A GENERALIZED KOSZUL THEORY AND ITS RELATION TO THE CLASSICAL THEORY

LIPING LI

Abstract. Let A = ⨁_{i⩾0} A_{i} be a graded locally finite k-algebra where A_{0} is a finite-dimensional algebra whose finitistic dimension is 0. In this paper we develop a generalized Koszul theory preserving many classical results. Moreover, we define a quotient graded algebra Ā = ⨁_{i⩾0} Ā_{i} and show that A is a generalized Koszul algebra if and only if it is a projective A_{0}-module and Ā is a classical Koszul algebra. Some applications in representations of categories and extension algebras of standard modules of standardly stratified algebras are described.

1. Introduction

The classical Koszul theory plays an important role in the representation theory of graded algebras. However, there are a lot of structures (algebras, categories, etc) having natural gradings with non-semisimple degree 0 parts, to which the classical theory cannot apply. Particular examples of such structures include tensor algebras generated by non-semisimple algebras A_{0} and (A_{0}, A_{0})-bimodules A_{1}, extension algebras of finitely generated modules (among which we are most interested in extension algebras of standard modules of standardly stratified algebras studied in [?, ?]), skew group algebras defined by actions of a finite group on a locally finite graded algebras, category algebras of finite categories [?], and graded k-linear categories for which the endomorphism algebras of objects are not semisimple. Therefore, it is reasonable to develop a generalized Koszul theory which preserves classical results such as the Koszul duality, and can be used to study representations and homological properties of above structures.

In [?, ?, ?, ?] several generalized Koszul theories have been described, where the degree 0 part A_{0} of a graded algebra A is not required to be semisimple. In [?], A is supposed to be both a left projective A_{0}-module and a right projective A_{0}-module. However, in many cases A is indeed a left projective A_{0}-module, but not a right projective A_{0}-module. In Madsen’s paper [?], A_{0} is supposed to have finite global dimension. This requirement is too strong for us since in many applications A_{0} is a self-injective algebra or a direct sum of local algebras, and hence A_{0} has finite global dimension if and only if it is semisimple, falling into the framework of the classical theory. The theory developed by Green, Reiten and Solberg in [?] works in a very general framework, and we want to find some conditions which are easy to check in practice. The author has already developed a generalized Koszul theory in [?] under the assumption that A_{0} is self-injective, and have used it to study representations and homological properties of categories.

The goal of the work described in this paper is to loose the assumption that A_{0} is self-injective (as required in [?]) and replace it by a weaker condition so that the generalized theory can apply to more situations. Specifically, since we are interested in the extension algebras of modules, category algebras of finite categories, and graded k-linear categories for which the endomorphism algebra of each object is a finite dimensional local algebra, this weaker condition should be satisfied by selfinjective algebra and finite dimensional local algebras. On the other hand, we also expect that many classical results as the Koszul duality can be preserved. Moreover, we hope to get a close relation between this generalized theory and the classical theory as well.

A trivial observation tells us that in the classical setup A_{0} is semisimple if and only if gl. dim A_{0}, the global dimension of A_{0}, is 0. Therefore, it is natural to consider the condition that fin. dim A_{0}, the finitistic dimension of A_{0}, is 0. Obviously, finite dimensional local algebras and self-injective algebras do have this property. It turns out that this weaker condition is suitable for our applications, and many classical results still hold.

Explicitly, let A = ⨁_{i⩾0} A_{i} be a graded locally finite k-algebra generated in degrees 0 and 1, i.e., dim_{k} A_{i} < ∞ and A_{1} · A_{i} = A_{i+1} for all i ⩾ 0. We assume that both fin. dim A_{0} and fin. dim A_{0}^{op} are 0, where A_{0}^{op} is the opposite algebra of A_{0}. We then define generalized Koszul modules and generalized Koszul algebras by linear projective resolutions, as people did for classical Koszul modules and classical Koszul algebras.

It is well known that in the classical theory linear modules (defined by linear projective resolutions) and Koszul modules (defined by a certain extension property) coincide. We have a similar result:

Theorem 1.1. Let A = ⨁_{i⩾0} A_{i} be a locally finite graded algebra with fin. dim A_{0} = fin. dim A_{0}^{op} = 0. If A is a projective A_{0}-module, then a graded module M is generalized Koszul if and only if it is a projective A_{0}-module and the graded Γ = Ext_{A}^{∗}(A_{0}, A_{0})-module Ext_{A}^{∗}(M, A_{0}) is generated in degree 0, i.e.,

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

We also have the generalized Koszul duality as follows:

Theorem 1.2. Let A = ⨁_{i⩾1} A_{i} be a locally finite graded algebra with fin. dim A_{0} = fin. dim A_{0}^{op} = 0. If A is a generalized Koszul algebra, then E = Ext_{A}^{∗}(−, A_{0}) gives a duality between the category of generalized Koszul A-modules and the category of generalized Koszul Γ = Ext_{A}^{∗}(A_{0}, A_{0})-modules. That is, if M is a generalized Koszul A-module, then E(M) is a generalized Koszul Γ-module, and E_{Γ}EM = Ext_{Γ}^{∗}(EM, Γ_{0}) ≅ M as graded A-modules.

Let 𝔯 be the radical of A_{0} and define ℜ = A𝔯A to be the two-sided ideal generated by 𝔯. For a graded A-module M = ⨁_{i⩾0} M_{i}, we then define a quotient algebra Ā = A/A𝔯A = ⨁_{i⩾0} A_{i}/(A𝔯A)_{i} and M̄ = M/ℜM = ⨁_{i⩾0} M_{i}/(ℜM)_{i}. Clearly, M̄ is a graded Ā-module, and the graded A-module M is generated in degree 0 if and only if the corresponding graded Ā-module M̄ is generated in degree 0. Consequently, we get the following correspondence between our generalized Koszul theory and the classical theory:

Theorem 1.3. Let A = ⨁_{i⩾1} A_{i} be a locally finite graded algebra and M be a graded A-module. Suppose that both A and M are projective A_{0}-modules. Then M is generalized Koszul if and only if the corresponding graded Ā-module M̄ is classical Koszul. In particular, A is a generalized Koszul algebra if and only if Ā is a classical Koszul algebra.

We then apply the generalized Koszul theory to some finite categories. They play an important role in group theory and group representation theory. By the correspondence established in the previous theorem, we show that the generalized Koszul property of the category algebras of these categories is equivalent to the classical Koszul property of quotient algebras of some finite dimensional hereditary algebras.

Extension algebras of standard modules of standardly stratified algebras have been widely studied in [?, ?, ?, ?, ?, ?, ?]. Different from the usual approach, we do not assume that the standardly stratified algebra is graded, the degree 0 part is semisimple, and its grading is compatible with the filtration by standard modules. Instead, using some combinatorial properties of the filtration by standard modules, we show that the extension algebra of standard modules has generalized Koszul property.

The paper is organized as follows: In the next section we develop the generalized Koszul theory and prove the first two theorems. In Section 3 we describe the relation between the generalized theory and the classical theory, and prove the third theorem. Applications of this theory and the correspondence to some categories are described in Section 4. In the last section we discuss the generalized Koszul property of extension algebras of standard modules.

