REPRESENTATIONS OF FINITE DIRECTED CATEGORIES

LIPING LI

Abstract. A finite directed category is a k-linear category with finitely many objects and an underlying poset structure, where k is an algebraically closed field. This concept unifies structures such as k-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, estimate their global dimensions and finitistic dimensions, and discuss their stratification properties. We also show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.

1. Introduction

It is worth to point out that in representation theory many structures people are interested in have underlying posets. Specific examples include posets, directed quivers, quotient algebras of finite-dimensional hereditary algebras (in particular, piecewise hereditary algebras, see [10, 11]), Auslander algebras of representationdirected algebras, triangular matrix algebras, transporter categories (see [25]), orbit categories ([22]), fusion systems ([17]), and skeletal finite EI categories (i.e., finite categories such that every endomorphism is an isomorphism, see [6, 12, 13, 18, 21, 22, 24]). Therefore, it makes sense to define a concept unifying these structures, study their representations and homological properties, and generalize many existed but sporadic results.

This concept has been defined in [14, 15], which we call finite directed categories. By definition, a finite directed category A is a k-linear category with finitely many objects, where k is an algebraically closed field, satisfying the following properties: A is locally finite, i.e., for two objects x, y ∈ Ob A, A(x, y) is a finite-dimensional vector space; there is a partial order ⩽ on Ob A such that A(x, y) ≠ 0 implies x ⩽ y. Note that we can extend this partial order to a linear order with respect to which A is still directed. Indeed, let O_{1} be the set of all minimal objects in Ob A; let O_{2} be the set of all minimal objects in Ob A\O_{1}, and so on. Define an arbitrary linear order ⩽_{i} for each set O_{i}. For two objects x, y ∈ Ob A, we then define x < y if x <_{i} y for some i, or x ∈ O_{i}, y ∈ O_{j}, and i < j. The order defined in this way is indeed linear, and A is directed with respect to it. Therefore, without loss of generality we assume that the partial order ⩽ is linear. We also suppose that A is connected. That is, for x, y ∈ Ob A, there is a sequence of objects x = x_{0}, x_{1}, …, x_{n} = y such that either A(x_{i}, x_{i+1}) ≠ 0 or A(x_{i+1}, x_{i}) ≠ 0, 0 ⩽ i ⩽ n − 1.

A representation R of A is a k-linear covariant functor from A to k-vec, the category of finite-dimensional vector spaces. Note that by Gabriel’s construction ([4]), A (more precisely, the space of all morphisms in A) can be viewed as a finitedimensional algebra A, and the category of representations of A can be identified with A-mod, the category of finitely generated A-modules. ^{1} We call A the associated algebra of A, and call A the associated category of A. By abuse of notation, we identify the category A with the algebra A, and call R an A-module.

It is clear from this definition that k-linearizations of finite posets, transporter categories, fusion systems, orbit categories, and skeletal finite EI categories are indeed directed categories. Furthermore, finite-dimensional hereditary algebras and their quotient algebras, and triangular matrix algebras can be viewed as directed categories in an obvious way. It is also clear from the definition that every directed category A is skeletal. However, the corresponding algebra A might not be basic since for x ∈ Ob A, the endomorphism algebra A(x, x) might not be basic. In the case that A(x, x) is a local algebra, we call x a primitive object. If every object in A is primitive, then the associated algebra A is basic.

In the next section we introduce some elementary results on representations of of directed categories, describe the indecomposable projective modules and simple modules, and study the induction and restriction functors with respect to full subcategories (which are also directed). Corresponding results for finite EI categories have been explored in [21, 24].

We then study the homological dimensions of A. This has been explored by by Xu ([24]) and Dietrich ([7]) when A is the k-linearization of a finite EI category. We show that gldim A, the global dimension of A, is of finite if and only if so are the global dimensions of endomorphism algebras of all objects in A. Moreover, if the finitistic dimensions of endomorphism algebras of all objects are finite, so is the finitistic dimension of A. Best possible upper bounds of these homological dimensions are obtained, based on the underlying poset structure of A and the corresponding homological dimensions of endomorphisms algebras of objects. In particular, we answer a question described in [11].

Directed categories have nice stratification properties. Explicitly, every directed category is stratified with respect to a preorder ≼ determined by the given linear order ⩽ on Ob A, and standard modules with respect to ≼ coincide with indecomposable summands of endomorphism algebras of objects. Directed categories standardly stratified with respect to ≼ have been characterized in [14]. In Section 4 we give more properties. In particular, we prove that the associated category of an arbitrary finite-dimensional algebra is a directed category with respect to a linear order if and only if the composition factors of every standard module with respect to this linear order are all isomorphic, if and only if all proper standard modules are simple. We also show that when every object in A is primitive, and A is standardly stratified with respect to ⩽, then an A-module M has finite projective dimension if and only if it has a filtration by standard modules, if and only if its value M(x) on each object x ∈ Ob A is a free A(x, x)-module. In other words, under the assumptions F(Δ), the category of all finitely generated A-modules with filtrations by standard modules, coincide with P^{f}(A), the category of finitely generated A-modules with finite projective dimension. The problem whether these two important subcategories of A-mod coincide has been considered by Platzeck and Reiten in [19].

Let Λ be a finite-dimensional basic k-algebra and suppose that it has a simple projective module S. The APR tilting module is defined in [3] as Q⊕τ^{−1}S, where τ is the Auslander-Reiten translation, and Q is the direct sum of all indecomposable projective A-modules (up to isomorphism) except S. An observation tells us that

1This result is true for all locally finite k-linear categories with finitely many objects, even if it is not directed.

under the given assumption Λ ≅ [\begin{matrix}Λ_{1} & 0 \\ M & k \end{matrix}] is a triangular matrix algebra, and hence

can be viewed as a directed category. It is natural to ask whether general APR

tilting modules exist for arbitrary triangular matrix algebras [\begin{matrix}Λ_{1} & 0 \\ M & Λ_{2} \end{matrix}] where Λ_{2}

is a local algebra. In the last section we show the existence of such APR tilting modules under suitable conditions.

We introduce the notation and convention here. Throughout this paper A is a connected directed category with respect to a fixed linear order ⩽ on Ob A, and its associated algebra is denoted by A. Sometimes we consider an arbitrary algebra and denote it by Λ to distinguish it from A. For every x ∈ Ob A, we let 1_{x} be the identity morphism, which is also an idempotent in A. The symbol [n] is the set of all positive integers from 1 to n. All modules we consider in this paper are left finitely generated modules if we make no other claim. Composite of maps, morphisms and actions is from right to left. To simplify the expression of statements, we view the zero module as a projective or a free module.

2. Preliminaries

We first give some examples of directed categories. Let Λ be a quotient algebra of a finite-dimensional hereditary algebra, and let Q be the ordinary quiver. Then Λ can be regarded as a finite directed category. Objects are just the vertices of Q, and morphisms from vertex v to vertex w are elements in 1_{w}Λ1_{v}.

By definition, a finite EI category E is a small category with finitely many morphisms such that every endomorphism is an isomorphisms. Examples of finite EI categories includes finite posets, transporter categories [25], orbit categories [22], and fusion systems [17]. When E is skeletal, we can define a partial order ≼ on Ob E as follows: for x, y ∈ Ob E such that E(x, y) ≠ ∅, we let x ≼ y. As we did in the introduction, we can extend this partial order to a linear order ⩽, with respect to which the k-linearization of E is a directed category.

Let A_{1} and A_{2} be two finite-dimensional k-algebras and let M be a (A_{2}, A_{1})-

bimodule. Then we can construct the triangular matrix algebra A = [\begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix}].

The elements of A are 2×2 matrices [\begin{matrix}a & 0 \\ v & b \end{matrix}], where a ∈ A_{1}, b ∈ A_{2}, v ∈ M. Addition

and multiplication are defined by the usual operations on matrices. For details, see [2]. The associated category of A is a directed category with the following structure:

[figure: A :; x; A_{1}; M; ...; y; A_{2}].

