LEFT SECTIONS AND THE LEFT PART OF AN ARTIN ALGEBRA

IBRAHIM ASSEM

Introduction

Let A be an artin algebra. We are interested in studying the representation theory of A, thus the category mod A of finitely generated right A-modules. For this purpose, we fix a full subcategory ind A of mod A having as objects exactly one representative from each isomorphism class of indecomposable modules. Following Happel, Reiten and Smalø [19], we define the left part ℒ_{A} to be the full subcategory of ind A with objects those modules whose predecessors have projective dimension at most one. The right part is defined dually. These classes, whose definition suggests the interplay between homological properties of an algebra and representation theoretic ones, were heavily investigated and applied (see, for instance, the survey [4]).

The initial motivation for this paper comes from the observations, made in [5, 2, 1], that the left part of an arbitrary artin algebra closely resembles that of a tilted algebra. Tilted algebras, introduced by Happel and Ringel in [20], are among the most important and best understood classes of algebras. Many criteria allow to recognise whether a given algebra is tilted or not. Most of them revolve around the existence of a combinatorial configuration, called “complete slice” or “section” inside the module category, see [20, 15, 22, 13, 28, 29, 7]. Perhaps the most efficient is the Liu-Skowroński criterion: they define (combinatorially) a so-called section in an Auslander-Reiten component and prove that, if there exists a section satisfying reasonable algebraic conditions, then the algebra is tilted (see [26, 30] or [9, (Chapter VIII)]). Surprisingly, however, as is shown in [1], none of the known criteria seems to apply directly to the tilted algebras arising from the study of the left part.

The first aim of this paper is to derive a more suitable version of the Liu-Skowroński criterion, easier to apply in our case. For this purpose, we define a notion of left section in a translation quiver by weakening one of the Liu-Skowroński axioms for section (see (2.1)). Several known results for sections carry over to left sections, sometimes in a restricted form (see, for instance, (2.2) and (3.2)). We thus obtain our first main theorem.

Theorem A. Let A be an artin algebra, and Σ be a left section in a component Γ of the Auslander-Reiten quiver of A such that Hom_{A} (τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ in Σ, then A/Ann_{A}Σ is a tilted algebra having Σ as complete slice.

If Σ is a section, then the condition that Hom_{A}(τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ in Σ is equivalent to several other conditions, notably that the component Γ which contains it is generalised standard (see [26, 30]). This is not true for left sections. However, if Σ is a left section, then this condition implies (but is not equivalent to say) that the full translation subquiver Γ_{⩽Σ} of Γ consisting of the predecessors of Σ in Γ is generalised standard.

As corollaries of the above theorem, we obtain, not only the Liu-Skowroński criterion, but also the statements necessary for the study of the left part. If, in particular, Σ is a left section which is convex in ind A, then the condition of the theorem is satisfied so A/Ann_{A}Σ is tilted (4.3). Also, if A is an algebra over an algebraically closed field, then A/Ann_{A}Σ coincides with the support algebra of Σ, which is a full convex subcategory of A, see (4.5).

We next apply our criterion to the study of the left part. As shown in [5, 6, 2, 1], the main tool in the proofs of the known results is the description of the Ext-injectives (in the sense of [12]) in the left part ℒ_{A}. Here, we rather work with a full subcategory 𝒞 of ℒ_{A} which is closed under predecessors, and we prove that the most useful statements about Ext-injectives in ℒ_{A} carry over to this context. This approach allows to work with connected subcategories of ℒ_{A} (which is not connected in general). Also, this hypothesis is optimal: easy examples show that the known techniques about ℒ_{A} do not carry over to subcategories closed under predecessors which are not contained in ℒ_{A} (this more general situation is addressed in a forthcoming work with Coelho and Trepode). This leads to our second main theorem.

Theorem B. Let A be an artin algebra, and 𝒞 ⊆ ℒ_{A} be a full subcategory closed under predecessors, having ℰ as subcategory of Ext-injectives. Let Γ be a component of the Auslander-Reiten quiver of A. Then:

(a) If Γ ∩ ℰ = ∅, then either Γ ⊆ 𝒞 or Γ ∩ 𝒞 = ∅.

(b) If Σ = Γ ∩ ℰ ≠ ∅, then Σ is a left section of Γ, convex in ind A. Moreover, A/Ann_{A}Σ is a tilted algebra having Σ as complete slice.

As corollaries, we obtain the first two main results of [1]. Following this line, we define the support algebra of a subcategory 𝒞 as above, thus generalising the notion of left support algebra [5, 32]. As a consequence of the theorem, we describe completely the Auslander-Reiten components which lie entirely inside 𝒞 and, for those which intersect 𝒞, the part which precedes the left section Σ. This is contained in (7.4)(7.5)(7.6), which generalise the remaining results of [1]. In the last section, we introduce a new class of algebras, called 𝒞 -supported, modeled after the left supported algebras of [5] and we obtain, in (8.2)(8.8), generalisations of the results of [5, 2].

Clearly, the dual results, for right sections and the right part, hold as well. For the sake of brevity, we refrain from stating them, leaving the primal-dual translation to the reader.

1. Preliminaries

1.1. Notation. Throughout this paper, all our algebras are basic and connected artin algebras. For an algebra A, we denote by mod A its category of finitely generated right modules and by ind A a full subcategory of mod A consisting of one representative from each isomorphism class of indecomposable modules. Whenever we speak about a module (or an indecomposable module), we always mean implicitly that it belongs to mod A (or to ind A, respectively). Also, all subcategories of mod A are full and so are identified with their object classes. We sometimes consider an algebra A as a category, in which the object class A_{0} is a complete set {e_{1}, e_{2}, …, e_{n}} of primitive orthogonal idempotents and the set of morphisms from e_{i} to e_{j} is e_{i}Ae_{j}. An algebra B is a full subcategory of A if there is an idempotent e ∈ A, sum of some of the distinguished idempotents e_{i}, such that B = eAe. It is convex in A if, for any sequence e_{i} = e_{i_{0}}, e_{i_{1}}, …, e_{i_{t}} = e_{j} of objects in A such that e_{i_{ℓ}}Ae_{i_{ℓ+1}} ≠ 0 (with 0 ≤ ℓ < t) and e_{i}, e_{j} objects in B, then all e_{i_{ℓ}} lie in B. We denote by P_{x} (or I_{x}, or S_{x}) the indecomposable projective (or injective, or simple, respectively) A-module corresponding to the idempotent e_{x}.

A subcategory 𝒞 of ind A is called finite if it has only finitely many objects. We sometimes write M ∈ 𝒞 to express that M is an object in a subcategory 𝒞. We denote by add 𝒞 the subcategory of mod A with objects the direct sums of summands of modules in 𝒞. Given a module M, we let pd M (or id M) stand for its projective (or injective, respectively) dimension. The global dimension of A is denoted by gl.dim. A and its Grothendieck group by K_{0}(A). For a module M, the support Supp (M, −) (or Supp (−, M)) of the functor Hom_{A}(M, −) (or Hom_{A}(−, M)) is the subcategory of ind A consisting of all X such that Hom_{A}(M, X) ≠ 0 (or Hom_{A}(X, M) ≠ 0, respectively). We denote by Gen M (or Cogen M) the subcategory of mod A having as objects all modules generated (or cogenerated, respectively) by M.

For an algebra A, we denote by Γ(mod A) its Auslander-Reiten quiver and by τ_{A} = DTr , τ_{A}^{−1} = Tr D its Auslander-Reiten translations. For further definitions and facts on mod A or Γ(mod A), we refer to [9, 11]. For tilting theory, we refer to [9].

1.2. Paths. Let A be an algebra. Given M, N ∈ ind A, a path from M to N in ind A (denoted by M ⇝ N) is a sequence of non-zero morphisms

(∗)  \xymatrix{M = X_{0} \ar[r]^{f_{1}} & X_{1} \ar[r]^{f_{2}} & ⋯ \ar[r] & X_{t−1} \ar[r]^{f_{t}} & X_{t} = N}

(t ≥ 1) where X_{i} ∈ ind A for all i. We then say that M is a predecessor of N and N is a successor of M. A path from M to M involving at least one non-isomorphism is a cycle. A module M ∈ ind A which lies on no cycle is directed. If each f_{i} in (∗) is irreducible, we say that (∗) is a path of irreducible morphisms, or path in Γ(mod A). A path (∗) of irreducible morphisms is sectional if τ_{A}X_{i+1} ≠ X_{i−1} for all i with 0 < i < t. A refinement of (∗) is a path in ind A

\xymatrix{M = X′_{0} \ar[r] & X′_{1} \ar[r] & ⋯ \ar[r] & X′_{s−1} \ar[r] & X′_{s} = N}

such that there exists an order-preserving injection σ : {1, …, t − 1} → {1, …, s − 1} satisfying X_{i} = X′_{σ(i)} for all i with 0 < i < t.

A subcategory 𝒞 is closed under predecessors if, whenever M ⇝ N is a path in ind A with N ∈ 𝒞, then M ∈ 𝒞. Equivalently, add 𝒞 is the torsion-free class of a split torsion pair. We define dually subcategories closed under successors, which generate torsion classes of split torsion pairs.

Important examples are the left and right parts of mod A, defined in [19]. The left part is the full subcategory of ind A with object class

ℒ_{A} = {M ∈ ind A | for any L with L ⇝ M, we have pd L ≤ 1} .

