Dualities on generalized Koszul algebras

Edward L. Green

Idun Reiten

Øyvind Solberg

Author address:

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA

E-mail address: green@math.vt.edu

Institutt for matematiske fag, NTNU, Lade, N–7491 Trondheim, Norway

E-mail address: idunr@math.ntnu.no

Institutt for matematiske fag, NTNU, Lade, N–7491 Trondheim, Norway

E-mail address: oyvinso@math.ntnu.no

Contents

Introduction iv Acknowledgment vii

Notation and general preliminary results viii

Chapter I. Main results and examples 1 1. Two classical cases 1 2. Statements of main results 4 3. T -Koszul algebras concentrated in degrees 0 and 1 7 4. Further examples and classes of T -Koszul algebras 10

Chapter II. Main results with proofs 17 1. Preliminaries 17 2. Duality 23 3. T -Koszul algebras and quadratic algebras 28

Chapter III. Generalized T -Koszul algebras 30 1. Natural properties of Koszul modules 30 2. Generalized T -Koszul algebras 35 3. The classical cases revisited 37 4. Necessity of the ⊗-Tor-conditions 38 5. Properties of the Ext-functor 39 6. Duality 44

Chapter IV. Further results and questions 48 1. Modules generated in 0 48 2. T -Koszul algebras in degrees 0 and 1, revisited 51 3. Ext-algebras for injective cotilting modules 53 4. rej_{T} = 0 and corej_{T} = 0 58 5. Questions 59

Bibliography 65

Abstract

Koszul rings are graded rings which have played an important role in algebraic topology, algebraic geometry, noncommutative algebraic geometry, and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules. When Λ is an artin algebra and T is a cotilting Λ-module, the functor Hom_{Λ}( , T) induces a duality between certain subcategories of the finitely generated modules over Λ and End_{Λ}(T).

The purpose of this paper is to develop a unified approach to both the Koszul duality and the duality for cotilting modules. This theory specializes to these two cases as well as containing new interesting examples. The starting point for the theory is a positively ℤ-graded ring Λ = Λ_{0} + Λ_{1} + Λ_{2} + ⋯ and a (Wakamatsu) cotilting Λ_{0}-module T, satisfying additional assumptions. The theory gives a duality between certain subcategories of the finitely generated graded modules generated in degree zero over Λ and the extension algebra ⊕_{i≥0} Ext_{Λ}^{i}(T, T).

1991 Mathematics Subject Classification. Primary 16W50, 16S37, 16D90; Secondary 18E05, 16E05.

Introduction

Let Λ be a ring and M a (left) Λ-module. Let Γ = End_{Λ}(M). Then M is a Λ-Γ-bimodule, and we have a natural ring homomorphism Λ ⟶^{α} End_{Γ}(M). The case when α is an isomorphism is of special interest, and in this case M is said to be faithfully balanced. Denote by Mod Λ the category of left Λ-modules. For X in Mod Λ, the homomorphism set Hom_{Λ}(M, X) is a Γ^{op}-module. Hence we have a natural functor (M, ): Mod Λ → Mod Γ^{op}. Here we use the notation (M, X) for Hom_{Λ}(M, X), since no confusion can arise. Then M is taken to Γ and Λ is taken to the Γ^{op}-module (M, Λ). So the pair (Λ, M) is taken to the pair (Γ, (M, Λ)). This correspondence with a pair consisting of a ring and a module over the ring was investigated by M. Auslander for finitely generated modules over artin algebras, and was called Wedderburn correspondence in [1]. It has been of interest to find appropriate subcategories of Mod Λ and of Mod Γ^{op} such that (M, ) induces an equivalence of categories, as generalization of the Morita equivalence theorem.

We also have the functor ( , M) from Mod Λ to Mod Γ. In this case M is taken to Γ and Λ is taken to M, so that the pair (Λ, M) is taken to the pair (Γ, M) via the natural correspondence of pairs. Hence from the point of view of correspondence of such pairs, the contravariant functor ( , M) is more natural than the covariant functor (M, ). It is now of interest to find natural subcategories of Mod Λ and of Mod Γ such that ( , M) induces a duality.

Examples in the representation theory of artin algebras come from tilting and cotilting modules, and there are also generalizations to other rings. A module T is a tilting module over Λ if pd_{Λ}T < ∞, Ext_{Λ}^{i}(T, T) = (0) for i > 0, and there is an exact sequence 0 → Λ → T_{0} → T_{1} → T_{2} → ⋯ → T_{n} → 0 with T_{i} in add T. Then T is also a tilting Γ-module, and End_{Γ}(T) ≃ Λ. A module T is a cotilting Λ-module if id_{Λ}T < ∞, Ext_{Λ}^{i}(T, T) = (0) for i > 0, and there is an exact sequence 0 → T_{n} → ⋯ → T_{1} → T_{0} → D(Λ^{op}) → 0 with T_{i} in add T. Then T is a cotilting Γ-module, and the contravariant functor ( , T) induces a duality between _{Λ}^{⊥}T and _{Γ}^{⊥}T, where ^{⊥}X = {M ∈ mod Λ | Ext_{Λ}^{i}(M, X) = (0) for all i > 0}.

Instead of associating with the pair (Λ, M) the endomorphism ring End_{Λ}(M), we can associate the Ext-algebra Γ = ⊕ Ext_{Λ}^{i}(M, M). Since Γ is in a natural way a graded ring, it is natural to also start with a graded ring Λ and a graded Λ-module M. Then M is again a Γ-module, as before. So assume we have the following setting: Λ is a graded ring Λ = Λ_{0} +Λ_{1} +⋯+Λ_{i} +⋯, where Λ_{0} is an artin algebra and the Λ_{i} are finitely generated as left and as right Λ_{0}-modules, and let M = Λ_{0} and let Γ be the Ext-algebra E(Λ) = ⊕ Ext_{Λ}^{i}(Λ_{0}, Λ_{0}). Here Ext_{Λ}^{i}( , ) denotes the ordinary i-th ungraded extension group. Then we can consider the condition that Λ → E(E(Λ)) is an isomorphism. Let E denote the functor ⊕ Ext_{Λ}^{i}( , M). This functor has been investigated in the following setting: Λ_{0} is semisimple (not necessarily an artin algebra) and M = Λ_{0}. Then Λ is said to be Koszul if there is an isomorphism Λ → ⊕ Ext_{Γ}^{i}(Γ_{0}, Γ_{0}). Let Gr Λ denote the subcategory of Mod Λ consisting of graded modules with degree 0 morphisms. Denote by gr Λ the full subcategory of Gr Λ consisting of the finitely generated modules. A natural functor from Gr Λ to Gr Γ is in this case E = ⊕ Ext_{Λ}^{i}( , M), and when Λ is Koszul, then there are natural subcategories of Gr Λ and of Gr Γ, which are called the Koszul modules, where E induces a duality.

The two cases discussed above are analogous, since for a cotilting module the endomorphism algebra and the Ext-algebra coincide. Actually, one of the motivations for this paper was to provide a general framework covering both the cotilting situation and the case of classical Koszul algebras, with respect to duality of subcategories.

We also point out that a more general framework is natural. To explain this we need to recall some constructions concerning relative homology from [6].

Given an additive subfunctor F of Ext_{Λ}^{1}( , ) an exact sequence η: 0 → A → B → C → 0 is F -exact if η is in F (C, A). A module P is called F -projective or relative projective if 0 → (P, A) → (P, B) → (P, C) → 0 is exact for all for F -exact sequences 0 → A → B → C → 0. We denote the full subcategory of relative projective modules by P(F). An additive subfunctor F of Ext_{Λ}^{1}( , ) is said to have enough projectives if for every module C in Mod Λ (or mod Λ, depending on the context we work in) there exists an F -exact sequence 0 → K → P → C → 0 with P relative projective.

For a subcategory X in Mod Λ let F_{X} be the additive subfunctor of Ext_{Λ}^{1}( , ) given by

F_{X} (C, A) = {0 → A → B → C → 0 | Hom_{Λ}( , B)|_{X} → Hom_{Λ}( , C)|_{X} → 0 is exact}

for all pairs of modules A and C in Mod Λ. Then F_{X} defines a family of exact sequences which is closed under pushouts, pullbacks and Baer sums. Dually we have that F^{X} given by

F^{X} (C, A) = {0 → A → B → C → 0 | Hom_{Λ}(B, )|_{X} → Hom_{Λ}(A, )|_{X} → 0 is exact}

for all pairs of modules A and C in Mod Λ is an additive subfunctor of Ext_{Λ}^{1}( , ).

If X is a contravariantly finite subcategory of Mod Λ closed under summands and containing the projective Λ-modules, it is straightforward to see that P(F_{X}) = X and that F_{X} has enough projectives. In this case we can construct the relative derived functors of Hom_{Λ}( , ) using relative projective resolutions, and we denote them by Ext_{F_{X}}^{i} ( , ).

The general framework is described by quadruples (Λ, T, C, F_{X}), where Λ is a ring, T a Λ-module, C a subcategory of M (which is mod Λ or Mod Λ) containing T and F_{X} a subfunctor of Ext_{Λ}^{1}( , ) given by a contravariantly finite subcategory X of M closed under summands and containing P(Λ), where P(Λ) denotes the subcategory of projective Λ-modules. Then we define E_{T} (Λ) to be the Ext-algebra ⊕_{i≥0} Ext_{F_{X}}^{i} (T, T) and E the functor ⊕_{i≥0} Ext_{F_{X}}^{i} ( , T): Gr Λ → Gr E_{T} (Λ). From the quadruple (Λ, T, C, F_{X}) we construct the quadruple (E_{T} (Λ), E(Λ), E(X), F_{E(C)}), where it is clear from the assumptions that E(Λ) is in E(X) and that P(E_{T} (Λ)) is in E(C). However it is not necessarily true in general that E(C) is a contravariantly finite subcategory in the subcategory we would be interested in.

Now we explain how the above constructions can be viewed in this framework. We start by considering the construction of an endomorphism algebra.

Consider the pair (Λ, M) where Λ is an artin algebra, and M is a finitely generated Λ-module. Then let C = add(Λ ⊕ M) and X = C. As we pointed out above, P(F_{X}) = X, and therefore Ext_{F_{X}}^{i} (M, M) = (0) for i ≥ 1. Hence the quadruple corresponding to (Λ, M, C, F_{X}) is given by (Γ = End_{Λ}(M),_{Γ}M, add(Γ ⊕ ΓM), F_{add(Γ⊕_{Γ}M)}). We see that we can do the construction again, and we get back to the first quadruple if M is a faithful and balanced module.

Also associated with the pair (Λ, T) when T is a cotilting module, there is a natural quadruple (Λ, T, add T, F_{P(Λ)}), where F_{P(Λ)} = Ext_{Λ}^{1}( , ). This induces the quadruple (Γ = End_{Λ}(T),_{Γ}T , add_{Γ}T , F_{P(Γ)}). Repeating the construction, it is well known that we get back to the quadruple we started with. We also point out that the construction of the endomorphism algebra of a relative cotilting module can be viewed in this framework.

Now we discuss the classical Koszul situation from this point of view. Let Λ = ⊕_{i≥0}Λ_{i} be a positively ℤ-graded ring generated in degrees 0 and 1, where Λ_{0} is semisimple artin algebra and Λ_{i} is a Λ_{0}-Λ_{0}-bimodule of finite length over Λ_{0} for all i ≥ 1. Then consider the quadruple (Λ, Λ_{0}, add Λ_{0}, F_{P(Λ)}), which induces the quadruple (Γ = ⊕_{i≥0} Ext_{Λ}^{i}(Λ_{0}, Λ_{0}), Γ_{0}, add Γ_{0}, F_{P(Γ)}). Clearly we can do the construction again, and we return to the original quadruple if and only if Λ is a Koszul algebra.

The main object of study in this paper is to give a natural framework for a duality theory in a setting containing both the duality given by cotilting modules for artin algebras and the duality for Koszul modules over Koszul rings Λ = Λ_{0}+Λ_{1}+⋯ when the semisimple ring Λ_{0} is an artin algebra. So we want to identify appropriate pairs (Λ, T) for this purpose.

We start with a positively ℤ-graded ring Λ = ⊕_{i≥0}Λ_{i} generated in degrees 0 and 1, where Λ_{0} is an artin algebra and Λ_{i} is a Λ_{0}-Λ_{0}-bimodule of finite length over Λ_{0} for all i ≥ 1. For a Λ-module T we want to study the functors

E = ⊕_{i≥0} Ext_{Λ}^{i}( , T): Gr Λ → Gr E_{T} (Λ)

and

E′ = ⊕_{i≥0} Ext_{E_{T} (Λ)}^{i}( , T′): Gr E_{T} (Λ) → Gr E_{T′}(E_{T} (Λ)),

where T′ = E(Λ).

In generalizing the classical cases one needs to decide which of the characteristic properties one wants to keep. We have already emphasized the duality property. In addition we would like to have as many essential properties of the category of Koszul modules as possible. We consider operations under which this category is closed, and compare with properties of the subcategory ^{⊥}T associated with a cotilting module. Actually, we consider a generalization of a cotilting module, investigated by Wakamatsu, which we call a Wakamatsu cotilting module.

We first give an approach to this problem in Chapters I and II, which follows closely the classical approach. To start with we assume some conditions on Λ, which we call the ⊗-Tor-conditions. They are trivially satisfied in the two special cases, and may look artificial. However, we show later that they are almost necessary in order to have a duality theory. We prove a duality theorem in this setting. For the benefit of the reader we collect in Chapter I definitions and properties which are essential from our point of view, from each of the two classical cases.

A second approach, which turns out to be more general, is treated in Chapter III. Here we emphasize which categorical properties we would like to have for the category of Koszul modules, by analyzing the two motivating cases. At the same time we have in mind that we would like to have a good duality theory.

In Chapter IV we discuss some further results, along with some open questions. There is another generalization of a graded ring Λ = Λ_{0} + Λ_{1} + ⋯ being

Koszul, due to Woodcock [18]. He assumes that Λ is projective as a left and as a right Λ_{0}-module, i.e. Λ_{i} is projective as a left and as a right Λ_{0}-module. This is not necessarily the case in our setting. Moreover he has no restrictions on Λ_{0}. His point of view is considering the Koszul complex as the defining property he wants to generalize. However, assuming that Λ is a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra, and choosing T = D(Λ_{0}), one can show that Λ is a T -Koszul algebra in the sense of this paper.

Acknowledgment

The first author wants to thank NSF, NSA and the American Scandinavian Foundation for support and his coauthors for their kind hospitality and support during his visits in Trondheim.

The last two authors would like to thank their coauthor and Virginia Tech for kind hospitality and support during their visits in Blacksburg. They also thank the Norwegian Research Council and NTNU for support during the preparation of this paper.

Notation and general preliminary results

In this chapter we try to make this paper as selfcontained as possible, as we attempt to include all the different notation, definitions and preliminary results that we need in the succeeding chapters.

We start by recalling some notions, definitions and results from the representation theory of artin algebras. For a general reference on this we refer the reader to [4].

A quiver is an oriented graph Γ, where Γ_{0} denotes the set of vertices and Γ_{1} denotes the set of arrows. A path in the quiver Γ is an ordered sequence of arrows a_{n}a_{n−1} ⋯ a_{1} or v for some v in Γ_{0}, where the arrow a_{i} ends where the arrow a_{i+1} starts for i = 1, 2, …, n − 1. The paths v for v in Γ_{0} is called the trivial paths. Now, let k be any field. Let kΓ denote the vector space over k with the paths of Γ as a k-basis. The path algebra of Γ over k is kΓ, where the multiplication is defined as follows. It is enough to define the multiplication of two paths p and q. We define

p · q = \begin{cases}pq & if q ends where p starts and p and q are both nontrivial paths, \\ p & if q = v for the vertex v where p starts in Γ, \\ q & if p = v for the vertex v where q ends in Γ, \\ 0 & otherwise. \end{cases}

A finite dimensional algebra Λ over a field k is said to be basic if Λ ≃ ⊕_{i=1}^{n}P_{i} for some indecomposable projective Λ-modules P_{i}, then P_{i} ≄ P_{j} for i ≠ j.

It is known that for a finite dimensional basic algebra Λ over an algebraically closed field k, there exists a finite quiver Γ and an ideal I in kΓ, such that Λ ≃ kΓ/I. Moreover, the ideal I contains some power of the ideal J in kΓ generated by the arrows in Γ and I is contained in J^{2}. An ideal I satisfying these conditions is called an admissible ideal. The radical of kΓ/I is J/I and kΓ/J ≃ k^{n}, where n is the number of vertices in Γ.

The following is a list of definitions related to subcategories of the category mod Σ of finitely generated Σ-modules for an artin algebra Σ.

Definition 0.1. Let Σ be an artin algebra, and let X be a full subcategory of mod Σ.

(a) The subcategory X is called extension closed if for all exact sequence 0 → A → B → C → 0 with A and C in X, then B is in X.

(b) The subcategory X is called resolving if X is closed under kernel of epimorphisms and extensions and X contains the finitely generated projective Σ-modules.

(c) The full subcategory X̂ consists of all modules C in mod Σ such that there exists an exact sequence 0 → X_{n} → X_{n−1} → ⋯ → X_{2} → X_{1} → X_{0} → C → 0 for some n and with X_{i} in X for all i ≥ 0.

(d) A morphism f: X_{C} → C with X_{C} in X is called a left X -approximation if for any morphism g: X → C for X in X, there exists a morphism h: X → X_{C} such that fh = g. A left X -approximation f: X_{C} → C is called minimal if for any morphism g: X_{C} → X_{C} with fg = f implies that g is an isomorphism.

(d) A morphism f: A → X^{A} with X^{A} in X is called a right X -approximation if for any morphism g: A → X for X in X, there exists a morphism h: X^{A} → X such that hf = g. A right X -approximation f: A → X^{A} is called minimal if for any morphism g: X^{A} → X^{A} with gf = f implies that g is an isomorphism.

(e) The subcategory X is called contravariantly finite in mod Σ if each module C in mod Σ has a left X -approximation. The subcategory X is called covariantly finite in mod Σ if each module C in mod Σ has a right X - approximation.

(f) The subcategory X is called funtorically finite in mod Σ if X is both covariantly and contravariantly finite in mod Σ.

Next we recall some notions concerning tilting and cotilting theory for artin algebras.

Definition 0.2. Let Σ be an artin algebra, and let X be a full subcategory of mod Σ.

(a) A Σ-module T is called selforthogonal if Ext_{Σ}^{i}(T, T) = (0) for all i > 0.

(b) Let C be a subcategory of mod Σ. Then ^{⊥}C denotes the full subcategory {X ∈ mod Σ | Ext_{Σ}^{i}(X, C) = (0) for all C ∈ C}.

(c) Let C be a subcategory of mod Σ. Then C^{⊥} denotes the full subcategory {X ∈ mod Σ | Ext_{Σ}^{i}(C, X) = (0) for all C ∈ C}.

(d) For a Σ-module M the full subcategory Sub(M) of mod Σ consists of all submodules of a finite direct sum of direct summands of M.

(e) For a Σ-module M the full subcategory add M of mod Σ consists of all direct summands of finite direct sums of copies of M.

For a ring Σ and a Σ-module M we denote by M^{∗} the Σ^{op}-module Hom_{Σ}(M, Σ). Recall that an artin algebra Σ is said to satisfy the Generalized Nakayama

conjecture if Ext_{Σ}^{i}(M, M ⊕ Σ) = (0) for all i > 0 for a Σ-module M implies that M is projective.

We need to recall the following notion from [6]. Let X be a finitely generated Λ_{0}-module. A Λ_{0}-module U is said to be dualized

by X if the natural homomorphism

U → Hom_{Σ}(Hom_{Λ_{0}}(U, X), X)

is an isomorphism, where Σ = End_{Λ_{0}}(X). It is shown in [6] that U is dualized by X if and only if there exists an exact sequence

0 → U ⟶^{f} X_{0} → X_{1}

with X_{0} and X_{1} in add X, where f is a (minimal) left add X-approximation.

Recall that we introduced the subfunctor F^{X} of Ext_{Λ}^{1}( , ) for a ring Λ and a subcategory X of Mod Λ in the introduction. When the subcategory X is equal to add T for some module T, then we write F^{T} for F^{add T}.

We now recall some notions and definitions for graded rings.

Definition 0.3. Let Λ = ⊕_{i≥0}Λ_{i} be a positively ℤ-graded ring.

(a) Λ is called a left finite graded ring if Λ_{i} is a finitely generated left Λ_{0}-module for all i ≥ 1.

(b) Λ is said to be generated in degrees 0 and 1 if Λ_{i} = Λ_{1}Λ_{i−1} for all i ≥ 1.

(c) Let M be a graded Λ-module, then (i) M[n] is the graded Λ-module such that M[n]_{i} = M_{−n+i} for all i in ℤ. (ii) M_{≥n} is the graded submodule of M such that

(M_{≥n})_{i} = \begin{cases}(0) & for i < n, \\ M_{i} & otherwise. \end{cases}

(iii) ⟨M_{n}⟩ denotes the graded submodule of M generated by M_{n}.

(d) Let M be a graded Λ-module. If U is a Λ_{0}-module of M, then ⟨U⟩ denotes the (graded) Λ-submodule of M generated by U.

Let Λ be a positively ℤ-graded ring, with Λ_{0} an artin algebra over some commutative artin ring R. Assume that Λ_{i} is finitely generated as a left Λ_{0}-module. Then Λ_{i} is finitely generated as a module over R. This implies that Λ_{i} is finitely generated as a right Λ_{0}-module. Hence being a left finite graded and a right finite graded ring is equivalent when Λ_{0} is an artin algebra.

Let Λ be a positively ℤ-graded ring. Denote by Gr Λ the category of graded Λ-modules with degree zero morphisms. The full subcategory of Gr Λ consisting of the finitely generated modules is denoted by gr Λ. We denote by gr_{0} Λ the full subcategory of gr Λ consisting of the modules generated in degree 0. Denote by fgsyz Λ the full subcategory of gr Λ consisting of all modules X such that Ω_{Λ}^{i}(X) is finitely generated for all i ≥ 1.

For any ring Λ we sometimes abbreviate Hom_{Λ}(A, B) as _{Λ}(A, B) or just (A, B). For a positively ℤ-graded ring Λ we denote by Ext_{Λ}^{i}(C, A) the ordinary i-th ungraded extension group and by Ext_{gr Λ}^{i}(C, A) the i-th graded extension group.

We end this chapter with an elementary result on graded modules, where we leave the proof to the reader.

Lemma 0.4. Let Λ = ⊕_{i≥0}Λ_{i} be a positively ℤ-graded ring. Let X be a graded Λ-module generated in positive degrees, and let T be a graded Λ-module concentrated in degree 0. Then the exact sequence 0 → ⟨X_{0}⟩X → X/⟨X_{0}⟩ → 0 induces a split exact sequence

0 → Hom_{Λ}(X/⟨X_{0}⟩, T) → Hom_{Λ}(X, T) → Hom_{Λ_{0}}(X_{0}, T) → 0.

CHAPTER I Main results and examples

In order to get an understanding for the background for our theory of T -Koszul algebras, we first give an account of some of the main results in the two classical cases. We then state some of the central results of this paper. We want to quickly get to a variety of examples, so we postpone the proofs of these results to the next chapter.

1. Two classical cases

In this section we recall relevant properties for cotilting modules and Koszul algebras, in particular with respect to existence of dualities for subcategories.

Tilting theory has played a central role in the representation theory of artin algebras [3, 12, 13, 17]. One of the aspects of the theory deals with equivalence or duality between appropriate subcategories of module categories. We deal here with artin algebras, even though there are analogous results in more general contexts.

Let Λ be an artin algebra and denote by mod Λ the category of finitely generated (left) Λ-modules. Let T in mod Λ be selforthogonal. We say T is a tilting module if pd_{Λ}T < ∞ and there is an exact sequence 0 → Λ → T_{0} → T_{1} → ⋯ → T_{n} → 0 with the T_{i} in add T. Dually, T is a cotilting module if id_{Λ}T < ∞ and there is an exact sequence 0 → T_{n} → ⋯ → T_{1} → T_{0} → DΛ → 0 with the T_{i} in add T, where D(Λ) is an injective generator.

Associated with a cotilting module T is the full subcategory ^{⊥}T of mod Λ, whose objects are modules C with Ext_{Λ}^{i}(C, T) = (0) for i > 0. We have the following important properties for ^{⊥}T.

Theorem 1.1. If T is a cotilting module, ^{⊥}T has the following properties.

(a) ^{⊥}T is resolving; that is, contains Λ and is closed under extensions and kernels of epimorphisms.

(b) ^{⊥}T is functorially finite.

(c) ^{⊥}T has almost split sequences.

Let Γ = End_{Λ}(T) for a cotilting Λ-module T. Then T is a cotilting module over Γ, the algebra Λ is isomorphic to End_{Λ}(T), and the module theories for mod Λ and mod Γ are closely connected in the following way.

Theorem 1.2. If T is a cotilting module, the functor Hom_{Λ}( , T): mod Λ → mod Γ induces a duality Hom_{Γ}( , T): _{Λ}^{⊥}T → _{Γ}^{⊥}T.

Note that D(Λ) is clearly a cotilting module. In this case _{Λ}^{⊥}D(Λ) = mod Λ, the algebra Γ is isomorphic to Λ^{op}, the subcategory _{Γ}^{⊥}D(Λ) is mod Λ^{op}, and we recover the duality D = Hom_{Λ}( , D(Λ)): mod Λ → mod Λ^{op}.

In the general case T is an injective object in^{⊥}T in the sense that Ext_{Λ}^{i}(C, T) = (0) for i > 0 and C in ^{⊥}T. In addition we have the following.

Proposition 1.3. Let T be a cotilting Λ-module. Then for each C in ^{⊥}T there

is an exact sequence 0 → C → T_{0} ⟶^{f_{0}} T_{1} ⟶^{f_{1}} T_{2} → ⋯ → T_{n} → ⋯ with T_{i} in

add T and Ext_{Λ}^{1}(Ker f_{i}, T) = (0) for i > 0.

We shall mainly deal with the following common generalization of a tilting and cotilting module considered by Wakamatsu [17]. Let T be selforthogonal module, and denote by X_{_{Λ}T} or, simply by X_{T}, the full subcategory of ^{⊥}T whose objects are the modules C for which there is an exact sequence 0 → C → T_{0} ⟶^{f_{0}} T_{1} ⟶^{f_{1}} T_{2} → ⋯ with T_{i} in add T and Ext_{Λ}^{1}(Ker f_{i}, T) = (0) for all i > 0. We have the following property for X_{T}.

Proposition 1.4. X_{T} is closed under extensions and kernels of epimorphisms.

We say that T is a Wakamatsu cotilting module if Λ is in X_{T} or, equivalently, if X_{T} is resolving. Replacing ^{⊥}T by X_{T} in the more general context we have some similar results. The next result shows that this more general concept is preserved by duality, while this is not the case for cotilting (or tilting) modules.

Theorem 1.5. Let T be a Wakamatsu cotilting Λ-module and Γ = End_{Λ}(T).

(a) T is a Wakamatsu cotilting Γ-module.

(b) D(T) is a Wakamatsu cotilting Λ^{op}-module.

(c) End_{Γ}(T) ≃ Λ.

(d) There is a duality Hom_{Λ}( , T): X_{_{Λ}T} → X_{_{Γ}T}.

We shall need the following, which is a direct consequence of the definitions.

Proposition 1.6. Let T be a Wakamatsu cotilting Λ-module.

(a) If 0 → A → B → C → 0 is an exact sequence in mod Λ such that 0 → (C, T) → (B, T) → (A, T) → 0 is exact, and A and B are in X_{T}, then C is in X_{T}.

(b) X_{T} ⊂ Sub T.

There are examples of Wakamatsu cotilting modules T where both the projective and injective dimension is infinite. For example, this is the case for the path algebra of the quiver

\xymatrix{1 \ar[r] & 2 \ar[l] & 3 \ar[l] & 4 \ar[l] \ar[r] & 5 \ar[l]}

over a field k, modulo the relations which are compositions of two arrows [W]. Let T = I_{2} ⊕ P_{2} ⊕ P_{4} ⊕ P_{5} ⊕ S_{3}, where I_{j}, P_{j} and S_{j} denote the indecomposable injective, the indecomposable projective and the simple module over the path algebra associated to vertex j. In this example we have X_{T} = ^{⊥}T, even though T is not a cotilting module.

In addition to the definition there are the following criteria for a selforthogonal (in particular, Wakamatsu cotilting) module to be a cotilting module.

Theorem 1.7. Let T be a selforthogonal Λ-module.

(a) T is a cotilting module if and only if X̂_{T} = mod Λ.

(b) T is a cotilting module if and only if ^{⊥}T is contravariantly finite and ^{ ˆ⊥}T = mod Λ.

Koszul algebras and Koszul duality have been of interest in algebraic topology, algebraic geometry, noncommutative algebraic geometry, and in the theory of quantum groups. We now recall definitions and relevant results for Koszul algebras (see [9], [11] and [10]).

Let k be a field and Λ = Λ_{0} + Λ_{1} + ⋯ + Λ_{i} + ⋯ a left finite graded k-algebra, where we assume Λ_{0} ≃ k^{n} for some n. Consider the Ext-algebra E(Λ) = ⊕_{i≥0} Ext_{Λ}^{i}(Λ_{0}, Λ_{0}). Note that E(Λ) is a graded k-algebra with E(Λ)_{i} = Ext_{Λ}^{i}(Λ_{0}, Λ_{0}). We say that Λ is generated in degrees 0 and 1 if for each i ≥ 1 we have Λ_{i} = Λ_{1}Λ_{i−1}. The algebra Λ is Koszul if Λ and E(Λ) are generated in degrees 0 and 1.

A graded projective resolution

⋯ → P_{i} → ⋯ → P_{1} → P_{0} → C → 0

of C in gr Λ is said to be linear if each P_{i} is generated in degree i. We denote the full subcategory of gr Λ whose objects have linear projective resolutions by K(Λ), and call these modules the Koszul modules.

We recall some characterizations of Koszul algebras and Koszul modules. Denote by E = ⊕_{i≥0} Ext_{Λ}^{i}( , Λ_{0}): Gr Λ → Gr E(Λ) the natural functor.

Theorem 1.8. Let k be a field and Λ = Λ_{0} + Λ_{1} + ⋯ + Λ_{i} + ⋯ a left finite graded k-algebra, where we assume Λ_{0} ≃ k^{n} for some n. Then the following are equivalent when Λ is generated in degrees 0 and 1.

(a) Λ is a Koszul algebra.

(b) Λ_{0} has a linear projective resolution.

(c) E(E(Λ)) ≃ Λ as graded algebras.

Theorem 1.9. Let Λ be a Koszul algebra, and let C be in gr Λ. Then C is a Koszul module if and only if C and E(C) are generated in degree 0.

Of special interest for this paper is the fact that there is an associated duality.

Theorem 1.10. Let Λ be a Koszul algebra.

(a) The functor E: Gr Λ → Gr E(Λ) induces a duality E: K(Λ) → K(E(Λ)) between the Koszul modules.

(b) The category K(Λ) contains all the finitely generated projective and semisimple modules generated in degree 0.

(c) The module C is a projective module if and only if the E(Λ)-module E(C) is semisimple, and C is semisimple if and only if E(C) is projective.

(d) If C is in K(Λ), then Ω_{Λ}(C)[−1] and C_{≥_{1}}[−1] are in K(Λ).

We have the following basic properties for the category of Koszul modules.

Proposition 1.11. Let Λ be a Koszul algebra, and 0 → A → B → C → 0 an exact sequence of graded Λ-modules with A, B and C generated in degree 0.

(a) If A and C are in K(Λ), then B is in K(Λ).

(b) If A and B are in K(Λ), then C is in K(Λ).

