EXTENSION ALGEBRAS OF STANDARD MODULES

LIPING LI

Abstract. Let A be a basic finite-dimensional k-algebra standardly stratified for a partial order ⩽ and Δ be the direct sum of all standard modules. In this paper we study the extension algebra Γ = Ext_{A}^{∗}(Δ, Δ) of standard modules, characterize the stratification property of Γ for ⩽ and ⩽^{op}, and obtain a sufficient condition for Γ to be a generalized Koszul algebra (in a sense which we define).

Let A be a basic finite-dimensional k-algebra standardly stratified with respect to a poset (Λ, ⩽) indexing all simple modules (up to isomorphism), Δ be the direct sum of all standard modules, and F(Δ) be the category of finitely generated A-modules with Δ-filtrations. That is, for each M ∈ F(Δ), there is a chain 0 = M_{0} ⊆ M_{1} ⊆ … ⊆ M_{n} = M such that M_{i}/M_{i−1} is isomorphic to an indecomposable summand of Δ, 1 ⩽ i ⩽ n. Since standard modules of A are relative simple in F(Δ), we are motivated to exploit the extension algebra Γ = Ext_{A}^{∗}(Δ, Δ) of standard modules. These extension algebras were studied in [1, 6, 11, 19, 23]. In this paper, we are interested in the stratification property of Γ with respect to (Λ, ⩽) and (Λ, ⩽^{op}), and its Koszul property since Γ has a natural grading. A particular question is that in which case it is a generalized Koszul algebra, i.e., Γ_{0} has a linear projective resolution.

By Gabriel’s construction, we associate a locally finite k-linear category E to the extension algebra Γ such that the category Γ-mod of finitely generated left Γmodules is equivalent to the category of finitely generated k-linear representations of E. We show that the category E is a directed category with respect to ⩽. That is, the morphism space E(x, y) = 0 whenever x ⩽̸ y. With this terminology, we have:

Theorem 0.1. If A is standardly stratified for (Λ, ⩽), then E is a directed category with respect to ⩽ and is standardly stratified for ⩽^{op}. Moreover, E is standardly stratified for ⩽ if and only if for all λ, μ ∈ Λ and s ⩾ 0, Ext_{A}^{s}(Δ_{λ}, Δ_{μ}) is a projective End_{A}(Δ_{μ})-module.

In particular, if A is a quasi-hereditary algebra, then Γ is quasi-hereditary with respect to both ⩽ and ⩽^{op}. We also generalize the above theorem to abstract stratifying systems and Ext-projective stratifying systems (EPSS) described in [7, 17, 18, 24].

In the case that the standardly stratified algebra A is a graded algebra and A_{0} is semisimple, its Koszul duality has been studied in [2, 3, 20, 21, 22]. Since the extension algebra Γ = Ext_{A}^{∗}(Δ, Δ) has a natural grading, and Γ_{0} = End_{A}(Δ) in general is not semisimple, we study the general Koszul property of Γ by using some generalized Koszul theories developed in [10, 12, 15, 16], where the degree 0 parts of graded algebras are not required to be semisimple.

2000 Mathematics Subject Classification. 16G10, 16E40.

Key words and phrases. Extension algebras, standardly stratified, Koszul, Quasi-hereditary.

The author would like to thank his thesis advisor, Professor Peter Webb, for proposing to study the the extension algebras of standard modules, and carefully checking the manuscript. He also thanks Prof. Mazorchuk for pointing out many related works, which are unknown to the author before.

Take a fixed EPSS (\underline{Θ}, \underline{Q}). As an analogue to linear modules of graded algebras, we define linearly filtered modules in this system. With this terminology, a sufficient condition can be obtained for Γ to be a generalized Koszul algebra.

Theorem 0.2. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽) such that Ext_{A}^{i}(Q, Θ) = 0 for all i ⩾ 1 and Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ). Suppose that all Θ_{λ} are linearly filtered for λ ∈ Λ. If M ∈ F(Θ) is linearly filtered, then the graded Γ-module Ext_{A}^{∗}(M, Θ) has a linear projective resolution. In particular, Γ = Ext_{A}^{∗}(Θ, Θ) is a generalized Koszul algebra.

The paper is organized as follows: in section 1 we characterize the stratification property of Γ; in section 2 we define linearly filtered modules, study their basic properties and prove the second theorem.

Throughout this paper A is a finite-dimensional basic associative k-algebra with identity 1, where k is algebraically closed. We only consider finitely generated modules and denote by A-mod the category of finitely generated left A-modules. Maps and morphisms are composed from right to left.

1. Stratification property of extension algebras

Let (Λ, ⩽) be a finite preordered set parameterizing all simple A-modules S_{λ} (up to isomorphism). This preordered set also parameterizes all indecomposable projective A-modules P_{λ} (up to isomorphism). According to [4], the algebra A is standardly-stratified with respect to (Λ, ⩽) 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. In some literatures the preordered set (Λ, ⩽) is supposed to be a poset ([5]) or even a linearly ordered set ([2, 3]). Algebras standardly stratified in this sense are called strongly standardly stratified ([8, 9]). In this paper (Λ, ⩽) is supposed to be a poset.

If A is standardly stratified, then standard modules can be defined as ^{1}:

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}. See [5, 24] for more details. 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. Clearly, since A is standardly stratified for ⩽, _{A}A ∈ F(Δ), or equivalently, every indecomposable projective A-module has a filtration by standard modules.

Throughout this section we suppose that A is standardly stratified with respect to ⩽ if it is not specified. We also remind the reader that F(Δ) is closed under extensions, kernels of epimorphisms, and direct summands, but is not closed under cokernels of monomorphisms.

Given M ∈ F(Δ) and a fixed filtration 0 = M_{0} ⊆ M_{1} ⊆ … ⊆ M_{n} = M, we define the filtration multiplicity m_{λ} = [M : Δ_{λ}] to be the number of factors isomorphic to Δ_{λ} in this filtration. By Lemma 1.4 of [7], The filtration multiplicities defined above are independent of the choice of a particular filtration. Moreover, since each standard module has finite projective dimension, we deduce that every Amodule contained in F(Δ) has finite projective dimension. Therefore, the extension algebra Γ = Ext_{A}^{∗}(Δ, Δ) is finite-dimensional.

1In [2, 3] standard modules are defined as Δ_{λ} = P_{λ}/ ∑_{μ>λ} tr_{P_{μ}}(P_{λ}). Note that in their setup ⩽ is a linear order, so this description of standard modules coincides with ours.

Lemma 1.1. Let Δ_{λ}, Δ_{μ} be standard modules. Then Ext_{A}^{n}(Δ_{λ}, Δ_{μ}) = 0 if λ ⩽̸ μ for all n ⩾ 0.

Proof. First, we claim [Ω^{i}(Δ_{λ}) : Δ_{ν}] = 0 whenever λ ⩽̸ ν for all i ⩾ 0, where Ω is the Heller operator. Indeed, for i = 0 the conclusion holds clearly. Suppose that it is true for all i ⩽ n and consider Ω^{n+1}(Δ_{λ}). We have the following exact sequence:

\xymatrix{0 \ar[r] & Ω^{n+1}(Δ_{λ}) \ar[r] & P \ar[r] & Ω^{n}(Δ_{λ}) \ar[r] & 0.}

By the induction hypothesis, [Ω^{n}(Δ_{λ}) : Δ_{ν}] = 0 whenever λ ⩽̸ ν. Therefore, [P : Δ_{ν}] = 0 whenever λ ⩽̸ ν, and hence [Ω^{n+1}(Δ_{λ}) : Δ_{ν}] = 0 whenever λ ⩽̸ ν. The claim is proved by induction.