Thus, ℒ_{A} is closed under predecessors. The right part ℛ_{A} is defined dually and is closed under successors. For properties of ℒ_{A} and ℛ_{A}, we refer to [4, 19].

2. Left sections in translation quivers

2.1. In this section, (Γ, τ), or briefly Γ, denotes a translation quiver. Given x, y ∈ Γ_{0}, a path from x to y (denoted by x ⇝ y) is a sequence of arrows

(∗)  \xymatrix{x = x_{0} \ar[r] & x_{1} \ar[r] & ⋯ \ar[r] & x_{t−1} \ar[r] & x_{t} = y} .

We say that x is a predecessor of y, or y is a successor of x. If y = x and t ≥ 1, this path is a cycle. A full subquiver Σ of Γ is acyclic if it contains no cycle. It is convex (in Γ) if, for any path (∗) with x, y ∈ Σ_{0}, we have x_{i} ∈ Σ_{0} for all i.

Definition. A full subquiver Σ of a translation quiver Γ is called a left section if: (LS1) Σ is acyclic; (LS2) for any x ∈ Γ_{0} such that there exist y ∈ Σ_{0} and a path x ⇝ y, there exists a

unique n ≥ 0 such that τ^{−n}x ∈ Σ_{0}; (LS3) Σ is convex in Γ.

Examples. (a) A full connected subquiver Σ of Γ is a section (see [25, 30] or else [9]) if it satisfies (LS1), (LS3) and: (S2) For any x ∈ Γ_{0}, there exists a unique n ∈ ℤ such that τ^{n}x ∈ Σ_{0}.

Thus, any section is a left section.

(b) Our second example is the motivating one: let A be an artin algebra, ℒ_{A} be the left part of mod A, ℰ_{A} be the class of indecomposable Ext-injectives in add ℒ_{A} and Γ be a component of Γ(mod A) such that Γ ∩ ℰ_{A} ≠ ∅. Then Γ ∩ ℰ_{A} is a left section in Γ, but generally not a section [1].

(c) Other examples can be found in the directed part of a semiregular tube (or coil) containing projectives. Let A be given by the quiver

\xymatrix{∘_{1} & ∘_{2} \ar[l]_{β} \ar[l] & ∘_{3} \ar[l]_{α} \ar[r] & ∘_{4} \ar[r] & ∘_{5}}

bound by αβ = 0. The projective module P_{5} lies in a tube of the form

[figure: 5; 4; 3 2 1; 2 1; 4 5; 3 2 4 1; 3 2 2 1 1; 3 2 4 1 5; 3 2 2 4 1 1; ·; 2 1; 3 2 2 4 1 1 5; ·; 2 2 1 1; ·; ·]

where modules are represented by their Loewy series and one identifies along the vertical dotted lines. The full subquiver with points \{ \begin{matrix}4 \\ 5 \end{matrix} , ^{4} \} is a left section, but not a section.

Lemma. Let Σ be a left section in a translation quiver Γ. Then:

(a) Σ intersects at most once each τ-orbit in Γ;

(b) every path between two points of Σ is sectional;

(c) if x ∈ Γ_{0} is injective and precedes Σ, then x ∈ Σ_{0};

(d) if x → y, x ∈ Σ_{0} and y is non-projective, then y ∈ Σ_{0} or τy ∈ Σ_{0};

(e) if x → y, y ∈ Σ_{0}, then x ∈ Σ_{0} or τ^{−1}x ∈ Σ_{0};

(f) if x ∈ Σ_{0} and y precedes Σ, then every path from x to y is sectional and y ∈ Σ_{0}.

Proof. (a) Follows from the uniqueness in (LS2).

(b) Follows from (a).

(c) There exists n ≥ 0 such that τ^{−n}x ∈ Σ_{0}. Since x is injective, n = 0.

(d) Since y is non-projective, there is an arrow τy → x. By (LS2), there exists a unique n ≥ 0 such that τ^{−n}(τy) = τ^{1−n}y ∈ Σ_{0}. If n > 1, the path x → y ⇝ τ^{1−n}y and convexity yield y ∈ Σ_{0}. Since τ^{1−n}y ∈ Σ_{0}, this contradicts (a). Hence n ∈ {0, 1}, as required.

(e) This is clear if x is injective. Otherwise, τ^{−1}x is non-projective and we apply (d) to the arrow y → τ^{−1}x.

(f) Since y precedes Σ, there exists n ≥ 0 such that τ^{−n}y ∈ Σ_{0}. Thus, a path x ⇝ y induces a path x ⇝ y ⇝ τ^{−n}y. By convexity, y ∈ Σ_{0}. Hence n = 0 and the sectionality of the path follows from (b).

□

2.2. Lemma. Let Σ be a left section in Γ. The full subquiver Γ_{⩽Σ} of Γ consisting of all predecessors of Σ in Γ is isomorphic to a full translation subquiver of ℤΣ (and, in particular, is acyclic).

Proof. Repeat the proof of [25, (3.2)] (or [9, (VIII.1.5)]) with the obvious changes. □

2.3. We give necessary and sufficient conditions for a left section to be a section.

Proposition. Let Σ be a left section in Γ. The following are equivalent:

(a) Σ is a section;

(b) every projective in Γ precedes Σ;

(c) for any projective p ∈ Γ_{0} such that there exist x ∈ Σ_{0} and a path x ⇝ p, we have p ∈ Σ_{0}.

Proof. (a) implies (b). Since Σ is a section, Γ is fully embedded in ℤΣ, and Σ cuts

each τ-orbit of Γ (see [25, (3.2)] or [9, (VII.1.5)]).

(b) implies (c). This follows from convexity.

(c) implies (a). We must show that Σ cuts each τ-orbit of Γ. For this, it suffices to prove that, if x ∈ Σ_{0} and z ∈ Γ_{0} are in two neighbouring orbits, then Σ cuts the τ-orbit of z (the statement then follows by induction). Assume that there exist m ∈ ℤ and y in the τ-orbit of x such that we have an arrow τ^{m}x → y or y → τ^{m}x. Assume also, without loss of generality, that |m| is minimal. There are three cases:

1) Suppose m > 0. If there is an arrow y → τ^{m}x, then there is a path y → τ^{m}x ⇝ x, so Σ cuts the τ-orbit of y. If, on the other hand, there is an arrow τ^{m}x → y, then there is an arrow y → τ^{m−1}x, contradicting minimality.

2) Suppose m < 0. If there is an arrow y → τ^{m}x, then there is an arrow τ^{m+1}x → y, contradicting minimality. If, on the other hand, there is an arrow τ^{m}x → y, then we have two cases. If y is projective, then the path x ⇝ τ^{m}x → y and the hypothesis imply y ∈ Σ_{0}, hence m = 0, a contradiction. If y is non-projective, then there is an arrow τy → τ^{m}x, hence an arrow τ^{m+1}x → τy, contradicting minimality.

3) Suppose m = 0. If there is an arrow y → x, then y ∈ Σ_{0} or τ^{−1}y ∈ Σ_{0}. If, on the other hand, there is an arrow x → y, then we have two cases. If y is projective, then y ∈ Σ_{0} by hypothesis. If y is non-projective, there is an arrow τy → x which, by (2.1)(e), yields τy ∈ Σ_{0} or y ∈ Σ_{0}.

□

3. Left sections and tilted algebras

3.1. Let A be an artin algebra and Γ be a component of Γ(mod A). Recall that the annihilator of a full subcategory 𝒞 of ind A is defined by Ann_{A}𝒞 = ⋂_{X∈𝒞} Ann_{A}X.

Lemma. If Σ is a finite left section of Γ, then:

(a) Ann_{A}Σ = Ann_{A}Γ_{⩽Σ};

(b) Σ cogenerates Γ_{⩽Σ}.

Proof. (a) Repeat the proof of [26, (2.1)], [30, (Lemma 3)], with the obvious changes.

(b) Let X ∈ Γ_{⩽Σ} and j : X ↪ I be an injective envelope. Since no indecomposable

summand of I is a proper predecessor of Σ, then j factors through Σ.

□

3.2. The following is a “left” version of [26, (1.3)], [30, (Theorem 2)].

Proposition. Let Σ be a left section of Γ. The following are equivalent:

(a) Hom_{A}(E′, τ_{A}E′′) = 0 for all E′, E′′ ∈ Σ_{0};

(b) |Σ_{0}| ≤ rk K_{0}(A) and rad_{A}^{∞}(E′, E′′) = 0 for all E′, E′′ ∈ Σ_{0};

(c) Γ_{⩽Σ} is generalised standard.

Proof. (a) implies (b). The first statement follows from Skowroński’s lemma [9, (VIII.5.3)], [31, (Lemma 1)], and the second from the fact that, by [9, (VIII.5.4)], any non-zero morphism in rad_{A}^{∞}(E′, E′′) factors through τ_{A}Σ.

(b) implies (c). Let X, Y ∈ Γ_{⩽Σ} be such that rad_{A}^{∞}(X, Y) ≠ 0. For each i ≥ 0, there exists a path in ind A

\xymatrix{X = X_{0} \ar[r]^{f_{1}} & X_{1} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{i}} & X_{i} \ar[r]^{g_{i}} & Y}

with all f_{j} irreducible and g_{i} ∈ rad_{A}^{∞}(X_{i}, Y) such that g_{i}f_{1} ⋯ f_{1} ≠ 0 (see [27]). We claim that there exists i such that X_{i} ∈ Σ_{0}. Indeed, since Σ is finite, there exists m ≥ 0 such that X_{m} ∈ Γ_{⩽Σ} and X_{m+1} ∉ Γ_{⩽Σ}. We show that X_{m} ∈ Σ. If X_{m} is injective, this follows from (2.1)(c). If not, consider the almost split sequence

\xymatrix{0 \ar[r] & X_{m} \ar[r] & X_{m+1} ⊕ Z \ar[r] & τ_{A}^{−1}X_{m} \ar[r] & 0}.