Another important class of algebras are the quadratic algebras. Again let Λ_{0} = k^{n} and let V be a Λ_{0}-bimodule which is a Λ_{0}-module of finite length. Let K in V ⊗_{Λ_{0}} V be a Λ_{0}-sub-bimodule. Then the factor algebra Λ = T_{Λ_{0}}(V)/⟨K⟩, where T_{Λ_{0}}(V) denotes the tensor algebra, is said to be a quadratic algebra. Associated with Λ is the algebra Λ^{!} = T_{Λ_{0}}(V^{∗})/⟨K^{⊥}⟩, where V^{∗} = Hom_{Λ_{0}}(V, Λ_{0}) and K^{⊥} ⊂ V^{∗} ⊗_{Λ_{0}} V^{∗} = (V ⊗_{Λ_{0}} V)^{∗} is perpendicular to K in V ⊗_{Λ_{0}} V. We recall the following.

Theorem 1.12. Let Λ be a Koszul algebra.

(a) Λ is quadratic.

(b) E(Λ) ≃ Λ^{!}.

2. Statements of main results

In this section we state some of the main results of this paper, and give the definitions necessary for this.

Throughout this section, Λ = Λ_{0}+Λ_{1}+⋯ will be a positively ℤ-graded algebra generated in degrees 0 and 1. Let gr_{0} Λ denote the full subcategory of gr(Λ) whose objects consist of graded modules generated in degree 0. We will freely view the category of Λ_{0}-modules and its various subcategories as embedded in gr_{0} Λ by viewing a Λ_{0}-module M_{0} as the graded module ∑_{i} N_{i} with N_{i} = (0) if i ≠ 0 and N_{0} = M_{0}.

We make the following assumptions throughout the section unless otherwise stated. First we will always assume that Λ_{0} is an artin algebra and that Λ is left finite. By T we will denote a Wakamatsu cotilting Λ_{0}-module.

Let Γ = ⊕_{n≥0} Ext_{Λ}^{n}(T, T). We define E: gr Λ → gr Γ by

E(M) = ⊕_{n≥0} Ext_{Λ}^{n}(M, T),

where E(M) has a natural graded Γ-module structure via the Yoneda product. We note that if Λ = Λ_{0}, then E, when restricted to X_{T}, is the Wakamatsu cotilting duality described in the previous section. Next we note that if Λ_{0} is a product of n copies of a field k, then the only Wakamatsu Λ_{0}-cotilting module is T = Λ_{0} up to multiplicity of direct summands. If Λ is a Koszul algebra, then the functor E restricts to the Koszul duality on the category of Koszul modules K(Λ). In this section, we define a general setting where Wakamatsu cotilting duality and Koszul duality fit in as special cases.

We shall define and introduce two conditions on Λ which will be important for our definition of an algebra which is Koszul with respect to T.

Definition 2.1. Let Y be a subcategory of mod Λ_{0}.

(a) We say that a Λ_{0}-bimodule B satisfies the ⊗-condition with respect to Y if B ⊗_{Λ_{0}} Y ⊆ Y.

(b) A Λ_{0}-bimodule B satisfies the Tor-condition with respect to Y, if Tor_{i}^{Λ_{0}}(B, Y) = (0) for each i ≥ 1.

(c) When T is a Wakamatsu cotilting Λ_{0}-module, we say that the graded ring Λ satisfies the ⊗-condition and/or the Tor-condition with respect to T if each Λ_{i} satisfies the corresponding condition with respect to X_{T}.

When both the ⊗-condition and the Tor-condition hold, we say that the ⊗- Tor-conditions are satisfied.

These conditions may seem artificial, but in Section 4 of Chapter III, they will be motivated by showing that conditions like ⊗-Tor-conditions are a necessary consequence of the existence of a duality theory.

Note that in both Wakamatsu cotilting theory and Koszul algebra theory, the ⊗-Tor-conditions are trivially satisfied, and these conditions will be part of our definition of an algebra being Koszul with respect to T. In Chapter III we give a more general setting for a duality theory to hold, but we postpone the general rather technical conditions for clarity of exposition.

Next we define a class of modules which will play the role of projective modules in the duality theory.

Definition 2.2. Assume that Λ satisfies the ⊗-Tor-conditions with respect to T. We say a graded Λ-module M is fat (with respect to T) if there is Λ_{0}- module X_{0} in X_{T} such that M is isomorphic to Λ⊗_{Λ_{0}} X_{0} as graded modules, where (Λ ⊗_{Λ_{0}} X_{0})_{n} = Λ_{n} ⊗_{Λ} X_{0}. In particular, if M is a fat module, then M is in gr_{0} Λ and M_{i} = Λ_{i} ⊗_{Λ_{0}} M_{0}.

We note that if Λ = Λ_{0}, then the fat modules are just the modules in X_{T}. If Λ is a Koszul algebra, then the fat modules are just the projective modules since the modules in X_{T} are just the finitely generated projective Λ_{0}-modules.

Next we define a type of resolution that will play the role that linear projective resolutions play in Koszul theory.

Definition 2.3. Let M be in gr Λ. We say that M has a linear fat resolution if there is an exact sequence of graded Λ-modules

⋯ → F_{n} → F_{n−1} → ⋯ → F_{1} → F_{0} → M → 0,

where for each n ≥ 0 the module F_{n}[−n] is fat.

Note that if Λ is a Koszul algebra, then the concept of linear fat resolutions coincides with the concept of linear projective resolutions.

We now define a Koszul algebra with respect to T.

Definition 2.4. Let Λ = Λ_{0} + Λ_{1} + ⋯ be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting Λ_{0}-module and assume that Λ satisfies the ⊗-Tor-conditions with respect to T. Then Λ is a Koszul algebra with respect to T if Λ_{0} has a linear fat resolution. For simplicity we say that Λ is a T -Koszul algebra.

Under this definition and because of our earlier remarks, it is clear that if Λ = Λ_{0} and T is a Wakamatsu cotilting module, then Λ is T -Koszul. If Λ is a left finite graded algebra generated in degrees 0 and 1 with Λ_{0} a finite product of a field, then T = Λ_{0} is the only Wakamatsu cotilting module up to multiplicity of direct summands. This implies that Λ is a Koszul algebra with respect to T if and only if it is a Koszul algebra (in the classical case).

The full subcategory of gr Λ whose objects are the graded modules M with linear fat resolutions, and where M_{i} is in X_{T} for all i ≥ 0, is denoted by K_{T} (Λ). If Λ is a Koszul algebra with respect to T, then the objects of the category K_{T} (Λ) will be called T -Koszul modules.

We now state some of the main results of the paper. The proofs of these results can be found in the next chapter.

Theorem 2.5. Let Λ be a Koszul algebra with respect to T. Then E_{T} (Λ) is a Koszul algebra with respect to T = E(Λ).

Note that E(Λ) = ⊕_{i≥0} Ext_{Λ}^{i}(Λ, T) ≃ Hom_{Λ}(Λ, T) ≃ T. Thus T has a natural Γ-module structure. It will be clear from the context whether we view T as a Λor Γ-module.

Theorem 2.6. Let Λ be a Koszul algebra with respect to T. Then, as graded algebras, Λ is isomorphic to ⊕_{n≥0} Ext_{Γ}^{n}(T, T).

We view this isomorphism as an identification when Λ is a T -Koszul algebra, and in this case we let E′: gr Γ → gr Λ denote the functor given by E′(M) = ⊕_{n≥0} Ext_{Γ}^{n}(M, T). The next theorem is the main duality result. If M is in K_{T} (Λ) and ⋯ → F_{2} → F_{1} → F_{0} → M → 0 is a linear fat resolution of M, then we denote the kernel of F_{0} → M by M^{1}.

Theorem 2.7. Let Λ be a Koszul algebra with respect to T.

(a) The functor E: gr Λ → gr Γ specializes to a duality E: K_{T} (Λ) → K_{T} (Γ), with inverse given by E′: K_{T} (Γ) → K_{T} (Λ).

(b) If M_{0} is in X_{T}, then Λ ⊗_{Λ_{0}} M_{0} and M_{0} are T -Koszul modules.

(c) If M is a T -Koszul module, then M_{≥1}[−1] and M^{1}[−1] are also T -Koszul modules.

(d) (i) E(M_{≥1}[−1]) = (E(M))^{1}[−1].

(ii) E(M^{1}[−1]) = (E(M))_{≥1}[−1].

(iii) If F = Λ ⊗_{Λ_{0}} M_{0} is a fat module, then E(F) = E(F)_{0}.

(iv) E(M_{0}) = Γ ⊗_{Γ_{0}} (E(M))_{0}.

To see that this duality specializes to the two situations discussed in the last section, we note that if Λ = Λ_{0}, it is immediate that K_{T} (Λ) = X_{T} and E(M) = Hom_{Λ}(M, T) if M is in X_{T}. Thus, Wakamatsu cotilting duality is a special case of Theorem 2.7. When Λ_{0} is a finite product of copies of a field, then fat modules are projective modules generated in degree 0, the category X_{T} is mod Λ_{0}, and linear fat resolutions are just linear projective resolutions. Hence, in this case, the category K_{T} (Λ) is the category of Koszul modules, and the duality described in Theorem 2.7 is Koszul duality.

The statements that fat modules and Λ_{0}-modules are T -Koszul modules correspond to the fact that for Koszul algebras, projective modules and semisimple modules are Koszul modules. The statement about M_{≥1} and M^{1} correspond to the statement that if M is a Koszul module, then so are the negative −1-shift of the graded radical of M and the first syzygy of M. Finally, using these correspondences, we see that the T -Koszul duality extends the usual Koszul duality.

Our duality result implies that if Λ is a T -Koszul algebra and M is in K_{T} (Λ) then, since E(M) is in K_{T} (Γ), we see that E(M) is generated in degree 0. Unlike what happens in the Koszul algebra case, E(M) being generated in degree 0 is not sufficient for M to be in K_{T} (Λ). Example 4.1 below provides an example of this. These observations lead to the following definitions.

Definition 2.8. Let Λ be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting Λ_{0}-module. We denote by G_{T} (Λ) the full subcategory of gr_{0} Λ consisting of modules M such that Ext_{Λ}^{1}(T, T) Ext_{Λ}^{i}(M, T) = Ext_{Λ}^{i+1}(M, T) for all i ≥ 0.

If Γ is generated in degrees 0 and 1, for example if Λ is T -Koszul, then G_{T} (Λ) consists of the modules M in gr_{0} Λ such that E(M) is generated in degree 0 as a Γ-module. We shall see that for a T -Koszul algebra Λ we always have that K_{T} (Λ) is contained in G_{T} (Λ). The case when we have equality is especially interesting, so it is convenient to give the following definition.

Definition 2.9. A T -Koszul algebra Λ is said to be strongly T -Koszul if K_{T} (Λ) = G_{T} (Λ).

3. T -KOSZUL ALGEBRAS CONCENTRATED IN DEGREES 0 AND 1 7

Many of the examples at the end of this section are strongly T -Koszul. In Example 4.1 in section 4 we give a T -Koszul algebra which is not a strongly T - Koszul algebra.

Next we note that the category of T -Koszul modules has some nice properties.

Proposition 2.10. Let Λ be a Koszul algebra with respect to T. Suppose that 0 → A → B → C → 0 is a short exact sequence of graded Λ-modules generated in degree 0. Suppose that the sequence is F^{T} -exact. Then we have the following.

(a) If A and C are T -Koszul modules, then B is a T -Koszul module.

(b) If A and B are T -Koszul modules, then C is a T -Koszul module.

The next result continues the strong analogy between Koszul algebras and T - Koszul algebras. It is immediate that Λ is generated in degrees 0 and 1 if and only if

Λ is a quotient of the tensor algebra T_{Λ_{0}}(Λ_{1}) = Λ_{0} +Λ_{1} +(Λ_{1} ⊗_{Λ_{0}} Λ_{1})+(Λ_{1}^{⊗_{Λ_{0}}^{3}} )+⋯

by an ideal contained in (Λ_{1} ⊗_{Λ_{0}} Λ_{1}) + (Λ_{1}^{⊗_{Λ_{0}}^{3}} ) + · · ·. We have the following result which generalizes the fact that Koszul algebras are quadratic algebras and that the Ext-algebra Γ has orthogonal quadratic relations.

Theorem 2.11. Let Λ be a Koszul algebra with respect to T, and let Γ = E_{T} (Λ). Let X_{2}^{2} be the kernel of the multiplication map Λ_{1} ⊗_{Λ_{0}} Λ_{1} → Λ_{2}. Then

(a) Λ is isomorphic to T_{Λ_{0}}(Λ_{1})/I where I is the ideal generated by X_{2}^{2}.

(b) Γ_{1} is isomorphic to Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T).

(c) Identify Γ_{0} and Γ_{1} with End_{Λ_{0}}(T) and Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T), respectively. Then Γ is isomorphic to T_{Γ_{0}}(Γ_{1})/J where J is the ideal generated by {∑ β_{i}⊗ α_{i} ∈ Γ_{1} ⊗_{Γ_{0}} Γ_{1} | ∑ β_{i}(id_{Λ_{1}} ⊗ α_{i})|_{X_{2}^{2}⊗_{Λ_{0}} T} = 0}.

3. T -Koszul algebras concentrated in degrees 0 and 1

We now turn our attention to the question of the existence of T -Koszul algebras beyond the classical cases discussed in Section 1. In general, most graded algebras Λ are not T -Koszul. Theorem 2.11 shows that they must be quadratic in a natural sense. But, as with the classical Koszul algebras, it is not clear which quadratic algebras are T -Koszul. This section deals with the “easiest” case where Λ = Λ_{0} + Λ_{1} + 0 + 0 + · · ·. The following gives a criterion for such an algebra to be T -Koszul and gives a description of the T -Koszul modules.

Theorem 3.1. Let Λ = Λ_{0} + Λ_{1} + 0 + 0 + ⋯ be a left finite graded algebra with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting Λ_{0}-module. Then

(a) Λ is a Koszul algebra with respect to T if and only if the ⊗-Tor-conditions with respect to T are satisfied.

(b) If Λ is a Koszul algebra with respect to T, then Γ = ⊕_{n≥0} Ext_{Λ}^{n}(T, T) is isomorphic to the tensor algebra T_{End_{Λ_{0}} (T)}(Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T)).

(c) If Λ is a Koszul algebra with respect to T, then the category of T -Koszul modules is the full subcategory of gr_{0} Λ consisting of modules M_{0}+M_{1}+0+⋯ such that M_{0} and M_{1} are in X_{T}.

(d) If T is a cotilting Λ_{0}-module and Λ is a Koszul algebra with respect to T, then Λ is a strongly T -Koszul algebra.

Proof. (a) From the definition of a T -Koszul algebra, we need to show that if Λ satisfies the ⊗-Tor-conditions then Λ_{0} is a T -Koszul module. For this we note that if M_{0} is a Λ_{0}-module with M_{0} in X_{T} then, viewing M_{0} as a Λ-module, (M_{0})^{1} = 0 + (Λ_{1} ⊗_{Λ_{0}} M_{0}) + 0 + · · ·. By the ⊗-condition, (Λ_{1} ⊗_{Λ_{0}} M_{0}) is in X_{T}. It follows that M^{1}[−1] is again a Λ_{0}-module in X_{T} in degree 0 and (0) in degree 1. Hence, applying this observation repeatedly, we see that M_{0} has a linear fat resolution of the form:

⋯ → Λ ⊗_{Λ_{0}} Λ_{1}^{⊗_{Λ_{0}}^{2}} ⊗_{Λ_{0}} M_{0} → Λ⊗_{Λ_{0}}Λ_{1} ⊗_{Λ_{0}} M_{0} → Λ ⊗_{Λ_{0}} M_{0} → M_{0} → 0

Setting M_{0} = Λ_{0}, we see that Λ_{0} has a linear fat resolution and hence Λ is a Koszul algebra with respect to T.

(b) Assume that Λ is a T -Koszul algebra. Since Λ_{2} = (0), the kernel of Λ_{1} ⊗_{Λ_{0}} Λ_{1} → Λ_{2} is Λ_{1} ⊗_{Λ_{0}} Λ_{1}. By Theorem 2.11, the statement in (b) immediately follows.

(c) We begin by assuming that M = M_{0} + M_{1} + 0 + ⋯ is a graded module generated in degree 0 with M_{i} in X_{T}. Consider the short exact sequence 0 → M^{1} → Λ ⊗_{Λ_{0}} M_{0} → M → 0. Then M^{1} has support in degree 1. At degree 1, we obtain a short exact sequence of Λ_{0}-modules 0 → M_{1}^{1} → Λ_{1} ⊗_{Λ_{0}} M_{0} → M_{1} → 0. By the ⊗-condition, Λ_{1} ⊗_{Λ_{0}} M_{0} is in X_{T}. Since X_{T} is closed under kernels of epimorphisms by Proposition 1.4, we conclude that M_{1}^{1} is in X_{T}. By our remarks earlier in the proof, M_{1}^{1}[−1] has a linear fat resolution. It now follows that M does. We have shown that M is a T -Koszul module.

Conversely, assume that M is a T -Koszul module. Then by definition M is generated in degree 0 with M_{i} in X_{T} for all i. This completes the proof of (c).

The proof of (d) will be postponed (see Theorem 2.3 in Chapter IV).

The following example provides an interesting class of Koszul algebras with respect to T, where Λ_{0} is an arbitrary finite dimensional algebra of the form kQ/I for a field k and the quiver of Λ = Λ_{0} + Λ_{1} has quiver Q together arrows from the ”top” of Λ_{1}. These extra arrows can be arbitrary as shown in the next example. Thus the quiver of Λ as a finite dimensional algebra can be viewed as a directed graph with arrows of degree 0 (from Q) and arrows of degree 1 from Λ_{1}.

Example 3.2. Let Λ_{0} be a k-algebra of the form kQ/I where kQ is a path algebra and I is an admissible ideal. Let X be a finite dimensional semisimple Λ_{0}- bimodule. Let r denote the Jacobson radical of Λ_{0}. Set Λ_{1} = Λ_{0} ⊗_{Λ_{0}/r} X ⊗_{Λ_{0}/r} Λ_{0}. By Theorem 3.1, we see that Λ = Λ_{0} + Λ_{1} is a Koszul algebra with respect to any Wakamatsu cotilting Λ_{0}-module T. One can see this since (Λ_{i})_{Λ_{0}} is projective and Λ_{1} ⊗_{Λ_{0}} Y is a left projective Λ_{0}-module for any left Λ_{0}-module Y. Hence the ⊗-Tor-conditions hold. Note that X is arbitrary and the bimodule decomposition of X as a semisimple Λ/r-module gives the ”degree 1” arrows in the quiver of Λ.

We obtain two immediate corollaries to Theorem 3.1 that will allow constructing other classes of examples.

Corollary 3.3. Let Λ = Λ_{0} + Λ_{1} + 0 + 0 + ⋯ and T = Λ_{0}. Assume also that Λ_{0} has finite global dimension. If Λ_{1} is projective as a left Λ_{0}-module, then Λ is a Koszul algebra with respect to T.

Proof. Since Λ_{0} has finite global dimension, Λ_{0} is a cotilting module and X_{T} is the category of finitely generated projective modules. By Theorem 3.1, to show that Λ is a T -Koszul algebra, we need only show that the ⊗-Tor-conditions hold. Let M be in X_{T}. Then by projectivity of Λ_{1} and M, we see that Λ_{1} ⊗_{Λ_{0}} M is left projective and hence the ⊗-condition follows. Since T is left projective Λ_{0}-module, the Tor-condition is clear.

Examples of this can be obtained if we have Λ_{0} = Λ′_{0} ⊕ R_{0} as left modules, Λ_{0} of finite global dimension, Hom_{Λ_{0}}(Λ′_{0}, R_{0}) = (0) and choose Λ_{1} = R_{0}. It is easy to see that R_{0} in this case is a Λ_{0}-bimodule which is left projective. We provide two concrete examples of these types of T -Koszul algebras.

Example 3.4. Let Λ_{0} be the path algebra kQ where k is a field and Q is

the quiver \xymatrix{v_{0} \ar[r]^{a} & v_{2} & v_{1} \ar[l]_{b}} . We write kQ = (kQ)(v_{0} + v_{1}) ⊕ (kQ)v_{2}. Let R_{0} = (kQ)v_{2}. Note that Λ_{1} = R_{0} is the Λ_{0}-bimodule _{2}k_{2}; that is, k as a vector space and the simple module corresponding to the vertex 2 on both the left and right. We note that Λ_{1} is a left projective but not a right projective Λ_{0}-module. It is easy to see the conditions of Corollary 3.3 are satisfied.

It is immediate that the quiver Q′ of Λ is given by [figure: v_{0}; a; v_{2}; c; v_{1}; b] and

that Λ = KQ′/J^{2} where J is the ideal generated by the arrows. Note that the grading is obtained by giving the arrows a and b degree 0 and c degree 1.

The indecomposable T -Koszul modules are Λ_{0}v_{i} + 0 for i = 0, 1, 2 and Λ_{0}v_{2} + Λ_{0}v_{2}.

Example 3.5. Let Q be a quiver with a set of relations ρ = {r_{1}, …, r_{n}} such that we can partition the vertices Q_{0} = V_{1} ∪ V_{2} to satisfy the following conditions:

(i) There are no arrows from a vertex in V_{2} to a vertex in V_{1}.

(ii) kQ/⟨ρ⟩ is of finite global dimension.

Let I be the ideal in kQ generated by the set of relations ρ. Set Λ_{0} = kQ/I, and let v_{i} be the sum of vertices in V_{i}, for i = 1, 2. Then Λ_{0} = Λ_{0}v_{1} + Λ_{0}v_{2} and Hom_{Λ_{0}}(Λ_{0}v_{2}, Λ_{0}v_{1}) = (0) by the condition above. Taking Λ_{1} = R_{0} = Λ_{0}v_{1} and T = Λ_{0}, we obtain T -Koszul algebras of the form Λ_{0} + Λ_{1} + 0 + · · ·.

Another class of algebras with Λ = Λ_{0} + Λ_{1} + 0 + 0 + ⋯ can be obtained by using the next corollary of Theorem 3.1.

Corollary 3.6. Let Λ_{0} be a finite dimensional k-algebra, where k is a field, and let T be a Wakamatsu cotilting module. Suppose, for i = 1, …, n, we choose left Λ_{0}-modules U_{i} in X_{T} and Q_{i} in X_{T}^{⊥}. Let Λ_{1} = ⊕_{i=1}^{n}U_{i} ⊗_{k} D(Q_{i}), where D is the usual artin algebra duality. Then Λ = Λ_{0} + Λ_{1} + 0 + 0 + ⋯ is a Koszul algebra with respect to T.

Proof. By Theorem 3.1, we must show that the ⊗-Tor-conditions hold. Let X be in X_{T}. For each i, the module U_{i} ⊗_{k} D(Q_{i}) ⊗_{Λ_{0}} X is a direct sum of copies of U_{i} and hence is in X_{T}. Hence the ⊗-condition follows. Next consider Tor_{1}^{Λ_{0}}(U_{i} ⊗_{k} D(Q_{i}), X). As a right Λ_{0}-module, U_{i} ⊗_{k} D(Q_{i}) is a direct sum of copies of D(Q_{i}). Since Q_{i} is in X_{T}^{⊥}, we have that Ext_{Λ_{0}}^{1}(X, D(U_{i} ⊗_{k} D(Q_{i}))) = (0). But

Tor_{1}^{Λ_{0}}(U_{i} ⊗_{k} D(Q_{i}), X) = D Ext_{Λ_{0}}^{1}(X, D(U_{i} ⊗_{k} D(Q_{i})))

by [16, Theorem 9.51]. Hence the Tor-condition holds and we are done.

Note that if Q_{i} is a left injective Λ_{0}-module, Q_{i} is in X_{T}^{⊥}, and if U_{i} is a left projective Λ_{0}-module, then U_{i} is in X_{T}. This allows an easy construction of T - Koszul algebras where Λ_{0} need not be of finite global dimension and T can be an arbitrary Wakamatsu cotilting module. We also note that U_{i} need only to be in X_{T} and not necessarily projective. In such cases, the Λ_{1} that we obtain is not left projective.

There are examples of T -Koszul algebras of the form Λ_{0} + Λ_{1} + 0 + 0 + ⋯ that do not satisfy the hypothesis of the above two corollaries. One such class is the following. If Λ_{0} is a selfinjective artin algebra, we may take Λ_{1} = Λ_{0} together with T = Λ_{0}. The ⊗-Tor-conditions are clearly satisfied under these hypotheses. We describe the T -Koszul modules and the duality in a special case.

Since T is an injective cogenerator, we see that X_{T} is mod Λ_{0}. It follows that K_{T} (Λ) is gr_{0} Λ in this case by Theorem 3.1. Furthermore, since Γ_{1} = Hom_{Λ_{0}}(Λ_{1}⊗_{Λ_{0}} T, T), we see that Γ_{1} can be identified with Λ_{0}^{op}. By Theorem 3.1 it follows that Γ is isomorphic to the tensor algebra T_{Λ_{0}^{op}}(Λ_{0}^{op}).

Lest one begins to believe that most graded algebras of the form Λ_{0} + Λ_{1} + 0 + 0 + ⋯ is a T -Koszul algebra we give some ’nonexamples’. This is different from the Koszul algebra case where all algebras of the form Λ_{0} + Λ_{1} + 0 + ⋯ with Λ_{0} a product of a field are Koszul algebras. We omit the details from the following examples and just state the results.

Example 3.7. a) Let Λ_{0} be the path algebra kQ associated to the quiver Q = \xymatrix{v_{1} \ar[r]^{a} & v_{2}} . Let Λ_{1} = D(Λ_{0}) and T = Λ_{0}. Then Λ_{0} + D(Λ_{0}) + 0 + 0 + ⋯ is not a T -Koszul algebra. This shows that trivial extensions Λ_{0} ⋉ D(Λ_{0}) of an artin algebra Λ_{0} need not be a Koszul algebra with respect to T = Λ_{0}.

b) Let Λ_{0} = k[x]/(x^{2}), T = Λ_{0} and let Λ_{1} be the trivial Λ_{0}-bimodule k for a field k. Then k[x]/(x^{2}) + k + 0 + ⋯ is not a T -Koszul algebra.

c) Let Λ_{0} be the path algebra as in part a) above. Let T = Λ_{0}, and let Λ_{1} = _{1}k_{1}; that is, Λ_{1} is the bimodule that is the simple corresponding to the vertex v_{1} on the left and on the right. Then Λ_{0} + Λ_{1} + 0 + ⋯ is not a T -Koszul algebra. Note that by taking Λ_{1} to be _{2}k_{2}, the algebra Λ_{0} + Λ_{1} is a T -Koszul algebra by Corollary 3.3.

d) Let Λ be the radical square zero algebra with quiver \xymatrix{v_{1} \ar[r]^{a} & v_{2} \ar[l]^{b}} . We view

Λ as Λ_{0} + Λ_{1} + 0 + ⋯ by putting the arrow a in degree 0 and the arrow b in degree 1. Then Λ is not a T -Koszul algebra for T = Λ_{0}. We note that Λ is a preprojective algebra (see [15] for definition of a preprojective algebra).

4. Further examples and classes of T -Koszul algebras

In this section we give various examples of T -Koszul algebras which are not concentrated in degrees 0 and 1. We give an example of a T -Koszul algebra that is not strongly T -Koszul, and generalize results on quadratic monomial algebras from classical Koszul theory.

We start with giving an example of a T -Koszul algebra which is not strongly T -Koszul.

Example 4.1. Let k be a field and Λ_{0} the path algebra associated to the quiver

\xymatrix{v_{1} & v_{2} & v_{2′} & v_{3} \ar[r]^{a} & v_{4}}

4. FURTHER EXAMPLES AND CLASSES OF T -KOSZUL ALGEBRAS 11

We let P_{i} denote the left projective Λ_{0}-module associated to vertex i, and let P_{i}^{∗} denote the right projective Λ_{0}-module associated to vertex i. Next let Λ_{1} = (P_{2} ⊗_{k} P_{1}^{∗}) ⊕ (P_{2′} ⊗_{k} P_{1}^{∗}) ⊕ (P_{3} ⊗_{k} P_{2}^{∗}) ⊕ (P_{4} ⊗_{k} P_{2′}^{∗}). It is now easy to see that the quiver Q of the tensor algebra T_{Λ_{0}}(Λ_{1}) is

\xymatrix{v_{1} \ar[r]^{x} \ar[dr]^{z} & v_{2} \ar[r]^{y} & v_{3} \ar[d]^{a} \\  & v_{2′} \ar[r]^{w} & v_{4}}

The tensor algebra is isomorphic to the path algebra kQ. The grading is given by the arrow a being in degree 0, and the arrows x, y, z, and w are in degree 1. Let I be the ideal in kQ generated by ayx−wz. Note that this is a homogeneous element of degree 2. We let Λ = kQ/I = Λ_{0} + Λ_{1} + Λ_{2} + 0 + 0 + · · ·. Next, we note that Λ_{2} is isomorphic to P_{3} ⊗_{k} P_{1}^{∗}.

First we claim that Λ is a Koszul algebra with respect to T = Λ_{0}. Since Λ_{0} is an hereditary algebra, we see that T is a cotilting module. Furthermore, in this case, X_{T} is the category of finitely generated projective Λ_{0}-modules. To show that the ⊗-Tor-conditions hold, we note that both Λ_{1} and Λ_{2} are direct sums of Λ_{0}- bimodules of the form P ⊗_{K} P′ where P is a left projective Λ_{0}-module and P′ is a right projective Λ_{0}-module. The ⊗-Tor-conditions are then immediate. It remains to show that Λ_{0} has a linear fat resolution. For this we describe the graded structure of the indecomposable projective Λ-modules, P̂_{1}, P̂_{2}, P̂_{2′}, P̂_{3}, and P̂_{4}, where P̂_{i} is the projective module associated to vertex i in the quiver Q. The module P̂_{1} has basis v_{1} in degree 0, basis x and z in degree 1, and basis yx and ayx = wz in degree 2. The module P̂_{2} has basis v_{2} in degree 0, y and ay in degree 1. The module P̂_{2′} has basis v_{2′} in degree 0, and w in degree 1. The module P̂_{3} has basis v_{3} and a in degree 0. Finally, P̂_{4} has basis v_{4} in degree 0.

With these preliminaries we give a linear fat resolution of Λ_{0}, projective by projective (all tensors are over Λ_{0}).

0 → Λ ⊗ P_{3}[2] \underrightarrow{( \begin{matrix}ay \\ w \end{matrix})} ((Λ ⊗ P_{2}) ⊕ (Λ ⊗ P_{2′})[1] ⟶^{(x,z)} Λ ⊗ P_{1} → P_{1} → 0

0 → Λ ⊗ P_{3}[1] ⟶^{y} Λ ⊗ P_{2} → P_{2} → 0

0 → (Λ ⊗ P_{4})[1] ⟶^{(w)} Λ ⊗ P_{2′} → P_{2′} → 0

0 → Λ ⊗ P_{3} → P_{3} → 0
0 → Λ ⊗ P_{4} → P_{4} → 0

This completes the proof that Λ is a T -Koszul algebra.

Next we show that there is a graded Λ-module M such that M is in gr_{0} Λ, M = M_{1} + M_{2} + 0 + ⋯ with M_{i} in X_{T}, E(M) is a Γ-module generated in degree 0, but M is not in K_{T} (Λ). Thus, although E(M) is generated in degree 0, M is not a T -Koszul module.

We let M = P̂_{1}/L where L is the submodule of P̂_{1} with basis z, yx, and ayx = wz. Thus M has a basis with two elements, v_{1} + L in degree 0 and x + L in degree 1. First we show that M in not in K_{T} (Λ). We begin by noting that both M_{0} and M_{1} are in X_{T}, since they are respectively isomorphic to the simple projective Λ_{0}-modules P_{1} and P_{2}. We show that M does not have a linear fat resolution. Consider the kernel N of Λ ⊗ P_{1} → M. We identify Λ ⊗ P_{1} with Q_{1}. If M were to be in K_{T} (Λ), the module N would need to be generated in degree 1. But N has two generators, z and yx, and yx is in degree 2. Then we conclude that M is not a T -Koszul module.