We introduce some notation here. Throughout this paper k is an algebraically closed field. Let A = ⨁_{i⩾0} A_{i} be a locally finite graded algebra generated in degrees 0 and 1, i.e., dim_{k} A_{i} < ∞ and A_{i+1} = A_{1} · A_{i} for all i ⩾ 0. An A-module M = ⨁_{i⩾0} M_{i} is graded if A_{i} · M_{j} ⊆ M_{i+j}. It is said to be generated in degree s if M = A · M_{s}. It is locally finite if dim_{k} M_{i} < ∞ for all i ⩾ 0. In this paper all graded modules are supposed to be locally finite.

Given two graded A-modules M and N, Hom_{A}(M, N) and hom_{A}(M, N) are the spaces of all module homomorphisms and of all graded module homomorphisms respectively. The composition of maps f : L → M and g : M → N is denoted by gf. The degree shift functor [−] is defined by letting M[i]_{s} = M_{s−i} for i, s ∈ ℤ. Denote 𝔍 = ⨁_{i⩾1} A_{i}, which is a two-sided ideal of A. We identify A_{0} with the quotient module A/𝔍 and view it as a graded A-module concentrated in degree 0. We view the zero module 0 as a projective module since this can simplify the expression of some statement.

2. A generalized Koszul theory

We start with some preliminary results, most of which are generalized from those described in [?, ?, ?, ?, ?]. The reader is also suggested to look at other generalized Koszul theories described in [?, ?, ?, ?].

The following lemmas are proved in [?], where we did not use the condition that A_{0} is self-injective (Remark 2.8 in [?]).

Lemma 2.1. (Lemma 2.1 in [?]) Let A be as above and M be a graded A-module. Then:

(1) 𝔍 is contained in the graded radical of A;

(2) M has a graded projective cover;

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

Lemma 2.2. (Lemma 2.2 in [?]) Let 0 → L → M → N → 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.

Now we define generalized Koszul modules and generalized Koszul algebras.

Definition 2.3. A graded A-module M is called a generalized Koszul module if it has a (minimal) linear projective resolution

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

such that P^{i} is generated in degree i for all i ⩾ 0. The graded algebra A is called a generalized Koszul algebra if A_{0} viewed as an A-module is generalized Koszul.

The reader can easily see that M is a generalized Koszul A-module if and only if M is generated in degree 0 and each syzygy Ω^{i}(M) is generated in degree i for every i ⩾ 1. Moreover, from the above projective resolution, we deduce that M_{0} ≅ P_{0}^{0} and Ω^{i}(M)_{i} ≅ P_{i}^{i} are projective A_{0}-modules for all i ⩾ 1.

Recall for a finite dimensional algebra Λ, the finitistic dimension fin. dim Λ is defined as the supremum of projective dimensions of all indecomposable Λ-modules having finite projective dimension ([?, ?]). In particular, if the global dimension gl. dim Λ is finite, then fin. dim Λ = gl. dim Λ. It is well known that fin. dim Λ = 0 if Λ is a finite dimensional local algebras or a self-injective algebra. The famous finitistic dimension conjecture asserts that the finitistic dimension of any finite dimensional algebra is finite.

From now on we assume that fin. dim A_{0} = 0 = fin. dim A_{0}^{op}. It is easy to see that this assumption is equivalent to the following splitting condition:

(S): Every exact sequence 0 → P → Q → R → 0 of left (right, resp.) Λ-modules splits if P and Q are left (right, resp.) projective Λ-modules.

Indeed, from the short exact sequence we deduce that pd_{Λ} R ⩽ 1. If fin. dim Λ = 0 = fin. dim Λ^{op}, then pd_{Λ} R = 0, and hence the exact sequence splits. Conversely, suppose that every such exact sequence splits. If there is some Λ-module M such that pd_{Λ} M = n ⩾ 1, we consider R = Ω^{n−1}(M) and deduce that pd_{Λ} R = 1. Consequently, the projective resolution of R is non-splitting, contradicting the splitting condition.

Proposition 2.4. Let 0 → L → M → N → be an exact sequence of graded Amodules such that L is generalized Koszul. Then M is generalized Koszul if and only if N is generalized Koszul.

Proof. This is Proposition 2.9 in [?]. The proof is almost the same except replacing the self-injective property of A_{0} by the splitting property (S). For the sake of completeness we give a brief proof here.

By the second statement of the previous lemma, M is generated in degree 0 if and only if N is generated in degree 0. Consider the following diagram in which all rows and columns are exact:

\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.}

Here P and Q are graded projective covers of L and N respectively. We claim M′ ≅ ΩM. Indeed, the given exact sequence induces an 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} is a projective A_{0}-module. If N is generalized Koszul, then N_{0} is a projective A_{0}-module since N_{0} ≅ Q_{0}^{0}, and the above sequence splits. If M is generalized Koszul, then M_{0} is a projective A_{0}-module, and this sequence splits as well by the splitting property (S). In either case we have M_{0} ≅ L_{0} ⊕ N_{0}. Thus P ⊕ Q is a graded projective cover of M, and hence M′ ≅ 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 them are generalized Koszul) respectively in the short exact sequence. Repeating the above procedure we prove the conclusion by recursion. □

If M is a generalized Koszul module, its truncations (with suitable degree shifts) are generalized Koszul as well:

Proposition 2.5. Let A be a generalized Koszul algebra and M be a generalized Koszul module. Then 𝔍^{i}M[−i] is also generalized Koszul for each i ⩾ 1.

Proof. This is Proposition 2.13 in [?]. For the convenience of the reader we include a brief proof here. Consider 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}

Since M_{0} is a projective A_{0}-module and A_{0} is generalized Koszul, Ω(M_{0})[−1] is also generalized Koszul. Similarly, ΩM[−1] is generalized Koszul since so is M. Therefore, 𝔍M[−1] is generalized Koszul by the previous proposition. Now replacing M by 𝔍M[−1] and using recursion, we conclude that 𝔍^{i}M[−i] is a generalized Koszul A-module for every i ⩾ 1. □

From this proposition we immediately deduce that if A is a generalized Koszul algebra, then it is a projective A_{0}-module. We now focus on graded algebras with this property.

Proposition 2.6. If A is a projective A_{0}-module, then every generalized Koszul module M is a projective A_{0}-module.

Proof. Clearly, it suffices to show that M_{i} is a projective A_{0}-module for each i ⩾ 0. Since M is generalized Koszul, M_{0} is a projective A_{0}-module. Now suppose i ⩾ 1. The minimal linear projective resolution of M gives rise to exact sequences of A_{0}- modules:

\xymatrix{0 \ar[r] & Ω^{s+1}(M)_{i} \ar[r] & P_{i}^{s} \ar[r] & Ω^{s}(M)_{i} \ar[r] & 0,} 0 ⩽ s ⩽ i.

If s = i, we have Ω^{i+1}(M)_{i} = 0 since Ω^{i+1}(M) is generated in degree i + 1. Thus Ω^{i}(M)_{i} ≅ P_{i}^{i} is a projective A_{0}-module. Now let s = i − 1. We claim that the first term Ω^{i}(M)_{i} is a projective A_{0}-module. Indeed, Ω^{i}(M)[−i] is a generalized Koszul module, so (Ω^{i}(M)[−i])_{0} is a projective A_{0}-module. But Ω^{i}(M)_{i} ≅ (Ω^{i}(M)[−i])_{0}. This proves the claim. Since the first two terms are projective A_{0}-modules, by the splitting property (S), we deduce that Ω^{i−1}(M)_{i} is a projective A_{0}-module. By recursion, we conclude that M_{i} is a projective A_{0}-module for every i > 0. □

The following lemma will be used in the proof of Theorem 1.1.