Conversely, given a directed category A, its associated algebra A is a triangular matrix algebra. Indeed, let x be a maximal object in A with respect to ⩽, and let ϵ = ∑_{x≠z∈Ob A} 1_{z}. Define A_{1} = ϵAϵ, A_{2} = 1_{x}A1_{x}, and M = 1_{x}Aϵ. Note that ϵA1_{x} = ϵA1_{x} = 0 since there is no nonzero morphisms from objects different from

x to x. Consequently, A = [\begin{matrix}A_{1} & 0 \\ M & A_{2} \end{matrix}] is a triangular matrix algebra.

Let Λ be a finite-dimensional algebra standardly stratified with respect to a linear order on isomorphism classes of simple modules, and let Γ be the extension algebra of standard modules. In [15] we show that the associated k-linear category of Γ is a directed category with respect to this linear order. In [16] we show that if Λ is standardly stratified with respect to all linear orders on isomorphism classes of simple modules, then the associated category of Λ is directed.

Let Λ be a finite-dimensional algebra. A path in Λ-mod is a sequence

\xymatrix{M_{0} \ar[r]^{f_{1}} & M_{1} \ar[r]^{f_{2}} \ar[r] & … \ar[r]^{f_{t}} & M_{t}}

of nonzero nonisomorphisms f_{1}, …, f_{t}, where all modules in this sequence are indecomposable. It is a cycle if M_{0} ≅ M_{t}. A Λ-module is called a directed module if it appears in no cycles. The algebra Λ is called representation-directed if every indecomposable Λ-module is directed. It is known that every representation-directed algebra has finite representation type. Conversely, if Λ is a hereditary or tilted algebra of finite representation type, then it is representation-directed (see Lemma 1.1 and Corollary 3.4 in Chapter IX, [1].

The following proposition gives us a good relation between representation-directed algebras and directed categories. Recall that for an algebra Λ of finite representation type, its Auslander algebra is the endomorphism algebra of the direct sum of all indecomposable Λ-modules (up to isomorphism).

Proposition 2.1. Let Λ be a finite-dimensional algebra of finite representation type and let A be its Auslander algebra. Then the following are equivalent:

(1) Λ is a representation-directed algebra.

(2) The associated category A of A is a directed category.

(3) A is a quotient algebra of a finite-dimensional hereditary algebra.

Proof. Let M = ⨁_{i∈[n]} M_{i} be the direct sum of all indecomposable Λ-modules (up to isomorphism). Note that the associated category of A = End_{Λ}(M) has the following structure. Its objects are are indexed by M_{i}. By abuse of notation, we still denote these objects by M_{i}, i ∈ [n]. For two objects M_{i} and M_{j}, A(M_{i}, M_{j}) = Hom_{Λ}(M_{i}, M_{j}), which is an (End_{Λ}(M_{j}), End_{Λ}(M_{i}))-bimodule. Now it is straightforward to see that A is a directed category if and only if there is no sequences of nonisomorphic indecomposable Λ-modules M_{1}, …, M_{t} such that Hom_{Λ}(M_{1}, M_{2}) ≠ 0, …, Hom_{Λ}(M_{t}, M_{1}) ≠ 0, i.e., every indecomposable Λ-module is not in a cycle. Therefore, (1) is equivalent to (2).

Clearly (3) implies (2). We finish the proof by showing (1) implies (3). We already know that A is a directed category. By Proposition 1.4 in Chapter IX [1], the endomorphism algebra of every directed module is one-dimensional, so End_{Λ}(M_{i}) ≅ k for all i ∈ [n]. Therefore, A is indeed a quotient algebra of a finite-dimensional hereditary algebra. □

Already given enough examples, we turn to study representations of A. Recall a representation R of A is a k-linear covariant functor from A to k-vec. For x ∈ Ob A, the value of R on x is defined as R(x). The support of R is defined to be the set of objects x such that R(x) ≠ 0, and is denoted by supp(R). We say R is generated by its value on x_{1}, …, x_{n} if R = A ∑_{i∈[n]} R(x_{i}) and call {x_{i}}_{i∈[n]} a generating set of R. If the set {x_{i}}_{i∈[n]} is contained in every generating set of R, it is called a minimal generating set of R.

Note that the identity morphisms 1_{x}, when x ranges over all objects in A, form a set of orthogonal idempotents in A, although they might not be primitive. Therefore, we have an A-module decomposition _{A}A ≅ ⨁_{x∈Ob A} A1_{x}, where A1_{x} is the space of all morphisms starting from x. Let E_{x} be a chosen set of orthogonal primitive idempotents in A(x, x) such that ∑_{e∈E_{x}} e = 1_{x}. Then E = ⊔_{x∈Ob A}E_{x} is a set of primitive orthogonal idempotents of A with ∑_{e∈E} e = 1. Furthermore, the space constituted of all non-endomorphisms in A is a two-sided ideal of A. Therefore, the space constituted of all endomorphisms in A is a quotient algebra of A, and can be viewed as an A-module. Also observe that for every x ∈ Ob A, a simple A(x, x)-module can be lifted to a simple A-module supported on x. These observations give us a description of indecomposable projective A-modules and simple A-modules.

Proposition 2.2. Let A be a connected finite directed category and A be the associated algebra. Let R be a representation of A. Then:

(1) Every indecomposable projective A-module is isomorphic to Ae with e ∈ E_{x} for some x ∈ Ob A.

(2) Every simple A-module can be identified with a simple A(x, x)-module for some x ∈ Ob A.

(3) For every x ∈ Ob A, the value R(x) = 1_{x}R ≅ Hom_{A}(A1_{x}, R).

(4) The minimal generating set of R exists, and is unique.

(5) If R is an indecomposable projective A-module generated by R(x), then R(x) is a projective A(x, x)-module.

Proof. The first two statements are straightforward. The third statement follows from the equivalence between the category of representations of A and the category A-mod. It is clear that the minimal generating set of R coincide with the support of the Top(R), where Top(R) = R/ rad R, which clearly exists and is unique since the generating set of every simple module is a set containing a single object. If R is indecomposable and projective, then R ≅ Ae, where e is a primitive idempotent in A(x, x). Therefore, R(x) ≅ 1_{x}Ae is a summand of A(x, x) = 1_{x}A1_{x} up to isomorphism. □

In the rest of this section we consider the behaviors of induction and restriction functors. Let B be a subcategory of A, and let V and W be an A-module and a B-module respectively. The induction functor is ↑_{B}^{A}= A ⊗_{B} −, sending W to A ⊗_{B} W. Since the associated algebra B of B is a subalgebra of A, this functor is well defined. On the other hand, the restriction functor ↓_{B}^{A} sends V to 1_{B} · V, which is a B-module.

Suppose that B is a full subcategory of A. We say B is an ideal of A if whenever x ∈ Ob B, then every y ∈ Ob A with y ⩽ x is also contained in Ob B. Dually, we define co-ideals of A. It is not hard to see that if B is an ideal of A, then the associated algebra B is a right ideal of A. Dually, if B is a co-ideal of A, then the associated algebra B is a left ideal of A.

Proposition 2.3. Suppose that B is a (connected) full subcategory of A. Let V and V′ be A-modules, and W be a B-module. We have:

(1) W ↑_{B}^{A}↓_{B}^{A}≅ W.

(2) If W is indecomposable, then W ↑_{B}^{A} is indecomposable.

(3) If B is an ideal of A, then ↓_{B}^{A} preserves left projective modules.

(4) If B is a co-ideal of A, then ↓_{B}^{A} preserves right projective modules.

(5) If supp(V) is contained in Ob B, and B is a co-ideal of A, then we have Ext_{A}^{i}(V, V′) ≅ Ext_{B}^{i}(V ↓_{B}^{A}, V′ ↓_{B}^{A}) for i ⩾ 0.

Proof. These results have been described in [24] in the context of finite EI categories, and the proofs are essentially the same. For details, please refer to that paper.

