A GENERALIZED KOSZUL THEORY AND ITS APPLICATION

LIPING LI

Abstract. Let A be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that A_{0} is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.

1. Introduction

The work described in this paper originated in the exploration of homological properties of finite EI categories. We want to apply Koszul theory, which has been proved to be very useful in the representation theory of algebras, to study the Ext groups of representations of finite EI categories. Examples of such applications can be found in [19] and [21], where Koszul theory has been applied to incidence algebras of posets. In general this theory applies to graded algebras, and we do not assume that the degree 0 part of the algebra is semisimple, unlike the classical Koszul theory described in [4] [9] [10] [15]. This generalization is necessary so that we can apply the theory to finite EI categories.

There do already exist several generalized Koszul theories where the degree 0 part A_{0} of a graded algebra A is not required to be semisimple, see [11] [13] [14] and [21]. Each Koszul algebra A defined by Woodcock in [21] is supposed to satisfy that A is both a left projective A_{0}-module and a right projective A_{0}-module. This requirement is too strong for us. Indeed, even the category algebra kE of a standardly stratified finite EI category E (studied in [20]) does not satisfy this requirement: kE is a left projective kE_{0}-module but in general not a right projective kE_{0}-module. In Madsen’s paper [14], A_{0} is supposed to have finite global dimension. But for a finite EI category E, this happens in our context if and only if kE_{0} is semisimple. The theory developed by Green, Reiten and Solberg in [11] works in a very general framework, but some efforts are required to fill the gap between their theory and our applications.

Thus we want to develop a generalized Koszul theory which can inherit many useful results of the classical theory, and can be applied to structures with nice properties such as finite EI categories. Let A be a graded k-algebra with A_{0} being self-injective instead of being semisimple. Then we define generalized linear modules (or Koszul modules), Koszul algebras in a similar way to the classical case. That is, a graded A-module M is Koszul if M has a linear projective resolution, and A is a Koszul algebra if A_{0} viewed as a graded A-module is Koszul. We also define quasi-Koszul modules and quasi-Koszul algebras: M is quasi-Koszul if the Ext_{A}^{∗}(A_{0}, A_{0})- module Ext_{A}^{∗}(M, A_{0}) is generated in degree 0, and A is a quasi-Koszul algebra if A_{0} is a quasi-Koszul A-module. It turns out that this generalization works nicely for our goal. Many classical results described in [4] [9] [10] and [15] generalize to our context. In particular, we obtain the Koszul duality both on the category of linear modules and on the derived category.

The author wants to express great appreciation to his thesis advisor, Professor Peter Webb, for the proposal to develop a generalized Koszul theory, and the invaluable suggestions and contributions provided in numerous discussions.

We then focus on the applications of this generalized Koszul theory. First we define directed categories. A directed category C is a k-linear category equipped with a partial order ⩽ on Ob C such that for each pair of objects x, y ∈ Ob C, the space of morphisms C(x, y) is non-zero only if x ⩽ y. Directed categories include the k-linearizations of finite EI categories as special examples. This partial order determines a canonical pre-order ≼ on the isomorphism classes of simple representations. Following the technique in [20], we develop a stratification theory for directed categories, describe the structures of standard modules and characterize every directed category C standardly stratified with respect to the canonical preorder:

Theorem 1.1. Let C be a directed category with respect to a partial order ⩽. Then C is standardly stratified for the canonical pre-order ≼ if and only if the morphism space C(x, y) is a projective C(y, y)-module for every pair of objects x, y ∈ Ob C.

By the correspondence between graded k-linear categories and graded algebras described in [16], we can view a directed category as a directed algebra and viceversa. Therefore, all of our results on graded algebras can be applied to graded directed categories. In particular, the following theorem relates Koszul theory to stratification theory:

Theorem 1.2. Let C be a graded directed category with C_{0} = ⨁_{x∈Ob C} C(x, x) being a self-injective algebra . Then:

(1) C is standardly stratified with respect to the canonical pre-order ≼ if and only if C is a projective C_{0}-module.

(2) C is a Koszul category if and only if C is a quasi-Koszul category standardly stratified for ≼.

(3) If C is standardly stratified for ≼, then a graded C-module M generated in degree 0 is linear if and only if it is a quasi-Koszul C-module and a projective C_{0}-module.

Applying the homological dual functor E = Ext_{C}^{∗}(−, C_{0}) to a graded directed category C, we construct the Yoneda category E(C_{0}) = Ext_{C}^{∗}(C_{0}, C_{0}). We prove that if C is a Koszul directed category with C_{0} being self-injective, then E(C_{0}) is also a Koszul directed category.

We acquire a very nice correspondence between the classical Koszul theory and our generalized Koszul theory for directed categories.

Theorem 1.3. Let C be a graded directed category with C_{0} being a self-injective algebra. Define D to be the subcategory of C obtained by replacing each endomorphism ring by k · 1, the span of the identity endomorphism. Then:

(1) C is a Koszul category in our sense if and only if C is standardly stratified for the canonical pre-order ≼ and D is a Koszul category in the classical sense.

(2) If C is a Koszul category, then a graded C-module M is linear if and only if M is a projective C_{0}-module, and M ↓_{D}^{C} is a linear D-module.

Finite EI categories have nice combinatorial properties. These properties can be used to define length gradings on the sets of morphisms. We discuss the possibility to put such a grading on an arbitrary finite EI category, and prove the following result for finite free EI categories (defined in [12]):

Theorem 1.4. Let E be a finite free EI category. Then the following are equivalent: (1) pd kE_{0} < ∞;

(2) pd kE_{0} ⩽ 1;

(3) E is standardly stratified in a sense defined in [20];

(4) kE is a Koszul algebra.

Moreover, kE is a quasi-Koszul algebra if and only if Ext_{kE}^{i}(kE_{0}, kE_{0}) = 0 for each i ⩾ 2.

We then give a sufficient condition for the category algebra of a finite EI category E to be quasi-Koszul. An object x ∈ Ob E is called left regular if for every morphism α : z → x, the stabilizer of α in Aut_{E}(y) has order invertible in k. Similarly we define right regular objects. Then:

Theorem 1.5. Let E be a finite free EI category. If every object in E is either left regular or right regular, then kE is a quasi-Koszul algebra.

Motivated by the fact that the Yoneda category E(C_{0}) of a directed Koszul category C is still a directed Koszul category, and hence is standardly stratified, we ask whether the Koszul dual algebra Γ = Ext_{A}^{∗}(A_{0}, A_{0}) of a graded algebra A standardly stratified for a partial order ⩽ is still standardly stratified for ⩽ (or ⩽^{op}). This question has only been studied for the case that A_{0} is semisimple, see [1] [2] [8]

[17] and [18]. By assuming that A_{0} is a self-injective algebra, and supposing that all standard modules are concentrated in degree 0 and linear, we get a sufficient condition for Γ to be standardly stratified with respect to ⩽^{op}.

The layout of this paper is as follows. The generalized Koszul theory is developed in the first three sections. In Section 2 we define Koszul modules and quasi-Koszul modules, which generalize linear modules and Koszul modules in the classical theory, and describe their basic properties. Since Koszul modules and quasi-Koszul modules do not coincide in our context, we also give a relation between these two concepts. Koszul algebras are studied in Section 3. The Koszul dualities are proved in Section 4. Most results in these three sections are generalized from works in [4]

[9] [10] [15]. Some results can be deduced from the paper [11] of Green, Reiten and Solberg, who worked in a more general context, but we present full arguments for the sake of completeness.

The last three sections are on the application of the general theory developed before. Directed categories are defined in Section 5. Their stratification properties and Koszul properties are discussed in details in this section as well. The main content of Section 6 to apply the Koszul theory and stratification theory to finite EI categories, which have nice combinatorial structures. In Section 7 we modify the technique of [1] to study standardly stratified algebras with linear standard modules.

Here are the notation and conventions we use in this paper. All algebras are k-algebras with k being algebraically closed. All modules are finitely generated left modules. Let A be a graded algebra. A graded A-module M is said to be locally finite if dim_{k} M_{i} < ∞ for each degree i ∈ ℤ. By A-mod and A-gmod we denote the category of all finitely generated A-modules and the category of all locally finite graded A-modules. Let M and N be two A-modules. By Hom_{A}(M, N) we denote the space of module homomorphisms from M to N. If furthermore M and N are graded, we use hom_{A}(M, N) to denote all graded module homomorphisms from M to N, i.e., the homomorphisms φ ∈ Hom_{A}(M, N) such that φ(M_{i}) ⊆ N_{i} for all i ∈ ℤ. When φ is said to be a homomorphism between two graded modules, it is supposed to be a graded homomorphism. If M is a graded A-module, its s-th shift M[s] is defined in the following way: M[s]_{i} = M_{i−s} for all i ∈ ℤ. If M is generated in degree s, then ⨁_{i⩾s+1} M_{i} is a graded submodule of M, and M_{s} ≅ M/ ⨁_{i⩾s+1} M_{i} as vector spaces. We then view M_{s} as an A-module by identifying it with this quotient module. We also regard the zero module 0 as a projective module since with this convention the expressions of many results can be simplified.

2. Generalized Koszul Modules

Throughout this section A is a non-negatively graded and locally finite associative k-algebra with identity 1 generated in degrees 0 and 1, i.e., A = ⨁_{i=0}^{∞} A_{i} such that A_{i} · A_{j} = A_{i+j} for all i, j ⩾ 0; each A_{i} is finite-dimensional. We also suppose that A_{0} is a self-injective algebra, i.e., every projective A_{0}-module is injective as well. Define J = ⨁_{i=1}^{∞} A_{i}, which is a two-sided ideal of A. An A-module M is called graded if M = ⨁_{i∈ℤ} M_{i} such that A_{i} · M_{j} ⊆ M_{i+j}. We say M is generated in degree s if M = A · M_{s}. It is clear that M is generated in degree s if and only if JM ≅ ⨁_{i⩾s+1} M_{i}, which is equivalent to J^{l}M ≅ ⨁_{i⩾s+l} M_{i} for all l ⩾ 1.

Most results in this section are generalized from [9] [10] and [15]. We suggest that the reader refer to these papers.

We collect some preliminary results in the following lemma.

Lemma 2.1. Let A be as above and M be a locally finite graded A-module. Then:

(1) J = ⨁_{i⩾1} A_{i} is contained in the graded radical of A;

(2) M has a graded projective cover;

(3) the graded syzygy ΩM is also locally finite.

Proof. By definition, the graded radical grad A is the intersection of all maximal proper graded submodules of A. Let L ⫋ A be a maximal proper graded submodule. Then L_{0} is a proper subspace of A_{0}. We claim that A_{i} = L_{i} for all i ⩾ 1. Otherwise, we can define L̃ ⊆ A in the following way: L̃_{0} = L_{0} and L̃_{i} = A_{i} for i ⩾ 1. Then L ⫋ L̃ ⫋ A, so L is not a maximal proper graded submodule of A. This contradiction tells us that L_{i} = A_{i} for all i ⩾ 1. Therefore, J ⊆ grad A, and the first statement is proved.

We use the following fact to prove the second statement: every primitive idempotent of A_{0} (as an algebra) can be lifted to a primitive idempotent of A. Consequently, a projective A_{0}-module concentrated in some degree d can be lifted to a graded projective A-module generated in degree d.

Define M̄ = M/JM, which is also a locally finite graded A-module. Write M̄ = ⨁_{i⩾0} M̄_{i}. Then each M̄_{i} is a finite-dimensional graded A-module since JM̄ = 0 and A_{0}M̄_{i} = M̄_{i} for all i ⩾ 0. Therefore, there is a decomposition of M̄ in which each indecomposable summand (which must be of finite dimension) is concentrated in a certain degree. Moreover, for each i ∈ ℤ, there are only finitely many summands which are concentrated in degree i.

Take L to be such an indecomposable summand and without loss of generality suppose that it is concentrated in degree 0. As an A_{0}-module, L is finitely generated and has a finitely generated projective cover P_{0}. By the lifting property, P_{0} can be lifted to a finitely generated (hence locally finite) graded projective module P generated in degree 0.

We claim that P is a graded projective cover of L. Indeed, there is a graded surjective homomorphism φ : P → L by mapping P_{0} onto L = L_{0} and JP to 0. Furthermore, this surjection is minimal. That is, if Q is a proper summand of P, then the induced map φ′ : Q → L is not surjective. Otherwise, the proper submodule Q_{0} ⫋ P_{0} can be mapped onto L_{0}. But this contradicts the fact that P_{0} is a projective cover of L as an A_{0}-module. In conclusion, we proved that P is a graded projective cover of L.

Take the direct sum of these projective covers P when L ranges over all indecomposable summands of M̄. In this way we obtain a graded projective cover P̃ of M̄. We claim that P̃ is also a graded projective cover of M. On one hand, the surjection p : P̃ → M̄ can be factored through the quotient homomorphism q : M → M̄ and induces a graded homomorphism p′ : P̃ → M such that p = qp′, as showed in the following diagram:

\xymatrix{ & M \ar[d]^{q} \\ P \ar[ur]^{p′} \ar[r]^{p} & M̄}.

Define M′ to be the image of p′. Viewed as vector spaces, M̄ = M/JM ⊆ M′ and M = M̄ + JM. Therefore, M = JM + M′ as graded A-modules. By the graded Nakayama’s lemma and the first statement, we deduce that M′ = M, and hence p′ is a surjection. On the other hand, if there is a proper summand Q̃ ⫋ P̃ such that the induced map

\xymatrix{Q̃ \ar[r]^{ι} & P̃ \ar[r]^{p′} & M}

is a graded surjection, where ι is the inclusion, then the map ιp′q = ιp is a graded surjection from Q̃ to M̄ = M/JM. This contradicts the fact that P̃ is a graded projective cover of M̄. In conclusion, P̃ is a graded projective cover of M, and the second statement is proved.

Now we turn to the third statement. By the above proof, the graded projective cover P̃ of M can be written as a direct sum ⨁_{i⩾0} P^{i} of graded projective modules, where P^{i} is generated in degree i. For each fixed degree i ⩾ 0, there are only finitely many indecomposable summands L of M̄ which are concentrated in degree i, and the graded projective cover of each L is finitely generated. Consequently, P^{i} as the direct sum of graded projective covers of those L concentrated in degree i is finitely generated, and hence locally finite.

For a fixed n ⩾ 0, we have P̃_{n} = ⨁_{i⩾0} P_{n}^{i} = ⨁_{0⩽i⩽n} P_{n}^{i}. Since each P^{i} is locally finite, dim_{k} P_{n}^{i} < ∞. Therefore, dim_{k} P̃_{n} < ∞, and P̃ is locally finite as well. As a submodule (up to isomorphism) of P̃, the graded syzygy ΩM is also locally finite. □

The results in the previous lemma will be used frequently in the rest of this paper without being mentioned.

Lemma 2.2. Let 0 → L → M → N → 0 be an exact sequence of graded A-modules. Then:

(1) If M is generated in degree s, so is N.

(2) If L and N are generated in degree s, so is M.

(3) If M is generated in degree s, then L is generated in degree s if and only if JM ∩ L = JL.

Proof. (1): This is obvious.

(2): Let P and Q be graded projective covers of L and N respectively. Then P and Q, and hence P ⊕ Q are generated in degree s. In particular, each graded projective cover of M, which is isomorphic to a direct summand of P ⊕ Q, is generated in degree s. Thus M is also generated in degree s.

(3): We always have JL ⊆ JM ∩ L. Let x ∈ L ∩ JM be a homogeneous element of degree i. Since M is generated in degree s, we have i ⩾ s + 1. If L is generated in degree s, then x ∈ J^{i−s}L ⊆ JL. Thus L ∩ JM ⊆ JL, so JL = L ∩ JM.

Conversely, the identity JL = L ∩ JM gives us the following commutative diagram where all rows and columns are exact:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & JL \ar[r] \ar[d] & JM \ar[r] \ar[d] & JN \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 \ar[r] & L/JL \ar[r] \ar[d] & M/JM \ar[r] \ar[d] & N/JN \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

Consider the bottom sequence. Notice that (M/JM) ≅ M_{s} is concentrated in degree s. Thus L/JL is also concentrated in degree s, i.e., L/JL ≅ L_{s}. Let I = A · L_{s}. Then L ⊆ I + JL ⊆ L, so I + JL = L. Notice that J is contained in the graded Jacobson radical of A. Therefore, by the graded Nakayama lemma, I = A · L_{s} = L, so L is generated in degree s. □

Corollary 2.3. Suppose that each graded A-module in the short exact sequence 0 → L → M → N → 0 is generated in degree 0. Then J^{i}M ∩ L = J^{i}L for all i ⩾ 0.

Proof. Since all modules L, M and N are generated in degree 0, all J^{s}L, J^{s}M and J^{s}N are generated in degree s for s ⩾ 0. The exactness of the above sequence implies JL = L ∩ JM, which in turns gives the exactness of 0 → JL → JM → JN → 0. By the above lemma, J^{2}M ∩ JL = J^{2}L and this implies the exactness of 0 → J^{2}L → J^{2}M → J^{2}N → 0. The conclusion follows from induction. □

Now we introduce generalized linear modules (or called Koszul modules).

Definition 2.4. A graded A-module M generated in degree 0 is called a linear module (or a Koszul module) if it has a (minimal) projective resolution

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

such that P^{i} is generated in degree i for all i ⩾ 0.

A direct consequence of this definition and the previous lemma is:

Corollary 2.5. Let M be a linear module. Then Ω^{i}(M)/JΩ^{i}(M) ≅ Ω^{i}(M)_{i} is a projective A_{0}-module for each i ⩾ 0, or equivalently, Ω^{i}(M) ⊆ JP^{i−1} where P^{i−1} is a graded projective cover of Ω^{i−1}(M), where Ω is the Heller operator.

Proof. Since M is linear, Ω^{i}(M) is generated in degree i, and Ω^{i}(M)/JΩ^{i}(M) ≅ Ω^{i}(M)_{i}. Moreover, all Ω^{i}(M)[−i] are linear A-modules as well. By induction, it is sufficient to prove ΩM ⊆ JP^{0}. But this is obvious since ΩM is generated in degree 1. From the following commutative diagram we deduce that ΩM ⊆ JP^{0} if and only if the bottom sequence is exact, or equivalently M/JM ≅ P^{0}/JP^{0} ≅ P_{0}^{0} is a projective A_{0}-module.

[figure: 0; ΩM; JP^{0}; JM; 0; 0; ΩM; P^{0}; M; 0; 0; 0; P^{0}/JP^{0}; M/JM; 0]

□

There are several characterizations of linear modules.

Proposition 2.6. Let M be a graded A-module generated in degree 0. Then the following are equivalent:

(1) M is linear.

(2) The syzygy Ω^{i}(M) is generated in degree i for every i ⩾ 0.

(3) For all i > 0, Ω^{i}(M) ⊆ JP^{i−1} and Ω^{i}(M) ∩ J^{2}P^{i−1} = JΩ^{i}(M), where P^{i−1} is a graded projective cover of Ω^{i−1}(M).

(4) Ω^{i}(M) ⊆ JP^{i−1} and Ω^{i}(M) ∩ J^{s+1}P^{i−1} = J^{s}Ω^{i}(M) for all i > 0, s ⩾ 0.

Proof. The equivalence of (1) and (2) is clear. It is also obvious that (3) is the special case of (4) for s = 1. Now we show (1) implies (4). Indeed, if M is a linear module, then both JP^{0} and ΩM are generated in degree 1 and ΩM ⊆ JP^{0}. Therefore we have the following exact sequence

\xymatrix{0 \ar[r] & ΩM \ar[r] & JP^{0} \ar[r] & JM \ar[r] & 0}

in which all modules are generated in degree 1. By Corollary 2.5 J^{s+1}P^{0} ∩ ΩM = J^{s}ΩM for all s > 0. Notice that all syzygies of M are also linear with suitable grade shifts. Replacing M by Ω^{i}(M)[−i] and using induction we get (4).

Finally we show (3) implies (2) to finish the proof. Since ΩM ⊆ JP^{0} we still have the above exact sequence. Notice that both JM and JP^{0} are generated in degree 1 and J^{2}P^{0} ∩ ΩM = JΩM, by Lemma 2.2, ΩM is generated in degree 1 as well. Now the induction procedure gives us the required conclusion. □

The condition that Ω^{i}(M) ⊆ JP^{i−1} (or equivalently, Ω^{i}(M)/JΩ^{i}(M) ≅ Ω^{i}(M)_{i} is a projective A_{0}-module) in (3) of the previous proposition is necessary, as shown by the following example:

Example 2.7. Let G be a finite cyclic group of prime order p and k be an algebraically closed field of characteristic p. Let the group algebra kG be concentrated on degree 0, so J = 0. Consider the trivial kG-module k. Obviously, k is not a linear module. But since J = 0, the condition JΩ^{i}(k) = J^{2}P^{i−1} ∩ Ω^{i}(k) holds trivially.

Remark 2.8. We do not use the property that A_{0} is a self-injective algebra up to now. Therefore, all results described before still hold for a non-negatively graded, locally finite graded algebra A with A_{0} being an arbitrary finite-dimensional algebra.