Lemma 2.7. Let M be a graded A-module generated in degree 0. Suppose that both A and M are projective A_{0}-modules. Then ΩM is generated in degree 1 if and only if every A-module homomorphism ΩM → A_{0} extends to an A-module homomorphism 𝔍P → A_{0}, where P is a graded projective cover of M.

Proof. This is a varied version of Lemma 2.17 in [?]. The exact sequence 0 → ΩM → P → M → 0 induces an exact sequence 0 → (ΩM)_{1} → P_{1} → M_{1} → 0 of A_{0}-modules, which splits since M_{1} is a projective A_{0}-module. Applying the functor Hom_{A_{0}}(−, A_{0}) we get another split 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.

Note that (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 M_{1}, P_{1} and (ΩM)_{1} are projective A_{0}-modules. But the above sequence is isomorphic to

0 → Hom_{A}(𝔍M, A_{0}) → Hom_{A}(𝔍P, A_{0}) → Hom_{A}(ΩM, A_{0}) → 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}. □

Let M be a graded A-module and Γ = Ext_{A}^{∗}(A_{0}, A_{0}). Then Ext_{A}^{∗}(M, A_{0}) is a graded Γ-module. Now we restate and prove Theorem 1.1.

Theorem 2.8. Let A = ⨁_{i⩾0} A_{i} be a locally finite graded algebra satisfying the splitting property (S). If A is a projective A_{0}-module, then a graded module M is generalized Koszul if and only if it is a projective A_{0}-module and the graded Γ = Ext_{A}^{∗}(A_{0}, A_{0})-module Ext_{A}^{∗}(M, A_{0}) is generated in degree 0, i.e.,

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

Proof. This is a varied version of Theorem 2.16 in [?]. Since the proof is almost the same, we only give a sketch. Please refer to [?] for details.

The only if part. Let M be a generalized Koszul A-module. Without loss of generality we can suppose that M is indecomposable. By Lemma 2.6 M is a projective A_{0}-module. As in the original proof, it suffices to show that the given identity is true for i = 1, i.e.,

Ext_{A}^{1}(M, A_{0}) = Ext_{A}^{1}(A_{0}, A_{0}) · Hom_{A}(M, A_{0}).

The proof of this identity is completely the same as the original proof. We omit the details.

The if part. As in the original proof, we only need to show that ΩM is generated in degree 1. By the previous lemma, it suffices to show that each (non-graded) Amodule homomorphism g : ΩM → A_{0} extends to 𝔍P^{0}, where P^{0} is a graded projective cover of M. The proof of this fact is completely the same as the original proof. □

An immediate corollary of the above theorem is:

Corollary 2.9. The graded algebra A is generalized Koszul if and only if A is a projective A_{0}-module and Γ = Ext_{A}^{∗}(A_{0}, A_{0}) is generated in degrees 0 and 1.

Now we can prove a generalized Koszul duality.

Theorem 2.10. Let A = ⨁_{i⩾1} A_{i} be a locally finite graded algebra satisfying the splitting condition (S). If A is a generalized Koszul algebra, then E = Ext_{A}^{∗}(−, A_{0}) gives a duality between the category of generalized Koszul A-modules and the category of generalized Koszul Γ = Ext_{A}^{∗}(A_{0}, A_{0})-modules. That is, if M is a generalized Koszul A-module, then E(M) is a generalized Koszul Γ-module, and E_{Γ}EM = Ext_{Γ}^{∗}(EM, Γ_{0}) ≅ M as graded A-modules.

Proof. This is a varied version of Theorem 4.1 in [?]. We give a detailed proof for the convenience of the reader. Since A_{0} is a generalized Koszul module and M is a projective A_{0}-module, M_{0} is generalized Koszul as well. By Proposition 2.5, 𝔍M[−1] is also generalized Koszul. Furthermore, we have the following short exact sequence of generalized Koszul 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 2.4, this sequence induces exact sequences of generalized Koszul modules recursively:

\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.}

Take a fixed sequence for a certain i > 0. It gives a splitting exact sequence 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 Hom_{A_{0}}(−, A_{0}) to it 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,

which 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, E(M_{0}) is a projective Γ-module. But 𝔍M[−1] is generalized Koszul, so E(𝔍M[−1]) is generated in degree 0 by the previous theorem. Thus E(𝔍M[−1])[1] is generated in degree 1, and E(M_{0}) is a graded projective cover of EM. This proves the claim. Consequently, Ω(EM) is generated in degree 1. Moreover, replacing M by 𝔍M[−1] (which is also generalized Koszul) and using the claimed identity, we have that

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

is generated in degree 2. By recursion, Ω^{i}(EM) ≅ E(𝔍^{i}M[−i])[i] is generated in degree i for all i ⩾ 0. Thus EM is a generalized Koszul Γ-module (note that Γ_{0} ≅ A_{0}^{op} also satisfies the splitting property (S)!). In particular for M =_{A} A,

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

is a generalized Koszul Γ-module.

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

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 also 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 M_{i} is a projective A_{0}-module and Γ_{0} ≅ A_{0}^{op}. Therefore, we get

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 such that Γ_{0} ≅ A_{0}^{op} is self-injective as an algebra and generalized Koszul 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 Koszul Γ-module N. Thus E is a dense functor.

Let L be another generalized Koszul A-module. Since L, M, EL, EM are all generated in degree 0, we have

hom_{Γ}(EL, EM) ≅ Hom_{Γ_{0}}((EL)_{0}, (EM)_{0})
≅ Hom_{Γ_{0}}(Hom_{A}(L, A_{0}), Hom_{A}(M, A_{0}))
≅ Hom_{A_{0}^{op}}(Hom_{A_{0}}(L_{0}, A_{0}), Hom_{A_{0}}(M_{0}, A_{0}))
≅ Hom_{A_{0}}(L_{0}, M_{0}) ≅ hom_{A}(L, M).

Consequently, E is a duality between the category of generalized Koszul A-modules and the category of generalized Koszul Γ-modules. □

It is well known that a locally finite graded algebra A is classical Koszul if and only if the opposite algebra A^{op} is also classical Koszul. Unfortunately, this result does not hold in the generalized theory. Here is an example.

Example 2.11. Let A be the path algebra of the following quiver with relations δ^{2} = αδ = 0. Put A_{0} = ⟨1_{x}, 1_{y}, δ⟩ and A_{1} = ⟨α⟩.

[figure: x; δ; α; y]

Then the (graded) left indecomposable projective modules and right indecomposable projective modules are described as follows, where the indices mean the degree.

LP_{x} = \begin{aligned}& x_{0} \\ x_{0} & & y_{1} \end{aligned}, _{L}P_{y} = y_{0}; _{R}P_{x} = \begin{matrix}x_{0} \\ x_{0} \end{matrix}, _{R}P_{y} = \begin{matrix}y_{0} \\ x_{1} \end{matrix}.

It is not hard to check that A is a generalized Koszul algebra, but A^{op} is not. Actually, it is because that A is a projective as a left A_{0}-module, but A^{op} is not a left projective A_{0}^{op} ≅ A_{0}-module.