The above short exact sequence induces a surjection Hom_{A}(Ω^{n}(Δ_{λ}), Δ_{μ}) → Ext_{A}^{n}(Δ_{λ}, Δ_{μ}). Thus it suffices to show Hom_{A}(Ω^{n}(Δ_{λ}), Δ_{μ}) = 0 for all n ⩾ 0 if λ ⩽̸ μ. By the above claim, all filtration factors Δ_{ν} of Ω^{n}(Δ_{λ}) satisfy ν ⩾ λ, and hence ν ⩽̸ μ. But Hom_{A}(Δ_{ν}, Δ_{μ}) = 0 whenever ν ⩽̸ μ. The conclusion follows. □

Gabriel’s construction gives rise to a bijective correspondence between finitedimensional algebras and locally finite k-linear categories with finitely many objects. Explicitly, To each finite-dimensional k-algebra A with a chosen set of orthogonal primitive idempotents {e_{λ}}_{λ∈Λ} satisfying ∑_{λ∈Λ} e_{λ} = 1 we define a k-linear category A with Ob A = {e_{λ}}_{λ∈Λ} and A(e_{λ}, e_{μ}) = e_{μ}Ae_{λ} ≅ Hom_{A}(Ae_{μ}, Ae_{λ}). Conversely, given a locally finite k-linear category A with finitely many objects, we define A = ⨁_{x,y∈Ob A} A(x, y), and the multiplication in A is induces by the composition of morphisms in A in an obvious way. Clearly, A-mod is Morita equivalent to the category of all finite-dimensional k-linear representations of A. We then call A the associated category of A and A the associated algebra of A. The category A is called directed if there is a partial order ≼ on Ob A such that A(x, y) = 0 unless x ≼ y.

Now let Γ = Ext_{A}^{∗}(Δ, Δ). This is a graded finite-dimensional algebra equipped with a natural grading. In particular, Γ_{0} = End_{A}(Δ). For each λ ∈ Λ, Δ_{λ} is an indecomposable A-module. Therefore, up to isomorphism, the indecomposable projective Γ-modules are exactly those Ext_{A}^{∗}(Δ_{λ}, Δ), λ ∈ Λ.

The associated k-linear category E of Γ has the following structure: Ob E = {Δ_{λ}}_{λ∈Λ}; the morphism space E(Δ_{λ}, Δ_{μ}) = Ext_{A}^{∗}(Δ_{λ}, Δ_{μ}). The partial order ⩽ induces a partial order on Ob E which we still denote by ⩽, namely, Δ_{λ} ⩽ Δ_{μ} if and only if λ ⩽ μ.

Proposition 1.2. The associated category E of Γ is directed with respect to ⩽. In particular, Γ is standardly stratified with respect to ⩽^{op} and all standard modules are projective.

Proof. The first statement follows from the previous lemma. The second statement is also clear. Indeed, since Γ is directed with respect to ⩽, e_{μ}Γe_{λ} ≅ Hom_{Γ}(Q_{μ}, Q_{λ}) = 0 if μ ⩾̸ λ, where Q_{μ}, Q_{λ} are projective Γ-modules. Thus tr_{Q_{μ}}(Q_{λ}) = 0 whenever μ ⩾̸ λ, or equivalently, tr_{Q_{μ}}(Q_{λ}) = 0 whenever μ ⩽̸^{op} λ. Therefore, all standard modules with respect to ⩽^{op} are projective. □

The following proposition characterizes the stratification property of a k-linear category directed with respect to ⩽.

Proposition 1.3. Let C be a locally finite k-linear category directed with respect to a partial order ⩽ on Ob C. Then it is stratified for this order. The standard modules are isomorphic to indecomposable summands of ⨁_{x∈Ob C} C(x, x). Moreover, this stratification is standard if and only if for each pair of objects x, y ∈ Ob C, C(x, y) is a projective C(y, y)-module.

Proof. This is just a collection of results in [12]. The first statement is Corollary 5.4; the second statement comes from Proposition 5.5; and the last statement is Theorem 5.7. □

Now we restate and prove the first theorem.

Theorem 1.4. If A is standardly stratified for (Λ, ⩽), then E is a directed category with respect to ⩽ and is standardly stratified for ⩽^{op}. Moreover, E is standardly stratified for ⩽ if and only if for all λ, μ ∈ Λ and s ⩾ 0, Ext_{A}^{s}(Δ_{λ}, Δ_{μ}) is a projective End_{A}(Δ_{μ})-module.

Proof. The first statement follows from Proposition 1.2 and the second statement follows from Proposition 1.3. □

In the case that A is quasi-hereditary, we have:

Corollary 1.5. If A is a quasi-hereditary algebra with respect to ⩽, then Γ is quasi-hereditary with respect to both ⩽ and ⩽^{op}.

Proof. We have showed that Γ is standardly stratified with respect to ⩽^{op} and the corresponding standard modules _{Γ}Δ_{λ} ≅ Γ1_{λ} for λ ∈ Λ. Therefore,

End_{Γ}(_{Γ}Δ_{λ}) = End_{Γ}(Γ1_{λ}) ≅ 1_{λ}Γ1_{λ} = Ext_{A}^{∗}(Δ_{λ}, Δ_{λ}) = End_{A}(Δ_{λ}) ≅ k

since A is quasi-hereditary. So Γ is also quasi-hereditary with respect to ⩽^{op}.

Now consider the stratification property of Γ with respect to ⩽. The associated category E is directed with respect to ⩽. Since Ext_{A}^{∗}(Δ_{μ}, Δ_{μ}) = End_{A}(Δ_{μ}) ≅ k for all μ ∈ Λ, E(Δ_{λ}, Δ_{μ}) = Ext_{A}^{∗}(Δ_{λ}, Δ_{μ}) is a projective k-module for each pair λ, μ ∈ Λ. Therefore, E is standardly stratified for ⩽ by the previous theorem. Moreover, by Proposition 1.3, the standard modules of E (or the standard modules of Γ) are precisely indecomposable summands of ⨁_{λ∈Λ} Ext_{A}^{∗}(Δ_{λ}, Δ_{λ}) ≅ ⨁_{λ∈Λ} k_{λ}. Clearly, for λ ∈ Λ, End_{Γ}(k_{λ}, k_{λ}) ≅ k, so Γ is quasi-hereditary with respect to ⩽. □

The following example from 8.2 in [9] illustrates why we should assume that ⩽ is a partial order rather than a preorder. Indeed, in a preordered set (Λ, ⩽) we cannot deduce x = y if x ⩽ y and y ⩽ x.

Example 1.6. Let A be the path algebra of the following quiver with relations α_{1}β_{1} = α_{2}β_{2} = α_{2}α_{1} = β_{1}β_{2} = 0. Define a preorder ⩽ by letting x ⩽ y < z and y ⩽ x < z.

\xymatrix{x \ar[r]^{α_{1}} & y \ar[r]^{α_{2}} \ar[l]^{β_{1}} & z \ar[l]^{β_{2}}}.

Projective modules and standard modules are described as follows:

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

Then the associated category E of Γ = Ext_{A}^{∗}(Δ, Δ) is not a directed category since both Hom_{A}(Δ_{x}, Δ_{y}) and Hom_{A}(Δ_{y}, Δ_{x}) are nonzero.