Since X_{m+1} ∉ Γ_{⩽Σ}, τ_{A}^{−1}X_{m} ∉ Γ_{⩽Σ}. The conclusion follows from the fact that X_{m} ∈ Γ_{⩽Σ}, and so Σ cuts the τ_{A}-orbit of X_{m}.

We thus have rad_{A}^{∞}(X_{m}, Y) ≠ 0. By (3.1)(b), Y is cogenerated by Σ. Therefore there exists E′ ∈ Σ such that rad_{A}^{∞}(X_{m}, E′) ≠ 0, a contradiction.

(c) implies (a). Suppose that there exists a non-zero morphism E′ → τ_{A}E′′, with E′, E′′ ∈ Σ. The hypothesis implies the existence of a path E′ ⇝ τ_{A}E′′ in Γ, hence a path E′ ⇝ τ_{A}E′′ → ∗ → E′′. The convexity of Σ in Γ yields τ_{A}E′′ ∈ Σ, a contradiction.

□

Remark. If Σ is a section then, by [26, (1.3)], [30, (Theorem 2)], the conditions of the proposition are equivalent to saying that Hom_{A}(τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ ∈ Σ, or to the condition that Γ be generalised standard. However, there exist left sections lying in non-generalised standard components but satisfying the condition of the proposition. Let A be given by the quiver

\xymatrix{ &  &  & ∘^{4} \\ ∘_{1} & ∘_{2} \ar[l]_{β} \ar[l] & ∘_{3} \ar[l]_{α} \ar[ur] \ar[dr] \\  &  &  & ∘^{5}}

bound by αβ = 0. The component containing the projective P_{3} is not generalised standard, by [10, (3.2)], but the simple modules {S_{4}, S_{5}} form a left section in that component satisfying the conditions of the proposition.

3.3. Clearly, if Γ is generalised standard, so is Γ_{⩽Σ}, hence Σ satisfies the equivalent conditions of (3.2). We also have the following lemma.

Lemma. Let Σ be a left section of Γ such that Hom_{A}(τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ ∈ Σ, then Σ satisfies the equivalent conditions of (3.2).

Proof. Indeed, Skowroński’s lemma [9, (VIII.5.3)], [31, (Lemma 1)], ensures that |Σ_{0}| ≤ rk K_{0}(A). Let E′, E′′ ∈ Σ. Since, by [9, (VIII.5.4)], any non-zero morphism in rad_{A}^{∞}(E′, E′′) factors through τ_{A}^{−1}Σ, we infer that rad_{A}^{∞}(E′, E′′) = 0. □

3.4. Lemma. Let Σ be a left section of Γ satisfying the equivalent conditions of (3.2) and let C = A/I, where I ⊆ Ann_{A}Σ. Then all indecomposables in Σ lie in the same component Γ′ of Γ(mod C), Σ is a left section of Γ′ and Γ′_{⩽Σ} = Γ_{⩽Σ} is generalised standard in mod C.

Proof. Since I ⊆ Ann_{A}Σ, then all indecomposables in Σ are C-modules. Now, recall that, if C is a quotient of A and X, Y are two indecomposable C-modules such that there is an irreducible morphism f : X → Y in mod A, then f remains irreducible in mod C. Therefore, all indecomposables in Σ lie on the same component Γ′ of Γ(mod C). On the other hand, by (3.1), Γ_{⩽Σ} ⊆ ind C hence Γ_{⩽Σ} = Γ′_{⩽Σ} and consequently Σ is a left section in Γ′ as well. The last statement follows from the fact that rad_{C}^{∞} is contained in rad_{A}^{∞}. □

3.5. We recall the Liu-Skowroński criterion (see [26, (3.2)], [30, (Theorem 3)] or [9, (VIII.5.6)]. Let A be an artin algebra having a section Σ such that Hom_{A}(E′, τ_{A}E′′) = 0 for all E′, E′′ ∈ Σ, then A/Ann_{A}Σ is a tilted algebra having Σ as complete slice. If in particular, Σ is faithful, then A is tilted having Σ as complete slice, and the component in which Σ lies as connecting component.

Theorem. Let Σ be a left section in a component Γ of Γ(mod A) such that Hom_{A}(τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ in Σ. Then B = A/Ann_{A}Σ is a tilted algebra having Σ as complete slice.

Proof. By (3.3), Σ satisfies the conditions of (3.2). In particular, Σ is finite so we can set E = ⨁_{U∈Σ} U. By (3.4), all indecomposables in Σ lie in the same component Γ′ of Γ(mod B) in which Σ is a left section such that Γ′_{⩽Σ} = Γ_{⩽Σ} and moreover, by (3.2), Hom_{B}(E, τ_{B}E) = 0. We also have Hom_{B}(τ_{B}^{−1}E, E) = 0: indeed, assume to the contrary that there exist E′, E′′ ∈ Σ and a non-zero morphism τ_{B}^{−1}E′ → E′′ then, since by [11, (p. 186–7)], there exists an epimorphism τ_{A}^{−1}E′ → τ_{B}^{−1}E′, we get upon composing a nonzero morphism τ_{A}^{−1}E′ → E′′, a contradiction. In order to complete the proof, it suffices to show that E_{B} is a tilting B-module with H = EndE_{B} hereditary. This is done as in [9, (VIII.5.6)], but we include the proof for the benefit of the reader.

Since E_{B} is faithful, then pd E_{B} ≤ 1 and id E_{B} ≤ 1, by [9, (VIII.5.1)]. Moreover, Ext_{B}^{1}(E, E) = 0, whence E is a partial tilting module. Let f_{1}, …, f_{d} be a generating set of the B-module Hom_{B}(B, E). Setting f = [f_{1}, …, f_{d}] : B → E^{d}, we have an exact sequence

\xymatrix{0 \ar[r] & B_{B} \ar[r]^{f} & E^{d} \ar[r] & X \ar[r] & 0}.

We claim that E⊕X is a tilting B-module. Since B_{B} is projective, we have pd X ≤ 1. Applying successively Hom_{B}(−, E), Hom_{B}(X, −) and Hom_{B}(E, −) to the preceding exact sequence yields respectively Ext_{B}^{1}(X, E) = 0, Ext_{B}^{1}(X, X) = 0 and Ext_{B}^{1}(E, X) = 0. This establishes our claim.

Assume now that Y is an indecomposable summand of X such that Y ∉ add E. The exact sequence above yields a non-zero morphism E → Y. By [9, (VIII.5.4)], Hom_{B}(τ_{B}^{−1}E, Y) ≠ 0, hence Ext_{B}^{1}(Y, T) ≠ 0, a contradiction. This shows that X ∈ add E and therefore E_{B} is a tilting module.

We now prove that H is hereditary. Let P_{H} be an indecomposable projective H-module and f : M → P be a monomorphism with M indecomposable. The tilting module E determines a torsion pair (𝒯 (E), ℱ (E)) in mod B and another (𝒳 (E), 𝒴 (E)) in mod H. Since P ∈ 𝒴 (E), then M ∈ 𝒴 (E) hence there exist g : V → E′ with V ∈ 𝒯 (E), E′ ∈ Σ ⊆ 𝒯 (E) such that M ≅ Hom_{B}(E, V), P ≅ Hom_{B}(E, E′) and f ≅ Hom_{B}(E, g). Now, since M ≠ 0, there exist an indecomposable projective H-module P′ and a non-zero morphism f′ : P′ → M. Again, there exist E′′ ∈ Σ and g′ : E′′ → V such that P′ ≅ Hom_{B}(E, E′′), f′ ≅ Hom_{B}(E, g′). Since ff′ ≠ 0, then gg′ ≠ 0. If now V ∉ Σ, then by [9, (VIII.5.4)], gg′ factors through τ_{A}E. But then Hom_{H}(E′′, τ_{A}E) ≠ 0, a contradiction which shows that V ∈ Σ, and thus completes the proof. □

3.6. Clearly, the Liu-Skowroński criterion follows directly from the above theorem. We also have the following easy corollary.

Corollary. Let Σ be a left section in a generalised standard component Γ of Γ(mod A). Then A/Ann_{A}Σ is a tilted algebra having Σ as complete slice.

3.7. Corollary. An algebra A is tilted if and only if it admits a faithful left section Σ such that Hom_{A}(τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ ∈ Σ.

3.8. Corollary. Let Σ be a left section in a component Γ of Γ(mod A) such that every projective in Γ precedes Σ and moreover Hom_{A}(E′, τ_{A}E′′) = 0 for all E′, E′′ ∈ Σ. Then A/Ann_{A}Σ is a tilted algebra having Σ as complete slice and Γ as connecting component.

Proof. By (2.3), Σ is a section. We apply the Liu-Skowroński criterion. □

4. Left sections convex in ind A

4.1. A full subcategory 𝒞 of ind A is called convex in ind A if, for any path

\xymatrix{X = X_{0} \ar[r] & X_{1} \ar[r] & ⋯ \ar[r] & X_{t} = Y}

in ind A, with X, Y ∈ 𝒞, then X_{i} ∈ 𝒞 for all i.