3. A relation between the generalize theory and the classical theory

In this section we describe a correspondence between the generalized Koszul theory we just developed and the classical theory. As before, let A = ⨁_{i⩾0} A_{i} be a locally finite graded algebra generated in degrees 0 and 1. At this moment we do not assume the condition that fin. dim A_{0} = fin. dim A_{0}^{op} = 0.

Let 𝔯 be the radical of A_{0}, and ℜ = A𝔯A be the two-sided ideal generated by 𝔯. We then define the quotient graded algebra Ā = A/ℜ = ⨁_{i⩾0} A_{i}/ℜ_{i}. Clearly, Ā is a locally finite graded algebra for which the grading is induced from that of A, and ℜ_{s} = ∑_{i=0}^{s} A_{i}𝔯A_{s−i}. Note that Ā_{0} = A_{0}/𝔯 is a semisimple algebra. Given an arbitrary graded A-module M = ⨁_{i⩾0} M_{i}, define M̄ = M/ℜM = ⨁_{i⩾0} M_{i}/(ℜM)_{i}. Then M̄ is a graded Ā-module and M̄ ≅ Ā ⊗_{A} M.

We use an example to show our construction.

Example 3.1. Let A be the path algebra of the following quiver with relations: δ^{2} = θ^{2} = 0, θα = αδ. Put A_{0} = ⟨1_{x}, 1_{y}, δ, θ⟩ and A_{1} = ⟨α, θα⟩.

[figure: x; δ; α; y; θ]

The structures of graded indecomposable projective A-modules are:

P_{x} = \begin{aligned}& x_{0} \\ x_{0} & & y_{1} \\ & y_{1} \end{aligned} P_{y} = \begin{matrix}y_{0} \\ y_{0} \end{matrix}.

We find 𝔯 = ⟨δ, θ⟩, ℜ = ⟨δ, θ, θα⟩. Then the quotient algebra Ā is the path algebra of the following quiver with a natural grading:

\xymatrix{x \ar[r]^{α} & y}.

Let M = rad P_{x} = ⟨δ, α, αδ⟩ which is a graded A-module. It has the following structure and is not generated in degree 0:

M = \begin{aligned}x_{0} & & y_{1} \\ & y_{1} \end{aligned}.

Then M̄_{0} = M_{0}/𝔯M_{0} = ⟨δ̄⟩ ≅ S_{x}, the simple Ā-module corresponding to x; M̄_{1} = M_{1}/(𝔯M_{1} + A_{1}𝔯M_{0}) = ⟨ᾱ⟩ ≅ S_{y}[1]. Therefore, M̄ ≅ S_{x} ⊕ S_{y}[1] is a direct sum of two simple Ā-modules, and is not generated in degree 0 either.

The following proposition is crucial to prove the main result in this section.

Proposition 3.2. A graded A-module M is generated in degree 0 if and only if the corresponding graded Ā-module M̄ is generated in degree 0.

Proof. If M is generated in degree 0, then A_{i}M_{0} = M_{i} for all i ⩾ 0. By our construction, it is clear that Ā_{i}M̄_{0} = M̄_{i}. That is, M̄ is generated in degree 0.

Conversely, suppose that M̄ is generated in degree 0. We want to show A_{i}M_{0} = M_{i} for i ⩾ 0. We use induction to prove this identity. Clearly, it holds for i = 0. So we suppose that it is true for all 0 ⩽ i < n and consider M_{n}.

Take v_{n} ∈ M_{n} and consider its image v̄_{n} in M̄_{n} = M_{n}/ ∑_{i=0}^{n} A_{i}𝔯M_{n−i}. Since M̄ is generated in degree 0, we can find some a_{n} ∈ A_{n} and v_{0} ∈ M_{0} such that v̄_{n} = ā_{n}v̄_{0}. Thus \overline{v_{n} − a_{n}v_{0}} = v̄_{n} − ā_{n}v̄_{0} = 0. This means

v_{n} − a_{n}v_{0} ∈ ∑_{i=0}^{n} A_{i}𝔯M_{n−i} = 𝔯M_{n} + ∑_{i=1}^{n} A_{i}𝔯M_{n−i} = 𝔯M_{n} + ∑_{i=1}^{n} A_{i}𝔯A_{n−i}M_{0},

where the last identity follows from the induction hypothesis. But it is clear A_{n}M_{0} ⊇ ∑_{i=1}^{n} A_{i}𝔯A_{n−i}M_{0}, so v_{n} − a_{n}v_{0} ∈ 𝔯M_{n} + A_{n}M_{0}. Consequently, v_{n} ∈ 𝔯M_{n} + A_{n}M_{0}. Since v_{n} ∈ M_{n} is arbitrary, we have M_{n} ⊆ 𝔯M_{n} + A_{n}M_{0}. Applying Nakayama’s lemma to these A_{0}-modules, we conclude that M_{n} = A_{n}M_{0} as well. The conclusion then follows from induction. □

Lemma 3.3. Let M be a graded A-module generated in degree 0. If P is a grade projective cover of M, then P̄ is a graded projective cover of M̄.

Proof. Clearly, P̄ is a graded projective module. Both P̄ and M̄ are generated in degree 0 by the previous proposition. To show that P̄_{0} is a graded projective cover of M̄_{0}, it suffices to show that P̄_{0} is a projective cover of M̄_{0} as Ā_{0}-modules. But this is clearly true since P̄_{0} = P_{0}/𝔯P_{0} ≅ M_{0}/𝔯M_{0} ≅ M̄_{0}. □

The procedure of sending M to M̄ preserves exact sequences of graded A-modules which are projective regarded as A_{0}-modules.

Lemma 3.4. Let 0 → L → M → N → 0 be a short exact sequence of graded A-modules such that all terms are projective A_{0}-modules. Then the corresponding sequence 0 → L̄ → M̄ → N̄ → 0 is also exact.

Proof. For each i ⩾ 0, the given exact sequence induces a short exact sequence of A_{0}-modules 0 → L_{i} → M_{i} → N_{i} → 0. Since all terms are projective A_{0}-modules, this sequence splits. Thus we get an exact sequence of Ā_{0}-modules 0 → L̄_{i} → M̄_{i} → N̄_{i} → 0. Let the index i vary and take direct sum. Then we get an exact sequence of graded Ā-modules 0 → L̄ → M̄ → N̄ → 0 as claimed. □

The condition that all terms are projective A_{0}-modules cannot be dropped, as shown by the following example.

Example 3.5. Let A = A_{0} = k[t]/(t^{2}) and S be the simple module and consider a short exact sequence of graded A-modules 0 → S → A → S → 0. We have Ā ≅ k. But the corresponding sequence 0 → S̄ → Ā → S̄ → 0 is not exact. Actually, the first map S̄ → Ā is 0 since the image of S is contained in 𝔯A_{0}.

Now we can prove the main result of this section.

Theorem 3.6. Let A = ⨁_{i⩾1} A_{i} be a locally finite graded algebra and M be a graded A-module. Suppose that both A and M are projective A_{0}-modules. Then M is generalized Koszul if and only if the corresponding grade Ā-module M̄ is classical Koszul. In particular, A is a generalized Koszul algebra if and only if Ā is a classical Koszul algebra.