Proposition 2.9. Let 0 → L → M → N → 0 be a short exact sequence of graded A-modules such that L is a linear A-module. Then M is linear if and only if N is linear.

Proof. We verify the conclusion by using statement (2) in the last proposition. That is, given that Ω^{i}(L) is generated in degree i for each i ≥ 0, we want to show that Ω^{i}(M) is generated in degree i if and only if so is Ω^{i}(N).

Consider the given exact sequence. By Lemma 2.2, M is generated in degree 0 if and only if N is generated in degree 0. Therefore we have the following diagram in which all rows and columns are exact:

(2.1)  \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & ΩL \ar[r] \ar[d] & M′ \ar[r] \ar[d] & ΩN \ar[r] \ar[d] & 0 \\ 0 \ar[r] & P \ar[r] \ar[d] & P ⊕ Q \ar[r] \ar[d] & Q \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}

where P and Q are graded projective covers of L and N respectively. In general P ⊕Q is not a graded projective cover of M, and hence M′ ≇ ΩM. However, under the hypothesis of this proposition we claim that P ⊕Q is indeed a graded projective cover of M and M′ ≅ ΩM. To see this, we point out that the given exact sequence induces a short exact sequence of A_{0}-modules:

\xymatrix{0 \ar[r] & L_{0} \ar[r] & M_{0} \ar[r] & N_{0} \ar[r] & 0.}

Observe that L_{0} = Ω^{0}(L)_{0} is projective (by Corollary 2.5) and hence injective (since A_{0} is self-injective) as an A_{0}-module. Therefore, the above sequence splits, and M_{0} ≅ L_{0} ⊕ N_{0}. On one hand, a graded projective cover P̃ of M should be isomorphic to a direct summand of P ⊕ Q. On the other hand, since P̃ induces a projective cover P̃_{0} of M_{0} ≅ L_{0} ⊕ N_{0}, P̃ should contain a summand isomorphic to Q ⊕ P. This forces P̃ ≅ P ⊕ Q, and our claim is proved.

Now let us consider the top row of diagram 2.1. Since M′ ≅ ΩM, and ΩL is generated in degree 1, ΩM is generated in degree 1 if and only if ΩN is generated in degree 1 by Lemma 2.2. Replace L, M and N by (ΩL)[−1], (ΩM)[−1] and (ΩN)[−1] (all of which are linear) respectively in the short exact sequence. Repeating the above procedure and using the fact that the Heller operator Ω and the grade shift functor [−] commute, we conclude that Ω^{2}(M)[−1] is generated in degree 1 if and only if Ω^{2}(N)[−1] is generated in degree 1, i.e., Ω^{2}(M) is generated in degree 2 if and only if Ω^{2}(N) is generated in degree 2. By induction, M is linear if and only if N is linear. □

The condition that L is linear in this proposition is necessary. Indeed, quotient modules of a linear module might not be linear.

Lemma 2.10. Let M be a graded A-module generated in degree s. If M_{s} is a projective A_{0}-module, then Ext_{A}^{i}(M, A_{0}) ≅ Ext_{A}^{i−1}(ΩM, A_{0}) for all i ⩾ 1.

Proof. It is true for i > 1. When i = 1, consider the following exact sequence:

0 → Hom_{A}(M, A_{0}) → Hom_{A}(P, A_{0}) → Hom_{A}(ΩM, A_{0}) → Ext_{A}^{1}(M, A_{0}) → 0.

As a graded projective cover of M, P is also generated in degree s. Since M_{s} is a projective A_{0}-module, P_{s} ≅ M_{s}. So

Hom_{A}(M, A_{0}) ≅ Hom_{A_{0}}(M_{s}, A_{0}) ≅ Hom_{A_{0}}(P_{s}, A_{0}) ≅ Hom_{A}(P, A_{0}).

Thus Hom_{A}(ΩM, A_{0}) ≅ Ext_{A}^{1}(M, A_{0}). □

Proposition 2.11. Let 0 → L → M → N → 0 be an exact sequence of linear modules. Then it induces the following short exact sequence:

\xymatrix{0 \ar[r] & Ext_{A}^{∗}(N, A_{0}) \ar[r] & Ext_{A}^{∗}(M, A_{0}) \ar[r] & Ext_{A}^{∗}(L, A_{0}) \ar[r] & 0.}

Proof. As in the proof of Proposition 2.9, the above exact sequence gives exact sequences 0 → Ω^{i}(L) → Ω^{i}(M) → Ω^{i}(N) → 0, i ⩾ 0. For a fixed i, the sequence 0 → Ω^{i}(L)_{i} → Ω^{i}(M)_{i} → Ω^{i}(N)_{i} → 0 splits since all terms are projective A_{0}- modules by Corollary 2.5. Applying the exact functor Hom_{A_{0}}(−, A_{0}) we get an exact sequence

0 → Hom_{A_{0}}(Ω^{i}(N)_{i}, A_{0}) → Hom_{A_{0}}(Ω^{i}(M)_{i}, A_{0}) → Hom_{A_{0}}(Ω^{i}(L)_{i}, A_{0}) → 0

which is isomorphic to

0 → Hom_{A}(Ω^{i}(N), A_{0}) → Hom_{A}(Ω^{i}(M), A_{0}) → Hom_{A}(Ω^{i}(L), A_{0}) → 0

since all modules are generated in degree i. By Lemma 1.9, it is isomorphic to

\xymatrix{0 \ar[r] & Ext_{A}^{i}(N, A_{0}) \ar[r] & Ext_{A}^{i}(M, A_{0}) \ar[r] & Ext_{A}^{i}(L, A_{0}) \ar[r] & 0.}

Putting them together we have:

\xymatrix{0 \ar[r] & Ext_{A}^{∗}(N, A_{0}) \ar[r] & Ext_{A}^{∗}(M, A_{0}) \ar[r] & Ext_{A}^{∗}(L, A_{0}) \ar[r] & 0.}
□

If a graded A-module M is linear, so are all syzygies Ω^{i}(M)[−i], i ⩾ 0. The next example shows that the linear property of M in general does not imply the linear property of J^{i}M[−i]:

Example 2.12. Let E be a finite EI category with two objects x and y such that: Aut_{E}(x) = ⟨g⟩ ≅ Aut_{E}(y) = ⟨h⟩ are cyclic groups of order 2; Hom_{E}(x, y) has one element α on which both Aut_{E}(x) and Aut_{E}(y) act trivially; and Hom_{E}(y, x) = ∅. Let k be an algebraically closed field of characteristic 2. We put the following grade on the category algebra A = kE: A_{0} is spanned by {g, 1_{x}, h, 1_{y}} and A_{1} is spanned by α. Consider the projective kE-module P_{x} = kE1_{x}. Obviously, P_{x} is linear, but JP_{x} ≅ k_{y} is not linear.

However, for some special cases, we can get a conclusion as follows.

Proposition 2.13. Suppose that A_{0} is a linear A-module. If M is a linear Amodule, then J^{i}M[−i] and M_{0} are also linear A-modules. In particular, M is projective viewed as an A_{0}-module (or equivalently, M_{i} is a projective A_{0}-module for every i ⩾ 0).

Proof. Without loss of generality we assume that M is indecomposable. Since M is linear, M_{0} is a projective A_{0}-module and is contained in add(A_{0}), the category of all A-modules each of which is isomorphic to a direct summand of A_{0}^{⊕m} for some m ⩾ 0. But A_{0} is linear, so is M_{0}.

Notice that M and M_{0} have the same graded projective cover (up to isomorphism) as A-modules. Thus we have the following commutative diagram:

\xymatrix{ & 0 \ar[r] & ΩM \ar[r] \ar[d] & Ω(M_{0}) \ar[r] \ar[d] & JM \ar[r] & 0 \\  & 0 \ar[r] & P^{0} \ar[r]^{id} \ar[d] & P^{0} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & JM \ar[r] & M \ar[r] & M_{0} \ar[r] & 0}

Consider the top sequence. The modules M and M_{0} are linear, so are ΩM[−1] and Ω(M_{0})[−1]. Therefore, JM[−1] is also linear by Proposition 2.9. Now replacing M by JM[−1] and using recursion, we conclude that J^{i}M[−i] is a linear A-module for every i ⩾ 0. This proves the first statement.

Since J^{i}M[−i] is a linear A-module, (J^{i}M[−i])_{0} ≅ (J^{i}M)_{i} ≅ M_{i} is a projective A_{0}-module for each i ⩾ 0 by Corollary 2.5. Since M = ⨁_{i⩾0} M_{i} as an A_{0}-module, we deduce that M is a projective A_{0}-module if and only if M_{i} is a projective A_{0}- module for every i ⩾ 0. □

The following lemma is very useful.

Lemma 2.14. Let M be a non-negatively graded A-module and suppose that A is a projective A_{0}-module. Then the following are equivalent:

(1) all Ω^{i}(M)_{j} are projective A_{0}-modules, i, j ⩾ 0;

(2) all Ω^{i}(M)_{i} are projective A_{0}-modules, i ⩾ 0;
(3) all M_{i} are projective A_{0}-modules, i ⩾ 0.

Proof. It is clear that (1) implies (2).

(3) implies (1): Suppose that M_{i} is a projective A_{0}-module for all i ⩾ 0. Let P be a graded projective cover of M. The surjective homomorphism φ : P → M gives a surjective homomorphism φ_{j} : P_{j} → M_{j} with kernel (ΩM)_{j}. By the hypothesis, P_{j} and M_{j} are projective A_{0}-modules for all j ⩾ 0. Then P_{j} ≅ M_{j} ⊕ (ΩM)_{j}, so all (ΩM)_{j} are projective A_{0}-modules for j ⩾ 0. Replacing M by ΩM and using recursion, we conclude that all Ω^{i}(M)_{j} are projective A_{0}-modules for i, j ⩾ 0. In particular, all Ω^{i}(M)_{i} are projective A_{0}-modules.

(2) implies (3): Conversely, suppose that Ω^{i}(M)_{i} is a projective A_{0}-modules for every i ⩾ 0. We use contradiction to show that all M_{i} are projective A_{0}-modules. If this not the case, we can find the minimal number n ⩾ 0 such that M_{n} is not a projective A_{0}-module. As above, consider φ_{n} : P_{n} → M_{n} with kernel (ΩM)_{n}. We claim that this kernel is not a projective A_{0}-module. Indeed, if it is a projective A_{0}-module, then it is injective as well, so P_{n} ≅ (ΩM)_{n} ⊕ M_{n}. Consequently, M_{n} is isomorphic to a summand of the projective A_{0}-module P_{n} and must be a projective A_{0}-module, too. This is impossible. Therefore, (ΩM)_{n} is not a projective A_{0}- module. Now replacing M by ΩM and using induction, we deduce that Ω^{n}(M)_{n} is not a projective A_{0}-module. This contradicts our assumption. Therefore, all M_{i} are projective A_{0}-modules. □

Now we define quasi-Koszul modules over the graded algebra A.

Definition 2.15. A non-negatively graded A-module M is called quasi-Koszul if

Ext_{A}^{1}(A_{0}, A_{0}) · Ext_{A}^{i}(M, A_{0}) = Ext_{A}^{i+1}(M, A_{0})

for all i ≥ 0. The algebra A is called a quasi-Koszul algebra if A_{0} as an A-module is quasi-Koszul.

A graded A-module M is quasi-Koszul if and only if as a graded Ext_{A}^{∗}(A_{0}, A_{0})- module Ext_{A}^{∗}(M, A_{0}) is generated in degree 0. The graded algebra A is a quasi-Koszul algebra if and only if the cohomology ring Ext_{A}^{∗}(A_{0}, A_{0}) is generated in degree 0 and degree 1.

The quasi-Koszul property is preserved by the Heller operator. Explicitly, if M is a quasi-Koszul A-module with M_{0} being a projective A_{0}-module, then its syzygy ΩM is also quasi-Koszul. This is because for each i ⩾ 1, we have:

Ext_{A}^{i}(ΩM, A_{0}) ≅ Ext_{A}^{i+1}(M, A_{0})
= Ext_{A}^{1}(A_{0}, A_{0}) · Ext_{A}^{i}(M, A_{0})
= Ext_{A}^{1}(A_{0}, A_{0}) · Ext_{A}^{i−1}(ΩM, A_{0}).

The identity Ext_{A}^{i}(M, A_{0}) ≅ Ext_{A}^{i−1}(ΩM, A_{0}) is proved in Lemma 2.10.

If A_{0} is a semisimple k-algebra, quasi-Koszul modules generated in degree 0 coincide with linear modules. This is not true if A_{0} is only self-injective. Actually, by the following theorem, every Koszul module is quasi-Koszul, but the converse does not hold in general. For example, let kG be the group algebra of a finite group concentrated in degree 0. The reader can check that every kG-module generated in degree 0 is quasi-Koszul, but only the projective kG-modules are Koszul. If |G| is not invertible in k, then all non-projective kG-modules generated in degree 0 are quasi-Koszul but not Koszul.

The following theorem gives us a close relation between quasi-Koszul modules and Koszul modules.

Theorem 2.16. A graded A-module M generated in degree 0 is Koszul if and only if it is quasi-Koszul and Ω^{i}(M)_{i} is a projective A_{0}-module for every i ⩾ 0.

The following lemma will be used in the proof of this theorem.

Lemma 2.17. Let M be a graded A-module generated in degree 0 with M_{0} being a projective A_{0}-module. Then ΩM is generated in degree 1 if and only if every Amodule homomorphism ΩM → A_{0} extends to an A-module homomorphism JP → A_{0}, where P is a graded projective cover of M.

Proof. The short exact sequence 0 → ΩM → P → M → 0 induces an exact sequence 0 → (ΩM)_{1} → P_{1} → M_{1} → 0. Applying the exact functor Hom_{A_{0}}(−, A_{0}) we get another exact sequence

0 → Hom_{A_{0}}(M_{1}, A_{0}) → Hom_{A_{0}}(P_{1}, A_{0}) → Hom_{A_{0}}((ΩM)_{1}, A_{0}) → 0.

Since M_{0} is a projective A_{0}-module, M_{0} ≅ P_{0}, so (ΩM)_{0} = 0. Therefore, ΩM is generated in degree 1 if and only if ΩM/J(ΩM) ≅ (ΩM)_{1}, if and only if the above sequence is isomorphic to

0 → Hom_{A_{0}}(M_{1}, A_{0}) → Hom_{A_{0}}(P_{1}, A_{0}) → Hom_{A_{0}}(ΩM/JΩM, A_{0}) → 0.

Here we use the fact that A_{0} is self-injective and the functor Hom_{A_{0}}(−, A_{0}) is a dual functor. But the above sequence is isomorphic to

\xymatrix{0 \ar[r] & Hom_{A}(JM, A_{0}) \ar[r] & Hom_{A}(JP, A_{0}) \ar[r] & Hom_{A}(ΩM, A_{0}) \ar[r] & 0}

since JM and JP are generated in degree 1. Therefore, ΩM is generated in degree 1 if and only if every (non-graded) A-module homomorphism ΩM → A_{0} extends to a (non-graded) A-module homomorphism JP → A_{0}. □

Now let us prove the theorem.

Proof. The only if part. Let M be a Koszul A-module. Without loss of generality we can suppose that M is indecomposable. Notice that all syzygies Ω^{i}(M)[−i] are also Koszul. Therefore, Ω^{i}(M)_{i} ≅ (Ω^{i}(M)[−i])_{0} is a projective A_{0}-module for all i ⩾ 0.

Now we show that M is quasi-Koszul, i.e.,

Ext_{A}^{i+1}(M, A_{0}) = Ext_{A}^{1}(A_{0}, A_{0}) · Ext_{A}^{i}(M, A_{0})

for all i ⩾ 0. By Lemma 2.10, we have Ext_{A}^{i+1}(M, A_{0}) ≅ Ext_{A}^{1}(Ω^{i}(M), A_{0}) and Ext_{A}^{i}(M, A_{0}) ≅ Hom_{A}(Ω^{i}(M), A_{0}). Therefore, it suffices to show Ext_{A}^{1}(M, A_{0}) = Ext_{A}^{1}(A_{0}, A_{0}) · Hom_{A}(M, A_{0}) since the conclusion follows immediately if we replace M by (ΩM)[−1] recursively.

To prove this identity, we first identify Ext_{A}^{1}(M, A_{0}) with Hom_{A}(ΩM, A_{0}) by Lemma 2.10. Take an element x ∈ Ext_{A}^{1}(M, A_{0}) and let g : ΩM → A_{0} be the corresponding homomorphism. Since M is linear, M_{0} is a projective A_{0}-module, and ΩM is generated in degree 1. Thus by the previous lemma, g extends to JP^{0}, and hence there is a homomorphism g̃ : JP^{0} → A_{0} such that g = g̃ι, where P^{0} is a graded projective cover of M and ι : ΩM → JP^{0} is the inclusion.

\xymatrix{ΩM \ar[r]^{ι} \ar[d]^{g} & JP^{0} \ar[dl]^{g̃} \\ A_{0}}

We have the following commutative diagram:

[figure: 0; ΩM; ι; P^{0}; M; p; 0; 0; JP^{0}; P^{0}; P_{0}^{0}; 0]

where the map p is defined to be the projection of M onto M_{0} ≅ P_{0}^{0}.

The map g̃ : JP^{0} → A_{0} gives a push-out of the bottom sequence. Consequently, we have the following commutative diagram:

[figure: 0; ΩM; ι; P^{0}; M; p; 0; 0; JP^{0}; g̃; P^{0}; P_{0}^{0}; 0; 0; A_{0}; E; P_{0}^{0}; 0.]

Since P_{0} ∈ add(A_{0}), we can find some m such P_{0} can be embedded into A_{0}^{⊕m}. Thus the bottom sequence y ∈ Ext_{A}^{1}(P_{0}, A_{0}) ⊆ ⨁_{i=1}^{m} Ext_{A}^{1}(A_{0}, A_{0}) and we can write y = y_{1} + … + y_{m} where y_{i} ∈ Ext_{A}^{1}(A_{0}, A_{0}) is represented by the sequence

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

Composed with the inclusion ϵ : P_{0} → A_{0}^{⊕m}, the map ϵ ∘ p = (p_{1}, …, p_{m}) where each component p_{i} is defined in an obvious way. Consider the pull-backs:

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

Let x_{i} be the top sequence. Then x = ∑_{i=1}^{m} x_{i} = ∑_{i=1}^{m} y_{i}p_{i} ∈ Ext_{A}^{1}(A_{0}, A_{0}) · Hom_{A}(M, A_{0}) and hence Ext_{A}^{1}(M, A_{0}) ⊆ Ext_{A}^{1}(A_{0}, A_{0}) · Hom_{A}(M, A_{0}). The other inclusion is obvious.

The if part. By Proposition 2.6, it suffices to show that Ω^{i}(M) is generated in degree i, i > 0. But we observe that if M is quasi-Koszul and Ω^{i}(M)_{i} are projective A_{0}-modules for all i ⩾ 0, then each Ω^{i}(M) has these properties as well. Thus we only need to show that ΩM is generated in degree 1 since the conclusion follows if we replace M by ΩM recursively. By the previous lemma, it suffices to show that each (non-graded) A-module homomorphism g : ΩM → A_{0} extends to JP^{0}.

The map g gives a push-out x ∈ Ext_{A}^{1}(M, A_{0}) as follows:

[figure: 0; ΩM; g; P^{0}; M; 0; 0; A_{0}; E; M; 0]

Since M is quasi-Koszul, x is contained in Ext_{A}^{1}(A_{0}, A_{0}) · Hom_{A}(M, A_{0}). Thus x = ∑_{i} y_{i}h_{i} with y_{i} ∈ Ext_{A}^{1}(A_{0}, A_{0}) and h_{i} ∈ Hom_{A}(M, A_{0}), and each y_{i}h_{i} gives the following commutative diagram, where the bottom sequence corresponds to y_{i}:

[figure: 0; A_{0}; E_{i}; M; h_{i}; 0; 0; A_{0}; F_{i}; A_{0}; 0] (2.2)

By the natural isomorphism Ext_{A}^{1}(M, A_{0}) ≅ Hom_{A}(ΩM, A_{0}) (see Lemma 2.10), each y_{i}h_{i} corresponds an A-homomorphism g_{i} : ΩM → A_{0} such that the following diagram commutes:

[figure: 0; ΩM; g_{i}; P^{0}; M; 0; 0; A_{0}; E_{i}; M; 0] (2.3)

Diagrams 2.2 and 2.3 give us:

[figure: 0; ΩM; ι; JP^{0}; j̃; JM; j; 0; 0; ΩM; g_{i}; P^{0}; h̃_{i}; M; h_{i}; 0; 0; A_{0}; ρ; F_{i}; A_{0}; 0]

Since JM is sent to 0 by h_{i}j, there is a homomorphism φ_{i} from JP_{0} to the first term A_{0} of the bottom sequence such that ρφ_{i} = h̃_{i}j̃. Then g_{i} factors through φ_{i}, i.e., g_{i} = φ_{i}ι. Since g = ∑_{i} g_{i}, we know that g extends to JP^{0}. This finishes the proof. □

An easy corollary of the above theorem is:

Corollary 2.18. Suppose that A is projective viewed as an A_{0}-module. Then a graded A-module M is linear if and only if it is quasi-Koszul as an A-module and projective as an A_{0}-module (or equivalently all M_{i} are projective A_{0}-modules).

