CLEFT EXTENSIONS OF ABELIAN CATEGORIES AND APPLICATIONS TO RING THEORY

Apostolos Beligiannis

Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece.

Current Address: Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece. Email: abeligia@aegean.gr.

1. Introduction

Let Γ be an associative ring and let M be a Γ-Γ-bimodule. Gerstenhaber, studying deformations of rings, introduced in [16] the notion of a Γ-structure on the bimodule M by means of a short exact sequence:

0 ⟶ M ⟶ Λ ⟶^{ε} Γ ⟶ 0 (E)

where ε is a split epimorphism of rings together with a specified splitting of ε.

The purpose of this paper is to study the module category Mod(Λ) for a given Γ-structure (E) on a Γ-Γ-bimodule M. Given a Γ−structure (E) on M, it follows that M becomes a two sided ideal of Λ and in case M satisfies M^{2} = 0, then Λ is the well known trivial extension Γ⋉M of Γ by the Γ-Γ-bimodule M, studied extensively by Fossum-Griffith-Reiten in [15], and also by Palmer-Roos in [35]. In the general case there exists an associative Γ-Γ-bimodule morphism ϑ : M ⊗_{Γ} M ⟶ M, i.e. (1_{M} ⊗ ϑ) ∘ ϑ = (ϑ ⊗ 1_{M}) ∘ ϑ, and then Λ is isomorphic to the ϑ−extension ring Γ ⋉_{ϑ}M, which additively is Γ ⊕ M and the multiplication is given by:

(γ_{1}, m_{1})(γ_{2}, m_{2}) = (γ_{1}γ_{2}, γ_{1}m_{2} + m_{1}γ_{2} + ϑ(m_{1} ⊗ m_{2})).

We call the rings Λ arising from a given Γ−structure on M, cleft extension rings of Γ by the Γ-Γ-bimodule M. The study of cleft extension rings is justified by the fact that this class of rings is very large and it is omnipresent in ring theory: it includes free rings, symmetric rings, polynomial rings, exterior rings, tensor algebras, basic semiperfect rings, tic-tac-toe rings in the sense of Mitchell [33], positively graded rings, group rings, supplemented rings, universal enveloping algebras of Lie algebras, trivial extension rings (especially generalized matrix rings), truncated rings, quotients of quiver algebras, and all rings satisfying some version of the Principal Theorem of Wedderburn. Hence the study of the module category of a cleft extension ring unifies large parts of ring and module theory.

The model categories of the form Mod(Λ) for a cleft extension ring Λ of a ring Γ, are the so called η-extension categories D_{F} (η) of a given abelian category D by a right exact functor F : D → D, introduced by Marmaridis [29]. Here η : F^{2} → F is an associative natural morphism, i.e. ηF ∘ η = F η ∘ η. An object of D_{F} (η) is of the form (X, f) where f : F X ⟶ X is a morphism in D satisfying the relation η_{X} ∘ f = F f ∘ f. More generally we define an arbitrary category C to be a cleft extension of a category D, if there exists a tripleable functor [7], U : C → D and a functor Z : D → C such that: UZ ≅ Id_{D}. We show that an additive category C is a cleft extension of an additive category D iff C is equivalent to an η-extension category D_{F} (η). From the point of view of representation theory a cleft extension C of the abelian category D is a natural generalization of the module category of the ring kQ/<ρ> where Q is a quiver and ρ is a set of relations [4]. Indeed a cleft extension of an abelian category D is a full exact reflective subcategory of a free cleft extension D(F), which is a model of the module category of a tensor ring; the morphisms f : F X ⟶ X can be regarded as generators, and the relations are of the form η_{X} ∘ f − F f ∘ f. In all these cases it is very useful to view objects, resp. modules, as morphisms satisfying relations given by the multiplication. As expected the complexity of the multiplication η : F^{2} ⟶ F, resp. ϑ : M ⊗_{Γ} M ⟶ M, measures the complexity of the cleft extension category, resp. ring.