Proof. Let

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

be a minimal projective resolution of M. Note that all terms in this resolution and all syzygies are projective A_{0}-modules. By Lemmas 3.3 and 3.4, M̄ has the following minimal projective resolution

(3.2)  \xymatrix{… \ar[r] & \overline{P^{2}} \ar[r] & \overline{P^{1}} \ar[r] & \overline{P^{0}} \ar[r] & \overline{M} \ar[r] & 0}.

Moreover, this resolution is linear if and only if the resolution (3.1) is linear by Proposition 3.2. That is, M is generalized Koszul if and only if M̄ is classical Koszul. This proves the first statement. Applying it to the graded A-module A_{0} we deduce the second statement immediately. □

If fin. dim A_{0} = fin. dim A_{0}^{op} = 0, i.e., A_{0} has the splitting property (S), we have the following corollary:

Corollary 3.7. Let A = ⨁_{i⩾1} A_{i} be a locally finite graded algebra with fin. dim A_{0} = fin. dim A_{0}^{op} = 0. Then:

(1) A is a generalized Koszul algebra if and only it is a projective A_{0}-module and Ā is a classical Koszul algebra.

(2) Suppose that A is a projective A_{0}-module. A graded A-module M is generalized Koszul if and only if it is a projective A_{0}-module and the corresponding grade Ā-module M̄ is classical Koszul.

Proof. If A is a generalized Koszul algebra, then applying Proposition 2.5 to _{A}A we conclude that it is a projective A_{0}-module. Moreover, Ā is a classical Koszul algebra by the previous theorem. The converse statement also follows from the previous theorem. This proves the first statement.

If A is a projective A_{0}-module and M is generalized Koszul, by Proposition 2.6 M is a projective A_{0}-module. Moreover, M̄ is a classical Koszul module by the previous theorem. The converse statement follows from the previous theorem as well. □

We cannot drop the condition that A is a projective A_{0}-module in the above theorem, as shown by the following example.

Example 3.8. Let A be the path algebra of the following quiver with relations: δ^{2} = θ^{2} = 0, θα = αδ = 0. Put A_{0} = ⟨1_{x}, 1_{y}, δ, θ⟩ and A_{1} = ⟨α⟩.

[figure: x; δ; α; y; θ]

The structures of graded indecomposable projective A-modules are:

P_{x} = \begin{aligned}& x_{0} \\ x_{0} & & y_{1} \end{aligned} P_{y} = \begin{matrix}y_{0} \\ y_{0} \end{matrix}.

We find 𝔯 = ⟨δ, θ⟩ = ℜ. Then the quotient algebra Ā is the path algebra of the following quiver:

\xymatrix{x \ar[r]^{α} & y}.

Let Δ_{x} = P_{x}/S_{y} = ⟨δ, 1_{x}⟩ which is a graded A-module concentrated in degree 0. The first syzygy Ω(Δ_{x}) ≅ S_{y}[1] is generated in degree 1, but the second syzygy Ω^{2}(Δ_{x}) ≅ S_{y}[1] is not generated in degree 2. Therefore, Δ_{x} is not generalized Koszul. However, Δ̄_{x} ≅ S̄_{x} is obviously a classical Koszul Ā-module. Moreover, we can check that A is not a generalized Koszul algebra, but Ā is a classical Koszul algebra.

From the conclusions of Propositions 2.5, 2.6 and Theorem 3.6, the reader may have the feeling that the generalized Koszul algebra A is a tensor product of a classical Koszul algebra and a finite dimensional algebra with finitistic dimension 0 concentrated in degree 0, i.e., A ≅ A_{0} ⊗Ā_{0} Ā, or A ≅ Ā ⊗Ā_{0} A_{0}, or A ≅ Λ ⊗_{k} Ā, where Λ is a finite dimensional algebra concentrated in degree 0. This is not true as explained in the following example. Therefore, our generalized Koszul theory works in situations much more general than this special case.

Example 3.9. Let A be the path algebra of the following quiver with relations δ^{2} = 0. Put A_{0} = ⟨1_{x}, 1_{y}, δ⟩ and A_{1} = ⟨α, αδ⟩.

[figure: x; δ; α; y]

Then the quotient graded algebra Ā is spanned by 1_{x}, 1_{y}, α, and Ā_{0} = ⟨1_{x}, 1_{y}⟩ is semisimple. Clearly, A ≇ Λ ⊗_{k} Ā for any finite dimensional algebra Λ.

Consider the tensor product A_{0} ⊗Ā_{0} Ā. It is spanned by 1_{x} ⊗ 1_{x}, 1_{y} ⊗ 1_{y}, 1_{y} ⊗ α, δ ⊗ 1_{x} since 1_{x} ⊗ 1_{y} = δ ⊗ 1_{y} = 1_{y} ⊗ 1_{x} = 1_{x} ⊗ α = δ ⊗ α = 0. Thus this tensor product cannot be isomorphic to A.

Consider the tensor product Ā ⊗Ā_{0} A_{0}. We have 1_{y} ⊗ 1_{x} = 1_{x} ⊗ 1_{y} = α ⊗ 1_{y} = 1_{y} ⊗ δ = 0. Moreover,

α ⊗ 1_{x} = 1_{y}α ⊗ 1_{x}1_{x} = (1_{y} ⊗ 1_{x}) · (α ⊗ 1_{x}) = 0 · (α ⊗ 1_{x}) = 0.

Similarly, α ⊗ δ = 0. Therefore, this tensor product is actually spanned by 1_{x} ⊗ 1_{x}, 1_{y} ⊗ 1_{y} and 1_{x} ⊗ δ, and is not isomorphic to A either.

Indeed, there exist finite categories C whose graded category algebras kC are projective kC_{0}-modules, but these category algebras kC are not isomorphic to tensor products of two algebras. We will see some such examples in Section 4.

The correspondence described in Theorem 3.6 and Corollary 3.7 can be used to show a generalized Koszul algebra A has some properties similar to those of a classical Koszul algebra. The following proposition gives us an example. Recall A = ⨁_{i⩾1} A_{i} is quadratic if A ≅ A_{0}(A_{1})/I, where A_{0}(A_{1}) = A_{0} ⊕A_{1} ⊕(A_{1} ⊗_{A_{0}} A_{1})⊕… is the tensor algebra generated by A_{0} and A_{1}, and I can be generated by some elements in A_{1} ⊗_{A_{0}} A_{1}.

Proposition 3.10. Let A = ⨁_{i⩾1} A_{i} be a locally finite graded algebra and suppose that A viewed as an A_{0}-module is projective. If A is generalized Koszul, then it is a quadratic algebra.

Proof. Let Ā = ⨁_{i⩾0} Ā_{i} ≅ A/ℜ be the quotient graded algebra as in Corollary 3.7, and let I be the kernel of the surjective algebra homomorphism A_{0}(A_{1}) → A. Since Ā is a classical Koszul algebra, it is a quadratic algebra. That is, Ā ≅ Ā_{0}(Ā_{1})/Ī, where Ī is generated by some elements in Ā_{1} ⊗Ā_{0} Ā_{1}.

Observe that there is a surjective graded algebra homomorphism A_{0}(A_{1}) → Ā_{0}(Ā_{1}). Denote the kernel of this algebra homomorphism by K. We check that K_{n}, the degree n component, is ∑_{i=1}^{n} A_{1}^{⊗i−1} ⊗ (𝔯A_{1} + A_{1}𝔯) ⊗ A_{1}^{⊗n−i}.