Proof. By Lemma 2.14, all Ω^{i}(M)_{i} are projective A_{0}-modules for i ⩾ 0 if and only if M_{s} is a projective A_{0}-module for every s ⩾ 0. The conclusion follows from the previous theorem. □

In particular, if A_{0} is is a linear A-module, then by letting M = A in Proposition 2.13, we deduce that all A_{i} are projective A_{0}-modules for i ⩾ 0.

3. Generalized Koszul Algebras

In this section we generalize to our context some useful results on classical Koszul algebras which appear in [4]. As before, throughout this section A is a non-negatively graded, locally finite associative k-algebra with A_{0} being a selfinjective algebra. For two graded A-modules M and N, we use Hom_{A}(M, N) and hom_{A}(M, N) to denote the space of all module homomorphisms and the space of graded module homomorphisms respectively. The derived functors Ext and ext correspond to Hom and hom respectively.

Recall that A a quasi-Koszul algebra if A_{0} is quasi-Koszul as an A-module. In particular, if A_{0} is a linear A-module, then A is a quasi-Koszul algebra.

Theorem 3.1. The graded algebra A is quasi-Koszul if and only if the opposite algebra A^{op} is quasi-Koszul.

Proof. Since the quasi-Koszul property is invariant under the Morita equivalence, without loss of generality we can suppose that A is a basic algebra. Therefore, A_{0} is also a basic algebra. Let M and N be two graded A-modules. We claim Ext_{A}^{i}(M, N) ≅ Ext_{A^{op}}^{i}(DN, DM) for all i ⩾ 0, where D is the duality functor Hom_{k}(−, k). Indeed, Let

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

be a projective resolution of M. Applying the functor Hom_{A}(−, N) we get the following chain complex C^{∗}:

\xymatrix{0 \ar[r] & Hom_{A}(P^{0}, N) \ar[r] & Hom_{A}(P^{1}, N) \ar[r] & ….}

Using the natural isomorphism Hom_{A}(P^{i}, N) ≅ Hom_{A^{op}}(DN, DP^{i}), we get another chain complex E^{∗} isomorphic to the above one:

\xymatrix{0 \ar[r] & Hom_{A^{op}}(DN, DP^{0}) \ar[r] & Hom_{A^{op}}(DN, DP^{1}) \ar[r] & ….}

Notice that all DP^{i} are injective A^{op}-modules. Thus

Ext_{A}^{i}(M, N) ≅ H^{i}(C^{∗}) ≅ H^{i}(E^{∗}) ≅ Ext_{A^{op}}^{i}(DN, DM)

which is exactly our claim.

Now let M = N = A_{0}. Then Ext_{A}^{i}(A_{0}, A_{0}) ≅ Ext_{A^{op}}^{i}(DA_{0}, DA_{0}). Since A_{0} is self-injective and basic, it is a Frobenius algebra. Therefore, DA_{0} is isomorphic to A_{0}^{op} as a left A_{0}^{op}-module (and hence as a left A^{op}-module). Consequently, A_{0} is a quasi-Koszul A-module if and only if A_{0}^{op} is a quasi-Koszul A^{op}-module. □

However, if A_{0} is a linear A-module, A_{0}^{op} need not be a linear A^{op}-module, as shown by the following example.

Example 3.2. Let E be a finite EI category with two objects x and y such that: Aut_{C}(x) = ⟨g⟩ is a cyclic group of order 2; Aut_{C}(y) is a trivial group; Hom_{E}(x, y) = {α} (thus α ∘ g = α) and Hom_{E}(y, x) = ∅. Let k be an algebraically closed field of characteristic 2. The category algebra A = kE is of dimension 4. Let A_{0} be the space spanned by 1_{x}, g and 1_{y}, and let A_{1} be the one-dimensional space spanned by α. The reader can check that A_{0} is a linear A-module. The opposite algebra A^{op} = kE^{op} can also be graded in a similar way, but A_{0}^{op} is not a linear A^{op}-module. However, we will show in Section 6 that A^{op} is a quasi-Koszul algebra.

Proposition 3.3. The graded algebra A is Koszul if and only if Ω^{i}(A_{0})_{i} are projective A_{0}-modules for all i ⩾ 0, and whenever ext_{A}^{i}(A_{0}, A_{0}[n]) ≠ 0 we have n = i.

Proof. If A is a Koszul algebra, then A_{0} is a linear A-module, and by Corollary 2.5 all Ω^{i}(A_{0})_{i} are projective A_{0}-modules. Moreover, there is a linear projective resolution

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

with P^{i} being generated in degree i. Applying hom_{A}(−, A_{0}[n]) we find that all terms in this complex except hom_{A}(P^{n}, A_{0}[n]) are 0. Consequently, ext_{A}^{i}(A_{0}, A[n]) ≠ 0 unless i = n.

Conversely, suppose that Ω^{i}(A_{0})_{i} is a projective A_{0}-module for each i ⩾ 0 and ext_{A}^{i}(A_{0}, A_{0}[n]) = 0 unless n = i, we want to show that Ω^{i}(A_{0}) is generated in degree i by induction. Obviously, Ω^{0}(A_{0}) = A_{0} is generated in degree 0. Suppose that Ω^{j}(A_{0}) is generated in degree j for 0 ⩽ j ⩽ i. Now consider Ω^{i+1}(A_{0}). By applying the graded version of Lemma 2.10 recursively, we have

hom_{A}(Ω^{i+1}(A_{0}), A_{0}[n]) = ext_{A}^{i+1}(A_{0}, A_{0}[n]).

The right-hand side is 0 unless n = i + 1, so Ω^{i+1}(A_{0}) is generated in degree i + 1. By induction we are done. □

The reader can check that the conclusion of this proposition is also true for linear modules. i.e., M is a linear A-module if and only if ext_{A}^{i}(M, A_{0}[n]) ≠ 0 implies n = i.

We can define a tensor algebra T (A) generated by A_{1}, which is a (A_{0}, A_{0})- bimodule. Explicitly,

T (A) = A_{0} ⊕ A_{1} ⊕ (A_{1} ⊗ A_{1}) ⊕ (A_{1} ⊗ A_{1} ⊗ A_{1}) ⊕ …,

where all tensors are over A_{0} and we use ⊗ rather than ⊗_{A_{0}} to simplify the notation. This tensor algebra has a natural grading. Clearly, A is a quotient algebra of T (A). Let I be the kernel of the quotient map q : T (A) → A. We say that A is a quadratic algebra if the ideal I has a set of generators contained in A_{1} ⊗ A_{1}.

Theorem 3.4. If A is a Koszul algebra, then it is a quadratic algebra.

Proof. This proof is a modification of the proofs of Theorem 2.3.2 and Corollary 2.3.3 in [4]. First, consider the exact sequence

\xymatrix{0 \ar[r] & W \ar[r] & A ⊗ A_{1} \ar[r] & A \ar[r] & A_{0} \ar[r] & 0}

where W is the kernel of the multiplication. Clearly, Ω(A_{0}) ≅ J = ⨁_{i⩾1} A_{i}. Since the image of (A⊗A_{1})_{1} = A_{0} ⊗A_{1} under the multiplication is exactly A_{1} = Ω(A_{0})_{1}, A ⊗ A_{1} is a projective cover of Ω(A_{0}) and Ω^{2}(A_{0}) = W ⊆ J ⊗ A_{1}. Therefore, W is generated in degree 2, and hence W/JW ≅ W_{2} is concentrated in degree 2. Observe that A is a quotient algebra of T (A) with kernel I. Let R_{n} be the kernel of the quotient map A_{1}^{⊗n} → A_{n}.

If A is not quadratic, we can find some x ∈ R_{n} with n > 2 such that x is not contained in the two-sided ideal generated by ∑_{i=2}^{n−1} R_{i}. Consider the following composite of maps:

\xymatrix{A_{1}^{⊗n} = A_{1}^{⊗n−1} ⊗ A_{1} \ar[r]^{p} & A_{n−1} ⊗ A_{1} \ar[r]^{m} & A_{n}} .

Clearly p(x) ∈ W since m(p(x)) = 0. We show p(x) ∉ JW by contradiction.

Indeed, if p(x) ∈ JW, then p(x) ∈ A_{1}W since JW ≅ ⨁_{i⩾3} W_{i} = A_{1}W (notice that W is generated in degree 2). Therefore, we can express p(x) as a linear combination of vectors of the form λ·w with λ ∈ A_{1} and w ∈ W. But W ⊆ J ⊗A_{1}, so each w can be expressed as ∑_{i} w′_{i} ⊗ λ′_{i} with w′_{i} ∈ A_{n−2}, λ′_{i} ∈ A_{1} such that ∑_{i=1}^{s} w′_{i} · λ′_{i} = 0 by the definition of W.

Since there is a surjective product map φ : A_{1}^{⊗n−2} ↠ A_{n−2}, we can choose a pre-image v_{i}^{1} ⊗ … ⊗ v_{i}^{n−2} ∈ φ^{−1}(w′_{i}) for each i and define

w̃ = ∑_{i=1}^{s} v_{i}^{1} ⊗ … ⊗ v_{i}^{n−2} ⊗ λ′_{i}

which is contained in R_{n−1} clearly. Observe that p(λ ⊗ w̃) = λ · w. Since p(x) is a linear combination of vectors of the form λ · w, by the above process we can get some y which is a linear combination of vectors of the form λ ⊗ w̃ such that p(y) = p(x). Clearly, p(x − y) = 0 and y ∈ A_{1} ⊗ R_{n−1}.

Consider the following short exact sequence

\xymatrix{0 \ar[r] & R_{n−1} \ar[r] & A_{1}^{⊗n−1} \ar[r] & A_{n−1} \ar[r] & 0.}

Since A_{0} is a linear A-module, so is J[−1] = Ω(A_{0})[−1]. Therefore, by Corollary 2.5, A_{1} ≅ J[−1]_{0} is a projective A_{0}-module. Therefore, the following sequence is also exact:

\xymatrix{0 \ar[r] & R_{n−1} ⊗ A_{1} \ar[r] & A_{1}^{⊗n} \ar[r]^{p} & A_{n−1} ⊗ A_{1} \ar[r] & 0.}

Thus x − y ∈ R_{n−1} ⊗ A_{1} since p(x − y) = 0. It follows x ∈ A_{1} ⊗ R_{n−1} + R_{n−1} ⊗ A_{1}, which contradicts our choice of x.

We proved x ∉ JW. Then p(x) ∈ W/JW ≅ W_{2} is of degree 2. But this is impossible since p as a graded homomorphism sends x ∈ R_{n} with n > 2 to an element of degree n. □

We can define the Koszul complex for A in a similar way to the classical situation. Let A ≅ T (A)/(R) be quadratic with R ⊆ A_{1} ⊗ A_{1} being a set of relations.

Define P_{n}^{n} = ⋂_{i=0}^{n−2} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−2} ⊆ A_{1}^{⊗n}. In particular, P_{0}^{0} = A_{0}, P_{1}^{1} = A_{1} and P_{2}^{2} = R. Let P^{n} = A ⊗ P_{n}^{n} such that A_{0} ⊗ P_{n}^{n} ≅ P_{n}^{n} is in degree n. Define d^{n} : P^{n} → P^{n−1} to be the restriction of A⊗A_{1}^{⊗n} → A⊗A_{1}^{⊗n−1} by a⊗v_{1}⊗…⊗v_{n} ↦ av_{1} ⊗ v_{2} ⊗ … ⊗ v_{n}. The reader can check d^{n−1}d^{n} = 0 for n ⩾ 1. Therefore we get the following Koszul complex K^{∗}:

\xymatrix{… \ar[r] & P^{3} \ar[r]^{d^{3}} & A ⊗ R \ar[r]^{d^{2}} & A ⊗ A_{1} \ar[r]^{d^{1}} & A \ar[r] & 0.}

Theorem 3.5. Let A ≅ T (A)/(R) be a quadratic algebra. Then A is a Koszul algebra if and only if the Koszul complex is a projective resolution of A_{0}.

Proof. One direction is trivial. Now suppose that A_{0} is a linear A-module. The Koszul complex K^{∗} of A has the following properties: (1). Let Z^{n} be the kernel of d^{n} : P^{n} → P^{n−1}. The restricted map d_{n}^{n} :

P_{n}^{n} = A_{0} ⊗ ( ⋂_{i=0}^{n−2} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−2}) → P_{n}^{n−1} = A_{1} ⊗ ( ⋂_{i=0}^{n−3} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−3})

is injective. Therefore Z_{i}^{n} = 0 for every i ⩽ n. (2). Z_{n+1}^{n}, the kernel of the map d_{n}^{n+1} :

P_{n+1}^{n} = A_{1} ⊗ ( ⋂_{i=0}^{n−2} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−2}) → P_{n+1}^{n−1} = A_{2} ⊗ ( ⋂_{i=0}^{n−3} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−3})

is

A_{1} ⊗ ( ⋂_{i=0}^{n−2} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−2}) ∩ (R ⊗ A_{1}^{⊗n−1}) = ⋂_{i=0}^{n−1} A_{1}^{⊗i} ⊗ R ⊗ A_{1}^{⊗n−i−1}

which is exactly P_{n+1}^{n+1} (or d_{n+1}^{n+1}(P_{n+1}^{n+1}) since d_{n+1}^{n+1} is injective by the last property).

We claim that each P^{n} = A ⊗ P_{n}^{n} is a projective A-module. Clearly, it is enough to show that each P_{n}^{n} = Z_{n}^{n−1} is a projective A_{0}-module. We prove the following stronger conclusion. That is, Z_{i}^{n} are projective A_{0}-modules for i ∈ ℤ and n ⩾ 0. We use induction on n.

Since A_{0} is a linear A-module, by Proposition 2.13, A_{i} are projective A_{0}-modules for all i ⩾ 0. The conclusion is true for Z^{0} ≅ J since J_{0} = 0 and J_{m} = A_{m} for m ⩾ 1. Suppose that it is true for l ⩽ n. That is, all Z_{i}^{l} are projective A_{0}-modules for l ⩽ n and i ∈ ℤ. Consider l = n + 1. By the second property described above, P_{n+1}^{n+1} = Z_{n+1}^{n}, which is a projective A_{0}-module by the induction hypothesis.

Therefore, P^{n+1} = A ⊗ P_{n+1}^{n+1} is a projective A-module, so P_{i}^{n+1} are all projective A_{0}-modules for i ∈ ℤ. But the following short exact sequence of A_{0}-modules splits

\xymatrix{0 \ar[r] & Z_{i}^{n+1} \ar[r] & P_{i}^{n+1} \ar[r] & Z_{i}^{n} \ar[r] & 0}

since Z_{i}^{n} is a projective A_{0}-module by the induction hypothesis. Now as a direct summand of P_{i}^{n+1} which is a projective A_{0}-module, Z_{i}^{n+1} is a projective A_{0}-module as well. Our claim is proved by induction.

We claim that this complex is acyclic. First, the sequence

\xymatrix{P^{1} = A ⊗ A_{1} \ar[r] & P^{0} = A ⊗ A_{0} \ar[r] & A_{0} \ar[r] & 0}

is right exact. By induction on n > 1

ext_{A}^{n+1}(A_{0}, A_{0}[m]) = coker(hom_{A}(P^{n}, A_{0}[m]) → hom_{A}(Z^{n}, A_{0}[m])).

By Property (1), Z_{m}^{n} = 0 if m < n + 1. If m > n + 1, hom_{A}(P^{n}, A_{0}[m]) = 0 since P^{n} is generated in degree n, so ext_{A}^{n+1}(A_{0}, A_{0}[m]) = hom_{A}(Z^{n}, A_{0}[m]) by the above identity. But the left-hand side of this identity is non-zero only if m = n + 1 since A_{0} is linear. Therefore, hom_{A}(Z^{n}, A_{0}[m]) = 0 for m > n + 1. Consequently, Z^{n} is generated in degree n + 1. By property (2), Z_{n+1}^{n} = d_{n+1}^{n+1}(P_{n+1}^{n+1}), so Z^{n} = d^{n+1}(P^{n+1}) since both modules are generated in degree n+1. Therefore, the Koszul complex is acyclic, and hence is a projective resolution of A_{0}. □

4. Generalized Koszul Duality

In this section we prove the Koszul duality. As before, A is a non-negatively graded, locally finite algebra with A_{0} being a self-injective algebra. Define Γ = Ext_{A}^{∗}(A_{0}, A_{0}) which has a natural grading. Notice that Γ_{0} ≅ A_{0}^{op} is also a selfinjective algebra. Let M be a graded A-module. Then Ext_{A}^{∗}(M, A_{0}) is a graded Γ-module. Thus we can define a functor E = Ext_{A}^{∗}(−, A_{0}) from A-gmod to Γ-gmod.

Theorem 4.1. If A is a Koszul algebra, then E = Ext_{A}^{∗}(−, A_{0}) gives a duality between the category of linear A-modules and the category of linear Γ-modules. That is, if M is a linear A-module, then E(M) is a linear Γ-module, and E_{Γ}EM = Ext_{Γ}^{∗}(EM, Γ_{0}) ≅ M.

Proof. Since M and A_{0} both are linear, by Proposition 2.13 M_{0} and JM[−1] are linear, where J = ⨁_{i⩾1} A_{i}. Furthermore, we have the following short exact sequence of linear modules:

\xymatrix{0 \ar[r] & ΩM[−1] \ar[r] & Ω(M_{0})[−1] \ar[r] & JM[−1] \ar[r] & 0.}

As in the proof of Proposition 2.9, this sequence induces exact sequences recursively (see diagram 2.1):

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

and gives exact sequences of A_{0}-modules:

\xymatrix{0 \ar[r] & Ω^{i}(M)_{i} \ar[r] & Ω^{i}(M_{0})_{i} \ar[r] & Ω^{i−1}(JM[−1])_{i−1} \ar[r] & 0.}

Applying the exact functor Hom_{A_{0}}(−, A_{0}) and using the following isomorphism for a graded A-module N generated in degree i

Hom_{A}(N, A_{0}) ≅ Hom_{A}(N_{i}, A_{0}) ≅ Hom_{A_{0}}(N_{i}, A_{0},)

we get:

0 → Hom_{A}(Ω^{i−1}(JM[−1]), A_{0}) → Hom_{A}(Ω^{i}(M_{0}), A_{0}) → Hom_{A}(Ω^{i}M, A_{0}) → 0.

By Lemma 2.10, this sequence is isomorphic to

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

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

\xymatrix{0 \ar[r] & E(JM[−1])[1] \ar[r] & E(M_{0}) \ar[r] & EM \ar[r] & 0.}

Let us focus on this sequence. We claim Ω(EM) ≅ E(JM[−1])[1]. Indeed, since M_{0} is a projective A_{0}-module and the functor E is additive, E(M_{0}) is a projective Γ-module. Since JM[−1] is Koszul, JM[−1] is quasi-Koszul and hence E(JM[−1]) as a Γ-module is generated in degree 0. Thus E(JM[−1])[1] is generated in degree 1, and E(M_{0}) is minimal. This proves the claim. Consequently, Ω(EM) is generated in degree 1 as a Γ-module. Moreover, replacing M by JM[−1] (which is also linear) and using the claimed identity, we have

Ω^{2}(EM) = Ω(E(JM[−1])[1]) = Ω(E(JM[−1])[1] = E(J^{2}M[−2])[2],

which is generated in degree 2. By recursion, we know that Ω^{i}(EM) ≅ E(J^{i}M[−i])[i] is generated in degree i for all i ⩾ 0. Thus EM is a linear Γ-module. In particular for M = A,

EA = Ext_{A}^{∗}(A, A_{0}) = Hom_{A}(A, A_{0}) = Γ_{0}

is a linear Γ-module.

Since Ω^{i}(EM) is generated in degree i and

Ω^{i}(EM)_{i} ≅ E(J^{i}M[−i])[i]_{i} ≅ E(J^{i}M[−i])_{0}
= Hom_{A}(J^{i}M[−i], A_{0}) ≅ Hom_{A}(M_{i}, A_{0}),

we have

Hom_{Γ}(Ω^{i}(EM), Γ_{0}) ≅ Hom_{Γ_{0}}(Ω^{i}(EM)_{i}, Γ_{0})
≅ Hom_{Γ_{0}}(Hom_{A}(M_{i}, A_{0}), Γ_{0})
≅ Hom_{Γ_{0}}(Hom_{A_{0}}(M_{i}, A_{0}), Γ_{0})
≅ M_{i}.

The last isomorphism holds because A_{0} is self-injective and Γ_{0} ≅ A_{0}^{op}.

We have proved that EM is a linear Γ-module. Therefore, (Ω^{i}(EM))_{i} is a projective Γ_{0}-module for every i ⩾ 0. Applying Lemma 2.10 recursively, Ext_{Γ}^{i}(EM, Γ_{0}) ≅ Hom_{Γ}(Ω^{i}(EM), Γ_{0}) ≅ M_{i} for every i ⩾ 0. Adding them together, E_{Γ}E(M) ≅ ⨁_{i=0}^{∞} M_{i} ≅ M.