We point out that there is also the dual notion of a cleft coextension of a category, which corresponds to the ζ-coextension category of [29]. In some cases, for instance in the setting of module categories, cleft extensions are also cleft coextensions. This is very useful in the study of homological questions [8]. All the results about cleft extensions of rings and categories have a prime dual translation to results about cleft coextensions of corings and categories. We study only cleft extensions and we leave to the reader to state and prove the dual results.

In this paper we prove basic structure results for cleft extensions generalizing some of the corresponding results of [15], [29], [35]. For a detailed study of the relative homological behaviour of a cleft extension of a ring or of an abelian category, we refer to [8].

The organization of the article is as follows. In section 2 we define the notion of cleft extension C of an arbitrary category D and we prove that this concept is equivalent to the concept of an η−extension D_{F} (η) of D as described above. We give conditions for C to be abelian or Grothendieck category and we study the various adjoint functors relating C and D. Finally we analyze a connection between cleft extensions of D and triples [7] defined on D and we study the relations between C and the induced Eilenberg-Moore categories.

In sections 3, 4 we characterize cleft extensions of abelian categories which are of module type and we indicate the connection with the cleft extensions of rings. We give methods for constructing cleft extensions starting with an abelian category equipped with a family of right exact endofunctors. In this way we obtain the free, symmetric, polynomial, exterior, repetitive categories, categories of complexes, etc. Apart from constructing cleft extensions we present examples of module categories which are cleft extensions for structural reasons.

In section 5 we study intrinsic properties of a cleft extension C of an abelian category D. In particular we characterize under mild conditions the simple and projective objects of C in terms of those of D, and we study the relations between homological invariants of C, D, for instance K−theory, minimal projective presentations, extension functors and global dimension.

In section 6 we study briefly the consequences of the fact that a cleft extension C of D is also a cleft coextension. For instance we determine the structure of the injective objects of C and as an application we prove that almost split sequences [4] in D can be lifted to almost split sequences in C.

Finally in section 7 we present some applications to ring theory. In general if a ring Γ belongs to a class of rings defined by a ring theoretic property (e.g. Γ is Noetherian, perfect, Nakayama, etc.), then it is very interesting to know when a cleft extension Λ of Γ, belongs to the same class. We devote this section to the investigation of this question and we obtain some partial results.

A general convention used in the paper, is that the composition of morphisms in a given category is meant in the diagrammatic order, i.e. f ∘ g means first f then g. There are two exceptions. We use the usual anti-diagrammatic order when we compose functors and when we apply elements to morphisms in concrete categories. If f : X ⟶ A, g : Y ⟶ A and h : A ⟶ Z, k : A ⟶ W are morphisms in an additive category A, then we denote by ^{t}(f, g) : X ⊕ Y ⟶ A and (h, k) : A ⟶ Z ⊕ W, the uniquely induced morphisms.

2. Cleft Extensions And Triples Without Unit

Let C, D be two categories, and let Z : D → C be a fully faithful functor. Classifying the above situation we call the category C an extension of D. Such an extension splits iff there exist a functor U : C → D, such that UZ ≅ Id_{D}. Our purpose is to study in the additive setting, an important class of split extensions of categories, namely the class of cleft extensions defined below. For the concepts and results we use from the theory of triples, we refer to [7], [28], [39].

Definition 2.1. A cleft extension of the category D is a category C together with functors Z : D → C (cleaving ) and U : C → D (projection), such that:

(i) U is tripleable [7]. (ii) UZ ≅ Id_{D}.

If in the above definition the functor U is cotripleable, then we say that C is a cleft coextension of D. Since the notions of cleft extension and cleft coextension are dual, we shall study only cleft extensions of a given category. However in some cases we shall state the dual versions of the results, leaving the proofs to the reader.

Let C be a cleft extension of the category D with projection U and cleaving Z. We denote by T : D → C the left adjoint of U (T exists since U is tripleable, see [7]). We fix throughout unit δ : Id_{D} → UT and counit ε : TU → Id_{C} of the adjoint pair (T, U) and we view the natural isomorphism UZ ≅ Id_{D} as an identification. The quantuple {T, U, Z, δ, ε} is called the cleaving data of the cleft extension.

The following observation justifies the term extension in the above definition.

Lemma 2.2. The functor Z is fully faithful, the functor T is faithful, and the functor U is dense and faithful.