Now we generalize the above results to Ext-Projective Stratifying Systems (EPSS). From now on the algebra A is finite-dimensional and basic, but we do not assume that it is standardly stratified for some partial order, as we did before. The EPSS we describe in this paper is indexed by a finite poset (Λ, ⩽) rather than a linearly ordered set as in [17, 18]. However, this difference is not essential and all properties described in [17, 18] can be applied to our situation with suitable modifications.

Definition 1.7. (Definition 2.1 in [18]) Let \underline{Θ} = \{Θ_{λ}\}_{λ∈Λ} be a set of nonzero Amodules and \underline{Q} = \{Q_{λ}\}_{λ∈Λ} be a set of indecomposable A-modules, both of which are indexed by a finite poset (Λ, ⩽). We call (\underline{Θ}, \underline{Q}) an EPSS if the following conditions are satisfied:

(1) Hom_{A}(Θ_{λ}, Θ_{μ}) = 0 if λ ⩽̸ μ;

(2) for each λ ∈ Λ, there is an exact sequence 0 → K_{λ} → Q_{λ} → Θ_{λ} → 0 such that K_{λ} has a filtration only with factors isomorphic to Θ_{μ} satisfying μ > λ;

(3) for every A-module M ∈ F(\underline{Θ}) and λ ∈ Λ, Ext_{A}^{1}(Q_{λ}, M) = 0.

We denote Θ and Q the direct sums of all Θ_{λ}’s and Q_{λ}’s respectively, λ ∈ Λ. Given an EPSS (\underline{Θ}, \underline{Q}) indexed by (Λ, ⩽), (\underline{Θ}, ⩽) is a stratifying system (SS):

Hom_{A}(Θ_{λ}, Θ_{μ}) = 0 if λ ⩽̸ μ, and Ext_{A}^{1}(Θ_{λ}, Θ_{μ}) = 0 if λ ≮ μ. Conversely, given a stratifying system (\underline{Θ}, ⩽), we can construct an EPSS (\underline{Θ}, \underline{Q}) unique up to isomorphism. See [18] for more details. Moreover, as described in [18], the algebra B = End_{A}(Q)^{op} is standardly stratified, and the functor e_{Q} = Hom_{A}(Q, −) gives an equivalence of exact categories between F(Θ) and F(_{B}Δ).

To study the extension algebra Γ = Ext_{A}^{∗}(Θ, Θ), one may want to use projective resolutions of Θ. However, different from the situation of standardly stratified algebras, the regular module _{A}A in general might not be contained in F(Θ). If we suppose that _{A}A is contained in F(Θ) (in this case the stratifying system (\underline{Θ}, ⩽) is said to be standard) and F(Θ) is closed under the kernels of surjections, then by Theorem 2.6 in [17] A is standardly stratified for ⩽ and those Θ_{λ}’s coincide with standard modules of A. This situation has been completely discussed previously. Alternately, we use the relative projective resolutions whose existence is guaranteed by the following proposition.

Proposition 1.8. (Corollary 2.11 in [18]) Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽). Then for each M ∈ F(Θ), there is a finite resolution

\xymatrix{0 \ar[r] & Q^{d} \ar[r] & … \ar[r] & Q^{0} \ar[r] & M \ar[r] & 0}

such that each kernel is contained in F(Θ), where 0 ≠ Q^{i} ∈ add(Q) for 0 ⩽ i ⩽ d.

The number d in this resolution is called the relative projective dimension of M.

Proposition 1.9. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽) and d be the relative projective dimension of Θ. If Ext_{A}^{s}(Q, Θ) = 0 for all s ⩾ 1, then for M, N ∈ F(Θ) and s > d, Ext_{A}^{s}(M, N) = 0.

Proof. Since both M and N are contained in F(Θ), it is enough to show that Ext_{A}^{s}(Θ, Θ) = 0 for all s > d. If d = 0, then Q = Θ and the conclusion holds trivially. So we suppose d ⩾ 1. Applying the functor Hom_{A}(−, Θ) to the exact sequence

\xymatrix{0 \ar[r] & K_{1} \ar[r] & Q \ar[r] & Θ \ar[r] & 0}

we get a long exact sequence. In particular, from the segment

\xymatrix{Ext_{A}^{s−1}(Q, Θ) \ar[r] & Ext_{A}^{s−1}(K_{1}, Θ) \ar[r] & Ext_{A}^{s}(Θ, Θ) \ar[r] & Ext_{A}^{s}(Q, Θ)}

of this long exact sequence we deduce that Ext_{A}^{s}(Θ, Θ) ≅ Ext_{A}^{s−1}(K_{1}, Θ) since the first and last terms are 0. Now applying Hom_{A}(−, Θ) to the exact sequence

\xymatrix{0 \ar[r] & K_{2} \ar[r] & Q^{1} \ar[r] & K_{1} \ar[r] & 0}

we get Ext_{A}^{s−1}(K_{1}, Θ) ≅ Ext_{A}^{s−2}(K_{2}, Θ). Thus Ext_{A}^{s}(Θ, Θ) ≅ Ext_{A}^{s−d}(K_{d}, Θ) by induction. But K_{d} ≅ Q^{d} ∈ add(Q). The conclusion follows. □

Thus Γ = Ext_{A}^{∗}(Θ, Θ) is a finite-dimensional algebra under the given assumption. There is a natural partition on the finite poset (Λ, ⩽) as follows: let Λ_{1} be the

subset of all minimal elements in Λ, Λ_{2} be the subset of all minimal elements in Λ\Λ_{1}, and so on. Then Λ = ⊔_{i⩾1}Λ_{i}. With this partition, we can introduce a height function h : Λ → ℕ in the following way: for λ ∈ Λ_{i} ⊆ Λ, i ⩾ 1, we define h(λ) = i.

For each M ∈ F(Θ), we define supp(M) to be the set of elements λ ∈ Λ such that M has a Θ-filtration in which there is a factor isomorphic to Θ_{λ}. For example, supp(Θ_{λ}) = {λ}. By Lemma 2.6 in [18], the multiplicities of factors of M is independent of the choice of a particular Θ-filtration. Therefore, supp(M) is well defined. We also define min(M) = min({h(λ) | λ ∈ supp(M)}).

Lemma 1.10. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽). For each M ∈ F(Θ), there is an exact sequence 0 → K_{1} → Q^{0} → M such that K_{1} ∈ F(Θ) and min(K_{1}) > min(M), where Q^{0} ∈ add(Q).

Proof. This is Proposition 2.10 in [18] which deals with the special case that Λ is a linearly ordered set. The general case can be proved similarly by observing the fact that Ext_{A}^{1}(Θ_{λ}, Θ_{μ}) = 0 if h(λ) = h(μ). □

By this lemma, the relative projective dimension of every M ∈ F(Θ) cannot exceed the length of the longest chain in Λ.

As before, we let E be the k-linear category associated to Γ = Ext_{A}^{∗}(Θ, Θ).

Theorem 1.11. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽). such that Ext_{A}^{i}(Q, Θ) = 0 for all i ⩾ 1. Then E is a directed category with respect to ⩽ and is standardly stratified for ⩽^{op}. Moreover, it is standardly stratified for ⩽ if and only if for all s ⩾ 0, Ext_{A}^{s}(Θ_{λ}, Θ_{μ}) is a projective End_{A}(Θ_{μ})-module, λ, μ ∈ Λ.