(1): By definition, we have

W ↑_{B}^{A}↓_{B}^{A} = 1_{B} · (A ⊗_{B} W) = 1_{B}A ⊗_{B} 1_{B}W
= 1_{B}A1_{B} ⊗_{B} W = B ⊗_{B} W ≅ W

since B is a full subcategory and 1_{B}A1_{B} can be identified with B.

(2): Suppose W ↑_{B}^{A} is decomposable. Then we can write W ↑_{B}^{A}= M_{1} ⊕ M_{2}, where both M_{1} and M_{2} are nonzero. But then

W ≅ W ↑_{B}^{A}↓_{B}^{A}= M_{1} ↓_{B}^{A} ⊕M_{2} ↓_{B}^{A},

so either M_{1} ↓_{B}^{A}= 0 or M_{2} ↓_{B}^{A}= 0. Without loss of generality we assume M_{2} ↓_{B}^{A}= 0.

Let G be the minimal generating set of W, so G ⊆ Ob B and W = B·∑_{x∈G} W (x). Since

W ↑_{B}^{A}= A ⊗_{B} W = A ⊗_{B} (B · ∑_{x∈G} W (x)) = A · ∑_{x∈G}(1_{B} ⊗_{B} W (x)),

W ↑_{B}^{A} and hence M_{2} are generated by its values on elements in G. But M_{2} ↓_{B}^{A}= 0 implies that the values of M_{2} on all objects in G ⊆ Ob B are all 0. Therefore, M_{2} = 0. This contradiction tells us that W ↑_{B}^{A} is indecomposable.

(3): Let P ≅ Ae be a projective A-module. Without loss of generality we can assume that P is indecomposable, so P ≅ Ae, where by the previous proposition e is a primitive idempotent in A(x, x) for some x ∈ Ob A.

By definition, P ↓_{B}^{A}≅ 1_{B}Ae. Note that 1_{B}A constitutes of all morphisms in A ending at some object y ∈ Ob B. Since B is an ideal, by definition, there is no nonzero morphism in A staring from an object in Ob A \ Ob B and ending at an object in Ob B. Therefore, 1_{B}A = 1_{B}A1_{B} = B, so P ↓_{B}^{A}≅ 1_{B}Ae = Be. If y ∉ Ob B, the last term in the above identity is 0. Otherwise, it is a nonzero projective B-module.

(4): This is a dual statement of (3).

(5): First, since B is a co-ideal, every B-module can be viewed as an A-module, whose values on objects not contained in Ob B are all zero. Conversely, given an A-module whose values on objects not in Ob B are all zero, it can be regarded as a B-module.

By Eckmann-Shapiro Lemma, Ext_{B}^{i}(V ↓_{B}^{A}, V′ ↓_{B}^{A}) ≅ Ext_{A}^{i}(V ↓_{B}^{A}↑_{B}^{A}, V′) for i ⩾ 0. By the above observation, V ↓_{B}^{A}↑_{B}^{A}≅ V, and the conclusion follows. □

An immediate result of this proposition is:

Corollary 2.4. If A is of finite representation type, so is every full subcategory.

Proof. Let B be a full subcategory of A. If B has infinitely many non-isomorphic indecomposable representations, applying the induction functor, we get infinitely many non-isomorphic indecomposable representations by (2) of the above proposition. These induced indecomposable representations are non-isomorphic by (1) of the above proposition since restricted to B they are non-isomorphic. The conclusion follows. □

3. Homological dimensions

In this section we consider the homological dimensions of A. Specifically, we are interested in gldim A, the global dimension of A, and fidim A, the finitistic dimension of A. These dimensions have been studied in [24] and [7] for finite EI categories. We generalize their results to the context of directed categories. By definition, fidim A = sup{pd_{A} M | M ∈ A-mod and pd_{A} M < ∞}, where pd_{A} M is the projective dimension of M. The finitistic dimension conjecture asserts that the finitistic dimension of every finite-dimensional algebra is finite. See [26] for details.

Lemma 3.1. Let V be an A-module and x be a minimal object in supp(V) with respect to the linear order inherited from ⩽. Suppose that

\xymatrix{… \ar[r] & P^{2} \ar[r] & P^{1} \ar[r] & P^{0} \ar[r] & V \ar[r] & 0}.

is a minimal projective resolution of V. Then the exact sequence

\xymatrix{… \ar[r] & 1_{x}P^{1} = P^{1}(x) \ar[r] & 1_{x}P^{0} = P^{0}(x) \ar[r] & 1_{x}V = V (x) \ar[r] & 0}.

obtained by applying the functor Hom_{A}(A1_{x}, −) is a projective resolution of the A(x, x)-module V (x). In particular, if pd_{A} V < ∞, so is pd_{A(x,x)} V (x).

Proof. Let B be the full subcategory of A constituted of all objects y satisfying that there is some z ∈ supp(V) with z ⩽ y. The reader can check that B is a co-ideal of A, and is the minimal co-ideal of A contains all objects in supp(V). Since V is supported on supp(V) and A is a directed category, supp(P^{0}) is contained Ob B, so is the first syzygy ΩV. By induction, all P^{i} are supported on Ob B. Therefore, for each i ⩾ 0, we can write P^{i} = P_{1}^{i} ⊕ P_{2}^{i}, where P_{1}^{i} is a projective A-module generated by its value P_{1}^{i}(x) on x, and P_{2}^{i} is a projective A-module generated by its values on objects in B except x. In other words, P_{2}^{i}(x) = 0 since x is a minimal object in B.

Applying the exact functor Hom_{A}(A1_{x}, −) to the given projective resolution, we get an exact sequence of A(x, x)-modules

\xymatrix{… \ar[r] & 1_{x}P^{1} = P^{1}(x) \ar[r] & 1_{x}P^{0} = P^{0}(x) \ar[r] & 1_{x}V = V (x) \ar[r] & 0}.

Using the above decomposition and the fact that P_{2}^{i}(x) = 0, we deduce that this exact sequence is actually isomorphic to

\xymatrix{… \ar[r] & P_{1}^{1}(x) \ar[r] & P_{1}^{0}(x) \ar[r] & V (x) \ar[r] & 0}.

The first statement follows by observing that for every i ⩾ 0, P_{1}^{i}(x) is a projective A(x, x)-module. If pd_{A(x,x)} V (x) = ∞, then P^{i}(x) ≠ 0, so P^{i} ≠ 0 for every i ⩾ 0. Therefore, pd_{A} V = ∞. This proves the second statement. □

To describe the main result of this section, we need to introduce a few more notations. Consider the poset (Ob A, ⩽). A chain C in this poset is a sequence of objects x_{1} < x_{2} < … < x_{n}. It length l(C) is n − 1. We say C is a maximal chain if it is not properly contained in another chain. Note that a maximal chain in our sense does not necessarily have the maximal length.

To each chain C : x_{1} < x_{2} < … < x_{n} we assign two numbers g(C) and f(C) in the following way:

g(C) = ∑_{i=1}^{n} gldim A(x_{i}, x_{i}) + n − 1; f(C) = ∑_{i=1}^{n} fidim A(x_{i}, x_{i}) + n − 1.

Note that f(C) and g(C) can be ∞. We also define:

g(A) = max{g(C) | C is a chain in (Ob A, ⩽)},
f(A) = max{f(C) | C is a chain in (Ob A, ⩽)}.

which might be ∞ as well. Note that to find g(A) and f(A), we only need to check maximal chains. Indeed, for every chain C we can always embed it into a maximal chain C̃. Clearly, f(C) ⩽ f(C̃) and g(C) ⩽ g(C̃).

With these notation, we have:

Theorem 3.2. Let A be a connected finite directed category. Then:

(1) The global dimension of A is finite if and only if so is the endomorphism algebra of each object in A. Moreover, if gldim A < ∞, gldim A ⩽ g(A).

(2) If the finitistic dimension of the endomorphism algebra of every object in A is finite, so is fidim A, and fidim A ⩽ f(A).