Proof. First U is faithful since any tripleable functor is faithful, and U is dense since UZ ≅ Id_{D}. If a ∈ C(Z(X), Z(Y)), then we have U(a) = UZU(a), and hence a = ZU(a) since U is faithful. Thus Z is full and since UZ ≅ Id_{D} we have that Z is also faithful. From the adjunction relation δU ∘ Uε = 1_{U}, it follows that δUZ ∘ UεZ = 1_{UZ}, i.e. δ ∘ UεZ = 1_{Id_{D}} and δ is a pointwise split monic. Hence if a ∈ D(X, Y) with T(a) = T(b) then: UT(a) = UT(b) ⇒ δ_{X} ∘ UT(a) = δ_{X} ∘ UT(b) ⇒ a ∘ δ_{Y} = b ∘ δ_{Y} ⇒ a = b, so T is faithful.

We are mainly interested in the case of cleft extensions of additive categories, so from now on we assume that all categories and functors we deal with are additive.

Proposition 2.3. Let C be a cleft extension of D, with cleaving data {T, U, Z, δ, ε}. Then for the following statements we have (i) ⇒ (ii) ⇔ (iii):

(i) D has cokernels. (ii) The functor Z has a left adjoint C. (iii) The functor Z is tripleable. These statements are equivalent if C has cokernels. In case C exists, then we have: CZ ≅ Id_{D}, CT ≅ Id_{D}.

Proof. (i) ⇒ (ii) Suppose that D has cokernels. Consider the natural morphism Uε − UεZU : UTU → U and set C := Coker(Uε − UεZU). Hence the functor C is defined by the exact sequence:

UTU ⟶^{Uε−UεZU} U ⟶^{κ} C ⟶ 0 (1)

Since: (Uε − UεZU)Z = UεZ − UεZUZ = UεZ − UεZ = 0 : UT → Id_{D}, calculating the exact sequence (1) on Z, we have that the natural morphism κZ : Id_{D} → CZ is the identity. Since U is tripleable we have an exact sequence:

TUTU ⟶^{TUε−εTU} TU ⟶^{ε} Id_{D} ⟶ 0 (2)

Then using that UZ ≅ Id_{D}, we have: (TUε−εTU)∘εZU∘Zκ = TUε∘εZU∘Zκ−εTU∘εZU∘Zκ = TUε∘Tκ∘εZC− TUTκ ∘ εTC ∘ εZC = T(Uε ∘ κ) ∘ εZC − TUTκ ∘ εTC ∘ εZC = T(UεZU ∘ κ) ∘ εZC−TUTκ∘εTC∘εZC = TUεZU∘Tκ∘εZC−TUTκ∘εTC∘εZC = TUTκ∘ TUεZC∘εZC−TUTκ∘εTC∘εZC = TUTκ∘εTC∘εZC−TUTκ∘εTC∘εZC = 0.

Hence since ε = coker(TUε − εTU), there exists a unique natural morphism λ : Id_{C} → ZC with ε ∘ λ = εZU ∘ Zκ. We claim that λ is the unit and κZ is the counit of an adjoint pair (C, Z). It suffices to show that Cλ = 1_{C} , and λZ = 1_{Z}. But ε ∘ λ = εZU ∘ Zκ ⇒ εZ ∘ λZ = εZUZ ∘ ZκZ ⇒ εZ ∘ λZ = εZ ∘ ZκZ ⇒ λZ = ZκZ ⇒ λZ = 1_{Z}. Similarly ε ∘ λ = εZU ∘ Zκ ⇒ Uε ∘ Uλ = UεZU ∘ κ ⇒ δU ∘ Uε ∘ Uλ = δU ∘ UεZU ∘ κ ⇒ Uλ = κ, since from the adjoint pair (T, U) we have δU ∘ Uε = 1_{U} and δU ∘ UεZU = δUZU ∘ UεZU = 1_{UZU} = 1_{U}. Then κ ∘ Cλ = Uλ ∘ κZC ⇒ κ ∘ Cλ = κ ⇒ Cλ = 1_{C}, because κ is epic.