To complete the example, we need to compute E(M) and show that it is generated in degree 0 as a Γ-module. First we describe Γ. We have Γ_{0} = End_{Λ_{0}}(Λ_{0}) since T = Λ_{0}. The algebra Γ_{0} is the path algebra associated to the quiver

\xymatrix{v_{1} & v_{2} & v_{2′} & v_{3} & v_{4} \ar[l]_{a^{o}}}

where the vertices correspond the obvious simple modules. Next, using Theorem 2.11 or by directly computing the minimal Λ-projective resolution of Λ_{0} given above, and then computing the Ext-algebra, we see that Γ is the quotient of the path algebra kQ^{o}/J where Q^{o} is the quiver

\xymatrix{v_{1} \ar[r]^{y^{o}} \ar[dr]^{x^{o}} & v_{2′} \ar[r]^{w^{o}} & v_{4} \ar[d]^{a^{o}} \\  & v_{2} \ar[r]^{z^{o}} & v_{3}}

and where J is the ideal generated by a^{o}w^{o}y^{o} + z^{o}x^{o}. Here a^{o} is in degree 0 and x^{o}, y^{o}, z^{o}, and w^{o} are in degree 1.

To continue our computation of E(M), we provide the minimal graded Λprojective resolution of M below.

0 → Q_{4}[2] \underrightarrow{( \begin{matrix}a \\ w \end{matrix})} (Q_{3}[2]) ⊕ (Q_{2′}) ⟶^{(zx,y)} Q_{1} → M → 0.

Now it is easy to compute that E(M) = E(M)_{0} +E(M)_{1} +E(M)_{2} +0+⋯ = Γv_{1}/U where U is the left submodule of Γv_{1} generated by x^{0}. Thus, E(M) is generated in degree 0 even though M does not have a linear fat resolution.

We do remark though that the duality we have described on K_{T} (Λ) does extend to include M. We also note that although E(M) has a linear fat resolution, E(M)_{2} is isomorphic to the simple Γ_{0}-module associated to vertex v_{4} and is not in X_{T}. Thus, E(M), as must be, is not in K_{T} (Γ).

Some of the motivation for this paper was to provide a common framework for Koszul and cotilting theory. The next class of examples, pointed out to us by Lutz Hille, is obtained from the two classical cases using tensor product.

Proposition 4.2. Let Σ = Σ_{0} + Σ_{1} + ⋯ be a Koszul k-algebra, where k is a field. Suppose that A is a finite dimensional k-algebra, and choose any Wakamatsu cotilting A-module T′. Then the left finite graded ring Λ = Σ ⊗_{k} A is a Koszul algebra with respect to the Σ_{0} ⊗_{k} A-module T = Σ_{0} ⊗_{k} T′.

Moreover, E_{T} (Λ) ≃ E(Σ) ⊗_{k} End_{A}(T′) as graded algebras, where E(Σ) is the Koszul dual of Σ.

Proof. It is easy to see that

Hom_{Σ_{0}⊗_{k}A}(P ⊗_{k} X, Q ⊗_{k} Y) ≃ Hom_{Σ_{0}}(P, Q) ⊗_{k} Hom_{A}(X, Y),

where P and Q are Σ_{0}-modules and X and Y are in mod A. This implies that

Ext_{Σ_{0}⊗_{k}A}^{i}(Σ_{0} ⊗_{k} X, Σ_{0} ⊗_{k} Y) ≃ Σ_{0} ⊗_{k} Ext_{A}^{i}(X, Y)

for all modules X and Y in mod A. From this we obtain that T = Σ_{0} ⊗_{k} T′ is a selforthogonal module and that X_{T} = add(Σ_{0} ⊗_{k} X_{T′}). Hence T is a Wakamatsu cotilting Λ_{0}-module.

Since Λ_{i} = Σ_{i}⊗_{k}A is a projective right Λ_{0}-module, Λ satisfies the Tor-condition with respect to T.

By [8, page 165] we have that

(Σ_{i} ⊗_{k} X) ⊗_{Σ_{0}⊗_{k}A} (Σ_{0} ⊗_{k} Y) ≃ Σ_{i} ⊗_{Σ_{0}} (X ⊗_{A} Y)

for all right A-modules X and left A-modules Y. Hence we infer that Λ satisfies the ⊗-condition.

Since Σ is a Koszul algebra, Σ_{0} has a linear projective resolution. Tensoring this resolution with A over k yields a linear fat resolution of Λ_{0} = Σ_{0} ⊗_{k} A, and by definition Λ is a T -Koszul algebra with respect to T.

Similar as above one can show that

Hom_{Σ⊗_{k}A}(P ⊗_{k} X, Q ⊗_{k} Y) ≃ Hom_{Σ}(P, Q) ⊗_{k} Hom_{A}(X, Y),

where P is a projective Σ-module, Q is a Σ-module and X and Y are in mod A. Using this and the description of Γ = E_{T} (Λ) in Theorem 2.11 (c), it follows that Γ_{0} ≃ Σ_{0}^{op} ⊗_{k} End_{A}(T′), the degree one part Γ_{1} ≃ Hom_{Σ_{0}}(Σ_{1}, Σ_{0})⊗_{k} End_{A}(T′), and E_{T} (Λ) ≃ T_{Γ_{0}}(Γ_{1})/⟨A_{2}⟩, where A_{2} = A′_{2} ⊗_{k} End_{A}(T′) with

E(Σ) = T_{Σ_{0}^{op}}(Hom_{Σ_{0}}(Σ_{1}, Σ_{0}))/⟨A′_{2}⟩.

Since tensoring over the field k is an exact functor and

T_{Σ_{0}^{op}⊗_{k}End_{A}(T′)}(Hom_{Σ_{0}}(Σ_{1}, Σ_{0})⊗_{k}End_{A}(T′)) ≃ T_{Σ_{0}^{op}}(Hom_{Σ_{0}}(Σ_{1}, Σ_{0}))⊗_{k}End_{A}(T′),

we obtain that E_{T} (Λ) ≃ E(Σ) ⊗_{k} End_{A}(T′) as graded algebras.

The following result is a useful result for constructing further examples.

Proposition 4.3. Suppose Λ is a graded algebra generated in degrees 0 and 1. Let T = D(Λ_{0}) (where Λ_{0} is viewed as a right projective Λ_{0}-module). If Λ_{i} is a right projective Λ_{0}-module for each i ≥ 0, then Λ is T -Koszul if and only if Λ_{0} is in K_{T} (Λ). Moreover, if Λ is a T -Koszul algebra, then Λ is a strongly T -Koszul algebra.

Proof. By definition, if Λ is T -Koszul, then Λ_{0} is in K_{T} (Λ). Assuming that Λ_{0} is in K_{T} (Λ), we need to show that the ⊗-Tor-conditions hold. Since each Λ_{i} is a right projective Λ_{0}-module the Tor-condition holds. Since T is an injective cogenerator of mod Λ_{0}, it follows that X_{T} = mod(Λ_{0}). Hence, the ⊗-condition holds. We postpone the last part of the proof until Proposition 3.3 of Chapter IV.

An easy example of a T -Koszul algebra obtained by applying the previous proposition is the following.

Example 4.4. Let Λ_{0} be a self-injective artin algebra, T = Λ_{0}, and Λ = Λ_{0} + Λ_{0} + Λ_{0} + · · ·. Then Λ_{0} is in K_{T} (Λ) since

0 → (Λ ⊗_{Λ_{0}} Λ_{0})[1] → Λ ⊗_{Λ_{0}} Λ_{0} → Λ_{0} → 0

is a linear fat resolution of Λ_{0}. By Proposition 4.3 the algebra Λ is a T -Koszul algebra. It follows from Theorem 2.11 that Γ = Λ_{0}^{op} + Λ_{0}^{op} + 0 + · · ·.

A second example is the following.

Example 4.5. Let Q the quiver

[figure: v_{1}; a; v_{2}; v_{3}; v_{4}]

and Λ_{0} = kQ/⟨a^{2}⟩. Thus Λ_{0} is isomorphic to k[x]/(x^{2})×k×k×k and is selfinjective. Let Λ_{1} = _{1}k_{2} +_{3}k_{2} +_{4}k_{2} +_{3}k_{3} +_{4}k_{4}. Let Λ_{2} = (Λ_{1} ⊗_{Λ_{0}} Λ_{1})/(_{3}k_{3} ⊗_{3}k_{2} +_{3}k_{3} ⊗_{3}k_{3}). Thus, as a Λ_{0}-bimodule, Λ_{2} = _{4}k_{2} +_{4}k_{4}. Setting Λ_{i} = Λ_{1} ⊗_{Λ_{0}} Λ_{i−1} for i ≥ 3, we see that Λ_{i} = Λ_{2} as Λ_{0}-bimodules for i ≥ 3. If we set T = Λ_{0}, noting that each Λ_{i} is a right projective Λ_{0}-module, we see that the conditions of Proposition 4.3 are met and Λ is T -Koszul if and only if Λ_{0} is in K_{T} (Λ).

We must show that Λ_{0} is in K_{T} (Λ). For this, we provide a linear fat resolution for each indecomposable projective summand of Λ_{0}. We let Q_{i} denote the left projective Λ_{0}-module Λ_{0}v_{i} and S_{i} denote the associated simple module. Then we have

0 → Λ ⊗_{Λ_{0}} Q_{1} → Q_{1} → 0
Λ ⊗_{Λ_{0}} (S_{1} ⊕ Q_{3} ⊕ Q_{4}) → Λ ⊗_{Λ_{0}} Q_{2} → Q_{2} → 0
⋯ → Λ ⊗_{Λ_{0}} (S_{1} ⊕ Q_{3})[2] → Λ ⊗_{Λ_{0}} (S_{1} ⊕ Q_{3})[1] →
⋯ → Λ ⊗_{Λ_{0}} Q_{3}[2] → Λ ⊗_{Λ_{0}} Q_{3}[1] →→ Λ ⊗_{Λ_{0}} Q_{3} → Q_{3} → 0
0 → Λ ⊗_{Λ_{0}} Q_{4}[1] → Λ ⊗_{Λ_{0}} Q_{4} → Q_{4} → 0

We remark that Λ_{1} is not a left projective Λ_{0}-module.

We now provide an interesting class of Koszul algebras with respect to T whose construction is related to the construction given in Example 3.2. As in that example, we give a method for constructing quotients of path algebras where the arrows occur in degree 0 or 1. This method is quite general, and given any finite dimensional k-algebra Λ_{0} of the form kQ, we provide a method of ’adding’ an arbitrary finite number of arrows in degree 1 with no restrictions. Thus, the ’quiver’ of Λ will be the quiver of Λ_{0} plus the degree one arrows.

Example 4.6. Let Λ_{0} be a k-algebra of the form kQ/I where kQ is a path algebra and I is an admissible ideal. Let {v_{1}, …, v_{n}} be the vertices of Q. Let X be a finite dimensional semisimple Λ_{0}-bimodule. Thus X is a direct sum of modules of the form _{i}k_{j} where _{i}k_{j} is Λ_{0}-bimodule which is isomorphic to the simple left Λ_{0}- module associated to the vertex v_{i} and is isomorphic to the simple right Λ_{0}-module associated to the vertex v_{j}. Let r denote the Jacobson radical of Λ_{0}. Since I is an admissible ideal, we may view Λ_{0} as a left or right Λ_{0}/r-module.

Set Λ_{1} = Λ_{0}⊗_{Λ_{0}/r}X ⊗_{Λ_{0}/r}Λ_{0}. We now take Λ to be the tensor algebra T_{Λ_{0}}(Λ_{1}). First we show that the ⊗-Tor-conditions are satisfied. Let n ≥ 1 and consider Λ_{n}. As a right Λ_{0}-module it is of the form Y ⊗_{Λ_{0}/r} Λ_{0} for some Λ_{0}-module Y. But such a module is a right projective Λ_{0}-module, and hence the Tor-condition holds. Next, if X is in X_{T}, the module Λ_{n} ⊗_{Λ_{0}} X is of the form Λ_{0} ⊗_{Λ_{0}/r} Z for some Λ_{0}/r-module Z. Hence Λ_{n} ⊗_{Λ_{0}} X is a left projective Λ_{0}-module. Since projective Λ_{0}-modules are in X_{T}, we see that the ⊗-condition is satisfied.

It remains to show that Λ_{0} has a linear fat resolution. We see that the kernel of Λ ⊗_{Λ_{0}} Λ_{0} → Λ_{0} is just Λ_{≥1}. But by the construction of Λ we see that Λ_{≥1} = Λ ⊗_{Λ_{0}} Λ_{1}[1], which is a fat module. Thus, the linear fat resolution is

0 → Λ ⊗_{Λ_{0}} Λ_{1}[1] → Λ ⊗_{Λ_{0}} Λ_{0} → Λ_{0} → 0.

Hence Λ is a T -Koszul algebra for any Wakamatsu cotilting module T.

Example 4.7. Let Λ_{0} and Λ_{1} be as above. If we set Λ = Λ_{0} + Λ_{1} + 0 + ⋯, then by some of the above arguments we have that Λ satisfies the ⊗-Tor-conditions for any Wakamatsu cotilting module T. Hence, by Theorem 3.1, it follows that Λ is a T -Koszul algebra.

Our next class of examples generalizes the fact that quadratic monomial algebras are Koszul algebras. We begin with an arbitrary artin algebra Λ_{0} and T a Wakamatsu cotilting Λ_{0}-module. Let Λ_{1} be a Λ_{0}-bimodule which satisfies the ⊗-Tor-conditions. Note, that as a left Λ_{0}-module, Λ_{1} is in X_{T}. Let Λ_{1} = ⊕_{i=1}^{n}B_{i} be a decomposition of Λ_{1} into indecomposable Λ_{0}-bimodules. We introduce some notational conventions. Let S(q) denote the set of length q sequences of integers taken from the set of integers between 1 and n. If α is in S(q) with α = (a_{1}, …, a_{q}), then we define B_{α} to be the Λ_{0}-bimodule B_{a_{1}} ⊗_{Λ_{0}} B_{a_{2}} ⊗_{Λ_{0}} ⋯ ⊗_{Λ_{0}} B_{a_{q}}. Thus, for

example, Λ_{1}^{⊗_{Λ_{0}}^{q}} = ⊕_{α∈S(q)}B_{α}.

Our assumption that Λ_{1} satisfies the ⊗-Tor-conditions implies that for each q ≥ 1 and each α in S_{q}, as a left module, B_{α} is in X_{T} and B_{α} satisfies the same conditions. We now fix a subset G of S(2). We let G(m) = {α = (a_{1}, …, a_{m}) ∈ S(m) | for some i, 1 ≤ i < m, (a_{i}, a_{i+1}) ∈ G} for m ≥ 2. Thus, G(2) = G. Let L(m) = S(m) \ G(m) for m ≥ 2. Thus, L(m) are the sequences (a_{1}, …, a_{m}) of length m such that for each i where 1 ≤ i < m, the pair (a_{i}, a_{i+1}) is not in G. Also of interest is the set K(m) = {(a_{1}, …, a_{m}) ∈ S(m) | for each i, 1 ≤ i < m, (a_{i}, a_{i+1}) ∈ G(2)}.

Let I_{G} denote the ideal in the tensor algebra T_{Λ_{0}}(Λ_{1}) generated by {B_{g} | g ∈ G}. Note that I_{G} is graded ideal in T_{Λ_{0}}(Λ_{1}) generated in degree 2 where (I_{G})_{m} = ⊕_{α∈G(m)}B_{α}. Let Λ = T_{Λ_{0}}(Λ_{1})/I_{G} = Λ_{0} + Λ_{1} + Λ_{2} + Λ_{3} + ⋯ where

Λ_{m} = Λ_{1}^{⊗_{Λ_{0}}^{m}} /(I_{G})_{m}. It is easy to see that Λ_{m} is isomorphic to ⊕_{α∈L(m)}B_{α} and that we have exact sequences

0 → ⊕_{α∈G(m)}B_{α} → ⊕_{α∈S(m)}B_{α} → Λ_{m} → 0.

We note that the ⊗-condition implies that if α is in S(m), then B_{α} is in X_{T}. We will show that Λ is a T -Koszul algebra.

We begin by noting that our earlier remarks imply that Λ satisfies the ⊗-Tor-conditions. Thus, we must show that Λ_{0} has a linear fat resolution. For ease of notation we let ⊗ mean ⊗_{Λ_{0}} for the remainder of this class of examples. We first consider the fat Λ-module F_{m} = Λ ⊗ ⊕_{α∈K(m)}B_{α}[m]. In degree p ≥ m, (F_{m})_{p} = Λ_{p−m} ⊗ ⊕_{α∈K(m)}B_{α}. Using the observation that Λ_{k} = ⊕_{α∈L(k)}B_{α}, we see that

(F_{m})_{p} = ⊕_{\begin{matrix}α=(a_{1},...,a_{p}) \\ (a_{1},...,a_{p−m})∈L(p−m) \\ (a_{p−m+1},...,a_{p})∈K(m) \end{matrix}} B_{α}.

We now give a linear fat resolution of Λ_{0}. Consider

⋯ ⟶^{μ_{m+1}} F_{m} ⟶^{μ_{m}} ⋯ → F_{3} ⟶^{μ_{3}} F_{2} ⟶^{μ_{2}} Λ ⊗ Λ_{1}[1] ⟶^{μ_{1}} Λ ⊗ Λ_{0} ⟶^{μ_{0}} Λ_{0} → 0, (∗∗)

where μ_{0} and μ_{1} are the obvious maps and μ_{m} for m ≥ 2 is described as follows. By the description of (F_{m})_{p} for p ≥ m, we need to define μ_{m} in degree p from the summands of (F_{m})_{p} to the (F_{m−1})_{p}. A summand of (F_{m})_{p} is B_{α} where α = (a_{1}, …, a_{d}) where (a_{1}, …, a_{p−m}) is in L(p − m) and (a_{p−m+1}, …, a_{p}) is in K(m). We have that B_{α} is a term in the direct sum for (F_{m−1})_{p} if and only if (a_{p−m}, a_{p−m+1}) is not in G(2). We define μ_{m} in degree p on B_{α} to be the identity if B_{α} is a summand in (F_{m−1})_{p} and to be 0 on B_{α} otherwise. It is now a straightforward computation that (∗∗) is a linear fat resolution.

We now turn our attention to Γ = ⊕_{n≥0} Ext_{Λ}^{n}(T, T). Let (∗∗) be the linear fat resolution of Λ_{0} and let X^{n} = Im(μ_{n}: F^{n} → F^{n−1}). It follows from the definition of a linear fat resolution that F^{n} = Λ ⊗_{Λ_{0}} X_{n}^{n}. We will show that Γ_{n} = Hom_{Λ_{0}}(X_{n}^{n} ⊗ T, T) for n ≥ 1. Thus, Γ_{1} = ⊕_{i=1}^{n} Hom_{Λ_{0}}(B_{i} ⊗ T, T). Setting C_{i} = Hom_{Λ_{0}}(B_{i} ⊗ T, T), we see that Γ_{1} decomposes into a direct sum of bimodules ⊕_{i=1}^{n}C_{i}. By Theorem 2.5 the algebra Γ is a T -Koszul algebra with respect to T = E(Λ) and hence is a quadratic algebra by Theorem 2.11. Since Γ is generated in degrees 0 and 1, we see that Γ is isomorphic to a quotient of the tensor algebra T_{Γ_{0}}(Γ_{1})/J by an ideal J generated by elements of degree 2. The ideal J is generated by all C_{i} ⊗_{Γ_{0}} C_{j} where (i, j) is in L(2) = S(2) \ G(2). This corresponds to the ’orthogonal’ relations to the generators B_{i} ⊗ B_{j} with (i, j) in G(2) of I_{G}.

We summarize the above discussion.

Example 4.8. Let Λ_{0} be an artin algebra and T a Wakamatsu cotilting Λ_{0}- module. Let Λ_{1} be a Λ_{0}-bimodule such that Λ_{1} = ⊕_{i=1}^{n}B_{i} with each B_{i} an indecomposable Λ_{0}-bimodule. Assume that Λ_{1} satisfies the ⊗-Tor-conditions and let I be an ideal in the tensor algebra T_{Λ_{0}}(Λ_{1}) generated by some of the summands of Λ_{1} ⊗_{Λ_{0}} Λ_{1} of the form B_{i} ⊗_{Λ_{0}} B_{j}. Then T_{Λ_{0}}(Λ_{1})/I is a T -Koszul algebra. Furthermore, Γ is isomorphic to the tensor algebra T_{Γ_{0}}(Γ_{1})/J, the bimodule Γ_{1} = ⊕_{i=1}^{n}C_{i} where the C_{i} are defined above, and J is generated by the summands C_{i} ⊗_{Γ_{0}} C_{j} for the pairs (i, j) such that B_{i} ⊗ B_{j} is not a generator of I.

We end this section with an example that does not fit into the definitions of this section. The following algebra is T -Koszul in the more general sense we define later, but does not satisfy the Tor-condition.

Example 4.9. Let Λ_{0} = k[x]/(x^{2}) × k[y]/(y^{2}) for a field k. Let e_{1} = (1, 0) and e_{2} = (0, 1). Let r denote the Jacobson radical of Λ_{0}. Let S = (Λ_{0}/r)e_{1} and S′ = e_{2}(Λ_{0}/r). We let Λ_{1} = S ⊗_{k} S′, so Λ_{1} is isomorphic to S as a left Λ_{0}-module and to S′ as a right Λ_{0}-module. Let Λ = Λ_{0} + Λ_{1} + 0 + · · ·. Let T = Λ_{0}. Then X_{T} = mod(Λ_{0}). But Tor_{1}^{Λ_{0}}(Λ_{1}, T) ≠ 0. Thus the Tor-condition fails. We will see that there is a more general theory in which Λ is a Koszul algebra with respect to T.

CHAPTER II Main results with proofs

The aim of this chapter is to prove a duality theorem for the Koszul modules of a T -Koszul algebra and its Ext-algebra. We have collected various preliminary results in the first section, and in the last section we deal with quadratic algebras. These results were stated without proof in Chapter I.

1. Preliminaries

In this section we give some preliminary results needed in the proof of the duality theorem. It is natural to study subcategories of the category G_{T} (Λ) of modules which are generated in degree zero and such that the modules in the image of E are also generated in degree zero as a module over the subalgebra of E_{T} (Λ) generated by E_{T} (Λ)_{0} and E_{T} (Λ)_{1}. We show that for a T -Koszul algebra Λ the category K_{T} (Λ) of Koszul modules is a subcategory of G_{T} (Λ). But first we investigate when bimodules satisfy the ⊗-Tor-conditions with respect to a resolving subcategory. We show that such bimodules are closed under tensor products and kernels of epimorphisms of bimodules.

Lemma 1.1. Let Σ be an artin algebra, and let Y be a resolving subcategory of mod Σ. Let X and Y be two Σ-bimodules satisfying the ⊗-Tor-conditions with respect to Y.

(a) X ⊗_{Σ} Y satisfies the ⊗-Tor-conditions with respect to Y.

(b) If 0 → K → Y → X → 0 is an exact sequence of Σ-bimodules with X and Y as above, then K also satisfies the ⊗-Tor-conditions with respect to Y.

Proof. (a) Let Z be in Y. Since Tor_{i}^{Σ}(X, Y) = (0) and Tor_{i}^{Σ}(Y, Z) = (0) for all i ≥ 1, it follows essentially from [8, Theorem 2.8] that Tor_{i}^{Σ}(X ⊗_{Σ} Y, Z) ≃ Tor_{i}^{Σ}(X, Y ⊗_{Σ} Z), which is zero by assumption. Hence X ⊗_{Σ} Y satisfies the Torcondition with respect to Y. Clearly X ⊗_{Σ} Y satisfies the ⊗-condition with respect to Y.

(b) The long exact sequence in Tor-functors immediately implies that K satisfies the Tor-condition with respect to Y.

Let Z be in Y. Since X satisfies the Tor-condition, the sequence 0 → K ⊗_{Σ}Z → Y ⊗_{Σ}Z → X ⊗_{Σ}Z → 0 is exact. By assumption X ⊗_{Σ}Z and Y ⊗_{Σ}Z are in Y, hence K ⊗_{Σ} Z is in Y, since Y is resolving. Therefore K satisfies the ⊗-Tor-conditions with respect to Y.

Now we formalize and generalize the ⊗-Tor-conditions for graded rings introduced in the previous chapter (see Definition 2.1 of Chapter I).

Definition 1.2. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, and let Y be a subcategory of mod Λ_{0}. Let X be a Λ-Λ_{0}-bimodule generated as a left Λ-module in positive degrees. Then X is said to satisfy the ⊗- Tor-conditions with respect to Y if X_{i} satisfies the ⊗-Tor-conditions with respect to Y for all i ≥ 0.

The next result is used to show that for a T -Koszul algebra Λ the kernels X^{i} in the fat resolution of Λ_{0} satisfy the ⊗-Tor-conditions with respect to X_{T}.

Proposition 1.3. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, and let Y be a resolving subcategory of mod Λ_{0}. Suppose Λ satisfies the ⊗-Tor-conditions with respect to Y. Let X be in gr_{0} Λ with X a Λ-Λ_{0}-bimodule satisfying the ⊗-Tor-condition with respect to Y.

Then X^{1} is a Λ-Λ_{0}-bimodule satisfying the ⊗-Tor-conditions with respect to Y, where we recall that X^{1} is given by the exact sequence 0 → X^{1} → Λ ⊗_{Λ_{0}} X_{0} → X → 0.

Proof. This follows directly from the definitions involved and Lemma 1.1.

Corollary 1.4. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, and let Y be a resolving subcategory of mod Λ_{0}. Suppose that Λ satisfies the ⊗-Tor-conditions with respect to Y and that

⋯ → Λ ⊗_{Λ_{0}} X_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗_{Λ_{0}} X_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗_{Λ_{0}} Λ_{0} ⟶^{d_{0}} Λ_{0} → 0

is exact, with X^{i} = Ker d_{i−1}.

Then the Λ-Λ_{0}-bimodules X^{i} satisfy the ⊗-Tor-conditions with respect to Y for all i ≥ 1.

Proof. This is a direct consequence of Proposition 1.3 and the assumptions on Λ, since Λ_{0} is a Λ-Λ_{0}-bimodule that satisfies the ⊗-Tor-conditions with respect to Y.

In order to show that K_{T} (Λ) is a subcategory of G_{T} (Λ), the following criterion for computation of Ext-groups is essential.

Lemma 1.5. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting module over Λ_{0}. Let M be a module in gr_{0} Λ with M_{0} in _{Λ_{0}}^{⊥}T.

(a)

Ext_{Λ}^{1}(M, T) ≃ Hom_{Λ}(M^{1}, T)

as Γ_{0} = End_{Λ_{0}}(T)-modules, where M^{1} as before is given by the exact sequence 0 → M^{1} → Λ ⊗_{Λ_{0}} M_{0} → M → 0.

(b) If M^{1} is generated in degree 1, then Ext_{Λ}^{1}(M, T) ≃ Hom_{Λ_{0}}(M_{1}^{1}, T) and the isomorphism is given by sending a map f: M_{1}^{1} → T to the extension given by the pushout

[figure: 0; M^{1}; fp; Λ ⊗_{Λ_{0}} M_{0}; M; 0; 0; T; E; M; 0],

where p: M^{1} → M_{1}^{1} is the natural epimorphism.

Proof. (a) Let η: 0 → Ω_{Λ}(M) ⟶^{g} Λ⊗_{Λ_{0}} P_{Λ_{0}}(M_{0}) → M → 0 be the projective cover of M. From Lemma 0.4 we have that

Hom_{Λ}(Ω_{Λ}(M), T) ≃ Hom_{Λ_{0}}(Ω_{Λ_{0}}(M_{0}), T) ⊕ Hom_{Λ}(M^{1}, T),

since Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩ = M^{1}. Applying the functor Hom_{Λ}( , T) to η then gives rise to the following exact sequence, using in addition that Hom_{Λ}(X, T) ≃ Hom_{Λ_{0}}(X_{0}, T) for all X in gr_{0} Λ.

0 → Hom_{Λ_{0}}(M_{0}, T) → Hom_{Λ_{0}}(P_{Λ_{0}}(M_{0}), T) →

Hom_{Λ_{0}}(Ω_{Λ_{0}}(M_{0}), T) ⊕ Hom_{Λ}(M^{1}, T) → Ext_{Λ}^{1}(M, T) → 0.

Since the image of the map g^{∗}: Hom_{Λ}(Λ ⊗_{Λ_{0}} P_{Λ_{0}}(M_{0}), T) → Hom_{Λ}(Ω_{Λ}(M), T) is contained in Hom_{Λ_{0}}(Ω_{Λ_{0}}(M_{0}), T) and M_{0} is in _{Λ_{0}}^{⊥}T, we infer that Hom_{Λ}(M^{1}, T) ≃ Ext_{Λ}^{1}(M, T) as Γ_{0}-modules.

(b) This is a straightforward consequence of (a) and the definition of the connecting homomorphism for the long exact sequence.

Let Λ and T be as above, and let Γ = E_{T} (Λ). A central ingredient for proving that K_{T} (Λ) and K_{T} (Γ) are dual categories is the following result, here formulated in a more general setting.

Lemma 1.6. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1 with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting module over Λ_{0}. If M is in G_{T} (Λ), then the sequence

0 → E(M_{≥1})[1] → E(M_{0}) → E(M) → 0

is exact.

Proof. The sequence η: 0 → M_{≥1} → M ⟶^{p} M_{0} → 0 is clearly an exact sequence of graded Λ-modules. Hence we obtain an induced homomorphism E(p): E(M_{0}) → E(M). Since E(p) is onto in degree 0 and M is in G_{T} (Λ), we infer that E(p) is an epimorphism. In addition Hom_{Λ}(M_{0}, T) ≃ Hom_{Λ}(M, T), so that the long exact sequence induced from η implies that the sequence

0 → E(M_{≥1})[1] → E(M_{0}) → E(M) → 0

is exact.

The proof of the duality involves identifying the groups Ext_{Λ}^{i}(M, T) with homomorphism groups over Λ_{0} for a module M in K_{T} (Λ) over a T -Koszul algebra Λ. For this it is useful to give a orthogonality property of the Λ_{0}-module Hom_{Λ_{0}}(X, T) for a Λ_{0}-bimodule X satisfying the ⊗-Tor-conditions with respect to X_{T}. This is done in the next result. Furthermore, given the Tor-condition this result reduces the verification of the ⊗-condition for a bimodule to checking a property of only one module.

Lemma 1.7. Let Σ be an artin algebra, and let T be a Wakamatsu cotilting module over Σ. Let X be a Σ-bimodule with _{Σ}X in X_{T}, and let Y be a resolving subcategory of X_{T}. Assume that X satisfies the Tor-condition with respect to Y. Then X ⊗_{Σ} Y is in ^{⊥}T for all Y in Y if and only if Hom_{Σ}(X, T) is in Y^{⊥}.

Proof. Let Y be in Y, and let ⋯ → P^{1} → P^{0} → Y → 0 be a projective resolution of Y over Σ. By the Tor-condition with respect to Y the sequence

⋯ → X ⊗ P^{1} → X ⊗ P^{0} → X ⊗ Y → 0

is exact. Since _{Σ}X is in X_{T}, the module X ⊗ P^{j} is in add_{Σ}X for all j and hence in X_{T}. Using these observations it follows that the homology of the upper complex in the following commutative diagram

[figure: 0; (X ⊗ P^{0}, T); ≀; (X ⊗ P^{1}, T); ≀; (X ⊗ P^{2}, T); ≀; ⋯; 0; (P^{0}, (X, T)); (P^{1}, (X, T)); (P^{2}, (X, T)); ⋯]

is Ext_{Σ}^{j}(X ⊗Y, T) and the homology of the lower complex is Ext_{Σ}^{j}(Y, (X, T)). Hence X ⊗ Y is in ^{⊥}T for all Y in Y if and only if Hom_{Σ}(X, T) is in Y^{⊥}.

We have seen that T being a cotilting module gives nice properties for T -Koszul algebras. That this is also the case for the ⊗-Tor-conditions is an easy consequence of Lemma 1.7.

Corollary 1.8. Let Σ be an artin algebra, and let T be a cotilting module over Σ. Let X be a Σ-bimodule with _{Σ}X in X_{T}. Assume that X satisfies the Torcondition with respect to X_{T}. Then X satisfies the ⊗-Tor-conditions with respect to X_{T} if and only if Hom_{Σ}(X, T) is in X_{T}^{⊥}.

