δ to be the maximum of i(p) + 1 for all paths p in Δ. Observe that if Δ is a section [9] (respectively, double section [11]) in C then w(Δ) = 1 (respectively, w(Δ) = 2). Observe also that Δ is directed if and only if w(Δ) < ∞. We say that a multisection Δ in C is an n−section if n = w(Δ).

The following examples show that for any n ∈ N ∪ {∞} there exists an AR-component C having a multisection Δ with w(Δ) = n.

EXAMPLE 2.4. Let Λ the bound quiver algebra KQ/I, where KQ is the path algebra of the quiver

Q : 1 → 2 → ⋯ → n → n + 1 → n + 2

over a field K and I the ideal in KQ generated by all paths in Q of length 2. Then the AR-quiver Γ_{Λ} of Λ is of the form

S_{n+2} → P_{n+2} → S_{n+1} → P_{n+1} → ⋯ → P_{2} → S_{1} where S_{i} and P_{i}, 1 ≤ i ≤ n + 2, denote the simple and indecomposable projective module at the vertex i, respectively. Then the full subquiver Δ of Γ_{Λ} formed by all vertices of Γ_{Λ} except S_{n+2} and S_{1} is a multisection with w(Δ) = n. We also note that gl.dim Λ = n + 1.

EXAMPLE 2.5. Let K be a field and Λ = K[x]/(x^{2}). Then Γ_{Λ} is of the form

S ⇄ Λ

where S = Λ/ rad Λ, Δ = Γ_{Λ} is a multisection in Γ_{Λ}, w(Δ) = ∞ and gl.dim Λ = ∞.

Our next example shows that an almost directed AR-component admits usually many multisections.

EXAMPLE 2.6. Let Λ be the bound quiver algebra KQ/I, where KQ is the path algebra of the quiver

1\underrightarrow{α}2\underrightarrow{β}3\underleftarrow{γ}4\underleftarrow{σ}5\underleftarrow{δ}6\underleftarrow{ε}7\underleftarrow{η}8

over a field K and I is the ideal in KQ generated by γσ, σδ and δε. Then Γ_{Λ} is of the form

where S_{i}, P_{i}, I_{i}, 1 ≤ i ≤ 8, denote the simple, projective, injective module given by the vertex i. Then the following families of modules form all multisections of Γ_{Λ}:

Δ^{(1)} = X ∪ {P_{4}, rad I_{3}, I_{6}}, Δ^{(2)} = X ∪ {P_{4}, rad I_{3}, S_{8}}, Δ^{(3)} = X ∪ {S_{2}, rad I_{3}, I_{6}}, Δ^{(4)} = X ∪ {S_{2}, rad I_{3}, S_{8}},

Δ^{(5)} = X ∪ {S_{2}, I_{2}, I_{6}}, Δ^{(6)} = X ∪ {S_{2}, I_{2}, S_{8}},

Δ^{(7)} = X ∪ {S_{1}, I_{2}, I_{6}}, Δ^{(8)} = X ∪ {S_{1}, I_{2}, S_{8}},

where X = {I_{3}, S_{4}, P_{5}, S_{5}, P_{6}, S_{6}, P_{7}}. Observe that, for each i ∈ {1, 2, …, 8}, we have w (Δ^{(i)}) = 3 and Δ_{c}^{(i)} = {S_{5}}.

The following general fact shows that the width and the core of a multisection of an almost directed component C are in fact invariants of C.

PROPOSITION 2.7. Let C be a component of Γ_{Λ} and Δ, Σ multisections in C. Then w(Δ) = w(Σ) and Δ_{c} = Σ_{c}.

Proof. Applying arguments as in the proof of (iii) in Proposition 2.1, we obtain that if O is a τ_{Λ} − orbit in C with | Δ ∩ O |≥ 2 (respectively, | Σ∩O |≥ 2), then O ⊆ Δ∩Σ and consequently Δ_{c} = Σ_{c}. Then the equality w(Δ) = w(Σ) also easily follows.

4 Generalized double tilted algebras

We introduce here the class of generalized double tilted algebras, containing the classes of tilted algebras [6], double tiltled algebras [11], and all algebras of finite representation type.

A connected artin algebra Λ is said to be a generalized double tilted algebra if the following conditions are satisfied:

(1) Γ_{Λ} admits a component C with a faithful multisection Δ.