(ii) ⇔ (iii). If Z has a left adjoint, then by Lemma 2.2, D is reflective in C, and then Z is tripleable by [7]. The converse is trivial.

Suppose now that Z has a left adjoint C. Composing the adjoint pairs (C, Z), (T, U) we have an adjoint pair (CT, UZ) : D → D. Since UZ ≅ Id_{D} we have CT ≅ Id_{D}. If C has cokernels then since CZ ≅ Id_{D} and since C preserves cokernels as a left adjoint of Z, it follows trivially that D has cokernels.

Suppose that in the cleft extension C of D the left adjoint C of Z exists. We denote by KerC the full subcategory of C with objects A ∈ C such that C(A) = 0. If λ : Id_{C} → ZC is the unit of the adjoint pair (C, Z), then we say that λ is a minimal epimorphism, if ∀0 ≠ A ∈ C, the morphism λ_{A} is a minimal epimorphism, i.e. a : B → A is an epimorphism whenever a ∘ λ_{A} is so.

Proposition 2.4. Let C be a cleft extension of D with cleaving data {T, U, Z, δ, ε}. If D has cokernels, then the following are equivalent for the left adjoint C of Z:

(i) KerC = 0. (ii) The unit λ : Id_{C} → ZC of the adjoint pair (C, Z) is a minimal epimorphism.

Proof. Since D has cokernels, by Proposition 2.3 we have an adjoint pair (C, Z), the unit λ : Id_{C} → ZC of (C, Z) is an epimorphism and Cλ = 1_{C}.

(ii) ⇒ (i) Let A be in C with C(A) = 0. Then the minimal epimorphism λ_{A} : A → 0 shows that A = 0, hence KerC = 0.

(i) ⇒ (ii) Suppose that 0 ≠ A ∈ C, a : B → A is a morphism in C such that a∘λ_{A} is an epimorphism, and let E := Coker(a). Then C(a) ∘ C(λ_{A}) is an epimorphism; but obviously C(λ_{A}) is an isomorphism, hence C(a) is an epimorphism and C(E) = 0 since C preserves cokernels. Since by hypothesis KerC = 0 we have that E = 0 and thus a is an epimorphism. We conclude that λ_{A} is a minimal epimorphism.

2.1. η−extensions. The archetypical cleft extensions of a given category are the so called η−extensions of abelian categories introduced constructively by Marmaridis in [29], trying to generalize the notion of the trivial extension of abelian categories (see [15]). We recall from [29] the construction of the η-extension of an additive category D. Let F : D → D be an endofunctor. A natural morphism η : F^{2} → F is called associative if F η ∘ η = ηF ∘ η, i.e. if the following diagram commutes:

F^{3} ⟶^{F η} F^{2}
ηF ↓  ^{η}↓
F^{2} ⟶^{η} F.

Let D_{F} (η) be the category with objects (X, f) where X ∈ D and f : F X → X is a morphism in D with F f ∘ f = η_{X} ∘ f. A morphism a : (X, f) → (Y, g) in D_{F} (η) is a morphism a : X → Y in D with F a ∘ g = f ∘ a. Obviously D_{F} (η) is additive and the categories D_{F} (η), D are connected through the following additive functors:

U : D_{F} (η) → D defined by U(X, f) = X , U(a) = a.

Z : D → D_{F} (η) defined by Z(X) = (X, 0) , Z(a) = a.

T : D → D_{F} (η) defined by T(X) = (X ⊕ F X, t_{X}) ,

where t_{X} = ( \begin{matrix}0 & 1_{F X} \\ 0 & η_{X} \end{matrix} ) , T(a) = ( \begin{matrix}a & 0 \\ 0 & F a \end{matrix} ) .

Lemma 2.5. Any η-extension is a cleft extension.

Proof. Since trivially UZ = Id_{D}, it suffices to show that U is tripleable. First observe (see also [29]), that T is a left adjoint of U with unit δ = (1_{Id_{D}}, 0) : Id_{D} → Id_{D} ⊕F = UT and counit ε : TU → Id_{D_{F} (η)}, where ε_{(X,f)} = ^{t}(1_{X}, f). Obviously U is faithful. Let F be the triple generated on D by the adjoint pair (T, U). The Eilenberg-Moore comparison functor Θ : D_{F} (η) → D^{F} is given by Θ(X, f) = (X,^{t}(1_{X}, f)), Θ(a) = a and obviously Θ is an isomorphism. This means that U is tripleable.

