ALGEBRAS STRATIFIED FOR ALL LINEAR ORDERS

LIPING LI

Abstract. In this paper we describe several characterizations of basic finitedimensional 𝕜-algebras A stratified for all linear orders, and classify their graded algebras as tensor algebras satisfying some extra property. We also discuss whether for a given linear order ≼, F(_{≼}Δ), the category of A-modules with _{≼}Δ-filtrations, is closed under cokernels of monomorphisms, and classify quasi-hereditary algebras satisfying this property.

Let A be a basic finite-dimensional 𝕜-algebra where 𝕜 is an algebraically closed filed, and A-mod be the category of finitely generated left A-modules. Dlab and Ringel showed in [3] that A is quasi-hereditary for all linear orders if and only if A is a hereditary algebra. In [5] stratification property of A for different orders was studied. These results motivate us to classify algebras standardly stratified or properly stratified for all linear orders, which include hereditary algebras as special cases.

Choose a fixed set of orthogonal primitive idempotents {e_{λ}}_{λ∈Λ} for A such that ∑_{λ∈Λ} e_{λ} = 1. We can define a 𝕜-linear category A as follows: Ob A = {e_{λ}}_{λ∈Λ}; for λ, μ ∈ Λ, the morphism space A(e_{λ}, e_{μ}) = e_{μ}Ae_{λ}. The category A is skeletal, and the endomorphism algebra of each object is local. A 𝕜-linear representation of A is a 𝕜-linear functor from A to the category of finite-dimensional 𝕜-vector spaces. It is clear that A-rep, the category of finite-dimensional 𝕜-linear representations of A, is equivalent to A-mod. Call A the associated category of A.

we prove that if A is standardly stratified for all linear orders, then its associated category is a directed category, i.e., there is a partial order ⩽ on Λ such that λ ⩽ μ whenever A(e_{λ}, e_{μ}) = e_{μ}Ae_{λ} ≠ 0. Please see [8, 9] for details of directed categories. With this terminology, we get:

Theorem 0.1. Let A be a basic finite-dimensional algebra and A be its associated category. Then the following are equivalent:

(1) A is standardly stratified for all linear orders.

(2) A is a directed category and J = ⨁_{λ≠μ∈Λ} e_{μ}Ae_{λ} is a projective A-module.

(3) The trace tr_{P_{λ}}(P_{μ}) is a projective module for λ, μ ∈ Λ.

(4) The projective dimension pd_{A} M ⩽ 1 for all M ∈ F(_{≼}Δ) and linear orders ≼.

Note that J = ⨁_{λ≠μ∈Λ} e_{μ}Ae_{λ} can be viewed as a two-sided ideal of A with respect to this chosen set of orthogonal primitive idempotents since A is a directed category. Moreover, we have A = A_{0} ⊕ J as vector spaces, where A_{0} = ⨁_{x∈Ob A} A(x, x) constitutes of all endomorphisms in A. Thus we consider the associated graded algebra Ǎ with Ǎ_{0} = A_{0} and Ǎ_{i} = J^{i}/J^{i+1} for i ⩾ 1, and obtain the following classification:

2000 Mathematics Subject Classification. 16G10, 16G20.

Key words and phrases. Standardly stratified, directed categories, quasi-hereditary.

The author would like to thank his thesis advisor, Professor Peter Webb, for many invaluable suggestions in numerous discussions. He also thanks Professor Mazorchuk for pointing out some works on relevant topics, which were unknown to the author before. The referee is appreciated for the careful reading and many nice suggestions.

Theorem 0.2. Let A be a basic finite-dimensional 𝕜-algebra whose associated category A is directed. Then the following are equivalent:

(1) A is standardly stratified (resp., properly stratified) for all linear orders;

(2) the associated graded algebra Ǎ is standardly stratified (resp., properly stratified) for all linear orders;

(3) Ǎ is the tensor algebra generated by A_{0} = ⨁_{λ∈Λ} A(e_{λ}, e_{λ}) = e_{λ}Ae_{λ} and a left (resp., left and right) projective A_{0}-module Ǎ_{1}.

We also describe a combinatorial construction for Ǎ, the associated graded category of Ǎ.

Let ≼ be a particular linear order for which A is standardly stratified. It is well known that F(_{≼}Δ) is an additive category closed under extensions, direct summands, and kernels of epimorphisms (see [1, 2, 11]). But in general it is not closed under cokernels of monomorphisms. However, if A is standardly stratified with respect to all linear orders, there exists a particular linear order (not necessarily unique) ≼ for which the corresponding category F(_{≼}Δ) is closed under cokernels of monomorphisms. It motivates us to give a criterion for whether F(_{≼}Δ) has this property. We obtain the following result:

Theorem 0.3. Let A be a finite-dimensional basic algebra standardly stratified for a linear order ≼. Then:

(1) F(_{≼}Δ) is closed under cokernels of monomorphisms if and only if the cokernel of every monomorphism ι : Δ_{λ} → P is contained in F(_{≼}Δ), where P is an arbitrary projective module and λ ∈ Λ.

(2) If F(_{≼}Δ) is closed under cokernels of monomorphisms and A is standardly stratified for another linear order ≼′, then F(_{≼′}Δ) ⊆ F(_{≼}Δ).

(3) If A is quasi-hereditary, then F(_{≼}Δ) is closed under cokernels of monomorphisms if and only if A is a quotient of a finite-dimensional hereditary algebra and all standard modules are simple.

Thus if A is standardly stratified for ≼ and ≼′ such that both F(_{≼}Δ) and F(_{≼′}Δ) are closed under cokernels of monomorphisms, then F(_{≼}Δ) = F(_{≼′}Δ).

In practice it is hard to determine whether there exists a linear order ≼ for which A is standardly stratified and the corresponding category F(_{≼}Δ) is closed under cokernels of monomorphisms. Certainly, checking all linear orders is not an ideal way to do this. We then describe an explicit algorithm to construct a set L of linear orders with respect to all of which A is standardly stratified. Moreover, if there exists a linear order ≼ such that A is standardly stratified and the corresponding category F(_{≼}Δ) is closed under cokernels of monomorphisms, then ≼∈ L.

This paper is organized as follows. In Section 1 we describe some equivalent conditions for A to be standardly stratified with respect to all linear orders, and prove the first theorem. The classification is described in Section 2 with a proof of the second theorem. The problem that whether F(_{≼}Δ) is closed under cokernels of monomorphisms is considered in Section 3. In the last section we describe the algorithm and give some examples.

We introduce some conventions here. All algebras in this paper are finitedimensional basic 𝕜-algebras, and all modules are finitely generated left modules if we do not make other assumptions. Composition of morphisms is from right to left. The field 𝕜 is supposed to be algebraically closed. We view the zero module 0 as a projective (or free) module since this will simplify the expressions and proofs of many statements.

1. Characterizations of algebras stratified for all linear orders

We start with some preliminary knowledge on stratification theory, for which the reader can find more details in [1, 2, 3, 11, 12]. Throughout this section A is a basic finite-dimensional 𝕜-algebra with a chosen set of orthogonal primitive idempotents {e_{λ}}_{λ∈Λ} indexed by a set Λ such that ∑_{λ∈Λ} e_{λ} = 1. Let P_{λ} = Ae_{λ} and S_{λ} = P_{λ}/ rad P_{λ}. According to [1], A is standardly stratified with respect to a fixed preorder ≼ if there exist modules Δ_{λ}, λ ∈ Λ, such that the following conditions hold:

(1) the composition factor multiplicity [Δ_{λ} : S_{μ}] = 0 whenever μ ⪯̸ λ; and (2) for every λ ∈ Λ there is a short exact sequence 0 → K_{λ} → P_{λ} → Δ_{λ} → 0

such that K_{λ} has a filtration with factors Δ_{μ} where μ ≻ λ. The standard module Δ_{λ} is the largest quotient of P_{λ} with only composition factors S_{μ} such that μ ≼ λ. Similarly, the proper standard module Δ̄_{λ} can be defined as the largest quotient of P_{λ} such that all composition factors S_{μ} satisfy μ ≻ λ except for one copy of S_{λ}. Costandard module ∇_{λ} and proper costandard module ¯∇_{λ} are defined dually. Since by Proposition 1.1 A is standardly stratified for all preorders if and only if it is a direct sum of local algebras, in this paper we only deal with linear orders instead of general preorders as in [6, 7, 11] to get a class of algebras whose size is big enough to contain hereditary algebras.