Note under the assumption of the theorem, both g(A) and f(A) are finite. Thus these two numbers f(A) and g(A) give upper bounds for the finitistic dimension and global dimension respectively. It is clear that these upper bounds are best possible by considering the trivial example A = k.

Proof. We use the same idea to show the conclusions of (1) and (2). Therefore, we give a detailed proof for (1) and a sketched proof for (2).

(1): Suppose that there is some x ∈ Ob A such that gldim A(x, x) = ∞. Then there is a simple A(x, x)-module S_{x} such that the projective dimension of S_{x} is infinity. We can regard S_{x} as a simple A-module. Therefore, by the above lemma, pd_{A} S_{x} = ∞. In particular, gldim A = ∞.

Conversely, suppose that for every x ∈ Ob A, gldim A(x, x) < ∞. We show gldim A ⩽ g(A) by induction on | Ob A|, the size of Ob A, which clearly implies gldim A < ∞. It suffices to show the projective dimension of every simple A-module is bounded by this number.

The conclusion is trivially true for directed categories with one object. Suppose that it is also true for directed categories with at most n objects, and let A be a finite directed category with n + 1 objects. Take an arbitrary simple module S_{x} supported on the object x. We have two cases.

Case I: x is not a minimal object in the poset (Ob A, ⩽). Then we can find a minimal object y such that y > x. Let B be the full subcategory of A constituted of all objects except y. In particular, x ∈ Ob B. Note that B is a co-ideal of A. By (5) of Proposition 2.3, Ext_{A}^{i}(S_{x}, V) ≅ Ext_{B}^{i}(S_{x} ↓_{B}^{A}, V ↓_{B}^{A}) for every B-module V and every i ⩾ 0. Since gldim B ⩽ g(B) < ∞ by the induction hypothesis, we conclude that Ext_{A}^{i}(S_{x}, V) = 0 for every i > g(B). Clearly, g(B) ⩽ g(A) since every chain in (Ob B, ⩽) is also a chain in (Ob A, ⩽). Consequently, pd_{A} S_{x} ⩽ g(B) ⩽ g(A) < ∞.

Case II: x is a minimal object in the poset (Ob A, ⩽). Let B be the full subcategory of A constituted of all objects y with y ⩾ x, and let C be the full subcategory of A constituted of all objects y with y > x. The reader can check that both B and C are co-ideals of A, and C is a co-ideal of B.

Again, consider a minimal projective resolution P^{•} of the simple A-module S_{x}:

\xymatrix{… \ar[r] & P^{2} \ar[r] & P^{1} \ar[r] & P^{0} \ar[r] & S_{x} \ar[r] & 0}.

Let Ω^{i}S_{x} be the i-th syzygy, i ⩾ 0. Since S_{x} is supported on Ob B, so are P^{0} and ΩS_{x}. By induction, we can show that all projective modules and syzygies in this sequence are supported on Ob B.

Applying the exact functor Hom_{A}(A1_{x}, −), we get a projective resolution of the simple A(x, x)-module S_{x}:

\xymatrix{… \ar[r] & P^{2}(x) \ar[r] & P^{1}(x) \ar[r] & P^{0}(x) \ar[r] & S_{x} \ar[r] & 0}.

Since pd_{A(x,x)} S_{x} ⩽ gldim A(x, x) = s < ∞, we conclude that Ω^{s}S_{x}(x), the value of Ω^{s}S_{x} on x, is a projective A(x, x)-module. Again, for every i ⩾ 0, write P^{i} = P_{1}^{i} ⊕ P_{2}^{i}, where P_{1}^{i} is generated by its value on x, and P_{2}^{i} is generated by its values on other objects different from x. Then P_{2}^{i}(x) = 0 since x is a minimal object. Since P^{•} is a minimal resolution, we deduce P^{s}(x) ≅ P_{1}^{s}(x) ≅ Ω^{s}S_{x}(x) as A(x, x)-modules, so Ω^{s+i}S_{x} and hence P^{s+i} are only supported on Ob C for i ⩾ 1. Using (5) of Proposition 2.3 and the same reasoning as in Case I, we deduce that

pd_{A} Ω^{s+1}S_{x} = pd_{B} Ω^{s+1}S_{x} = pd_{C} Ω^{s+1}S_{x} ⩽ g(C).

Therefore, pd_{A} S_{x} ⩽ s + 1 + g(C).

Note that g(C) < ∞ by the induction hypothesis. Therefore, by our definition, we can find a chain C : x_{1} < x_{2} < … < x_{n} such that g(C) = g(C). Extending C to a new chain C̃ : x < x_{1} < … < x_{n} in (Ob A, ⩽), we find that

g(C̃) = gldim A(x, x) + 1 + g(C) = s + 1 + g(C),

so pd_{A} S_{x} ⩽ s + 1 + g(C) = g(C̃) ⩽ g(A). The conclusion follows from induction. This proves (1).

(2): The conclusion is trivially true if | Ob A| = 1. Suppose that it is also true for finite directed categories with at most n objects, and let A be a finite directed category with n + 1 objects. Taking an A-module V such that pd_{A} V < ∞, we want to show that pd_{A} V is bounded by f(A). We still have two cases.

If there is a minimal object x in A such that V (x) = 0, then let B be the full subcategory constituted of all objects in A except x. Again, B is a co-ideal of A, and we have

pd_{A} V = pd_{B} V ⩽ f(B) ⩽ f(A).

Otherwise, all minimal objects in A lie in supp(V). Let x = x_{1}, x_{2} …, x_{r} be all minimal objects in A. Without loss of generality we can assume

s = fidim A(x, x) ⩽ fidim A(x_{2}, x_{2}) ⩽ … ⩽ fidim A(x_{r}, x_{r}).

Let B be the full subcategory of A formed by all objects in A except x_{1}. Again, B is a co-ideal of A.

Take a minimal projective resolution P^{•} of V and apply Hom_{A}(A1_{x}, −) on it. We get a projective resolution P^{•}(x) of V (x) with syzygies Ω^{i}V (x). As we proved in the second case of (1), for r > s, Ω^{r}V and hence P^{r} are only supported on Ob B since pd_{A(x,x)} V (x) < ∞ and actually pd_{A(x,,x)} V (x) ⩽ s. By induction hypothesis,

pd_{A} V = s + 1 + pd_{A} Ω^{s+1}V = s + 1 + pd_{B} Ω^{s+1}V ⩽ s + 1 + f(B).

But since f(B) < ∞, we can find a chain C in Ob B such that f(C) = f(B). Since x_{1}, …, x_{r} are all minimal objects in Ob A, we can extend C to a new chain C̃ by adding some minimal object x_{i}, 1 ⩽ i ⩽ r. Then

pd_{A} V ⩽ s + 1 + f(B) = s + 1 + f(C) ⩽ fidim A(x_{i}, x_{i}) + 1 + f(C) = f(C̃) ⩽ f(A)

since by our choice s = fidim A(x, x) ⩽ fidim A(x_{i}, x_{i}). The conclusion follows from induction. □

In the case that A(x, x) is a group algebra for every x ∈ Ob A, this theorem generalizes the corresponding results of Xu and Dietrich in [24] and [7].

Let A be a basic piecewise hereditary algebra. Then it is a quotient algebra of a finite-dimensional hereditary algebra ([10]). In [11] Happel, Reiten and Smalø showed that for every A-module M, pd_{A} M + id_{A} M is bounded by n + 1, where id_{A} M is the injective dimension of M and n is the number of isomorphism classes of simple modules. They asked whether this estimate is best possible. Actually, we have the following result.

Corollary 3.3. Let A be a quotient algebra of a basic finite-dimensional hereditary k-algebra, and let M be an A-module. Then pd_{A} M + id_{A} M ⩽ n − 1, where n is the number of isomorphism classes of simple A-modules.