Proof. We only need to show that E is a directed category with respect to ⩽ since the other statements can be proved as in Theorem 1.4. We know Hom_{A}(Θ_{λ}, Θ_{μ}) = 0 if λ ⩽̸ μ and Ext_{A}^{1}(Θ_{λ}, Θ_{μ}) = 0 for all λ ≮ μ. Therefore, it suffices to show that for all s ⩾ 2, Ext_{A}^{s}(Θ_{λ}, Θ_{μ}) = 0 if λ ≮ μ.

By Proposition 1.8 and Lemma 1.10, Θ_{λ} has a relative projective resolution

\xymatrix{0 \ar[r] & Q^{d} \ar[r]^{f_{d}} & … \ar[r]^{f_{1}} & Q^{0} \ar[r]^{f_{0}} & Θ_{λ} \ar[r] & 0}

such that for each map f_{t}, min(K_{t}) > min(K_{t−1}), where K_{t} = Ker(f_{t}) and 1 ⩽ t ⩽ d. By Proposition 1.9, Ext_{A}^{s}(Θ_{λ}, Θ_{μ}) = 0 if s > d; if 2 ⩽ s ⩽ d, we have Ext_{A}^{s}(Θ_{λ}, Θ_{μ}) ≅ Ext_{A}^{1}(K_{s−1}, Θ_{μ}). But we have chosen

min(K_{s−1}) > min(K_{s−2}) > … > min(Θ_{λ}) = h(λ) ⩾ h(μ).

Thus each factor Θ_{ν} appearing in a Θ-filtration of K_{s−1} satisfies h(ν) > h(μ), and hence ν ⩽̸ μ. Since Ext_{A}^{1}(Θ_{ν}, Θ_{μ}) = 0 for all ν ⩽̸ μ, we deduce

Ext_{A}^{s}(Θ_{λ}, Θ_{μ}) ≅ Ext_{A}^{1}(K_{s−1}, Θ_{μ}) = 0.

This finishes the proof. □

The following corollary is a generalization of Corollary 1.5.

Corollary 1.12. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽). If for all s ⩾ 1 and λ ∈ Λ we have Ext_{A}^{s}(Q, Θ) = 0 and End_{A}(Θ_{λ}, Θ_{λ}) ≅ k, then Γ is quasi-hereditary with respect to both ⩽ and ⩽^{op}.

Proof. This can be proved as Corollary 1.5. □

2. Koszul Property of Extension Algebras

There is a well known duality related to the extension algebras: the Koszul duality. Explicitly, if A is a graded Koszul algebra with A_{0} being a semisimple algebra, then B = Ext_{A}^{∗}(A_{0}, A_{0}) is a Koszul algebra, too. Moreover, the functor Ext_{A}^{∗}(−, A_{0}) gives an equivalence between the category of linear A-modules and the category of linear B-modules.^{2} However, even if A is quasi-hereditary with respect to a partial order ⩽, B might not be quasi-hereditary with respect to ⩽ or ⩽^{op}. This problem has been considered in [2, 20].

On the other hand, if A is quasi-hereditary with respect to ⩽, we have showed that the extension algebra Γ = Ext_{A}^{∗}(Δ, Δ) is quasi-hereditary with respect to both ⩽ and ⩽^{op}. But Γ is in general not a Koszul algebra in a sense which we define later. In this section we want to get a sufficient condition for Γ to be a generalized Koszul algebra.

We work in the context of EPSS described in last section. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽); Q = ⨁_{λ∈Λ} Q_{λ} and Θ = ⨁_{λ∈Λ} Θ_{λ}. We insist the following conditions: Ext_{A}^{s}(Q, Θ) = 0 for all s ⩾ 1; each Θ_{λ} has a simple top S_{λ}; and S_{λ} ≇ S_{μ} for λ ≠ μ. These conditions are always true for the classical stratifying system of a standardly stratified basic algebra. In particular, in that case Q =_{A} A.

Proposition 2.1. Let 0 ≠ M ∈ F(Θ) and i = min (M). Then there is an exact sequence

(2.1)  \xymatrix{0 \ar[r] & M[1] \ar[r] & M \ar[r] & ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} \ar[r] & 0}

such that M[1] ∈ F(Θ) and min(M[1]) > min(M).

Proof. This is Proposition 2.9 in [18] which deals with the special case that Λ is a linearly ordered set. The general case can be proved similarly by observing the fact that Ext_{A}^{1}(Θ_{λ}, Θ_{μ}) = 0 if h(λ) = h(μ). □

It is clear that m_{λ} = [M : Θ_{λ}]. Based on this proposition, we make the following definition:

Definition 2.2. Let M ∈ F(Θ) with min(M) = i. We say M is generated in height i if in sequence (2.1) we have

Top M = M/ rad M ≅ Top ( ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}) = ⨁_{h(λ)=i} S_{λ}^{⊕m_{λ}}.

We introduce some notation: if M ∈ F(Θ) is generated in height i, then define M_{i} = ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} in sequence (2.1). If M[1] is generated in some height j, we can define M[2] = M[1][1] and M[1]_{j} in a similar way. This procedure can be repeated.

Proposition 2.3. Let 0 → L → M → N → 0 be an exact sequence in F(Θ). If M is generated in height i, so is N. Conversely, if both L and N are generated in height i, then M is generated in height i as well.

Proof. We always have Top N ⊆ Top M and Top M ⊆ Top L ⊕ Top N. The conclusion follows from these inclusions and the rightmost identity in the above definition. □

Notice that [Q_{λ} : Θ_{λ}] = 1 and [Q_{λ} : Θ_{μ}] = 0 for all μ ⩾̸ λ. We claim that Q_{λ} is generated in height h(λ) for λ ∈ Λ. Indeed, the algebra B = End_{A}(Q)^{op} is a standardly stratified algebra, with projective modules Hom_{A}(Q, Q_{λ}) and standard modules Hom_{A}(Q, Θ_{λ}), λ ∈ Λ. Moreover, the functor Hom_{A}(Q, −) gives an equivalence between F(Θ) ⊆ A-mod and F(_{B}Δ) ⊆ B-mod. Using this equivalence and the standard filtration structure of projective B-modules we deduce the conclusion.

2In [12] we generalized these results to the situation that A_{0} is a self-injective algebra.

Lemma 2.4. If M ∈ F(Θ) is generated in height i with [M : Θ_{λ}] = m_{λ}, then M has a relative projective cover Q^{i} ≅ ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}.

Proof. There is a surjection f : M → ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} by Proposition 2.1. Consider the following diagram:

\xymatrix{ & ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}} \ar[d]^{p} \ar@{-->}[dl]_{q} \\ M \ar[r]^{f} & ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} \ar[r] & 0.}

Since Q^{i} is projective in F(Θ), the projection p factors through the surjection f. In particular, Top ( ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}) = ⨁_{h(λ)=i} S_{λ}^{⊕m_{λ}} is in the image of fq. Since M is generated in height i, f induces an isomorphism between Top M and Top ( ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}). Thus Top M is in the image of q, and hence q is surjective.

It is clear that q is minimal, so Q^{1} = ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}} is a relative projective cover of M. The uniqueness follows from Proposition 8.3 in [24]. □

We use Ω_{Θ}^{i}(M) to denote the i-th relative syzygy of M. Actually, for every M ∈ F(Θ) there is always a relative projective cover by Proposition 8.3 in [24].

The following definition is an analogue of linear modules in the representation theory of graded algebras.