In the case that A is standardly stratified, standard modules Δ_{λ} have the following description:

Δ_{λ} = P_{λ}/ ∑_{μ≻λ} tr_{P_{μ}}(P_{λ}),

where tr_{P_{μ}}(P_{λ}) is the trace of P_{μ} in P_{λ} ([2, 11]). Let Δ be the direct sum of all standard modules and F(Δ) be the full subcategory of A-mod such that each object in F(Δ) has a filtration by standard modules. Similarly we define F(\overline{Δ}), F(∇) and F(\overline{∇}). According to Theorem 3.4 in [11], A is standardly stratified if and only if AA ∈ F(Δ). The algebra is called properly stratified if _{A}A ∈ F(Δ) ∩ F(\overline{Δ}). If the endomorphism algebra of each standard module is one-dimensional, then A is called quasi-hereditary.

Proposition 1.1. The algebra A is standardly stratified for all preorders if and only if A is a direct sum of local algebras.

Proof. If A is a direct sum of local algebras, it is standardly stratified for all preorders such that all standard modules coincide with indecomposable projective modules. Conversely, suppose that A is standardly stratified for all preorders. To show that A is a direct sum of local algebras, it is sufficient to show that Hom_{A}(P_{λ}, P_{μ}) ≅ e_{λ}Ae_{μ} = 0 for all pairs of distinct elements λ ≠ μ ∈ Λ.

Consider the preorder on Λ such that every two different elements cannot be compared. By the second condition in the definition of standardly stratified algebras, Δ_{λ} ≅ P_{λ} for all λ ∈ Λ. By the first condition, P_{λ} only has composition factors S_{λ}. Therefore, Hom_{A}(P_{λ}, P_{μ}) ≅ e_{λ}Ae_{μ} = 0 for all λ ≠ μ ∈ Λ. □

The following statement is an immediate corollary of Dlab’s theorem ([7, 11]).

Proposition 1.2. The algebra A is properly stratified for a linear order ≼ on Λ if and only if both A and A^{op} are standardly stratified with respect to ≼, in other words, A is both left and right standardly stratified.

Proof. The algebra A is properly stratified if and only if F(Δ) ∩ F(\overline{Δ}) contains all left projective A-modules. By duality, this is true if and only if F(Δ) contains all left projective A-modules and F(\overline{∇}_{A}) contains all right injective A-modules, where \overline{∇}_{A} is a right A-module. By Theorem 3.4 in [11], we conclude that A is properly stratified if and only if F(Δ) contains all left projective A-module and F(Δ_{A}) contains all right projective A-modules, here Δ_{A} is the direct sum of all right standard modules. That is, A is properly stratified if and only if it is both left and right standardly stratified. □

Recall the associated category A of A is a directed category if there is a partial order ⩽ on Λ such that A(e_{λ}, e_{μ}) = e_{μ}Ae_{λ} ≠ 0 implies λ ⩽ μ.

Proposition 1.3. If A is standardly stratified for all linear orders on Λ, then the associated category A is a directed category.

Proof. Suppose that the conclusion is not true. Then there is an oriented cycle e_{0} → e_{1} → … → e_{n} = e_{0} with n > 1 such that A(e_{i}, e_{i+1}) = e_{i+1}Ae_{i} ≠ 0 for all 0 ⩽ i ⩽ n − 1. Take some e_{s} such that dim_{𝕜} P_{s} ⩾ dim_{𝕜} P_{i} for all 0 ⩽ i ⩽ n − 1, where P_{s} = Ae_{s} coincides with the vector space formed by all morphisms starting from e_{s}. Then for an arbitrary linear order ≼ with respect to which e_{s} is maximal, we claim that A is not standardly stratified.

Indeed, if A is standardly stratified with respect to ≼, then tr_{P_{s}}(P_{i}) ≅ P_{s}^{m_{i}} for some m_{i} ⩾ 0. Consider P_{s−1} (if s = 0 we consider P_{n−1}). Since A(e_{s−1}, e_{s}) = e_{s}Ae_{s−1} ≠ 0, tr_{P_{s}}(P_{s−1}) ≠ 0, so m_{s−1} ⩾ 1. But on the other hand, since tr_{P_{s}}(P_{s−1}) ⊆ rad P_{s−1}, and dim_{𝕜} P_{s} ⩾ dim_{𝕜} P_{s−1}, we should have m_{s−1} = 0. This is absurd. Therefore, there is no oriented cycle in A, and the conclusion follows. □

The following proposition describes some properties of algebras with directed associated categories.

Proposition 1.4. Suppose that the associated category A of A is a directed category with respect to a partial order ⩽. Then

(1) The standard modules _{⩽}Δ_{λ} ≅ A(e_{λ}, e_{λ}) = e_{λ}Ae_{λ} for all λ ∈ Λ.

(2) A is standardly stratified for ⩽ if and only if for every pair λ, μ ∈ Λ, e_{μ}Ae_{λ} is a free e_{μ}Ae_{μ}-module.

(3) The category F(_{⩽}Δ) is closed under cokernels of monomorphisms.

Proof. The first statement is Proposition 5.5 in [8]. The second statement comes from Theorem 5.7 in that paper. We reminder the reader that A(e_{μ}, e_{μ}) is a local algebra, so projective modules coincide with free modules. It remains to show the last statement. That is, for an arbitrary exact sequence with L, M ∈ F(_{⩽}Δ), we need to show N ∈ F(_{⩽}Δ) as well:

\xymatrix{0 \ar[r] & L \ar[r] & M \ar[r] & N \ar[r] & 0.}

By the structure of standard modules, it is clear that an A-module K ∈ F(_{⩽}Δ) if and only if e_{λ}K is a free A(e_{λ}, e_{λ}) = e_{λ}Ae_{λ}-module for all λ ∈ Λ. For an arbitrary λ ∈ Λ, the above sequence induces an exact sequence of e_{λ}Ae_{λ}-modules

\xymatrix{0 \ar[r] & e_{λ}L \ar[r] & e_{λ}M \ar[r] & e_{λ}N \ar[r] & 0.}

Since L, M ∈ F(_{⩽}Δ), we know that e_{λ}L ≅ (e_{λ}Ae_{λ})^{l} and e_{λ}M ≅ (e_{λ}Ae_{λ})^{m} for some l, m ⩾ 0.

Notice that e_{λ}Ae_{λ} is a local algebra. The regular module over e_{λ}Ae_{λ} is indecomposable, has a simple top and finite length. Therefore, the top of e_{λ}L is embedded into the top of e_{λ}M since otherwise the first map cannot be injective. Therefore, the top of e_{λ}N is isomorphic to S_{λ}^{m−l} where S_{λ} ≅ (e_{λ}Ae_{λ})/ rad(e_{λ}Ae_{λ}) and e_{λ}N is the quotient module of (e_{λ}Ae_{λ})^{m−l}. But by comparing dimensions we conclude that e_{λ}N ≅ (e_{λ}Ae_{λ})^{m−l}. Consequently, e_{λ}N and hence N are contained in F(_{⩽}Δ) as well. This finishes the proof. □

The second statement of this proposition is the main condition of pyramid algebras mentioned in [10]. The third statement motivates us to study the problem of whether F(_{≼}Δ) is closed under cokernels of monomorphisms for an arbitrary linear order ≼. We will return to this topic in Section 3.