(2) There exists a tilted factor algebra Λ_{l} of Λ (not necessarily connected) such that Λ_{l} is a disjoint union of sections of connecting components of the connected parts of Λ_{l}, and the category of all predecessors of Λ_{l} in ind Λ coincides with the categories of all predecessors of Λ_{l} in ind Λ_{l}.

(3) There exists a tilted factor algebra Λ_{r} of Λ (not necessarily connected) such that Λ_{r} is a disjoint union of sections of connecting components of the connected parts of Λ_{r}, and the category of all successors of Δ_{r} in ind Λ coincides with the categories of all successors of Δ_{r} in ind Λ_{r}.

In the above notation, Λ_{l} and Λ_{r} are said to be the left tilted algebra and the right tilted algebra of Λ, respectively. Moreover, C is said to be a connecting component of Λ. If Δ is a section of C then Λ_{l} = Λ = Λ_{r} and hence Λ is a tilted algebra. Further, if w(Δ) ≥ 2, then C is the unique connecting component of Λ. We say a connected artin algebra Λ is an n−double tilted algebra if Γ_{Λ} admits a component C with an n−section Δ satisfying the above conditions (1) − (3). It follows from Proposition 2.7 and properties of tilted algebra that n = w(Δ) does not depend on the choice of Δ and C. Observe that 1− double tilted algebras and 2− double tilted algebras are exactly tilted algebras [6] and double tilted algebras [11], respectively. Finally, we note that every connected artin algebra of finite representation type is a generalized double tilted algebra.

We have the following characterization generalized double tilted algebra.

THEOREM 3.1. Let Λ be a basic connected artin algebra. The following conditions are equivalent:

(i) Λ is generalized double tilted. (ii) Γ_{Λ} admits a faithful generalized standard almost directed component. (iii) Γ_{Λ} admits a component C with a faithful multisection Δ such that Hom_{1}(X, τ_{Λ}Y) = 0 for all modules X from Δ_{r} and Y from Δ_{l}.

Proof. It is a straightforward extension of (invoking Theorem 2.2) of arguments applied in the proofs of [13, Theorem 2] and [11, Theorem 7.3]), where similar characterizations of tilted and double tilted algebras have been established .

We obtain the following consequences.

COROLLARY 3.2 Let Λ be a basic connected artin algebra. Then Λ is n−double tilted, n ≥ 2, if and only if Γ_{Λ} admits a faithful generalized standard almost directed component with a nonsectional path from an injective module to projective module.

COROLLARY 3.3 Let Λ be a basic connected artin algebra. Then Λ is n−double tilted, n ≥ 3, if and only if Γ_{Λ} admits a faithful generalized standard component with a multisection Δ such that Δ_{c} ≠ ∅.

We have also the following results on the structure of category of a generalized double tilted algebra of infinite representation type.

THEOREM 3.4 Let Λ be a basic connected n−double tilted algebra of infinite representation type which is not tilted, and let C be the connecting component of Γ_{Λ}. Then there are a hereditary artin algebra _{∞}H and a tilting _{∞}H−module T_{∞} without nonzero preprojective directed summands such that the following statements hold:

(i) The tilted algebras _{∞}Λ = End_{_{∞}H} (_{∞}T)^{op} and Λ_{∞} = End_{H_{∞}} (T_{∞})^{op} are factor algebras of Λ. (ii) The torsion-free part Y(_{∞}T) of mod_{∞} Λ is a full exact subcategory of mod Λ which is closed under predecessors of indecomposable modules. (iii) The torsion part X (T_{∞}) of mod Λ_{∞} is a full exact subcategory of mod Λ which is closed under successors of indecomposable modules. (iv) _{∞}Λ is a factor algebra of _{l}Λ and _{∞}C = Y(_{∞}T) ∩ C is the torsion-free part of the family C_{_{∞}T} of the connecting component of _{∞}Λ and also a full translation subquiver of C which is closed under predecessors in C.