Now we have E_{Γ}(E(A)) = E_{Γ}(Γ_{0}) ≅ A. Moreover, Γ is a graded algebra with Γ_{0} ≅ A_{0}^{op} being self-injective as an algebra and linear as a Γ-module. Using this duality, we can exchange A and Γ in the above reasoning and get EE_{Γ}(N) ≅ N for an arbitrary linear Γ-module N. Thus E induces an equivalence between these two categories. □

Remark 4.2. We can also use hom_{A}(−, A_{0}) to define the functor E on the category of linear A-modules, namely E := ⨁_{i⩾0} ext_{A}^{i}(−, A_{0}[i]). Indeed, for a linear Amodule M, we have:

Ext_{A}^{∗}(M, A_{0}) = ⨁_{i⩾0} Ext_{A}^{i}(M, A_{0})

= ⨁_{i⩾0} ⨁_{j∈ℤ} ext_{A}^{i}(M, A_{0}[j])

= ⨁_{i⩾0} ext_{A}^{i}(M, A_{0}[i])

since ext_{A}^{i}(M, A_{0}[j]) = 0 for i ≠ j, see the paragraph after Proposition 3.3.

Now we want to study the Koszul duality on the level of derived categories. We introduce the following notation: let D_{gr}^{b}A be the bounded derived category of Agmod, and define T to be the category formed by objects T [i] where T ranges over indecomposable summand of A_{0} and i ∈ ℤ.

Lemma 4.3. Let A be a locally finite graded algebra with A_{0} being a self-injective algebra and a linear A-module. Let Γ = Ext_{A}^{∗}(A_{0}, A_{0}). Then we have:

(1) If pd_{A} A_{0} < ∞, then dim_{k} Γ < ∞.

(2) If the global dimension gldim A < ∞, then gldim A_{0} < ∞, pd_{A} A_{0} < ∞;

(3) Conversely, if gldim A_{0} < ∞, pd_{A} A_{0} < ∞, and dim_{k} A < ∞, then gldim A <

∞.

Proof. Notice that Γ_{i} = Ext_{A}^{i}(A_{0}, A_{0}) = 0 for all i ⩾ m + 1. If 0 ⩽ i ⩽ m, then

Γ_{i} = Ext_{A}^{i}(A_{0}, A_{0}) ≅ Hom_{A}(Ω^{i}(A_{0}), A_{0})

which is finite-dimensional since Ω^{i}(A_{0}) is a finitely generated A-module and A is locally finite. Therefore, dim_{k} Γ < ∞. This proves the first statement.

If gldim A = m < ∞, clearly pd_{A} A_{0} ⩽ m. Take an arbitrary A_{0}-module M. Viewed as a A-module concentrated in degree 0, A has a projective resolution

\xymatrix{0 \ar[r] & P^{m} \ar[r] & P^{n−1} \ar[r] & … \ar[r] & P^{0} \ar[r] & M \ar[r] & 0,}

which induces

\xymatrix{0 \ar[r] & P_{0}^{m} \ar[r] & P_{0}^{n−1} \ar[r] & … \ar[r] & P_{0}^{0} \ar[r] & M_{0} \ar[r] & 0.}

This is a projective resolution of the A_{0}-module M = M_{0}. Therefore, pd_{A_{0}} M ⩽ m. So gldim A_{0} ⩽ m, and (2) is proved.

Conversely, assume that pd_{A} A_{0} < ∞, gldim A_{0} < ∞ and dim_{k} A < ∞. To prove gldim A < ∞, it suffices to show pd_{A} S < ∞ for each simple A-module S since every A-module has a composition series of finite length. Clearly, S is still simple viewed as a A_{0}-module. Therefore, there exists a number m ⩾ 0 such that Ω_{A_{0}}^{m+1}(S) = 0 but Ω_{A_{0}}^{m}(S) is a non-zero projective A_{0}-module. Consider the exact sequence of A_{0}-modules:

\xymatrix{0 \ar[r] & Ω_{A_{0}}^{m}(S) \ar[r] & P \ar[r] & Ω_{A_{0}}^{m−1}(S) \ar[r] & 0.}

Each module appearing in this sequence can be viewed as an A-module concentrated in degree 0. Notice that the first two terms are projective A_{0}-modules and hence have finite projective dimensions as A-modules. Therefore, pd_{A} Ω_{A_{0}}^{m−1}(S) < ∞. By induction, pd_{A} S < ∞. □

Theorem 4.4. Suppose that A is a Koszul algebra such that gldim A_{0}, pd_{A} A_{0}, gldim Γ_{0} and pd_{Γ} Γ_{0} are all finite. Then D_{gr}^{b}A ≅ D_{gr}^{b}Γ^{op} as triangulated categories.

Proof. By the above lemma, both A and Γ are finite-dimensional algebras. Therefore, gldim A and gldim Γ are also finite. Consequently, the category T generates D_{gr}^{b}A as a triangulated category, and the category of all indecomposable graded projective Γ-modules generates D_{gr}^{b}Γ. Now the conclusion can be deduced from Theorem 4.3.4 in [14] by letting T = A_{0}. We remind the reader that functor G_{A_{0}} in [14] is Ext_{A}^{∗}(A_{0}, −) instead of E = Ext_{A}^{∗}(−, A_{0}) used by us throughout this paper, and the algebra Γ in his paper is set to be Ext_{A}^{∗}(A_{0}, A_{0})^{op}. □

5. Application to Directed Categories

In this section we will apply the general theory to a type of structures called directed categories, for which there exist very nice relations between stratification theory and Koszul theory and a nice correspondence between our generalized Koszul theory and the classical theory. All categories C we consider in this section are locally finite k-linear categories with finitely many objects, that is, for x, y ∈ Ob C, the set of morphisms C(x, y) is a finite-dimensional k-vector space. To simplify the technical part, we suppose furthermore that C is skeletal, i.e., x ≅ y implies x = y for x, y ∈ Ob C.

Definition 5.1. A locally finite k-linear category C is a directed category if there is a partial order ⩽ on Ob C such that C(x, y) ≠ 0 only if x ⩽ y.

Correspondingly, we define directed algebras.

Definition 5.2. A finite-dimensional algebra A is called a directed algebra with respect to a partially ordered set of orthogonal idempotents {e_{i}; ⩽}_{i=1}^{n} if ∑_{i=1}^{n} e_{i} = 1 and Hom_{A}(Ae_{i}, Ae_{j}) ≅ e_{i}Ae_{j} ≠ 0 implies e_{j} ⩽ e_{i}.

Notice that in the above definition we do not require the idempotents e_{i} to be primitive. Clearly, every algebra A is always directed with respect to the trivial set {1}.

There is a bijective correspondence between directed categories and directed algebras. Let A be a directed algebra with respect to a poset of orthogonal idempotents ({e_{i}}_{i=1}^{n}, ⩽). Then we can construct a directed category A in the following way: Ob A = {e_{i}}_{i=1}^{n} with the same partial order, and A(e_{i}, e_{j}) = e_{j}Ae_{i} ≅ Hom_{A}(Ae_{j}, Ae_{i}). The reader can check that A is indeed a directed category. We call A the associated category of A.

Conversely, given a directed category A with the poset (Ob A, ⩽), we obtain an algebra A which is directed with respect to the poset of orthogonal idempotents ({1_{x}}_{x∈Ob A}, ⩽), namely, 1_{x} ⩽ 1_{y} if and only if x ⩽ y. As a k-vector space, A = ⨁_{x,y∈Ob A} A(x, y). For two morphisms α : x → y and β : z → w, the product β · α = 0 if y ≠ z, otherwise it is the composite morphism βα. Since every vector in A is a linear combination of morphisms in A, the multiplication of morphisms can be extended linearly to a well defined product in A. The reader can check that the algebra A we get in this way is indeed a directed algebra, which is called the associated algebra of A.

It is well known that A-mod is equivalent to the category of finite-dimensional k-linear representations of A. If one of A and A is graded, then the other one can be graded as well. Moreover, A-gmod is equivalent to the category of finite-dimensional graded k-linear representations of A. For more details, see [16]. Because of these facts, we may view a directed category A as a directed algebra with respect to the set of idempotents {1_{x} | x ∈ Ob A} and abuse notation and terminologies. For example, we may say idempotents in A, ideals of A and so on. We hope this would not cause confusions to the reader and point out that all results in previous sections can be applied to directed categories.

Directed categories generalize k-linearizations of finite EI categories. Explicitly, let E be a skeletal finite EI category. Consider the category algebra kE with a set of idempotents {1_{x}}_{x∈Ob E} on which there is a partial order ≤ such that 1_{x} ≤ 1_{y} if and only if E(x, y) ≠ ∅. Then the category algebra kE is directed with respect to this poset of idempotents, so we can construct a direct category Ẽ by the above correspondence. Actually, Ẽ is precisely the k-linearization of E.

Let C be a directed category. A C-module (or a representation of C) is defined to be a k-linear functor from C to the category of finite-dimensional k-vector spaces.

The morphism space of C can be decomposed as the direct sum of C1_{x} with x ranging over all objects, where by C1_{x} we denote the vector space formed by all morphisms with source x. Therefore, each C1_{x} is a projective C-module, and every indecomposable projective C-module is isomorphic to a direct summand of a certain C1_{x}. The isomorphism classes of simple C(x, x)-modules with x varying within Ob C give rise to isomorphism classes of simple C-modules. Explicitly, let V be a simple C(x, x)-module for some object x, we can construct a simple C-module S: S(x) = V and S(y) = 0 for y ≠ x. These results are well known for finite EI categories, see [20].

Our next task is to translate some results on finite EI categories in Section 2 of [20] to directed categories. First, we want to show that every directed category is stratified with respect to the given partial order.

Proposition 5.3. Let D and E be full subcategories of a directed category C such that Ob D ∪ Ob E = Ob C, Ob D ∩ Ob E = ∅, and C(x, y) = 0 for x ∈ Ob D and y ∈ Ob E. Let e = ∑_{x∈Ob D} 1_{x} and I = CeC. Then I is a stratified ideal of C.

Proof. The proof is similar to that of Proposition 2.2 in [20]. Clearly I is idempotent. Notice that Ce is the space spanned by all morphisms with sources contained in Ob D and eCe is the space spanned by all morphisms with both sources and targets contained in Ob D. Since C(x, y) = 0 for x ∈ Ob D and y ∈ Ob E, these two spaces coincide, i.e., Ce = eCe. In particular, Ce is projective eCe-module, here eCe is an algebra for which the associated directed category is precisely D. Therefore, Tor_{n}^{eCe}(Ce, eC) = 0 for n ⩾ 1. Furthermore,

Ce ⊗_{eCe} eC = eCe ⊗_{eCe} eC ≅ eC.

We claim eC = CeC. Clearly, eC ⊆ CeC. On the other hand, since we just proved Ce = eCe, we have CeC = eCeC ⊆ eC. Therefore, eC = CeC as we claimed. In conclusion, I is indeed a stratified ideal of C. □

Corollary 5.4. Every directed category C is stratified with respect to the give partial order on Ob C.

Proof. The partial order ⩽ on Ob C gives a filtration on Ob C in the following way: let S_{1} be a set containing a maximal object in Ob C, S_{2} is formed by adding a maximal object in Ob C \ S_{1} into S_{1}, S_{3} is formed by adding a maximal object in Ob C \ S_{2} into S_{2}, and so on. Consider the full subcategories D_{i} formed by S_{i} and let e_{i} = ∑_{x∈S_{i}} 1_{x}. Then the ideals Ce_{i}C give a stratification of C by the previous proposition. □

Now we want to describe standard modules and give a characterization of standardly stratified directed categories with respect to a particular pre-order. Before doing that, we need to define this pre-order on a complete set of primitive idempotents of C (or precisely, primitive idempotents of the assciated algebra). For every object x, C(x, x) = 1_{x}C1_{x} is a finite-dimensional k-algebra, so we can choose a complete set of orthogonal primitive idempotents E_{x} = {e_{λ}}_{λ∈Λ_{x}} with ∑_{λ∈Λ_{x}} e_{λ} = 1_{x}. In this way we get a complete set of orthogonal primitive idempotents ⨆_{x∈Ob C} E_{x}. The partial order ⩽ on Ob C can be applied to define a pre-ordered set (Λ, ⪯) to index all these primitive idempotents, namely for e_{λ} ∈ E_{x} and e_{μ} ∈ E_{y}, e_{λ} ⪯ e_{μ} if and only if x ⩽ y. We can check that ⪯ is indeed a pre-order. We denote e_{λ} ≺ e_{μ} if e_{λ} ⪯ e_{μ} but e_{μ} ⪯̸ e_{λ} for λ, μ ∈ Λ. Notice that indecomposable summands of C (viewed as an algebra) can be indexed by these primitive idempotents in a obvious way, namely P_{λ} = Ce_{λ}. Therefore, the pre-ordered set (Λ, ⪯) can also be used to index all indecomposable summands of C.

We define standard C-modules in the following way:

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

where tr_{P_{μ}}(P_{λ}) is the trace of P_{μ} in P_{λ}. The following proposition gives a description of standard C-modules with respect to the above pre-order.

Proposition 5.5. The standard C-module Δ_{λ} is only supported on x with value Δ_{λ}(x) ≅ 1_{x}Ce_{λ}, where x ∈ Ob C and e_{λ} ∈ E_{x}.

Proof. Let us first analyze the structure of P_{λ} = Ce_{λ}. Since e_{λ} ∈ E_{x}, P_{λ} is a direct summand of C1_{x}. The value of C1_{x} on an arbitrary object y is 1_{y}C1_{x}, the space of all morphisms from x to y. Therefore, the value of P_{λ} on y is 1_{y}Ce_{λ}. By our definition of the partial order on Ob C, if x ⩽̸ y, then there is no nontrivial morphisms from x to y. Therefore, 1_{y}C1_{x} and hence 1_{y}Ce_{λ} are 0. We deduce immediately that Δ_{λ} is only supported on objects y satisfying x ⩽ y.

Let y be an object such that y > x. Then every e_{μ} ∈ E_{y} satisfies e_{μ} ≻ e_{λ}. Since ∑_{e_{μ}∈E_{y}} e_{μ} = 1_{y}, by taking the sum we find that tr_{C1_{y}}(P_{λ}) is contained in ∑_{μ≻λ, μ∈Λ} tr_{P_{μ}}(P_{λ}). The value on y of tr_{C1_{y}}(C1_{x}) is 1_{y}C1_{x}. Since P_{λ} = Ce_{λ} is a direct summand of C1_{x}, the value on y of tr_{C1_{y}}(P_{λ}) is 1_{y}Ce_{λ}. Consequently, the value of ∑_{μ≻λ, μ∈Λ} tr_{P_{μ}}(P_{λ}) on y contains 1_{y}Ce_{λ}, which equals the value of P_{λ} on y. Therefore, the value of ∑_{μ≻λ, μ∈Λ} tr_{P_{μ}}(P_{λ}) on y is precisely 1_{y}Ce_{λ}, so the value of Δ_{λ} on y is 0.

We have proved that Δ_{λ} is only supported on x. Clearly, its value on x is 1_{x}Ce_{λ}. □

This proposition tells us that standard modules are exactly indecomposable direct summands of ⨁_{x∈Ob C} C(x, x) (viewed as a C-module by identifying it with the quotient module ⨁_{x,y∈Ob C} C(x, y)/ ⨁_{x≠y} C(x, y)).

Definition 5.6. A directed category C is said to be standardly stratified if every (indecomposable) projective module P_{λ} has a Δ-filtration by standard modules.

To simplify the expression, we stick to the following convention frow now on:

Convention: When we say a directed category is standardly stratified, we always refer to the preorder ⪯ induced by the given partial order ⩽ on the set of objects.

This definition is very simple compared to the definition of standardly stratified algebras (for example, the definition in [6]). However, from the previous proposition we find that every standard module Δ_{λ} of C satisfies the following condition: if S_{μ} and S_{ν} are two different composition factors of Δ_{λ}, then both S_{μ} ⪯ S_{ν} and S_{ν} ⪯ S_{μ} (but in general S_{μ} ≇ S_{ν}). Moreover, If S_{μ} is a composition factor of the kernel K_{λ} of the surjection P_{λ} → Δ_{λ}, then S_{μ} ≻ S_{λ} since K_{λ} is only supported on objects y > x, here x is the object where S_{λ} is supported. Therefore, if C satisfies the requirement in the above definition, then the associated algebra is standardly stratified as well.

The next theorem characterizes standardly stratified directed categories.

Theorem 5.7. Let C be a directed category. Then C is standardly stratified if and only if the morphism space C(x, y) is a projective C(y, y)-module for every pair of objects x, y ∈ Ob C. Moreover, if C is standardly stratified, then ⨁_{x∈ObC} C(x, x) as a C-module has finite projective dimension.

Proof. Suppose that C is standardly stratified and take two arbitrary objects x and y in C. Since 0 is regarded as a projective module, we can assume C(x, y) ≠ 0 and want to show that it is a projective C(y, y)-module. Consider the projective C-module C1_{x}, which has a filtration with factors standard modules. Since each standard module is only supported on one object, the value of C1_{x} on y is exactly the sum of these standard modules with non-zero values on y. This sum is direct since standard modules supported on y are non-comparable with respect to the pre-order and therefore there is no extension between them (or because by the previous proposition each of these standard modules is projective viewed as a C(y, y)-module). Therefore, the value of C1_{x} on y is a projective C(y, y)-module. But the value of C1_{x} on y is precisely C(x, y), so the only if part is proved.

Conversely, let P_{λ} = Ce_{λ} be an indecomposable projective C-module. Its value on an arbitrary object y is 1_{y}Ce_{λ} ≅ 1_{y}C1_{x} which is either 0 or isomorphic to a direct summand of C(x, y). If C(x, y) is a projective C(y, y)-module, then the value of P_{λ} on y is a projective C(y, y)-module as well. This value can be expressed as a direct sum of standard modules supported on y since standard modules are exactly indecomposable direct summands of ⨁_{x∈Ob C} C(x, x). Therefore we can get a filtration of P_{λ} by standard modules.

It is well known that the projective dimension of a standard module is finite if the algebra is standardly stratified. Since ⨁_{x∈ObC} C(x, x) as a C-module is a direct sum of standard modules, the last statement follows from this fact immediately. □

If the directed category C is standardly stratified, then all standard modules have finite projective dimensions. But the converse is not true in general. However, we will prove later that for a finite EI category, all standard modules have finite projective dimension if and only if this category is standardly stratified with respect to the canonical pre-order.

From now on we suppose that C is a graded category, that is, there is a grading on the morphisms in C such that C_{i} · C_{j} ⊆ C_{i+j}, where we denote the subspace spanned by all morphisms with grade i by C_{i}. Furthermore, C is supposed to satisfy the following condition: C_{i} · C_{j} = C_{i+j}. Every vector in C_{i} is a linear combination of morphisms with degree i. Clearly, C_{i} = ⨁_{x,y∈Ob C} C(x, y)_{i}. We always suppose C_{i} = 0 for i < 0 and C_{0} = ⨁_{x∈Ob C} C(x, x). This is equivalent to saying that C_{0} is the direct sum of all standard C-modules by Proposition 5.5.

Given a graded directed category C, we can apply the functor E = Ext_{C}^{∗}(−, C_{0}) to construct the Yoneda category E(C_{0}): Ob E(C_{0}) = Ob C and E(C_{0})(x, y)_{n} = Ext_{C}^{n}(C(x, x), C(y, y)). This is precisely the categorical version of Yoneda algebras. By the correspondence between graded algebras and graded categories, we can define Koszul categories, quasi-Koszul categories, quadratic categories, Koszul modules, quasi-Koszul modules for graded categories as well. We do not repeat these definitions here but emphasize that all results described in the previous sections can be applied to graded categories.

A corollary of Theorem 5.7 and Corollary 2.18 relates stratification theory to Koszul theory in the context of directed categories.

Theorem 5.8. Let C be a graded directed category with C_{0} = ⨁_{x∈Ob C} C(x, x) being a self-injective algebra . Then:

(1) C is standardly stratified if and only if C is a projective C_{0}-module.

(2) C is a Koszul category if and only if C is standardly stratified and quasi-Koszul.

(3) If C is standardly stratified, then a graded C-module M generated in degree 0 is Koszul if and only if it is a quasi-Koszul C-module and a projective C_{0}-module.

Proof. Take an arbitrary pair of objects x, y ∈ Ob C. If C is standardly stratified, then C(x, y) is either 0 (a zero projective module) or a projective C(y, y)-module by the previous theorem. Notice that each C(x, y)_{i} is a C(y, y)-module since C(y, y) ⊆ C_{0}, and we have the decomposition C(x, y) = ⨁_{i⩾0} C(x, y)_{i}. Therefore, C(x, y)_{i} is a projective C(y, y)-module, and hence a projective C_{0}-module since only the block C(y, y) of C_{0} acts on C(x, y)_{i} nontrivially. In conclusion, C_{i} = ⨁_{x,y∈Ob C} C(x, y)_{i} is a projective C_{0}-module. Conversely, if C_{i} are projective C_{0}-modules for all i ⩾ 0, then each C(x, y)_{i}, and hence C(x, y) are projective C(y, y)-modules, so C is standardly stratified again by the previous theorem. The first statement is proved.