In some sense the property of being standardly stratified for all linear orders is inherited by quotient algebras. Explicitly,

Lemma 1.5. If A is standardly stratified for all linear orders, then for an arbitrary primitive idempotent e_{λ}, the quotient algebra Ā = A/Ae_{λ}A is standardly stratified for all linear orders.

Proof. Let ≼ be a linear order on Λ\{λ}. Then we can extend it to a linear order ˜≼ on Λ by letting λ be the unique maximal element. Since A is standardly stratified for ˜≼, we conclude that Ā is standardly stratified for ≼ as well. □

Now suppose that the associated category A of A is directed. Then we define

J = ⨁_{λ≠μ∈Λ} A(e_{λ}, e_{μ}) = ⨁_{λ≠μ∈Λ} e_{μ}Ae_{λ}, A_{0} = ⨁_{λ∈Λ} A(e_{λ}, e_{λ}) = ⨁_{λ∈Λ} e_{λ}Ae_{λ}.

Clearly, J is a two-sided ideal of A with respect to this chosen set of orthogonal primitive idempotents, and A_{0} ≅ A/J as A-modules.

Proposition 1.6. If A is standardly stratified for all linear orders on Λ, then J is a projective A-module.

Proof. Notice that the associated category A is directed by Proposition 1.3, so J is a well defined A-module. Let ⩽ be a partial order on Λ with respect to which A is directed, i.e., A(e_{λ}, e_{μ}) = e_{μ}Ae_{λ} ≠ 0 implies λ ⩽ μ. We can extend this partial order to a linear order with respect to which A is still directed. Indeed, let O_{1} be the maximal objects in A with respect to ⩽, O_{2} be the maximal objects in Ob A \ O_{1} with respect to ⩽, and so on. In this way we get Ob A = ⨆_{i=1}^{n} O_{i}. Then define arbitrary linear orders ⩽_{i} on O_{i}, 1 ⩽ i ⩽ n. Take x, y ∈ Ob A and suppose that x ∈ O_{i} and y ∈ O_{j}, 1 ⩽ i, j ⩽ n. Define x ˜<y if i < j or i = j but x <_{i} y. Then ˜⩽ is a linear order extending ⩽, and A is still directed with respect to ˜⩽. Therefore, without loss of generality we assume that ⩽ is linear.

We prove this conclusion by induction on |Λ|, the size of Λ. It holds for |Λ| = 1 since J = 0. Now suppose that it holds for |Λ| = s and consider |Λ| = s + 1.

Let λ be the minimal element in Λ with respect to ⩽. Therefore, A(e_{μ}, e_{λ}) = e_{λ}Ae_{μ} = 0 for all μ ≠ λ ∈ Λ. Let ≼ be a linear order on Λ such that λ is the unique maximal element with respect to ≼. Consider the quotient algebra Ā = A/Ae_{λ}A, which is standardly stratified for all linear orders on Λ\{λ} by the previous lemma. Thus the associated category Ā of Ā is directed, and we can define J̄ as well.

For μ ≠ λ, we have Hom_{A}(P_{λ}, P_{μ}) ≅ e_{λ}Ae_{μ} = 0 since λ is minimal with respect to ⩽ and A is directed. Therefore, tr_{P_{λ}}(P_{μ}) = 0, and the corresponding indecomposable projective Ā-module P̄_{μ} is isomorphic to P_{μ}. Let J_{μ} = Je_{μ}. Then we have:

J_{μ} = ⨁_{ν≠μ∈Λ} e_{ν}Ae_{μ} = ⨁_{ν>μ} e_{ν}Ae_{μ} ≅ ⨁_{ν>μ} Hom_{A}(P_{ν}, P_{μ})

≅ ⨁_{ν>μ} HomĀ(P̄_{ν}, P̄_{μ}) ≅ ⨁_{ν>μ} e_{ν} Āe_{μ} = J̄_{μ}

By the induction hypothesis, J_{μ} ≅ J̄_{μ} is a projective Ā-module. By our choice of λ, it is actually a projective A-module since e_{λ} acts on J_{μ} as 0. Therefore, J_{μ} is a projective A module.

Since J = ⨁_{ν∈Λ} J_{ν} and we have proved that all J_{ν} are projective for ν ≠ λ, it remains to prove that J_{λ} is a projective module. To achieve this, we take the element μ ∈ Λ which is minimal in Λ\{λ} with respect to ⩽. Therefore, A(e_{ν}, e_{μ}) = e_{μ}Ae_{ν} ≠ 0 only if ν = μ or ν = λ.

Now define another linear order ≼′ on Λ such that μ is the unique maximal element with respect to ≼′. Similarly, for all ν ≠ μ, λ, we have P̄′_{ν} ≅ P_{ν}, where Ā′ = A/Ae_{μ}A and P̄′_{ν} = Ā′e_{ν}. As before, the associated category of Ā′ is directed so we can define J̄′. Moreover, we have M = tr_{P_{μ}}(P_{λ}) ⊆ J_{λ}. Thus we get the following commutative diagram:

[figure: 0; 0; 0; M; J_{λ}; J̄′_{λ}; 0; 0; M; P_{λ}; P̄′_{λ}; 0; e_{λ}Ae_{λ}; e_{λ}Ae_{λ}; 0; 0; 0]

where J̄′_{λ} = J̄′e_{λ}. By the induction hypothesis on the quotient algebra Ā′, J̄′_{λ} is a projective Ā′-module. Clearly, each indecomposable summands of J̄′_{λ} is isomorphic to a certain P̄′_{ν} with ν ≠ λ, μ. But P̄′_{ν} ≅ P_{ν}. Therefore, J̄′_{λ} is actually a projective A-module, and the top sequence in the above diagram splits, i.e., J_{λ} ≅ M ⊕ J̄′_{λ}. But M = tr_{P_{μ}}(P_{λ}) is also a projective A-module (which is actually isomorphic to a direct sum of copies of P_{μ}) since A is standardly stratified with respect to ≼′ and μ is maximal with respect to this order. Thus J_{λ} is a projective A-module as well. The proof is completed. □

Proposition 1.7. Suppose that the associated category A of A is directed. If J is a projective A-module then for an arbitrary pair λ, μ ∈ Λ, tr_{P_{λ}}(P_{μ}) ≅ P_{λ}^{m_{μ}} for some m_{μ} ⩾ 0.

Proof. Let ⩽ be a linear order on Λ with respect to which A is directed. We can index all orthogonal primitive idempotents: e_{n} > e_{n−1} > … > e_{1}. Let P_{i} = Ae_{i}. Suppose that e_{λ} = e_{s} and e_{μ} = e_{t}. If s < t, tr_{P_{s}}(P_{t}) = 0 and the conclusion is trivially true. For s = t, the conclusion holds as well. So we assume s > t and conduct induction on the difference d = s − t. Since it has been proved for d = 0, we suppose that the conclusion holds for all d ⩽ l.

Now suppose d = s − t = l + 1. Let E_{t} = A(e_{t}, e_{t}) ≅ P_{t}/J_{t} ≅ e_{t}Ae_{t}, which can be viewed as an A-module. We have the following exact sequence:

\xymatrix{0 \ar[r] & J_{t} \ar[r] & P_{t} \ar[r] & E_{t} \ar[r] & 0}.

Since J is projective, so is J_{t} = Je_{t}. Moreover, since J_{t} = ⨁_{m≠t} e_{m}Ae_{t} and e_{m}Ae_{t} = 0 if m ≯ t, we deduce that J_{t} has no summand isomorphic to P_{m} with m < t. Therefore, J_{t} ≅ ⨁_{t+1⩽i⩽n} P_{i}^{m_{i}}, where m_{i} is the multiplicity of P_{i} in J_{t}.