Definition 2.5. An A-module M ∈ F(Θ) is said to be linearly filtered if there is some i ∈ ℕ such that Ω_{Θ}^{s}(M) is generated in height i + s for s ⩾ 0.

Equivalently, M ∈ F(Θ) is linearly filtered if and only if it is generated in height i and has a relative projective resolution

\xymatrix{0 \ar[r] & Q^{l} \ar[r] & Q^{l−1} … \ar[r] & Q^{i+1} \ar[r] & Q^{i} \ar[r] & M \ar[r] & 0}

such that each Q^{s} is generated in height s, i ⩽ s ⩽ l.

We remind the reader that there is a common upper bound for the relative projective dimensions of modules contained in F(Θ), which is the length of the longest chains in the finite poset (Λ, ⩽). It is also clear that if M is linearly filtered, so are all relative syzygies and direct summands. In other words, the subcategory LF(Θ) constituted of linearly filtered modules contains all relative projective modules, and is closed under summands and relative syzygies. But in general it is not closed under extensions, kernels of epimorphisms and cokernels of monomorphisms.

Proposition 2.6. Let 0 → L → M → N → 0 be an exact sequence in F(Θ) such that all terms are generated in height i. If L is linearly filtered, then M is linearly filtered if and only if N is linearly filtered.

Proof. Let m_{λ} = [M : Θ_{λ}], l_{λ} = [L : Θ_{λ}] and n_{λ} = [N : Θ_{λ}]. By the previous lemma, we get the following commutative diagram:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Ω_{Θ}(L) \ar[r] \ar[d] & Ω_{Θ}(M) \ar[r] \ar[d] & Ω_{Θ}(N) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ⨁_{h(λ)=i} Q_{λ}^{⊕l_{λ}} \ar[r] \ar[d] & ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}} \ar[r] \ar[d] & ⨁_{h(λ)=i} Q_{λ}^{⊕n_{λ}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & L \ar[r] \ar[d] & M \ar[r] \ar[d] & N \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

Since Ω_{Θ}(L) is generated in height i + 1, by Proposition 2.3, Ω_{Θ}(M) is generated in height i + 1 if and only if Ω_{Θ}(N) is generated in height i + 1. Replacing L, M and N by Ω_{Θ}(L), Ω_{Θ}(M) and Ω_{Θ}(N) respectively, we conclude that Ω_{Θ}^{2}(M) is generated in height i + 2 if and only if Ω_{Θ}^{2}(N) is generated in height i + 2. The conclusion follows from induction. □

Corollary 2.7. Suppose that M ∈ F(Θ) is generated in height i and linearly filtered. If ⨁_{h(λ)=i} Θ_{λ} is linearly filtered, then M[1] is generated in height i+1 and linearly filtered.

Proof. Clearly ⨁_{h(λ)=i} Θ_{λ} is generated in height i. Let m_{λ} = [M : Θ_{λ}]. Notice that both M and ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} have projective cover ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}. Thus the exact sequence

\xymatrix{0 \ar[r] & M[1] \ar[r] & M \ar[r] & ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} \ar[r] & 0}

induces the following diagram:

[figure: Ω_{Θ}(M); Ω_{Θ}( ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}); M[1]; ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}; ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}; M[1]; M; ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}.]

Consider the top sequence. Since both Ω_{Θ}(⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}) and Ω_{Θ}(M) are generated in height i + 1 and linearly filtered, M[1] is also generated in height i + 1 and linearly filtered by Propositions 2.3 and 2.6. □

These results tell us that linearly filtered modules have properties similar to those of linear modules of graded algebras.

Lemma 2.8. Let M ∈ F(Θ) be generated in height i and m_{λ} = [M : Θ_{λ}]. If Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ), then

Hom_{A}(M, Θ) ≅ Hom_{A}( ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}, Θ) ≅ Hom_{A}( ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}, Θ).

Proof. We claim that Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ) as End_{A}(Θ)-modules implies Hom_{A}(Q_{λ}, Θ) ≅ Hom_{A}(Θ_{λ}, Θ) for every λ ∈ Λ. Then the second isomorphism follows immediately. First, notice that Hom_{A}(Θ, Θ) is a basic algebra with n non-zero indecomposable summands Hom_{A}(Θ_{λ}, Θ), λ ∈ Λ, where n is the cardinal number of Λ. But Hom_{A}(Q, Θ) ≅ ⨁_{λ∈Λ} Hom_{A}(Q_{λ}, Θ) has at least n non-zero indecomposable summands. If Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ), by Krull-Schmidt theorem, Hom_{A}(Q_{λ}, Θ) must be indecomposable, and is isomorphic to some Hom_{A}(Θ_{μ}, Θ).

If λ ∈ Λ is maximal, Hom_{A}(Θ_{λ}, Θ) ≅ Hom_{A}(Q_{λ}, Θ) since Q_{λ} ≅ Θ_{λ}. Define Λ_{1} to be the subset of maximal elements in Λ and consider λ_{1} ∈ Λ \ Λ_{1} which is maximal. We have

Hom_{A}(Q_{λ_{1}}, Θ) ≇ Hom_{A}(Θ_{λ}, Θ) ≅ Hom_{A}(Q_{λ}, Θ)

for every λ ∈ Λ_{1} since End_{A}(Θ) is a basic algebra. Therefore, Hom_{A}(Q_{λ_{1}}, Θ) must be isomorphic to some Hom_{A}(Θ_{μ}, Θ) with μ ∈ Λ \ Λ_{1}. But Hom_{A}(Θ_{μ}, Θ) contains a surjection from Θ_{μ} to the direct summand Θ_{μ} of Θ, and Hom_{A}(Q_{λ_{1}}, Θ) contains a surjection from Q_{λ_{1}} to Θ_{μ} if and only if λ_{1} = μ. Thus we get λ_{1} = μ. Repeating the above process, we have Hom_{A}(Q_{λ}, Θ) ≅ Hom_{A}(Θ_{λ}, Θ) for every λ ∈ Λ.

Applying Hom_{A}(−, Θ) to the surjection M → ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}} we get

Hom_{A}( ⨁_{h(λ)=i} Θ_{λ}^{⊕m_{λ}}, Θ) ⊆ Hom_{A}(M, Θ).

Similarly, from the relative projective covering map ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}} → M we have

Hom_{A}(M, Θ) ⊆ Hom_{A}( ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}, Θ).

Comparing these two inclusions and using the second isomorphism, we deduce the first isomorphism. □

The reader may aware that the above lemma is an analogue to the following result in representation theory of graded algebras: if A is a graded algebra and M is a graded module generated in degree 0, then Hom_{A}(M, A_{0}) ≅ Hom_{A}(M_{0}, A_{0}).

Lemma 2.9. Suppose that Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ). If M ∈ F(Θ) is generated in height i, then Ext_{A}^{s}(M, Θ) ≅ Ext_{A}^{s−1}(Ω_{Θ}(M), Θ) for all s ⩾ 1.

Proof. Let m_{λ} = [M : Θ_{λ}]. Applying Hom_{A}(−, Θ) to the exact sequence

\xymatrix{0 \ar[r] & Ω_{Θ}(M) \ar[r] & ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}} \ar[r] & M \ar[r] & 0}

we get a long exact sequence. In particular, for all s ⩾ 2, by observing the segment

0 = Ext_{A}^{s−1}( ⨁_{h(λ)=i} Q^{⊕m_{λ}}, Θ) → Ext_{A}^{s−1}(Ω_{Θ}(M), Θ)

→ Ext_{A}^{s}(M, Θ) → Ext_{A}^{s}( ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}, Θ) = 0

we conclude Ext_{A}^{s−1}(Ω_{Θ}(M), Θ) ≅ Ext_{A}^{s}(M, Θ).