In making further identifications involving Ext_{Λ}^{1}(M, T) the following result is crucial in the proof of the duality.

Proposition 1.9. Let Σ be an artin algebra, and let T be a Wakamatsu cotilting module over Σ. Let Y be a subcategory of X_{T} which is closed under cokernels of left add T -approximations. Let Z = {Z ∈ mod Σ | Ext_{Σ}^{1}(Y, Z) = (0)}, and let Δ = End_{Σ}(T). Then there is an isomorphism

ϕ: Hom_{Σ}(T, ω) ⊗_{Δ} Hom_{Σ}(X, T) → Hom_{Σ}(X, ω)

given by ϕ(f ⊗ g) = f ∘ g for f in Hom_{Σ}(T, ω) and g in Hom_{Σ}(X, T), where X is in Y and ω is in Z. The homomorphism ϕ is functorial in both X and ω.

Proof. There is a homomorphism

ϕ: Hom_{Σ}(T, ω) ⊗_{Δ} Hom_{Σ}(X, T) ≃ Hom_{Σ}(X, ω)

given by ϕ(f ⊗ g) = f ∘ g for f: T → ω and g: X → T. The homomorphism ϕ is clearly an isomorphism for X = T. Since ϕ commutes with direct sums, it follows that ϕ is an isomorphism for any X in add T. It is easy to check that ϕ is natural in both X and ω.

Let X be in Y. Then we have an exact sequence 0 → X → T_{0} ⟶^{h} T_{1} of left add T -approximations, where by assumption Im h = X′ and Coker h = X′′ are in Y. This gives rise to the exact sequence (T_{1}, T) → (T_{0}, T) → (X, T) → 0. Tensoring this sequence with (T, ω) we obtain the following commutative diagram

\xymatrix{(T, ω) ⊗_{Δ} (T_{1}, T) \ar[r] \ar[d] & (T, ω) ⊗_{Δ} (T_{0}, T) \ar[r] \ar[d] & (T, ω) ⊗_{Δ} (X, T) \ar[r] \ar[d] & 0 \\ (T_{1}, ω) \ar[r] & (T_{0}, ω) \ar[r] & (X, ω) \ar[r] & Ext_{Σ}^{1}(X′, ω)}

The two leftmost vertical maps are isomorphisms, and Ext_{Σ}^{1}(X′, ω) = (0) by assumption. It follows from this that we have our desired isomorphism.

The following step is also essential.

Lemma 1.10. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting module over Λ_{0}. Suppose Λ satisfies the ⊗-Tor-conditions with respect to T.

If M is in gr_{0} Λ with M_{i} in X_{T} for i = 0, 1 and M^{1}[−1] is in gr_{0} Λ, then

Ext_{Λ}^{1}(T, T) Hom_{Λ}(M, T) = Ext_{Λ}^{1}(M, T).

Proof. For T we have that T^{1} = Λ_{≥1} ⊗_{Λ_{0}} T, and this module is generated in the highest degree. Therefore by Lemma 1.5 we have that Ext_{Λ}^{1}(T, T) ≃ Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) and that Ext_{Λ}^{1}(M, T) ≃ Hom_{Λ_{0}}(M_{1}^{1}, T). Note that Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) is a right Γ_{0}-module via the Γ_{0}-module structure on T in the tensor product Λ_{1} ⊗_{Λ_{0}} T, where Γ_{0} = End_{Λ_{0}}(T). We want to show that the map ϕ: Ext_{Λ}^{1}(T, T)⊗_{Γ_{0}}Hom_{Λ}(M, T) → Ext_{Λ}^{1}(M, T) given by multiplication is onto.

Let θ: M_{1}^{1} → Λ_{1} ⊗_{Λ_{0}} M_{0} be the inclusion. Then ϕ is given by ϕ(f ⊗ g) = f ∘(id_{Λ_{1}} ⊗ g) ∘ θ: M_{1}^{1} → T when using the above identifications for f in Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) and g in Hom_{Λ_{0}}(M_{0}, T).

Since Ext_{Λ_{0}}^{1}(X_{T} , Hom_{Λ_{0}}(Λ_{1}, T)) = (0) by Lemma 1.7, we have the following isomorphisms by Proposition 1.9

Ext_{Λ}^{1}(T, T) ⊗_{Γ_{0}} Hom_{Λ}(M, T) ≃ Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) ⊗_{Γ_{0}} Hom_{Λ_{0}}(M_{0}, T)

≃ Hom_{Λ_{0}}(T, Hom_{Λ_{0}}(Λ_{1}, T)) ⊗_{Γ_{0}} Hom_{Λ_{0}}(M_{0}, T)

≃ Hom_{Λ_{0}}(M_{0}, Hom_{Λ_{0}}(Λ_{1}, T))
≃ Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} M_{0}, T)

This implies that we have the following commutative diagram

\xymatrix{Ext_{Λ}^{1}(T, T) ⊗_{Γ_{0}} Hom_{Λ}(M, T) \ar[r] \ar[d]^{≀} & Ext_{Λ}^{1}(M, T) \ar[d]^{≀} \\ (Λ_{1} ⊗_{Λ_{0}} M_{0}, T) \ar[r]^{θ^{∗}} & (M_{1}^{1}, T) \ar[r] & Ext_{Λ_{0}}(M_{1}, T) \ar[r] & 0}

where θ^{∗} is onto, since Ext_{Λ_{0}}(M_{1}, T) = (0) by assumption. This completes the proof.

Now proving that K_{T} (Λ) is contained in G_{T} (Λ) is an easy induction argument.

Proposition 1.11. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting module over Λ_{0}. Suppose Λ satisfies the ⊗-Tor-conditions with respect to T.

(a) K_{T} (Λ) is a subcategory of G_{T} (Λ).

(b) Let M be in K_{T} (Λ), and let

⋯ → Λ ⊗ M_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗ M_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗ M_{0} ⟶^{d_{0}} M → 0

be the linear fat resolution of M, where M^{i} = Ker d_{i−1} for i ≥ 1. Then Ext_{Λ}^{i}(M, T) ≃ Hom_{Λ_{0}}(M_{i}^{i}, T) for all i ≥ 0.

(c) Λ ⊗_{Λ_{0}} X_{T} is in K_{T} (Λ) and E(Λ ⊗_{Λ_{0}} X) ≃ Hom_{Λ}(X, T) for all X in X_{T}.

Proof. Assume that M is in K_{T} (Λ), and let M have a linear fat resolution as above. Since Λ_{i} ⊗ X_{T} ⊂ X_{T} and X_{T} is resolving, the Λ_{0}-module M_{i}^{1} is in X_{T} for all i ≥ 1. Moreover, given the resolution of M via the modules Λ ⊗_{Λ_{0}} M_{i}^{i}[i] and that Λ ⊗_{Λ_{0}} M_{i}^{i}[i] is in _{Λ}^{⊥}T for all i, we see that

Ext_{Λ}^{i}(M, T) = Hom_{Λ}(M^{i}, T)
= Hom_{Λ_{0}}(M_{i}^{i}, T)

for all i ≥ 0, where M^{0} = M. Then

Ext_{Λ}^{1}(T, T) Ext_{Λ}^{i}(M, T) = Ext_{Λ}^{1}(T, T) Hom_{Λ}(M^{i}, T)
= Ext_{Λ}^{1}(M^{i}, T)
= Ext_{Λ}^{i+1}(M, T)

This proves (a) and (b).

The claim in (c) is a direct consequence of the definition of K_{T} (Λ) and (b).

We end this section with some useful properties of the category K_{T} (Λ) of Koszul modules.

The functor E is far from being an exact functor, but the next result shows that in some situations it does preserve exactness. This is used in Proposition 2.2 to show that the image of X_{T} under E consists of fat modules.

Proposition 1.12. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting module over Λ_{0}.

(a) If 0 → A → B → C → 0 is an F^{T} -exact sequence in gr_{0} Λ with A in G_{T} (Λ), then 0 → E(C) → E(B) → E(A) → 0 is an exact sequence.

(b) If 0 → A → B → C → 0 is an F^{T} -exact sequence in gr_{0} Λ with A and C in G_{T} (Λ), then B is in G_{T} (Λ).

Proof. (a) Denote the map A → B by f. Then we get an induced map E(f): E(B) → E(A). Since the sequence 0 → A → B → C → 0 is F^{T} -exact and all modules are generated in degree 0, the map E(f)_{0} = Hom_{Λ}(f, T): Hom_{Λ}(B, T) → Hom_{Λ}(A, T) is onto. Since A is in G_{T} (Λ) and E(f) is onto in degree 0, the map E(f) is onto. The exactness of 0 → E(C) → E(B) → E(A) → 0 then follows immediately.

(b) This is an easy consequence of (a).

Proposition 1.13. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting module over Λ_{0}. Assume that Λ satisfies the ⊗-Tor-conditions with respect to T.

(a) Suppose that 0 → A → B → C → 0 is an exact sequence in gr_{0} Λ with C_{i} in X_{T} for all i ≥ 0. If A and B are in K_{T} (Λ), then C is in K_{T} (Λ).

(b) If 0 → A → B → C → 0 is an exact sequence with A and C in K_{T} (Λ), then B is in K_{T} (Λ).

Proof. (a) It is enough to show that C has a linear fat resolution. Consider the following commutative diagram.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & A^{1} \ar[r] \ar[d] & B^{1} \ar[r] \ar[d] & C^{1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Λ ⊗_{Λ_{0}} A_{0} \ar[r] \ar[d] & Λ ⊗_{Λ_{0}} B_{0} \ar[r] \ar[d] & Λ ⊗_{Λ_{0}} C_{0} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A \ar[r] \ar[d] & B \ar[r] \ar[d] & C \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

The middle row is exact, since C_{0} is in X_{T} and Λ satisfies the ⊗-Tor-conditions with respect to T. Hence the upper row, which is the sequence of kernels, is exact. It follows that C^{1} is generated in degree 1. Since Λ satisfies the ⊗-Tor-conditions with respect to T, we infer that C_{i}^{1} is in X_{T} for all i ≥ 1. By assumption A^{1}[−1] and B^{1}[−1] are in K_{T} (Λ), so that by repeating the above arguments it follows that C has a linear fat resolution.

(b) Using the same diagram as in the proof of (a), the claim follows directly.

2. Duality

In this section we show that if Λ is Koszul with respect to T, then E_{T} (Λ) has the same property. Further we show that E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras, and that there is a duality between the T -Koszul modules.

We first want to show that the ⊗-Tor-conditions hold for E_{T} (Λ) when Λ is Koszul with respect to T.

For any M in gr_{0} Λ denote as before by M^{1} the graded Λ-module given by the exact sequence 0 → M^{1} → Λ ⊗_{Λ_{0}} M_{0} → M → 0.

Proposition 2.1. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting module over Λ_{0}. Let Γ = ⊕_{i≥0} Ext_{Λ}^{i}(T, T). Suppose that Λ is Koszul with respect to T.

(a) Γ_{i} ≃ Hom_{Λ_{0}}(X_{i}^{i} ⊗_{Λ_{0}} T, T) as Γ_{0}-bimodules, where

⋯ → Λ ⊗_{Λ_{0}} X_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗_{Λ_{0}} X_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗_{Λ_{0}} Λ_{0} ⟶^{d_{0}} Λ_{0} → 0

is the linear fat resolution of Λ_{0}.

(b) Γ satisfies the ⊗-Tor-conditions with respect to _{Γ_{0}}T.

Proof. (a) Let

η: ⋯ → Λ ⊗_{Λ_{0}} X_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗_{Λ_{0}} X_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗_{Λ_{0}} Λ_{0} ⟶^{d_{0}} Λ_{0} → 0

be the linear fat resolution of Λ_{0} over Λ. Since Λ satisfies the ⊗-Tor-conditions with respect to T, the tensor product η ⊗_{Λ_{0}} T is a linear fat resolution of T over Λ. By Proposition 1.11 we have that Γ_{i} = Ext_{Λ}^{i}(T, T) ≃ Hom_{Λ_{0}}(X_{i}^{i} ⊗_{Λ_{0}} T, T) as Γ_{0}-bimodules.

(b) Let C be in X_{T}. Then we have the following isomorphisms

Γ_{i} ⊗_{Γ_{0}} Hom_{Λ_{0}}(C, T) ≃ Hom_{Λ_{0}}(X_{i}^{i} ⊗_{Λ_{0}} T, T) ⊗_{Γ_{0}} Hom_{Λ_{0}}(C, T)
≃ Hom_{Λ_{0}}(T, (X_{i}^{i}, T)) ⊗_{Γ_{0}} Hom_{Λ_{0}}(C, T)
≃ Hom_{Λ_{0}}(C, (X_{i}^{i}, T))
≃ Hom_{Λ_{0}}(X_{i}^{i} ⊗_{Λ_{0}} C, T)

by Proposition 1.9. Since X_{i}^{i} satisfies the ⊗-condition with respect to T by Corollary 1.4, it follows that Γ satisfies the ⊗-condition with respect to T.

Let C be in X_{T}. Then the sequence of left add T -approximations

0 → C → T_{0} ⟶^{d_{0}} T_{1} ⟶^{d_{1}} T_{2} ⟶^{d_{2}} ⋯

is an exact sequence, where in addition C_{i} = Ker d_{i} is in X_{T} for all i ≥ 0. Then the sequence

⋯ → _{Λ_{0}}(T_{2}, T) → _{Λ_{0}}(T_{1}, T) → _{Λ_{0}}(T_{0}, T) → _{Λ_{0}}(C, T) → 0

is a projective resolution of Hom_{Λ_{0}}(C, T) over Γ_{0}. Applying the functor Γ_{i}⊗_{Γ_{0}} to this exact sequence, we obtain the following commutative diagram using the same isomorphism as above.

\xymatrix{Γ_{i} ⊗_{Γ_{0}} (T_{1}, T) \ar[r] \ar[d]^{≀} & Γ_{i} ⊗_{Γ_{0}} (T_{0}, T) \ar[r] \ar[d]^{≀} & Γ_{i} ⊗_{Γ_{0}} (C, T) \ar[r] \ar[d]^{≀} & 0 \\ (X_{i}^{i} ⊗_{Λ_{0}} T_{1}, T) \ar[r] & (X_{i}^{i} ⊗_{Λ_{0}} T_{0}, T) \ar[r] & (X_{i}^{i} ⊗_{Λ_{0}} C, T) \ar[r] & 0}

Since X_{i}^{i} satisfies the ⊗-Tor-conditions with respect to T by Corollary 1.4, the sequence 0 → X_{i}^{i} ⊗_{Λ_{0}} C_{j} → X_{i}^{i} ⊗_{Λ_{0}} T_{j} → X_{i}^{i} ⊗_{Λ_{0}} C_{j+1} → 0 is exact for all i ≥ 0 and j ≥ 0. Hence the lower row in the above diagram is exact, and therefore Tor_{j}^{Γ_{0}}(Γ_{i}, X_{_{Γ_{0}} T}) = (0) for all j ≥ 1, and Γ satisfies the ⊗-Tor-conditions with respect to _{Γ_{0}}T.

The missing piece in order to show that E_{T} (Λ) is Koszul with respect to T when Λ is Koszul with respect to T, is to show that E_{T} (Λ)_{0} has a linear fat resolution. Lemma 1.6 and Proposition 1.13 are our preliminary stepping stones to prove this.

Proposition 2.2. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting module over Λ_{0}. Assume that Λ is T -Koszul.

(a) If M is in K_{T} (Λ), then M_{≥1}[−1] is in K_{T} (Λ).

(b) If X is in X_{T}, then X is in K_{T} (Λ) and E(X) ≃ Γ ⊗_{Γ_{0}} Hom_{Λ_{0}}(X, T), where Γ = E_{T} (Λ). In particular, E(X) is a fat module.

Proof. (a) Let M be in K_{T} (Λ). Then the sequence 0 → M^{1} → Λ_{≥1}⊗_{Λ_{0}} M_{0} → M_{≥1} → 0 is exact. By assumption M^{1}[−1] is in K_{T} (Λ). Since Λ_{0} is in K_{T} (Λ), we have that (Λ_{0})^{1}[−1] = Λ_{≥1}[−1] is in K_{T} (Λ). Since Λ satisfies the ⊗-Tor-conditions with respect to T, tensoring the linear fat resolution of Λ_{≥1} with T gives rise to a linear fat resolution of Λ_{≥1} ⊗_{Λ_{0}} T. Hence Λ_{≥1} ⊗_{Λ_{0}} T is also in K_{T} (Λ). Since C_{i} is in X_{T} for all i ≥ 1, Proposition 1.13 implies that M_{≥1}[−1] is in K_{T} (Λ).

(b) Let X be in X_{T}. Since Λ_{0} is in K_{T} (Λ) and Λ satisfies the ⊗-Tor-conditions with respect to T, tensoring the linear fat resolution of Λ_{0} with X gives rise to a linear fat resolution of X. Hence X_{T} is in K_{T} (Λ).

Let ϕ_{X}: Γ ⊗_{Γ_{0}} Hom_{Λ_{0}}(X, T) → E(X) be the map given by multiplication. The homomorphism ϕ clearly commutes with finite direct sums, and ϕ_{T} is an isomorphism. It follows that ϕ_{T′} is an isomorphism for all X = T′ in add T. For X in X_{T} we have an exact sequence of left add T -approximations 0 → X → T_{0} ⟶^{h} T_{1}, where Im h and Coker h are in X_{T}. By Proposition 1.12 the sequence E(T_{1}) → E(T_{0}) → E(X) → 0 is exact. The sequence (T_{1}, T) → (T_{0}, T) → (X, T) → 0 is exact, and tensoring this sequence with Γ gives rise to the following commutative diagram

\xymatrix{E(T_{1}) \ar[r] \ar[d]^{≀} & E(T_{0}) \ar[r] \ar[d]^{≀} & E(X) \ar[r] \ar[d] & 0 \\ Γ ⊗_{Γ_{0}} (T_{1}, T) \ar[r] & Γ ⊗_{Γ_{0}} (T_{0}, T) \ar[r] & Γ ⊗_{Γ_{0}} (X, T) \ar[r] & 0}

We infer that E(X) ≃ Γ ⊗_{Γ_{0}} Hom_{Λ_{0}}(X, T).

Now we have the necessary tools to show that Λ Koszul with respect to T implies that E_{T} (Λ) is Koszul with respect to T.

Theorem 2.3. If Λ is Koszul with respect to T, then Γ = E_{T} (Λ) is Koszul with respect to T.

Proof. By Proposition 2.1 we infer that Γ = E_{T} (Λ) satisfies the ⊗-Tor-conditions with respect to _{Γ_{0}}T.

We have that Γ_{0} = E(Λ ⊗_{Λ_{0}} T) and Λ ⊗_{Λ_{0}} T is in K_{T} (Λ) by Proposition 1.11. By Lemma 1.6 we have that the sequence 0 → E((Λ ⊗_{Λ_{0}} T)_{≥1})[1] → E(T) = Γ → E(Λ ⊗_{Λ_{0}} T) = Γ_{0} → 0 is exact. Since (Λ ⊗_{Λ_{0}} T)_{≥1} is in K_{T} (Λ), we inductively infer that the sequences

0 → E((Λ ⊗_{Λ_{0}} T)_{≥i+1})[1] → E(Λ_{i} ⊗_{Λ_{0}} T) → E((Λ ⊗_{Λ_{0}} T)_{≥i}) → 0

are exact. Since E(Y) ≃ Γ⊗_{Γ_{0}} (Y, T) for all Y in X_{T} by Proposition 2.2, we conclude that Γ_{0} has a linear fat resolution. Hence Γ is Koszul with respect to T.

Whenever E_{T} (Γ) ≃ Λ as graded algebras for Γ = E_{T} (Λ), we have the natural functors E = ⊕_{i≥0} Ext_{Λ}^{i}( , T): Gr Λ → Gr Γ and E′ = ⊕_{i≥0} Ext_{Γ}^{i}( , T): Gr Γ → Gr Λ. Our next aim is to show that the functors E and E′ induce inverse dualities on K_{T} (Λ) and K_{T} (Γ) when Λ is T -Koszul. We first show that the compositions of the functors E and E′ are identities on objects, and for this we need the following.

Lemma 2.4. Assume that Λ is Koszul with respect to T. If M is in K_{T} (Λ), then

(a) E(M) is in K_{T} (Γ).

(b) E′E(M)_{i} ≃ M_{i} as a Λ_{0}-bimodule.

(c) The identification ϕ_{i}: M_{i} → E′E(M)_{i} is given in the following way. Let η_{i}(M): 0 → E(M_{≥i+1})[1] → E(M_{i}) → E(M_{≥i}) → 0. Then ϕ_{i}(m_{i}) = (E(a_{m_{i}})·η_{i}(M))·η_{i−1}(M) ⋯ η_{1}(M)·η_{0}(M), where m_{i} is in M_{i} and a_{m_{i}}: Λ → M_{≥i} is given by a_{m_{i}}(λ) = λm_{i}.

Proof. (a) Let M be in K_{T} (Λ). By Lemma 1.6, Proposition 2.1 and Proposition 1.11 (c) the sequence

⋯ → E(M_{2}) → E(M_{1}) → E(M_{0}) → E(M) → 0

is a linear fat resolution of E(M).

(b) Using Lemma 1.5 and the linear fat resolution of E(M) above it follows that

Ext_{Γ}^{i}(E(M), T) ≃ Ext_{Γ}^{i−1}(E(M_{≥1}), T)
≃ ⋯
≃ Ext_{Γ}^{1}(E(M_{≥i−1}), T)
≃ Hom_{Γ}(E(M_{≥i}), T)
≃ Hom_{Γ_{0}}(Hom_{Λ_{0}}(M_{i}, T), T)
≃ M_{i}

(c) It follows from (b) that the identification of Ext_{Λ}^{i}(M, T) with M_{i} is given by the composition of i − 1 applications of the connecting homomorphism given by the sequences η_{i}(M) and the identification between Ext_{Λ}^{1}(E(M^{i−1}, T) and Hom_{Γ_{0}}(E(M_{≥i})_{0}, T) given in Lemma 1.5. The identification of M_{i} with Hom_{Γ_{0}}(Hom_{Λ}(M_{i}, T), T) and Hom_{Γ}(E(M_{≥i}), T) is given by sending an element m_{i} in M_{i} to the map E(a_{m_{i}}): E(M_{≥i}) → T = E(Λ) where a_{m_{i}}: Λ → M_{≥i} is given by a_{m_{i}}(λ) = λm_{i}.

We can now prove the following.

Theorem 2.5. Assume that Λ is a Koszul algebra with respect to T.

(a) E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras.

(b) E′(E(M)) ≃ M as graded left Λ-modules for all M in K_{T} (Λ).

Proof. Before we start the proof of this result we introduce a functor that we need. Let M and N be in Gr(Λ) generated in degree 0 or higher, and let f: M → N be a morphism in Gr(Λ). Define rej_{T} (M) = M_{≥1} and rej_{T} (f) = f|_{M_{≥1}}. Inductively define rej_{T}^{i} = rej_{T} ∘ rej_{T}^{i−1} for i ≥ 2.

Let M be in K_{T} (Λ) and let η_{i}(M) denote the exact sequence 0 → E(M_{≥i+1}) → E(M_{i}) → E(M_{≥i}) → 0 for i ≥ 0. Let φ_{i}(M): M_{i} → E′E(M)_{i} be the identifications given in Lemma 2.4 for all i ≥ 0.

Let λ_{r} be in Λ_{r} for r ≥ 1 and m_{n} in M_{n} for n ≥ 1. Then

φ_{r}(Λ)(λ_{r}) = (E(a_{λ_{r}})η_{r}(Λ))η_{r−1}(Λ)η_{r−2}(Λ) ⋯ η_{0}(Λ)

and

φ_{n}(M)(m_{n}) = (E(a_{m_{n}})η_{n}(M))η_{n−1}(M)η_{n−2}(M) ⋯ η_{0}(M).

We want to show that φ_{r}(Λ)(λ_{r})φ_{n}(M)(m_{n}) = φ_{n+r}(M)(λ_{r}m_{n}). The product φ_{r}(Λ)(λ_{r})φ_{n}(M)(m_{n}) can be represented as

((E(a_{λ_{r}})η_{r−1}(Λ))E(rej_{T}^{r−1}(a_{m_{n}})))η_{n+r−1}(M) ⋯ η_{0}(M).

The sequence (E(a_{λ_{r}})η_{r−1}(Λ))E(rej_{T}^{r−1}(a_{m_{n}})) is the same as

(E(a_{λ_{r}})E(rej_{T}^{r} (a_{m_{n}})))η_{r−1}(M) = E(a_{λ_{r}}a_{m_{n}})η_{r−1}(M) = φ_{n+r}(M)(λ_{r}m_{n}).

Let λ_{0} be in Λ_{0} and m_{n} in M_{n} for n ≥ 1. Then φ_{0}(Λ)(λ_{0})φ_{n}(M)(m_{n}) is given by

(E(a_{λ_{0}})(E(a_{m_{n}})η_{n}(M)))η_{n−1}(M) ⋯ η_{0}(M).

Since E(a_{λ_{0}})(E(a_{m_{n}})η_{n}(M)) is equal to E(a_{λ_{0}}a_{m_{n}})η_{n}(M) = E(a_{λ_{0}m_{n}})η_{n}(M), it follows that φ_{0}(Λ)(λ_{0})φ_{n}(M)(m_{n}) = φ_{n}(M)(λ_{0}m_{n}).

Let λ_{r} be in Λ_{r} for r ≥ 1 and m_{0} in M_{0}. Then φ_{r}(Λ)(λ_{r})φ_{0}(M)(m_{0}) is given by

(E(a_{λ_{r}})η_{r}(Λ))η_{r−1}(Λ) ⋯ (η_{0}(Λ)E(a_{m_{0}})).

Using the same arguments as above this element can be represented as

((E(a_{λ_{r}})η_{r}(Λ))E(rej_{T}^{r−1}(a_{m_{0}})))η_{r−1}(M) ⋯ η_{0}(M).

Since ((E(a_{λ_{r}})η_{r}(Λ))E(rej_{T}^{r−1}(a_{m_{0}}))) is equal to E(a_{λ_{r}})(E(rej_{T}^{r} (a_{m_{0}})η_{r}(M))) = E(a_{λ_{r}m_{0}})η_{r}(M), it follows that φ_{r}(Λ)(λ_{r})φ_{0}(M)(m_{0}) = φ_{r}(M)(λ_{r}m_{0}).

Let λ_{0} be in Λ_{0} and m_{0} in M_{0}. Then φ_{0}(Λ)(λ_{0})φ_{0}(M)(m_{0}) = E(a_{λ_{0}})E(a_{m_{0}}) = E(a_{λ_{0}m_{0}}) = φ_{0}(M)(λ_{0}m_{0}).

This shows that Λ ≃ E_{T} (E_{T} (Λ)) = E′(T) = E′(E(Λ)) as graded algebras, and that M ≃ E′(E(M)) as graded left Λ-modules if M is in K_{T} (Λ).

To complete the proof that the functors E: K_{T} (Λ) → K_{T} (Γ) and E′: K_{T} (Γ) → K_{T} (Λ) are inverse dualities, we need to show that they are full and faithful.

Theorem 2.6. Assume that Λ is Koszul with respect to T and let Γ = E_{T} (Λ). Then we have the following assertions.

(a) Γ is Koszul with respect to T, and Λ ≃ E_{T} (E_{T} (Λ)) as graded algebras.

(b) The functors

E = ⊕_{i≥0} Ext_{Λ}^{i}( , T): Gr Λ → Gr Γ

and

E′ = ⊕_{i≥0} Ext_{Γ}^{i}( , T): Gr Γ → Gr Λ

induce inverse dualities between K_{T} (Λ) and K_{T} (Γ).

(c) If X is in X_{T}, then E(X) is a fat Γ-module. If Z is a fat Λ-module, then E(Z) is in X_{_{Γ_{0}} T}.

Proof. The statements in (a) and (c) have been proved in Theorem 2.5, Proposition 2.2 (b) and Proposition 1.11 (c).

(b) By Theorem 2.5 (b) we have that E′ ∘ E ≃ id_{K_{T} (Λ)} and E ∘ E′ ≃ id_{K_{T} (Γ)} on objects.

Let M and N be in K_{T} (Λ). Then we have the following commutative diagram

[figure: _{Gr(Λ)}(M, N); E; _{Gr(E_{T} (Λ))}(E(N), E(M)); E′; _{Gr(Λ)}(E′(E(M)), E′(E(N))); _{Λ_{0}}(M_{0}, N_{0}); _{E_{T} (Λ)_{0}}(E(N)_{0}, E(M)_{0}); _{Λ_{0}}(M_{0}, N_{0})] ,

where the vertical maps are monomorphisms. It follows that E′ ∘ E induces a monomorphism from Hom_{Gr(Λ)}(M, N) to Hom_{Gr(Λ)}(E′(E(M)), E′(E(N))) = Hom_{Gr(Λ)}(M, N) for M and N in K_{T} (Λ). Since this homomorphism group is of finite length over the center of Λ_{0}, the composition E′ ∘ E induces an isomorphism on Hom_{Gr(Λ)}(M, N) for all M and N in K_{T} (Λ). Similarly, we can show that E ∘ E′ induces an isomorphism on Hom_{E_{T} (Λ)}(X, Y) for all X and Y in K_{T} (Γ). This shows that E and E′ are full and faithful, so that E and E′ are inverse dualities.

3. T -Koszul algebras and quadratic algebras

In this section we prove that T -Koszul algebras are quadratic, and, that the relations for the Ext-algebra are orthogonal to those of the original algebra. We start by proving that any T -Koszul algebra is quadratic.

Proposition 3.1. Let Λ = ⊕_{i≥0}Λ_{i} be a Koszul algebra with respect to T.

(a) Λ ≃ T_{Λ_{0}}(Λ_{1})/I with I contained in T_{Λ_{0}}(Λ_{1})_{≥2}.

(b) I is a quadratic ideal in T_{Λ_{0}}(Λ_{1}).

Proof. (a) Let Λ = ⊕_{i≥0}Λ_{i} be a T -Koszul algebra. Then Λ is generated in degrees 0 and 1. Hence it is well-known that Λ ≃ T_{Λ_{0}}(Λ_{1})/I, for some ideal I in the tensor algebra T_{Λ_{0}}(Λ_{1}), with I contained in T_{Λ_{0}}(Λ_{1})_{≥2}.

(b) Let L = T_{Λ_{0}}(Λ_{1})_{≥1} = T_{Λ_{0}}(Λ_{1}) ⊗_{Λ_{0}} Λ_{1} and J = L/I. Then Λ_{0} = Λ/J, and we have the following exact sequence

0 → I/IL → L/IL → Λ → Λ_{0} → 0.

We have that

L/IL = (T_{Λ_{0}}(Λ_{1}) ⊗_{Λ_{0}} Λ_{1})/I(T_{Λ_{0}}(Λ_{1}) ⊗_{Λ_{0}} Λ_{1})
≃ (T_{Λ_{0}}(Λ_{1})/I) ⊗_{T_{Λ_{0}} (Λ_{1})} (T_{Λ_{0}}(Λ_{1}) ⊗_{Λ_{0}} Λ_{1})
= ((T_{Λ_{0}}(Λ_{1})/I) ⊗_{T_{Λ_{0}} (Λ_{1})} T_{Λ_{0}}(Λ_{1})) ⊗_{Λ_{0}} Λ_{1}
≃ T_{Λ_{0}}(Λ_{1})/I ⊗_{Λ_{0}} Λ_{1} = Λ ⊗_{Λ_{0}} Λ_{1}

This shows that the above exact sequence can be rewritten as

0 → I/IL → Λ ⊗_{Λ_{0}} Λ_{1} → Λ ⊗_{Λ_{0}} Λ_{0} → Λ_{0} → 0.