(v) Λ_{∞} is a factor algebra of Λ_{r} and C_{∞} = X (T_{∞}) ∩ C in the torsion part of the family C_{T_{∞}} of the connecting component of Λ_{∞} and also a full translation subquiver of C which is closed under predecessors in C. (vi) Y(_{∞}T) and X (T_{∞}) have no common nonzero modules. (vii) The family of indecomposable Λ−modules which are neither in Y(_{∞}T) nor in X (T_{∞}) is finite and coincides with the family of indecomposable modules in C which are neither in _{∞}C nor in C_{∞}.

Proof. It is a direct extension of arguments applied in the proof of [13, Theorem 1], where the structure of the module categories of algebras having faithful generalized standard directed components has been established .

We note that if the left tilted algebra Λ_{l} (respectively, the right tilted algebra Λ_{r}) of Λ has a finite torsion-free part Y(T_{l}) (respectively finite torsion part X (T_{r})) then _{∞}Λ (respectively Λ_{∞}) is zero. The known structure of AR-components of tilted algebras (see [6], [7], [8], [10], [12], [17]) and the above theorem lead to the following description of the AR-quivers of nontilted generalized double tilted algebras of infinite representation type.

COROLLARY 3.5. Let Λ be a basic connected generalized double tilted algebra and assume Λ is not tilted and of infinite representation type. Then we have the following.

(i) Every component of Γ_{Λ} different from the the connecting component either lies entirely in Y(_{∞}T) or lies entirely in X (T_{∞}). (ii) Every component of Γ_{Λ} contained in Y(_{∞}T) is either preprojective, a stable tube ZA_{∞}/ (τ^{m}), for m ≥ 1, of the form ZA_{∞} , or can be obtained from a stable tube or a component of type ℤA_{∞} by a finite number of ray insertions. (iii) Every component of Γ_{Λ} contained in X (T_{∞}) is either preinjective, a stable tube ℤA_{∞}/ (τ^{m}), for m ≥ 1, of the form ZA_{∞} , or can be obtained from a stable tube or a component of type ZA_{∞} by a finite number of coray insertions.

We illustrate the above considerations by the following example.

EXAMPLE 3.6. Let Q be a quiver

KQ the path algebra of Q over a field K, I the ideal in KQ generated by all paths in Q of length 2, and Λ = KQ/I. Then Γ_{Λ} admits a component C of the form

Observe that the full convex subquiver Δ of C given by the modules P_{2}, S_{2}, P_{3}, S_{3}, I_{5}, S_{5}, P_{6}, I_{3}, I_{5}, S_{6}, P_{7}, S_{7}, P_{8}, P_{9}, P_{10}, P_{11} is the unique multisection in C and w(Δ) = 5 = i(p) + 1, for p being the path S_{2} → P_{3} → S_{3} → P_{6} → I_{5} → S_{6} → P_{7} → S_{7}. Further, Δ′_{l} (respectively Δ′_{r}) is the subquiver given by the modules P_{2}, S_{2}, P_{3}, S_{3}, I_{4}, S_{5}, P_{6}, I_{3}, I_{5}, S_{6} (respectively

P_{3}, S_{3}, P_{6}, I_{3}, I_{5}, S_{6}, P_{7}, S_{7}, P_{8}, P_{9}, P_{10}, P_{11}), and hence Δ_{c} = Δl^{′} ∩ Δr^{′′} is the

subquiver given by P_{3}, S_{3}, P_{6}, I_{3}, I_{5}, S_{6}. Moreover, Δl^{′′} (respectively Δr^{′′}) is given by the modules P_{2}, P_{3}, I_{4}, P_{6}, I_{3}, I_{5}, S_{6} (respectively P_{3}, S_{3}, P_{6}, I_{3}, P_{7}, P_{8}, P_{9}, P_{10}, P_{11}). Hence Δ_{l} = (Δ\Δl^{′}) ∪ τ_{Λ}Δr^{′′} is given by the modules P_{2}, I_{2}, I_{4}, S_{5} and Δ_{r} = (Δ\Δ′_{l}) ∪ τ_{Λ}^{−}Δ′′_{l} is given by the modules P_{7}, S_{7}, P_{8}, P_{9}, P_{10}, P_{11}. Therefore, the left tilted algebra Λ_{l} of Λ is the product H_{1} × H_{2}, where H_{1} is the path algebra of the subquiver of Q given by 1 and 2 and H_{2} is the path algebra of the subquiver of Q given by 4 and 5, and the right tilted algebra Λ_{r} of Λ is the bound quiver algebra KΣ/J with Σ being the full subquiver of Q given by the vertices 6, 7, 8, 9, 10, 11 and J the ideal in KΣ generated by all paths in Σ of length 2. Finally, _{∞}Λ = H_{2} and Λ_{∞} is the path algebra KΩ of the full subquiver Ω of Q given by the vertices 7, 8, 9, 10, 11. Thus the AR-quiver Γ_{Λ} of Λ consists of a preprojective component, an infinite family of components of type ZA_{∞}, the connecting component C, a preinjective component and an infinite family of stable tubes (3 of them of the form ℤA_{∞}/ (τ^{2}) and the remaining ones of the form ZA_{∞}/ (τ)).