The above sequence induces an exact sequence

\xymatrix{0 \ar[r] & ⨁_{t+1⩽i⩽n} tr_{P_{s}}(P_{i})^{m_{i}} \ar[r] & tr_{P_{s}}(P_{t}) \ar[r] & tr_{P_{s}}(E_{t}) \ar[r] & 0}.

Clearly, tr_{P_{s}}(E_{t}) = 0, so

⨁_{t+1⩽i⩽n} tr_{P_{s}}(P_{i})^{m_{i}} ≅ tr_{P_{s}}(P_{t}).

But for each t + 1 ⩽ i ⩽ n, we get s − i < s − t = l + 1. Thus by the induction hypothesis, each tr_{P_{s}}(P_{i}) is a projective module isomorphic to a direct sum of P_{s}. Therefore, tr_{P_{s}}(P_{t}) is a direct sum of P_{s}. The conclusion follows from induction. □

The condition that A is directed is required in the previous propositions since otherwise J might not be well defined.

Proposition 1.8. If for each pair λ, μ ∈ Λ, tr_{P_{λ}}(P_{μ}) is a projective module, then the associated category A is a directed category. Moreover, A is standardly stratified for all linear orders.

Proof. The first statement comes from the proof of Proposition 1.3. Actually, if A is not directed, in that proof we find some P_{λ} and P_{μ} such that tr_{P_{λ}}(P_{μ}) ≠ 0 is not projective.

To prove the second statement, we use induction on |Λ|. It is clearly true if |Λ| = 1. Suppose that the conclusion holds for |Λ| ⩽ l and suppose that |Λ| = l + 1. Let ≼ be an arbitrary linear order on Λ and take a maximal element λ with respect to this linear order. Since tr_{P_{λ}}(P_{μ}) is a projective A-module for all μ ∈ Λ by the given condition, it is enough to show that the quotient algebra Ā = A/Ae_{λ}A has the same property. That is, trP̄_{μ}(P̄_{ν}) is a projective Ā-module for all μ, ν ∈ Λ\{λ}, where P̄_{μ} = Āe_{μ} and P̄_{ν} is defined similarly. Then the conclusion will follow from induction hypothesis.

Let ⩽ be a linear order on Λ with respect to which A is directed. It restriction on Λ \ {λ} gives a linear order with respect to which Ā, the associated category of Ā, is directed. Consider trP̄_{μ}(P̄_{ν}). If μ ⩽ ν, this trace is 0 or P̄_{ν}. Thus we assume that μ > ν. Since tr_{P_{μ}}(P_{ν}) is projective, it is isomorphic to a direct sum of of P_{μ}, and we get the following exact sequence:

\xymatrix{0 \ar[r] & tr_{P_{μ}}(P_{ν}) ≅ P_{μ}^{m} \ar[r] & P_{ν} \ar[r] & M \ar[r] & 0}

with e_{μ}M = 0.

Since tr_{P_{λ}}(P_{μ}) and tr_{P_{λ}}(P_{ν}) are also isomorphic to direct sums of P_{λ}, by considering the traces of P_{λ} in the modules in the above sequence, we get a commutative diagram as follows:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & P_{λ}^{s} ≅ tr_{P_{λ}}(P_{μ}^{m}) \ar[r] \ar[d] & P_{λ}^{t} ≅ tr_{P_{λ}}(P_{ν}) \ar[r] \ar[d] & P_{λ}^{t−s} ≅ tr_{P_{λ}}(M) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & P_{μ}^{m} \ar[r] \ar[d] & P_{ν} \ar[r] \ar[d] & M \ar[r] \ar[d] & 0 \\ 0 \ar[r] & _{P̄_{μ}_{m}} \ar[r] \ar[d] & P̄_{ν} \ar[r] \ar[d] & M̄ \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0 & .}

Since e_{μ}M = 0, we get e_{μ}M̄ = 0 as well. Thus from the bottom row we conclude that trP̄_{μ}(P̄_{ν}) ≅ P̄_{μ}^{m} is projective. This finishes the proof. □

Now we can prove the first theorem.

Proof. (1) ⇒ (2): by Propositions 1.3 and 1.6.

(2) ⇒ (3): by Proposition 1.7.

(3) ⇒ (1): by Proposition 1.8.

(2) ⇒ (4): By the assumption, there is a linear order ⩽ on Λ such that A is directed with respect to it. By Proposition 1.4, we know that _{⩽}Δ ≅ A_{0} = ⨁_{λ∈Λ} e_{λ}Ae_{λ}. Thus the projective dimension of _{⩽}Δ is at most 1 since we have the exact sequence 0 → J → A → A_{0} → 0 and J is projective. Therefore, every M ∈ F(_{⩽}Δ) has projective dimension at most 1.

Note that F(_{⩽}Δ) is closed under cokernels of monomorphisms by Proposition 1.4. Let ≼ be an arbitrary linear order on Λ. By the second statement of Theorem 0.3 (which will be proved later), F(_{≼}Δ) ⊆ F(_{⩽}Δ). Thus every M ∈ F(_{≼}Δ) ⊆ F(_{⩽}Δ) has projective dimension at most 1.

(4) ⇒ (3): Take an arbitrary pair λ, μ ∈ Λ. We want to show that tr_{P_{λ}}(P_{μ}) is a projective A-module. Clearly, there exists a linear order ≼ on Λ such that λ is the maximal element in Λ and μ is the maximal element in Λ \ {λ}. Therefore, by the definition, we have a short exact sequence

\xymatrix{0 \ar[r] & tr_{P_{λ}}(P_{μ}) \ar[r] & P_{μ} \ar[r] & _{≼}Δ_{μ} \ar[r] & 0}.

Clearly, _{≼}Δ_{μ} ∈ F(_{≼}Δ). Therefore, it has projective dimension at most 1, which means that tr_{P_{λ}}(P_{μ}) is projective. □

This immediately gives us the following characterization of algebras properly stratified for all linear orders.

Corollary 1.9. Let A, A and J be as in the previous theorem. Then A is properly stratified for all linear orders if and only if A is a directed category and J is a left and right projective A-module.

Proof. By Proposition 1.2 and Theorem 0.1. □

The following example describes an algebra which is standardly stratified (but not properly stratified) for all linear orders.

Example 1.10. Let A be the algebra defined by the following quiver with relations: δ^{2} = βδ = 0, αδ = γβ.

[figure: y; γ; x; δ; β; α; z]

The indecomposable left projective modules are described as follows:

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

It is easy to see that J ≅ P_{y} ⊕ P_{z} ⊕ P_{z} is a left projective A-module, so A is standardly stratified for all linear orders.

On the other hand, the indecomposable right projective modules have the following structures:

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

Thus J is not a right projective A-module, and hence A is not properly stratified for all linear orders.

2. The classification

Throughout this section let A be a finite-dimensional basic algebra with a chosen set of orthogonal primitive idempotents {e_{λ}}_{λ∈Λ} such that ∑_{λ∈Λ} e_{λ} = 1. We also suppose that the associated category A is directed. Thus we can define A_{0} and J as before, and consider its associated graded algebra Ǎ = ⨁_{i⩾0} J^{i}/J^{i+1}, where we set J^{0} = A. Correspondingly, for a finitely generated A-module M, its associated graded Ǎ-module M̌ = ⨁_{i⩾0} J^{i}M/J^{i+1}M. Clearly, we have Ǎ_{i} · Ǎ_{j} = Ǎ_{i+j} for i, j ⩾ 0.

Lemma 2.1. Let M be an A-module and M̌ be the associated graded Ǎ-module. Then M̌ is a graded projective Ǎ-module if and only if M is a projective A-module.