If C is Koszul, then it is quasi-Koszul. Moreover, each C_{i} is a projective C_{0} module, see the last paragraph of Section 2. Therefore, C is a projective C_{0}-module, and hence is standardly stratified. Conversely, if C is standardly stratified and quasi-Koszul, then C is a projective C_{0}-module, so C_{0} is linear by Corollary 2.18, and hence C is Koszul. This proves the statement.

The last part is an immediate result of Corollary 2.18. □

To each graded category C we can associate an associated quiver Q in the following way: the vertices of Q are exactly the objects in C; if C(x, y)_{1} ≠ 0, then we put an arrow from x to y with x, y ranging over all objects in C. Clearly, the associated quiver of C is completely determined by C_{0} and C_{1}. There is no loop in Q since C(x, x)_{1} = 0 for each x ∈ Ob C.

Proposition 5.9. Let C be a graded category with C_{0} = ⨁_{x∈Ob C} C(x, x) and Q be its associated quiver. Then C is a directed category if and only if Q is an acyclic quiver.

Proof. Assume that C is directed. By the definition, there is a partial order ⩽ on Ob C such that C(x, y) ≠ 0 only if x ⩽ y for x, y ∈ Ob C. In particular, C(x, y)_{1} ≠ 0 only if x < y. Therefore, an arrow x → y exists in Q only if x < y. If there is an oriented cycle

x_{1} → x_{2} → … → x_{n} → x_{1}

in Q, then x_{1} < x_{2} < … < x_{n} < x_{1}, which is impossible. Hence Q must be acyclic.

Conversely, if Q is acyclic, we then define x ⩽ y if and only if there is a directed path (including trivial path with the same source and target) from x to y in Q for x, y ∈ Ob C. This gives rise to a well defined partial order on Ob C. We claim that C is a directed category with respect to this partial order, i.e., C(x, y) ≠ 0 implies x ⩽ y. Since it holds trivially for x = y, we assume that x ≠ y. Take a morphism 0 ≠ α ∈ C(x, y) with a degree n (this is possible since C(x, y) is a non-zero graded space). Since C_{n} = C_{1}·. . .·C_{1}, we can express α as a linear combination of composite morphisms

\xymatrix{x = x_{0} \ar[r]^{α_{1}} & x_{1} \ar[r]^{α_{2}} & … \ar[r]^{α_{n}} & x_{n} = y}

with each α_{i} ∈ C_{1} and all x_{i} being distinct (since endomorphisms in C are contained in C_{0}). Therefore, there is a nontrivial directed path

x = x_{0} → x_{1} → x_{2} → … → x_{n} = y

in Q, and we have x < x_{1} < x_{2} < … < y, which proves our claim. □

Let C be a graded directed category. We define the free cover Ĉ of C by using the associated quiver Q. Explicitly, Ĉ has the same objects and endomorphisms as C. For each pair of objects x ≠ y we construct Ĉ(x, y) as follows. let Γ_{x,y} be the set of all paths from x to y in Q. In the case that Γ_{x,y} = ∅ we let Ĉ(x, y) = 0. Otherwise, take an arbitrary path γ ∈ Γ_{x,y} pictured as below

x → x_{1} → x_{2} → … → x_{n−1} → y,

and define (x, y)_{γ} to be

C(x_{n−1}, y)_{1} ⊗_{C(x_{n−1},x_{n−1})} C(x_{n−2}, x_{n−1})_{1} ⊗_{C(x_{n−2},x_{n−2})} … ⊗_{C(x_{1},x_{1})} C(x, x_{1})_{1}.

Finally, we define

Ĉ(x, y) = ⨁_{γ∈Γ_{x,y}}(x, y)_{γ}.

It is clear that Ĉ is also a graded category with Ĉ_{0} = C_{0} and Ĉ_{1} = C_{1}. Therefore, Ĉ has the same associated quiver as that of C and is also a directed category by the above lemma. Actually, if two grade categories C and D have the same degree 0 and degree 1 components, then one is a directed category if and only if so is the other.

Theorem 5.10. Let C be a directed Koszul category with C_{0} being a self-injective algebra, then the Yoneda category E = E(C_{0}) is also directed and Koszul.

Proof. Applying the Koszul duality (Theorem 4.1) we know that E is a Koszul category. What we need to show is that E is a directed category as well. Since C is standardly stratified, pd_{C} C_{0} < ∞. Therefore, all morphisms in E spans a finitedimensional space by Lemma 4.3. In particular, for x, y ∈ Ob E, dim_{k} E(x, y) < ∞. Therefore, E is a locally finite k-linear category.

Let ⩽ be the partial order on Ob C with respect to which C is directed. This partial order gives a partial order on Ob E as well because Ob E = Ob C. We claim that E is directed with respect to this partially ordered set, i.e., if x ≰ y are two distinct objects in E, then E(x, y) = 0.

Since E is the Yoneda category of C, E(x, y) = 1_{y}E1_{x} ≅ Ext_{C}^{∗}(C_{0}1_{x}, C_{0}1_{y}). But C is a Koszul category, so it is standardly stratified by the second statement of Theorem 5.8. Therefore, C_{i} are projective C_{0}-modules for all i ⩾ 0 by the first part of this theorem. By Lemma 2.14, all Ω^{i}(C_{0}1_{x})_{i} are projective C_{0}-modules. By Lemma 2.10, we have

E(x, y)_{i} ≅ Ext_{C}^{i}(C_{0}1_{x}, C_{0}1_{y}) ≅ Hom_{C}(Ω^{i}(C_{0}1_{x}), C_{0}1_{y}).

Observe that C_{0}1_{y} is only supported on the object y and y ≱ x. If we can prove the statement that each Ω^{i}(C_{0}1_{x}) is only supported on objects z with z ⩾ x, then our claim is proved.

Clearly, Ω^{0}(C_{0}1_{x}) = C_{0}1_{x} = C(x, x) is only supported on x, so the statement is true for i = 0. Now suppose that Ω^{n}(C_{0}1_{x}) is only supported on objects z ⩾ x and consider Ω^{n+1}(C_{0}1_{x}). Let S be the set of objects z such that the value 1_{z}Ω^{n}(C_{0}1_{x}) of Ω^{n}(C_{0}1_{x}) on z is non-zero. Then we can find a short exact sequence:

\xymatrix{0 \ar[r] & N \ar[r] & ⨁_{z∈S}(C1_{z})^{m_{z}} \ar[r]^{p} & Ω^{n}(C_{0}1_{x}) \ar[r] & 0}

such that the map p gives a surjection p_{z} : (1_{z}C1_{z})^{m_{z}} → 1_{z}Ω^{n}(C_{0}1_{x}) for z ∈ S. Thus p is a surjection and Ω^{n+1}(C_{0}1_{x}) is a direct summand of N. Notice that all C1_{z} are supported only on objects w ⩾ z, and z ⩾ x by the induction hypothesis. Therefore, the submodule Ω^{n+1}(C_{0}1_{x}) ⊆ N ⊆ ⨁_{z∈S}(C1_{z})^{m_{z}} is only supported on objects w ⩾ x. Our statement is proved by induction. This finishes the proof. □

Let A be the category of directed Koszul categories C with C_{0} = ⨁_{x∈Ob C} C(x, x) being self-injective algebras. This theorem tells us that the homological dual functor E is also a dual functor from A to itself. For a fixed category C ∈ Ob A, since standard modules of C are exactly indecomposable summands of C_{0}, E interchanges standard (indecomposable projective, resp.) C-modules and indecomposable projective (standard, resp.) E(C_{0})-modules bijectively.

The condition that C_{0} is exactly the space spanned by endomorphisms in C is crucial since it implies that standard modules are precisely indecomposable summands of C_{0}. The following example tells us that without this assumption, the Yoneda category E(C_{0}) might not be directed even if C is a Koszul directed category.

Example 5.11. Let C be the following category. Put an order x < y on the objects and the following grading on morphisms: C_{0} = ⟨1_{x}, 1_{y}, β⟩, C_{1} = ⟨α⟩.

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

This category is directed obviously. It is standardly stratified (actually hereditary) with Δ_{x} ≅ k_{x} and Δ_{y} ≅ k_{y}. By the exact sequence

\xymatrix{0 \ar[r] & k_{y}[1] \ar[r] & C \ar[r] & C_{0} \ar[r] & 0,}

C_{0} is a linear module. But Δ_{x} ⊕Δ_{y} ≇ C_{0}. Furthermore, Δ_{x} ⊕Δ_{y} is not linear since from the short exact sequence

\xymatrix{0 \ar[r] & k_{y}[1] ⊕ k_{y} \ar[r] & C \ar[r] & Δ_{x} ⊕ Δ_{y} \ar[r] & 0}

we find that Ω(Δ_{x} ⊕ Δ_{y}) is not generated in degree 1.

By computation we get the Yoneda category D = E(C_{0}) pictured as below, with relation α · β = 0.

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

This is not a directed category with respect to the order x < y. However, we check: P_{y} = Δ′_{y} = D_{0}1_{y} and Δ′_{x} = k_{x} ≅ P_{x}/P_{y}. Therefore, Δ′_{x} ⊕ Δ′_{y} ≅ D_{0}, and D is standardly stratified. The exact sequence

\xymatrix{0 \ar[r] & P_{y}[1] \ar[r] & D \ar[r] & D_{0} \ar[r] & 0}

tells us that D_{0} is a linear D-module. Therefore, D is standardly stratified and Koszul, but not directed.

The k-linearization of a finite EI category is a directed category by definition. However, this theorem does not hold in the context of finite EI categories. That is, if E is a Koszul finite EI category, E(kE_{0}) might not to be the category algebra of a finite EI category, as illustrated in the next example:

Example 5.12. Let E be the following finite EI category with three objects whose endomorphism groups are all trivial. Put a grading on E with E_{0} = {1_{x}, 1_{y}, 1_{z}}, E_{1} = {α, β} and E_{2} = {βα}. Then (kE_{0}) ≅ kE/(βα) is not the category algebra of a finite EI category.

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

There is a close relation between the classical Koszul theory and our generalized Koszul theory in the context of graded directed categories. Let C be a graded directed category. We then define a subcategory D of C by replacing all endomorphism rings in C by k · 1, the span of the identity endomorphism. Explicitly, Ob D = Ob C; for x, y ∈ Ob D, D(x, y) = k⟨1_{x}⟩ if x = y and D(x, y) = C(x, y) otherwise. Clearly, D is also a graded directed category with D_{i} = C_{i} for every i ⩾ 1. Observe that the degree 0 component D_{0} is semisimple.

Theorem 5.13. Let C be a graded directed Koszul category and define the subcategory D as above. If M is a linear C-module, then M ↓_{D}^{C} is a linear D-module. In particular, D is a Koszul category in the classical sense.

Proof. We prove the conclusion by induction on the size of Ob C. If the size of Ob C is 1, the conclusion holds trivially. Now suppose that the conclusion is true for categories with at most n objects and let C be a graded directed category with n + 1 objects. Take x to be a minimal object in C and define C_{x} (D_{x}, resp.) to be the full subcategory of C (D, resp.) formed by removing x from it. Clearly C_{x} and D_{x} have n objects.

The following fact, which is well known in the context of finite EI categories (see [22]), is essential in the proof.

Fact: Every graded C_{x}-module N can be viewed as a C-module with N(x) = 0 by induction. Conversely, every graded C-module M with M(x) = 0 can also be viewed as a C_{x}-module by restriction. Furthermore, if M(x) = 0, then Ω^{i}(M)(x) = 0 for all i ⩾ 0. The above induction and restriction preserves projective modules: a projective C_{x}-module is still projective when viewed as a C-module; conversely, a projective C-module P with P (x) = 0 is still projective viewed as a C_{x}-module. Therefore, a graded C-module M with M(x) = 0 is linear if and only if it is linear as a C_{x}-module. All these results hold for the pair (D, D_{x}) similarly.

By this fact, we only need to handle linear C-modules M with M(x) ≠ 0. Indeed, if M(x) = 0, then M is also linear regarded as a C_{x}-module. By the induction hypothesis, M ↓_{D_{x}}^{C_{x}} is a linear D_{x}-module. By the above fact, M ↓_{D}^{C} is a linear D-module. Thus the conclusion is true for linear C-modules M with M(x) = 0.

Firstly we consider the special case M = C_{0}1_{x} = C(x, x) which is concentrated on x when viewed as a C-module. It is clear that

Ω(C_{0}1_{x}) ↓_{D}^{C}= Ω(D_{0}1_{x})

as vector spaces since for each pair u ≠ v ∈ Ob C, C(u, v) = D(u, v), and

C_{0}1_{x} ↓_{D}^{C}≅ (D_{0}1_{x})^{m} ≅ k_{x}^{m},

where m = dim_{k} C(x, x). Since C_{0}1_{x} is a linear C-module, Ω(C_{0}1_{x})[−1] is a linear C-module supported on Ob C_{x}. By the induction hypothesis, Ω(D_{0}1_{x})[−1] = Ω(C_{0}1_{x}) ↓_{D}^{C} [−1] is a linear D_{x}-module, and hence a linear D-module. Therefore, D_{0}1_{x}, and hence C_{0}1_{x} ↓_{D}^{C}≅ (D_{0}1_{x})^{m} are linear D-modules. In the case that y ≠ x, D_{0}1_{y} is a direct summand of C_{0}1_{y} ↓_{D}^{C}. It is linear viewed as a D_{x}-module by the induction hypothesis, and hence is a linear D-module. Consequently, D_{0} is a linear D-module, so D is a Koszul category in the classical sense.

Now let M be an arbitrary linear C-module with M(x) ≠ 0. Consider the exact sequence

(5.1)  \xymatrix{0 \ar[r] & ΩM ↓_{D}^{C} \ar[r] & P ↓_{D}^{C} \ar[r]^{p} & M ↓_{D}^{C} \ar[r] & 0}

induced by

\xymatrix{0 \ar[r] & ΩM \ar[r] & P \ar[r]^{p} & M \ar[r] & 0.}

The structures of C and D give the following exact sequence:

\xymatrix{0 \ar[r] & _{D}D \ar[r] & _{D}C \ar[r] & ⨁_{x∈Ob C} k_{x}^{m_{x}} \ar[r] & 0}

where k_{x} ≅ D_{0}1_{x} and m_{x} = dim_{k} C(x, x) − 1. Since P ↓_{D}^{C}∈ add(_{D}C), the above sequence gives us a corresponding sequence for P ↓_{D}^{C}:

(5.2)  \xymatrix{0 \ar[r] & P′ \ar[r]^{ι} & P ↓_{D}^{C} \ar[r] & T \ar[r] & 0,}

here P′ is a projective D-module and T ∈ add(D_{0}). Both of them are generated in degree 0.

Putting sequences 5.1 and 5.2 together we get

\xymatrix{0 \ar[r] & Ω(M′) \ar[r] \ar[d]^{φ} & P′ \ar[r]^{p∘ι} \ar[d]^{ι} & M′ \ar[d] \ar[r] & 0 \\ 0 \ar[r] & Ω(M) ↓_{D}^{C} \ar[r] & P ↓_{D}^{C} \ar[r]^{p} \ar[d] & M ↓_{D}^{C} \ar[r] \ar[d] & 0 \\  &  & T \ar[r] & M ↓_{D}^{C} /M′}

where M′ = (p ∘ ι)(P′). Notice that p and ι both are injective restricted to degree 0 components. Therefore p ∘ ι is also injective restricted to the degree 0 component of P′ (actually it is an isomorphism restricted to the degree 0 component). Consequently, P′ is a projective cover of M′ and the kernel of p∘ι is indeed Ω(M′).

We claim that φ is an isomorphism and hence Ω(M′) ≅ (ΩM) ↓_{D}^{C}. It suffices to show T ≅ M ↓_{D}^{C} /M′ by the snake lemma. First, since T is concentrated in degree 0 in sequence 5.2, (P ↓_{D}^{C})_{i} = P′_{i} and

M′_{i} = (p ∘ ι)(P′_{i}) = p((P ↓_{D}^{C})_{i}) = (M ↓_{D}^{C})_{i}, i > 0.

Therefore, M ↓_{D}^{C} /M′ is concentrated in degree 0. Furthermore, since M is linear, (P ↓_{D}^{C})_{0} = P_{0} = M_{0} = (M ↓_{D}^{C})_{0}, and P′_{0} = M′_{0} because p ∘ ι restricted to P′_{0} is an isomorphism as well. We deduce that

T ≅ (P ↓_{D}^{C})_{0}/P′_{0} ≅ (M ↓_{D}^{C})_{0}/M′_{0} = M ↓_{D}^{C} /M′,

exactly as we claimed.

Now consider the rightmost column of the above diagram. Clearly, the bottom term M ↓_{D}^{C} /M′ ≅ T ∈ add(D_{0}) is linear since we just proved that D_{0} is linear. The C-module (ΩM)[−1] is linear since M is supposed to be linear. Moreover, because x is minimal and M is generated in degree 0, M(x) ⊆ M_{0} and hence (ΩM)[−1](x) = (ΩM)(x) = 0. Therefore, (ΩM)[−1] is a linear C-module supported on Ob C_{x}, so it is also a linear C_{x}-module. By the induction hypothesis, (ΩM′)[−1] ≅ (ΩM)[−1] ↓_{D}^{C} is linear viewed as a D_{x}-module, and hence linear as a D-module. Thus the top term M′ is a linear D-module since as a homomorphic image of P′ (which is generated in degree 0) it is generated in degree 0 as well. By Proposition 2.9, M ↓_{D}^{C} is also a linear D-module since D_{0} is semisimple by our construction. The conclusion follows from induction. □

The converse of the above theorem is also true.

Theorem 5.14. Let C be a graded directed category and construct the subcategory D as before. Suppose that D is Koszul in the classical sense. Let M be a graded C-module generated in degree 0 such that Ω^{i}(M)_{i} are projective C_{0}-modules for all i ⩾ 0. Then M is a linear C-module whenever M ↓_{D}^{C} is a linear D-module.

Proof. We use the similar technique to prove the conclusion. Notice that we always assume that C_{0} = ⨁_{x∈Ob C} C(x, x). If C has only one object, then linear modules are exactly projective modules generated in degree 0 and the conclusion holds. Suppose that it is true for categories with at most n objects. Let C be a category of n + 1 objects and take a minimal object x. Define C_{x} and D_{x} as before. As in the proof of last theorem, a graded C-module M with M(x) = 0 is linear if and only if it is linear viewed as a C_{x}-module by restriction. and the same result holds for the pair (D, D_{x}). In particular, D_{x} is a Koszul category. Therefore, we only need to show that an arbitrary graded C-module M which is generated in degree 0 and satisfies the following conditions is linear: Ω^{i}(M)_{i} is a projective C_{0}-module for each i ⩾ 0; M(x) ≠ 0; and M ↓_{D}^{C} is a linear D-module.

Let M be such a C-module and consider the commutative diagram:

(5.3)  [figure: K; K; 0; 0; Ω(M ↓_{D}^{C}); P; φ; M ↓_{D}^{C}; 0; 0; (ΩM) ↓_{D}^{C}; P̃ ↓_{D}^{C}; M ↓_{D}^{C}; 0]

where P and P̃ are projective covers of M ↓_{D}^{C} and M respectively. Since M_{0} is a projective C_{0}-module, P_{0} = M_{0} = (M ↓_{D}^{C})_{0} = (P̃ ↓_{D}^{C})_{0} as vector spaces, and the induced map φ restricted to P_{0} is an isomorphism. Therefore φ is surjective since both P and P̃ ↓_{D}^{C} are generated in degree 0. Let K be the kernel of φ.

We have the following exact sequence similar to sequence 5.2:

\xymatrix{0 \ar[r] & P′ \ar[r] & P̃ ↓_{D}^{C} \ar[r]^{p̃} & T \ar[r] & 0,}

where P′ is a projective D-module such that P′_{i} = (P̃ ↓_{D}^{C})_{i} for every i ⩾ 1, and T ∈ add(D_{0}).

Let P′′ be a projective cover of T (as a D-module). Then we obtain:

(5.4)  [figure: K; K; 0; P′; α; P; p; φ; P′′; p′′; 0; 0; P′; P̃ ↓_{D}^{C}; p̃; T; 0].

We give some explanations here. Since P is a projective D-module and the map p′′ is surjective, the map p̃ ∘ φ factors through p′′ and gives a map p : P → P′′. Restricted to degree 0 components, p′′ and φ (see diagram 5.3) are isomorphisms and p̃ is surjective. Thus p restricted to the degree 0 components is also surjective. But P′′ is generated in degree 0, so p is surjective. Since P_{0} = (P̃ ↓_{D}^{C})_{0} and P′′_{0} = T_{0} = T, α restricted to the degree 0 components is an isomorphism, and hence an isomorphism of projective kD-modules (notice that the middle row splits since P′′ is a projective D-module, so the kernel should be a projective D-module generated in degree 0). By the snake Lemma, the kernel of p′′ is also K up to isomorphism.