Lemma. Let Σ be a left section convex in ind A. Then Hom_{A}(τ_{A}^{−1}E′, E′′) = 0 for all E′, E′′ ∈ Σ.

Proof. If E′, E′′ ∈ Σ are such that Hom_{A}(τ_{A}^{−1}E′, E′′) ≠ 0, then we have a path E′ → ∗ → τ_{A}^{−1}E′ → E′′ in ind A. Convexity yields τ_{A}^{−1}E′ ∈ Σ, a contradiction to (LS2). □

Remark. It is easy to find examples of left sections (even of sections) which satisfy the conditions of (3.2), but are not convex in ind A. For instance, let A be the radical square zero algebra with quiver

\xymatrix{ & ∘^{2} \ar[dr] \ar[dr] \\ ∘_{3} \ar[ur] \ar[ur] &  & ∘_{1} \ar[ll] \ar[ll]}

and E = I_{1} ⊕ S_{2} ⊕ P_{3}. The set {I_{1}, S_{2}, P_{3}} is a section in its component, and satisfies Hom_{A}(τ_{A}^{−1}E, E′) = 0, but is not convex in ind A.

4.2. Corollary. Let A be an algebra having a left section convex in ind A. Then A/Ann_{A}Σ is tilted having Σ as complete slice.

4.3. Recall from [21] that an A-module L (not necessarily indecomposable) is called directed if there do not exist two indecomposable summands L′, L′′ of L, an indecomposable non-projective module Nand a path in ind A of the form

L′ ⇝ τ_{A}N → ∗ → N ⇝ L′′.

Lemma. Let Σ be a left section convex in ind A, and E = ⨁_{U∈Σ} U. Then:

(a) E is a directed A-module;

(b) if E is sincere, then Σ is a section (and, actually, a complete slice);

(c) E is sincere if and only if it is faithful.

Proof. (a) If there exist E′, E′′ ∈ Σ, an indecomposable non-projective module N and a path in ind A of the form

E′ ⇝ τ_{A}N → ∗ → N ⇝ E′′,

then convexity implies N, τ_{A}N ∈ Σ, a contradiction.

(b) To show that Σ is a section, it suffices, by (2.3), to prove that, if P is a projective module such that there exist E′ ∈ Σ and a path of irreducible morphisms E′ ⇝ P, then P ∈ Σ. Since E is sincere, there exists E′′ ∈ Σ such that Hom_{A}(P, E′′) ≠ 0. Thus we have a path E′ ⇝ P → E′′ in ind A and convexity forces P ∈ Σ. Hence Σ is a section. The second statement follows from the observation that any sincere section which is convex in ind A is a complete slice (see [28, 29]).

(c) This follows from (b).

□

4.4. Let Σ be a finite left section and E = ⨁_{U∈Σ} U. The support of Σ is the full subcategory Supp Σ = eAe where e is the sum of those primitive idempotents e_{x} of A such that Ee_{x} ≠ 0.

Lemma. Assume A is a finite dimensional algebra over an algebraically closed field, and Σ is a left section convex in ind A. Then Supp Σ is a full convex subcategory of A.

Proof. We slightly modify Bongartz’ convexity argument [16]. If Supp Σ is not convex,

there exists a path \xymatrix{x_{0} \ar[r]^{α_{1}} & x_{1} \ar[r] & ⋯ \ar[r] & x_{m−1} \ar[r]^{α_{m}} & x_{m}} in the quiver of A such that m ≥ 2, x_{0}, x_{m} ∈ Supp Σ and x_{i} ∉ Supp Σ for 1 ≤ i < m. Let α_{1} = β_{1}, …, β_{s} be all the arrows from x_{0} to x_{1} and α_{m} = γ_{1}, …, γ_{t} be all those from x_{m−1} to x_{m}. Let J be the twosided ideal of Supp Σ generated by all paths of the forms β_{i}δ or δγ_{j}, then consider A′ = Supp Σ/J. Since EJ = 0, then E is an A′-module. Denoting by P′_{x_{0}}, I′_{x_{m}}, respectively, the indecomposable projective A′-module at x_{0} and injective A′-module at x_{m}, we have Hom_{A′}(P′_{x_{0}}, E) ≠ 0 and Hom_{A′}(E, I′_{x_{m}}) ≠ 0. Let ( \begin{matrix}S_{y} \\ S_{z} \end{matrix} ) be the uniserial module of length two having the simple S_{y} as top and S_{z} as socle. We get E′, E′′ ∈ Σ and a path in ind A′ (hence in ind A) of the form

[figure: E′; I′_{x_{m}}; S_{x_{m−1}}; ( \begin{matrix}S_{x_{m−2}} \\ S_{x_{m−1}} \end{matrix} ); ⋯; ( \begin{matrix}S_{x_{1}} \\ S_{x_{2}} \end{matrix} ); S_{x_{1}}; P′_{x_{0}}; E′′] .

Since Σ is convex in ind A, we get S_{x_{i}} ∈ Σ for all i, a contradiction. □

4.5. Theorem. Let A be a finite dimensional algebra over an algebraically closed field, and Σ be a left section convex in ind A. Then Supp Σ ≅ A/Ann_{A}Σ is a tilted algebra having Σ as complete slice and is a full convex subcategory of A.

Proof. By (4.2), A/Ann_{A}Σ is tilted and has Σ as complete slice. Also, by (4.4), Supp Σ is a full convex subcategory of A. In particular, Supp Σ ≅ A/I(Σ), where I(Σ) is the twosided ideal of A generated by those primitive idempotents e_{x} such that Ee_{x} = 0. Since I(Σ) ⊆ Ann_{A}Σ, it follows from (3.4) that all points of Σ lie in the same component Γ′ of Γ(mod A/I(Σ)) and Σ is a left section in Γ′. Further, Σ is convex in ind A/I(Σ), because it is so in ind A. By (4.2), Σ is a complete slice in A/I(Σ). We have established that each of A/Ann_{A}Σ and Supp Σ ≅ A/I(Σ) is tilted and that these algebras have Σ as common complete slice. Therefore they are isomorphic (see, for instance, [9, (VIII.5.6)]). □

5. Subcategories closed under predecessors

5.1. Throughout this section, 𝒞 is a full subcategory of ind A, closed under predecessors. We first characterise the relative projectives and injectives in add 𝒞.

Lemma. Let P_{0} ∈ 𝒞. The following are equivalent:

(a) P_{0} is Ext-projective in add 𝒞 (that is, Ext_{A}^{1}(P_{0}, −)|_{𝒞} = 0).

(b) If 0 → L → M → N → 0 is a short exact sequence lying in add 𝒞, then the induced sequence 0 → Hom_{A}(P_{0}, L) → Hom_{A}(P_{0}, M) → Hom_{A}(P_{0}, N) → 0 is exact.

(c) Every short exact sequence of the form 0 → L → M → P_{0} → 0 splits.

(d) P_{0} is a projective A-module.

Proof. We prove that (c) implies (d) (the other implications are trivial). Let f : P → P_{0} be

a projective cover in mod A. By hypothesis, the sequence \xymatrix{0 \ar[r] & Ker f \ar[r] & P \ar[r]^{f} & P_{0} \ar[r] & 0} splits, so P_{0} is projective. □

5.2. Lemma. Let E_{0} ∈ 𝒞. The following are equivalent:

(a) E_{0} is Ext-injective in add 𝒞 (that is Ext_{A}^{1}(−, E_{0})|_{𝒞} = 0).

(b) If 0 → L → M → N → 0 is a short exact sequence lying in add 𝒞, then the induced sequence 0 → Hom_{A}(N, E_{0}) → Hom_{A}(M, E_{0}) → Hom_{A}(L, E_{0}) → 0 is exact.

(c) Every short exact sequence of the form 0 → E_{0} → M → N → 0 with N ∈ add 𝒞 splits.

(d) τ_{A}^{−1}E_{0} ∉ 𝒞.

Proof. The equivalence of (a), (b), (c) is trivial and that of (a), (d) follows from [12, (3.4)]. □

5.3. While the Ext-projectives in add 𝒞 are perfectly characterised above, the same is not true of the Ext-injectives. Let ℰ denote the full subcategory of 𝒞 (hence of ind A) with objects the Ext-injectives in add 𝒞. Clearly, any injective A-module lying in 𝒞 belongs to ℰ, but the converse is generally not true. Note also that we may have ℰ = ∅ (take, for instance, A hereditary and representation-infinite, and 𝒞 consisting of all postprojective modules).

Lemma. (a) For every E′, E′′ ∈ ℰ, we have Hom_{A}(τ_{A}^{−1}E′, E′′) = 0.

(b) |ℰ | ≤ rk K_{0}(A).

(c) ℰ cuts each τ_{A}-orbit in Γ(mod A) at most once.

(d) Every path of irreducible morphisms contained in ℰ is sectional.

(e) ℰ contains no cycle of irreducible morphisms.

Proof. (a) Indeed, E′′ ∈ 𝒞, while τ_{A}^{−1}E′ ∉ 𝒞.

(b) Follows from (a) and Skowroński’s lemma [9, (VIII.5.3)], [31, (Lemma 1)].

(c) Assume, E′, τ_{A}^{−t}E′ ∈ ℰ (with t > 0). We have a path of irreducible morphisms E′ → ∗ → τ_{A}^{−1}E′ ⇝ τ_{A}^{−t}E′. Since τ_{A}^{−t}E′ ∈ 𝒞, we have τ_{A}^{−1}E′ ∈ 𝒞, a contradiction.