Proof. Without loss of generality we can assume that M is indecomposable. Since the free module _{A}A is sent to the graded free module ǍǍ by the grading process, we know that M̌ is a graded projective Ǎ-module generated in degree 0 if M is a projective A-module. Now suppose that M is not a projective A-module but M̌ is a projective Ǎ-module, and we want to get a contradiction.

Let p : P → M be a projective covering map of M. Then dim_{𝕜} M < dim_{𝕜} P since M is not projective. Moreover, Top P = P/ rad P ≅ Top M = M/ rad M. The surjective map p gives a graded surjective map p̌ : P̌ → M̌. Since M̌ is supposed to be projective and P̌ is projective, we get a splitting exact sequence of graded projective Ǎ-modules generated in degree 0 as follows:

\xymatrix{0 \ar[r] & Ľ \ar[r] & P̌ \ar[r] & M̌ \ar[r] & 0}.

Notice that Ľ ≠ 0 since dim_{𝕜} M̌ = dim_{𝕜} M < dim_{𝕜} P = dim_{𝕜} P̌. Consider the degree 0 parts. We have a splitting sequence of A_{0}-modules

\xymatrix{0 \ar[r] & Ľ_{0} \ar[r] & P̌_{0} \ar[r] & M̌_{0} \ar[r] & 0}

which by definition is isomorphic to

\xymatrix{0 \ar[r] & Ľ_{0} \ar[r] & P/JP \ar[r] & M/JM \ar[r] & 0}.

View them as A-modules on which J acts as 0. Observe that J is contained in the radical of A. Thus Top P ≅ Top(P/JP) and Top M ≅ Top(M/JM). But the above sequence splits, hence Top P ≅ Top Ľ_{0} ⊕ Top M, contradicting the fact that Top P ≅ Top M. This finishes the proof. □

The above lemma still holds if we replace left modules by right modules. It immediately implies the following result, which is a part of Theorem 0.2:

Proposition 2.2. Let A be a finite-dimensional basic algebra whose associated category is directed. Let Ǎ be the associated graded algebra. Then A is standardly (resp., properly) stratified for all linear orders if and only if so is Ǎ.

Proof. The algebra A is standardly stratified for all linear orders if and only if the associated category A is a directed category and J is a projective A-module. This happens if and only if the graded category Ǎ is a directed category and J̌ is a projective Ǎ-module by the previous lemma, and if and only if Ǎ is standardly stratified for all linear orders. □

In general Ǎ (when viewed as a non-graded algebra) is not isomorphic to A, as shown by the following example.

Example 2.3. Let A be the algebra described in Example 1.10, where we proved that it is standardly stratified for all linear orders. It is easy to see J/J^{2} = ⟨ᾱ, β̄, γ̄⟩, J^{2}/J^{3} = ⟨γ̄β̄⟩ and J^{3} = 0. Therefore, Ǎ is defined by the following quiver with relations ᾱδ = β̄δ̄ = 0, which is not isomorphic to A.

[figure: y; γ̄; x; δ; β̄; ᾱ; z]

The indecomposable left projective Ǎ-modules have the following structures:

\xymatrix{ & x \ar[dr]_{δ̄} \ar[dl]_{ᾱ} \ar[d]_{β̄} \\ z & y \ar[d]_{γ̄} & x \\ & z} \xymatrix{y \ar[d]_{γ̄} \\ z} z

It still holds that J̌ ≅ P_{y} ⊕P_{z} ⊕P_{z}, so Ǎ is standardly stratified for all linear orders. The indecomposable right projective Ǎ-modules are as follows:

\xymatrix{x \ar[d]_{δ̄} & y \ar[d]_{β̄} &  & z \ar[dl]_{γ̄} \ar[dr]_{ᾱ} \\ x & x & y \ar[d]_{β̄} &  & x \\  &  & x}

We deduce that Ǎ is not properly stratified for all linear orders since J̌ is not a right projective module.

Let X be an (A_{0}, A_{0})-bimodule. We denote the tensor algebra generated by A_{0} and X to be A_{0}[X]. That is, A_{0}[X] = A_{0} ⊕ X ⊕ (X ⊗_{A_{0}} X) ⊕ . . .. With this notation, we have:

Lemma 2.4. Let A = ⨁_{i⩾1} A_{i} be a finite-dimensional graded algebra with A_{i} · A_{j} = A_{i+j}, i, j ⩾ 0. Then J = ⨁_{i⩾1} A_{i} is a projective A-module if and only if A ≅ A_{0}[A_{1}], and A_{1} is a projective A_{0}-module.

Proof. Suppose that A = A_{0}[A_{1}] and A_{1} is a projective A_{0}-module. Observe that A is a finite-dimensional algebra, so there exists a minimal number n ⩾ 0 such that A_{n+1} ≅ \underbrace{A_{1} ⊗_{A_{0}} … ⊗_{A_{0}} A_{1}}_{n+1} = 0. Without loss of generality we assume that n > 0 since otherwise J = 0 is a trivial projective module. Therefore,

J = ⨁_{i=1}^{n} A_{i} = A_{1} ⊕ (A_{1} ⊗_{A_{0}} A_{1}) ⊕ … ⊕ (\underbrace{A_{1} ⊗_{A_{0}} … ⊗_{A_{0}} A_{1}}_{n})

= A_{1} ⊕ (A_{1} ⊗_{A_{0}} A_{1}) ⊕ … ⊕ (\underbrace{A_{1} ⊗_{A_{0}} … ⊗_{A_{0}} A_{1}}_{n+1})

≅ (A_{0} ⊕ A_{1} ⊕ … ⊕ (\underbrace{A_{1} ⊗_{A_{0}} … ⊗_{A_{0}} A_{1}}_{n})) ⊗_{A_{0}} A_{1}

= A ⊗_{A_{0}} A_{1},

which is projective since A_{1} is a projective A_{0}-module.

Conversely, suppose that J = ⨁_{i⩾1} A_{i} is a projective A-module. Since J is generated in degree 1, A_{1} must be a projective A_{0}-module. Moreover, we have a minimal projective resolution of A_{0} as follows

\xymatrix{… \ar[r] & A ⊗_{A_{0}} A_{1} \ar[r] & A \ar[r] & A_{0} \ar[r] & 0}.

But we also have the short exact sequence 0 → J → A → A_{0} → 0. Since J is projective, we deduce that J ≅ A ⊗_{A_{0}} A_{1}. Therefore, A_{2} ≅ A_{1} ⊗_{A_{0}} A_{1}, A_{3} ≅ A_{1} ⊗_{A_{0}} A_{1} ⊗_{A_{0}} A_{1}, and so on. Thus A ≅ A_{0}[A_{1}] as claimed. □

Now we prove Theorem 0.2.

Proof. For standardly stratified algebras, the equivalence of (1) and (2) has been established in Proposition 2.2, and the equivalence of (2) and (3) comes from the previous lemma and Theorem 0.1. Since all arguments work for right modules, we also have the equivalence of these statements for properly stratified algebras. □

We end this section with a combinatorial description of Ǎ, the associated graded category of Ǎ stratified for all linear orders. Let Q = (Q_{0}, Q_{1}) be a finite acyclic quiver, where both the vertex set Q_{0} and the arrow set Q_{1} are finite. We then define a quiver of bimodules Q̃ = (Q_{0}, Q_{1}, f, g): to each vertex v ∈ Q_{0} the map f assigns a finite-dimensional local algebra A_{v}, i.e., f(v) = A_{v}; for each arrow α : v → w, g(α) is a finite-dimensional (A_{w}, A_{v})-bimodule.