Since Λ is T -Koszul, the above exact sequence is the start of a fat resolution of Λ_{0}. By definition Λ is in K_{T} (Λ), in particular, Λ_{0} is in K_{T} (Λ) and therefore Λ_{0} has a linear fat resolution. In particular, there is an epimorphism Λ ⊗_{Λ_{0}} (I/IL)_{2}[2] → I/IL, and therefore I/IL is generated by elements in

(I/IL)_{2} = (I/IL) ∩ (Λ ⊗_{Λ_{0}} Λ_{1})_{2} = (I/IL) ∩ (Λ_{1} ⊗_{Λ_{0}} Λ_{1}).

Let x_{i} be in Λ_{1} ⊗_{Λ_{0}} Λ_{1} for i = 1, …, n, such that {x_{1}, x_{2}, …, x_{n}} are generators of the left Λ-module I/IL. The ideal ⟨x_{1}, x_{2}, …, x_{n}⟩ in T_{Λ_{0}}(Λ_{1}) is a graded ideal since it is generated by homogeneous elements. Therefore the T_{Λ_{0}}(Λ_{1})- module I/⟨x_{1}, x_{2}, …, x_{n}⟩ is a graded T_{Λ_{0}}(Λ_{1})-module. It follows immediately that I/⟨x_{1}, x_{2}, …, x_{n}⟩ = (I/⟨x_{1}, x_{2}, …, x_{n}⟩)L. The graded Nakayama Lemma (the proof is similar to the ungraded case, see [14] Lemma I.7.4) now gives that {x_{1}, x_{2}, …, x_{n}} is a generating set for I as a twosided ideal in T_{Λ_{0}}(Λ_{1}), hence I is a quadratic ideal in T_{Λ_{0}}(Λ_{1}).

When Λ is T -Koszul, then we have shown that Γ = E_{T} (Λ) is T -Koszul. Hence Γ is also quadratic in the above sense. Furthermore we give a closed formula for Γ in terms of Λ.

Theorem 3.2. Let Λ be a T -Koszul algebra. Let 0 → X^{2} → Λ ⊗_{Λ_{0}} Λ_{1} → Λ ⊗_{Λ_{0}} Λ_{0} → Λ_{0} → 0 be the start of a linear fat resolution of Λ_{0}.

(a) Λ ≃ T_{Λ_{0}}(Λ_{1})/⟨X_{2}^{2}⟩.

(b) Denote by Γ_{0} and Γ_{1} the endomorphism ring End_{Λ_{0}}(T) and Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T), respectively. Then

E_{T} (Λ) = T_{Γ_{0}}(Γ_{1})/⟨A_{2}⟩,

3. T -KOSZUL ALGEBRAS AND QUADRATIC ALGEBRAS 29

where A_{2} = {∑ β_{i} ⊗ α_{i} ∈ Γ_{1} ⊗_{Γ_{0}} Γ_{1} | ∑ β_{i}(id_{Λ_{1}} ⊗ α_{i})|_{X_{2}^{2}⊗_{Λ_{0}} T} = 0}.

Proof. (a) By Proposition 3.1 we have that Λ ≃ T_{Λ_{0}}(Λ_{1})/I for some quadratic ideal I in T_{Λ_{0}}(Λ_{1}). The ideal I is generated by the kernel of the multiplication map Λ_{1} ⊗_{Λ_{0}} Λ_{1} → Λ_{2}. The kernel is exactly given by X_{2}^{2}.

(b) Since Λ is T -Koszul, we know by Theorem 2.3 and Proposition 3.1 that E_{T} (Λ) = T_{E_{T} (Λ)_{0}}(E_{T} (Λ)_{1})/⟨A_{2}⟩ for some submodule A_{2} of E_{T} (Λ)_{1}⊗_{E_{T} (Λ)_{0}} E_{T} (Λ)_{1}.

We also have that E_{T} (Λ)_{i} ≃ Γ_{i} for i = 0, 1. Using these identifications the submodule A_{2} is given as the kernel of the multiplication map Γ_{1} ⊗_{Γ_{0}} Γ_{1} → E_{T} (Λ)_{2}. We have that E_{T} (Λ)_{2} ≃ Hom_{Λ_{0}}(X_{2}^{2} ⊗_{Λ_{0}} T, T). Using this identification in addition and that Λ_{1} satisfies the Tor-condition we obtain the following commutative diagram by Proposition 1.9 and Lemma 1.7

[figure: E_{T} (Λ)_{1} ⊗_{E_{T} (Λ)_{0}} E_{T} (Λ)_{1}; ≀; E_{T} (Λ)_{2}; ≀; (Λ_{1} ⊗_{Λ_{0}} T, T) ⊗_{Γ_{0}} (Λ_{1} ⊗_{Λ_{0}} T, T); ≀; (X_{2}^{2} ⊗_{Λ_{0}} T, T); (T, (Λ_{1}, T)) ⊗_{Γ_{0}} (Λ_{1} ⊗_{Λ_{0}} T, T); ≀; (X_{2}^{2} ⊗_{Λ_{0}} T, T); (Λ_{1} ⊗_{Λ_{0}} T, (Λ_{1}, T)); ≀; (X_{2}^{2} ⊗_{Λ_{0}} T, T); (Λ_{1} ⊗_{Λ_{0}} Λ_{1} ⊗_{Λ_{0}} T, T); (X_{2}^{2} ⊗_{Λ_{0}} T, T)]

where the identification ψ: (Λ_{1}⊗_{Λ_{0}} T, T)⊗_{Γ_{0}} (Λ_{1}⊗_{Λ_{0}} T, T) → (Λ_{1}⊗_{Λ_{0}} Λ_{1}⊗_{Λ_{0}} T, T) is given by ψ(β⊗α) = β(id_{Λ_{1}} ⊗α) for α and β in Γ_{1} = Hom_{Λ_{0}}(Λ_{1}⊗_{Λ_{0}} T, T). Moreover the last horizontal map is given by θ^{∗} = Hom_{Λ_{0}}(θ, T), where θ: X_{2}^{2} ⊗_{Λ_{0}} T → Λ_{1} ⊗_{Λ_{0}} Λ_{1} ⊗_{Λ_{0}} T is the inclusion. We have that

Ker θ^{∗} = {∑ β_{i} ⊗ α_{i} ∈ Γ_{1} ⊗_{Γ_{0}} Γ_{1} | ∑ β_{i}(id_{Λ_{1}} ⊗ α_{i})|_{X_{2}^{2}⊗_{Λ_{0}} T} = 0}.

This completes the proof.

CHAPTER III Generalized T -Koszul algebras

In this chapter we discuss the second approach to our generalization of Koszul algebras, which we call generalized T -Koszul algebras. Our point of view is to focus on essential common properties of the category of Koszul modules for a Koszul algebra and the category X_{T} associated with a Wakamatsu cotilting module. At the same time we have in mind to get a duality theorem, which is the main result of this chapter. It turns out that the new class contains the class of T -Koszul algebras discussed in the previous chapter, along with the example at the end of Chapter I.

Let Λ = ⊕_{i≥0}Λ_{i} be a positively ℤ-graded ring, and let T be a Λ-module. Our main requirement on the pair (Λ, T) is that the functor E = ⊕_{i≥0} Ext_{Λ}^{i}( , T) induces a duality between “nice” subcategories of Gr Λ containing Λ and T, and a corresponding type of subcategory of Gr E_{T} (Λ), where E_{T} (Λ) = ⊕_{i≥0} Ext_{Λ}^{i}(T, T). In order for this to be true, it is easy to see that T must be concentrated in degree 0, since T = E′(E(T)) = E′(E_{T} (Λ)) = Hom_{E_{T} (Λ)}(E(T), E(Λ)) ⊕ 0 ⊕ · · ·. Hence it is no restriction to assume that T is concentrated in degree 0.

In Section 1 we identify some essential properties of the subcategories we are interested in, and give some necessary conditions for having a reasonable duality theory.

We introduce the notion of generalized T -Koszul algebras in section 2, and show the relationship to the T -Koszul algebras discussed in Chapter II. In section 3 we compare with the classical cases. We see in section 4 that the ⊗-Tor-conditions, which occurred in the definition of a T -Koszul algebra, are almost necessary in order to have a good duality theory. In section 5 we investigate correspondences between various types of modules via the functor E. The main duality theorem is discussed in section 6.

1. Natural properties of Koszul modules

In this section we identify essential common properties of the category of Koszul modules over a Koszul algebra and the subcategories X_{T} of mod Λ_{0} associated with Wakamatsu cotilting Λ_{0}-modules T. We also give some necessary conditions in order to have a duality theory.

For the category of Koszul modules K(Λ) over a Koszul algebra, we have that if M is in K(Λ), then rad M[−1] and Ω_{Λ}(M)[−1] are again in K(Λ). When T is a Wakamatsu cotilting Λ-module, and M is in X_{T}, then Coker f is again in X_{T}, when f: M → M_{T} is a minimal left add T -approximation, and Ω_{Λ}(M) is also in X_{T}. For a Koszul algebra Λ = Λ_{0} +Λ_{1} +⋯, the module T = Λ_{0} is a cotilting Λ_{0}-module. In this case the natural epimorphism f: M → M/ rad M for a Koszul module M is a minimal left add T -approximation, and in this case it is the kernel rad M which, up to shift, is again a Koszul module. There is then a common formulation involving minimal left add T -approximations. So we first need to recall and develop some background on this topic.

Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1 with Λ_{0} an artin algebra. We will assume throughout that T is concentrated in degree 0, and we assume that T is a finitely generated module over Λ_{0} to ensure that E_{T} (Λ)_{0} = End_{Λ_{0}}(T) is an artin algebra.

Let X be a category and C a subcategory of X. For X in X a morphism f: X → C_{X} with C_{X} in C is called a left C-approximation of X, if for any morphism g: X → C′ with C′ in C there exists a morphism h: C_{X} → C′ such that hf = g. A left approximation f: X → C_{X} is called minimal if whenever gf = f for a morphism g: C_{X} → C_{X}, then g is an isomorphism.

Let ψ_{Λ} : Gr Λ → Mod Λ denote the forgetful functor. In the following let T be in gr Λ such that T = T_{0}. Moreover, let σT = add{T [j]}_{j∈ℤ}. Next we show that the modules in gr Λ have minimal left σT -approximations.

Lemma 1.1. (a) If M is in gr Λ, then ψ_{Λ}(M) has a minimal left add T -

approximation in mod Λ.

(b) If M is in gr Λ, then M has a minimal left σT -approximation M ⟶^{α}

M_{T} in gr Λ, such that ψ(M) ⟶^{ψ(α)} ψ(M_{T}) is a minimal left add ψ(T)- approximation of ψ(M).

Proof. (a) Suppose M is generated in the degrees i_{0}, i_{1}, …, i_{t}, where i_{0} < i_{1} < ⋯ < i_{t}. Let M_{i_{−1}} = (0). For each Λ_{0}-submodule U of M let ⟨U⟩ denote the Λ-submodule of M generated by U. Then by using Lemma 0.4 we have that

(1)  Hom_{Λ}(M, T) = ⊕_{j=0}^{t} Hom_{Λ_{0}}((M/⟨M_{i_{−1}} + ⋯ + M_{i_{j−1}}⟩)_{i_{j}}, T)

Let X_{j} be the Λ_{0}-module (M/⟨M_{i_{−1}} + ⋯ + M_{i_{j−1}}⟩)_{i_{j}}. Let X_{j} → T (j) be the minimal left add T -approximation in mod Λ_{0} for j = 0, …, t. Let f_{j} denote the composition M → (M/⟨M_{i_{−1}} + ⋯ + M_{i_{j−1}}⟩)_{i_{j}} = X_{j} → T (j) for j = 0, …, t and f = (f_{j})_{j=0}^{t}: M → ⊕_{j=0}^{t}(T (j))[i_{j}] in mod Λ. Given any morphism g: M → T, the morphism g can be decomposed according to the decomposition in (1), and therefore g factors through f. Hence we have constructed a left add T -approximation in mod Λ.

Choose a left add T -approximation f: M → M_{T} in mod Λ with M_{T} of minimal length as a Λ_{0}-module. Assume there is a morphism g: M_{T} → M_{T} such that gf = f. Then g^{n}f = f for all n ≥ 1. Since M_{T} has finite length over Λ_{0}, the Fitting Lemma applies, and therefore there exists an n such that M_{T} = Ker g^{n} ⊕ Im g^{n}. Then it follows that all maps h: M → T factor through M ⟶^{g^{n}f} Im g^{n}, and it is a left add T -approximation in mod Λ. If Ker g^{n} ≠ (0), we have a contradiction. Therefore Ker g^{n} = (0), and g is an isomorphism.

(b) We continue to use the notation from (a). As in (a), construct the maps X_{j} → T (j). We view X_{j} and T (j) as graded Λ-modules concentrated in degree i_{j}, and hence we can view the Λ_{0}-morphism X_{j} → T (j) as a degree 0 map. As above let f_{j}: M → T (j) be the induced map in Hom_{gr Λ}(M, T (j)). Let f = (f_{j})_{j=0}^{t}: M → ⊕_{j=0}^{t}T (j).

32 III. GENERALIZED T -KOSZUL ALGEBRAS

It suffices to show that any degree 0 homomorphism g: M → T [i] factors through f: M → ⊕_{j=0}^{t}T (j). Since M is generated in the degrees i_{0}, …, i_{t}, the map g is 0 if i ≠ i_{j} for some j, and hence factors through f. If i = i_{j} for some j, then g: M → T [i_{j}] factors through M → X_{j} → T [i_{j}] (since T [i_{j}] has support in degree i_{j}). But as Λ_{0}-modules X_{j} → ψ(T [i_{j}]) factors through X_{j} → T (j), and hence we get

\xymatrix{M \ar[rr]^{g} \ar[dr]^{f_{j}} \ar[dd]^{f} &  & T [i_{j}] \\  & T (j) \ar[ur] \\ ⊕_{j=1}^{t}T (j) \ar[ur]}

in gr Λ. The minimality we prove similarly as in (a).

Since ψ(M) and ψ(M_{T}) are finitely generated, then we have the following exact commutative diagram

\xymatrix{Hom_{Λ}(ψ(M_{T}), T) \ar[r] \ar[d]^{≀} & Hom_{Λ}(ψ(M), T) \ar[d]^{≀} \\ ⊕_{j∈ℤ} Hom_{gr Λ}(M_{T} , T [j]) \ar[r] & ⊕_{j∈ℤ} Hom_{gr Λ}(M, T [j]) \ar[r] & 0}

It follows directly from this diagram that ψ(M) → ψ(M_{T}) is a left add T - approximation in mod Λ.

Remark 1.2. Note that the approximation of a module M in gr Λ has support in at most the degrees of the generators of M.

We now go back to discussing common properties of K(Λ) and X_{T}. Let T be a module in gr Λ concentrated in degree zero, and let M be a module in gr Λ. Denote by rej_{T} (M) the submodule ∩_{f∈Hom_{gr Λ}(M,T)} Ker f of M and by \overline{M} the factor module M/ rej_{T} (M) of M. Let α_{M}: M → M_{T} denote the minimal left σT -approximation of M in gr Λ. Furthermore, denote by corej_{T} (M) the module M_{T} /\overline{M}. Then we have the exact sequences

0 → rej_{T} (M) → M → \overline{M} → 0

and

0 → \overline{M} → M_{T} → corej_{T} (M) → 0.

The last of the above sequences has an additional property which we now explain.

As in the introduction, let F^{σT} be the additive subfunctor of Ext_{Λ}^{1}( , ) for the subcategory σT of gr Λ. For simplicity we denote F^{σT} by F^{T}. Then one observes that the sequence 0 → \overline{M} → M_{T} → corej_{T} (M) → 0 is F^{T} -exact. By construction all these modules and sequences are in Gr Λ. The following observation follows directly from the previous result.

Lemma 1.3. Let M be in gr Λ. Then

ψ(rej_{T} (M)) = rej_{T} ψ(M) = ∩_{f∈Hom_{Λ}(ψ(M),ψ(T))} Ker f,
ψ(\overline{M}) = \overline{ψ(M)}

and

ψ(corej_{T} (M)) = corej_{T} (ψ(M)).

When Λ is Koszul, we have rej_{T} (M) = rad M and corej_{T} (M) = (0), so that rej_{T} (M)[−1] and corej_{T} (M) are Koszul modules when M is. Also \overline{M} = M/ rad M is a Koszul module. When T is a Wakamatsu cotilting module for an artin algebra Λ, then, as we have observed, corej_{T} (M) is in X_{T} when M is, and rej_{T} (M) = (0) in this case. Further \overline{M} = M.

If M is in K(Λ) or in X_{T}, the property of syzygies can be formulated in a common way to say that ⟨Ω(M)_{0}⟩ and Ω(M)/⟨Ω(M)_{0}⟩[−1] are also in K(Λ) or X_{T}, respectively. For in the case of Koszul modules, ⟨Ω(M)_{0}⟩ = (0) and Ω(M)/⟨Ω(M)_{0}⟩[−1] = Ω(M)[−1], and in the second case ⟨Ω(M)_{0}⟩ = Ω(M) and Ω(M)/⟨Ω(M)_{0}⟩ = (0). In addition, any Λ-module of the form Λ ⊗_{Λ_{0}} N for a Λ_{0}- module N is projective, and is hence in K(Λ), and in the second case Λ ⊗_{Λ_{0}} N ≃ N since Λ = Λ_{0}.

We shall use the above observations as inspiration for our definitions in the next section. We end this section by deducing some necessary conditions in order to have a reasonable duality theory. In particular we give necessary conditions on the modules in the exact sequences associated with M given above. In order for a duality to make sense, we must assume that E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras.

Lemma 1.4. Assume that E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras. Suppose M is in G_{T} (Λ) and that E′E(M) ≃ M as graded Λ-modules. Then

(a) M_{0} is dualized by T as a Λ_{0}-module.

(b) corej_{T} (M) is in Sub(T).

(c) \overline{M} = M_{0}.

(d) rej_{T} (M) = M_{≥1}.

Proof. We have that

M_{0} ≃ Hom_{Γ}(E(M), Hom_{Λ}(Λ, T)) ≃ Hom_{Γ_{0}}(E(M)_{0}, Hom_{Λ_{0}}(Λ_{0}, T))
≃ Hom_{Γ_{0}}(Hom_{Λ_{0}}(M, T), T)

Consider the minimal left add T -approximation corej_{T} (M) → corej_{T} (M)_{T} = T^{1}. Then the sequence

(T^{1}, T) → (M_{T} , T) → (\overline{M}, T) ≃ (M, T) → 0

is exact. This gives rise to the following exact commutative diagram

[figure: 0; ((M, T), (Λ, T)); ≀; ((M_{T} , T), (Λ, T)); ≀; ((T^{1}, T), (Λ, T)); ≀; 0; M_{0}; M_{T}; T^{1}; 0; \overline{M}; f; M_{T}; corej_{T} (M); 0]

It follows that f is a monomorphism, and that M_{0} is in Sub(T).

Let p: M → M_{0} be the natural epimorphism. The map M → \overline{M} factors through p. Hence \overline{M} is a factor module of M_{0}. Since \overline{M} is also a submodule of M_{0} and the modules have finite length over Λ_{0}, it follows that \overline{M} and M_{0} are isomorphic.

It is an immediate consequence of the above exact commutative diagram that corej_{T} (M) is in Sub(T). Recall from Notation and general preliminary results, that M_{0} is then said to be dualized by T as a Λ_{0}-module.

We shall now see that assuming T to be a Wakamatsu cotilting module is natural in order to have a reasonable duality theory. Assuming E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras we would like to have subcategories X and Y of G_{T} (Λ) and G_{T} (E_{T} (Λ)) respectively, such that the functor E induces a duality between X and Y. Due to the central role of the modules Λ and T, it is natural to require that they lie in X. Under one extra assumption, that X and Y are closed under taking coreject with respect to T, we obtain the following necessary condition for having duality.

Proposition 1.5. Assume that E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras, and that there are subcategories X in G_{T} (Λ) and Y in G_{T} (Γ) such that they are dual subcategories via the functors E and E′. Furthermore, suppose that X and Y are closed under taking coreject with respect to _{Λ}T and _{Γ}T, respectively, and that Λ and T are in X. Then

(a) _{Λ_{0}}T is a Wakamatsu cotilting module.

(b) _{Γ_{0}}T is a Wakamatsu cotilting module.

(c) If M is in X, then M_{0} is in X_{_{Λ_{0}} T}.

(d) If N is in Y, then N_{0} is in X_{_{Γ_{0}} T}.

Proof. Let M be in X. Since X is closed under corejects, then by Lemma 1.4 there exists an exact sequence

0 → M_{0} → T^{0} → T^{1} → T^{2} → ⋯

of minimal left add T -approximations.

We have that E(M) is in Y. Using that T is in X, we infer that Γ = E(T) is in Y, and therefore EE′(Γ) ≃ Γ. Hence we can apply Lemma 1.4 and use that Y is closed under corejects to obtain an exact sequence

0 → E(M)_{0} → T^{0} → T^{1} → T^{2} → ⋯

of minimal left add T -approximations. Applying Hom_{Γ_{0}}( , T) gives rise to the exact sequence

⋯ → (T^{2}, T) → (T^{1}, T) → (T^{0}, T) → (E(M)_{0}, T) → 0,

which is a projective Λ_{0}-resolution of (E(M)_{0}, T). Since M is generated in degree 0, we have that

(E(M)_{0}, T) ≃ (Hom_{Λ_{0}}(M_{0}, T), Hom_{Λ_{0}}(Λ_{0}, T)) ≃ M_{0}

by Lemma 1.4. Applying Hom_{Λ_{0}}( , T) we obtain the following commutative diagram

[figure: 0; ((E(M)_{0}, T), T); ≀; ((T^{0}, T), T); ≀; ((T^{1}, T), T); ≀; ⋯; 0; E(M)_{0}; T^{0}; T^{1}; ⋯]

The lower row is exact, hence Ext_{Λ_{0}}^{i}(M_{0}, T) = (0) for all i > 0. This shows the statements in (a) and (c), and the claims in (b) and (d) follow by symmetry.

2. GENERALIZED T -KOSZUL ALGEBRAS 35

2. Generalized T -Koszul algebras

Motivated by the connections between the categories K(Λ) and X_{T} discussed in the previous section, we introduce our notion of generalized T -Koszul algebras, and the corresponding category of Koszul modules. We also compare with our earlier definitions.

We have pointed out six operations under which the categories K(Λ) and X_{T} are closed. Actually, the first three conditions are related to the last three via duality, as now discuss.

Let M be in gr Λ and generated in degree zero. Then we have the following commutative exact diagram

[figure: 0; 0; V_{M}; V_{M}; 0; Ω_{Λ}(M); P (M) = Λ ⊗_{Λ_{0}} P (M)_{0}; M; 0; 0; M^{1}; Λ ⊗_{Λ_{0}} M_{0}; M; 0; 0; 0]

where P (M)_{0} is the projective cover of M_{0} as a Λ_{0}-module. The module V_{M} is generated in degree 0, since Λ ⊗_{Λ_{0}} Ω_{Λ_{0}}(M_{0}) → V_{M} is an epimorphism. The module M^{1} is generated in degree 1 and higher, since (Λ ⊗_{Λ_{0}} M_{0})_{0} = Λ_{0} ⊗_{Λ_{0}} M_{0} ≃ M_{0}. It follows from this that V_{M} = ⟨Ω_{Λ}(M)_{0}⟩ ⊂ Ω_{Λ}(M) and M^{1} = Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩. We show in Section 5 (see Proposition 5.4 and Proposition 5.5) that the concept of rej_{T} (M) is dual to Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩, \overline{M} is dual to Λ ⊗_{Λ_{0}} M_{0}, and corej_{T} (M) is dual to ⟨Ω_{Λ}(M)_{0}⟩ via the functor E, under some additional assumptions.

Let Λ be a ℤ-graded ring as above, and let now T be some module in mod Λ_{0}. Denote as usual ⊕_{i≥0} Ext_{Λ}^{i}(T, T) by E_{T} (Λ). For ease of notation we also denote E_{T} (Λ) by Γ. Define E: Gr Λ → Gr Γ by E( ) = ⊕_{i≥0} Ext_{Λ}^{i}( , T). Then E(Λ) = Hom_{Λ}(Λ, T), which we also denote by T, is a module also over Γ. Define E′: Gr Γ → Gr E_{T} (Γ) by E′ = ⊕_{i≥0} Ext_{Γ}^{i}( , T).

We now define the concept of generalized T -Koszul algebras.

Definition 2.1. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting Λ_{0}- module.

(a) We define K_{T} (Λ) to be the largest full subcategory of G_{T} (Λ) consisting of modules M with M_{0} in X_{T} such that K_{T} (Λ) is closed under the operations rej_{T} ( )[−1], \overline{( )}, corej_{T} ( ) Ω_{Λ}( )/⟨Ω_{Λ}( )_{0}⟩[−1], Λ ⊗_{Λ_{0}} ( )_{0} and ⟨Ω_{Λ}( )_{0}⟩.

(b) The modules in K_{T} (Λ) are called the Koszul modules over Λ with respect to T.

(c) The algebra Λ is said to be a generalized T -Koszul algebra if Λ and T are in K_{T} (Λ).

(d) The algebra Λ is said to be a generalized strongly T -Koszul algebra if K_{T} (Λ) = G_{T} (Λ).

For any subcategory Y of X_{T} define FR_{Λ}(Y) to be the full subcategory of Gr(Λ) consisting of all graded modules M such that M_{i} is in Y for all i and such that there is an exact sequence

⋯ → F_{n} → F_{n−1} → ⋯ → F_{1} → F_{0} → M → 0,

where for each n ≥ 0 the modules F_{n} ≃ Λ ⊗_{Λ_{0}} Y_{n}[n] for some Y_{n} in Y. Note that for a T -Koszul algebra Λ, the category of T -Koszul modules is FR_{Λ}(X_{T}). Immediately from the definition of K_{T} (Λ) for a generalized T -Koszul algebra we show that in this more general situation the category of T -Koszul modules has similar properties as the category of modules with linear fat resolutions FR_{Λ}(X_{T}) studied in the previous chapter.

Lemma 2.2. If M is in K_{T} (Λ), then the following assertions hold.

(a) M_{i} is in K_{T} (Λ) ∩ X_{T} for all i ≥ 0.

(b) There exists an exact sequence

⋯ → Λ ⊗_{Λ_{0}} M_{2}^{2}[2] → Λ ⊗_{Λ_{0}} M_{1}^{1}[1] → Λ ⊗_{Λ_{0}} M_{0} → M → 0

where M_{2}^{2} is in K_{T} (Λ) ∩ X_{T} for all i ≥ 0.

Proof. (a) Let M be in K_{T} (Λ). Then it follows immediately from the definition that \overline{M} = M_{0} and that M_{0} is in K_{T} (Λ) ∩ X_{T}. A consequence of this is that rej_{T} (M) = M_{≥1}, where, by definition of K_{T} (Λ), the module rej_{T} (M)[−1] is in K_{T} (Λ). Hence by using the same arguments as above, rej_{T} (M)[−1]_{0} = M_{1} is in K_{T} (Λ) ∩ X_{T}. Continuing this way we infer that M_{i} is in K_{T} (Λ) ∩ X_{T} for all i ≥ 0.

(b) Let M be in K_{T} (Λ). Then the sequence 0 → M^{1} → Λ ⊗_{Λ_{0}} M_{0} → M → 0 is exact with M^{1}[−1] in K_{T} (Λ) and M_{0} in K_{T} (Λ) ∩ X_{T} by (a). The claim in (b) then follows by induction.

Next we show that a T -Koszul algebra as defined in the previous chapter is a special case of the notion of a generalized T -Koszul algebra.

Proposition 2.3. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T be a Wakamatsu cotilting Λ_{0}- module. Then Λ is a T -Koszul algebra if and only if Λ satisfies the ⊗-Tor-conditions with respect to T and Λ is a generalized T -Koszul algebra.

Proof. Assume now that Λ is a T -Koszul algebra as defined in the previous chapter. We need to show that the category FR_{Λ}(X_{T}) is closed under our six operations. For the reject it follows from Proposition 2.2. We leave the other cases to the reader. Hence we have that FR_{Λ}(X_{T}) is contained in K_{T} (Λ). Since T and Λ ≅ Λ ⊗_{Λ_{0}} Λ_{0} are in FR_{Λ}(X_{T}), they are in K_{T} (Λ), so that Λ is a generalized T -Koszul algebra. By the previous result receive that FR_{Λ}(X_{T}) is contained in K_{T} (Λ), and consequently K_{T} (Λ) = FR_{Λ}(X_{T}).

Assume that Λ satisfies the ⊗-Tor-conditions with respect to X_{T} and that Λ is a generalized T -Koszul algebra. Since Λ is a generalized T -Koszul algebra, Λ is in K_{T} (Λ). By definition we have that Λ_{0} = \overline{Λ} is in K_{T} (Λ). Then by Lemma 2.2 we immediately get that Λ_{0} has a linear fat resolution. Therefore Λ_{0} is in FR_{Λ}(X_{T}) and Λ is T -Koszul.

3. The classical cases revisited

In this section we take a closer look at how the classical cases fit in with the concept of generalized T -Koszul algebras.

We start with the classical Koszul case. Note that for a semisimple ring Λ_{0} the only Wakamatsu cotilting module up to multiplicity of direct summands is Λ_{0}.

Proposition 3.1. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, where Λ_{0} is semisimple. Let T = Λ_{0}. Then the following are equivalent.

(a) Λ is a generalized Koszul algebra with respect to T.

(b) Λ is a generalized strongly Koszul algebra with respect to T.

(c) Λ is a Koszul algebra.

Proof. Given the assumptions on Λ, it is immediate that Λ satisfies the ⊗- Tor-conditions with respect to ^{⊥}T = add Λ_{0} = mod Λ_{0}. Using Proposition 2.3 we conclude that Λ is a generalized T -Koszul algebra if and only if Λ is a T -Koszul algebra. We already observed in Proposition 4.3 that under these assumptions that Λ is a T -Koszul algebra if and only if Λ is a Koszul algebra. This proves that (a) and (c) are equivalent.

We clearly have that (b) implies (a).

We have that G_{T} (Λ) is closed under all six operations, and hence K_{T} (Λ) = G_{T} (Λ), so that Λ is strongly Λ_{0}-Koszul. This shows that (a) implies (b), and this completes the proof of the proposition.

We next treat the (Wakamatsu) cotilting case.

Proposition 3.2. Let Λ = Λ_{0} be an artin algebra, and let T be a Wakamatsu cotilting module.

(a) Λ is a generalized T -Koszul algebra.

(b) The following are equivalent.

(i) T is a cotilting module.

(ii) Λ is a generalized strongly T -Koszul algebra with id_{Λ}T < ∞.

Proof. (a) When Λ is an artin algebra and T is a Wakamatsu cotilting module, G_{T} (Λ) is clearly ^{⊥}T. We have observed that the subcategory X_{T} is closed under the six operations. Since K_{T} (Λ) is supposed to be closed under corejects, it follows from the definition of X_{T} that K_{T} (Λ) = X_{T}. Since T and Λ are in X_{T}, we have that Λ is a generalized T -Koszul algebra.

(b) If T is a cotilting module, then id_{Λ}T < ∞ by definition. By the above K_{T} (Λ) = X_{T} and G_{T} (Λ) = ^{⊥}T. If T is a cotilting module, then X_{T} = ^{⊥}T. Hence Λ is strongly T -Koszul.

Assume that Λ is generalized strongly T -Koszul with id_{Λ}T < ∞. Then by assumption K_{T} (Λ) = X_{T} = ^{⊥}T = G_{T} (Λ). The subcategory X_{T} = ^{⊥}T is a resolving subcategory of mod Λ with an Ext-injective cogenerator T. Since id_{Λ}T < ∞, we infer that ^{ ˆ⊥}T = mod Λ. Then by [2, Theorem 2.3] ^{ ˆ⊥}T is contravariantly finite in mod Λ and (^{⊥}T)^{⊥} = { \atop add T}. It is clear that ^{⊥}T ∩ (^{⊥}T)^{⊥} = add T, which by [3] is the additive category generated by a cotilting module. Hence T is a cotilting module.

4. Necessity of the ⊗-Tor-conditions

In our original definition of a T -Koszul algebra the ⊗-Tor-conditions appeared, which at first sight look artificial. In this section we try to remove this mystery by showing that they are almost necessary in order to have a duality between nice subcategories. We start with the ⊗-condition.