(d) Follows from (c).

(e) Let E_{0} → E_{1} → ⋯ → E_{t} = E_{0} be a cycle of irreducible morphisms in ℰ. By the Bautista-Smalø theorem [14, 17], it is not sectional. Hence E_{1} = τ_{A}^{−1}E_{t−1} or there exists i with 1 ≤ i < t such that E_{i+1} = τ_{A}^{−1}E_{i−1}, thus contradicting (a).

□

5.4. Lemma.

(a) Let \xymatrix{E′ = M_{0} \ar[r]^{f_{1}} & M_{1} \ar[r] & ⋯ \ar[r]^{f_{t}} & M_{t}} be a path in ind A, with E′ ∈ ℰ and M_{t} ∈ 𝒞. If no f_{i} factors through an injective module, then M_{i} ∈ ℰ for all i.

(b) Let Γ be a component of Γ(mod A) not containing injectives and such that Γ∩ℰ ≠ ∅. Then, for every M ∈ 𝒞, there exists a unique m ≥ 0 such that τ_{A}^{−m}M ∈ ℰ.

Proof. (a) Since M_{t} ∈ 𝒞, then M_{i} ∈ 𝒞 for all i. Also, no M_{i} is injective and the Auslander-Reiten isomorphism \underline{Hom}_{A}(τ_{A}^{−1}M_{i−1}, τ_{A}^{−1}M_{i}) ≅ \overline{Hom}_{A}(M_{i−1}, M_{i}) yields a non-zero morphism τ_{A}^{−1}M_{i−1} → τ_{A}^{−1}M_{i} (for each i). This yields a path τ_{A}^{−1}E′ = τ_{A}^{−1}M_{0} → τ_{A}^{−1}M_{1} → ⋯ → τ_{A}^{−1}M_{t}. Since τ_{A}^{−1}E′ ∉ 𝒞, then τ_{A}^{−1}M_{i} ∉ 𝒞 for all i. Consequently, M_{i} ∈ ℰ for all i.

(b) Since Γ contains no injectives (hence is right stable) but contains at least one Extinjective in add 𝒞 (the set of which is finite, by (5.3)(b)), we may take ℓ > 0 so that τ_{A}^{−ℓ}M succedes τ_{A}^{−1}E′ ∉ 𝒞 (for every E′ ∈ Γ ∩ ℰ), then τ_{A}^{−ℓ}M ∉ 𝒞. Hence there exists m ≥ 0 such that τ_{A}^{−m}M ∈ 𝒞 but τ_{A}^{−m−1}M ∉ 𝒞. Thus τ_{A}^{−m}M ∈ ℰ. Uniqueness of m follows from (5.3)(c).

□

5.5. Proposition. Let Γ be a component of Γ(mod A) not containing injectives and such that Σ = Γ ∩ ℰ ≠ ∅. Then Σ is a left section in Γ and A/Ann_{A}Σ is a tilted algebra having Σ as complete slice.

Proof. We check the axioms for a left section: (LS1) follows from (5.3)(e) and (LS3) from (5.4)(a), then let M be a predecessor of Σ in Γ, we have M ∈ Γ ∩ 𝒞 and so (LS2) follows from (5.4)(b). The statement then follows from (5.3)(a) and (3.5). □

Remark. The statement is false if Γ contains injectives. Let A be given by the quiver

\xymatrix{∘_{1} & ∘_{2} \ar[l]_{δ} & ∘_{3} \ar[l]_{β} \ar[l]^{γ} & ∘_{4} \ar[l]_{α}}

bound by αβ = 0, βδ = 0 and 𝒞 be the full subcategory of ind A having as objects all non-preinjective modules. Then the only Ext-injective in add 𝒞 is the projective-injective module P_{4} = I_{1} which lies in a tube

[figure: 4 3 2 1; 3 2; 3 2 1; 4 3 2; 3 2; 3 3 2 2 1; 3 2; 4 3 3 2 2; ·; ·; ·; ·]

Clearly, Γ ∩ ℰ satisfies neither (LS2) nor (LS3).

This raises the question of finding the right conditions so that the statement of (5.5)stays valid in a component containing injectives. As is easy to see, the condition is that every object in 𝒞 should have projective dimension at most one (or, equivalently, 𝒞 ⊆ ℒ_{A}).

6. Full subcategories of ℒ_{A}

6.1. In this section, we assume that 𝒞 is a full subcategory of ℒ_{A}, closed under predecessors. The first result generalises [3, (1.5)].

Lemma. Let I be an indecomposable injective A-module.

(a) There exist at most rk K_{0}(A) indecomposable modules N ∈ 𝒞 such that there exists a path I ⇝ N in ind A.

(b) Every such path is refinable to a path of irreducible morphisms.

(c) Every such path of irreducible morphisms is sectional.

(d) Every such module N ∈ 𝒞 is Ext-injective in add 𝒞.

Proof. If I ∉ 𝒞, then no successor of I lies in 𝒞 and the statement holds trivially. Assume that I ∈ 𝒞 has infinitely many successors in 𝒞. Then, for each s ≥ 0, there exists a path in ind A

\xymatrix{I = L_{0} \ar[r] & L_{1} \ar[r] & ⋯ \ar[r] & L_{s−1} \ar[r] & L_{s}}

with L_{i} ∈ 𝒞 for all i. An easy induction (as in [3, (1.5)]) shows that this path induces another one

(∗)  \xymatrix{I = M_{0} \ar[r]^{f_{1}} & M_{1} \ar[r] & ⋯ \ar[r]^{f_{i}} & M_{i} \ar[r] & L_{j}}

with j ≤ i, M_{ℓ} ∈ 𝒞 and f_{ℓ} irreducible for all ℓ, and f′_{i} ≠ 0.

We prove that (∗) is sectional. If this is not the case, there exists a least i such that the subpath I = M_{0} ⇝ M_{i} is sectional and M_{i+1} = τ_{A}^{−1}M_{i−1}. In particular, by [17], Hom_{A}(I, M_{i−1}) ≠ 0, hence pd τ_{A}^{−1}M_{i−1} > 1. But M_{i+1} ∈ 𝒞 implies pd M_{i+1} ≤ 1, a contradiction.

Since I is injective, it is Ext-injective in add 𝒞. Moreover, Hom_{A}(I, M_{i}) ≠ 0 implies τ_{A}^{−1}M_{i} ∉ 𝒞 (because either M_{i} is injective or pd τ_{A}^{−1}M_{i} > 1). Since M_{i} ∈ 𝒞, we infer that M_{i} is Ext-injective for each i. Invoking (5.3)(b) finishes the proof. □

6.2. A module M ∈ 𝒞 is called Ext-injective of the first kind if there exist an injective module I and a path I ⇝ M in ind A. We denote by ℰ_{1} the class of Ext-injectives of the first kind. An Ext-injective which is not of the first kind is of the second kind, and the class of Ext-injectives of the second kind is denoted by ℰ_{2} (= ℰ \ℰ_{1}). The following result generalises part of [5, (3.1)], [6, (3.1)].

Corollary. Let M ∈ 𝒞. The following are equivalent:

(a) M is Ext-injective of the first kind;

(b) there exists an injective module I and a path of irreducible morphisms I ⇝ M;

(c) there exists an injective module I and a sectional path of irreducible morphisms I ⇝ M;

(d) there exists an injective module I such that Hom_{A}(I, M) ≠ 0.

Proof. That (a) implies (b) and that (b) implies (c) follow from (6.1), that (c) implies (d) follows from [17] and finally (d) implies (a) is trivial. □

Remark. If 𝒞 = ℒ_{A}, then Ext-injectives of the second kind are characterised as being for instance, those M ∈ ℒ_{A} \ ℰ_{1} such that there exists a projective module P ∉ ℒ_{A} satisfying Hom_{A}(P, τ_{A}^{−1}M) ≠ 0 (see [5, (3.1)], [6, (3.1)]). No such characterisation is known in general.

6.3. The next result generalises [5, (3.4)].

Proposition. Assume that E′ ∈ ℰ and M ∈ 𝒞 are such that there exists a path E′ ⇝ M in ind A. Then this path can be refined to a sectional path of irreducible morphisms and M ∈ ℰ. In particular, ℰ is convex in ind A.

Proof. Let \xymatrix{E′ = X_{0} \ar[r]^{f_{1}} & X_{1} \ar[r] & ⋯ \ar[r]^{f_{t}} & X_{t} = M} be the given path. We first show that X_{i} ∈ ℰ for each i. If no f_{i} factors through an injective, this follows from (5.4)(a). Otherwise, let i be minimal such that f_{i} : X_{i−1} → X_{i} factors through an injective I. We thus have a subpath I → X_{i} ⇝ X_{t} = M. By (6.1), X_{j} ∈ ℰ for j ≥ i. But, on the other hand, we have another subpath E′ = X_{0} ⇝ X_{i−1} where, because of the minimality of i, none of the morphisms factors through an injective. Since, for every j < i, we have X_{j} ∈ 𝒞, we deduce from (5.4)(a) that X_{j} ∈ ℰ.

This establishes the convexity of ℰ in ind A. There remains to show that each f_{i} lies in a finite power of the radical of mod A. Indeed, if this is not the case for some f_{i}, then, for every s ≥ 1, the given path has a refinement

E′ = X_{0} ⇝ X_{i−1} = Y_{0} → Y_{1} → ⋯ → Y_{s} = X_{i} ⇝ X_{t} = M.