The quiver of bimodules Q̃ determines a category C. Explicitly, Ob C = Q_{0}. The morphisms between an arbitrary pair of objects v, w ∈ Q_{0} are defined as follows. Let

\xymatrix{γ : v = v_{0} \ar[r]^{α_{1}} \ar[r] & v_{1} \ar[r]^{α_{2}} & … \ar[r]^{α_{n−1}} & v_{n−1} \ar[r]^{α_{n}} & v_{n} = w}

be an oriented path in Q. We define

M_{γ} = g(α_{n}) ⊗_{f(v_{n−1})} g(α_{n−1}) ⊗_{f(v_{n−2})} … ⊗_{f(v_{1})} g(α_{1}).

This is a (f(w), f(v))-bimodule. Then

C(v, w) = ⨁_{γ∈P (v,w)} M_{γ},

Where P (v, w) is the set of all oriented paths from v to w. The composite of morphisms is defined by tensor product. We call a category defined in this way a free directed category. It is left regular if for every arrow α : v → w, g(α) is a left projective f(w)-module. Similarly, we define right regular categories. It is regular if this category is both left regular and right regular.

Using these terminologies, we get the following combinatorial description of Ǎ.

Theorem 2.5. Let A be a finite-dimensional basic algebra whose associated category is directed. Then A is standardly (resp., properly) stratified for all linear orders if and only if the graded category Ǎ is a left regular (resp., regular) free directed category.

Proof. It is straightforward to check that if Ǎ is a left regular free directed category, then Ǎ satisfies (3) in Theorem 0.2. So A is standardly stratified for all linear orders. Conversely, if A is standardly stratified for all linear orders, then Ǎ is a directed category, and Ǎ is a tensor algebra generated by A_{0} = ⨁_{λ∈Λ} e_{λ}Ae_{λ} and a projective left A_{0}-module Ǎ_{1}. Define Q = (Q_{0}, Q_{1}, f, g) in the following way: Q_{0} = Λ, and f(λ) = e_{λ}A_{0}e_{λ}. Arrows and the map g are defined as follows: for λ ≠ μ ∈ Q_{0}, we put an arrow ϕ : λ → μ if e_{μ}Ǎ_{1}e_{λ} ≠ 0 and define g(ϕ) = e_{μ}Ǎ_{1}e_{λ}. In this way Q defines a left regular free directed category which is isomorphic to Ǎ. Since all arguments work for right modules, the proof is completed. □

3. Whether F(_{≼}Δ) is closed under cokernels of monomorphisms?

In Proposition 1.4 we proved that if A is directed and standardly stratified with respect to a linear order ⩽, then the corresponding category F(_{⩽}Δ) is closed under cokernels of monomorphisms. This result motivates us to study the general situation. Suppose A is standardly stratified with respect to a fixed linear ordered set (Λ, ≼). In the following lemmas we describe several equivalent conditions for F(_{≼}Δ) to be closed under cokernels of monomorphisms.

Lemma 3.1. The category F(_{≼}Δ) is closed under cokernels of monomorphisms if and only if for each exact sequence 0 → L → M → N → 0, where L, M ∈ F(_{≼}Δ) and L is indecomposable, N is also contained in F(_{≼}Δ).

Proof. The only if direction is trivial, we prove the other direction: for each exact sequence 0 → L̃ → M → N → 0 with L̃, M ∈ F(_{≼}Δ), we have N ∈ F(_{≼}Δ) as well.

We use induction on the number of indecomposable direct summands of L̃. If L̃ is indecomposable, the conclusion holds obviously. Now suppose that the if part is true for L̃ with at most l indecomposable summands. Assume that L̃ has l + 1 indecomposable summands. Taking an indecomposable summand L_{1} of L̃ we have the following diagram:

[figure: 0; 0; 0; L_{1}; L_{1}; 0; 0; L̃; M; N; 0; 0; L̄; M̄; N; 0; 0; 0]

Considering the middle column, M̄ ∈ F(_{≼}Δ) by the given condition. Therefore, we conclude that N ∈ F(_{≼}Δ) by using the induction hypothesis on the bottom row. □

Lemma 3.2. The category F(_{≼}Δ) is closed under cokernels of monomorphisms if and only if for each exact sequence 0 → L → P → N → 0, where L, P ∈ F(_{≼}Δ) and P is projective, N is also contained in F(_{≼}Δ).

Proof. It suffices to show the if part. Let 0 → L → M → N → 0 be an exact sequence with L, M ∈ F(_{≼}Δ). We want to show N ∈ F(_{≼}Δ) as well. Let P be a projective cover of M. Then we have a commutative diagram by the Snake Lemma:

[figure: 0; Ω(M); N′; L; 0; P; P; 0; L; M; N; 0,]

where all rows and columns are exact. Clearly, Ω(M) ∈ F(_{≼}Δ), so is N′ since F(_{≼}Δ) is closed under extension. Considering the last column, we conclude that N ∈ F(_{≼}Δ) by the given condition. □

Every M ∈ F(_{≼}Δ) has a_{≼}Δ-filtration ξ and we define [M : Δ_{λ}] to be the number of factors isomorphic to Δ_{λ} in ξ. This number is independent of the particular ξ (see [2] and [4]). We then define l(M) = ∑_{λ∈Λ}[M : Δ_{λ}] and call it the filtration length of M, which is also independent of the choice of ξ.

Lemma 3.3. The category F(_{≼}Δ) is closed under cokernels of monomorphisms if and only if for each exact sequence 0 → Δ_{λ} → M → N → 0, where M ∈ F(_{≼}Δ) and λ ∈ Λ, N is also contained in F(_{≼}Δ).

Proof. We only need to show the if part. Let 0 → L → M → N → 0 be an exact sequence with L, M ∈ F(_{≼}Δ). We use induction on the filtration length of L.

If l(L) = 1, then L ≅ Δ_{λ} for some λ ∈ Λ and the conclusion holds clearly. Suppose that the conclusion is true for all objects in F(_{≼}Δ) with filtration length at most s and assume l(L) = s + 1. Then we have an exact sequence 0 → L′ → L → Δ_{λ} → 0 for some λ ∈ Λ and l(L′) = s. Then we have a commutative diagram by the Snake Lemma:

[figure: 0; L′; L; Δ_{λ}; 0; M; M; 0; Δ_{λ}; N′; N; 0.]

Consider the first column. By induction hypothesis, N′ ∈ F(_{≼}Δ). By considering the bottom row we conclude that N ∈ F(_{≼}Δ) from the given condition. □

Now we prove Theorem 0.3.

Proof. The first statement follows immediately from the above lemmas.

Now we prove the second statement. The claim is clear if |Λ| = 1. So we assume Λ has more than one elements. Take an element λ ∈ Λ maximal with respect to ≼′. Clearly, Δ′_{λ} ≅ P_{λ} ∈ F(_{≼}Δ). Consider the quotient algebra Ā = A/Ae_{λ}A. It is standardly stratified with respect to the restricted linear order ≼′ on Λ \ {λ}. We claim Ā ∈ F(_{≼}Δ) as well. Indeed, consider the exact sequence

\xymatrix{0 \ar[r] & Ae_{λ}A \ar[r] & A \ar[r] & Ā \ar[r] & 0}.

Note that Ae_{λ}A ∈ F(_{≼}Δ) as a direct sum of P_{λ}. Therefore, Ā ∈ F(_{≼}Δ) as well since F(_{≼}Δ) is closed under cokernels of monomorphisms. Taking a maximal element ν ∈ Λ \ {λ} with respect to ≼′ and repeating the above procedure, we conclude that _{≼′}Δ_{ν} ≅ Āe_{ν} ∈ F(_{≼}Δ) and ¯ Ā = Ā/Āe_{ν} Ā ∈ F(_{≼}Δ). Recursively, we proved that _{≼′}Δ_{λ} ∈ F(Δ) for all λ ∈ Λ, so F(_{≼′}Δ) ⊆ F(_{≼}Δ).