5 Global dimension

Let Λ be a generalized double tilted algebra, C a connecting component of Γ_{Λ} and Δ a faithful multisection in C. Then every predecessor of Λ_{l} in ind Λ has injective dimension at most one. In particular, for all but finitely many indecomposable Λ− modules X we have pd_{Λ} X ≤ 1 or id_{Λ} X ≤ 1. On the other hand as we have seen in Section 2, there are generalized double tilted algebras of arbitrary global dimension. The aim of this section is to establish a bound for the global dimension of generalized double tilted algebra. We need some notations.

For an indecomposable module N in the multisection Δ, we denote by i(N) the maximum of the numbers i(p) , for all paths in Δ ending at N as defined in Section 2. We note that if Δ is a section then i(N) = 0 for all modules N in Δ. We have the following easy observation.

LEMMA 4.1. Let N be a module in Δ with i(N) > 0. Then there exists a path p in Δ starting with an injective module and ending at N such that i(N) = i(p).

Proof. Since i(N) > 0, there exists a path p′ in Δ ending at N such that i(p′) = i(N). Let M, Z^{(1)}, τ_{Λ}^{−1}M, …, Z^{(t)}, τ_{Λ}^{−t}M be the first maximal hook subpath of p′. Then by the minimality of Δ there is some sectional path in Δ from an injective module to M, and composing it with the path p′ we obtain the required path p.

THEOREM 4.2. Let Λ be a n−double tilted algebra. Then gl.dim Λ ≤ n + 1.

Proof. Let Δ be n−section in connecting component C of Γ_{Λ}. We first prove that for any indecomposable module N in Δ we have pd_{Λ} N ≤ i(N)+1. Let N be a module from Δ. Assume first i(N) = 0. Then any path in Δ ending at N is sectional. Since every injective predecessor of a module from C in ind Λ lies in C, this implies that Hom_{Λ}(D(Λ), τ_{Λ}N) = 0, and consequently pd_{Λ} N ≤ 1 (see [1,]) because otherwise there would be a path in Δ of the form I → ⋯ → τ_{Λ}N → X → N, contradicting i(N) = 0. Assume i(N) > 0. Consider the exact sequence

O → ΩN → P_{N} → N → 0

where P_{N} → N is the projective cover of N, and let P_{N} = P_{1} ⊕ ⋯ ⊕ P_{t} for some indecomposable modules P_{i}. Let L be an indecomposable direct summand of ΩM. Assume L belongs to Δ. Then they path L → P_{i} → N can be refined to a path of irreducible morphisms. Since the sum of these paths is zero, not all of them are sectional (see [1,]), and consequently i(L) < i(N). Then by the inductive assumption we have pd_{Λ} L ≤ i(L) + 1. If L is not in Δ, then L is a predecessor of Δ_{l} in ind Λ and hence pd_{Λ} L ≤ 1. Therefore we obtain pd_{Λ} N ≤ i(N) + 1.

Let X be an arbitrary indecomposable Λ−module, and let Y be an indecomposable direct summand of ΩX. Then Y is a predecessor of an indecomposable projective module in ind Λ, and hence lies in Δ or is a predecessor of Δ_{l} in Λ. In the first case we have pd_{Λ} X ≤ pd_{Λ} Y +1 ≤ i(Y)+2 ≤ w(Δ)+1, and in the second case pd_{Λ} X ≤ pd_{Λ} Y + 1 ≤ 2. In any case we have gl.dim Λ ≤ w(Δ) + 1 = n + 1.