By Proposition 2.3 and Lemma 2.5 it follows that if D has cokernels then the functor Z has a left adjoint C : D_{F} (η) → D defined by C(X, f) = Coker(f), and for a morphism a : (X, f) → (Y, g) in D_{F} (η), C(a) is the unique morphism: Coker(f) → Coker(g), with: coker(f) ∘ C(a) = a ∘ coker(g), (see also [29]). The unit λ : Id_{D_{F} (η)} → ZC is given by λ_{(X,f)} = coker(f) : (X, f) → ZC(X, f).

Using the notation introduced above we can prove the main result of this section.

Theorem 2.6. Suppose that C is an additive category closed under kernels of split epics. Then the following are equivalent: (α) C is a cleft extension of an additive category D. (β) There exists an additive category D, an additive endofunctor F : D → D, an

associative natural morphism η : F^{2} → F, and an equivalence of categories

Ω : C ⟶^{≈} D_{F} (η).

Moreover if C is a cleft extension of D with cleaving data {T, U, Z, δ, ε}, then:

(i) Under the equivalence Ω the functors U, Z, T correspond with the standard functors U, Z, T connecting the categories D_{F} (η), D.

(ii) The following are equivalent: (a) ∀(X, f) ∈ D_{F} (η), the cokernel coker(f) exists.

(b) The functor Z has a left adjoint C.

(c) The functor Z is tripleable. If C exists, then we have CZ = Id_{D}, CT = Id_{D}, and the functor C corresponds under the equivalence Ω with the functor C.

Proof. (α) ⇒ (β) Suppose that the category C is a cleft extension of D with cleaving data {T, U, Z, δ, ε}. Then we have the adjunction relations:

Tδ ∘ εT = 1_{T} , δU ∘ Uε = 1_{U} (1)

and as in Lemma 2.2: δUZ ∘ UεZ = δ ∘ UεZ = 1_{Id_{D}}. Setting γ := UεZ, we have a split short exact sequence of additive endofunctors of D:

0 ⟶ F ⟶^{μ} UT ⟶^{γ} Id_{D} ⟶ 0 (2)

Since C is closed under kernels of split epics, the same is true for D. So setting: μ := ker(γ), ν := coker(δ), the endofunctor UT of D admits a direct sum decomposition

UT = Id_{D} ⊕ F (3)
μ : F → UT , ν : UT → F , γ : UT → Id_{D} , δ : Id_{D} → UT

μ ∘ γ = 0 , δ ∘ γ = 1_{Id_{D}} , δ ∘ ν = 0 , μ ∘ ν = 1_{F} , ν ∘ μ + γ ∘ δ = 1_{UT} (4) We define a natural morphism η : F^{2} → F as the composite:

F^{2} ⟶^{μF} UTF ⟶^{UTμ} UTUT ⟶^{UεT} UT ⟶^{ν} F.

Hence setting φ := Tμ ∘ εT, we have: η = μF ∘ Uφ ∘ ν.

Let now T = (UT , δ , UεT) be the triple generated on D from the adjoint pair (T, U). Since U is tripleable, the Eilenberg-Moore comparison functor Θ : C → D^{T} is an equivalence, where D^{T} is the induced Eilenberg-Moore category. Using the natural isomorphisms ^{t}(δ, μ) : Id_{D} ⊕ F ⟶^{≅} UT, (γ, ν) : UT ⟶^{≅} Id_{D} ⊕ F, we get an isomorphic triple T^{∗} = (Id_{D} ⊕ F , δ^{∗} , η^{∗}), where: η^{∗} = ^{t}(δ, μ)(Id_{D} ⊕ F) ∘ UT^{t}(δ, μ) ∘ UεT ∘ (γ, ν) and δ^{∗} = (γ, ν) ∘ δ. Using (1), (4) we see easily that:

η^{∗} = ( \begin{matrix}1_{Id_{D}} & 0 \\ δF ∘ Uφ ∘ γ & δF ∘ Uφ ∘ ν \\ 0 & 1_{F} \\ μF ∘ Uφ ∘ γ & η \end{matrix} ) , δ^{∗} = (1_{Id_{D}}, 0).