The third statement can be proved by induction on |Λ| as well. If |Λ| = 1, the claim is clear since A ≅ 𝕜. Suppose that the conclusion holds for |Λ| = s and let Λ be a linear ordered set with s + 1 elements. Take a maximal element λ in Λ. Then Ā = A/Ae_{λ}A is also a quasi-hereditary algebra. Moreover, F(ĀΔ) is still closed under cokernels of monomorphisms. By the induction hypothesis, Ā is the quotient of a finite-dimensional hereditary algebra, and all standard modules are simple. Note that these standard Ā-modules can be viewed as standard A-modules.

Choose a composition series of Δ_{λ} ≅ P_{λ}: 0 = M_{0} ⊆ M_{1} ⊆ … ⊆ M_{t} = Δ_{λ}. It is clear that M_{i}/M_{i−1} ≅ S_{λ} if and only if i = t since A is quasi-hereditary. But F(_{≼}Δ) is closed under cokernels of monomorphisms and Δ_{μ} ≅ S_{μ} ≅ P_{μ}/ rad P_{μ} for all μ ∈ Λ \ {λ}, we deduce that S_{λ} ∈ F(_{≼}Δ). Thus S_{λ} ≅ Δ_{λ}.

It remains to show that the ordinary quiver of A has no oriented cycles. For P_{λ} = Ae_{λ} and P_{μ} = Ae_{μ} with λ ⪰̸ μ, we claim e_{λ}Ae_{μ} ≅ Hom_{A}(P_{λ}, P_{μ}) = 0. Indeed, since P_{μ} ∈ F(_{≼}Δ) and all standard modules are simple, the composition factors of P_{μ} are those S_{ν} with ν ∈ Λ and ν ≽ μ. Thus P_{μ} has no composition factors isomorphic to S_{λ}. Therefore, Hom_{A}(P_{λ}, P_{μ}) = 0. □

An immediate corollary is:

Corollary 3.4. If A is standardly stratified with respect to two different linear orders ≼ and ≼′ such that both F(_{≼}Δ) and F(_{≼}Δ) are closed under cokernels of monomorphisms, then F(_{≼}Δ) = F(_{≼′}Δ).

Proof. It is straightforward from the second statement of the previous theorem. □

4. Examples

In this section we describe an algorithm to determine whether there is a linear order ≼ with respect to which A is standardly stratified and F(_{≼}Δ) is closed under cokernels, as well as several examples.

4.1. An algorithm. Given an arbitrary algebra A, we want to check whether there exists a linear order ≼ for which A is standardly stratified and the corresponding category F(_{≼}Δ) is closed under cokernels. It is certainly not an ideal way to check all linear orders. In the rest of this paper we will describe an algorithm to construct a set L of linear orders for A satisfying the following property: A is standardly stratified for every linear order in L; moreover, if there is a linear order ≼ for which A is standardly stratified and F(_{≼}Δ) is closed under cokernels, then ≼∈ L.

As before, choose a set {e_{λ}}_{λ∈Λ} of orthogonal primitive idempotents in A such that ∑_{λ∈Λ} e_{λ} = 1 and let P_{λ} = Ae_{λ}. The algorithm is as follows:

(1) Define O_{1} = {λ ∈ Λ | ∀μ ∈ Λ, tr_{P_{λ}}(P_{μ}) ≅ P_{λ}^{m_{μ}}}. If O_{1} = ∅, the algorithm ends at this step. Otherwise, continue to the second step.

(2) Define a partial order ⩽′ on O_{1}: λ ⩽′ μ if and only if tr_{P_{μ}}(P_{λ}) ≠ 0 for λ, μ ∈ O_{1}. We can check that this partial order ⩽′ is well defined.

(3) Take e_{s_{1}} ∈ O_{1} which is maximal with respect to ⩽′. Let Ā = A/Ae_{s_{1}}A.

(4) Repeat the above steps for Ā recursively until the algorithm ends. Thus we get a chain of t idempotents e_{s_{1}}, e_{s_{2}}, …, e_{s_{t}} and define e_{s_{1}} ≻ e_{s_{2}} ≻ … ≻ e_{s_{t}}.

Let L̃ be the set of all linear orders obtained from the above algorithm, and let L ⊆ L̃ be the set of linear orders with length n = |Λ|.

Proposition 4.1. The algebra A is standardly stratified for every ≼∈ L.

Proof. We use induction on |Λ|. The conclusion is clearly true if |Λ| = 1. Suppose that it holds for |Λ| ⩽ n and consider the case that |Λ| = n + 1.

Let ≼ be an arbitrary linear order in L and take the unique maximal element λ ∈ Λ with respect to ≼. Consider the quotient algebra Ā = A/Ae_{λ}A. Then Ā, by the induction hypothesis and our algorithm, is standardly stratified with respect to the restricted order on Λ \ {λ}. It is clear from our definition of ≼ that Ae_{λ}A = ⨁_{μ∈Λ} tr_{P_{λ}}(P_{μ}) is projective. Thus A is standardly stratified for ≼. □

The next proposition tells us that it is enough to check linear orders in L to determine whether there exists a linear order ≼ for which A is standardly stratified and the corresponding category F(_{≼}Δ) is closed under cokernels of monomorphisms.

Proposition 4.2. Let ≼ be a linear order on Λ such that A is standardly stratified and the corresponding category F(_{≼}Δ) is closed under cokernels of monomorphisms. Then ≼∈ L.

Proof. The proof relies on induction on |Λ|. The claim is clear if |Λ| = 1. Suppose that the conclusion is true for |Λ| ⩽ l and let Λ be a set with n = l + 1 elements. Note that our algorithm is defined recursively. Furthermore, the quotient algebra Ā = A/Ae_{λ}A is also standardly stratified for the restricted linear order ≼ on Λ\{λ}, where λ is maximal with respect to ≼; Δ̄ = ⨁_{μ∈Λ\{λ}} Δ_{μ}; and F(Δ̄) is also closed under cokernels of monomorphisms. Thus by induction it suffices to show that λ ∈ O_{1}, and is maximal in O_{1} with respect to ⩽′ (see the second step of the algorithm).

Consider P_{λ} ≅ Δ_{λ}. Since A is standardly stratified for ≼, for each μ ∈ Λ, tr_{P_{λ}}(P_{μ}) is a projective module. Thus λ ∈ O_{1}.

If λ is not maximal in O_{1} with respect to ⩽′, then we can choose some μ ∈ O_{1} such that μ >′ λ, i.e., tr_{P_{μ}}(P_{λ}) ≠ 0. Since μ ∈ O_{1}, by definition, tr_{P_{μ}}(P_{λ}) ≅ P_{μ}^{m} for some m ⩾ 1. Now consider the exact sequence:

\xymatrix{0 \ar[r] & tr_{P_{μ}}(P_{λ}) \ar[r] & P_{λ} \ar[r] & P_{λ}/tr_{P_{μ}}(P_{λ}) \ar[r] & 0.}

Since tr_{P_{μ}}(P_{λ}) ≅ P_{μ}^{m} ∈ F(_{≼}Δ), P_{λ} ∈ F(_{≼}Δ), and F(_{≼}Δ) is closed under cokernels of monomorphism, we conclude that P_{λ}/tr_{P_{μ}}(P_{λ}) ∈ F(_{≼}Δ). This is impossible. Indeed, since P_{λ}/tr_{P_{μ}}(P_{λ}) has a simple top S_{λ} ≅ P_{λ}/ rad P_{λ}, if it is contained in F(_{≼}Δ), then it has a filtration factor Δ_{λ} ≅ P_{λ}. This is absurd.

We have proved by contradiction that λ ∈ O_{1} and is maximal in O_{1} with respect to ⩽′. The conclusion follows from induction. □

4.2. Example illustrating the above algorithm. The following example illustrates our algorithm.

Let A be the following algebra with relation α^{2} = δ^{2} = βα = δγ = ρ^{2} = ρφ = 0.