Proof. It is sufficient to show the inequality for simple A-modules. Let S be an arbitrary simple A-module. Note that the corresponding k-linear category A of A is a directed category. Then the simple A-module S is supported on a certain object x in A. Let B (resp., C) be the full subcategory of A formed by all objects y in A with y ⩾ x (resp., y ⩽ x). Therefore, B (resp., C) is a co-ideal (resp., an ideal) of A, and

pd_{A} S = pd_{A} S = pd_{B} S ⩽ gldim B = g(B) ⩽ | Ob B| − 1.

On the other hand, id_{A} S equals the right projective dimension of S. By considering C dually, we get

id_{A} S = id_{A} S = id_{C} S ⩽ gldim C = g(C) ⩽ | Ob C| − 1.

Putting these two inequalities together, we have

pd_{A} S + id_{A} S ⩽ | Ob B| + | Ob C| − 2 ⩽ | Ob A| + 1 − 2 = n − 1

since | Ob A| = n and Ob B ∩ Ob C = {x}. □

4. Stratification properties

In this section we study the stratification properties of directed categories. First we introduce some background knowledge on stratification theory. For more details, see [5, 8, 9, 23].

Let Λ be a finite-dimensional algebra and suppose that _{Λ}Λ has n indecomposable summands. Let ≼ be a preorder on the set [n] = {i | 1 ⩽ i ⩽ n}. For i ∈ [n], we let P_{i} be the corresponding indecomposable projective Λ-module, and let S_{i} be its top. According to [5], Λ is standardly stratified with respect to ≼ if there exist indecomposable modules Δ_{i}, called standard modules, such that the following conditions hold:

(1) the number of composition factors [Δ_{i}, S_{j}] = 0 unless j ≼ i for i, j ∈ [n];

(2) there is an exact sequence 0 → K_{i} → P_{i} → Δ_{i} → 0 for every i ∈ [n] such that K_{i} has a filtration by standard modules Δ_{j} with j ≻ i.

If furthermore the endomorphism algebra of every standard module has dimension 1, then Λ is called a quasi-hereditary algebra.

Actually, the i-th standard module Δ_{i} can be defined as the largest quotient of P_{i} all of whose composition factors S_{j} satisfy j ≼ i. This works for arbitrary algebras Λ even if it is not standardly stratified. The i-th proper standard module \overline{Δ}_{i} is defined to be the largest quotient of P_{i} all of whose composition factors S_{j} satisfy j ≺ i except for a single copy of S_{i}, where j ≺ i means j ≼ i but i ⪯̸ j. By considering indecomposable injective modules and largest submodules, we can define dually costandard modules ∇_{i} and proper costandard modules \overline{∇}_{i}. Let F_{Λ}(Δ) be the full subcategory of A-mod such that each module in it has a filtration by standard modules. Similarly we define categories F_{Λ}(\overline{Δ}), F_{Λ}(∇), and F_{Λ}(\overline{∇}).

It is clear that if Λ is standardly stratified with respect to ≼, then _{Λ}Λ ∈ F_{Λ}(Δ). The converse of this statement is also true if the partial order associated to ≼ is a linear order, as explained in 2.2.3 of [5] and pp 12-13 of [22]. Since this condition holds in our context, we take the equivalent condition. That is, we say Λ is standardly stratified with respect to ≼ if _{Λ}Λ ∈ F_{Λ}(Δ). It is said to be properly stratified if _{Λ}Λ ∈ F_{Λ}(Δ) ∩ F_{Λ}(\overline{Δ}). The reader can see from the definition that quasi-hereditary algebras are properly stratified, and properly stratified algebras are standardly stratified.

Now let A be a connected finite directed category with the linear order ⩽ on Ob A. This linear order ⩽ induces a preorder ≼ on the set of isomorphism classes of simple A-modules as follows. Recall in Section 2 we have chosen a fixed set E_{x} of primitive orthogonal idempotents with ∑_{e∈E_{x}} e = 1_{x} for every object x ∈ Ob A, and defined E to be the disjoint union of these sets. Therefore, for e ∈ E_{x} and e′ ∈ E_{y}, we let e ≼ e′ if x ⩽ y. The reader can check that ≼ defined in this way is indeed a preorder, but in general is not a partial order. Moreover, if every object x in A is primitive, i.e., 1_{x} is a primitive idempotent, then ≼ coincide with ⩽. Therefore, (E, ≼) is a preordered set indexing all indecomposable summands of _{A}A. Note that A might have isomorphic indecomposable summands. This is allowed since if P = Ae and Q = Af are isomorphic indecomposable projective A-modules, then we can find an object x and the corresponding set E_{x} such that both e and f lie in E_{x}. Therefore, we have e ≼ f and f ≼ e.

Results in the following proposition have been described in [14] (see Section 4) and [16].

Proposition 4.1. Let A and ≼ be as above. Then:

(1) Every standard module is isomorphic to an indecomposable summand of A(x, x) for some x ∈ Ob A, where we identify ⨁_{x∈Ob A} A(x, x) with the quotient module A/J and J is the two-sided ideal constituted of all nonendomorphisms in A.

(2) A is standardly stratified with respect to ≼ if and only if A(x, y) is a projective A(y, y)-module for all x, y ∈ Ob A.

In the rest of this section we assume that every object x in A is primitive. By definition, the identity morphism 1_{x} is a primitive idempotent in the associated algebra A. Therefore, the endomorphism algebra A(x, x) is a finite-dimensional local algebra. Consequently, the associated algebra A of A is a basic algebra since {1_{x}}_{x∈Ob A} is a set of primitive orthogonal idempotents satisfying ∑_{x∈Ob A} 1_{x} = 1 and A1_{x} ≅ A1_{y} if and only if x = y. Moreover, the preorder ≼ we defined before coincides with the given linear order ⩽. Examples of these finite directed categories are described in [15, 16]. The following proposition asserts that these directed categories are characterized by their stratification properties.

Proposition 4.2. Let A be a basic finite-dimensional algebra with n isomorphism classes of simple modules. Let ⩽ be a linear order on [n]. Then the following are equivalent:

(1) Every standard module Δ_{i} has only composition factors isomorphic to S_{i}, i ∈ [n].

(2) Every proper standard modules \overline{Δ}_{i} is simple, i.e., isomorphic to S_{i}, i ∈ [n].

(3) The associated category A is a directed category with respect to ⩽.

Note that this is true even if A is not standardly stratified.

Proof. Suppose that A is a directed category. Note that every object is primitive. By the previous proposition, every standard module Δ_{i} is supported on one object. Equivalently, Δ_{i} has only composition factors isomorphic to S_{i}. Clearly, \overline{Δ_{i}} ≅ S_{i}. Thus (3) implies (1) and (2). It is also clear that if Δ_{i} has composition factors not isomorphic to S_{i}, then the top of rad Δ_{i} must have a simple summand not isomorphic to S_{i}. Consequently, \overline{Δ}_{i} has composition factors not isomorphic to S_{i}, and hence is not simple. Thus (2) implies (1).

Now we prove (1) implies (3) by induction. Without loss of generality we assume that n is the maximal element in [n] with respect to ⩽. The conclusion is trivially true for n = 1. If n > 1, take e_{n} to be a primitive idempotent in A such that P_{n} = Ae_{n} is a projective cover of S_{n}. Clearly, P_{n} ≅ Δ_{n}, so it has only composition factors isomorphic to S_{n} by the given condition. It is straightforward to see that A has the following description where A_{1} = (1 − e_{n})A(1 − e_{n}) and A_{2} = e_{n}Ae_{n}.

[figure: •; A_{1}; e_{n}A(1−e_{n}); •; A_{2}]

By induction hypothesis, the associated category A_{1} of A_{1} is directed with respect to the linear order on [n−1] inherited from ⩽. Therefore, A is directed with respect to ⩽. □

Since all proper standard modules are simple, A is actually properly stratified with respect to ⩽. It is also straightforward to see that A is quasi-hereditary with respect to ⩽ if and only if A is a quotient algebra of a finite-dimensional hereditary algebra. Moreover, the reader can check that all costandard modules of A are precisely indecomposable injective modules.