Proposition 4.1. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting Λ_{0}-module T, and let X_{T} and K_{T} (Λ) be defined as before. Then Λ_{i} ⊗_{Λ_{0}} (K_{T} (Λ) ∩ X_{T}) ⊂ K_{T} (Λ) ∩ X_{T} for all i ≥ 0.

Proof. Let X be in K_{T} (Λ) ∩ X_{T}. Then Λ ⊗_{Λ_{0}} X is in K_{T} (Λ), and hence (Λ ⊗_{Λ_{0}} X)_{i} is in K_{T} (Λ) ∩ X_{T} for all i by Lemma 2.2.

To treat the Tor-condition we need the following preliminary result.

Lemma 4.2. Let M_{0} be in K_{T} (Λ) ∩ X_{T}. Then the exact sequence

η: 0 → ⟨Ω_{Λ}(M)_{0}⟩ → P (M) → Λ ⊗_{Λ_{0}} M_{0} → 0

is F^{T} -exact.

Proof. Applying Hom_{Λ}( , T) to the sequence η induces the exact sequence

0 → Hom_{Λ}(Λ ⊗_{Λ_{0}} M_{0}, T) → Hom_{Λ}(P (M), T) → Hom_{Λ_{0}}(⟨Ω_{Λ}(M)_{0}⟩, T).

This sequence is isomorphic to the sequence

0 → Hom_{Λ_{0}}(M_{0}, T) → Hom_{Λ_{0}}(P (M)_{0}, T) → Hom_{Λ_{0}}(Ω_{Λ_{0}}(M_{0}), T).

The cokernel of the last morphism in the above sequence is Ext_{Λ_{0}}^{1}(M_{0}, T), which is zero since M_{0} is in X_{T}. Hence η is F^{T} -exact.

We now have the following.

Proposition 4.3. Let Λ = ⨁_{i≥0} Λ_{i} be a graded artin algebra and T a Wakamatsu cotilting module. Then Tor_{1}^{Λ_{0}}(Λ_{i}, X_{T} ∩ K_{T} (Λ)) = (0) for i ≥ 0, and hence Ext_{Λ_{0}}^{1}(X_{T} ∩ K_{T} (Λ), D(Λ_{i})) = (0).

Proof. Let M_{0} be in K_{T} (Λ) ∩ X_{T}. We have the exact sequences

0 → ⟨Ω_{Λ}(M)_{0}⟩ → Λ ⊗_{Λ_{0}} P_{Λ_{0}}(M_{0}) → Λ ⊗_{Λ_{0}} M_{0} → 0

and

0 → Tor_{1}^{Λ_{0}}(Λ, M_{0}) → Λ ⊗_{Λ_{0}} Ω_{Λ_{0}}(M_{0}) → ⟨Ω_{Λ}(M)_{0}⟩ → 0.

Since ⟨Ω_{Λ}(M)_{0}⟩ is in K_{T} (Λ) and ⟨Ω_{Λ}(M_{0})_{0}⟩_{0} = Ω_{Λ_{0}}(M_{0}), it follows that Tor_{1}^{Λ_{0}}(Λ, M_{0}) = ⟨Ω_{Λ}(M)_{0}⟩^{1}, which is in K_{T} (Λ). Applying Hom_{Λ}( , T) to the short exact sequence starting in Tor_{1}^{Λ_{0}}(Λ, M_{0}) we obtain the following exact sequence

0 → (⟨Ω_{Λ}(M_{0})_{0}⟩, T) ⟶^{f} (Λ ⊗_{Λ_{0}} Ω_{Λ_{0}}(M_{0}), T) →

(Tor_{1}^{Λ_{0}}(Λ, M_{0}), T) → Ext_{Λ}^{1}(⟨Ω_{Λ}(M_{0})_{0}⟩, T),

where f is an isomorphism, since ⟨Ω_{Λ}(M)_{0}⟩_{0} = (Λ ⊗_{Λ_{0}} Ω_{Λ_{0}}(M_{0}))_{0}. It follows from Lemma 4.2 that Ext_{Λ}^{1}(⟨Ω_{Λ}(M_{0})_{0}⟩, T) = (0). Therefore Hom_{Λ}(Tor_{1}^{Λ_{0}}(Λ, M_{0}), T) = (0).

Since Tor_{1}^{Λ_{0}}(Λ, M_{0})[−1] is in K_{T} (Λ) (and therefore generated in degree 0), the module (Tor_{1}^{Λ_{0}}(Λ, M_{0})[1])_{0} is in X_{T} ⊂ Sub(T) by definition of K_{T} (Λ). Hence we infer that Tor_{1}^{Λ_{0}}(Λ, M_{0}) = (0). In particular, Tor_{1}^{Λ_{0}}(Λ_{j}, M_{0}) = (0) for all j ≥ 0.

5. Properties of the Ext-functor

In this section we investigate the category K_{T} (Λ) of T -Koszul modules for generalized T -Koszul algebras Λ more closely, especially the behavior under the functor E. Motivated by the property that a module goes to a module concentrated in degree 0 under E, we introduce the notion of fat modules, generalizing the notion from Chapter I. This leads to linear fat resolutions, which we show that all modules in K_{T} (Λ) have.

Throughout this section let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring with Λ_{0} an artin algebra, and let T be a Wakamatsu cotilting module over Λ_{0}.

Now we turn to the behavior of modules under the functor E.

Lemma 5.1. The graded projective cover of the E_{T} (Λ)-module E(M) for M in K_{T} (Λ) is isomorphic to E(M_{T}).

Proof. Let M be in K_{T} (Λ). Then E(M) is generated in degree 0, and we have an induced map α_{M}^{∗}: E(M_{T}) → E(M). Since the map α_{M}: M → M_{T} is a left add T -approximation, the induced map

(α_{M}^{∗})_{0} : Hom_{Λ}(M_{T} , T) → Hom_{Λ}(M, T)

is surjective. By assumption E(M) is generated in degree 0. Therefore the map α_{M}^{∗}: E(M_{T}) → E(M) is surjective. Since α_{M}: M → M_{T} is a left minimal morphism, (α_{M}^{∗})_{0} is a projective cover as an End_{Λ}(T)-module. This implies that α_{M}^{∗}: E(M_{T}) → E(M) is a projective cover.

The next result is the dual of the above result, showing that the projective cover of a module M in K_{T} (Λ) induces a minimal add T -approximation of E(M).

Lemma 5.2. If M is a module in K_{T} (Λ), then E(M) → E(P (M)) is the minimal add T -approximation of E(M), where P (M) → M is the projective cover of M.

Proof. It is clear that E(P (M)) is in add T. First we show that E(M) → E(P (M)) is a left add T -approximation. For this it is enough to show that Hom_{Γ}(E(P (M)), T) → Hom_{Γ}(E(M), T) is onto. We have the following commutative diagram

[figure: Hom_{Γ}(E(P (M)), T); Hom_{Γ}(E(M), T); Hom_{Γ_{0}}(E(P (M))_{0}, T); Hom_{Γ_{0}}(E(M)_{0}, T); Hom_{Γ_{0}}((P (M)_{0}, T), (Λ_{0}, T)); Hom_{Γ_{0}}((M_{0}, T), (Λ_{0}, T)); P (M)_{0}; M_{0}; 0]

Hence E(M) → E(P (M)) is a left add T -approximation of E(M).

The minimal left add T -approximation of E(M) is a direct summand of E(P (M)). Therefore there is a direct summand Q of P (M) such that E(M) → E(Q) is the minimal left add T -approximation. This follows from the fact that add Λ is dual to add T when these subcategories are viewed as subcategories of the category of graded modules with degree 0 morphisms. Using the same isomorphisms and the diagram above, it follows that Q_{0} → M_{0} is an epimorphism. This is a contradiction unless Q_{0} = P (M)_{0} and therefore Q = P (M). We conclude that E(M) → E(P (M)) is the minimal left add T -approximation.

To see what E of \overline{M}, rej_{T} (M) and corej_{T} (M) are mapped to via the functor E, we need the following lemma.

Lemma 5.3. For any T_{0} in add T there is a natural isomorphism

Γ ⊗_{Γ_{0}} Hom_{Λ}(T_{0}, T) → E(T_{0}).

Proof. We clearly have a natural isomorphism Γ⊗_{Γ_{0}} Hom_{Λ}(T, T) ≃ Γ = E(T), and also we have Γ ⊗_{Γ_{0}} Hom_{Λ}(T^{n}, T) ≃ Γ^{n} = E(T^{n}). The morphism is additive, hence

Γ ⊗_{Γ_{0}} Hom_{Λ}(T_{0}, T) ≃ E(T_{0})

for any T_{0} in add T.

Next we show that the syzygy of E(M) for M in K_{T} (Λ) is an extension of E(corej_{T} (M)) and E(rej_{T} (M))[1], in addition to showing that corej, \overline{( )} and rej are mapped to ⟨Ω_{Γ}( )_{0}⟩, Γ ⊗_{Γ_{0}} E( )_{0} and Ω_{Γ}( )/⟨Ω_{Γ}( )_{0}⟩, respectively.

Proposition 5.4. Let M be in K_{T} (Λ). Then the following assertions hold.

(a) We have the following exact commutative diagram

[figure: 0; 0; E(corej_{T} (M)); E(corej_{T} (M)); 0; Ω_{E_{T} (Λ)}(E(M)); E(M_{T}); E(M); 0; 0; E(rej_{T} (M))[1]; E(\overline{M}); E(M); 0; 0; 0]

(b) (i) ⟨Ω_{Γ}(E(M))_{0}⟩ ≃ E(corej_{T} (M)). (ii) Γ ⊗_{Γ_{0}} E(M)_{0} = Γ ⊗_{Γ_{0}} Hom_{Λ}(M, T) ≃ E(\overline{M}). (iii) Ω_{Γ}(E(M))/⟨Ω_{Γ}(E(M))_{0}⟩ ≃ E(rej_{T} (M))[1], moreover is generated in

degree one.

Proof. (a) By Lemma 1.6 the sequence 0 → E(rej_{T} (M))[1] → E(M_{0}) → E(M) → 0 is exact. The sequence 0 → M_{0} → M^{T} → corej_{T} (M) → 0 is F^{T} -exact, so that by Proposition 1.12 the sequence 0 → E(corej_{T} (M)) → E(M^{T}) → E(M_{0}) → 0 is exact. Using that E(M^{T}) ≃ Γ ⊗_{Γ_{0}} Hom_{Λ_{0}}(M^{T} , T) is the projective cover of E(M) by Lemma 5.1, the desired diagram is just putting these sequences together.

(b) It follows directly from (a) that ⟨Ω_{Γ}(E(M))_{0}⟩ ≃ E(corej_{T} (M)), since E(corej_{T} (M)) and E(rej_{T} (M)) are generated in degree 0 by assumption. It follows also that Ω_{Γ}(E(M))/⟨Ω_{Γ}(E(M))_{0}⟩ ≃ E(rej_{T} (M))[1], and for the same reason as above, is generated in degree 1.

The sequence

0 → (corej_{T} (M), T) ⟶^{t} (M_{T} , T) → (M, T) → 0

is exact. Hence we obtain the following exact sequence

Γ ⊗_{Γ_{0}} (corej_{T} (M), T) ⟶^{id⊗t} Γ ⊗_{Γ_{0}} (M_{T} , T) → Γ ⊗_{Γ_{0}} (M, T) → 0.

Since Γ ⊗_{Γ_{0}} (M_{T} , T) ≃ E(M_{T}) by Lemma 5.3 and the image of the map id ⊗ t is ⟨Ω_{Γ}(E(M))_{0}⟩, the assertion in (a) implies that E(\overline{M}) ≃ Γ ⊗_{Γ_{0}} Hom_{Λ}(M, T) = Γ ⊗_{Γ_{0}} E(M)_{0}.

Now we consider the dual situation; namely, we show that ⟨Ω_{Λ}( )_{0}⟩, Λ ⊗_{Λ_{0}} ( )_{0} and Ω_{Λ}( )/⟨Ω_{Λ}( )_{0}⟩ are mapped to corej, \overline{( )} and rej, respectively.

Proposition 5.5. Let M be in K_{T} (Λ). Then we have the following.

(a) The exact sequences

η: 0 → ⟨Ω_{Λ}(M)_{0}⟩ → P (M) → Λ ⊗_{Λ_{0}} M_{0} → 0

and

η′: 0 → ⟨Ω_{Λ}(M)_{0}⟩ → Ω_{Λ}(M) → Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩ → 0

are F^{T} -exact.

(b) (i) E(Λ ⊗_{Λ_{0}} M_{0}) ≃ E(M)_{0} = \overline{E(M)}. (ii) E(⟨Ω_{Λ}(M)_{0}⟩) ≃ corej_{T} E(M). (iii) E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩) ≃ rej_{T} E(M)[−1]. (iv)

E(Ω_{Λ}(M)) ≃ E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩) ⊕ E(⟨Ω_{Λ}(M)_{0}⟩)
≃ rej_{T} E(M) ⊕ corej_{T} E(M),

Proof. (a) Applying Hom_{Λ}( , T) to the sequence η induces the following exact sequence.

0 → Hom_{Λ}(Λ ⊗_{Λ_{0}} M_{0}, T) → Hom_{Λ}(P (M), T) → Hom_{Λ}(⟨Ω_{Λ}(M)_{0}⟩, T).

This sequence is isomorphic to the sequence

0 → Hom_{Λ_{0}}(M_{0}, T) → Hom_{Λ_{0}}(P (M)_{0}, T) → Hom_{Λ_{0}}(Ω_{Λ_{0}}(M_{0}), T).

The cokernel of the last morphism in the above sequence is Ext_{Λ_{0}}^{1}(M_{0}, T), which is zero since M_{0} is in X_{T}. Hence η is F^{T} -exact, and therefore η′ is also F^{T} -exact since it is a pullback of η.

(b) (i) The sequence η: 0 → ⟨Ω_{Λ}(M)_{0}⟩ → P (M) → Λ ⊗_{Λ_{0}} M_{0} → 0 is F^{T} - exact, so that by Proposition 1.12 in Chapter II the sequence 0 → E(Λ ⊗_{Λ_{0}} M_{0}) → E(P (M)) → E(⟨Ω_{Λ}(M)_{0}⟩) → 0 is exact. Since E(P (M)) = Hom_{Λ}(P (M), T), we infer that E(Λ ⊗_{Λ_{0}} M_{0}) = Hom_{Λ}(Λ ⊗_{Λ_{0}} M_{0}, T) = Hom_{Λ_{0}}(M_{0}, T) = E(M)_{0}.

(ii) By Lemma 5.2 we have that E(Λ ⊗_{Λ_{0}} M_{0}) = E(M)_{0} → E(P (M)) is the minimal left add T -approximation of E(M). Hence E(⟨Ω_{Λ}(M)_{0}⟩) = corej_{T} E(M).

(iii) We have the exact sequence 0 → Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩ → Λ ⊗_{Λ_{0}} M_{0} → M → 0. This sequence gives rise to the exact sequence

0 → E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩)[1] → E(M) → E(Λ ⊗_{Λ_{0}} M_{0}) = E(M)_{0} → 0

Hence rej_{T} E(M) = E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩)[1].

(iv) Using the above sequences we have the following commutative diagram.

[figure: 0; 0; E(M)_{0}; ≀; E(M)_{0}; 0; E(Λ ⊗_{Λ_{0}} M_{0}); 0; E(P (M)); E(⟨Ω_{Λ}(M)_{0}⟩); 0; 0; E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩); E(Ω_{Λ}(M)); E(⟨Ω_{Λ}(M)_{0}⟩); 0]

This shows that the lower row is split, hence the claim in (iv) follows.

We have the following basic property of the category K_{T} (Λ).

Proposition 5.6. The category K_{T} (Λ) is closed under extensions.

Proof. Let 0 → A → B → C → 0 be an exact sequence with A and C in K_{T} (Λ). Since this sequence is F^{T} -exact and G_{T} (Λ) is closed under F^{T} -extensions, B is also in G_{T} (Λ). Moreover all the sequences

0 → A_{0} → B_{0} → C_{0} → 0
0 → corej_{T} A → corej_{T} B ⊕ T′ → corej_{T} C → 0
0 → rej_{T} A → rej_{T} B → rej_{T} C → 0
0 → Λ ⊗_{Λ_{0}} A_{0} → Λ ⊗_{Λ_{0}} B_{0} → Λ ⊗_{Λ_{0}} C_{0} → 0
0 → ⟨Ω_{Λ}(A)_{0}⟩ → ⟨Ω_{Λ}(B)_{0}⟩ → ⟨Ω_{Λ}(C_{0})⟩ → 0
0 → A^{1} → B^{1} → C^{1} → 0

are F^{T} -exact with the end terms in K_{T} (Λ). Since G_{T} (Λ) is closed under F^{T} - extensions, it follows that all the middle terms are in G_{T} (Λ). From this we infer that B is in K_{T} (Λ).

It will be convenient to introduce the following definition, which turns out to generalize the corresponding one from Chapter I.

Definition 5.7. A graded Λ-module X is called fat if X is in gr_{0} Λ and E(X) = Hom_{Λ}(X, T), or equivalently X is in _{Λ}^{⊥}T.

An immediate consequence of the definitions involved is that any fat module is in G_{T} (Λ). If Λ is a Koszul algebra and T = Λ_{0}, then the fat modules are nothing else than the projective Λ-modules generated in degree zero. If Λ = Λ_{0} and T is a cotilting module, then the full subcategory of fat modules is all of K_{T} (Λ) = _{Λ}^{⊥}T.

Proposition 5.5 (b) shows that Λ ⊗_{Λ_{0}} M_{0} is a fat module for all M in K_{T} (Λ). The fat modules inside K_{T} (Λ) have the following properties.

Proposition 5.8. Suppose that Λ and T are as above.

(a) Let X be in K_{T} (Λ). Then X is fat if and only if X ≃ Λ ⊗_{Λ_{0}} X_{0}.

(b) If X is in K_{T} (Λ), then Λ ⊗_{Λ_{0}} X_{0} is a fat module, and all fat modules in K_{T} (Λ) are of this form.

(c) The full subcategory C of fat modules in K_{T} (Λ) is contravariantly finite in K_{T} (Λ).

Proof. (a) Assume that X in K_{T} (Λ) is fat. Consider the exact sequence 0 → Ω_{Λ}(X)/⟨Ω_{Λ}(X)_{0}⟩ → Λ ⊗_{Λ_{0}} X_{0} → X → 0. We saw in the proof of Proposition 5.5 (b) (iii) that it gives rise to the following exact sequence

0 → E(Ω_{Λ}(X)/⟨Ω_{Λ}(X)_{0}⟩)[1] → E(X) → E(X)_{0} → 0.

By assumption E(X) = E(X)_{0}, and hence E(Ω_{Λ}(X)/⟨Ω_{Λ}(X)_{0}⟩) = (0). Since Ω_{Λ}(X)/⟨Ω_{Λ}(X)_{0}⟩[−1] is in K_{T} (Λ), it follows that Ω_{Λ}(X)/⟨Ω_{Λ}(X)_{0}⟩ = (0) and therefore X ≃ Λ ⊗_{Λ_{0}} X_{0}.

Assume that X is in K_{T} (Λ) and that X ≃ Λ ⊗_{Λ_{0}} X_{0}. Since X is in K_{T} (Λ), Proposition 1.9 (c) implies that E(X) = E(Λ ⊗_{Λ_{0}} X_{0}) = E(X)_{0} = Hom_{Λ}(X, T). Hence X is a fat module.

(b) The statement in (b) follows directly from (a) and Proposition 1.9 (c).

(c) Let M be in K_{T} (Λ). We claim that Λ⊗_{Λ_{0}} M_{0} → M is a left C-approximation of M. To show this we first prove that Hom_{gr Λ}(Λ ⊗_{Λ_{0}} X_{0}, Y) ≃ Hom_{Λ_{0}}(X_{0}, Y_{0}) for X_{0} in mod Λ_{0} and Y in gr Λ generated in degree 0.

Define φ: Hom_{gr Λ}(Λ ⊗_{Λ_{0}} X_{0}, Y) → Hom_{Λ_{0}}(X_{0}, Y_{0}) and ψ: Hom_{Λ_{0}}(X_{0}, Y_{0}) → Hom_{gr Λ}(Λ ⊗_{Λ_{0}} X_{0}, Y) by φ(f) = f_{0} and ψ(g) is the composition of the maps

Λ ⊗_{Λ_{0}} X_{0} ⟶^{Λ⊗_{Λ_{0}} g} Λ ⊗_{Λ_{0}} Y_{0} ⟶^{mult} Y. Then it is easy to see that φψ = id and ψφ = id, and the above isomorphism is proved.

Let X = Λ ⊗_{Λ_{0}} X_{0} be a fat module. Then we have the following commutative diagram

[figure: Hom_{gr Λ}(Λ ⊗_{Λ_{0}} X_{0}, Λ ⊗_{Λ_{0}} M_{0}); ≀; Hom_{gr Λ}(Λ ⊗_{Λ_{0}} X_{0}, M); ≀; Hom_{Λ_{0}}(X_{0}, M_{0}); Hom_{Λ_{0}}(X_{0}, M_{0})]

and therefore the map Λ ⊗_{Λ_{0}} M_{0} → M is a left C-approximation of M.

It is clear that the fat modules introduced in Section 3 also are fat modules in the above sense. There we saw that the notion a linear fat resolution was important. In the setting of this section we define a linear fat resolution as follows.

Definition 5.9. A module M in gr Λ is said to have a linear fat resolution if there is an exact sequence in gr Λ of the form

⋯ → X^{2}[2] → X^{1}[1] → X^{0} → M → 0,

where X^{i} is a fat module for all i.

Since Λ ⊗_{Λ_{0}} M_{0} is fat for any M in K_{T} (Λ), it is an immediate consequence of the definition of K_{T} (Λ) that any M in K_{T} (Λ) has a linear fat resolution.

Next we observe that E(M_{0}) for M in K_{T} (Λ) is a fat module over Γ = E_{T} (Λ). Recall that E′ = ⊕_{i≥0} Ext_{E_{T} (Λ)}^{i}( , T): Gr E_{T} (Λ) → Gr Λ.

Lemma 5.10. If M is in K_{T} (Λ), then E(M_{0}) is in _{Γ}^{⊥}T. In particular, E′E(M_{0}) ≃ M_{0} as abelian groups.

Proof. Since M is in K_{T} (Λ), then M_{0} is in X_{T} and we have an exact sequence of minimal left add T -approximations

0 → M_{0} → T_{0} ⟶^{d_{1}} T_{1} ⟶^{d_{2}} T_{2} ⟶^{d_{3}} T_{3} → · · · .

Then Im d_{i} is in K_{T} (Λ) for all i ≥ 1.

By Proposition 1.12 the sequence ⋯ → E(T_{2}) → E(T_{1}) → E(T_{0}) → E(M_{0}) → 0 is exact and a projective resolution of E(M_{0}) over Γ. Applying Hom_{Γ}( , T) to this sequence we obtain the following exact diagram using that Hom_{Γ}(E(X), T) ≃ Hom_{Γ_{0}}(Hom_{Λ_{0}}(X, T), T) ≃ X_{0} for any module X in K_{T} (Λ).

\xymatrix{0 \ar[r] & (E(M_{0}), T) \ar[r] \ar[d]^{≀} & (E(T_{0}), T) \ar[r] \ar[d]^{≀} & (E(T_{0}), T) \ar[r] \ar[d]^{≀} & ⋯ \\ 0 \ar[r] & M_{0} \ar[r] & T_{0} \ar[r] & T_{1} \ar[r] & ⋯}

Since the lower row is exact, we conclude that E′E(M_{0}) ≃ M_{0} and that E(M_{0}) is in _{Γ}^{⊥}T.

We have seen that a module M in K_{T} (Λ) has a linear fat resolution. Next we see how a linear fat resolution of E(M) is given when M is in K_{T} (Λ).

Corollary 5.11. Let M be in K_{T} (Λ). Then

⋯ → E(M_{2})[2] → E(M_{1})[1] → E(M_{0}) → E(M) → 0

is a linear fat resolution of E(M).

Proof. The claim follows directly from Lemma 1.6 in Chapter II and Lemma 5.10.

6. Duality

In this section we generalize the central results on T -Koszul algebras to generalized T -Koszul algebras. In particular we show that the property of generalized T -Koszul is transferred to the Ext-algebra, and that E induces a duality between the Koszul modules for the original algebra and its Ext-algebra.

We start with a preliminary result for proving the duality.

Lemma 6.1. Assume that Λ is generalized T -Koszul. If M is in K_{T} (Λ), then (a) E′E(M)_{i} ≃ M_{i} as Λ_{0}-bimodules.

(b) The identification ϕ_{i}: M_{i} → E′E(M)_{i} is given in the following way. Let η_{i}(M): 0 → E(M_{≥i+1})[1] → E(M_{i}) → E(M_{≥i}) → 0. Then ϕ_{i}(m_{i}) = (E(a_{m_{i}})·η_{i}(M))·η_{i−1}(M) ⋯ η_{1}(M)·η_{0}(M), where m_{i} is in M_{i} and a_{m_{i}}: Λ → M_{≥i} is given by a_{m_{i}}(λ) = λm_{i}.

Proof. By Corollary 5.11 the module E(M) has the following linear fat resolution

⋯ → E(M_{2})[2] → E(M_{1})[1] → E(M_{0}) → E(M) → 0.

Now the same proof as for the corresponding result (Lemma 2.4) can be carried out, and this completes the proof.

The proof of the following result is the same as the one presented in Section 4.

Theorem 6.2. Assume that Λ is a generalized T -Koszul algebra.

(a) E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras.

(b) E′(E(M)) ≃ M as graded left Λ-modules for all M in K_{T} (Λ). In particular, E(K_{T} (Λ)) is contained in G_{T} (Γ).

We now prove that the image under E of a Koszul module is again a Koszul module.

Proposition 6.3. If M is in K_{T} (Λ), then E(M) is in K_{T} (Γ).

Proof. Let M be in K_{T} (Λ). First we show that E(M) is in fgsyz Γ. By Proposition 5.4 we know that Ω_{Γ}(E(M)) is an extension of E(corej_{T} (M)) and E(rej_{T} (M))[1]. Since M is in K_{T} (Λ), we have that E(corej_{T} (M)) and E(rej_{T} (M)) are generated in degree zero, hence Ω_{Γ}(E(M)) is generated in degree 0 and 1. By induction we get that Ω_{Γ}^{i}(E(M)) is generated in at most the degrees in the set {0, 1, …, i}. Consequently E(M) is in fgsyz Γ. Moreover, by Theorem 6.2 (b) the module E(M) is in G_{T} (Γ). Since M_{0} is in X_{_{Λ_{0}} T}, the module E(M)_{0} = Hom_{Λ_{0}}(M_{0}, T) is in X_{_{Γ_{0}} T}.

By Proposition 5.4 (b) we have that

⟨Ω_{Γ}(E(M))_{0}⟩ ≃ E(corej_{T} (M))
Ω_{Γ}(E(M))/⟨Ω_{Γ}(E(M))_{0}⟩[1] ≃ E(rej_{T} (M))
Γ ⊗_{Γ_{0}} E(M)_{0} ≃ E(\overline{M})

which all are in G_{T} (Γ).

By Proposition 5.5 (b) we have that

rej_{T} E(M) ≃ E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩)
\overline{E(M)} ≃ E(Λ ⊗_{Λ_{0}} M_{0})
corej_{T} E(M) ≃ E(⟨Ω_{Λ}(M)⟩)

which all are in G_{T} (Γ).

Since the six operations only produce modules of the form E(X) for some X in K_{T} (Λ) and E(X) is in G_{T} (Λ), we infer that E(M) is in K_{T} (Γ) for all M in K_{T} (Λ).

Now we can prove that the property of being generalized T -Koszul is preserved by the Ext-algebra.

Theorem 6.4. If Λ is generalized T -Koszul, then Γ = E_{T} (Λ) is generalized T -Koszul.

Proof. Assume that Λ is T -Koszul. Since Λ and T are in K_{T} (Λ), it follows immediately from Proposition 6.3 that Γ and T are in K_{T} (Γ).

Since Γ_{0} = Hom_{Λ}(T, T) = Hom_{Λ_{0}}(T, T), we have that Γ_{0} is an artin algebra. By assumption T is in K_{T} (Λ), so that Γ is generated in degree 0 and 1 as an algebra. Moreover, T is a Wakamatsu cotilting Γ_{0}-module.

Let η: ⋯ → Λ ⊗_{Λ_{0}} X_{2}^{2}[2] → Λ ⊗_{Λ_{0}} X_{1}^{1}[1] → Λ ⊗_{Λ_{0}} Λ_{0} → Λ_{0} → 0 be the linear fat resolution of Λ_{0}. Since T is in K_{T} (Λ), tensoring η with T gives rise to a linear fat resolution of T. Then we get that Γ_{i} = Ext_{Λ}^{i}(T, T) ≃ Hom_{Λ_{0}}(X_{i}^{i} ⊗_{Λ_{0}} T, T) as left Γ_{0}-modules. Since X_{i}^{i} are finitely generated Λ_{0}-modules due to the fact that Λ_{i} is a finitely generated Λ_{0}-module for all i ≥ 0; that is, X_{i}^{i} ⊗_{Λ_{0}} T is a finitely generated Λ-module, and therefore Γ_{i} is a finitely generated left Γ_{0}-module for all i ≥ 0. Hence Γ is a left finite graded ring. This shows that Γ is a T -Koszul algebra, and this completes the proof.

We are now in the position to give the main result on duality between Koszul modules.

Theorem 6.5. Assume that Λ is generalized T -Koszul and let Γ = E_{T} (Λ). Then we have the following assertions.

(a) Λ ≃ E_{T} (E_{T} (Λ)) as graded algebras.

(b) The functors

E = ⊕_{i≥0} Ext_{Λ}^{i}( , T): Gr Λ → Gr Γ

and

E′ = ⊕_{i≥0} Ext_{Γ}^{i}( , T): Gr Γ → Gr Λ

induce inverse dualities between K_{T} (Λ) and K_{T} (Γ).

(c) The subcategory X_{_{Λ_{0}} T} ∩ K_{T} (Λ) and the subcategory of fat modules in K_{T} (Γ) are dual subcategories via the functor E. The subcategory of fat modules in K_{T} (Λ) and the subcategory X_{_{Γ_{0}} T} ∩ K_{T} (Γ) are dual subcategories via the functor E.

Proof. (a) This is a restatement of Theorem 6.2 (a).

(b) By Theorem 6.2, Proposition 6.3 and (a) it follows that the compositions E′E and EE′ are isomorphic to the identity functor on objects. To prove that E and E′ are full and faithful the arguments used in Theorem 2.6 in Chapter II can be applied.

(c) The claims follow from Lemma 5.10, (a) and (b).

We saw above that the subcategory of fat modules in K_{T} (Λ) and X_{_{Γ_{0}} T} ∩ K_{T} (Γ) are dual subcategories via the functor E. Next we identify this subcategory with yet another subcategory and find further subcategories of K_{T} (Λ) and K_{T} (Γ) which correspond to each other under the duality.

Proposition 6.6. Let Λ be a generalized T -Koszul algebra.

(a) The full subcategories

X = {M ∈ K_{T} (Λ) | Ω_{Λ}^{i}(M) generated in degree 0 for all i ≥ 0}

of K_{T} (Λ) and X_{_{Γ_{0}} T} ∩K_{T} (Γ) of K_{T} (Γ) are dual subcategories via the functors E and E′. In particular, X and the subcategory of fat modules in K_{T} (Λ) are equal.

(b) The full subcategories

Y = {M ∈ K_{T} (Λ) | Ω_{Λ}^{i}(M) generated in degree i for all i ≥ 0}

of K_{T} (Λ) and Y′ = {Y ∈ K_{T} (Γ) | Y_{i} ∈ add T for all i} of K_{T} (Γ) are dual subcategories via the functors E and E′.