Let J = ⨁_{i⩾1} D_{i}. Since D_{0} is supposed to be a linear D-module, J[−1] ≅ Ω(D_{0})[−1] is a linear D-module, too. Consider the leftmost column in diagram 5.3. The top term K[−1] is a linear D-module since K ≅ P′′/T ≅ P′′/P′′_{0} ∈ add(J). The middle term Ω(M ↓_{D}^{C})[−1] is linear as well since M ↓_{D}^{C} is supposed to be linear. By Proposition 2.9, the bottom term (ΩM)[−1] ↓_{D}^{C} must be linear.

Since M is generated in degree 0 and x is a minimal object, M(x) ⊆ M_{0}, so M(x) = (M ↓_{D}^{C})(x) ⊆ (M ↓_{D}^{C})_{0} as vector spaces. Similarly, P (x) ⊆ P_{0} and P (x) ≅ (M ↓_{D}^{C})(x), so Ω(M ↓_{D}^{C})(x) = 0. Consequently, (ΩM)[−1] is supported on Ob C_{x} by observing the leftmost column of diagram 5.3. Moreover, we can show as in the proof of Theorem 5.10 that all of its syzygies are supported on Ob C_{x}, and Ω^{i}((ΩM)[−1])_{i} = Ω^{i+1}(M)_{i+1} are projective (C_{x})_{0}-modules. Therefore, applying the induction hypothesis to (ΩM)[−1] supported on Ob C_{x} and (ΩM)[−1] ↓_{D}^{C} supported on Ob D_{x}, we conclude that (ΩM)[−1] is a linear C_{x}-module, and hence a linear C-module. Clearly, M is a linear C-module since it is generated in degree 0. The conclusion follows from induction. □

Remark 5.15. We remind the reader that in the previous two theorems we do not require C_{0} = ⨁_{x∈Ob C} C(x, x) to be a self-injective algebra. By our construction, D_{0} ≅ ⨁_{x∈Ob D} k_{x} is a semisimple algebra.

Assuming that C_{0} is self-injective, we get the following nice correspondence.

Theorem 5.16. Let C be a graded directed category with C_{0} being a self-injective algebra and construct D as before. Then:

(1) C is a Koszul category in our sense if and only if C is standardly stratified and D is a Koszul category in the classical sense.

(2) If C is a Koszul category, then a graded C-module M generated in degree 0 is linear if and only if M ↓_{D}^{C} is a linear D-module and M is a projective C_{0}-module.

Proof. If C is Koszul in our sense, then it is standardly stratified by (2) of Theorem 5.8, and D is Koszul in the classical sense by Theorem 5.13. Conversely, if D is Koszul in the classical sense, then C_{0} ↓_{D}^{C}∈ add(D_{0}) is a linear D-module. If C is furthermore standardly stratified, then it is a projective C_{0}-module by Theorem 5.8. Therefore, all Ω^{i}(C_{0})_{i} are projective C_{0}-modules according to Lemma 2.14. Thus C_{0} is a linear C-module by Theorem 5.14, and hence C is a Koszul category. This proves the first statement.

Now suppose that C is Koszul. Then C is a projective C_{0}-module. If M is a linear C-module, M ↓_{D}^{C} is a linear D-module by Theorem 5.13. Furthermore, M is a projective C_{0}-module by Corollary 2.18. Conversely, if M is a projective C_{0}- module and M ↓_{D}^{C} is a linear D-module, then by Lemma 2.14 Ω^{i}(M)_{i} are projective C_{0}-modules for all i ⩾ 0. By Theorem 5.14 M is a linear C-module. □

6. Finite EI Categories

When applying the generalized Koszul theory to a directed category C in the previous section, we take for granted that there is already a grading on the morphisms in C such that the degree 0 component is formed precisely by endomorphisms in C. But in practice it is very hard to find such a grading for C. Actually, we do not even know the existence of such gradings in general. In this section we will focus on finite EI categories, whose k-linearizations form a type of directed categories with combinatorial properties. These properties can be used to define a length grading on the set of morphisms and completely determine whether an arbitrary finite EI category can be graded by this length grading.

In this section we only consider skeletal and connected finite EI categories E, i.e., for every pair x, y ∈ Ob E, there is a chain of objects x_{0} = x, x_{1}, x_{2}, …, x_{n} = y such that either E(x_{i}, x_{i+1}) ≠ ∅ or E(x_{i+1}, x_{i}) ≠ ∅ for every 0 ⩽ i ⩽ n − 1.

First we introduce some results from [12]. A morphism α in E is called unfactorizable if α is not an automorphism, and whenever there is a decomposition α = α_{1} ∘ α_{2}, either α_{1} or α_{2} is an automorphism. The composite morphism of an unfactorizable morphism and an automorphism is still unfactorizable. Therefore, all unfactorizable morphisms from an object x to another object y form an (Aut_{E}(y), Aut_{E}(x)) bi-set. Every non-isomorphism can be expressed as a composite of unfactorizable morphisms. This decomposition is not unique in general. We say a finite EI category E satisfies the Unique Factorization Property (UFP) if whenever a non-isomorphism α has two decompositions into unfactorizable morphisms:

\xymatrix{x = x_{0} \ar[r]^{α_{1}} & x_{1} \ar[r]^{α_{2}} & … \ar[r]^{α_{m}} & x_{m} = y \\ x = x_{0} \ar[r]^{β_{1}} & y_{1} \ar[r]^{β_{2}} & … \ar[r]^{β_{n}} & y_{n} = y}

then m = n, x_{i} = y_{i}, and there are h_{i} ∈ Aut_{E}(x_{i}) such that the following diagram commutes, 1 ⩽ i ⩽ n − 1:

[figure: x_{0}; α_{1}; id; x_{1}; α_{2}; h_{1}; …; α_{...}; h_{...}; x_{n−1}; α_{n}; h_{n−1}; x_{n}; id; x_{0}; β_{1}; x_{1}; β_{2}; …; β_{...}; x_{n−1}; β_{n}; x_{n}]

Finite EI categories with this property are called finite free EI categories by us. For every finite EI category E there is a unique (up to isomorphism) finite free EI category Ê (called the free EI cover) and a covering functor F̂ : Ê → E such that F̂ is the identity map restricted to objects, isomorphisms and unfactorizable morphisms. The functor F̂ induces a surjective algebra homomorphism φ : kÊ → kE. Therefore, kE ≅ kÊ/I, where I is the kernel of φ. We have the following description of I:

Lemma 6.1. The kÊ-ideal I as a vector space is spanned by elements of the form α̂ − β̂, where α̂ and β̂ are morphisms in Ê with F̂(α̂) = F̂(β̂).

Proof. Let U be the vector space spanned by elements α̂ − β̂ such that F̂(α̂) = F̂(β̂). Clearly, U ⊆ I and we want to show the other inclusion. Let x ∈ U. By the definition of category algebras, x can be expresses uniquely as ∑_{i=1}^{n} λ_{i}α_{i} where α_{i} are pairwise different morphisms in Ê and λ_{i} ∈ k. Then φ(∑_{i=1}^{n} λ_{i}α_{i}) = ∑_{i=1}^{n} λ_{i}F̂(α_{i}) = 0. Those F̂(α_{i}) are probably not pairwise different in E. By changing the indices if necessary, we can write the set {α_{i}}_{i=1}^{n} as a disjoint union of l subsets: {α_{1}, …, α_{s_{1}}}, {α_{s_{1}+1}, …, α_{s_{2}}} and so on, until {α_{s_{l−1}+1}, …, α_{s_{l}}} such that two morphisms have the same image under F̂ if and only if they are in the same set.

Now we have:

φ(x) = (λ_{1} + … + λ_{s_{1}})F̂(α_{s_{1}}) + … + (λ_{s_{l−1}+1} + … + λ_{s_{l}})F̂(α_{s_{l}}) = 0.

Therefore,

λ_{1} + … + λ_{s_{1}} = … = λ_{s_{l−1}+1} + … + λ_{s_{l}} = 0,

and hence

x = [λ_{2}(α_{2} − α_{1}) + … + λ_{s_{1}}(α_{s_{1}} − α_{1})] + …
+ [λ_{s_{l−1}+2}(α_{s_{l−1}+2} − α_{s_{l−1}+1}) + … + λ_{s_{l}}(α_{s_{l}} − α_{s_{l−1}+1})]

is contained in U. □

If E is a finite free EI category, we can put a length grading on its morphisms as follows: automorphisms and unfactorizable morphisms are given grades 0 and 1 respectively; if α is a factorizable morphism, then it can be expressed (probably not unique) as a composite α_{n}α_{n−1} … α_{2}α_{1} with all α_{i} being unfactorizable and we assign α grade n. This grading is well defined by the Unique Factorization Property of finite free EI categories. It is clear that this length grading cannot be applied to an arbitrary finite EI category. We say a finite EI category can be graded if this length grading is well defined on it. The following proposition gives us criterions to determine whether an arbitrary finite EI category can be graded.

Proposition 6.2. Let E be a finite EI category. Then the following are equivalent:

(1) E is a graded finite EI category.

(2) For each factorizable morphism α in E, whenever it has two factorizations α_{1} ∘…∘α_{m} and β_{1} ∘…∘β_{n} into unfactorizable morphisms, we have m = n.

(3) Let Ê be the free EI cover of E and F̂ : Ê → E be the covering functor. If two morphisms α̂ and β̂ in Ê have the same image under F̂, then they have the same length in Ê.

Proof. It is easy to see that if condition (2) holds, our grading works for E, and hence (1) is true. Otherwise, if a factorizable morphism α has two decompositions α_{n} ∘ … ∘ α_{1} and β_{m} ∘ … ∘ β_{1} with m ≠ n, then α should be assigned a grade n by the first decomposition, and a grade m by the second decomposition. Thus our grading cannot be applied to E. This proves the equivalence of (1) and (2).

Now let α be an arbitrary morphism in E which has two different decompositions α_{n} ∘ … ∘ α_{1} and β_{m} ∘ … ∘ β_{1} into unfactorizable morphisms. Since Ê is the free EI cover of E, these unfactorizable morphisms are also unfactorizable morphisms in Ê. Let α̂ and β̂ be the composite morphisms of these α_{i}’s and β_{i}’s in Ê respectively. Thus α̂ − β̂ is contained in U since they have the same image α under F̂. If (3) is true, then m = n since α̂ and β̂ have lengths m and n respectively. Therefore (3) implies (2). We can check that (2) implies (3) in a similar way. □

The following two lemmas are from [12].

Lemma 6.3. Let E be a finite free EI category and α : x → y be an unfactorizable morphism. Define H = Aut_{E}(y) and H_{0} = Stab_{H}(α). If |H_{0}| is invertible in k, then the cyclic module kEα is projective.

Proof. This is Lemma 5.2 of [12], where we assumed that the automorphism groups of all objects are invertible in k but only used the fact that |H_{0}| is invertible in k. Here we give a sketch of the proof. Let e = \frac{1}{|H_{0}|} ∑_{h∈H_{0}} h. Then e is well defined since |H_{0}| is invertible in k, and is an idempotent in kE. Now define a map φ : kEe → kEα by sending re to rα for r ∈ kE. We can check that φ is an kE-module isomorphism. Thus kEα is projective. See [12] for a detailed proof. □

Lemma 6.4. Let E be a finite free EI category and α : x → y and α′ : x′ → y′ be two distinct unfactorizable morphisms in E. Then kEα ∩ kEα′ = 0 or kEα = kEα′.

Proof. This is Lemma 5.1 of [12]. We give a sketch of the proof. Notice that kEα is spanned by all morphisms of the form βα with β : y → z being a morphism starting at y. Similarly, kEα′ is spanned by all morphisms of the form β′α′ with β′ : y′ → z′ being a morphism starting at y′. If x ≠ x′ or y ≠ y′, then by the Unique Factorization Property of finite free EI categories we conclude that the set Eα ∩ Eα′ = ∅, and the conclusion follows. If x = x′ and y = y′, then the set Eα coincides with the set Eα′ if and only if there is an automorphism h ∈ Aut_{E}(y) such that hα = α′ again by the UFP. Otherwise, we must have Eα ∩ Eα′ = ∅. The conclusion follows from this observation. □

Remark 6.5. The reader can check that the conclusion of Lemma 6.3 is true for any non-isomorphisms α in E by using the UFP. Moreover, a direct check shows that it is also true for automorphisms. Similarly, we can also prove that Lemma 6.4 still holds if we assume that α and α′ are two morphisms with the same target and source.

Theorem 5.8 has a corresponding version for finite EI categories.

Proposition 6.6. Let E be a graded finite EI category. Then kE is a Koszul algebra if and only if kE is a quasi-Koszul algebra and E is a standardly stratified category (in a sense defined in [20]) with respect to the canonical partial order on Ob E.

Proof. By the decompositions

kE_{0} = ⨁_{x∈Ob E} kAut_{E}(x), kE_{i} = ⨁_{x≠y∈Ob E} kE(x, y)_{i}, i > 0

we conclude that all kE_{i} are projective kE_{0}-modules if and only if kE(x, y)_{i} are projective kAut_{E}(y)-modules for all i ⩾ 0, x, y ∈ Ob E. Notice that kE(x, y)_{i} is spanned by morphisms from x to y with length i, and these morphisms form several orbits under the action of Aut_{E}(y). Suppose that there are n distinct orbits and take a representative α_{j} from each orbit. Then we have a decomposition kE(x, y)_{i} ≅ ⨁_{j=1}^{n} kAut_{E}(y)α_{j}. Thus kE(x, y)_{i} is a projective kAut_{E}(y)-module if and only if each kAut_{E}(y)α_{j} is a projective kAut_{E}(y)-module, and if and only if the stabilizer of α_{j} in Aut_{E}(y) has an order invertible in k. This happens if and only if E is standardly stratified by Theorem 2.5 in [20]. In conclusion, all kE_{i} are projective kE_{0}-modules if and only if E is standardly stratified in a sense defined in [20].

Notice that kE_{0} is the direct sum of several group algebras, and hence is selfinjective. If kE is Koszul, then it is quasi-Koszul by Theorem 2.17. Moreover, all kE_{i} are projective kE_{0}-modules (see the last paragraph of Section 2). Therefore, E is standardly stratified.

Conversely, if kE is standardly stratified, then all kE_{i} are projective kE_{0}-modules. By Corollary 2.18, kE_{0} is a linear kE-module if kE is quasi-Koszul. □

In the first paragraph of the above proof we have showed that a graded finite EI category E is standardly stratified in the sense of [20] if and only if its k-linearization as a graded directed category is standardly stratified in our sense. Actually this is still true for an arbitrary finite EI category by comparing Theorem 5.7 in this paper and Theorem 2.5 in [20]. This is not surprising since the k-linearization of E is precisely the associated category of the algebra kE.

Proposition 6.7. Let E be a finite EI category which might not be graded. Then E is standardly stratified if and only if M = ⨁_{x∈Ob E} Aut_{E}(x) viewed as a kE-module has finite projective dimension.

Proof. Consider the k-linearization of E, which is a directed category. By Theorem 5.8 and the remark we made in the paragraph before this proposition, we conclude that pd M < ∞.

Conversely, suppose that E is not standardly stratified. Then there is a nonisomorphism γ : t → y such that the order of H_{γ} is not invertible in the field k by Theorem 2.5 in [20], where H = Aut_{E}(y) and H_{γ} = Stab_{H}(γ). For this object y, define S to be the set of objects w such that there is a non-isomorphism β : w → y satisfying that |H_{β}| is not invertible in k. This set S is nonempty since t ∈ S. It is a poset equipped with the partial order inherited from the canonical partial order on Ob E. Take a fixed object z which is maximal in this set and define I_{>z} = {x ∈ Ob E | x > z}.

By our definition, for an arbitrary object x ∈ I_{>z} and a non-isomorphism α : x → y (if it exists), the group H_{α} ⩽ H has an order invertible in k. Therefore, the kH-module kHα is projective. Since the value of kE1_{x} on y is 0 or is spanned by all non-isomorphisms from x to y, and these non-isomorphisms form a disjoint union of H-orbits, we conclude that the value of kE1_{x} on y is a projective kHmodule (notice that we always view 0 as a zero projective module). With the same reasoning, we know that the value of kE1_{z} on y is not a projective kH-module.

Consider the kE-module L = kAut_{E}(z). We claim that pd L = ∞. If this is true, then pd M = ∞ since L is a direct summand of M. We prove this claim by showing the following statement: for each i ⩾ 1, every projective cover of Ω^{i}(L) is supported on I_{>z}; the value Ω^{i}(L)(y) of Ω(L) on y is non-zero and is not a projective kH-module. Clearly, Ω(L) is spanned by all non-isomorphisms starting from z and is supported on I_{>z}; Ω(L)(y), spanned by all non-isomorphisms from z to y, is non-zero. Moreover, Ω(L)(y) coincides with the value of kE1_{z} on y and is not a projective kH-module. Therefore our statement is true for i = 1.

Suppose that this statement is true for n, and let P be a projective cover of Ω^{n}(L). The exact sequence

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

gives rise to an exact sequence

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

Let us focus on the above sequences. Since Ω^{n}(L) is supported on I_{>z}, so are P and Ω^{n+1}(L). By the induction hypothesis Ω^{n}(L)(y) ≠ 0, Thus P (y) ≠ 0. But P is supported on I_{>z}, so P ∈ add(⨁_{x∈I_{>z}} kE1_{x}). Notice that the value of each kE1_{x} on y is zero or a nontrivial projective kH-module. Therefore, P (y) is a projective kH-module. Again by the induction hypothesis, Ω^{n}(L)(y) is not a projective kHmodule, so Ω^{n}(L)(y) ≇ P (y), and Ω^{n+1}(L)(y) is non-zero. It cannot be a projective kH-module. Otherwise, Ω^{n+1}(L)(y) is also an injective kH-module and hence the above sequence splits, so Ω^{n}(L)(y) as a summand of P (y) is a projective kH-module, too. But this contradicts the induction hypothesis.

We proved the induction hypothesis for Ω^{n+1}(L). Thus our statement and claim are proved. Consequently, pd M = ∞. □

Now we can prove:

Theorem 6.8. Let E be a finite free EI category. Then the following are equivalent: (1) pd kE_{0} ⩽ 1;

(2) kE is a Koszul algebra;

(3) E is standardly stratified;

(4) pd kE_{0} < ∞.

Proof. It is clear that pd kE_{0} = 0 if and only if E is a finite EI category with a single object since we only consider connected categories. In this situation, kE = kE_{0}, J = 0, and all statements are trivially true. Thus without loss of generality we suppose that pd kE_{0} ≠ 0.

Observe that pd kE_{0} = 1 if and only if Ω(kE_{0}) = J = ⨁_{i⩾1} kE_{i} is projective. Since J is spanned by all non-isomorphisms in E and each non-isomorphism can be written as a composition of unfactorizable morphisms, it is generated in degree 1. Thus kE_{0} is a linear kE-module, and (1) implies (2). By Proposition 6.6, (2) implies (3). The statements (3) and (4) are equivalent by Proposition 6.7.

Now we prove that (3) implies (1). If E is standardly stratified, then for every morphism α : x → y in E, the order of Stab_{H}(α) is invertible in k, where H = Aut_{E}(y). By Lemma 6.4, J is a direct sum of some kE-modules kEα_{i}’s with each α_{i} being unfactorizable. By Lemma 6.3, each kEα_{i} is projective. Therefore, J is also projective, i.e., pd kE_{0} = 1. □

This theorem and Theorem 5.13 give us a way to construct Koszul algebras in the classical sense. Indeed, let E be a standardly stratified finite free EI category and define D to be the subcategory formed by removing all non-identity automorphisms.

Then by Theorem 5.13 kD is a Koszul algebra in the classical sense since kE_{0} is a linear kE-module in the generalized sense by the previous theorem.

Let us get more information about the projective resolutions of kE_{0} for arbitrary finite free EI categories. In general, Ω(kE_{0}) ≅ J = ⨁_{i⩾1} kE_{i} is not projective, but it is still a direct sum of some kE-modules kEα’s with each α being unfactorizable by Lemma 6.4. Thus the projective resolutions of kE_{0} is completely determined by the projective resolutions of those kEα’s.

Lemma 6.9. Let E be a finite free EI category and α : x → y be an unfactorizable morphism. Grade the kE-module kEα by putting α in degree 1, namely, (kEα)_{1} = kAut_{E}(y)α. Then Ω(kEα) is 0 or is generated in degree 1, and Ω(kEα)_{1} = Ω(kEα)(y), the value of Ω(kEα) on y.

Proof. Let H = Aut_{E}(y) and H_{0} = Stab_{H}(α). If |H_{0}| is invertible in k, then by Lemma 6.3, kEα is a projective kE-module, so Ω^{i}(kEα) = 0 for all i ⩾ 1, in particular Ω(kEα) = 0. The conclusion is trivially true. Thus we only need to deal with the case that |H_{0}| is not invertible.

Consider the projective presentation

\xymatrix{0 \ar[r] & N \ar[r] & kE1_{y}[1] \ar[r]^{p} & kEα \ar[r] & 0}

where p maps 1_{y} to α. Since Ω(kEα) is isomorphic to a direct summand of N, it is enough to show that N is generated in degree 1, and N_{1} = N(y).