For an arbitrary standardly stratified algebra Λ, it is well known that F_{Λ}(Δ) is closed under direct summands, extensions, kernels of epimorphisms, but in general it is not closed under cokernels of monomorphisms. Actually, F_{Λ}(Δ) has this property if and only if F_{Λ}(Δ) = P^{f}(Λ), where P^{f}Λ is the full subcategory of Λ-mod constituted of all objects with finite projective dimension. This simple observation gives a possible approach to answer the question of Platzeck and Reiten in [19]: under what conditions these two subcategories of Λ-mod coincide.

Proposition 4.3. Let Λ be as above. Then the following are equivalent: (1) F_{Λ}(Δ) = P^{f}(Λ).

(2) F_{Λ}(Δ) is closed under the cokernels of monomorphisms.

(3) The cokernel of every monomorphism ι : Δ_{i} → P is contained in F_{Λ}(Δ), where Δ_{i} is a standard module and P is an arbitrary projective module.

Proof. The equivalence of (2) and (3) is the first statement of Theorem 0.3 in [16]. Thus it is sufficient to show the equivalence of (1) and (2). If F_{Λ}(Δ) = P^{f}(Λ), then it is closed under cokernels of monomorphisms since P^{f}(Λ) has this property. Conversely, suppose that F_{Λ}(Δ) has this property. Take an arbitrary Λ-module M with pd_{Λ} M = n < ∞ and consider a minimal projective resolution P^{•} of M. Clearly, P^{s} = 0 for s > n, and Ω^{n}M ≅ P^{n} ∈ F_{Λ}(Δ). By considering the exact sequence 0 → Ω^{n}M → P^{n−1} → Ω^{n−1}M → 0 we deduce that Ω^{n−1}M ∈ F_{Λ}(Δ) since the first two terms lie in this category, and it is closed under cokernels of monomorphisms. Continuing this process we get M ∈ F_{Λ}(Δ). Therefore, P^{f}(Λ) ⊆ F_{Λ}(Δ). The other inclusion is clear. □

When A is a directed category, we have:

Proposition 4.4. If A is standardly stratified with respect to ⩽, then: (1) F_{A}(Δ) is closed under cokernels of monomorphisms.

(2) F_{A}(Δ) = P^{f}(A).

(3) An A-module M has finite projective dimension if and only if for every x ∈ Ob A, M(x) is a free A(x, x)-module.

Proof. The first statement is proved in Proposition 1.4 in [16], which implies the second one immediately. Now we prove (3).

Note that every standard module of A has the form of A(x, x) for some x ∈ Ob A. Therefore, if M(x) is a free module for each x ∈ Ob A, it has a filtration by standard modules, so is contained in F_{A}(Δ) = P^{f}(A). Conversely, if M ∈ F_{A}(Δ) = P^{f}(A), then it has a filtration by standard modules, and from the description of standard modules we see M(x) ≅ Δ_{x}^{s} ≅ A(x, x)^{s}, where s = [M : Δ_{x}] □

Therefore, if A is standardly stratified with respect to ⩽ and all objects are primitive, we have an explicit description for objects in P^{f}(A). Unfortunately, for arbitrary finite directed categories, we cannot find such a description. Indeed, in the following example we show that for every linear order with respect to which A is standardly stratified, the category of modules with filtrations by standard modules is always a proper subcategory of P^{f}(A).

Example 4.5. Let A be the path algebra of the following quiver with relations δ^{2} = δγ = δϵ = 0 and γα = ϵβ.

[figure: y; γ; x; α; β; w; δ; z; ϵ]

Indecomposable projective modules are described as follows:

P_{x} = \begin{aligned}& x \\ y & & z \\ & w \end{aligned}; P_{y} = \begin{matrix}y \\ w \end{matrix}; P_{z} = \begin{matrix}z \\ w \end{matrix}; P_{w} = \begin{matrix}w \\ w \end{matrix}.

But for every linear with respect to which A is standardly stratified, we can find an indecomposable module with finite projective dimension which does not have a filtration by standard modules. For example, if y > x > z > w, the standard modules are:

Δ_{x} = \begin{matrix}x \\ z \end{matrix}; Δ_{y} = \begin{matrix}y \\ w \end{matrix}; Δ_{z} = \begin{matrix}z \\ w \end{matrix}; Δ_{w} = \begin{matrix}w \\ w \end{matrix}.

Since 0 → P_{z} → P_{x} → M = \begin{matrix}x \\ y \end{matrix} → 0 is exact, pd_{A} M = 1. But M has no filtration

by standard modules.

The reader may want to know when a finite directed category A is quasihereditary with respect to the given linear order. The following proposition answer this question.

Corollary 4.6. Let A be a basic finite-dimensional algebra. Then the following are equivalent:

(1) A is a quotient algebra of a finite-dimensional hereditary algebra.

(2) A is standardly stratified with respect to a linear order ⩽, and all standard modules are simple.

(3) A is standardly stratified and F_{A}(Δ) = A-mod.

Proof. (1) ⇒ (2): If A is a quotient algebra of a finite-dimensional hereditary algebra, then A is a finite directed category, and the endomorphism algebra of every object is isomorphic to k. Clearly, A is standardly stratified with respect to ⩽ by (1) of Proposition 4.1. Moreover, all standard modules are simple by (2) of Proposition 4.1.

The equivalent of (2) and (3) is clear.

(3) ⇒ (1): Since F_{A}(Δ) = A-mod, it is closed under cokernels of monomorphisms. Then (1) is true by Theorem 0.3 in [16]. □

In [20] Ringel shows that for every finite-dimensional algebra Λ standardly stratified with respect to a linear order, there is a characteristic tilting module T, which is a generalized tilting module and is the Ext-injective object in the category F_{Λ}(Δ). That is, for every M ∈ F_{Λ}(Δ), Ext_{Λ}^{1}(M, T) = 0. We end this section by describing explicitly the structure of this characteristic tilting module for some finite directed categories A. Clearly, the indecomposable summands of T can be indexed by Ob A. That is, for x ∈ Ob A T_{x} is the indecomposable summand of T satisfying [T_{x} : Δ_{x}] = 1 and [T_{x} : Δ_{y}] = 0 for all y ⩽̸ x, and T = ⨁_{x∈Ob A} T_{x}.

Corollary 4.7. Let A be a finite directed category such that every object is primitive and the endomorphism algebra is self-injective, and suppose that for all x, y ∈ Ob A, A(y, x) is a right free A(y, y)-module. Then for every x ∈ Ob A, T_{x} ≅ ∇_{x} ≅ I_{x}, where I_{x} is the indecomposable injective A-module corresponding to x.

Proof. We already know ∇_{x} ≅ I_{x} (see the remark after Proposition 4.2). Also, since clearly Ext_{A}^{i}(M, I_{x}) = 0 for i ⩾ 1 and M ∈ F_{A}(Δ), it is enough to show I_{x} ∈ F_{A}(Δ). Note that I_{x} = D(Q_{x}), where D is the functor Hom_{A}(−, k), and Q_{x} = 1_{x}A is the space of morphisms ending at x. As a right A-module (or equivalently, a left A^{op}-module), the value Q_{x}(y) of Q_{x} on an arbitrary object y is A^{op}(x, y) = A(y, x), which is a right free A(y, y)-module. Therefore, the value I_{x}(y) is a left free D(A(y, y))-module. But A(y, y) is local and self-injective, so it is a Frobenius algebra. Therefore, I_{x}(y) is actually a left free A(y, y)-module. The conclusion then follows from Proposition 4.4. □

This result is trivially true if A is quasi-hereditary with respect to the given linear order ⩽. Indeed, in this case F_{A}(Δ) contains all A-modules.

The condition that for all x, y ∈ Ob A, A(y, x) is a right free A(y, y)-module is equivalent to saying that the opposite category A^{op} is standardly stratified with respect to the opposite linear order ⩽^{op}. It is also equivalent to saying that the right projective dimension of A^{op}(x, x) as an A^{op}-module is finite. This condition cannot be dropped, as explained by the following example. This example also tells us that the associated category of the Ringel dual End_{A}(T)^{op} might not be a directed category.