Note that 1−double tilted algebras are tilted algebras where we know that the global dimension is at most 2 [6]. The 2−double tilted algebras are the strict shod algebras [2] which have global dimension at most 3. Examples 2.4 and 2.5 show that there are n−double tilted algebras Λ (for any n ∈ ℕ ∪ {∞}) with gl. dim Λ = n + 1. But in general it is not the case.

EXAMPLE 4.3. Let Λ = KQI, where Q is the quiver

and I is the ideal in the path algebra KQ of Q (over a field) generated by all paths of length 2. Then Λ is an ∞−double tilted algebra with gl.dim Λ = 4 and Γ_{Λ} admits a connecting component of the form

Moreover, both tilted algebras Λ_{l} and Λ_{r} are of infinite representation type.

EXAMPLE 4.4. Let Λ = KQ/I, where Q is the quiver

and let I be the ideal in the path algebra KQ (over a field K) generated by the paths αγ, βα, σξ, ηω and ην. Then Λ is a 4−double tilted algebra, gl.dim Λ = 2, and Γ_{Λ} admits a connecting component of the form

6 Standard almost directed components

The aim of this section is to prove a homological characterization of faithful generalized standard almost directed components.

Let Λ be an artin algebra. Following [5], we define two subcategories L_{Λ} and R_{Λ} of ind Λ as follows. The category L_{Λ} is formed by all modules X in ind Λ such that for every predecessor Y of X in ind Λ we have pd_{Λ} Y ≤ 1. Dually, R_{Λ} is formed by all modules X in ind Λ such that for every successor Y of X in ind Λ we have id_{Λ} Y ≤ 1. It is known that [2, (2.1)] that ind Λ = L_{Λ} ∪ R_{Λ} if and only if for every indecomposable Λ−module X we have pd_{Λ} X ≤ 1 or id_{Λ} X ≤ 1 (Λ is a shod algebra). Moreover, if Λ is basic connected, then Λ is shod if and only if Λ is tilted, double tilted or quasitilted of canonical type (see [6], [11], [4]). In general for a generalized double tilted algebra Λ we have ind Λ\ (L_{Λ} ∪ R_{Λ}) finite but not necessarily empty, as the examples presented in the previous sections show. In fact it has been proved in [16] that L_{Λ} ∪ R_{Λ} is cofinite in ind Λ if and only if Λ is quasitilted or generalized double tilted.

Let Λ be an artin algebra and C a component of Γ_{Λ}. We define two full translation subquivers L_{C} and R_{C} in C as follows. The quiver L_{C} is formed by all modules X in C such that for any path Y = Y_{0} → Y_{1} → ⋯ → Y_{t} = X in ind Λ with Y_{1}, Y_{2}, …, Y_{t} from C we have pd_{Λ} Y ≤ 1. Dually, R_{C} is formed by all modules X in C such that for every path X = Z_{0} → Z_{1} → ⋯ → Z_{m} = Z in ind Λ with Z_{1}, Z_{2}, …, Z_{m} from C we have id_{Λ} Z ≤ 1. Observe that L_{C} is closed under predecessors, R_{C} is closed under successors in C, and Δ_{c} ⊆ C\ (L_{C} ∪ R_{C}).

THEOREM 5.1. Let Λ be a basic connected artin algebra, C a faithful component of Γ_{Λ} with a multisection Δ , and assume that C contains both a projective module and an injective module. Then C is generalized standard if and only if there is a decomposition C = L_{C} ∪ Δ_{c} ∪ R_{C}.

Proof. Assume C is a generalized standard. Then, by Theorem 3.1 Λ is generalized tilted algebra. Moreover, by Proposition 2.1 every indecomposable module in C belongs to Δ_{c}, is a predecessor of Δ_{l} or a successor of Δ_{r} in C. Since Λ is generalized double tilted, all predecessors of Δ_{l} belong to L_{Λ} ∩ C ⊆ L_{C} and R_{Λ} ∩ C ⊆ R_{C}. Therefore, the required decomposition C = L_{C} ∪ Δ_{c} ∪ R_{C} holds.