Notice that kE1_{y} is spanned by all morphisms in E with source y, and kEα is spanned by all morphisms in E of the form βα where β is a morphism in E with source y. We claim that N is spanned by vectors of the form β_{1}−β_{2} with β_{1}α = β_{2}α, where β_{1} and β_{2} are two morphisms with source y.

Clearly, every such difference is contained in N. Conversely, let v ∈ N. Then v can be written as ∑_{i=1}^{n} λ_{i}β_{i} with λ_{i} ∈ k and β_{i} being pairwise different morphisms with source y. By the definition of p, ∑_{i=1}^{n} λ_{i}β_{i}α = 0. Those β_{i}α might not be pairwise different in E. Now we apply the same technique used in the proof of Lemma 6.1. By changing the indices if necessary, we can group the same morphisms together and suppose that β_{1}α = … = β_{s_{1}}α, β_{s_{1}+1}α = … = β_{s_{2}}α and so on, until β_{s_{l−1}+1}α = … = β_{s_{l}}α.

We have:

p(v) = (λ_{1} + … + λ_{s_{1}})β_{s_{1}}α + … + (λ_{s_{l−1}+1} + … + λ_{s_{l}})β_{s_{l}}α = 0.

Therefore,

λ_{1} + … + λ_{s_{1}} = … = λ_{s_{l−1}+1} + … + λ_{s_{l}} = 0,

and hence

v = [λ_{2}(β_{2} − β_{1}) + … + λ_{s_{1}}(β_{s_{1}} − β_{1})] + …
+ [λ_{s_{l−1}+2}(β_{s_{l−1}+2} − β_{s_{l−1}+1}) + … + λ_{s_{l}}(β_{s_{l}} − β_{s_{l−1}+1})].

So v can be written as a sum of these differences.

Now we can prove the lemma. Take an arbitrary object z ∈ Ob E and consider the value N(z). If it is 0, the conclusion holds trivially. Suppose that N(z) ≠ 0. By the above description, N(z) is spanned by vectors β_{1} − β_{2} such that β_{1}, β_{2} are two morphisms from y to z, and β_{1}α = β_{2}α. By the equivalent definition of UFP described in Remark 6.5, there is an automorphism h ∈ Aut_{E}(y) such that β_{1} = β_{2}h and α = h^{−1}α. Therefore hα = α, and 1 − h ∈ N(y). Thus β_{1} − β_{2} = β(1 − h) ∈ kE · N(y). Since z is taken to be an arbitrary object, N is generated by N(y), which is clearly equal to N_{1}. □

From this lemma we can get:

Proposition 6.10. Let E be a finite free EI category, then Ext_{kE}^{2}(kE_{0}, kE_{0}) = 0.

Proof. Since Ω(kE_{0}) ≅ J, it is enough to show Ext_{kE}^{1}(J, kE_{0}) = 0. The conclusion holds trivially if J is projective. Otherwise, since J is the direct sum of some kEα’s with α being unfactorizable, by the above lemma we know that ΩJ is generated in degree 1.

Applying the functor Hom_{kE}(−, kE_{0}) to 0 → ΩJ → P → J → 0 we get

0 → Hom_{kE}(J, kE_{0}) → Hom_{kE}(P, kE_{0}) → Hom_{kE}(ΩJ, kE_{0}) → Ext_{kE}^{1}(J, kE_{0}) → 0.

Since all modules are generated in degree 1, the sequence

(6.1)  0 → Hom_{kE}(J, kE_{0}) → Hom_{kE}(P, kE_{0}) → Hom_{kE}(ΩJ, kE_{0})

is isomorphic to the sequence

0 → Hom_{kE_{0}}(J_{1}, kE_{0}) → Hom_{kE_{0}}(P_{1}, kE_{0}) → Hom_{kE_{0}}((ΩJ)_{1}, kE_{0})

obtained by applying the exact functor Hom_{kE_{0}}(−, kE_{0}) to the exact sequence 0 → (ΩJ)_{1} → P_{1} → J_{1} → 0. Thus the last map in sequence 6.1 is surjective, so Ext_{kE}^{1}(J, kE_{0}) = 0. □

The fact that ΩJ is generated in degree 1 implies Ext_{kE}^{2}(kE_{0}, kE_{0}) = 0. Actually the converse statement is also true. Indeed, consider the exact sequence 0 → ΩJ → P → J → 0. If Ext_{kE}^{2}(kE_{0}, kE_{0}) = 0, applying the exact functor Hom_{kE_{0}}(−, kE_{0}) we get the exact sequence

0 → Hom_{kE_{0}}(J, kE_{0}) → Hom_{kE_{0}}(P, kE_{0}) → Hom_{kE_{0}}(ΩJ, kE_{0}) → 0,

which is isomorphic to

0 → Hom_{kE_{0}}(J_{1}, kE_{0}) → Hom_{kE_{0}}(P_{1}, kE_{0}) → Hom_{kE_{0}}(ΩJ/J(ΩJ), kE_{0}) → 0

since both J and P are generated in degree 1. Applying the functor Hom_{kE_{0}}(−, kE_{0}) again, we recover 0 → ΩJ/J(ΩJ) → P_{1} → J_{1} → 0. Therefore, ΩJ/J(ΩJ) ≅ (ΩJ)_{1}, so ΩJ is generated in degree 1.

Finite free EI categories with quasi-Koszul category algebras have very special homological properties. For example:

Proposition 6.11. Let E be a finite free EI category. Then the following are equivalent:

(1) Ext_{kE}^{i}(kE_{0}, kE_{0}) = 0 for all i ⩾ 2;

(2) for every unfactorizable morphism α : x → y and i ⩾ 0, either Ω^{i}(kEα) are all 0, or they are all generated in degree 1 (in which case it is generated by Ω^{i}(kEα)(y));

(3) kE is a quasi-Koszul algebra.

Proof. If kE is a quasi-Koszul algebra, then

Ext_{kE}^{i}(kE_{0}, kE_{0}) = Ext_{kE}^{2}(kE_{0}, kE_{0}) · Ext_{kE}^{i−2}(kE_{0}, kE_{0})

for every i ⩾ 2. But Ext_{kE}^{2}(kE_{0}, kE_{0}) = 0 by Proposition 6.10, so (3) implies (1). Clearly, (1) implies (3).

Notice that kEα is a isomorphic to a direct summand of J ≅ Ω(kE_{0}). Thus we only need to prove the equivalence of the following two statements: (1’) Ext_{kE}^{i}(J, kE_{0}) = 0 for every i ⩾ 1; (2’) Ω^{i}(J) = 0 or is generated in degree 1 for every i ⩾ 1.

Since the technique we use is similar to that in the proof of Proposition 6.10, we only give a sketched proof. In the case that J is projective, i.e., E is standardly stratified, then (1’) and (2’) are trivially true, hence they are equivalent. Now suppose that J is not projective. From the proof of Proposition 6.10 and the paragraph after it we conclude that ΩJ is generated in degree 1 if and only if Ext_{kE}^{1}(J, kE_{0}) = 0. Replacing J by ΩJ (which is also generated in degree 1 either by the induction hypothesis or by the hypothesis Ext_{kE}^{1}(J, kE_{0}) = 0) and using the same technique, we get Ω^{2}(J) is generated in degree 1 if and only if Ext_{kE}^{2}(J, kE_{0}) = 0. The equivalence of (1’) and (2’), and hence the equivalence of (1) and (2), come from induction. □

The reader may guess that the category algebra of a finite free EI category is always quasi-Koszul in our sense because of the following reasons: Finite free EI categories generalize finite groups and acyclic quivers, for which the associated algebras are all quasi-Koszul; by Proposition 6.6 and Theorem 6.8, for an arbitrary finite EI category E, kE is Koszul if E is standardly stratified and one of the following condition holds: kE is quasi-Koszul, or E is a finite free EI category; and we have proved that Ext_{kE}^{2}(kE_{0}, kE_{0}) = 0 if E is a finite free EI category. Unfortunately, this conjecture is false, as shown by the following example.

Example 6.12. Let E be the following finite EI category where: Aut_{E}(x) = ⟨1_{x}⟩, Aut_{E}(z) = ⟨1_{z}⟩, Aut_{E}(y) = ⟨h⟩ is a group of order 2; E(x, y) = {α}, E(y, z) = {β} and E(x, z) = {βα}. The reader can check that E is a finite free EI category and then the length grading can be applied on it. Let k be an algebraically closed field with characteristic 2.

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

The indecomposable direct summands of kE and kE_{0} are:

P_{x} = \begin{matrix}x_{0} \\ y_{1} \\ z_{2} \end{matrix},  P_{y} = \begin{aligned}& y_{0} \\ y_{0} & & z_{1} \end{aligned},  P_{z} = z_{0},  kE_{0} ≅ x_{0} ⊕ z_{0} ⊕ \begin{matrix}y_{0} \\ y_{0} \end{matrix}.

We use indices to mark the degrees of composition factors. The reader should bear in mind that the two simple modules y appearing in P_{y} have the same degree.

Take the summand x_{0} of kE_{0}. By computation, we get

Ω(x_{0}) = \begin{matrix}y_{1} \\ z_{2} \end{matrix},  Ω^{2}(x_{0}) = y_{1},  Ω^{3}(x_{0}) = y_{1} ⊕ z_{2}.

Applying Hom_{kE}(−, kE_{0}) to the exact sequence

\xymatrix{0 \ar[r] & Ω^{3}(x_{0}) \ar[r] & P_{y}[1] \ar[r] & Ω^{2}(x_{0}) \ar[r] & 0}

we get Ext_{kE}^{3}(kE_{0}, kE_{0}) ≠ 0. Consequently, kE is not a quasi-Koszul algebra in our sense by the previous proposition.

We aim to characterize finite free EI categories with quasi-Koszul category algebras. For this goal, we make the following definition:

Definition 6.13. Let E be a finite EI category. An object x ∈ Ob E is called left regular if for every morphism α with target x, the stabilizer of α in Aut_{E}(x) has an order invertible in k. Similarly, x is called right regular if for every morphism β with source x, the stabilizer of β in Aut_{E}(x) has an order invertible in k.

Remark 6.14. We make some comments for this definition.

(1) If x ∈ Ob E is maximal, i.e., there is no non-isomorphisms with source x, then x is right regular by convention; similarly, if x is minimal, then it is trivially left regular.

(2) The category E is standardly stratified if and only if every object x ∈ Ob E is left regular; similarly, E^{op} is standardly stratified if and only if every object x ∈ Ob E is right regular (3) If E is a finite free EI category and x ∈ Ob E. Then x is left regular if and only if for every α with target x, the kE-module kEα is a left projective kEmodule. Similarly, x is right regular if and only if for every β with source x, the right kE-module β(kE) is a right projective kE-module.

Lemma 6.15. Let E be a finite free EI category and β : x → y be a morphism with x ∈ Ob E being right regular. Then there exists some idempotent e in kE such that β(kE) ≅ e(kE) as right kE-modules by sending e to β. In particular, β(kG)α ≅ e(kG)α as vector spaces for every morphism α with target x, where G = Aut_{E}(x).

Proof. Let G_{0} = Stab_{G}(α) and e = ∑_{g∈G_{0}} g/|G_{0}|. This is well defined since x is right regular. Then we can prove β(kE) ≅ e(kE) as right kE-modules in a way similar to the proof of Lemma 6.3. The isomorphism is given by sending er to βr for r ∈ kE. Since the image of e(kG)α ⊆ kE is exactly β(kE)α, we deduce that e(kG)α ≅ β(kG)α as vector spaces. □

Using these concepts, we can get a sufficient condition for the category algebra of a finite free EI category to be quasi-Koszul.

Theorem 6.16. Let E be a finite free EI category such that every object x ∈ Ob E is either left regular or right regular. Then kE is quasi-Koszul.

Proof. By the second statement of Proposition 6.11, it is enough to show that for each unfactorizable α : x → y and every i ⩾ 1, Ω^{i}(kEα) is 0 or generated by Ω^{i}(kEα)(y). Let H = Aut_{E}(y) and H_{0} = Stab_{H}(α). If |H_{0}| is invertible in k, then kEα is a projective kE-module, and the conclusion follows. So we only need to deal with the case that the order of H_{0} is not invertible in k.

By Lemma 6.9, Ω(kEα) is generated in degree 1, or equivalently, generated by its value Ω(kEα)(y) = 1_{y}Ω(kEα) on y. Now suppose that Ω^{i}(kEα) is also generated in degree 1, or equivalently, generated by its value Ω^{i}(kEα)(y) = 1_{y}Ω^{i}(kEα) on y, where i ⩾ 1. We claim that Ω^{i+1}(kEα) is generated by Ω^{i+1}(kEα)(y), which is clearly equal to Ω^{i+1}(kEα)_{1}. If this is true, then conclusion follows from Proposition 6.11.

Take an arbitrary object z ∈ Ob E such that E(y, z) ≠ ∅. (In the case E(y, z) = ∅, Ω^{s}(kEα)(z) = 0 for s ⩾ 0, and the claim is trivially true.) The morphisms in E(y, z) form a disjoint union of orbits under the right action of H. By taking a representative β_{i} from each orbit we have E(y, z) = ⨆_{i=1}^{n} β_{i}H. Since |H_{0}| is not invertible, y is not left regular. By the assumption, y must be right regular. Therefore, by the previous lemma, for each representative morphism β_{s}, 1 ⩽ s ⩽ n, there exist some idempotent e_{i} such that β_{s}(kE) ≅ e_{s}(kE) as right projective kEmodules, and β_{s}(kE)α ≅ e_{s}(kE)α as vector spaces.

Consider the exact sequence

\xymatrix{0 \ar[r] & Ω^{i+1}(kEα) \ar[r] & P^{i} \ar[r] & Ω^{i}(kEα) \ar[r] & 0,}

where we assume inductively that Ω^{i}(kEα) is generated in degree 1, or equivalently generated by its value on y. Thus P^{i} ∈ add(kE1_{y}[1]). Observe that the segment of a minimal projective resolution of the kE-module kEα

\xymatrix{P^{i+1} \ar[r] & P^{i} \ar[r] & … \ar[r] & P^{0} \ar[r] & kEα \ar[r] & 0}

induces a minimal projective resolution of the kH-module kHα:

\xymatrix{P^{i+1}(y) \ar[r] & P^{i}(y) \ar[r] & … \ar[r] & P^{0}(y) \ar[r] & kHα \ar[r] & 0.}

Thus Ω^{j}(kEα)_{1} = Ω^{j}(kEα)(y) = Ω_{kH}^{j}(kHα) for 1 ⩽ j ⩽ i + 1.

Applying the exact functor Hom_{kE}(kE1_{y}, −) to the exact sequence

(6.2)  \xymatrix{0 \ar[r] & Ω^{i+1}(kEα) \ar[r] & P^{i} \ar[r] & Ω^{i}(kEα) \ar[r] & 0},

we get an exact sequence

\xymatrix{0 \ar[r] & Ω^{i+1}(kEα)(y) \ar[r] & 1_{y}P^{i} \ar[r] & Ω^{i}(kEα)(y) \ar[r] & 0},

which can be identified with

\xymatrix{0 \ar[r] & Ω_{kH}^{i+1}(kHα) \ar[r] & P^{i}(y) \ar[r] & Ω_{kH}^{i}(kHα) \ar[r] & 0}.

Applying the exact functor Hom_{kH}(⨁_{s=1}^{n} kHe_{s}, −) to the above sequence, we have another exact sequence

(6.3)  0 → ⨁_{s=1}^{n} e_{s}Ω_{kH}^{i+1}(kHα) → ⨁_{s=1}^{n} e_{s}P^{i}(y) → ⨁_{s=1}^{n} e_{s}Ω_{kH}^{i}(kHα) → 0.

Since Ω^{i}(kEα) is generated by Ω^{i}(kEα)(y) = Ω_{kH}^{i}(kHα) by the induction hypothesis, the value of Ω^{i}(kEα) on z is ∑_{s=1}^{n} β_{s} · Ω_{kH}^{i}(kHα) (this is well defined as Ω_{kH}^{i}(kHα) ⊆ (kH)^{⊕m} for some m ⩾ 0). We check that this sum is actually direct by the UFP of E. In conclusion,

(6.4)  Ω^{i}(kEα)(z) = ⨁_{s=1}^{n} β_{s} · Ω_{kH}^{i}(kHα) ≅ ⨁_{s=1}^{n} e_{s}Ω_{kH}^{i}(kHα).

Similarly, the value of P^{i} on z is

(6.5)  P^{i}(z) = ⨁_{s=1}^{n} β_{s} · P^{i}(y) ≅ ⨁_{s=1}^{n} e_{s}P^{i}(y).

Restricted to z, sequence 6.2 gives rise to

(6.6)  \xymatrix{0 \ar[r] & Ω^{i+1}(kEα)(z) \ar[r] & P^{i}(z) \ar[r] & Ω^{i}(kEα)(z) \ar[r] & 0}.

On one hand, ⨁_{s=1}^{n} β_{s}Ω_{kH}^{i+1}(kHα) ⊆ Ω^{i+1}(kEα)(z). On the other hand, we have:

dim_{k} ⨁_{s=1}^{n} β_{s}Ω_{kH}^{i+1}(kHα) = dim_{k} ⨁_{s=1}^{n} e_{s}Ω_{kH}^{i+1}(kHα) by Lemma 6.15

= dim_{k} ⨁_{s=1}^{n} e_{s}P^{i}(y) − dim_{k} ⨁_{s=1}^{n} e_{s}Ω_{kH}^{i}(kHα) by sequence 6.3

= dim_{k} P^{i}(z) − dim_{k} Ω^{i}(kEα)(z) by identities 6.4 and 6.5

= dim_{k} Ω^{i+1}(kEα)(z) by sequence 6.6.

Therefore, Ω^{i+1}(kEα)(z) = ⨁_{s=1}^{n} β_{s}Ω_{kH}^{i+1}(kHα) = ⨁_{s=1}^{n} β_{s}Ω_{kE}^{i+1}(kEα)(y) since Ω_{kH}^{i+1}(kHα) = Ω_{kE}^{i+1}(kEα)(y). That is, the value of Ω^{i+1}(kEα) on z is generated by Ω^{i+1}(kEα)(y). Since z is arbitrary, our claim holds, and the conclusion follows from induction. □

7. Standardly Stratified Algebras with Linear Standard Modules

Theorem 5.10 tells us that the Yoneda category E(C_{0}) of a directed Koszul category C is still a directed Koszul category, so is standardly stratified as well. Moreover, the homological dual functor E interchanges standard modules and indecomposable projective modules. Let A be a Koszul algebra which is standardly stratified with respect to a poset of orthogonal primitive idempotents ({e_{λ}}_{λ∈Λ}, ⩽). We may ask a similar question: is the Koszul dual algebra Γ = Ext_{A}^{∗}(A_{0}, A_{0}) standardly stratified with respect to ({e_{λ}}_{λ∈Λ}, ⩽) (or ({e_{λ}}_{λ∈Λ}, ⩽^{op})) as well? (Here we identify the primitive idempotents of A and Γ in the following way: let e be a primitive idempotent of A. Then it is also a primitive idempotent of A_{0}. Therefore, A_{0}e is a projective A_{0}-module, and Ext_{A}^{∗}(A_{0}e, A_{0}) is an indecomposable summand of Γ. This summand corresponds to a primitive idempotent of Γ, which we still denote by e.) This question has been studied in [1] [2] [8] [17] [18]. However, in all these papers A_{0} is supposed to be a semisimple algebra. By modifying the technique used in [1], we get a sufficient condition for the Yoneda algebra Γ to be standardly stratified with respect to the opposite order.

Throughout this section A is a graded finite-dimensional basic k-algebra with A_{0} being self-injective. We choose a complete set of orthogonal primitive idempotents {e_{λ}}_{λ∈Λ} and let ⩽ be a partial order on this set.

Theorem 7.1. If A is standardly stratified with respect to ⩽ such that all standard modules are concentrated in degree 0 and linear. Then A_{0} ≅ Δ and Γ = Ext_{A}^{∗}(A_{0}, A_{0}) is standardly stratified with respect to the poset ({e_{λ}}_{λ∈Λ}, ⩽^{op}), where Δ is the direct sum of all standard modules.

We show the first statement since it is relatively easier and leave the proof of the second statement to the end of this section. Let Δ_{λ} be a standard module with graded projective cover P_{λ}. Since Δ_{λ} is a linear A-module and concentrated in degree 0, Δ_{λ} = (Δ_{λ})_{0} is a projective A_{0}-module by Corollary 2.5. The surjection P_{λ} → Δ_{λ} induces a surjection (P_{λ})_{0} → Δ_{λ}, so Δ_{λ} is a summand of (P_{λ})_{0}, and hence is isomorphic to (P_{λ})_{0} since (P_{λ})_{0} is indecomposable. Put all these standard modules together we find A_{0} ≅ Δ.

Take a minimal element μ ∈ Λ and let e = e_{μ}. Let Λ_{1} = Λ \ {μ} and ϵ = ∑_{λ∈Λ_{1}} e_{λ}. Viewed as an idempotent of Γ, e is maximal with respect to ⩽^{op}. The basic idea to prove the second statement is to show that ΓeΓ is a projective Γmodule and the quotient algebra Γ/ΓeΓ is standardly stratified with respect to the poset ({e_{λ}}_{λ∈Λ_{1}}, ⩽^{op}). Then the conclusion follows from induction.