Proof. (a) Let M be in X. Then it follows immediately from the definition of K_{T} (Λ) that M ≃ Λ ⊗_{Λ_{0}} M_{0}. By Proposition 5.5 (b) (ii) we have that E(Λ ⊗_{Λ_{0}} M_{0}) = Hom_{Λ_{0}}(M_{0}, T). Moreover by Proposition 6.3 the module Hom_{Λ_{0}}(M_{0}, T) is in K_{T} (Γ), and consequently it is in X_{_{Γ_{0}} T} ∩ K_{T} (Γ).

Let X_{0} be in X_{_{Γ_{0}} T} ∩ K_{T} (Γ). Then we have an F^{T} -exact sequence 0 → X_{0} ⟶^{f_{0}}

T^{0} ⟶^{f_{1}} T^{1} ⟶^{f_{2}} T^{2} ⟶^{f_{3}} ⋯ with T^{i} in add_{Γ_{0}}T. Since X_{0} is in K_{T} (Γ), Proposition 1.12 gives that the sequence

⋯ → E′(T^{2}) → E′(T^{1}) → E′(T^{0}) → E′(X_{0}) → 0

is exact. It is clearly a projective resolution of E′(X_{0}) and Ω_{Λ}^{i}(E′(X_{0})) is generated in degree 0 for all i ≥ 0. It follows from Theorem 6.5 that full subcategories X and X_{_{Γ_{0}} T} ∩ K_{T} (Γ) are dual subcategories. Also we get that the subcategory of fat modules in K_{T} (Λ) is equal to X.

(b) Let Y be in Y. Then we have that Ω_{Λ}(Y)_{0} = (0), so that Λ ⊗_{Λ_{0}} P (Y)_{0} ≃ Λ ⊗_{Λ_{0}} Y_{0} and in particular that Y_{0} is in P(Λ_{0}). We also have that Ω_{Λ}(Y)[−1] is in Y, and therefore by induction we have that Ω_{Λ}^{n}(Y)_{n} is in P(Λ_{0}). Given that Y has a linear projective resolution,

Ext_{Λ}^{n}(Y, T) ≃ Hom_{Λ}(Ω_{Λ}^{n}(Y), T)
≃ Hom_{Λ_{0}}(Ω_{Λ}^{n}(Y)_{n}, T),

where Hom_{Λ_{0}}(Ω_{Λ}^{n}(Y), T) is in add_{Γ_{0}}T by the above for all n ≥ 0. This shows that E(Y) is in Y′ for all Y in Y.

Let Y′ be in Y′. Then corej_{T} (Y) = (0), and by Proposition 5.4 (b) we have that ⟨Ω_{Λ}(E′(Y′))_{0}⟩ ≃ E′(corej_{T} (Y)) = (0). Hence Ω_{Λ}(E′(Y)) is generated in degree 1. Furthermore, Ω_{Λ}(E′(Y)) ≃ E′(rej_{T} (Y))[1]. Since rej_{T}^{i} (Y)[−i] is in Y′ for all i ≥ 0, we obtain by induction that Ω_{Λ}^{n}(E′(Y)) ≃ E′(rej_{T}^{n} (Y))[n] for all n ≥ 0. This shows that E(Y′) is in Y for all Y′ in Y′. Therefore the full subcategories Y and Y′ are dual to each other.

The following result have the same proof as in the special case treated in Section 4 of Chapter II.

Theorem 6.7. Let Λ = ⊕_{i≥0}Λ_{i} be a generalized T -Koszul algebra.

(a) Λ ≃ T_{Λ_{0}}(Λ_{1})/I with I contained in T_{Λ_{0}}(Λ_{1})_{≥2}.

(b) I is a quadratic ideal in T_{Λ_{0}}(Λ_{1}). Let 0 → X^{2} → Λ⊗_{Λ_{0}} Λ_{1} → Λ⊗_{Λ_{0}} Λ_{0} → Λ_{0} → 0 be the start of a linear fat resolution of Λ_{0}.

(c) Λ ≃ T_{Λ_{0}}(Λ_{1})/⟨X_{2}^{2}⟩.

(d) Denote by Γ_{0} and Γ_{1} the endomorphism ring End_{Λ_{0}}(T) and Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T), respectively. Then

E_{T} (Λ) = T_{Γ_{0}}(Γ_{1})/⟨A_{2}⟩,

where A_{2} = {∑ β_{i} ⊗ α_{i} ∈ Γ_{1} ⊗_{Γ_{0}} Γ_{1} | ∑ β_{i}(id_{Λ_{1}} ⊗ α_{i})|_{X_{2}^{2}⊗_{Λ_{0}} T} = 0}.

CHAPTER IV Further results and questions

The category of Koszul modules for Λ Koszul or generalized Koszul with respect to a Wakamatsu cotilting module T is by definition a subcategory of w-G_{T} (Λ), where w-G_{T} (Λ) is the full subcategory of fgsyz Λ consisting of modules M such that Ext_{Λ}^{1}(T, T) Ext_{Λ}^{i}(M, T) = Ext_{Λ}^{i+1}(M, T) for all i ≥ 0. The latter category is also an interesting object of study, and section 1 is devoted to giving some basic properties. The description of the Koszul modules when Λ is Koszul with respect to T is much nicer when Λ is strongly T -Koszul. In the next two sections we give sufficient conditions for a T -Koszul algebra to be strongly T -Koszul. In Section 4 we show that we recover are the two classical cases of T -Koszul by requiring rej_{T} = 0 or corej_{T} = 0 on the Koszul modules. Finally we pose some questions in Section 5, along with some results to put the questions into the proper context.

1. Modules generated in 0

Throughout this section let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, where Λ_{0} is an artin algebra. We assume that T is a Wakamatsu cotilting Λ_{0}-module. Many results in this section remain true even without this assumption on T.

In this section we give elementary properties of the subcategories w-G_{T} (Λ) and G_{T} (Λ). We start with giving a useful description of the modules in w-G_{T} (Λ), and for this we need the following definition.

Definition 1.1. A graded projective resolution

⋯ → P_{n} → P_{n−1} → ⋯ → P_{0} → M → 0

for a module M in fgsyz Λ is said to be T -linear if

Ext_{Λ}^{1}((Ω_{Λ}^{i}(M))_{T} , T) → Ext_{Λ}^{1}(Ω_{Λ}^{i}(M), T)

is surjective for all i ≥ 0.

Proposition 1.2. The following are equivalent for a module M in fgsyz Λ.

(a) M is in w-G_{T} (Λ).

(b) M has a T -linear resolution

⋯ → P_{n} ⟶^{g_{n}} P_{n−1} ⟶^{g_{n−1}} ⋯ ⟶^{g_{1}} P_{0} ⟶^{g_{0}} M → 0

(c) The following properties are satisfied for all i ≥ 0.

(i) The sequence 0 → Ω_{Λ}^{i+1}(M) → g_{i−1}^{−1}(rej_{T} (Ω_{Λ}^{i}(M))) → rej_{T} (Ω_{Λ}^{i}(M)) → 0 is F^{T} -exact.

(ii) Ext_{Λ}^{1}((Ω_{Λ}^{i}(M))_{T} , T) → Ext_{Λ}^{1}(\overline{Ω_{Λ}^{i}(M)}, T)/ Hom_{Λ}(rej_{T} (Ω_{Λ}^{i}(M)), T) is surjective.

We show that the above proposition follows immediately from the following lemma.

Lemma 1.3. The following are equivalent for a Λ-module M in gr Λ.

(a) Ext_{Λ}^{1}(T, T) · Hom_{Λ}(M, T) = Ext_{Λ}^{1}(M, T).

(b) Ext_{Λ}^{1}(M_{T} , T) → Ext_{Λ}^{1}(M, T) is surjective, where the map is induced by the homomorphism α_{M}: M → M_{T}.

(c) (i) The sequence 0 → Ω_{Λ}(M) → g_{0}^{−1}(rej_{T} (M)) → rej_{T} (M) → 0 is F^{T} -

exact. (ii) Ext_{Λ}^{1}(M_{T} , T) → Ext_{Λ}^{1}(\overline{M}, T)/ Hom_{Λ}(rej_{T} (M), T) is surjective.

Proof. (a) implies (b). Assume that Ext_{Λ}^{1}(T, T)·Hom_{Λ}(M, T) = Ext_{Λ}^{1}(M, T). Let η: 0 → T → E → M → 0 be in Ext_{Λ}^{1}(M, T). Then there exists an extension η′: 0 → T → X → T^{n} → 0 in Ext_{Λ}^{1}(T^{n}, T) for some n and a homomorphism f: M → T^{n} such that we have the following commutative diagram

[figure: 0; T; E; M; f; 0; 0; T; X; T^{n}; 0]

Since α_{M}: M → M_{T} is a left add T -approximation, there exists a homomorphism g: M_{T} → T^{n}, such that f = g ∘ α_{M}. This gives rise to the commutative diagram

[figure: Ext_{Λ}^{1}(T^{n}, T); g^{∗}; Ext_{Λ}^{1}(M_{T} , T); α_{M}^{∗}; Ext_{Λ}^{1}(T^{n}, T); f^{∗}; Ext_{Λ}^{1}(M, T)]

Hence, f^{∗}(η′) = η = α_{M}^{∗} ∘ g^{∗}(η) and therefore the claim follows.

(b) implies (a). Assume that Ext_{Λ}^{1}(M_{T} , T) ⟶^{α_{M}^{∗}} Ext_{Λ}^{1}(M, T) is surjective. Since M_{T} is in add T, the claim follows immediately.

(b) implies (c). Assume that Ext_{Λ}^{1}(M_{T} , T) ⟶^{α_{M}^{∗}} Ext_{Λ}^{1}(M, T) is surjective. Since α_{M} factors through the natural projection π_{M}: M → \overline{M}, the morphism Ext_{Λ}^{1}(\overline{M}, T) → Ext_{Λ}^{1}(M, T) is surjective. Hence the morphism

Ext_{Λ}^{1}(rej_{T} (M), T) → Ext_{Λ}^{2}(\overline{M}, T) ≃ Ext_{Λ}^{1}(g_{0}^{−1}(rej_{T} (M)), T)

is a monomorphism. Using long exact sequences, the claim in (i) follows. Since we have the exact sequence

Hom_{Λ}(rej_{T} (M), T) → Ext_{Λ}^{1}(\overline{M}, T) ⟶^{π_{M}^{∗}} Ext_{Λ}^{1}(M, T) → 0

and α_{M}^{∗} is surjective, the claim in (ii) is a direct consequence of this.

(c) implies (b). Assume that the conditions in (i) and (ii) are satisfied. The statement in (i) means that the morphism Ext_{Λ}^{1}(rej_{T} (M), T) ⟶^{φ} Ext_{Λ}^{2}(\overline{M}, T) ≃ Ext_{Λ}^{1}(g_{0}^{−1}(rej_{T} (M)), T) is a monomorphism. Using the long exact sequence induced from the exact sequence 0 → rej_{T} (M) → M → \overline{M} → 0 it follows that the map π_{M}^{∗}: Ext_{Λ}^{1}(\overline{M}, T) → Ext_{Λ}^{1}(M, T) is an epimorphism. By the condition (ii) the claim in (b) is immediate.

We now explain how Proposition 1.2 follows from the above lemma.

Proof of Proposition 1.2. The equivalence of (b) and (c) follows immediately from Lemma 1.3.

We have the following commutative diagram, where all the maps are the natural maps.

[figure: Ext_{Λ}^{1}(T, T) ⊗_{ℤ} Hom_{Λ}(Ω_{Λ}^{i}(M), T); Ext_{Λ}^{1}(Ω_{Λ}^{i}(M), T); ≀; Ext_{Λ}^{1}(T, T) ⊗_{ℤ} Ext_{Λ}^{i}(M, T); Ext_{Λ}^{i+1}(M, T)]

The equivalence of (a) and (b) follows directly from this commutative diagram and Lemma 1.3.

We have the following immediate consequences.

Proposition 1.4. (a) P(Λ) is contained in w-G_{T} (Λ).

(b) The module T is in w-G_{T} (Λ) if and only if E_{T} (Λ) = ⊕_{i≥0} Ext_{Λ}^{i}(T, T) as a graded algebra is generated in degrees 0 and 1 and T is in fgsyz Λ.

(c) If a module M is in w-G_{T} (Λ), then Ω_{Λ}(M) is in w-G_{T} (Λ).

(d) A pair of modules A and B are in w-G_{T} (Λ) if and only if A ⊕ B is in w-G_{T} (Λ).

Note that for a module M in fgsyz Λ it is not the case that M is in w-G_{T} (Λ) if and only if E(M) as an E_{T} (Λ)-module is generated in degree 0. This is only true if E_{T} (Λ) is generated in degrees 0 and 1 as a graded algebra.

The category w-G_{T} (Λ) is not necessarily closed under all extensions, but we see next that it is closed under all F^{T} -extensions.

Proposition 1.5. Let η: 0 → A → B → C → 0 be an F^{T} -exact sequence in gr Λ with A and C in w-G_{T} (Λ). Then B is in w-G_{T} (Λ). In particular, w-G_{T} (Λ) is closed under F^{T} -extensions.

Proof. It is clear that B is in fgsyz Λ. Since η is F^{T} -exact, we have the following exact commutative diagram

\xymatrix{0 \ar[r] & A \ar[r] \ar[d] & B \ar[r] \ar[d]^{g} & C \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A_{T} \ar[r]^{δ} & A_{T} ⊕ C_{T} \ar[r] & C_{T} \ar[r] & 0}

where g: B → A_{T} ⊕ C_{T} is an add T -approximation. Therefore there exists a homomorphism h: B_{T} → A_{T} ⊕ C_{T}, such that h ∘ α_{B} = g. Hence to show that α_{B}^{∗}: Ext_{Λ}^{1}(B_{T} , T) → Ext_{Λ}^{1}(B, T) is an epimorphism, it is enough to show that g^{∗}: Ext_{Λ}^{1}(A_{T} ⊕ C_{T} , T) → Ext_{Λ}^{1}(B, T) is an epimorphism. By an extended version of Proposition 1.12 in Chapter II and the above diagram we obtain the following exact commutative diagram.

\xymatrix{0 \ar[r] & Ext_{Λ}^{1}(C_{T} , T) \ar[r] \ar[d] & Ext_{Λ}^{1}(A_{T} ⊕ C_{T} , T) \ar[r] \ar[d]^{g^{∗}} & Ext_{Λ}^{1}(A_{T} , T) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ext_{Λ}^{1}(C, T) \ar[r] \ar[d] & Ext_{Λ}^{1}(B, T) \ar[r] & Ext_{Λ}^{1}(A, T) \ar[r] \ar[d] & 0 \\  & 0 &  & 0}

2. T -KOSZUL ALGEBRAS IN DEGREES 0 AND 1, REVISITED 51

By the Five Lemma the morphism g^{∗} is an epimorphism. Since the syzygy sequences induced by η have the same properties as η, the claim follows.

In Chapter II we introduced some fundamental sequences involving the projective cover and the minimal left add T -approximation for a module M in gr_{0} Λ. The correspondence between the projective cover and the minimal left add T - approximation we found in Lemma 5.1 and Lemma 5.2 in Chapter III for modules in K_{T} (Λ) is still true for modules in G_{T} (Λ). We leave it to the reader to state and prove the analogous results for G_{T} (Λ).

We give some additional properties of modules in w-G_{T} (Λ). First we show that for a module M in w-G_{T} (Λ) the degree zero part M_{0} is in ^{⊥}T.

Proposition 1.6. Let M be in w-G_{T} (Λ). Then M_{0} is in ^{⊥}T.

Proof. Let M be in w-G_{T} (Λ). Then we have that

Ext_{Λ}^{i}(M, T) = ⊕_{j∈ℤ} Ext_{gr(Λ)}^{i}(M, T [j]) = ⊕_{j≥0} Ext_{gr(Λ)}^{i}(M, T [j])

Since

(⊕_{j≥0} Ext_{gr Λ}^{1}(T, T [j])) · (⊕_{j≥0} Ext_{gr Λ}^{i}(M, T [j])) = ⊕_{j≥0} Ext_{gr Λ}^{i+1}(M, T [j])

and the product of two elements in degree j_{1} and j_{2} is in degree j_{1} + j_{2}, we have that

Ext_{gr Λ}^{1}(T, T) · Ext_{gr Λ}^{i}(M, T) = Ext_{gr Λ}^{i+1}(M, T).

Furthermore, since Ext_{gr Λ}^{1}(T, T) = Ext_{Λ_{0}}^{1}(T, T), Ext_{gr Λ}^{i}(M, T) = Ext_{Λ_{0}}^{i}(M, T) and Ext_{Λ_{0}}^{1}(T, T) = (0), it follows that Ext_{Λ_{0}}^{i}(M, T) = (0) for all i > 0. Hence M_{0} is in ⊥T.

The next result shows that the operations defining the subcategory K_{T} (Λ) are not independent.

Lemma 1.7. Assume that M is in G_{T} (Λ). Then Λ ⊗_{Λ_{0}} M_{0} is in G_{T} (Λ) if and only if ⟨Ω_{Λ}(M)_{0}⟩ is in G_{T} (Λ).

Proof. Assume that Λ⊗_{Λ_{0}} M_{0} is in G_{T} (Λ). The module ⟨Ω_{Λ}(M)_{0}⟩ is a syzygy of Λ ⊗_{Λ_{0}} M_{0}. Since ⟨Ω_{Λ}(M)_{0}⟩ is generated in degree 0, it is in G_{T} (Λ).

Conversely, assume that ⟨Ω_{Λ}(M)_{0}⟩ is in G_{T} (Λ). The sequence

0 → ⟨Ω_{Λ}(M)_{0}⟩ → P (M) → Λ ⊗_{Λ_{0}} M_{0} → 0

is F^{T} -exact by Proposition 1.12 in Chapter II. If ⟨Ω_{Λ}(M)_{0}⟩ is in G_{T} (Λ), then by Proposition 1.12 in Chapter II the sequence

0 → E(Λ ⊗_{Λ_{0}} M_{0}) → E(P (M)) → E(⟨Ω_{Λ}(M)_{0}⟩) → 0

is exact. Therefore E(Λ ⊗_{Λ_{0}} M_{0}) = Hom_{Λ}(Λ ⊗_{Λ_{0}} M_{0}, T), and in particular E(Λ ⊗_{Λ_{0}} M_{0}) is generated in degree 0. This completes the proof.

2. T -Koszul algebras in degrees 0 and 1, revisited

In this section we show that a T -Koszul algebra concentrated in degrees 0 and 1, with T being a cotilting Λ_{0}-module must be strongly T -Koszul.

For M = M_{0} + M_{1} generated in degree 0, the essential point will be to show that M_{1} is in _{Λ_{0}}^{⊥}T when E(M) is generated in degree 0. For this we start with the following.

Lemma 2.1. Suppose that Λ is a left finite graded algebra generated in degrees 0 and 1 with Λ_{0} an artin algebra, and let T be a cotilting Λ_{0}-module. Suppose that Λ satisfies the ⊗-Tor-conditions. If M is in G_{T} (Λ), then Ext_{Λ_{0}}^{1}(M_{1}, T) = (0).

Proof. Let M → M_{T} be the minimal add T -approximation of M and let P → M and Q → M_{T} be Λ-projective covers.

Then, after applying the functor Hom_{Λ}( , T) to the obvious diagram, we obtain the following exact commutative diagram.

\xymatrix{0 \ar[r] & _{Λ}(M_{T} , T) \ar[r] \ar[d] & _{Λ}(Λ ⊗ Q, T) \ar[r] \ar[d] & _{Λ}(Ω_{Λ}(M_{T}), T) \ar[r] \ar[d] & Ext_{Λ}^{1}(M_{T} , T) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & _{Λ}(M, T) \ar[r] & _{Λ}(Λ ⊗ P, T) \ar[r] & _{Λ}(Ω_{Λ}(M), T) \ar[r] & Ext_{Λ}^{1}(M, T) \ar[r] & 0}

Next we note that _{Λ}(Ω_{Λ}(M_{T} , T)) = _{Λ_{0}}(Ω_{Λ_{0}}(M_{T} , T)) ⊕ _{Λ_{0}}(M_{T}^{1}, T) and Λ(Ω_{Λ}(M), T) = _{Λ_{0}}(Ω_{Λ_{0}}(M_{0}), T) ⊕_{Λ_{0}}(M^{1}, T) using the ⊗-Tor-conditions. Furthermore we get short exact sequences

0 → _{Λ}(M_{T} , T) → _{Λ}(Λ ⊗ Q, T) → _{Λ_{0}}(Ω_{Λ_{0}}(M_{T}), T) → 0

0 → _{Λ}(M, T) → _{Λ}(Λ ⊗ P, T) → _{Λ_{0}}(Ω_{Λ_{0}}(M), T) → 0

using that if X is generated in degree 0, then Hom_{Λ}(X, T) = Hom_{Λ_{0}}(X_{0}, T). Hence, we conclude that _{Λ}(M_{T}^{1}, T) ≃ Ext_{Λ}^{1}(M_{T} , T) and _{Λ}(M^{1}, T) ≃ Ext_{Λ}^{1}(M, T).

Next we consider the commutative diagram

\xymatrix{_{Λ}(M_{T}^{1}, T) \ar[r]^{∼} \ar[d]^{b} & Ext_{Λ}^{1}(M_{T} , T) \ar[d]^{a} \\ _{Λ}(M^{1}, T) \ar[r]^{∼} & Ext_{Λ}^{1}(M, T)}

Since we are assuming that E(M) is generated in degree 0, by Lemma 1.3, the map a is an epimorphism. Since M_{T} is concentrated in degree 0, we have that M_{T}^{1} = Λ_{≥1} ⊗ M_{T}. Hence _{Λ}(M_{T}^{1}, T) ≃_{Λ_{0}} (Λ_{1} ⊗ M_{T} , T). We note that _{Λ}(M^{1}, T) = Λ_{0}(M_{1}^{1}, T) ⊕_{Λ}(M^{1}/⟨M_{1}^{1}⟩, T) by Lemma 0.4. Finally, we note that the image of b is in _{Λ_{0}}(M_{1}^{1}, T) and that a being an epimorphism implies that _{Λ}(M^{1}/⟨M^{1}⟩, T) = 0. We have proved that _{Λ_{0}}(Λ_{1} ⊗ M_{T} , T) ⟶^{b} _{Λ_{0}}(M_{1}^{1}, T) is an epimorphism.

Let L → M_{1} be a Λ_{0}-projective cover. Next consider the following commutative diagram.

[figure: 0; Ω_{Λ_{0}}(M_{1}); L; M_{1}; 0; 0; M_{1}^{1}; α; Λ_{1} ⊗ M_{0}; β; M_{1}; 0; Λ_{1} ⊗ M_{T}]

Here, of course, β is obtained by tensoring the add(T)-approximation M_{0} → M_{T}. Apply the functor Hom_{Λ}( , T) to get

[figure: (Λ_{1} ⊗ M_{T} , T); β^{∗}; (Λ_{1} ⊗ M_{T} , T); α^{∗}β^{∗}; 0; (M_{1}, T); (Λ_{1} ⊗ M_{0}, T); α^{∗}; (M_{1}^{1}, T); Ext_{Λ_{0}}^{1}(M_{1}, T); 0; 0; (M_{1}, T); (L, T); (Ω_{Λ_{0}}(M_{1}), T); Ext_{Λ_{0}}^{1}(M_{1}, T); 0]

The 0 at the right side of the middle row comes from the fact that Ext_{Λ_{0}}^{1}(Λ_{1} ⊗ M_{0}, T) = 0 since we are assuming that T is a cotilting module and that Λ_{1} satisfies the ⊗-condition. Also, α^{∗} denotes (α, T) and β^{∗} denotes (β, T).

We have shown that α^{∗}β^{∗}: _{Λ_{0}}(Λ_{1} ⊗ M_{T} , T) → _{Λ_{0}}(M_{1}^{1}, T) is an epimorphism. A diagram chase shows that Ext_{Λ_{0}}^{1}(M_{1}, T) = (0).

As an immediate consequence, we have the following result.

Proposition 2.2. Let Λ be a left finite graded algebra generated in degrees 0 and 1, with Λ_{0} an artin algebra, and let T be a cotilting Λ_{0}-module. Assume that Λ satisfies the ⊗-Tor-conditions. If M is in G_{T} (Λ), then M_{0} and M_{1} are in ^{⊥}T.

Proof. We have shown that M_{0} is in ^{⊥}T and that Ext_{Λ_{0}}^{1}(M_{1}, T) = (0). We have a short exact sequence 0 → M_{1}^{1} → Λ_{1} ⊗ M_{0} → M_{1} → 0. By an extension of Proposition 5.5 (b) (iii) in Chapter III the module M^{1} has the property that E(M^{1}) is generated in degree 0. By Proposition 1.6 the module M_{1}^{1} is in ^{⊥}T. Applying the long exact sequence obtained by applying the functor Hom_{Λ}( , T) yields the result.

Our main result in this section completes the proof of Theorem 3.1 in Chapter I.

Theorem 2.3. Suppose that Λ = Λ_{0} + Λ_{1} + 0 + 0 + ⋯ is a T -Koszul algebra with respect to a cotilting Λ_{0}-module T. Then Λ is strongly T -Koszul.

Proof. We show that the category G_{T} (Λ) contained in K_{T} (Λ). Let M be in G_{T} (Λ). By Proposition 2.2 the modules M_{0} and M_{1} are in ^{⊥}T. It now follows that \overline{M} = M_{0}, rej_{T} (M) = M_{1}, corej_{T} (M) and M^{1} = ker(Λ_{1} ⊗ M_{0} → M_{1}) are all in ^{⊥}T. But ^{⊥}T is a subcategory of G_{T} (Λ). Finally, using the ⊗-Tor-conditions, we see that Λ⊗M_{0} and Λ⊗Ω_{Λ_{0}}(M_{0}) are also in ^{⊥}T. Thus, G_{T} (Λ) is closed under the necessary operations and we have our result.

3. Ext-algebras for injective cotilting modules

In this section we give other sufficient conditions for a T -Koszul algebra to be strongly T -Koszul when T is an injective Λ_{0}-module. We also give sufficient conditions for Λ T -Koszul to be equivalent to E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras, when Λ_{0} is selfinjective.

The following preliminary result is useful.

Lemma 3.1. Let Λ = ⊕_{i≥0}Λ_{i} where Λ is generated in degrees 0 and 1, and let T = D(Λ_{0}). Let M be in fgsyz(Λ), and assume that the module M^{1} in the exact sequence

η: 0 → M^{1} → Λ ⊗_{Λ_{0}} M_{0} → M → 0

is generated in at least two degrees, 1 and r, where r is chosen smallest possible with r > 1. Then

Ext_{Λ}^{1}(T, T) Hom_{Λ}(M, T) ≠ Ext_{Λ}^{1}(M, T)

Proof. We first produce a nonsplit extension in Ext_{Λ}^{1}(M, T). We have that

Hom_{Λ}(M^{1}, T) ≃ Hom_{Λ_{0}}(M_{1}^{1}, T) ⊕ Hom_{Λ_{0}}((M^{1}/⟨M_{1}^{1}⟩)_{r}, T) ⊕ X

for some X. Let i: (M^{1}/⟨M_{1}^{1}⟩)_{r} → T_{0} be the injective envelope, and let f: M^{1} → M^{1}/⟨M_{1}^{1}⟩ → (M^{1}/⟨M_{1}^{1}⟩)_{r} → T_{0} be the composition of the natural projections and i. Then we have the following pushout diagram.

[figure: η: 0; M^{1}; f; Λ ⊗_{Λ_{0}} M_{0}; M; 0; η′: 0; T_{0}; E; M; 0]

The homomorphism f is a degree zero morphism when we view T_{0} as a graded module concentrated in degree r. Since η is an exact sequence in gr Λ, the pushout η′ is also an exact sequence in gr Λ. Assume that η′ is split exact. Then Hom_{gr Λ}(Λ⊗_{Λ_{0}} M_{0}, T_{0}) ≠ (0). But this is a contradiction, since T_{0} is concentrated in degree r > 1. Hence η′ is not split.

Next we show that η′ is not in Ext_{Λ}^{1}(T, T) Hom_{Λ}(M, T).

We have that Ext_{Λ}^{1}(T, T) = ⊕_{j≥0} Ext_{gr Λ}^{1}(T, T [j]), and that Ext_{gr Λ}^{1}(T, T [j]) = (0) for j ≠ 1, since T = D(Λ_{0}) for j = 0 and Λ is generated in degree 0 and 1 for j > 1. Hence Ext_{Λ}^{1}(T, T) = Ext_{gr Λ}^{1}(T, T [1]).

Let θ: 0 → T → E′ → T → 0 be any extension in Ext_{Λ}^{1}(T, T). Then by the above we can view θ as an exact sequence in gr Λ where the right end term T is viewed as a graded module concentrated in degree 0 and the other end term as a graded module concentrated in degree 1.

Since M is generated in degree 0, all Λ-homomorphisms g: M → T can be viewed as a homomorphism of degree 0, when we consider T as a graded Λ-module concentrated in degree 0. Hence the pullback of θ by g, 0 → T → E′ → M → 0, is a graded extension with M generated in degree 0 and T concentrated in degree 1. It follows from this that our original nonsplit sequence η can not be in Ext_{Λ}^{1}(T, T) Hom_{Λ}(M, T). Therefore

Ext_{Λ}^{1}(T, T) Hom_{Λ}(M, T) ≠ Ext_{Λ}^{1}(M, T).

The next result gives a sufficient condition for the Koszul modules to be described as the modules having fat resolutions.

Proposition 3.2. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded algebra generated in degrees 0 and 1, with Λ_{0} an artin algebra. Let T = D(Λ_{0}), and assume that (Λ_{i})_{Λ_{0}} is projective for all i ≥ 0. Let M be in fgsyz(Λ). Then M is in G_{T} (Λ) if and only if the complex

⋯ → Λ ⊗_{Λ_{0}} X_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗_{Λ_{0}} X_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗_{Λ_{0}} M_{0} ⟶^{d_{0}} M → 0

is exact, where X^{i} = Ker d_{i−1}.

Proof. The claim follows from the previous result and Lemma 1.10 in Chapter II.

We now get our main result on the relationship between T -Koszul and strongly T -Koszul algebras.

Theorem 3.3. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded algebra generated in degrees 0 and 1 with Λ_{0} an artin algebra. Let T = D(Λ_{0}), and assume that (Λ_{i})_{Λ_{0}} is projective for all i ≥ 0. Furthermore assume that Λ_{0} is in fgsyz(Λ). Then Λ is T -Koszul if and only if Λ_{0} is in G_{T} (Λ).

Moreover, if Λ is T -Koszul, then Λ is strongly T -Koszul.

Proof. If Λ is T -Koszul, then Λ_{0} must be in K_{T} (Λ) and therefore certainly in G_{T} (Λ).

Conversely, assume that Λ_{0} is in G_{T} (Λ). By Proposition 3.2 there exists an exact sequence

⋯ → Λ ⊗_{Λ_{0}} X_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗_{Λ_{0}} X_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗_{Λ_{0}} Λ_{0} ⟶^{d_{0}} Λ_{0} → 0.

As we observed in the proof of Proposition 4.3 in Chapter I the algebra Λ satisfies the ⊗-Tor-conditions. Since T = D(Λ_{0}), we have that X_{T} = mod Λ_{0} and therefore the above exact sequence is a linear fat resolution of Λ_{0}. This shows that Λ is a T -Koszul algebra by definition.

Assume now that Λ is T -Koszul. We want to prove that Λ is strongly T - Koszul. Since Λ satisfies the ⊗-Tor-conditions, tensoring the above exact sequence with any module M_{0} in X_{T} induces a linear fat resolution of M_{0}. By Proposition 3.2 the subcategory X_{T} is in G_{T} (Λ). Moreover it is also immediate that Λ ⊗_{Λ_{0}} X_{T} is contained in G_{T} (Λ).