Note that we can obtain Ā from A_{0}(A_{1}) by two different procedures. That is, we have the following diagram:

\xymatrix{A_{0}(A_{1}) \ar[d]^{/K} \ar[r]^{/I} & A \ar[d]^{/ℜ} \\ Ā_{0}(Ā_{1}) \ar[r]^{/Ī} & Ā}

For each n ⩾ 0, this diagram induces the following diagram of short exact sequences of A_{0}-modules. Here we remind the reader that the since A is supposed to be a projective A_{0}-module, all rows actually split, and all columns are indeed exact.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & L_{n} \ar[d] \ar[r] & K_{n} \ar[r] \ar[d] & ℜ_{n} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I_{n} \ar[d] \ar[r] & A_{1}^{⊗n} \ar[r] \ar[d] & A_{n} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ī_{n} \ar[d] \ar[r] & Ā_{1}^{⊗n} \ar[r] \ar[d] & Ā_{n} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

From this diagram we get L_{n} = K_{n} ∩ I_{n}. Therefore, L = I ∩ K. By the above description of K_{n}, and using the fact that I_{n} is a direct summand of A_{1}^{⊗n}, we deduce that L is contained in the graded radical of I. Since as vector spaces I ≅ Ī ⊕ L, applying the graded Nakayama Lemma, we conclude that I is generated by vectors in Ī, so I has a set of generators of degree 2. This finishes the proof. □

4. Applications to directed categories

In this section we describe some applications of the generalized Koszul theory to certain categories. For the convenience of the reader, let us give some background knowledge.

By the definition in [?, ?], a directed category A is a k-linear category such that there is a partial order ⩽ on Ob A satisfying the condition that x ⩽ y whenever A(x, y) ≠ 0. In this section all directed categories A are supposed to have the following conditions: A is skeletal and has only finitely many objects; A is locally finite, i.e., dim_{k} A(x, y) < ∞ for all x, y ∈ Ob A; the endomorphism algebra of every object is a local algebra. We also suppose that A is graded and A_{0} = ⨁_{x∈Ob A} A(x, x). Note that the space of all morphisms in A forms a graded algebra A whose multiplication is determined by composition of morphisms. We call it the associated algebra of A.

In [?] a close relation between the generalized theory and the classical theory for directed categories has been established. The explicit correspondence is described as follows. Let A be a graded directed category with respect to a partial order ⩽. We define B to be the graded subcategory of A formed by replacing the endomorphism algebra of every object by k · 1, the span of the identity endomorphism. That is, Ob B = Ob A; B(x, y) = A(x, y) if x ≠ y and B(x, x) = k · 1_{x}. Let A and B be the associated graded algebras of A and B respectively. Then we have ⨁_{i⩾1} A_{i} = ⨁_{i⩾1} B_{i}. Note that B_{0} is a semisimple algebra, so the classical theory can be applied. On the other hand, A_{0} as a direct sum of several finite-dimensional local algebras has finitistic dimension 0, so we can use the generalized theory.

Theorem 4.1. Let A and B be defined as above.

(1) Suppose that A is a generalized Koszul algebra. If M is a generalized Koszul A-module, then the restricted module M ↓_{B}^{A} is classical Koszul. In particular, B is a classical Koszul algebra.

(2) Suppose that B is a classical Koszul algebra. If M is a graded A-module satisfying that Ω^{i}(M)_{i} is a projective A_{0}-module for each i ⩾ 0 and M ↓_{B}^{A} is classical Koszul, then M is generalized Koszul.

Proof. These two statements are precisely Theorems 5.13 and 5.14 in [?]. In the original proofs we did not assume that A_{0} is self-injective, see Remark 5.15. □

Theorem 4.2. Let A and B be defined as above.

(1) A is a generalized Koszul algebra if and only if it is a projective A_{0}-module and B is a classical Koszul algebra.

(2) Suppose that A is a generalized Koszul algebra. Then a graded A-module M is generalized Koszul if and only if it is a projective A_{0}-module and M ↓_{B}^{A} is classical Koszul.

Proof. This is Theorem 5.16 in [?], but we drop the unnecessary condition that A_{0} is self-injective.

(1). If A is a generalized Koszul algebra, then it is a projective A_{0}-module, see the paragraph after Proposition 2.5. By (1) of the previous theorem, B is a classical Koszul algebra. Conversely, if B is a classical Koszul algebra, then A_{0} ↓_{B}^{A} is a classical Koszul B-module since it is a projective B_{0}-module. Thus by (2) of the previous theorem, A is a generalized Koszul algebra if we can show that Ω^{i}(A_{0})_{i} is a projective A_{0}-module for each i ⩾ 0. We prove a stronger statement, that is, Ω^{i}(A_{0}) is a projective A_{0}-module for each i ⩾ 0.

Clearly, Ω^{0}(A_{0}) = A_{0} is a projective A_{0}-module. Consider the exact sequence

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

By the induction hypothesis, Ω^{i}(A_{0}) is a projective A_{0}-module. Thus the above sequence splits as A_{0}-modules. But P^{i} is a projective A_{0}-module since we assume that A is a projective A_{0}-module, so is Ω^{i+1}(A_{0}). This proves (1).

(2). Since A is a generalized Koszul algebra, it is a projective A_{0}-module. If M is generalized Koszul, then it is a projective A_{0}-module (Proposition 2.6) and M ↓_{B}^{A} is classical Koszul (by (1) of Theorem 4.1). Conversely, if M ↓_{B}^{A} is classical Koszul, to prove that M is generalized Koszul, by (2) of Theorem 4.1 it suffices to show that Ω^{i}(M)_{i} is a projective A_{0}-module for every i ⩾ 0. This can be proved by a similar induction as we just did. □

In practice these two theorems can be used to prove that some complicated directed categories are generalized Koszul. Compared to Theorem 3.6, they have the following disadvantages: They can only be applied to associated algebras A of some directed categories. Moreover, the algebra B is still complicated since we only reduce the sizes of endomorphism algebras of objects in A, and do not remove any non-endomorphisms. Here is an example:

Example 4.3. Let A be the path algebra of the following quiver with relations δ^{2} = ρ^{2} = θ^{2} = 0, αδ = ρα, and βρ = θβ. Put a grading on A by letting A_{0} be the space spanned by all endomorphisms and letting A_{1} = ⟨α, β, αδ, βρ⟩. The reader can check that A is indeed a projective A_{0}-module.

[figure: x; δ; α; y; ρ; β; z; θ]

By removing all non-identity endomorphisms, we get an algebra B isomorphic to the path algebra of the following quiver with relation β_{2}α_{1} = β_{1}α_{2}.

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

Using the procedure described before Example 3.1, we get a quotient algebra Ā isomorphic to the path algebra of the following quiver:

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

It is not hard to check that both B and Ā are classical Koszul algebras. Therefore, A is a generalized Koszul algebra either by Theorem 3.6 or by Theorem 4.2. However, the algebra Ā is much simpler.

We end this section by introducing some possible applications of our theory to certain finite categories which are interesting to people studying finite groups and their representations. These categories are examples of finite EI categories, that is, categories with finitely many morphisms such that each endomorphism is an isomorphism.