By convexity in ind A, we have Y_{ℓ} ∈ ℰ for each ℓ. This contradicts (5.3)(b). □

6.4. Corollary. The modules in ℰ are directed in ind A.

Proof. Let E′ = M_{0} → M_{1} → ⋯ → M_{t} = E′ be a cycle with E′ ∈ ℰ. By (6.3), it can be refined to a cycle of irreducible morphisms contained in ℰ, contradicting (5.3)(e). □

6.5. Corollary. Let Γ be a component of Γ(mod A) containing an injective. Then every module of Γ ∩ 𝒞 is directed.

Proof. Let M_{0} → M_{1} → ⋯ → M_{t} = M_{0} be a cycle with M_{0} ∈ Γ ∩ 𝒞. By [3, (1.4)], there exist an injective module I ∈ Γ and a path I ⇝ M_{0}. We compose this path with two copies of the cycle to get a longer path from I to M_{0} which, by (6.3), is refinable to a sectional path of irreducible morphisms, yielding a contradiction to [14]. □

6.6. Proposition. Let Γ be a component of Γ(mod A) such that Γ ∩ ℰ ≠ ∅. Then: (a) for every M ∈ Γ ∩ 𝒞, there exists a unique m ≥ 0 such that τ_{A}^{−m}M ∈ ℰ;

(b) the number of τ_{A}-orbit of Γ ∩ 𝒞 equals |Γ ∩ ℰ | (hence is finite);

(c) Γ ∩ 𝒞 contains no modules on a cycle between modules in Γ.

Proof. (a) By (5.4), we may assume that Γ contains an injective. Suppose that, for any ℓ ≥ 0, we have τ_{A}^{−ℓ}M ∈ 𝒞. Then M is right stable (and not periodic, by (6.5)). Since Γ contains an injective, there is a walk from this injective to the τ_{A}-orbit of M. Among all such injectives, choose one, say I, such that there is a walk of least length from I to the τ_{A}-orbit of M. Minimality implies that all modules on this walk, except I, are right stable. Hence there exist s ≥ 0 and a path of irreducible morphisms I ⇝ τ_{A}^{−s}M. Since I ∈ ℰ, we have τ_{A}^{−s}M ∈ ℰ by (6.3), hence τ_{A}^{−s−1}M ∉ 𝒞, a contradiction. Thus there exists m ≥ 0 such that τ_{A}^{−m}M ∈ 𝒞 but τ_{A}^{−m−1}M ∉ 𝒞, so τ_{A}^{−m}M ∈ ℰ. Uniqueness of m follows from (5.3).

(b) Let n be the number of τ_{A}-orbits of Γ ∩ 𝒞. By (a), n ≤ |Γ ∩ ℰ | ≤ rk K_{0}(A). The statement follows because each element of Γ ∩ ℰ lies in exactly one τ_{A}-orbit of Γ ∩ 𝒞.

(c) By (6.5), we may assume that Γ contains no injectives. Let

M_{0} → M_{1} → ⋯ → M_{t} = M_{0}

be a cycle with M_{0} ∈ Γ ∩ 𝒞 and all M_{i} ∈ Γ. Clearly, M_{i} ∈ 𝒞 for all i. By (6.4), M_{i} ∉ ℰ for all i, and also no f_{i} factors through an injective. Thus this cycle induces a new one

τ_{A}^{−1}M_{0} → τ_{A}^{−1}M_{1} → ⋯ → τ_{A}^{−1}M_{t} = τ_{A}^{−1}M_{0}

with τ_{A}^{−1}M_{i} ∈ Γ ∩ 𝒞 for all i. Repeating this procedure indefinitely, we get that, for all m ≥ 0, the module τ_{A}^{−m}M_{0} lies in Γ∩𝒞 and in a cycle. This contradicts (a).

□

6.7. Theorem. Let A be an artin algebra, and 𝒞 ⊆ ℒ_{A} be a full subcategory closed under predecessors, having ℰ as subcategory of Ext-injectives. Let Γ be a component of Γ(mod A). Then:

(a) If Γ ∩ ℰ = ∅, then either Γ ⊆ 𝒞 or Γ ∩ 𝒞 = ∅.

(b) If Σ = Γ ∩ ℰ ≠ ∅, then Σ is a left section of Γ, convex in ind A. Moreover A/Ann_{A}Σ is tilted having Σ as complete section.

Proof. (a) Assume Γ ∩ ℰ = ∅. If Γ contains a module in 𝒞 and one not in 𝒞, then there exists an irreducible morphism X → Y with X ∈ Γ ∩ 𝒞 and Y ∈ Γ \ 𝒞. Since Γ ∩ ℰ = ∅, then X is not injective, so we have an irreducible morphism Y → τ_{A}^{−1}X. Since Y ∉ 𝒞, then τ_{A}^{−1}X ∉ 𝒞. Thus X ∈ ℰ, a contradiction.

(b) If Γ contains no injective, then Σ is a left section by (5.5) and is convex in ind A by (6.3). If Γ contains an injective, then (LS1) follows from (5.3)(e), convexity in ind A (hence (LS3)) follows from (6.3) and finally (LS2) follows easily from (6.6)(a) (for, if M precedes Σ in Γ, then M ∈ Γ ∩ 𝒞).

The last statement follows from (4.2).

□

6.8. Corollary. Let A be a finite dimensional algebra over an algebraically closed field, 𝒞 ⊆ ℒ_{A} be a full subcategory closed under predecessors, having ℰ as subcategory of Ext-injectives and Γ be a component of Γ(mod A) such that Σ = Γ ∩ ℰ ≠ ∅. Then Supp Σ ≅ A/Ann_{A}Σ is a tilted algebra having Σ as complete slice and is a full convex subcategory of A closed under successors.

Proof. By (4.5) and (6.7), it suffices to prove that Supp Σ is closed under successors. Let x → y be an arrow with x ∈ Supp Σ. Then we have a non-zero morphism P_{y} → P_{x}. Also, there exists an embedding P_{x} ↪ Ē, with Ē ∈ add Σ. This yields a non-zero morphism P_{y} → E′ for some E′ ∈ Σ. Thus y ∈ Supp Σ. □

6.9. The next result generalises [1, (Theorem A)].

Corollary. Let A be an artin algebra, and Γ be a component of Γ(mod A) which intersects the class ℰ_{A} of Ext-injective indecomposables in add ℒ_{A}. Then:

(a) each τ_{A}-orbit of Γ ∩ ℒ_{A} intersects ℰ_{A} exactly once;

(b) the number of τ_{A}-orbits of Γ ∩ ℒ_{A} equals |Γ ∩ ℰ_{A}|;

(c) Γ ∩ ℒ_{A} contains no modules lying on a cycle between modules in Γ;

(d) A/Ann_{A}(Γ ∩ ℰ_{A}) is a tilted algebra having Γ ∩ ℰ_{A} as complete slice. If, on the other hand, Γ ∩ ℰ_{A} = ∅ then either Γ ⊆ ℒ_{A} or Γ ∩ ℒ_{A} = ∅.

Proof. This follows from (6.6) and (6.7). □

6.10. The following corollary generalises [1, (Theorem B)].

Corollary. With the notation of (6.7), if Σ = Γ ∩ ℰ ≠ ∅ and all projectives in Γ belong to 𝒞, then:

(a) Σ is a section in Γ;

(b) Γ is generalised standard;

(c) A/Ann_{A}Σ is a tilted algebra having Γ as connecting component and Σ as complete slice.

Proof. By (6.7) and (2.3), Σ is a section in Γ. The rest follows. □

7. The support algebra

7.1. Definition. Let 𝒞 be a full subcategory of ℒ_{A}, closed under predecessors. Its support algebra A(𝒞) is the endomorphism algebra of the direct sum of all indecomposable projectives lying in 𝒞 (that is, A(𝒞) = End (⨁_{P_{x}∈𝒞} P_{x})).

Clearly, this generalises the left support of an artin algebra [5, (2.2)], [32, (3.1)]. Note that A(𝒞) is a full subcategory of A, closed under successors (and hence convex).

Lemma. The support algebra A(𝒞) is a direct product of connected quasi-tilted algebras.

Proof. By the above remark, A may be written in matrix form

[\begin{matrix}A(𝒞) & 0 \\ M & B \end{matrix}]

where M is a B-A(𝒞)-bimodule. Moreover, 𝒞 ⊆ ind A(𝒞) for, if L ∈ 𝒞 and P_{x} is an indecomposable projective such that Hom_{A}(P_{x}, L) ≠ 0, then P_{x} ∈ 𝒞. By [5, (2.1)], ℒ_{A} ⊆ ℒ_{A(𝒞)} . Hence 𝒞 ⊆ ℒ_{A(𝒞)} . The statement follows because any indecomposable projective A(𝒞)-module (= projective A-module lying in 𝒞) belongs to ℒ_{A(𝒞)} . □

7.2. Let ℰ denote the full subcategory of 𝒞 consisting of the Ext-injectives in add 𝒞 and E = ⨁_{U∈ℰ} U. Denote by F the direct sum of all indecomposable projectives which are not (!) in 𝒞, and set T = E ⊕ F. We call a partial tilting A-module convex if the class of its indecomposable summands is convex in ind A, see [7].