We collect a list of preliminary results in the following lemmas, where the algebra A is the same as in Theorem 7.1 if we do not specify it particularly.

Lemma 7.2. The algebra ϵAϵ is standardly stratified with respect to the poset ({e_{λ}}_{λ∈Λ_{1}}, ⩽) and has standard modules ϵA_{0}e_{λ}, λ ∈ Λ_{1}, which are all concentrated in degree 0. Moreover,

(ϵAϵ)_{0} = ⨁_{λ∈Λ_{1}} ϵA_{0}e_{λ} = ϵA_{0}ϵ = ϵA_{0}

is a self-injective algebra. If M is a linear A-module, then ϵM is a linear ϵAϵmodule. In particular, all standard modules of ϵAϵ are linear.

Proof. The algebra ϵAϵ has projective modules ϵAe_{λ}, λ ∈ Λ_{1}. Notice that each Ae_{λ} has a Δ-filtration and the standard module A_{0}e ≅ Δ_{μ} cannot appear in the filtration since e is a minimal primitive idempotent. We conclude that ϵAϵ has standard modules ϵΔ_{λ} ≅ ϵA_{0}e_{λ}, and ϵAϵ has a filtration formed by ϵA_{0}e_{λ}, λ ∈ Λ_{1}. This proves the first statement.

Clearly,

(ϵAϵ)_{0} = ϵA_{0}ϵ = ⨁_{λ∈Λ_{1}} ϵA_{0}e_{λ}.

We claim ϵA_{0}e = 0, which implies ϵA_{0} = ϵA_{0}ϵ + ϵA_{0}e = ϵA_{0}ϵ. Indeed,

ϵA_{0}e ≅ Hom_{A_{0}}(A_{0}ϵ, A_{0}e) ≅ ⨁_{λ∈Λ_{1}} Hom_{A}(Δ_{λ}, Δ_{μ}) = 0

Since A is standardly stratified and μ is minimal in Λ.

By Proposition 2.5 on page 35 of [3], the exact functor F = Hom_{A_{0}}(A_{0}ϵ, −) gives an equivalence between a subcategory M of A_{0}-mod and the category ϵA_{0}ϵ-mod. Since all projective (injective as well) modules A_{0}e_{λ} with λ ∈ Λ_{1} are contained in M, and F sends these projective (injective, resp) modules to projective (injective, resp) modules of ϵA_{0}ϵ. Consequently, ϵA_{0}ϵ = ϵA_{0} is self-injective.

Let M be a linear A-module and

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

be a linear projective resolution of M. That is, each P^{i} is generated in degree i. Applying the exact functor Hom_{A}(Aϵ, −) we get a linear projective resolution of ϵM as follows

\xymatrix{… \ar[r] & ϵP^{1} \ar[r] & ϵP^{0} \ar[r] & ϵM \ar[r] & 0.}

Thus ϵM is a linear ϵAϵ-module. Since the standard modules of ϵAϵ are indecomposable summands of ϵA_{0}, and A_{0} = Δ is a linear A-module, we conclude that every standard module of ϵAϵ is a linear ϵAϵ-module as well. □

Lemma 7.3. Let M be a linear A-module up to a degree shift. Then (1) M has a Δ-filtration.

(2) Ext_{A}^{i}(M, A_{0}e) = 0 for i > 0.

(3) [Ω^{i}(M) : Δ_{μ}] = 0 for each i ⩾ 1, where [Ω^{i}(M) : Δ_{μ}] is the number of Δ-filtration factors of M isomorphic to Δ_{μ}.

(4) M = AϵM if and only if [M : Δ_{μ}] = 0.

Proof. Without loss of generality we assume that M is linear. By Corollary 2.18 M is a projective A_{0}-module, and hence has a Δ-filtration since A_{0} = Δ.

To prove the second statement, it is enough to show Ext_{A}^{1}(M, A_{0}e) = 0. Indeed, for i > 1, by Corollary 2.5 and using Lemma 2.10 recursively we have

Ext_{A}^{i}(M, A_{0}e) ≅ Ext_{A}^{i−1}(ΩM, A_{0}e) ≅ … ≅ Ext_{A}^{1}(Ω^{i−1}(M), A_{0}e).

Since M is linear, Ω^{i−1}(M) is also linear up to a degree shift. Therefore we can replace M by Ω^{i−1}(M) and use induction.

The projective presentation 0 → ΩM → P → M → 0 gives a surjective map Hom_{A}(ΩM, A_{0}e) → Ext_{A}^{1}(M, A_{0}e). The syzygy ΩM is linear up to a degree shift and hence has a Δ-filtration. Since e is minimal and the algebra A is standardly stratified with respect to the poset ({e_{λ}}_{λ∈Λ}, ⩽), every Δ-filtration of JP = ⨁_{i⩾1} P_{i} has no factors isomorphic to A_{0}e. Therefore, the Δ-filtration of ΩM ⊆ ⨁_{i⩾1} P_{i} has no factors isomorphic to A_{0}e either. But Hom_{A}(Δ_{λ}, Δ_{μ}) = 0 for λ ≠ μ. Thus Hom_{A}(ΩM, Δ_{μ}) = 0, so Ext_{A}^{1}(M, A_{0}e) = 0. This proves the second statement.

To prove (3), it suffices to show [ΩM : Δ_{μ}] = 0 and the conclusion comes from induction. But this fact has been established in last paragraph.

Now we prove (4). Notice that AϵM is the trace of Aϵ in M. If [M : Δ_{μ}] = 0, then in particular M_{0} ∈ add(A_{0}ϵ), and M is in the trace of Aϵ since it is generated in degree 0. Conversely, if M = AϵM, then M is in the trace of Aϵ, i.e., M is a quotient module of some (Aϵ)^{⊕m}. Since [Aϵ : Δ_{μ}] = 0, we deduce that [M : Δ_{μ}] = 0. □

We define an operator Π on A-gmod as follows:

Π(M) = \{ \begin{matrix}AϵM & if M ≠ AϵM, \\ ΩM & if M = AϵM. \end{matrix}

Lemma 7.4. Suppose that M and AϵM are linear A-modules up to a common degree shift. Then

(1) For all i ⩾ 1,

Π^{i}(M) = \{ \begin{matrix}Ω^{i}(M) & if M = AϵM, \\ Ω^{i−1}(AϵM) & if M ≠ AϵM. \end{matrix}

(2) There is some l ∈ ℤ such that Π^{l}(M) = 0.

Proof. If M = AϵM, then Π(M) = ΩM. By the previous lemma, [ΩM : Δ_{μ}] = 0, so ΩM = AϵΩM and Π(ΩM) = Ω^{2}(M). Using induction we get Π^{i}(M) = Ω^{i}(M).

If M ≠ AϵM, then Π(M) = AϵM. Clearly, Aϵ(AϵM) = AϵM, so Π^{2}(M) = Ω(AϵM) by the definition. Applying statements (3) and (4) of the previous lemma recursively, we get Π^{i}(M) = Ω^{i−1}(AϵM).

To prove (2), it suffices to show that there is a number l ∈ ℤ such that Π^{l}(Π(M)) = Ω^{l}(Π(M)) = 0. Since Π(M) = ΩM or Π(M) = AϵM, both of which are linear up to a degree shift, Π(M) has a Δ-filtration by the above lemma. Since A is standardly stratified and finite-dimensional, A_{0} ≅ Δ has finite projective dimension and every Δ-filtration of Π(M) is of finite length. Thus the projective dimension of Π(M) is finite. □

Recall Γ = Ext_{A}^{∗}(A_{0}, A_{0}) and the indecomposable summands of Γ are precisely Ext_{A}^{∗}(A_{0}e_{λ}, A_{0}), λ ∈ Λ.

Lemma 7.5. Suppose that M and AϵM are linear A-modules up to a common degree shift. Then the trace ΓeExt_{A}^{∗}(M, A_{0}) of Γe in Ext_{A}^{∗}(M, A_{0}) is a projective Γ-module. Moreover,

dim_{k}Ext_{A}^{∗}(M, A_{0}) − dim_{k}ΓeExt_{A}^{∗}(M, A_{0}) = dim_{k}Ext_{ϵAϵ}^{∗}(ϵM, ϵA_{0}).

Proof. Without loss of generality we suppose that M is generated in degree 0 and prove this lemma by induction on the least number l such that Π^{l}(M) = 0, which always exists by Lemma 7.4. If l = 0, then M = 0 and the conclusion holds trivially. Now suppose that it holds for l ⩽ n − 1 and let M be a linear A-module for which this least number is n. There are two cases.

(1). If M ≠ AϵM, then Π(M) = AϵM. This happens if and only if [M : Δ_{μ}] ≠ 0. Since e = e_{μ} is a minimal primitive idempotent, those factors isomorphic to Δ_{μ} = A_{0}e can only appear as direct summands of M_{0} by the second statement of Lemma 7.3. Therefore we get an exact sequence 0 → AϵM → M → (A_{0}e)^{⊕a} → 0, where all terms are linear by our assumption. By Proposition 2.11, this sequence gives rise to:

0 → Ext_{A}^{∗}((A_{0}e)^{⊕a}, A_{0}) → Ext_{A}^{∗}(M, A_{0}) → Ext_{A}^{∗}(AϵM, A_{0}) → 0,

that is:

(7.1)  \xymatrix{0 \ar[r] & (Γe)^{⊕a} \ar[r] & Ext_{A}^{∗}(M, A_{0}) \ar[r]^{p} & Ext_{A}^{∗}(AϵM, A_{0}) \ar[r] & 0.}

By taking the trace of Γe in those terms appearing in this exact sequence, we get another exact sequence:

(7.2) \xymatrix{0 \ar[r] & (Γe)^{⊕a} \ar[r] & ΓeExt_{A}^{∗}(M, A_{0}) \ar[r] & ΓeExt_{A}^{∗}(AϵM, A_{0}) \ar[r] & 0.}

Notice that AϵM is linear, AϵAϵM = AϵM, and Π^{n−1}(AϵM) = Π^{n}(M) = 0. By the induction hypothesis, ΓeExt_{A}^{∗}(AϵM, A_{0}) is a projective Γ-module. Then the above sequence splits and ΓeExt_{A}^{∗}(M, A_{0}) is a projective Γ-module as well.

Comparing sequence 7.1 to sequence 7.2, we get:

dim_{k}Ext_{A}^{∗}(M, A_{0}) − dim_{k}ΓeExt_{A}^{∗}(M, A_{0})
= dim_{k}Ext_{A}^{∗}(AϵM, A_{0}) − dim_{k}ΓeExt_{A}^{∗}(AϵM, A_{0})
= dim_{k}Ext_{ϵAϵ}^{∗}(ϵAϵM, ϵA_{0})

by the induction hypothesis. But ϵAϵM = ϵM. Thus the conclusion is true for M by induction.

(2). If M = AϵM, i.e., [M : Δ_{μ}] = 0, then Π(M) = ΩM. Let P be a graded projective cover of M. Then P ∈ add(Aϵ) and P_{0} ∈ add(A_{0}ϵ). Since M is linear, M_{0} ≅ P_{0}, so Ext_{A}^{i}(M, A_{0}) = Ext_{A}^{i−1}(ΩM, A_{0}) for i > 0 by Lemma 2.10, and

Hom_{A}(M, A_{0}) ≅ Hom_{A_{0}}(M_{0}, A_{0}) ≅ Hom_{A_{0}}(P_{0}, A_{0}) ≅ Hom_{A}(P_{0}, A_{0}).

Therefore, the following exact sequence:

\xymatrix{0 \ar[r] & ⨁_{i⩾1} Ext_{A}^{i}(M, A_{0}) \ar[r] & Ext_{A}^{∗}(M, A_{0}) \ar[r] & Hom_{A}(M, A_{0}) \ar[r] & 0}

is isomorphic to

(7.3) \xymatrix{0 \ar[r] & Ext_{A}^{∗}(ΩM, A_{0}) \ar[r] & Ext_{A}^{∗}(M, A_{0}) \ar[r] & Hom_{A}(P_{0}, A_{0}) \ar[r] & 0.}

Applying the exact functor Hom_{Γ}(Γe, −) to the above sequence, we get

0 → eExt_{A}^{∗}(ΩM, A_{0}) → eExt_{A}^{∗}(M, A_{0}) → eHom_{A}(P_{0}, A_{0}) → 0.

Notice that eHom_{A}(P_{0}, A_{0}) = 0 since A_{0}e = Δ_{μ}, P_{0} ∈ add(A_{0}ϵ) and [A_{0}ϵ : Δ_{μ}] = 0. Thus eExt_{A}^{∗}(M, A_{0}) ≅ eExt_{A}^{∗}(ΩM, A_{0})[−1] (as graded Γ-modules) and

(7.4)  ΓeExt_{A}^{∗}(M, A_{0}) ≅ ΓeExt_{A}^{∗}(ΩM, A_{0})

Since [ΩM : Δ_{μ}] = 0, by Lemma 7.3, AϵΩM = ΩM. So ΓeExt_{A}^{∗}(ΩM, A_{0}) is a projective Γ-module by the induction hypothesis on Π(M) = ΩM, and ΓeExt_{A}^{∗}(M, A_{0}) is projective as well.

The exact sequence 0 → ΩM → P → M → 0 gives the exact sequence 0 → ϵΩM → ϵP → ϵM → 0. Clearly, ϵP is a projective cover of ϵM, so ϵΩM = Ω(ϵM). Since ϵM is a linear ϵAϵ-module by Lemma 7.2, we have an exact sequence similar to (7.3):

(7.5) 0 → Ext_{ϵAϵ}^{∗}(Ω(ϵM), ϵA_{0}) → Ext_{ϵAϵ}^{∗}(ϵM, ϵA_{0}) → Hom_{ϵAϵ}(ϵP_{0}, ϵA_{0}) → 0.

Therefore,

dim_{k}Ext_{A}^{∗}(M, A_{0}) − dim_{k}ΓeExt_{A}^{∗}(M, A_{0}) by (7.3) and (7.4)
= dim_{k}Ext_{A}^{∗}(ΩM, A_{0}) + dim_{k}Hom_{A}(P_{0}, A_{0}) − dim_{k}ΓeExt_{A}^{∗}(ΩM, A_{0})
= dim_{k}Ext_{ϵAϵ}^{∗}(ϵΩM, ϵA_{0}) + dim_{k}Hom_{A}(P_{0}, A_{0}) by induction on ΩM
= dim_{k}Ext_{A}^{∗}(ϵM, ϵA_{0}) + dim_{k}Hom_{A}(P_{0}, A_{0}) − dim_{k}Hom_{ϵAϵ}(ϵP_{0}, ϵA_{0}),

where the last identity comes from (7.5).

We establish the identity dim_{k}Hom_{A}(P_{0}, A_{0}) = dim_{k}Hom_{ϵAϵ}(ϵP_{0}, ϵA_{0}) and finish the proof by induction. Take an arbitrary indecomposable summand of P_{0}, which is isomorphic to a certain A_{0}e_{λ}. Since AϵM = M, P_{0} ≅ M_{0} has no summands isomorphic to A_{0}e. Therefore λ ∈ Λ_{1} and e_{λ}ϵ = e_{λ}. Now

Hom_{A}(A_{0}e_{λ}, A_{0}) ≅ Hom_{A_{0}}(A_{0}e_{λ}, A_{0}) ≅ e_{λ}A_{0} = e_{λ}ϵA_{0}
≅ Hom_{ϵA_{0}ϵ}(ϵA_{0}e_{λ}, ϵA_{0}) ≅ Hom_{ϵAϵ}(ϵA_{0}e_{λ}, ϵA_{0}).

Since P_{0} is a direct sum of these summands, the identity holds. □

Now we can prove the second statement of Theorem 7.1.

Proof. We use induction on the size of the poset Λ. If this number is 1, then A and Γ = Λ^{op} both are local algebra. Clearly Γ is standardly stratified. Suppose that the conclusion is true for posets with sizes at most m − 1 and let Λ be an poset with m elements. Take e be a minimal idempotent and define ϵ, Λ_{1} as before.

Let M = A_{0} in the previous lemma. We conclude that ΓeΓ is a projective Γmodule. By Lemma 7.2 ϵAϵ satisfies all the conditions in the theorem and Λ_{1} has only m−1 elements. Therefore, by the induction hypothesis, Γ′ = Ext_{ϵAϵ}^{∗}(ϵA_{0}, ϵA_{0}) is standardly stratified with respect to the poset ({e_{λ}}_{λ∈Λ_{1}}, ⩽^{op}). Thus it is enough to show Γ/ΓeΓ ≅ Γ′.

There is an algebra homomorphism φ : Γ → Γ′ induced by the exact functor F = Hom_{A}(Aϵ, −) in the following way: φ sends an n-fold exact sequence

\xymatrix{0 \ar[r] & A_{0} \ar[r] & M^{n} \ar[r] & … \ar[r] & M^{1} \ar[r] & A_{0} \ar[r] & 0}

representing an element g ∈ Γ_{n} to an n-fold exact sequence

\xymatrix{0 \ar[r] & ϵA_{0} \ar[r] & ϵM^{n} \ar[r] & … \ar[r] & ϵM^{1} \ar[r] & ϵA_{0} \ar[r] & 0}

representing an element g′ ∈ Γ′_{n} for all n ⩾ 0. Every element x ∈ (Γe)_{n} = Ext_{A}^{n}(A_{0}e, A_{0}) can be represented by an exact sequence:

\xymatrix{0 \ar[r] & A_{0} \ar[r] & M^{n} \ar[r] & … \ar[r] & M^{1} \ar[r] & A_{0}e \ar[r] & 0.}

Since Hom_{A}(Aϵ, −) sends A_{0}e to Hom_{A}(Aϵ, A_{0}e) = 0, φ maps every (Γe)_{n} and hence Γe to 0. Thus the ideal ΓeΓ generated by Γe is also sent to 0, and φ gives rise to an algebra homomorphism φ̄ : Γ/ΓeΓ → Γ′.

Clearly, φ maps Γ_{0} = End_{A_{0}}(A_{0}) onto End_{ϵAϵ}(ϵA_{0}) = ϵΓ_{0}ϵ. Moreover, we have

Ext_{A}^{1}(A_{0}, A_{0}) ≅ Hom_{A}(⨁_{i⩾1} A_{i}, A_{0}) ≅ Hom_{A_{0}}(A_{1}, A_{0}) ≅ Hom_{ϵA_{0}ϵ}(ϵA_{1}, ϵA_{0}),

where the last isomorphism is induced by the functor F (see the last paragraph of the proof of Lemma 7.5). But

Hom_{ϵA_{0}ϵ}(ϵA_{1}, ϵA_{0}) ≅ Hom_{ϵAϵ}(⨁_{i⩾1} ϵA_{i}, ϵA_{0}) ≅ Ext_{ϵAϵ}^{1}(ϵA_{0}, ϵA_{0}).

Thus, the homomorphism φ induced by F maps Ext_{A}^{1}(A_{0}, A_{0}) to Ext_{ϵAϵ}^{1}(ϵA_{0}, ϵA_{0}) surjectively. Since A_{0} (ϵA_{0}, resp.) is a linear A-module (ϵAϵ-module, resp.), both Γ = Ext_{A}^{∗}(A_{0}, A_{0}) and Γ′ = Ext_{ϵAϵ}^{∗}(ϵA_{0}ϵ, ϵA_{0}ϵ) are generated in degree 0 and degree 1 as algebras. Therefore, the map φ : Γ → Γ′ is a surjective algebra homomorphism, so φ̄ : Γ/ΓeΓ → Γ′ is surjective as well.

Let M = A_{0} in the previous lemma. We get dim_{k}Γ′ = dim_{k}Γ − dim_{k}ΓeΓ. Therefore, as a surjective homomorphism between two k-algebras with the same dimension, φ̄ must be an isomorphism. This completes the proof. □

There is a similar conclusion in the case that A is a quasi-hereditary algebra, the definition for which can be found in [7].

Corollary 7.6. Let A be the same as in the previous theorem. If A is quasihereditary with respect to the poset ({e_{λ}}_{λ∈Λ}, ⩽), then Γ = Ext_{A}^{∗}(A_{0}, A_{0}) is also quasi-hereditary with respect to the poset ({e_{λ}}_{λ∈Λ}, ⩽^{op}).

Proof. We already proved that Γ is standardly stratified with respect to the poset ({e_{λ}}_{λ∈Λ}, ⩽^{op}). It suffices to check that the ideal ΓeΓ is a hereditary ideal, i.e, the endomorphism algebra of Γe is one-dimensional.

Since e is maximal as an idempotent of Γ, the standard Γ-module corresponding to e is exactly Γe. Thus

Hom_{Γ}(Γe, Γe) ≅ eΓe ≅ Ext_{A}^{∗}(A_{0}e, A_{0}e) = Hom_{A}(A_{0}e, A_{0}e)

where the last identity follows from (2) of Lemma 7.3. Since A_{0}e is a standard module of the quasi-hereditary algebra A, Hom_{A}(A_{0}e, A_{0}e) ≅ k. The conclusion follows from induction. □

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

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