For s = 1, we have

0 → Hom_{A}(M, Θ) → Hom_{A}( ⨁_{h(λ)=i} Q_{λ}^{⊕m_{λ}}, Θ)

→ Hom_{A}(Ω_{Θ}(M), Θ) → Ext_{A}^{1}(M, Θ) → 0.

By the previous lemma, the first inclusion is an isomorphism. Thus Ext_{A}^{1}(M, Θ) ≅ Hom_{A}(Ω_{Θ}(M), Θ). □

We remind the reader that although Γ = Ext_{A}^{∗}(Θ, Θ) has a natural grading, the classical Koszul theory cannot be applied directly since Γ_{0} = End_{A}(Θ) may not be a semisimple algebra. Thus we introduce generalized Koszul algebras as follows:

Definition 2.10. Let R = ⨁_{i⩾0} R_{i} be a positively graded locally finite k-algebra, i.e., dim_{k} R_{i} < ∞ for each i ⩾ 0. A graded R-module M is said to be linear if it has a linear projective resolution

\xymatrix{… \ar[r] & P^{s} \ar[r] & … \ar[r] & P^{1} \ar[r] & P^{0} \ar[r] & M}

such that P^{s} is generated in degree s. The algebra R is said to be generalized Koszul if R_{0} viewed as a R-module has a linear projective resolution.

It is easy to see from this definition that M is linear if and only if Ω^{s}(M) is generated in degree s for all s ⩾ 0.

Proposition 2.11. Suppose that Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ), and Θ_{λ} are linearly filtered for all λ ∈ Λ. If M ∈ F(Θ) is linearly filtered, then

Ext_{A}^{i+1}(M, Θ) = Ext_{A}^{1}(Θ, Θ) · Ext_{A}^{i}(M, Θ)

for all i ⩾ 0, i.e., Ext_{A}^{∗}(M, Θ) as a graded Γ = Ext_{A}^{∗}(Θ, Θ)-module is generated in degree 0.

Proof. Suppose that M is generated in height d and linearly filtered. By Lemma 2.9,

Ext_{A}^{i+1}(M, Θ) ≅ Ext_{A}^{i}(Ω_{Θ}(M), Θ).

But Ω_{Θ} is generated in height d + 1 and linearly filtered. Thus by induction

Ext_{A}^{i+1}(M, Θ) ≅ Ext_{A}^{1}(Ω_{Θ}^{i}(M), Θ), Ext_{A}^{i}(M, Θ) ≅ Hom_{A}(Ω_{Θ}^{i}(M), Θ).

Therefore, it suffices to show

Ext_{A}^{1}(M, Θ) = Ext_{A}^{1}(Θ, Θ) · Hom_{A}(M, Θ)

since we can replace M by Ω_{Θ}^{i}(M), which is linearly filtered as well.

Let m_{λ} = [M : Θ_{λ}] and define Q^{0} = ⨁_{h(λ)=d} Q_{λ}^{⊕m_{λ}}, M_{0} = ⨁_{h(λ)=d} Θ_{λ}^{⊕m_{λ}}. We have the following commutative diagram:

[figure: 0; Ω_{Θ}(M); Q^{0}[1]; M[1]; 0; 0; Ω_{Θ}(M); Q^{0}; M; 0; M_{0}; M_{0}]

where Q^{0}[1] = Ω_{Θ}(M_{0}), see Proposition 2.1.

Observe that all terms in the top sequence are generated in height d + 1 and linearly filtered. For every λ ∈ Λ with h(λ) = d + 1, we have

[Ω_{Θ}(M) : Θ_{λ}] + [M[1] : Θ_{λ}] = [Q^{0}[1] : Θ_{λ}].

Let r_{λ}, s_{λ} and t_{λ} be the corresponding numbers in the last equality. Then we get a split short exact sequence

\xymatrix{0 \ar[r] & ⨁_{h(λ)=d+1} Θ_{λ}^{⊕r_{λ}} \ar[r] & ⨁_{h(λ)=d+1} Θ_{λ}^{⊕t_{λ}} \ar[r] & ⨁_{h(λ)=d+1} Θ_{λ}^{⊕s_{λ}} \ar[r] & 0}.

Applying Hom_{A}(−, Θ) to this sequence and using Lemma 2.8, we obtain the exact sequence

0 → Hom_{A}(M[1], Θ) → Hom_{A}(Q^{0}[1], Θ) → Hom_{A}(Ω_{Θ}(M), Θ) → 0.

Therefore, each map Ω_{Θ}(M) → Θ can extend to a map Q^{0}[1] → Θ.

To prove Ext_{A}^{1}(M, Θ) = Ext_{A}^{1}(Θ, Θ) · Hom_{A}(M, Θ), by Lemma 2.9 we first identify Ext_{A}^{1}(M, Θ) with Hom_{A}(Ω_{Θ}(M), Θ). Take an element x ∈ Ext_{A}^{1}(M, Θ) and let g : Ω_{Θ}(M) → Θ be the corresponding homomorphism. As we just showed, g can extend to Q^{0}[1], and hence there is a homomorphism g̃ : Q^{0}[1] → Θ such that g = g̃ι, where ι is the inclusion.

\xymatrix{Ω_{Θ}(M) \ar[r]^{ι} \ar[d]^{g} & Q^{0}[1] \ar[dl]^{g̃} \\ Θ.}

We have the following commutative diagram:

[figure: 0; Ω_{Θ}(M); ι; Q^{0}; M; p; 0; 0; Q^{0}[1]; Q^{0}; M_{0}; 0,]

where p is the projection of M onto M_{0}. The map g̃ : Q^{0}[1] → Θ gives a push-out of the bottom sequence:

[figure: 0; Ω_{Θ}(M); ι; Q^{0}; M; p; 0; 0; Q^{0}[1]; g̃; Q^{0}; M_{0}; 0; 0; Θ; E; M_{0}; 0.]

Since M_{0} ≅ ⨁_{h(λ)=d} Θ_{λ}^{⊕m_{λ}}, the bottom sequence represents some

y ∈ Ext_{A}^{1}(Θ^{⊕m}, Θ) = ⨁_{i=1}^{m} Ext_{A}^{1}(Θ, Θ)

where m = ∑_{h(λ)=d} m_{λ}. Therefore, we can write y = y_{1} + … + y_{m} where y_{i} ∈ Ext_{A}^{1}(Θ, Θ) is represented by the sequence

\xymatrix{0 \ar[r] & Θ \ar[r] & E_{i} \ar[r] & Θ \ar[r] & 0}.

Composed with the inclusions ϵ_{λ} : Θ_{λ} → Θ, we get the map (p_{1}, …, p_{m}) where each component p_{i} is defined in an obvious way. Consider the pull-backs:

[figure: 0; Θ; F_{i}; M; p_{i}; 0; 0; Θ; E_{i}; Θ; 0.]

Denote by x_{i} the top sequence. Then

x = ∑_{i=1}^{m} x_{i} = ∑_{i=1}^{m} y_{i} · p_{i} ∈ Ext_{A}^{1}(Θ, Θ) · Hom_{A}(M, Θ),

so Ext_{A}^{1}(M, Θ) ⊆ Ext_{A}^{1}(Θ, Θ) · Hom_{A}(M, Θ). The other inclusion is obvious. □

Now we can prove the main result.