Example 4.8. Let A be the following path algebra with relations αδ = δ^{2} = 0.

[figure: x; δ; α; y]

It is easy to check that A is standardly stratified with respect to the order x < y. Projective modules, standard modules, and injective modules are as follows:

P_{x} = \begin{aligned}& x \\ x & & y \end{aligned} P_{y} = y Δ_{x} = \begin{matrix}x \\ x \end{matrix} Δ_{y} = y I_{x} = \begin{matrix}x \\ x \end{matrix} I_{y} = \begin{matrix}x \\ y \end{matrix}

Clearly, I_{y} is not contained in F_{A}(Δ). Actually, the characteristic tilting module is T ≅ P_{x} ⊕ I_{x}.

Now let us consider Γ = End_{A}(T)^{op}. It is isomorphic to the path algebra of the following quiver with relation βαβ = 0, whose associated category is not directed.

\xymatrix{1 \ar[r]^{α} & 2. \ar[l]^{β}}

With respect to the order 1 < 2 the algebra Γ is standardly stratified. Its indecomposable projective modules, injective modules, and standard modules are listed below:

P_{1} = 1 2 1 2 P_{2} = 2 1 2 Δ_{1} = 1 Δ_{2} = 2 1 2 I_{1} = 1 2 1 I_{2} = 1 2 1 2

The characteristic tilting Γ-module T′ ≅ P_{1}⊕Δ_{1}. The opposite algebra of End_{Γ}(T′) is isomorphic to A, as claimed by Ringel’s duality.

5. Generalized APR tilting modules

Our main goal in this section is to prove the existence of generalized APR tilting modules for triangular matrix algebras. Let A be a basic finite-dimensional algebra. In [3] it is shown that if A has a simple projective module S, then T = Q ⊕ τ^{−1}S is a tilting module, called the APR tilting module, where Q is the direct sum of all indecomposable summands of _{A}A not isomorphic to S, and τ is the Auslander-Reiten translation. In particular, if A is a hereditary algebra, then the functor Hom_{A}(T, −) is precisely the BGP reflection functor ([1, 3]).

Let e be a primitive idempotent in A with Ae ≅ S and ϵ = 1 − e. A simple

observation tells us that A = [\begin{matrix}ϵAϵ & 0 \\ eAϵ & k \end{matrix}] is a triangular matrix algebra with the

following structure (called one-point trivial extension):

[figure: •; ϵAϵ; eAϵ; •]

Therefore, we may ask whether a generalized APR tilting module exists if A has a projective module P_{S} all of whose composition factors are isomorphic to the simple module S. In other words, there is a primitive idempotent e in A such that

A ≅ [\begin{matrix}ϵAϵ & 0 \\ eAϵ & eAe \end{matrix}] and P_{S} = Ae. Structure of A can be pictured as below:

[figure: •; ϵAϵ; eAϵ; •; eAe].

We introduce some notations here. Let A be the associated category of A. Then {1_{x}}_{x∈Ob A} is a set of primitive orthogonal idempotents in A. Let z be the object on which P_{S} = Ae is supported. That is, A1_{z} ≅ Ae ≅ P_{S}. Note that A need not be a directed category. However, we always have A(z, x) = 0 for z ≠ x ∈ Ob A. Let B be the full subcategory of A constituted of all objects x different from z. Then the associated algebra of B is exactly ϵAϵ, and A has the following description.

[figure: B; eAϵ; z; eAe]

Let A^{o} and A^{o} be the opposite algebra of A and the opposite category of A respectively. Let Q = Aϵ, which is the direct sum of all other indecomposable summands of _{A}A not isomorphic to P_{S}. Define T = Q ⊕ τ^{−1}P_{S}. Now the problem is to check under what conditions T is a tilting module. Since T has n indecomposable summands, it suffices to check pd_{A} T ⩽ 1 and Ext_{A}^{1}(T, T) = 0.

The following lemmas are crucial for the main result of this section.

Lemma 5.1. The projective dimension pd_{A} T ⩽ 1 if and only Hom_{A^{o}}(DP_{S}, A^{o}) = 0, where D = Hom_{k}(−, k).

Proof. Obviously, pd_{A} T ⩽ 1 if and only if pd_{A} τ^{−1}P_{S} ⩽ 1. Note A^{o} has the following structure:

[figure: A^{o} :; z; (eAe)^{o}; B^{o}] .

Take a projective presentation P^{1} → P^{0} → DP_{S} → 0, where all modules are left A^{o}-modules (or right A-modules). This presentation gives rise to the following exact sequence

(∗) : 0 → Hom_{A^{o}}(DP_{S}, A^{o}) → Hom_{A^{o}}(P^{0}, A^{o}) → Hom_{A^{o}}(P^{1}, A^{o}) → τ^{−1}P_{S} → 0.

Note the second term and the third term are projective A-modules. Therefore, pd_{A} τ^{−1}P_{S} ⩽ 1 if and only if the first term is a projective A-module. But DP_{S} = D(eAe) is only supported on z, so it cannot be a nonzero projective A^{o}-module by considering the structure of the connected category A^{o}. Therefore, pd_{A} τ^{−1}P_{S} ⩽ 1 if and only if the first term is 0. □

Lemma 5.2. The condition Hom_{A^{o}}(DP_{S}, A^{o}) = 0 holds if there is some x ∈ Ob A such that A(x, z) as a left A(z, z)-module has a free summand.

Proof. Note that DP_{S} as a left A^{o}-module is only supported on z, which is a minimal object in Ob A^{o}. Therefore, for all z ≠ x ∈ Ob A^{o},

Hom_{A^{o}}(DP_{S}, A^{o}1_{x}) = Hom_{A^{o}}(DP_{S}, A^{o}1_{x}) = 0

since A^{o}1_{x} is supported on objects different from z. Therefore, Hom_{A^{o}}(DP_{S}, A^{o}) = 0 if and only if Hom_{A^{o}}(DP_{S}, A^{o}1_{z}) = 0.

Let z ≠ x ∈ Ob A^{o} be an object such that A(x, z) has a free summand as a left A(z, z)-module. Take φ ∈ Hom_{A^{o}}(DP_{S}, A^{o}1_{z}). The homomorphism φ gives the following diagram by considering the values on x and z:

\xymatrix{DP_{S}(z) = D(eAe) \ar[r] \ar[d]^{φ_{z}} & DP_{S}(x) = 0 \ar[d]^{φ_{x}} \\ A^{o}1_{z}(z) = (eAe)^{o} \ar[r]^{ρ} & A^{o}1_{z}(x) = A(x, z).}

Note the given condition tells us that A^{o}(z, x) = A(x, z) has a free summand as a right A^{o}(z, z)-module, where the right action of A^{o}(z, z) on A^{o}(z, x) = A(x, z) is defined as follow: for δ ∈ A^{o}(z, z) and α ∈ A^{o}(z, x), α ∗ δ = δα. Write A^{o}(z, x) = ⨁_{i∈[s]} M_{i} as right A^{o}(z, z)-modules and without loss of generality suppose that M_{1} is a right free summand. Therefore, the map ρ is determined by s morphisms α_{1}, …, α_{s} in A^{o}(z, z) such that for every δ ∈ A^{o}(z, z),

ρ(δ) = ∑_{i∈[s]} α_{i} ∗ δ = ∑_{i∈[s]} δα_{i}

with δα_{i} ∈ M_{i} ⊆ A^{o}(z, x). In particular, since M_{1} is a right free A^{o}(z, z)-module, α_{1} induces a bijection between A^{o}(z, z) and M_{1}. Therefore, for 0 ≠ δ ∈ A^{o}(z, z), δα_{1} ≠ 0, so ρ(δ) ≠ 0, and hence ρ is injective. Consequently, from the above diagram we conclude φ_{z} = 0, so φ = 0. This finishes the proof. □

Now we are ready to prove the main result.