Lemma. (a) E is a convex partial tilting A-module.

(b) E is a convex partial tilting A(𝒞)-module. In particular, |ℰ | ≤ rk K_{0}(A(𝒞)).

(c) T is a partial tilting module.

(d) T is a tilting A-module if and only if |ℰ | equals the number of projectives in 𝒞 or if and only if E is a tilting A(𝒞)-module.

(e) If T is a tilting A-module, then the associated torsion pair (𝒯 (T), ℱ (T)) is given by ℱ (T) = add (𝒞 \ ℰ) and 𝒯 (T) = add (ind A \ ℱ (T)).

Proof. (a) Since 𝒞 ⊆ ℒ_{A}, then pd E ≤ 1. Clearly, Ext_{A}^{1}(E, E) = 0 so E is partial

tilting. Its convexity follows from (6.3).

(b) Since A(𝒞) is a full convex subcategory of A, and 𝒞 ⊆ ℒ_{A}, then E is a partial tilting A(𝒞)-module. It is convex because any path in ind A(𝒞) induces one in ind A.

(c) Since pd T ≤ 1, it suffices to observe that Ext_{A}^{1}(E, F) ≅ DHom_{A}(F, τ_{A}E) = 0, because τ_{A}E ∈ add 𝒞 while no indecomposable summand of F lies in 𝒞.

(d) This is clear.

(e) Assume M ∈ 𝒞 \ ℰ. If M ∉ ℱ (T), then Hom_{A}(T, M) ≠ 0. Since no summand of F lies in 𝒞, Hom_{A}(F, M) = 0. Hence there exists E_{0} ∈ ℰ such that Hom_{A}(E_{0}, M) ≠ 0. By (6.3), M ∈ ℰ, a contradiction. Thus 𝒞 \ ℰ ⊆ ℱ (T). Conversely, let M ∈ ℱ (T) = Cogen τ_{A}T. There exist E′ ∈ ℰ and a path M → τ_{A}E′ → ∗ → E′. In particular, M ∈ ℰ. On the other hand, M ∉ ℰ since ℰ ⊆ add T. This shows the first statement. The second follows.

□

7.3. Lemma. If an indecomposable injective A(𝒞)-module I precedes ℰ, then I ∈ ℰ.

Proof. This is clear if I is injective in mod A. Assume it is not. Since I precedes ℰ, then I ∈ 𝒞. But then τ_{A}^{−1}I ∉ 𝒞 (for, otherwise, τ_{A}^{−1}I = τ_{A(𝒞)}^{−1} I, a contradiction to the injectivity of I in mod A(𝒞)). Therefore I ∈ ℰ. □

7.4. The following result generalises [1, (Theorem C)].

Theorem. Let A be an artin algebra and 𝒞 ⊆ ℒ_{A} be a full convex subcategory closed under predecessors, having ℰ as subcategory of Ext-injectives. Let Γ be a component of Γ(mod A(𝒞)) such that Σ = Γ ∩ ℰ ≠ ∅. Then:

(a) Σ is a section in Γ, convex in ind A(𝒞);

(b) Γ is directed, and generalised standard;

(c) A(𝒞)/Ann_{A}Σ is a tilted algebra having Γ as connecting component and Σ as complete slice.

Proof. (a) First, we show that there exists a unique component Γ′ of Γ(mod A) which contains Σ. Let indeed E_{1} → E_{2} be an irreducible morphism in mod A(𝒞) with E_{1}, E_{2} ∈ Σ. By (6.3), it induces a path of irreducible morphisms E_{1} ⇝ E_{2} in mod A. In particular, E_{1} and E_{2} lie in the same component.

We next show that Γ′_{⩽Σ} = Γ_{⩽Σ}. Indeed, if X → Y is irreducible in Γ′_{⩽Σ}, it stays so in Γ_{⩽Σ}. Conversely, let X → Y be irreducible in Γ_{⩽Σ}. If X ∉ ℰ, then X is not injective and τ_{A}^{−1}X = τ_{A(𝒞)}^{−1} X. Therefore the almost split sequence starting with X is the same in both categories, and X → Y is irreducible in Γ′_{⩽Σ}. If X ∈ ℰ then, by (6.3), Y ∈ ℰ. If Y is not projective, the almost split sequence ending with Y is the same in both categories. If Y is projective, then radY is the same in both categories. In any case, X → Y remains irreducible in Γ′_{⩽Σ}. This establishes the claim.

By (6.7), Σ = Γ ∩ ℰ = Γ′ ∩ ℰ is a left section and, by (7.2)(b), is convex in ind A(𝒞). In order to show that Σ is a section, consider, according to (2.3), a projective P such that there exist E′ ∈ Σ and a path E′ ⇝ P in Γ. Since P is a projective A(𝒞)-module, it lies in 𝒞. But then (6.3) yields P ∈ ℰ.

(b) Since Γ contains a section, it is directed by [25, (3.2)], [9, (VIII.1.5)]. It is generalised standard because so are the directed Auslander-Reiten components of a quasi-tilted algebra [18].

(c) Follows from (a) and the Liu-Skowroński criterion.

□

7.5. The following result generalises [1, (4.6)].

Theorem. Let A be an artin algebra and 𝒞 ⊆ ℒ_{A} be a full convex subcategory closed under predecessors, having ℰ as subcategory of Ext-injectives. Let B be a connected component of A(𝒞) such that mod B ∩ ℰ ≠ ∅ and Γ be a component of Γ(mod B) such that Σ = Γ ∩ ℰ ≠ ∅. Then B is tilted having Γ as connecting component and Σ as complete slice.

If A is a finite dimensional algebra over an algebraically closed field, then B ≅ Supp Σ.

Proof. In order to show that Γ is a connecting component, we start by assuming that Γ is postprojective non-connecting. We claim that there exists an indecomposable projective B-module not in Γ. Otherwise, indeed, the number of τ_{B}-orbits in Γ equals rk K_{0}(B), so Γ is connecting, a contradiction which establishes the claim.

Suppose Q ∉ Γ is indecomposable projective. There exists a walk of indecomposable projective B-modules P_{0} P_{1} ⋯ P_{s} ≅ Q with P_{0} ∈ Γ and therefore there exist P_{i} ∈ Γ and P_{i+1} ∉ Γ such that Hom_{B}(P_{i}, P_{i+1}) ≠ 0. By [27], there exists, for each t > 0, a path

\xymatrix{P_{i} = M_{0} \ar[r]^{f_{1}} & M_{1} \ar[r] & ⋯ \ar[r]^{f_{t}} & M_{t} \ar[r]^{f} & P_{i+1}}

with all M_{i} indecomposable, all f_{i} irreducible and ff_{t} ⋯ f_{1} ≠ 0. Since t is arbitrary, we may assume M_{t} successor of τ_{B}^{−1}Σ. But then P_{i+1} ∉ 𝒞, a contradiction.

Assume now that Γ is preinjective non-connecting. The same argument yields an M ∈ Γ, proper predecessor of Σ, and an indecomposable injective I ∉ Γ such that Hom_{B}(I, M) ≠ 0. Since I precedes Σ, then I ∈ ℰ. By (6.3), M ∈ ℰ, a contradiction.

Since, by (7.4), Γ is directed, it is connecting by [18]. Moreover, again by (7.4), Σ is a section in Γ, convex in ind B. By the Liu-Skowroński criterion, B is tilted having Γ as connecting component and Σ as complete slice. The final statement is clear. □

7.6. The following corollary generalises [5, (5.4)], [1, (2.6)].

Corollary. Let A be representation-infinite and 𝒞 ⊆ ℒ_{A} be a full subcategory closed under predecessors. The following are equivalent:

(a) there exists a component Γ ⊆ 𝒞;

(b) 𝒞 is infinite;

(c) 𝒞 contains a postprojective component without injectives. If, moreover, A is not hereditary, then such a component Γ is postprojective or regular, or obtained from a stable tube or a component of type ℤ𝔸_{∞} by ray extensions.

Proof. (a) implies (b). Indeed, otherwise A would be representation-finite, a contradiction.

(b) implies (c). Since 𝒞 is infinite, there exists a connected component B of A(𝒞) such that mod B ∩ 𝒞 is infinite. Also, B is quasi-tilted, by (7.1). Let Γ be a postprojective component of Γ(mod B). Suppose Γ contains an injective. Then Γ is connecting, it is the only postprojective component and Γ ∩ 𝒞 is finite. Since 𝒞 is infinite, there exists X ∈ 𝒞 \ Γ and then one can easily find a morphism from a module in Γ \ 𝒞 to X, a contradiction. Therefore Γ has no injectives. Now if Γ ⊈ 𝒞, the existence of an M ∈ Γ ∩ 𝒞 and an N ∈ Γ \ 𝒞 implies the existence of an Ext-injective in Γ. Again, we get that 𝒞 is finite, a contradiction. Hence Γ ⊆ 𝒞.

Since (c) implies (a) trivially, and the last statement follows from the description of the components of quasi-tilted algebras, [23, 24, 18], the proof is complete. □

8. Algebras supported by subcategories

8.1. Definition. Let 𝒞 be a full subcategory of ind A, closed under predecessors. The algebra A is called 𝒞 -supported if add 𝒞 has enough Ext-injectives (that is, if add 𝒞 = Cogen E, where E = ⨁_{U∈ℰ} U).

This generalises the left supported algebras of [5] which, in this terminology, are ℒ_{A}- supported.