Let G be a finite group, and S be a set of subgroups of group G. The transporter category T_{S} has elements in S as objects. For H, K ∈ S, T_{S}(H, K) = N_{G}(H, K) = {g ∈ G | (^{g}H) ⊆ K}. Clearly, for each object H ∈ S, T_{S}(H, H) = N_{G}(H) is the normalizer of H in G. Therefore, T_{S} is a finite EI category. The orbit category O_{S} has all cosets G/H, where H ∈ S, as objects, and O_{S}(G/H, G/K) is the set of all G-equivariant maps. These categories play important roles in group theory. For details, see [?].

Let C be a transporter category or an orbit category. By Corollaries 2.6, 2.7 and their proofs in [?], C(y, y) acts freely on C(x, y) for x, y ∈ Ob C. Therefore, C(x, y) is the disjoint union of several orbits, each of which is isomorphic to a copy of C(y, y). This observation will play a crucial role in our application.

Now we put a grading on all morphisms in C. It would be reasonable to let C_{0} be the set of all endomorphisms in C, which is a direct sum of several finite groups. The set of all non-endomorphisms in C is closed under the action of C_{0}. Moreover, from [?] we know that every non-endomorphism can be factorized as a composite of several unfactorizable morphisms (see Definition 2.3 and Proposition 2.5 in [?]), and the set of all unfactorizable morphisms is closed under the action of C_{0} (Proposition 2.6 in [?]). Therefore, we can let C_{1} be the set of all unfactorizable morphisms.

It is subtle to define C_{2}. We cannot define C_{2} to be the set of all composites βα where α, β ∈ C_{1}, since it is possible that βα can be written as a composite of more than two morphisms in C_{1}. Therefore, we define C_{2} to be the set of morphisms each of which can be expressed as a composite of two unfactorizable morphisms, but cannot be expressed as a composite of more than two morphisms. Similarly, we define C_{3} be the set of all morphisms each of which can be expressed as a composite of three unfactorizable morphisms, but cannot be expressed as a composite of more than three morphisms, and so on. Note that since C(y, y) acts freely on C(x, y) for x, y ∈ Ob C, each C_{i} is invariant under the action of C_{0}.

The category algebra kC as a vector space has a basis all morphisms in C. The product of two basis elements is their composite (when this is defined in C) or 0 (otherwise). This algebra in general is not graded. However, note that all nonendomorphisms span a two-sided ideal J of kC. Moreover, kC/J ≅ kC_{0}. Therefore, using this two-sided ideal, we get an associated graded algebra A = ⨁_{i⩾0} A_{i} with A_{0} = kC_{0}, a direct sum of group algebras, and A_{i} = J^{i}/J^{i+1} for i ⩾ 1.

The following proposition tells us that our generalized Koszul theory can apply to the graded algebra A associated to the category algebra kC.

Proposition 4.4. Notation as above, we have:

(1) C_{i} is a basis of A_{i} for each i ⩾ 0;

(2) A is a projective A_{0}-module;

(3) Ā = A/A𝔯A is Morita equivalent to a quotient algebra of a finite dimensional hereditary algebra.

Proof. Clearly, for each i ⩾ 1, J^{i} (resp., J^{i+1}) is spanned by all morphisms in C which can be expressed as a composite of n ⩾ i (resp., n ⩾ i + 1) unfactorizable morphisms. Therefore, the quotient A_{i} = J^{i}/J^{i+1} is spanned by all morphisms which can be expresses as a composite of i unfactorizable morphisms, but can not be expressed as a composite of more than i unfactorizable morphisms. Therefore, A_{i} is precisely spanned by elements in C_{i}, which proves (1).

It is obvious that A_{0} is a projective A_{0}-module. For i ⩾ 1, we have a decomposition C_{i} = ⊔_{x,y∈Ob C}C(x, y)_{i}, where C(y, y) acts freely on C(x, y)_{i}. Therefore, kC(x, y)_{i} is a free kC(y, y)-module, so it is a projective kC_{0} = A_{0}-module. Consequently, A_{i} = J^{i}/J^{i+1} = kC_{i} = ⊕_{x,y∈Ob C}kC(x, y)_{i} is a projective A_{0}-module. Thus (2) holds.

Now we prove (3). Take a representative P_{i} from each isomorphism class of indecomposable projective kC-module, 1 ⩽ i ⩽ n. Since C is a finite EI category and these projective modules are pairwise non-isomorphic, there is a partial order ≼ on {i}_{i=1}^{n} such that Hom_{kC}(P_{i}, P_{j}) = 0 if i ⪯̸ j. Correspondingly, there is a set of indecomposable projective A-modules {Q_{i}}_{i=1}^{n}, one from each isomorphism class, such that Hom_{A}(Q_{i}, Q_{j}) = 0 if i ⪯̸ j. By taking quotients, Ā has a set of indecomposable projective modules {Q̄_{i}}_{i=1}^{n}, one from each isomorphism class, satisfying the same condition. Also note that EndĀ(Q̄_{i}, Q̄_{i}) ≅ k for 1 ⩽ i ⩽ n. Therefore, the endomorphism algebra of ⊕_{i=1}^{n}Q̄_{i} is a quotient algebra of a finite hereditary algebra, which is Morita equivalent to Ā. □

As explained in the following example, the correspondence described in Theorem 3.6 indeed gives us a feasible way to show a complicated category or algebra to be generalized Koszul.

Example 4.5. Let G = S_{3} be the symmetric group on 3 letters, and let S be the set of 4 subgroups: x = 1, w = S_{3}, y = ⟨(1, 2)⟩, and z = ⟨(1, 2, 3)⟩. The structure of the transporter category T_{S} is pictured as below:

[figure: w; G_{w}; y; G_{y}; B_{yw}; z; G_{z}; B_{zw}; x; G_{x}; B_{xz}; B_{xy}]

By computation, we find G_{x} ≅ G_{z} ≅ G_{w} ≅ S_{3}, and G_{y} ≅ ⟨(1, 2)⟩. Moreover, the four bisets B_{xz} ≅ B_{xy} ≅ B_{yw} ≅ B_{zw} ≅ S_{3}. Note that we also have B_{yw}B_{xy} = B_{zw}B_{xz} since they both equal the set of morphisms from x to w, which is also isomorphic to a copy of S_{3}. This category can be graded in an obvious way. Let A be the graded category algebra.

The k-linear representations of this category (which are the same as kT_{S}-modules) depends on the characteristic of k. Let us do the computation for an algebraically closed field of characteristic 3. Then kS_{3} is the direct sum of two non-isomorphic indecomposable projective modules, and kG_{y} is semisimple. An explicit computation tells us that Ā = A/A rad A_{0}A is a direct product of two graded algebras, each being isomorphic to the graded incidence algebra of the following poset with length grading:

\xymatrix{ & • \\ • \ar[ur] &  & • \ar[ul] \\  & • \ar[ur] \ar[ul]}

Since this graded incidence algebra is classical Koszul, by Theorem 3.6, the category algebra kC is generalized Koszul.

However, we do not know for what transporter categories or orbit categories, the category algebras or the associated graded algebras are generalized Koszul.

5. Applications to standardly stratified algebras

Let A be a basic finite-dimensional algebra whose simple modules S_{λ} (up to isomorphism) are indexed by a preordered set (Λ, ⩽). This preordered set also indexes all indecomposable projective A-modules P_{λ} up to isomorphism. According to [?], A is standardly stratified with respect to ⩽ if there exist modules Δ_{λ}, λ ∈ Λ, such that the following conditions hold:

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

such that K_{λ} has a filtration with factors Δ_{μ} where μ > λ.