Theorem 2.12. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽) such that Ext_{A}^{i}(Q, Θ) = 0 for all i ⩾ 1 and Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ). Suppose that all Θ_{λ} are linearly filtered for λ ∈ Λ. If M ∈ F(Θ) is linearly filtered, then the graded module Ext_{A}^{∗}(M, Θ) has a linear projective resolution. In particular, Γ = Ext_{A}^{∗}(Θ, Θ) is a generalized Koszul algebra.

Proof. Suppose that M is generated in height d. Define m_{λ} = [M : Θ_{λ}] for λ ∈ Λ, Q^{0} = ⨁_{h(λ)=d} Q_{λ}^{⊕m_{λ}}, and M_{0} = ⨁_{h(λ)=d} Θ_{λ}^{⊕m_{λ}}. As in the proof of the previous lemma, we have the following short exact sequence of linearly filtered modules generated in height d + 1:

\xymatrix{0 \ar[r] & Ω_{Θ}(M) \ar[r] & Ω_{Θ}(M_{0}) \ar[r] & M[1] \ar[r] & 0}

where Ω_{Θ}(M_{0}) = Q^{0}[1]. This sequence induces exact sequences recursively (see the proof of Proposition 2.6):

\xymatrix{0 \ar[r] & Ω_{Θ}^{i}(M) \ar[r] & Ω_{Θ}^{i}(M_{0}) \ar[r] & Ω_{Θ}^{i−1}(M[1]) \ar[r] & 0,}

where all modules are linearly filtered and generated in height d + i. Again as in the proof of the previous lemma, we get an exact sequence

0 → Hom_{A}(Ω_{Θ}^{i−1}(M[1]), Θ) → Hom_{A}(Ω_{Θ}^{i}(M_{0}), Θ) → Hom_{A}(Ω_{Θ}^{i}(M), Θ) → 0.

According to Lemma 2.9, the above sequence is isomorphic to:

0 → Ext_{A}^{i−1}(M[1], Θ) → Ext_{A}^{i}(M_{0}, Θ) → Ext_{A}^{i}(M, Θ) → 0.

Now let the index i vary and put these sequences together. We have:

\xymatrix{0 \ar[r] & E(M[1])⟨1⟩ \ar[r] & E(M_{0}) \ar[r]^{p} & E(M) \ar[r] & 0,}

where E = Ext_{A}^{∗}(−, Θ) and ⟨−⟩ is the degree shift functor of graded modules. That is, for a graded module T = ⨁_{i⩾0} T_{i}, T ⟨1⟩_{i} is defined to be T_{i−1}.

Since M_{0} ∈ add(Θ), E(M_{0}) is a projective Γ-module. It is generated in degree 0 by the previous lemma. Similarly, E(M[1]) is generated in degree 0, so E(M[1])⟨1⟩ is generated in degree 1. Therefore, the map p is a graded projective covering map. Consequently, Ω(E(M)) ≅ E(M[1])⟨1⟩ is generated in degree 1.

Replacing M by M[1] (since it is also linearly filtered), we have

Ω^{2}(E(M)) ≅ Ω(E(M[1])⟨1⟩) ≅ Ω(E(M[1])⟨1⟩ ≅ E(M[2])⟨2⟩,

which is generated in degree 2. By recursion, Ω^{i}(E(M)) ≅ E(M[i])⟨i⟩ is generated in degree i for all i ⩾ 0. Thus E(M) is a linear Γ-module.

In particular let M = Q_{λ} for a certain λ ∈ Λ. We get

E(Q_{λ}) = Ext_{A}^{∗}(Q_{λ}, Θ) = Hom_{A}(Q_{λ}, Θ)

is a linear Γ-module. Therefore,

⨁_{λ∈Λ} E(Q_{λ}, Θ) = ⨁_{λ∈Λ} Hom_{A}(Q_{λ}, Θ) ≅ Hom_{A}(⨁_{λ∈Λ} Q_{λ}, Θ)
= Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ) = Γ_{0}

is a linear Γ-module. So Γ is a generalized Koszul algebra. □

Remark 2.13. To get the above result we made some assumptions on the EPSS (\underline{Θ}, \underline{Q}). Firstly, each Θ_{λ} has a simple top S_{λ} and S_{λ} ≇ S_{μ} for λ ≠ μ; secondly, Ext_{A}^{s}(Q, Θ) = 0 for every s ⩾ 1. These two conditions always hold for standardly stratified basic algebras. We also suppose that Hom_{A}(Θ, Θ) ≅ Hom_{A}(Q, Θ). This may not be true even if A is a quasi-hereditary algebra.

Although Γ is proved to be a generalized Koszul algebra, in general it does not have the Koszul duality. Consider the following example:

Example 2.14. Let A be the path algebra of the following quiver with relation α · β = 0. Put an order x < y < z.

\xymatrix{x \ar[r]^{α} & y \ar[l]^{β} \ar[r]^{γ} & z.}

The projective modules and standard modules of A are described as follows:

P_{x} = \begin{aligned}& x \\ & y \\ x & & z \end{aligned}  P_{y} = \begin{aligned}& y \\ x & & z \end{aligned}  P_{z} = z

Δ_{x} = x  Δ_{y} = \begin{matrix}y \\ x \end{matrix}  Δ_{z} = z ≅ P_{z}.

This algebra is quasi-hereditary. Moreover, Hom_{A}(Δ, Δ) ≅ Hom_{A}(A, Δ), and all standard modules are linearly filtered. Therefore, Γ = Ext_{A}^{∗}(Δ, Δ) is a generalized Koszul algebra by the previous theorem.

We explicitly compute the extension algebra Γ. It is the path algebra of the following quiver with relation γ · α = 0.

\xymatrix{x \ar[r]^{α} \ar[r]_{β} & y \ar[r]^{γ} & z.}

ΓP_{x} = \begin{aligned}& x_{0} \\ y_{0} & & y_{1} \\ z_{1} \end{aligned}  _{Γ}P_{y} = \begin{matrix}y_{0} \\ z_{1} \end{matrix}  _{Γ}P_{z} = z_{0}

and

ΓΔ_{x} = x_{0}  _{Γ}Δ_{y} = y_{0}  _{Γ}Δ_{z} = z_{0}  Γ_{0} = \begin{matrix}x_{0} \\ y_{0} \end{matrix} ⊕ y_{0} ⊕ z_{0} ≇_{Γ} Δ.

Here we use indices to mark the degrees of simple composition factors. As asserted by the theorem, Γ_{0} has a linear projective resolution. But _{Γ}Δ is not a linear Γmodule (we remind the reader that the two simple modules y appearing in _{Γ}P_{x} lie in different degrees!).

By computation, we get the extension algebra Γ′ = Ext_{Γ}^{∗}(Γ_{0}, Γ_{0}), which is the path algebra of the following quiver with relation β · α = 0.

\xymatrix{x \ar[r]^{α} & y \ar[r]^{β} & z.}

Since Γ′ is a Koszul algebra in the classical sense, the Koszul duality holds in Γ′. It is obvious that the Koszul dual algebra of Γ′ is not isomorphic to Γ. Therefore, as we claimed, the Koszul duality does not hold in Γ.