[figure: x; β; α; y; γ; φ; z; δ; w; ρ].

The projective A-modules are:

\xymatrix{ & x \ar[dl]_{α} \ar[dr]_{β} &  &  &  & y \ar[dl]_{γ} \ar[dr]_{ϕ} &  & z \ar[d]_{δ} & w \ar[d]_{ρ} \\ x &  & y \ar[dl]_{γ} \ar[dr]_{ϕ} &  & z &  & w & z & w \\  & z &  & w}

Then by the above algorithm, O_{1} = {x, y} and x ⩽′ y in O_{1}, we should take y as the maximal element. But then O_{2} = {x, z, w} and all these elements are maximal in O_{2} with respect to ⩽′. Thus we get three choices for O_{3}. Similarly, the two elements in each O_{3} are maximal with respect to ⩽′. In conclusion, 6 linear orders are contained in L: y ≻ x ≻ z ≻ w, y ≻ x ≻ w ≻ z, y ≻ z ≻ x ≻ w, y ≻ z ≻ w ≻ x, y ≻ w ≻ z ≻ x, and y ≻ w ≻ x ≻ z. For all these six linear orders A is standardly stratified and has the same standard modules. Moreover, the category F(_{≼}Δ) is closed under cokernels of monomorphisms.

\xymatrix{x \ar[d]_{α} &  & y \ar[dl]_{γ} \ar[dr]_{ϕ} &  & z \ar[d]_{δ} & w \ar[d]_{ρ} \\ x & z &  & w & z & w}

4.3. Different orders in L determine different standard modules. In general, for different linear orders in L the corresponding standard modules are different.

Let A be the following category with relation: δ^{2} = δα = 0, βδ = β′.

[figure: x; α; z; δ; y; β; β′]

The reader can check that the following two linear orders are contained in L: x ≻ z ≻ y and y ≻ x ≻ z. The corresponding standard modules are:

\xymatrix{x \ar[d]_{α} \\ z} y \xymatrix{z \ar[d]_{δ} \\ z}

and

\xymatrix{x \ar[d]_{α} & y \ar[d]_{β} & z \ar[d]_{δ} \\ z & z \ar[d]_{δ} & z \\  & z}

It is easy to see that F(Δ) corresponding to the first order is closed under cokernels of monomorphisms.

4.4. Not all linear orders can be obtained by the algorithm. We reminder the reader that although A is standardly stratified for all linear orders in L, it does not imply that all linear orders for which A is standardly stratified are contained in L. Moreover, it is also wrong that for every linear order ≼∉ L, there exists some ˜≼ ∈ L such that F(_{≼}Δ) ⊆ F( ˜≼Δ). Consider the following example.

Let A be the path algebra of the following quiver with relations: δ^{2} = δγ = 0, δβ = β′.

[figure: x; α; γ; y; β; β′; z; δ]

The structures of indecomposable projective A-modules are:

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

Applying the algorithm, we get a unique linear order y ≻ x ≻ z contained in L, and the corresponding standard modules are:

Δ_{x} = \xymatrix{x \ar[d]_{γ} \\ z} Δ_{y} ≅ P_{y} Δ_{z} ≅ P_{z}.

But there is another linear order x ≻ z ≻ y for which A is also standardly stratified with standard modules

Δ′_{x} ≅ P_{x} Δ′_{y} = y Δ′_{z} ≅ P_{z}.

Both F(_{≼}Δ) and F(_{≼′}Δ) are not closed under cokernels of monomorphisms. Moreover, each of them is not contained in the other one.

4.5. F(_{≼}Δ) is maximal but not closed under cokernels of monomorphisms. In general it is possible that there are more than one linear orders with respect to which A is standardly stratified. However, among these categories F(_{≼}Δ), there is at most one which is closed under cokernels of monomorphisms. If this category exists, it is the unique maximal category and contains all other ones as subcategories. But the converse of this statement is not true by the next example. That is, if there is a linear order ≼ with respect to which A is standardly stratified and F(_{≼}Δ) is the unique maximal category, F(_{≼}Δ) might not be closed under cokernels of monomorphisms.

Let A be the path algebra of the following quiver with relations: γ^{2} = ρ^{2} = δγ = ρδ = 0, γα = α′, γβ = β′, and δα = δβ.

[figure: x; α; α′; β; β′; y; γ; δ; z; ρ]

The structures of indecomposable projective A-modules are described as follows:

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

It is not hard to check that if A is standardly stratified with respect to some linear order ≼, then all standard modules coincide with indecomposable projective modules. Therefore, F(_{≼}Δ) is the category of all projective modules and is the unique maximal category. However, we get an exact sequence as follows, showing that F(Δ) is not closed under cokernels of monomorphisms:

\xymatrix{0 \ar[r] & P_{y} \ar[r] & P_{x} \ar[r] & M \ar[r] & 0}

where M ∉ F(Δ) has the following structure:

\xymatrix{x \ar[d]_{α} \\ y \ar[d]_{γ} \\ y}

4.6. F(_{≼}Δ) is closed under cokernels of monomorphisms but A is not directed. The last statement of Theorem 0.3 tells us that if A is quasi-hereditary and F(_{≼}Δ) is closed under cokernels of monomorphisms, then A is a directed category. This is incorrect if A is supposed to be standardly stratified, as shown by the next example.

Let A be the path algebra of the following quiver with relations δ^{2} = δα = βδ = βα = γβ = 0. Let x ≻ z ≻ y.

[figure: x; α; z; γ; y; β; δ]

Indecomposable projective modules of A are described below:

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

The standard modules are:

\xymatrix{x \ar[d]_{α} \\ y} \xymatrix{y \ar[d]_{δ} \\ y} z

It is clear that A is standardly stratified. Moreover, by (1) of Theorem 0.3, F(Δ) is closed under cokernels of monomorphisms. But A is not a directed category.

References

[1] E. Cline, B. Parshall, and L. Scott, Stratifying Endomorphism Algebras, Mem. Amer. Math. Sco. 124 (1996), no. 591;

[2] V. Dlab, and C. Ringel, The module theoretical approach to Quasi-hereditary Algebras, Representations of algebras and related topics (Kyoto, 1990), 200-224, London Math. Soc. Lecture Note Ser. 168, Cambridge Univ. Press 1992;

[3] V. Dlab, and C. Ringel, Quasi-hereditary Algebras, Illinois J. Math. 33 (1989), 280-291;

[4] K. Erdmann, and C. Sáenz, On standardly stratified algebras, Comm. Algebras 31 (2003), 3429-3446;

[5] A. Frisk, On different stratifications of the same algebra, Preprint, Uppsala University, 2004;

[6] A. Frisk, Two-step tilting for standardly stratified algebras, Algebra Discrete Math. 2004, no. 3, 38-59;

[7] A. Frisk, Dlab’s theorem and tilting modules for stratified algebras, J. Algebra 314 (2007), 507-537;

[8] L. Li, A generalized Koszul theory and its application, ArXiv: 1109.5760;

[9] L. Li, Extension algebras of standard modules, accepted by Comm. Algebra, ArXiv: 1110.6502;

[10] M. Klucznik, and V. Mazorchuk, Parabolic decomposition for properly stratified algebras, J. Pure Appl. Algebra 171 (2002), 41-58;

[11] P. Webb, Standard Stratifications of EI Categories and Alperin’s Weight Conjecture, J. Algebra 320 (2008), 4073-4091;

[12] C. Xi, Standardly stratified algebras and cellular algebras, Math. Proc. Camb. Phil. Soc. 133 (2002), 37-53.

School of Mathematics, University of Minnesota, MN, 55455, USA

E-mail address: lixxx480@math.umn.edu