Assume now that C = L_{C} ∪ Δ_{c} ∪ R_{C}. In order to prove that C is generalized standard, it is enough to show that Hom_{Λ}(X, τ_{Λ}Y) = 0 for all modules X from Δ_{r} and Y from Δ_{l} (see Theorem 3.1). Suppose there exist X ∈ Δ_{r} and Y ∈ Δ_{l} such that Hom_{Λ}(X, τ_{Λ}Y) ≠ 0. Let D be the full translation subquiver of C formed by all predecessors of Δ_{l} in C and E the full translation subquiver of C formed by all successors of Δ_{r} in C. Since by Proposition 2.1, all oriented cycles of C are entirely contained in Δ_{c}, the translation quivers D and E are directed, and obviously disjoint. In particular, there is no path in C from X to τ_{Λ}Y. Since Hom_{Λ}(X, τ_{Λ}Y) ≠ 0 then there exists an infinite path

X = X_{0} → X_{1} → ⋯ → X_{i} → X_{i+1} → …

in C such that Hom_{Λ}(X, τ_{Λ}Y) ≠ 0 for all i ≥ 0. We claim that there are i ≥ 0 and a path in C of the form

I → τ_{Λ}Z → v → Z → ⋯ → X_{i}

with I injective. It follows from our assumption on C that C contains at least one injective module. Further, since Δ intersects any τ_{Λ} − orbit of C, every indecomposable injective module from C lies in Δ or in E. Moreover, every indecomposable projective module from C lies in Δ or in D. Invoking now the fact that E is directed we conclude that all but finitely many modules X_{i}, i ≥ 0, belong to one connected component Γ of the right stable part E_{r} of E, and there is a path in C of the form

I → τ_{Λ}Z → V → Z → ⋯ → U

with I injective and U in Γ. If Γ has only finitely many orbits then clearly there is in Γ a path U → ⋯ → X_{i}, and consequently the required path

I → τ_{Λ}Z → V → Z → ⋯ → X_{i}.

Assume Γ has infinitely many orbits. Since Γ is directed then there are j ≥ 0, m ≥ 0 and an infinite path in Γ of the form

U → ⋯ → τ_{Λ}^{−m}X_{j} → τ_{Λ}^{−m}X_{j+1} → τ_{Λ}^{−m}X_{j+2} → …

Applying now [3, Lemma 1.5] (see also [15, Lemma 4]) we infer that there are i ≥ j and a path in ind Λ of the form τ_{Λ}^{−m}X_{i} → ⋯ → X_{i}, consisting of modules from Γ, and we obtain a required path

I → τ_{Λ}Z → V → Z → … U → ⋯ → τ_{Λ}^{−m}X_{i} → ⋯ → X_{i}

Since Hom_{Λ}(X, τ_{Λ}Y) ≠ 0 and there is no path in C from X_{i} to τ_{Λ}Y, there exists an infinite path

⋯ → Y_{t+1} → Y_{t} → ⋯ → Y_{1} → Y_{0} = τ_{Λ}Y

in C such that Hom_{Λ}(X_{i}, Y_{t}) ≠ 0 for all t ≥ 0. Invoking our assumption that C contains at least one projective module and applying dual arguments to those above, we conclude that there are t ≥ 0 and a path ind Λ of the form

Y_{t} → ⋯ → N → W → τ_{Λ}^{−}N → P

with P projective and consisting of modules from C. Observe that pd_{Λ} Z ≥ 2 and id_{Λ} N ≥ 2 because Hom_{Λ}(I, τ_{Λ}Z) ≠ 0 and Hom_{Λ}(τ_{Λ}^{−}N, P) ≠ 0. Therefore, Y_{t} belongs to C\ (L_{C} ∪ Δ_{c} ∪ R_{C}), which contradicts the assumption. This finishes the proof.

ACKNOWLEDGEMENTS.

The research described in this paper was made during the visit of both authors in the Centre for Advance Study of the Norwegian Academy of Science and Letter in Oslo (August / September 2001) and supported by the project Noncommutative Phenomena in Mathematics and Theoretical Physics. The authors thank the Centre for the warm hospitality.

7 References

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43(1991), 652 − 672

Idun Reiten  Andrzej Skowronski
Departement of Mathematical Sciences  Faculty of Mathematics and Informatics
Norwegian University of Science and Technology Nicholas Copernicus University
N−7491 Trondheim  Chopina 12/18, 87 − 100 Torun
Norway  Poland
E-mail: idun@math.ntnu.no  E-mail:skowron@mat.uni.torun.p