The standard module Δ_{λ} is the largest quotient of P_{λ} with only composition factors S_{μ} such that μ ⩽ λ. It has the following description:

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

where tr_{P_{μ}}(P_{λ}) is the trace of P_{μ} in P_{λ} ([?, ?]).

Throughout this section we suppose that A is standardly stratified with respect to a partial order ⩽. Let Δ be the direct sum of all standard modules and F(Δ) be the full subcategory of A-mod such that each object in F(Δ) has a filtration by standard modules. For M ∈ F(Δ) and λ ∈ Λ, we take a particular Δ-filtration ξ and define the multiplicity [M : Δ_{λ}] to be the number of factors in ξ isomorphic to Δ_{λ}. It is well known that the multiplicity is independent of the choice of a particular filtration.

Since standard modules are relative simple in the category F(Δ) and have finite projective dimensions, the extension algebra Γ = Ext_{A}^{∗}(Δ, Δ) of standard modules is a graded finite-dimensional algebra, and provides us a lot information on the structures of indecomposable objects in F(Δ). The structure of Γ has been considered in [?, ?, ?, ?, ?, ?, ?, ?, ?]. However, most authors assumed that A has a grading such that A_{0} is semisimple, and this grading is compatible with the Δ-filtration. That is, the top of each standard module and the socle of each costandard module lie in degree 0. Using this assumption, they show that the Koszul dual algebra is also standardly stratified, or the Ringel dual algebra is also Koszul if so is A. With some extra assumptions, they can prove that the Ringel dual and the Koszul dual commute.

In this paper we study the extension algebras of standard modules from a different approach. In general we do not assume that the standardly stratified algebra is graded. Indeed, since the extension algebra has a natural grading, if the Δfiltration of indecomposable projective modules have nice combinatorial structure, we can still get a Koszul property for the extension algebra. Moreover, since the degree 0 part of the extension algebra of standard modules is exactly the endomorphism algebra of standard modules, it is in general not semisimple. Therefore, our generalized Koszul theory can apply to this situation.

We describe some preliminary results in [?], where ⩽ is supposed to be a partial order.

Theorem 5.1. If A is standardly stratified for a poset (Λ, ⩽), then the associated category E of Γ is a directed category with respect to ⩽ and is standardly stratified for ⩽^{op}. Moreover, E is standardly stratified for ⩽ if and only if for all λ, μ ∈ Λ and s ⩾ 0, Ext_{A}^{s}(Δ_{λ}, Δ_{μ}) is a projective End_{A}(Δ_{μ})-module.

Proof. This is Theorem 1.4 in [?]. □

We introduce some notations. Let Λ_{1} be the subset of all minimal elements in Λ, Λ_{2} be the subset of all minimal elements in Λ\Λ_{1}, and so on. Then Λ = ⊔_{i⩾1}Λ_{i}. With this partition, we can introduce a height function h : Λ → ℕ in the following way: for λ ∈ Λ_{i} ⊆ Λ, i ⩾ 1, we define h(λ) = i. For each M ∈ F(Δ), we define supp(M) to be the set of elements λ ∈ Λ such that [M : Δ_{λ}] ≠ 0. For example, supp(Δ_{λ}) = {λ}. We also define min(M) = min({h(λ) | λ ∈ supp(M)}). We say M is generated in height i if every simple summand of M/ rad M is isomorphic to some S_{λ} with h(λ) = i. For example, the standard module Δ_{λ} is generated in height h(λ).

The following definition is an analogue of generalized Koszul modules of graded algebras.

Definition 5.2. An A-module M ∈ F(Θ) is said to be linearly filtered if there is some i ∈ ℕ such that Ω_{Θ}^{s}(M) is generated in height i + s for s ⩾ 0. Equivalently, M ∈ F(Θ) is linearly filtered if and only if it is generated in height i and has a projective resolution

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

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

With this terminology, we have:

Theorem 5.3. Suppose that Δ ≅ Γ_{0} as a Γ_{0}-module and Δ_{λ} is linearly filtered for each λ ∈ Λ. Then:

(1) Γ is a projective Γ_{0}-module.

(2) If M ∈ F(Δ) is linearly filtered, then Ext_{A}^{∗}(M, Θ) is a generalized Koszul Γ-module.

(3) In particular, Γ is a generalized Koszul algebra.

Proof. The last two statements come from Theorem 2.12 in [?] by letting Θ = Δ and Q =_{A} A. Thus we only need to show the first statement. But by (2.2) in page 14 of [?], we know that Γ_{s} = ⨁_{λ∈Λ} Ext_{A}^{s}(Δ_{λ}, Δ) is a projective Γ_{0}-module for every s ⩾ 1. □

We remind the reader that although Γ is a generalized Koszul algebra, the Koszul duality in general does not hold since Γ_{0} might not satisfy the splitting contion (S). See Example 2.14 in [?].

As before, let 𝔯 = rad Γ_{0} and Γ̄ = Γ/Γ𝔯Γ. Then we have an immediate corollary.

Corollary 5.4. Suppose that Δ ≅ Γ_{0} as a Γ_{0}-module and Δ_{λ} is linearly filtered for each λ ∈ Λ. Then the quotient algebra Γ̄ is a classical Koszul algebra.

Proof. The conclusion follows from Theorems 5.3 and 3.6. □

Example 5.5. Let A be the path algebra of the following quiver with relations δ^{2} = δα = βδ = βα = γβ = 0. Let x > z > y.

[figure: x; α; z; γ; y; β; δ]

Indecomposable projective modules and standard modules of A are described below:

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

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

Clearly, A is standardly stratified. Moreover, all standard modules have projective dimension 1 and are linearly filtered. By direct computation we check that Δ ≅ End_{A}(Δ) as Γ_{0} = End_{A}(Δ)-modules.

Now we compute the extension algebra Γ: Γ_{s} = 0 for s ⩾ 2; Ext_{A}^{1}(Δ_{x}, Δ) = 0; Ext_{A}^{1}(Δ_{y}, Δ) ≅ End_{A}(Δ_{z}); and Ext_{A}^{1}(Δ_{z}, Δ) ≅ End_{A}(Δ_{x}). Therefore, we find Γ is the path algebra of the following quiver with relations δ^{2} = βδ = αδ = γβ = 0.

[figure: x; z; γ; y; β; α; δ]

We remind the reader that α is in the degree 0 part of Γ. Indeed, Γ_{0} = ⟨1_{x}, 1_{y}, 1_{z}, δ, α⟩ and Γ_{1} = ⟨β, γ⟩. In this case Γ_{0} does not satisfy the splitting condition (S) since we can find the following non-splitting exact sequence:

0 → x → \begin{aligned}& y \\ y & & x \end{aligned} → \begin{matrix}y \\ y \end{matrix} → 0.

Since 𝔯 = rad Γ_{0} = ⟨δ, α⟩ = Γ𝔯Γ, the quotient algebra Γ is the path algebra of the following quiver with relation γβ = 0, which is clearly a classical Koszul algebra.

\xymatrix{x & z \ar[l]_{γ} & y \ar[l]_{β}} .

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

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