Lemma. An algebra A is 𝒞 -supported if and only if add 𝒞 is contravariantly finite in mod A.

Proof. Assume A is 𝒞 -supported. Since add 𝒞 is a torsion-free class, then by [33], it is contravariantly finite. Conversely, if add 𝒞 is contravariantly finite, then by [33], there exists an Ext-injective N ∈ add 𝒞 such that add 𝒞 = Cogen N. Since N ∈ add E, we have

add 𝒞 = Cogen N ⊆ Cogen E ⊆ add 𝒞

and equality follows. □

Remarks. (a) If A is representation-finite, then A is 𝒞 -supported for any full subcate-

gory 𝒞 of ind A.

(b) In general, this property depends on the chosen subcategory: let A be tame hereditary, and 𝒞 consist of all postprojective modules, then A is not 𝒞 -supported. If, on the other hand, 𝒞′ is a finite subcategory of ind A consisting of postprojective modules, then A is 𝒞′-supported.

8.2. From now on, we assume again 𝒞 ⊆ ℒ_{A}. We denote by Pred E the full subcategory of ind A consisting of those X such that there exist E′ ∈ ℰ and a path X ⇝ E′ in ind A. Also, an A-module L is called almost directed [2, (2.2)] if there do not exist two indecomposable summands L′, L′′ of L and a path L′ ⇝ τ_{A}L′′ in ind A. The dual notion is that of almost codirected module.

Theorem. Let A be an artin algebra, and 𝒞 be a full subcategory of ℒ_{A} closed under predecessors. The following are equivalent:

(a) A is 𝒞 -supported;

(b) add 𝒞 is contravariantly finite;

(c) T = E ⊕ F is a tilting A-module;

(d) |ℰ | equals the number of projectives in 𝒞;

(e) E is a tilting A(𝒞)-module;

(f) E is a cotilting A(𝒞)-module;

(g) 𝒞 = Supp (−, E); (h) there exists an almost codirected module L_{A} such that 𝒞 = Supp (−, L);

(i) there exists a module L_{A} such that Hom_{A}(τ_{A}^{−1}L, L) = 0 and 𝒞 = Supp (−, L);

(j) 𝒞 = Pred E;

(k) E is a sincere A(𝒞)-module;

(l) for every connected component B of A(𝒞), we have mod B ∩ ℰ ≠ ∅;

(m) every connected component B of A(𝒞) is tilted and has mod B ∩ ℰ as complete slice;

(n) every projective A-module in 𝒞 precedes ℰ.

Proof. The proofs of [2, (Theorem A)], [1, (2.1)], [5, (Theorems A,B)] carry over with the obvious changes. □

8.3. Corollary. If A is 𝒞 -supported, then:

(a) the A(𝒞)-modules not in ℱ (T) are those of Gen E;

(b) F is the Bongartz complement of E.

Proof. The proof of [5, (5.3)] applies with the obvious changes. □

8.4. The following is a new characterisation of supported algebras.

Corollary. An algebra A is 𝒞 -supported if and only if every morphism f : L → M with L ∈ 𝒞 and M ∉ 𝒞 factors through add E.

Proof. Necessity. If A is 𝒞 -supported then, by (8.3), T = E ⊕F is a tilting module. Since M ∉ 𝒞, then M ∈ 𝒯 (T), by (7.2). Let {g_{1}, …, g_{d}} be a generating set of the EndT - module Hom_{A}(T, M), then g = [g_{1}, …, g_{d}] : T^{d} → M is surjective and K = Ker g belongs to 𝒯 (T) (see [9, (VI.2.5)]). Applying Hom_{A}(L, −) to the exact sequence

\xymatrix{0 \ar[r] & K \ar[r] & T^{d} \ar[r]^{g} & M \ar[r] & 0}

yields an exact sequence.

\xymatrix{Hom_{A}(L, T^{d}) \ar[r] & Hom_{A}(L, M) \ar[r] & Ext_{A}^{1}(L, K)}.

Now Ext_{A}^{1}(L, K) ≅ D\overline{Hom}_{A}(K, τ_{A}L) = 0 because K ∈ 𝒯 (T) while τ_{A}L ∈ 𝒞 ⊆ ℱ (T). Hence Hom_{A}(L, g) is surjective, so f factors through add T. Since Hom_{A}(L, F) = 0, we have that, in fact, f factors through add E.

Sufficiency. The inclusion of any L ∈ 𝒞 into its injective envelope factors through add E. Therefore add 𝒞 = Cogen E. □

8.5. A particular case of ℒ_{A}-supported algebras was studied in [8]. Recall that a full subcategory 𝒞 of ind A, closed under predecessors, is abelian exact if add 𝒞 is abelian and the inclusion add 𝒞 ↪ mod A is an exact functor.

Corollary. Let A and 𝒞 be such that 𝒞 ⊆ ℒ_{A} is closed under predecessors and abelian exact.

(a) A ≅ [\begin{matrix}A(𝒞) & 0 \\ X & B \end{matrix}], with A(𝒞) hereditary and X_{A(𝒞)} injective.

(b) A is 𝒞 -supported.

(c) If A is triangular, then A = A(𝒞) (in particular, is hereditary).

Proof. (a) By [8, (2.5)], A ≅ [\begin{matrix}C & 0 \\ X & B \end{matrix}] with C hereditary, X_{C} injective and add 𝒞 ≅

mod C. Therefore C = A(𝒞).

(b) Since mod A(𝒞) is cogenerated by its minimal injective cogenerator, it follows from [33] that add 𝒞 is contravariantly finite.

(c) This follows from [8, (3.2)].

□

8.6. Denote by 𝒞^{c} = ind A \ 𝒞 the complement of 𝒞 in ind A and by ℰ_{1} the subcategory of Ext-injectives of the first kind (see (6.2)). We define the almost complement of 𝒞 to be the full subcategory 𝒞^{∗} = 𝒞^{c} ∪ ℰ_{1} of ind A. If, for instance, 𝒞 = ℒ_{A}, then 𝒞^{∗} is the class ℛ_{0} consisting of all M ∈ ind A such that there exist an injective module I and a path I ⇝ M in ind A (see [2, (5.1)]).

Lemma. 𝒞^{∗} is closed under successors.

Proof. Assume X ∈ 𝒞^{∗} and we have a path X ⇝ Y in ind A. If X ∈ 𝒞^{c}, then Y ∈ ind A. If X ∈ ℰ_{1}, then there is an injective I and a path I ⇝ X. If Y ∈ 𝒞^{c} there is nothing to show while, if Y ∈ 𝒞, the composed path I ⇝ X ⇝ Y yields Y ∈ ℰ_{1}, by (6.1). □

8.7. We set E_{1} = ⨁_{X∈ℰ_{1}} X, E_{2} = ⨁_{Y ∈ℰ_{2}} Y and U = E_{1} ⊕ τ_{A}^{−1}E_{2} ⊕ F (note that, by definition, no summand of E_{2} is injective).

Lemma. (a) Let M ∈ ind A, then:

(i) M is Ext-projective in add 𝒞^{∗} if and only if M ∈ add U; (ii) M is Ext-injective in add 𝒞^{∗} if and only if M is injective.

(b) U is a partial tilting module.

(c) U is a tilting module if and only if T = E_{1} ⊕ E_{2} ⊕ F is a tilting module.

(d) If U is a tilting module, then the resulting torsion pair (𝒯 (U), ℱ (U)) is given by 𝒯 (U) = add 𝒞^{∗} and ℱ (U) = add (ind A \ 𝒞^{∗}).

Proof. The proofs of [2, (5.3)(5.4)(5.5)] apply with the obvious changes. □

8.8. We now generalise [2, (Theorem B)]. For a functor F on mod A, we denote by Ker F the full subcategory of mod A consisting of the modules M such that F M = 0.

Theorem. Let A be an artin algebra and 𝒞 ⊆ ℒ_{A} be a full subcategory closed under predecessors. The following are equivalent:

(a) A is 𝒞 -supported;

(b) add 𝒞^{∗} is covariantly finite;

(c) add 𝒞^{∗} = Gen U;

(d) U is a tilting A-module;

(e) 𝒞^{∗} = Supp (U, −);

(f) there exists an almost directed module R_{A} such that 𝒞^{∗} = Supp (R, −);

(g) there exists a module R_{A} such that Hom_{A}(R, τ_{A}R) = 0 and 𝒞^{∗} = Supp (R, −);

(h) add 𝒞^{∗} = Ker Ext_{A}^{1}(U, −);

(i) Ker Hom_{A}(U, −) = add (𝒞 \ ℰ_{1}).

Proof. The proof of [2, (5.6)] applies with the obvious changes. □

8.9. Corollary. Let 𝒞 ⊆ ℒ_{A} be a full subcategory closed under predecessors, then add 𝒞 is contravariantly finite if and only if add 𝒞^{c} is covariantly finite.

Proof. The proof of [2, (5.8)] applies with the obvious changes. □

8.10. Corollary. Let A be 𝒞 -supported, then F is the Bongartz complement of E_{1} ⊕ τ_{A}^{−1}E_{2}.

Proof. The proof of [2, (5.10)] applies with the obvious changes. □

Acknowledgment. The author gratefully acknowledges partial support from the NSERC of Canada and the Université de Sherbrooke. He also thanks Shiping Liu, David Smith and Sonia Trepode for useful discussions.

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Département de mathématiques, Faculté des sciences, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada

E-mail address: Ibrahim.Assem@USherbrooke.ca