Let us return to the question of whether Γ = Ext_{A}^{∗}(Θ, Θ) is standardly stratified with respect to ⩽. According to Proposition 1.3, this happens if and only if for each pair Θ_{λ}, Θ_{μ} and s ⩾ 0, Ext_{A}^{s}(Θ_{λ}, Θ_{μ}) is a projective End_{A}(Θ_{μ})-module. Putting direct summands together, we conclude that Γ is standardly stratified with respect to ⩽ if and only if Ext_{A}^{s}(Θ, Θ) is a projective ⨁_{λ∈Λ} End_{A}(Θ_{λ})-module. With the conditions in Theorem 2.12, Ext_{A}^{s}(Θ_{λ}, Θ) ≅ Hom_{A}(Ω_{Θ_{λ}}^{s}(Θ), Θ) for all s ⩾ 0 and λ ∈ Λ by Lemma 2.9. Notice that Ω_{Θ}^{s}(Θ_{λ}) is linearly filtered. Suppose that min(Ω_{Θ}^{s}(Θ_{λ})) = d and m_{μ} = [Ω_{Θ}^{s}(Θ_{λ}) : Θ_{μ}]. Then

(2.2)  Ext_{A}^{s}(Θ_{λ}, Θ) ≅ Hom_{A}(Ω_{Θ}^{s}(Θ_{λ}), Θ) ≅ Hom_{A}( ⨁_{h(μ)=d} Θ_{μ}^{⊕m_{μ}}, Θ),

which is a projective Γ_{0} = End_{A}(Θ)-module.

With this observation, we have:

Corollary 2.15. Let (\underline{Θ}, \underline{Q}) be an EPSS indexed by a finite poset (Λ, ⩽). Suppose that all Θ_{λ} are linearly filtered for λ ∈ Λ, and Hom_{A}(Q, Θ) ≅ Hom_{A}(Θ, Θ). Then Γ = Ext_{A}^{∗}(Θ, Θ) is standardly stratified for ⩽ if and only if End_{A}(Θ) is a projective ⨁_{λ∈Λ} End_{A}(Θ_{λ})-module.

Proof. If Γ is standardly stratified for ⩽, then in particular Γ_{0} = End_{A}(Θ) is a projective ⨁_{λ∈Λ} End_{A}(Θ_{λ})-module by Proposition 1.3. Conversely, if Γ_{0} = End_{A}(Θ) is a projective ⨁_{λ∈Λ} End_{A}(Θ_{λ})-module, then by the isomorphism in (2.2) Ext_{A}^{s}(Θ, Θ) = ⨁_{λ∈Λ} Ext_{A}^{s}(Θ_{λ}, Θ) is a projective Γ_{0}-module for all s ⩾ 0, so it is a projective ⨁_{λ∈Λ} End_{A}(Θ_{λ})-module as well. Again by Proposition 1.3, Γ is standardly stratified with respect to ⩽. □

If A is quasi-hereditary with respect to ⩽ such that all standard module are linearly filtered, then Γ = Ext_{A}^{∗}(Δ, Δ) is again quasi-hereditary for this partial order by Corollary 1.5, and Γ_{0} has a linear projective resolution by the previous theorem. Let _{Γ}Δ be the direct sum of all standard modules of Γ with respect to ⩽. The reader may wonder whether _{Γ}Δ has a linear projective resolution as well. The following proposition gives a partial answer to this question.

Proposition 2.16. With the above notation, if _{Γ}Δ has a linear projective resolution, then Γ_{0} ≅_{Γ} Δ, or equivalently Hom_{A}(Δ_{λ}, Δ_{μ}) ≠ 0 only if λ = μ, λ, μ ∈ Λ. If furthermore Hom_{A}(A, Δ) ≅ End_{A}(Δ), then Δ ≅ A/ rad A.

Proof. We have proved that the k-linear category associated to Γ is directed with respect to ⩽. By Proposition 1.3, standard modules of Γ for ⩽ are exactly indecomposable summands of ⨁_{λ∈Λ} End_{A}(Δ_{λ}), i.e., _{Γ}Δ ≅ ⨁_{λ∈Λ} End_{A}(Δ_{λ}) ≅ ⨁_{λ∈Λ} k_{λ}. Clearly, _{Γ}Δ ⊆ Γ_{0} = End_{A}(Δ). If _{Γ}Δ has a linear projective resolution, then by Corollary 2.4 and Remark 2.7 in [12], _{Γ}Δ is a projective Γ_{0}-module. Consequently, every summand k_{λ} is a projective Γ_{0}-module. Since both _{Γ}Δ and Γ_{0} have exactly |Λ| pairwise non-isomorphic indecomposable summands, we deduce ΓΔ ≅ Γ_{0} ≅ ⨁_{λ∈Λ} k_{λ}, or equivalently Hom_{A}(Δ_{λ}, Δ_{μ}) = 0 if λ ≠ μ.

If furthermore Hom_{A}(A, Δ) ≅ End_{A}(Δ), then

Δ ≅ Hom_{A}(A, Δ) ≅ End_{A}(Δ) ≅ ⨁_{λ∈Λ} k_{λ} ≅ A/ rad A.

□

References

[1] N. Abe, First extension groups of Verma modules and R-polynomials, ArXiv:1003.0169;

[2] I. Ágoston, V. Dlab, and E. Lukás, Quasi-hereditary Extension Algebras, Alg. Repn. Theorey 6 (2003), 97-117;

[3] I. Ágoston, V. Dlab, and E. Lukás, Standardly stratified extension algebras, Comm. Algebra 33 (2005), 1357-1368;

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

[5] 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;

[6] Y. Drozd, V. Mazorchuk, Koszul duality for extension algebras of standard modules, J. Pure Appl. Algebra 211 (2007), 484-496;

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

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

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

[10] E. L. Green, I. Reiten, and Ø. Solberg, Dualities on Generalized Koszul Algebras, Mem. Amer. Math. Soc. 159 (2002), xvi+67pp; [11] A. Klamt, C. Stroppel, On the Ext algebras of parabolic Verma modules and A_{∞}-structures,

J. Pure Appl. Algebra 216 (2012), 323-336;

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

[13] L. Li, Algebras stratified for all preorders, accepted by Alg. Repn. Theory, ArXiv: 1110.6501;

[14] D. Madsen, Ext-algebras and derived equivalences, Colloq. Math. 104 (2006), 113-140;

[15] D. Madsen, On a Common Generalization of Koszul Duality and Tilting Equivalence, Adv. Math. 227 (2011), 2327-2348;

[16] D. Madsen, Quasi-hereditary algebras and generalized Koszul duality, ArXiv: 1201.0441;

[17] E. Marcos, O. Mendoza, and C. Sáenz, Stratifying systems via relative simple modules, J. Algebra 280 (2004), 472-487;

[18] E. Marcos, O. Mendoza, and C. Sáenz, Stratifying systems via relative projective modules, Comm. Algebra 33 (2005), 1559-1573;

[19] V. Mazorchuk, Some homological properties of the category O, Pacific J. Math. 232 (2007), 313-341;

[20] V. Mazorchuk, S. Ovsienko, and C. Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), 1129-1172;

[21] V. Mazorchuk, Koszul duality for stratified algebras I: balanced quasi-hereditary algebras, Manuscripta Math. 131 (2010), 1-10;

[22] V.Mazorchuk, Koszul duality for stratified algebras II: standardly stratified algebras, J. Aust. Math. Soc. 89 (2010), 23-49;

[23] V. Miemietz, W. Turner, The Weyl extension algebra of GL_{2}(𝔽̄_{p}), ArXiv: 1106.5665;

[24] P. Webb, Stratifications and Mackey functors. I: functors for a single group, Proc. London Math. Soc. (3) 82 (2001), 299-336.

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

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