Given M in G_{T} (Λ) we have that

corej_{T} (M) = Ω_{Λ_{0}}^{−1}(M_{0})
\overline{M} = M_{0}
rej_{T} (M) = M_{≥1}
⟨Ω_{Λ}(M)_{0}⟩ = Λ ⊗_{Λ_{0}} Ω_{Λ_{0}}(M_{0})
Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩[1] = Ker(Λ ⊗_{Λ_{0}} M_{0} → M)[1]

By the above remarks we have that corej_{T} (M), \overline{M}, Λ ⊗_{Λ_{0}} M_{0} and ⟨Ω_{Λ}(M)_{0}⟩ are in G_{T} (Λ) again. The module Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩[1] is in G_{T} (Λ) by the previous result. We want to show that G_{T} (Λ) is closed under all these operations, and therefore we would have K_{T} (Λ) = G_{T} (Λ). So it remains to prove that M_{≥1} is in G_{T} (Λ).

We have the following commutative exact diagram

[figure: 0; 0; M^{1}; M^{1}; 0; Λ_{≥1} ⊗_{Λ_{0}} M_{0}; Λ ⊗_{Λ_{0}} M_{0}; M_{0}; 0; 0; M_{≥1}; M; M_{0}; 0; 0; 0]

Hence 0 → M^{1} → Λ_{≥1} ⊗_{Λ_{0}} M_{0} → M_{≥1} → 0 is exact, where M^{1} and Λ_{≥1} ⊗_{Λ_{0}} M_{0} are in G_{T} (Λ). This gives rise to the following commutative exact diagram.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & M^{2} \ar[r] \ar[d] & X^{1} \ar[d] \ar[r] & Y^{1} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & Λ_{≥1} ⊗_{Λ_{0}} M_{1}^{1} \ar[d] \ar[r] & Λ ⊗_{Λ_{0}} Λ_{1} ⊗_{Λ_{0}} M_{0} \ar[d] \ar[r] & Λ ⊗_{Λ_{0}} M_{1} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & M^{1} \ar[d] \ar[r] & Λ_{≥1} ⊗_{Λ_{0}} M_{0} \ar[d] \ar[r] & M_{≥1} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

where M^{2} and X^{1} are in G_{T} (Λ), the middle row is exact since (Λ_{j})_{Λ_{0}} is a projective Λ_{0}^{op}-module and the upper row is the exact sequence of the kernels. It now follows by induction that M_{≥1} has a fat resolution and therefore it is in G_{T} (Λ) by Proposition 3.2.

This shows that G_{T} (Λ) is closed under all the operations K_{T} (Λ) should be closed under. Moreover Λ and T are in G_{T} (Λ), hence K_{T} (Λ) = G_{T} (Λ) and Λ is strongly T -Koszul.

For classical Koszul algebras a characterization of a Koszul algebra Λ is that Λ and the Ext-algebra of the Ext-algebra are isomorphic as graded algebras. As a consequence of the next result, we obtain a generalization of this.

Proposition 3.4. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring with Λ_{0} selfinjective, where Λ is generated in degrees 0 and 1. Let T = Λ_{0}, and let Γ = E_{T} (Λ) = ⊕_{i≥0} Ext_{Λ}^{i}(T, T). Assume that E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras. Then T is in G_{T} (Λ).

Proof. Since Λ_{≥1} is generated in degree 1, it follows that Λ_{≥1} ⊗_{Λ_{0}} T is generated in degree 1. By Lemma 1.5 in Chapter II we get that Γ_{1} = Ext_{Λ}^{1}(T, T) = Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) and that E_{T} (Γ)_{1} = Ext_{Γ}^{1}(T, T) = Hom_{Γ}(Γ_{≥1} ⊗_{Γ_{0}} T, T).

We claim that Γ_{≥1} ⊗_{Γ_{0}} T/⟨Γ_{1} ⊗_{Γ_{0}} T ⟩ = (0). To this end consider the exact sequence

0 → ⟨Γ_{1} ⊗_{Γ_{0}} T ⟩ → Γ_{≥1} ⊗_{Γ_{0}} T → Γ_{≥1} ⊗_{Γ_{0}} T/⟨Γ_{1} ⊗_{Γ_{0}} T ⟩ → 0.

Applying Hom_{Γ}( , T) to this sequence and observing that the middle term by assumption is isomorphic to Λ_{1}, we get the following exact sequence

η: 0 → Hom_{Γ}(Γ_{i≥1} ⊗_{Γ_{0}} T/⟨Γ_{1} ⊗_{Γ_{0}} T ⟩, T) → Λ_{1} → Hom_{Γ_{0}}(Γ_{1} ⊗_{Γ_{0}} T, T) → 0,

which is a split exact sequence by Lemma 0.4. We have that

Hom_{Γ_{0}}(Γ_{1} ⊗_{Γ_{0}} T, T) ≃ Hom_{Γ_{0}}(Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) ⊗_{Γ_{0}} T, T)
≃ Hom_{Γ_{0}}(T, Hom_{Γ_{0}}(Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T), T))
≃ Hom_{Γ_{0}}(T, Λ_{1} ⊗_{Λ_{0}} T)

since Hom_{Γ_{0}}(Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T), T) ≃ Λ_{1} ⊗_{Λ_{0}} T.

Let P_{1}^{∗} ⟶^{f^{∗}} P_{0}^{∗} → Λ_{1} → 0 be a projective presentation of Λ_{1} as a right Λ_{0}- module, where f: P_{0} → P_{1} is in P(Λ_{0}). Then we have the following commutative diagram.

[figure: P_{1}^{∗} ⊗_{Λ_{0}} T; f^{∗}⊗id; P_{0}^{∗} ⊗_{Λ_{0}} T; Λ_{1} ⊗_{Λ_{0}} T; 0; Hom_{Λ_{0}}(P_{1}, T); Hom_{Λ_{0}} (f,T); Hom_{Λ_{0}}(P_{0}, T); Λ_{1} ⊗_{Λ_{0}} T; 0]

Since _{Γ_{0}}T = Hom_{Λ_{0}}(Λ_{0}, T) is projective, the above commutative diagram gives rise to the following commutative exact diagram.

\xymatrix{_{Γ_{0}}((Λ_{0}, T), (P_{1}, T)) \ar[r]^{((Λ_{0},T),(f,T))} \ar[d] & _{Γ_{0}}((Λ_{0}, T), (P_{0}, T)) \ar[r] \ar[d] & _{Γ_{0}}((Λ_{0}, T), Λ_{1} ⊗_{Λ_{0}} T)) \ar[r] \ar[d] & 0 \\ P_{1}^{∗} \ar[r]^{f^{∗}} & P_{0}^{∗} \ar[r] & Λ_{1} \ar[r] & 0}

where the two first vertical maps are isomorphisms. Hence it follows that Hom_{Γ_{0}}(T, Λ_{1}⊗_{Λ_{0}} T) ≃ Λ_{1}. Using the sequence η above we infer that Hom_{Γ}(Γ_{≥1}⊗_{Γ_{0}} T/⟨Γ_{1} ⊗_{Γ_{0}} T ⟩, T) = (0). Since T contains a copy of every indecomposable injective Λ_{0}-module, we have that Γ_{≥1} ⊗_{Γ_{0}} T/⟨Γ_{1} ⊗_{Γ_{0}} T ⟩ = (0).

Consider the sequence

0 → ⟨Γ_{1}⟩ → Γ_{≥1} → Γ_{≥1}/⟨Γ_{1}⟩ → 0,

which gives rise to the exact sequence

⟨Γ_{1}⟩ ⊗_{Γ_{0}} T → Γ_{≥1} ⊗_{Γ_{0}} T → (Γ_{≥1}/⟨Γ_{1}⟩) ⊗_{Γ_{0}} T → 0.

We have that ⟨Γ_{1}⟩ ⊗_{Γ_{0}} T = ⟨Γ_{1} ⊗_{Γ_{0}} T ⟩, so that (Γ_{≥1}/⟨Γ_{1}⟩) ⊗_{Γ_{0}} T = (0). Since Γ_{0}T = Γ_{0}, it follows that Γ_{≥1}/⟨Γ_{1}⟩ = (0). Then we have that Γ_{1}Γ_{1} = Γ_{2} and inductively Γ_{1}Γ_{i} = Γ_{i+1} for all i ≥ 1. Hence Γ is generated in degrees 0 and 1, and T is in G_{T} (Λ).

As a consequence we get sufficient conditions for E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras to imply Λ being T -Koszul.

Corollary 3.5. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring with Λ_{0} a selfinjective artin algebra, T = Λ_{0} and where Λ_{i} is projective as a right Λ_{0}-module for all i. Then Λ is T -Koszul if and only if E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras.

Proof. If Λ is T -Koszul, then by Theorem 2.6 (a) in Chapter II the algebras Λ and E_{T} (E_{T} (Λ)) are isomorphic as graded algebras.

Conversely, assume that E_{T} (E_{T} (Λ)) ≃ Λ as graded algebras. By Proposition 3.4 the module T = Λ_{0} is in G_{T} (Λ), which in turn by Theorem 3.3 implies that Λ is T -Koszul.

4. rej_{T} = 0 and corej_{T} = 0

In this section we show for a T -Koszul algebra Λ that the classical Koszul and cotilting case occur as special cases of algebras having the property that corej_{T} = 0 on K_{T} (Λ) and rej_{T} = 0 on K_{T} (Λ), respectively.

We start by considering the case corej_{T} = 0. Here we let P(Σ) denote the full subcategory of mod Σ consisting of the finitely generated projective Σ-modules for an artin algebra Σ.

Proposition 4.1. Let Λ be a generalized T -Koszul algebra, and let Γ = E_{T} (Λ).

(a) If corej_{T} = 0 on K_{T} (Λ) and Λ_{0} satisfies the Generalized Nakayama conjecture, then add T = P(Λ_{0}) and M_{i} is in P(Λ_{0}) for all M in K_{T} (Λ).

(b) Assume that Λ_{0} and Γ_{0} satisfy the Generalized Nakayama conjecture. Then

the following are equivalent. (i) corej_{T} = 0 on K_{T} (Λ). (ii) corej_{T} = 0 on K_{T} (Γ). (iii) All modules in K_{T} (Λ) have linear projective resolutions. (iv) All modules in K_{T} (Γ) have linear projective resolutions.

Proof. (a) Suppose that corej_{T} (M) = (0) for all M in K_{T} (Λ). Then M_{i} is in add T for all i ≥ 0 for a module M in K_{T} (Λ). In particular we get that Λ_{0} is in add T. By our assumption on Λ_{0} we must have that add T = P(Λ_{0}). Hence M_{i} is in P(Λ_{0}) for all M in K_{T} (Λ).

(b) We first prove that (i) implies (iii). Assume that corej_{T} = 0 on K_{T} (Λ). We have in general that all modules M in K_{T} (Λ) have a linear fat resolution of the form

⋯ → Λ ⊗_{Λ_{0}} M_{2}^{2}[2] ⟶^{d_{2}} Λ ⊗_{Λ_{0}} M_{1}^{1}[1] ⟶^{d_{1}} Λ ⊗_{Λ_{0}} M^{0} ⟶^{d_{0}} M → 0,

where M^{i} = Ker d_{i−1}. From (a) it follows that this is a projective resolution. Therefore all modules in K_{T} (Λ) have a linear projective resolution.

Assume that all modules in K_{T} (Λ) have a linear projective resolution. By Proposition 5.5 (b) (ii) in Chapter III we have that corej_{T} (E(M)) = E(⟨Ω_{Λ}(M)_{0}⟩). Since Λ is generalized T -Koszul it follows that corej_{T} = 0 in K_{T} (Γ). This shows that (iii) implies (ii).

By symmetry we have also proved the other implications, and this completes the proof of the proposition.

If follows from the above result that when Λ is generalized T -Koszul, corej_{T} = 0 on K_{T} (Λ) and Λ_{0} satisfies the Generalized Nakayama conjecture, then Λ_{i} is in P(Λ_{0}) for all i. The converse is not true, as we can see from Theorem 3.1 in Chapter I by choosing Λ = Λ_{0} + Λ_{1} + ⋯ and T = Λ_{0}, where Λ_{0} is a selfinjective artin algebra. In particular, in this case K_{T} (Λ) = gr_{0} Λ.

Now we address the case when rej_{T} = 0 on K_{T} (Λ).

Proposition 4.2. Let Λ be a T -Koszul algebra, and let Γ = E_{T} (Λ). Then the following are equivalent.

(i) rej_{T} = 0 on K_{T} (Λ).

(ii) rej_{T} = 0 on K_{T} (Γ).

(iii) Λ = Λ_{0}.

(iv) Γ = Γ_{0}.

Proof. Assume that rej_{T} = 0 on K_{T} (Λ). We have that Λ is in K_{T} (Λ), so that rej_{T} (Λ) = ⊕_{i≥1}Λ_{i} = (0) and therefore Λ = Λ_{0}. This show that (i) implies (iii).

Suppose Λ = Λ_{0}. Then it is clear that Ω_{Λ}(M) is generated in degree 0 for all modules M in K_{T} (Λ). By Proposition 5.5 (b) (iii) in Chapter III we have that rej_{T} (E(M))[−1] = E(Ω_{Λ}(M)/⟨Ω_{Λ}(M)_{0}⟩). Since Λ is T -Koszul, it follows that rej_{T} = 0 on K_{T} (Γ). This proves that (iii) implies (ii). Using the same arguments as above we leave it to the reader to complete the proof of the proposition.

We end this section with a result useful for showing that algebras are not T - Koszul.

Proposition 4.3. Let Λ = ⊕_{i≥0}Λ_{i} be a T -Koszul algebra with Λ_{1} ≠ (0). Then Λ_{1} ⊗_{Λ_{0}} T ≠ (0) or equivalently Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) ≠ (0).

Proof. Assume that Λ is T -Koszul. Then T is in K_{T} (Λ), and therefore 0 → Λ_{i≥1}⊗_{Λ_{0}} T → Λ⊗_{Λ_{0}} T → T → 0 is the start of a fat resolution of T and Λ_{i≥1}⊗_{Λ_{0}} T is again in K_{T} (Λ). Hence the Λ_{0}-module Λ_{1}⊗_{Λ_{0}} T is dualized by T, so that Λ_{1}⊗_{Λ_{0}} T = (0) if and only if Hom_{Λ_{)}}(Λ_{1}⊗_{Λ_{0}} T, T) = (0). Assume that Λ_{1}⊗_{Λ_{0}} T = (0). Given the start of the fat resolution of T, we get that Ext_{Λ}^{1}(T, T) ≃ Hom_{Λ_{0}}(Λ_{1} ⊗_{Λ_{0}} T, T) = (0). Since Λ is T -Koszul, this implies that Γ = E_{T} (Λ) = Hom_{Λ_{0}}(T, T) = Γ_{0}. Then it follows from Proposition 4.2 that Λ = Λ_{0}. This is a contradiction, hence Λ_{1} ⊗_{Λ_{0}} T ≠ (0).

5. Questions

In this section we discuss some open problems related to our presentation of Koszul algebras with respect to T. We have emphasized the study when the functor E: gr_{0} Λ → gr_{0} Γ where Γ is the Ext-algebra ⊕_{n≥0} Ext_{Λ}^{n}(T, T), restricts to a duality on a subcategory of gr_{0} Λ closed under corejects. In Chapter III, in the introduction and Proposition 1.5, we showed that T must be a graded module concentrated in degree 0 and a Wakamatsu cotilting module for Λ_{0}. Furthermore, we have seen (Theorem 6.7 in Chapter III) that Λ must be a quadratic algebra in a natural sense. What is less clear is when there is some duality for some subcategories.

In the first part of the paper, we assumed that the category X_{T} satisfied the ⊗- Tor-conditions. In this case we showed there was a duality between the categories of modules with linear fat resolutions. The definitions and results are closely related to the classical Koszul case. One difference was that the category of modules with linear fat resolutions need not be all the graded modules generated in degree 0 such that E(M) is generated in degree 0; see Example 4.1 in Chapter I . This leads to the following questions.

Question 5.1. Let Λ be a T -Koszul algebra such that X_{T} satisfies the ⊗-Tor-conditions. Is there a largest subcategory of gr_{0} Λ containing the Koszul modules with respect to T such that E restricts to a duality? Does E restrict to a duality on the subcategory G_{T} (Λ)?

Next we turn our attention to the more general case of generalized T -Koszul algebras. By Example 4.9 in Chapter I, we see that in this general setting, it may be the case that X_{T} ∩K_{T} (Λ) can be properly smaller than X_{T}. If X_{T} ∩K_{T} (Λ) were a resolving subcategory of mod Λ_{0}, then we can show that K_{T} (Λ) = FR(X_{T} ∩K_{T} (Λ)). Here, if Y is a subcategory of mod Λ_{0}, then FR(Y) is the full subcategory of gr_{0} Λ of graded modules M such that M_{i} is in Y for i ≥ 0 and M has a linear fat resolution by modules of the form Λ ⊗_{Λ_{0}} Y where Y is in Y.

Proposition 5.2. Let Λ be a generalized Koszul algebra with respect to a Wakamatsu cotilting module T. Suppose that X_{T} ∩ K_{T} (Λ) is resolving. Then K_{T} (Λ) = FR(X_{T} ∩ K_{T} (Λ)).

Proof. First we show that, in general, K_{T} (Λ) is a subcategory of FR(X_{T} ∩ K_{T} (Λ)). Let M be in K_{T} (Λ). Then M_{i} is in X_{T} ∩ K_{T} (Λ) for i ≥ 0 by definition. Then we have an exact sequence of graded modules 0 → X^{1} → Λ⊗_{Λ_{0}} M_{0} → M → 0. By definition, X^{1}[−1] is contained in K_{T} (Λ). Thus X_{n}^{1} is in X_{T} ∩ K_{T} (Λ) for all n ≥ 1. It now follows by induction that M is in FR(X_{T} ∩ K_{T} (Λ)).

Now assume that Y = X_{T} ∩ K_{T} (Λ) is resolving. We show that FR(Y) is a subcategory of K_{T} (Λ) to complete the proof. To do this, we use that K_{T} (Λ) is the largest subcategory X of gr_{0} Λ whose objects M have the following properties:

1. E(M) is generated in degree 0.

2. M_{0} is in X

3. M^{1}[−1] = ker(Λ ⊗_{Λ_{0}} M_{0} → M)[−1] is in X.

4. The submodule of Ω_{Λ}(M) generated in degree 0 is in X.

5. rej_{T} (M)[−1] is in X.

6. corej_{T} (M) is in X.

7. Im(M → M_{T}) is in X. We show that FR(Y) satisfies all the above properties. Let M be in FR(Y). Similarly as in Proposition 1.11 (a) in Chapter II we can show that E(M) is generated in degree 0. Directly from the definition of FR(Y) and since Y is resolving, it is clear that property 3 holds. We have that Λ_{0} is in FR(Y). Hence by the ⊗-Tor-conditions Y is a subcategory of FR(Y), and therefore properties 2, 4, 6, and 7 hold. It remains to show that rej_{T} (M)[−1] is in FR(Y). But we have a short exact sequence

0 → M^{1} → Λ_{≥1} ⊗_{Λ_{0}} M_{0} → rej_{T} (M) → 0,

where rej_{T} (M)_{i} = M_{i} is in Y for all i ≥ 1 and the modules M^{1} and Λ_{≥1} ⊗_{Λ_{0}} M_{0} are in FR(Y). To see that Λ_{≥1}⊗_{Λ_{0}} M_{0} is in FR(Y), observe that Λ_{≥1}⊗_{Λ_{0}} M_{0} = (M_{0})^{1}. As in Proposition 1.13 in Chapter II we can prove that rej_{T} (M) is in FR(Y). This shows that FR(Y) is contained in K_{T} (Λ), and consequently we have proved that FR(Y) = K_{T} (Λ).

The above result motivates the following question.

Question 5.3. If Λ is a generalized T -Koszul algebra, is X_{T} ∩K_{T} (Λ) resolving? If X_{T} ∩ K_{T} (Λ) is not resolving, is K_{T} (Λ) of the form FR(Y) for a reasonable subcategory Y of mod(Λ_{0}), perhaps resolving?

The difference between an algebra being a generalized Koszul algebra with respect to T and a Koszul algebra with respect to T is not well understood. We have shown that if Λ is T -Koszul, then Λ is generalized T -Koszul. Also it follows from this that if Λ is strongly T -Koszul, then Λ is generalized strongly T -Koszul. However, we do not know if being strongly T -Koszul is equivalent to being generalized strongly T -Koszul. We have the following information in this direction.

Proposition 5.4. If Λ is a generalized strongly T -Koszul for a Wakamatsu cotilting module T, then X_{T} ∩K_{T} (Λ) is resolving. In particular, K_{T} (Λ) = FR(X_{T} ∩ K_{T} (Λ)).

Proof. Assume that Λ is generalized strongly T -Koszul for a Wakamatsu cotilting module T. We want to show that X_{T} ∩ K_{T} (Λ) is resolving. By definition Λ is in K_{T} (Λ), and therefore Λ_{0} is in K_{T} (Λ). We have that Λ_{0} is in X_{T}, so that P(Λ_{0}) is contained in X_{T} ∩ K_{T} (Λ).

By assumption K_{T} (Λ) = G_{T} (Λ). Since the degree zero part of any module in G_{T} (Λ) is in X_{T}, any exact sequence in G_{T} (Λ) is F^{T} -exact. Then by Proposition 1.12 (b), the category G_{T} (Λ) is closed under any extensions. The category X_{T} is also closed under extensions, so that we infer that X_{T} ∩ K_{T} (Λ) is closed under extensions.

Finally we need to show that X_{T} ∩ K_{T} (Λ) is closed under kernels of epimorphisms. Let η: 0 → A → B → C → 0 be exact with B and C in X_{T} ∩ K_{T} (Λ). Since X_{T} is resolving, A is in X_{T}. Since Λ satisfies the Tor-condition with respect to X_{T} ∩ K_{T} (Λ) and C is X_{T} ∩ K_{T} (Λ), the sequence 0 → Λ ⊗_{Λ_{0}} A → Λ ⊗_{Λ_{0}} B → Λ⊗_{Λ_{0}} C → 0 is exact. The modules Λ⊗_{Λ_{0}} B and Λ⊗_{Λ_{0}} C are fat modules. Using the long exact sequence induced by η when applying the functor Hom_{Λ}( , T), it follows that E(Λ ⊗_{Λ_{0}} A) is concentrated in degree 0. Hence Λ ⊗_{Λ_{0}} A is in G_{T} (Λ) = K_{T} (Λ). Therefore (Λ ⊗_{Λ_{0}} A)_{0} = A is in K_{T} (Λ). This shows that A is in X_{T} ∩ K_{T} (Λ), and consequently that X_{T} ∩ K_{T} (Λ) is resolving.

The last claim is a direct consequence of Proposition 5.2.

In view of the above proposition the following question is natural.

Question 5.5. If Λ is a generalized T -Koszul algebra for a Wakamatsu cotilting module T, find necessary and sufficient conditions so that K_{T} (Λ) ∩ X_{T} = X_{T}, that is, find necessary and sufficient conditions for Λ to be a Koszul algebra with respect to T.

We can also ask similar questions to Question 5.1 in this general case. More precisely,

Question 5.6. Let Λ be a generalized T -Koszul algebra. Is there a largest subcategory of gr_{0}(Λ) containing the T -Koszul modules such that E restricts to a duality? Does E restrict to a duality on the subcategory of G_{T} (Λ)?

We have the following related question.

Question 5.7. Suppose that Λ is a generalized T -Koszul algebra with respect to a Wakamatsu cotilting Λ_{0}-module T. Is there a finite set of criteria to determine if a module M in gr_{0}(Λ) is in K_{T} (Λ)?

Proposition 3.4 motivates the following question.

Question 5.8. Suppose that Λ is a graded algebra generated in degrees 0 and 1, with Λ_{0} an artin algebra and Λ_{1} a finitely generated as a left Λ_{0}-module. Let T be a Wakamatsu cotilting module. Suppose that E′E(Λ) is isomorphic to Λ as graded algebras. Under what additional hypothesis, if any, can we conclude that Λ is a (generalized) Koszul algebra with respect to T?

Our next question can be asked of either T -Koszul or generalized T -Koszul algebras.

Question 5.9. Suppose Λ is either a T -Koszul or generalized T -Koszul algebra for a Wakamatsu cotilting Λ_{0}-module T. Is there a right Λ_{0}-module T′, for example D(T), such that Λ^{op} is either a T′-Koszul or generalized T′-Koszul algebra?

We have seen many examples of strongly T -Koszul algebras. The next question is related to this class.

Question 5.10. Let T be a Wakamatsu cotilting Λ_{0}-module. Under what additional conditions, if any, does Λ being strongly T -Koszul imply that Γ is strongly T -Koszul?

Another question relates to tensor products.

Question 5.11. Suppose that A and B are (generalized) T -Koszul k-algebras, for Wakamatsu cotilting modules U and V over A_{0} and B_{0} respectively, where k is a field. Is A ⊗_{k} B a (generalized) T -Koszul algebra for U ⊗_{k} V?

We see that generalized T -Koszul algebras display similar features to Koszul algebras. To further highlight this similarity we include a generalization of another result from the Koszul case.

Proposition 5.12. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degrees 0 and 1, where Λ_{0} is an artin algebra. Let T be a finitely generated Λ_{0}- module with T in fgsyz Λ. If M is in G_{T} (Λ), then Ext_{gr Λ}^{i}(M, T [j]) = (0) for j ≠ i and i > 0.

Proof. By definition Ext_{Λ}^{1}(T, T) Ext_{Λ}^{i}(M, T) = Ext_{Λ}^{i+1}(M, T) for all i ≥ 0. Since M is in fgsyz Λ and generated in degree 0, we have that Ext_{Λ}^{i}(M, T) = ⊕_{j≥0} Ext_{gr Λ}^{i}(M, T [j]). Similarly Ext_{Λ}^{1}(T, T) = ⊕_{j≥0} Ext_{gr Λ}^{1}(T, T [j]). We have that Ext_{gr Λ}^{1}(T, T [j]) = (0) for j ≠ 1. Introducing this decomposition into the above product it follows by induction that Ext_{gr Λ}^{i}(M, T [j]) = (0) for j ≠ i and i > 0.

This naturally raises the question.

Question 5.13. When are the modules M in G_{T} (Λ) characterized by the property that Ext_{gr Λ}^{i}(M, T [j]) = (0) for j ≠ i and i > 0?

One situation where this can be answered to the affirmative is given in the next result.

Proposition 5.14. Let Λ = ⊕_{i≥0}Λ_{i} be a left finite graded ring generated in degree 0 and 1 with Λ_{0} an artin algebra. Assume that (Λ_{i})_{Λ_{0}} is projective, and let T = D(Λ^{op}). Suppose M is in gr_{0}(Λ) such that Ext_{gr Λ}^{i}(M, T [j]) = (0) for j ≠ i and i > 0. Then M is in G_{T} (Λ).

Proof. Let M be in gr_{0}(Λ) such that Ext_{gr Λ}^{i}(M, T [j]) = (0) for j ≠ i and i > 0. We have the standard exact sequence

0 → M^{1} → Λ ⊗_{Λ_{0}} M_{0} → M → 0.

Applying the functor Hom_{gr Λ}( , T [j]) to this sequence induces the exact sequence

0 → _{gr Λ}(M, T [j]) → _{gr Λ}(Λ ⊗_{Λ_{0}} M_{0}, T [j]) → _{gr Λ}(M^{1}, T [j]) → Ext_{gr Λ}^{1}(M, T [j]) → 0,

since Ext_{gr Λ}^{1}(Λ ⊗_{Λ_{0}} M_{0}, T [j]) = (0) for all j ≥ 0. The two first are zero for j ≠ 0 and both are isomorphic Hom_{Λ_{0}}(M_{0}, T) for j = 0. Hence we get that Hom_{gr Λ}(M^{1}, T [j]) ≃ Ext_{gr Λ}^{1}(M, T [j]) for all j ≥ 0. It follows from the assumptions that M^{1} is generated in degree 1. Using that Ext_{gr Λ}^{2}(M, T [j]) = (0) for all j ≠ 2 and so on, we have that M has a linear fat resolution. Hence M is in G_{T} (Λ).

List of Notation

(A, B), x C^{⊥}, ix D(Λ), 1 F^{T}, ix F^{X}, v F_{X}, v M[n], x M^{∗}, ix M^{1}, 6 M_{T}, 32 M_{≥n}, x E, 3 E′, 6 Ext_{Λ}^{i}(C, A), x Ext_{gr Λ}^{i}(C, A), x Gr Λ, x K(Λ), 3 K_{T} (Λ), 5 Sub(M), ix X_{T}, 2 add M, ix FR(Y), 60 G_{T} (Λ), 6 corej_{T} (M), 32 fgsyz Λ, x gr Λ, x gr_{0} Λ, x gr_{0} Λ, 4 ⟨U⟩, x mod Λ, 1 ψ_{Λ}, 31 rej_{T} (M), 32 w-G_{T} (Λ), 48 ⊥C, ix ⊥T, 1 Λ(A, B), x

Bibliography

[1] M. Auslander, Artin algebras I, Comm. Algebra 1 (1974) 177–268.

[2] M. Auslander, R.-O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de Pierre Samuel (Orsay 1987), Mém. Soc. Math. France, (N.S.) No. 38 (1989) 5-37.

[3] M. Auslander, I. Reiten, Applications of homologically finite subcategories, Adv. in Math., 86, No. 1 (1991) 111–152.

[4] M. Auslander, I. Reiten, S. O. Smalø, Representation theory of artin algebras, Cambridge studies in advanced mathematics, 36, Cambridge University Press.

[5] M. Auslander, S. O. Smalø, Preprojective modules over artin algebras, J. Algebra 66, (1980) No. 1, 61–122.

[6] M. Auslander, Ø. Solberg, Representation theory and relative homology I: Relative homology and homologically finite subcategories, Comm. Algebra, 21 (9) (1993) 2995–3031.

[7] M. Auslander, Ø. Solberg, Relative homology and representation theory III: Cotilting modules and Wedderburn correspondence, Comm. Algebra, 21 (9) (1993) 3081–3097.

[8] H. Cartan, S. Eilenberg, Homological algebra, Oxford University Press, Princeton University Press.

[9] A. Belinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc., Vol. 9, No. 2 (1996) 473–527.

[10] E. L. Green, R. Martinéz-Villa, Koszul and Yoneda algebras, Representation theory of algebras (Cocoyoc, 1994), 247–297, CMS Conf. Proc., 18, Amer. Math. Soc., Providence, RI, 1996.

[11] E. L. Green, R. Martinéz-Villa, Koszul and Yoneda algebras II, Algebras and modules, II (Geiranger, 1996), 227–244, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.

[12] D. Happel, C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), No. 2, 399–443.

[13] Y. Miyashita, Generalized tilting modules and applications to module theory, Math. J. Okayama Univ. 34 (1992), 75–98 (1994).

[14] C. Nâstâsescu, F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, 28. North-Holland Publishing Co., Amsterdam-New York, 1982. ix+340 pp. ISBN: 0-444-86489-X

[15] C. M. Ringel, The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996), 467–480, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.

[16] J. J. Rotman, An introduction to homological algebra, Academic Press, 1979.

[17] T. Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, J. Algebra 134 (1990), No. 2, 298–325.

[18] D. Woodcock, Cohen-Macaulay complexes and Koszul rings, J. London Math. Soc. (2) 57 (1998), No. 2, 398–410.

Index

F -exact, v F -projective, v Tor-condition, 4 ⊗-Tor-condition

for bimodules, 18 ⊗-condition, 4

algebra

T -Koszul, 5 generalized strongly, 36 strongly, 6

basic, viii generalized T -Koszul, 35 generated in degrees 0 and 1, x, 3 Koszul, iv, 3 left finite, x path, viii quadratic, 3

approximation left, ix, 31 minimal, ix, 31 right, ix

Generalized Nakayama conjecture, ix

ideal admissible, viii

module T -Koszul, 5 cotilting, 1 dualized by, ix fat, 5, 42 Koszul, 3, 35 selforthogonal, ix tilting, 1 Wakamatsu cotilting, 2

projective module enough projectives, v

relative, v

quiver, viii path in a quiver, viii trivial paths, viii

resolution T -linear, 48 linear, 3 fat, 43

subcategory contravariantly finite, ix covariantly finite, ix extension closed, viii funtorially finite, ix resolving, viii