Theorem 5.3. Notation as before. Suppose that A(z, z) = eAe is a self-injective algebra, and there is some z ≠ x ∈ Ob A such that A(x, z) has a free summand as a left A(z, z)-module. Then T is a tilting module.

Proof. Under the given assumptions, we have shown that pd_{A} T ⩽ 1, so it suffices to show Ext_{A}^{1}(T, T) = 0. We have:

Ext_{A}^{1}(T, T) = Ext_{A}^{1}(Q ⊕ τ^{−1}P_{S}, Q ⊕ τ^{−1}P_{S}) = Ext_{A}^{1}(τ^{−1}P_{S}, Q ⊕ τ^{−1}P_{S})
≅ D\overline{Hom}_{A}(Q ⊕ τ^{−1}P_{S}, P_{S}),

where the last isomorphism follows from the Auslander-Reiten formula (see [2]), and \overline{Hom}_{A}(Q ⊕ τ^{−1}P_{S}, P_{S}) is the quotient space of Hom_{A}(Q ⊕ τ^{−1}P_{S}, P_{S}) modulo all homomorphisms factoring through injective A-modules. Therefore, it is sufficient to show Hom_{A}(Q ⊕ τ^{−1}P_{S}, P_{S}) = 0. Observing the structure of A we conclude Hom_{A}(Q, P_{S}) = Hom_{A}(Aϵ, Ae) ≅ ϵAe = 0. Therefore, it is enough to show Hom_{A}(τ^{−1}P_{S}, P_{S}) = 0.

In the proof of Lemma 5.1 we have actually constructed a projective resolution for τ^{−1}P_{S}:

(∗) : 0 → Hom_{A^{o}}(P^{0}, A^{o}) → Hom_{A^{o}}(P^{1}, A^{o}) → τ^{−1}P_{S} → 0.

Let Q′ = Hom_{A^{o}}(P^{1}, A^{o}). Thus the conclusion will follow if Hom_{A}(Q′, P_{S}) = 0.

Since P_{S} is only supported on z, so is DP_{S}. Moreover, DP_{S}(z) = D(eAe) ≅ (eAe)^{o} since eAe is local and self-injective. Therefore, P^{0} ≅ eA = 1_{z}A and P^{0}(z) ≅ DP_{S}(z), so the first syzygy ΩDP_{S} and hence P^{1} are supported on objects different from z. We can write P^{1} ≅ ⨁_{z≠x∈Ob A}(1_{x}A)^{n_{x}}, n_{x} ⩾ 0. Consequently,

Q′ = Hom_{A^{o}}(P^{1}, A^{o}) ≅ Hom_{A^{o}} ( ⨁_{z≠x∈Ob A}(1_{x}A)^{n_{x}}, A^{o}) ≅ ⨁_{z≠x∈Ob A}(A1_{x})^{n_{x}}

is a direct sum of summands of Q. But we have shown Hom_{A}(Q, P_{S}) = 0, so Hom_{A}(Q′, P_{S}) = 0. This finishes the proof. □

In the proof of this theorem we have shown Hom_{A}(τ^{−1}P_{S}, P_{S}) = 0. Therefore, Hom_{A}(τ^{−1}P_{S}, S) = 0. Indeed, since by the assumption P_{S} has only composition factors isomorphism to S, in particular its socle contains a simple summand isomorphic to S and there is an inclusion S → P_{S}. If Hom_{A}(τ^{−1}P_{S}, S) ≠ 0, then Hom_{A}(τ^{−1}P_{S}, P_{S}) ≠ 0 either. This is impossible. Therefore, as an A-module, the value of τ^{−1}P_{S} on z is 0. Consequentely, for any A-module M which is only supported on z, or equivalently, which only has composition factors isomorphic to S, we have Hom_{A}(τ^{−1}P_{S}, M) = 0.

The generalized APR tilting module T induces a torsion theory (T , F), where T constitutes of all quotient modules of T^{s} for some s ⩾ 0, and F is formed by all A-modules M such that Hom_{A}(T, M) = 0. We have:

Corollary 5.4. The category F constitutes of all A-modules M all of whose composition factors are isomorphic to S.

Proof. Suppose that M only has composition factors isomorphic to S. Clearly, Hom_{A}(Q, M) = 0. But we also have Hom_{A}(τ^{−1}P_{S}, M) = 0, so Hom_{A}(T, M) = 0, and M is contained in F. Conversely, for every A-module X having a composition factor T not isomorphic to S, there exists a summand Q_{T} of Q such that Hom_{A}(Q_{T} , X) ≠ 0, so X is not contained in F. □

The torsion theory (T , F) induced by the generalized APR tilting module T in general is neither separating nor splitting (See [1] for definitions). Indeed, By Theorem 5.6 in Page 230 [1], it is splitting if and only if the injective dimension id_{A} M ⩽ 1 for every M ∈ F. If A(z, z) = eAe is not isomorphic to k, then the simple module S (which is in F) has infinite injective dimension.

We end this section with two examples.

Example 5.5. Let A be pictured below with relations βρ = δβ, δ^{2} = ρ^{2} = δα = 0.

[figure: x; α; z; δ; y; ρ; β].

Left projective modules P_{x}, P_{y}, P_{z} are pictured as below:

\xymatrix{x \ar[d]^{α} &  & y \ar[dl]^{ρ} \ar[dr]^{β} &  & z \ar[d]^{δ} \\ z & y \ar[dr]^{β} &  & z \ar[dl]^{δ} & z \\  &  & z}

Right projective modules Q_{x}, Q_{y}, Q_{z} are pictured as below:

x  \xymatrix{y \ar[d]^{ρ} \\ y}  \xymatrix{ & z \ar[dl]^{δ} \ar[d]^{β} \ar[dr]^{α} \\ z \ar[dr]^{β} & y \ar[d]^{ρ} & x \\  & y}

By the above theorem, P_{x} ⊕ P_{y} ⊕ τ^{−1}P_{z} is a generalized APR tilting module.

Let A be the associated algebra of A. As a right A-module, DP_{z} has the following projective presentation:

\xymatrix{0 \ar[r] & Q_{x} ⊕ Q_{y} \ar[r] & Q_{z} \ar[r] & DP_{z} \ar[r] & 0}.

Applying the functor Hom_{A^{o}}(−, A^{o}), we get a projective resolution of τ^{−1}P_{z}:

\xymatrix{0 \ar[r] & P_{z} \ar[r] & P_{x} ⊕ P_{y} \ar[r] & τ^{−1}P_{z} \ar[r] & 0}.

Thus pd_{A} τ^{−1}P_{z} = 1. It is not hard to see that τ^{−1}P_{z} has the following structure:

[figure: y; x; y; z]

Clearly, Hom_{A}(τ^{−1}P_{z}, P_{z}) = 0.

If P_{S} is simple, then the almost split sequence starting at P_{S} is precisely the projective resolution of τ^{−1}P_{S} (see [1]). This is not true for generalized APR tilting modules, as shown by the following example:

Example 5.6. Let A be the path algebra of the following quiver with relations δα = αρ and δ^{2} = ρ^{2} = 0.

[figure: x; ρ; α; y; δ].

Then P_{x} ⊕ τ^{−1}P_{y} is a generalized APR tilting module. By computation, τ^{−1}P_{y} coincides with the injective module I_{x}. But the almost split sequence ending at τ^{−1}P_{y} is:

\xymatrix{0 \ar[r] & P_{y} \ar[r] & M \ar[r] & τ^{−1}P_{y} ≅ I_{x} \ar[r] & 0},

where M has the following structure:

[figure: x; y; x; y] .

Therefore, the almost split sequence is not the projective resolution of τ^{−1}P_{x}:

\xymatrix{0 \ar[r] & P_{y} \ar[r] & P_{x} \ar[r] & τ^{−1}P_{y} ≅ I_{x} \ar[r] & 0}.

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Department of Mathematics, University of California, Riverside, CA, 92507.

E-mail address: lipingli@math.ucr.edu