Let D^{T ^{∗}} be the Eilenberg-Moore category of the triple T^{∗}. A typical object of D^{T ^{∗}} is of the form (X,^{t}(f^{!}, f)) , where ^{t}(f^{!}, f) : X ⊕ F X → X with δ_{X}^{∗} ∘^{t}(f^{!}, f) = 1_{X} and [(Id_{D} ⊕ F)^{t}(f^{!}, f)] ∘^{t}(f^{!}, f) = η_{X}^{∗} ∘^{t}(f^{!}, f). Analyzing these relations we see that: (X,^{t}(f^{!}, f)) ∈ D^{T ^{∗}} iff f^{!} = 1_{X}, and the following relations hold:

μ_{F X} ∘Uφ_{X} ∘γ_{X} +η_{X}^{∗} ∘f = F f ∘f, δ_{F X} ∘Uφ_{X} ∘γ_{X} +δ_{F X} ∘Uφ_{X} ∘ν_{X} ∘f = f (5)

\underline{Claim} : (I) δF ∘ Uφ ∘ γ = 0, (II) μF ∘ Uφ ∘ γ = 0, (III) δF ∘ Uφ ∘ ν = 1_{F} . Proof of the Claim: We use throughout the relations (1), (4) and the naturality

of the involved morphisms.

(I) Since δF ∘ UTμ = μ ∘ δUT, we have: δF ∘ Uφ ∘ γ = δF ∘ U(Tμ ∘ εT) ∘ γ = δF ∘ UTμ ∘ UεT ∘ γ = μ ∘ δUT ∘ UεT ∘ γ = μ ∘ (δU ∘ εT) ∘ γ = μ ∘ 1_{U} ∘ γ = μ ∘ γ = 0.

(II) Since μF ∘ UTμ = F μ ∘ μUT, we have: μF ∘ Uφ ∘ γ = μF ∘ U(Tμ ∘ εT) ∘ γ = μF ∘ UTμ ∘ UεT ∘ γ = F μ ∘ μUT ∘ UεT ∘ γ = F μ∘μUT∘UεT∘UεZ = F μ∘μUT∘U(εT∘UεZ) = F μ∘μUT∘U(TUεZ∘εZ) = F μ∘μUT∘UTUεZ∘UεZ = F μ∘μUT∘UT(UεZ)∘UεZ = F μ∘μUT∘UTγ ∘γ = F μ ∘ F γ ∘ μ ∘ γ = 0.

(III) Since δU ∘ Uε = 1_{U}, we have δUT ∘ UεT = 1_{UT} ⇒ δUT ∘ UεT ∘ ν = ν ⇒ μ ∘ δUT ∘ UεT ∘ ν = μ ∘ ν = 1_{F}. But since δF ∘ UTμ = μ ∘ δUT, we have: δF ∘ UTμ ∘ UεT ∘ ν = δF ∘ U(Tμ ∘ εT) ∘ ν = δF ∘ Uφ ∘ ν = 1_{F}.

From the above claim it follows that the natural morphism η^{∗} equals:

η^{∗} = ( \begin{matrix}1_{Id_{D}} & 0 \\ 0 & 1_{F} \\ 0 & 1_{F} \\ 0 & η \end{matrix} ) : Id_{D} ⊕ F ⊕ F ⊕ F^{2} ⟶ Id_{D} ⊕ F .

Since T^{∗} = (Id_{D} ⊕ F , δ^{∗} , η^{∗}) is a triple on D, we have (Id_{D} ⊕ F)η^{∗} ∘ η^{∗} = η^{∗}(Id_{D} ⊕ F) ∘ η^{∗}. We see easily that the last relation holds iff ηF ∘ η = F η ∘ η.

Hence the natural morphism η : F^{2} → F is associative and we can define the ηextension D_{F} (η) of D by the endofunctor F. The relations (5) together with the claim shows that: (X,^{t} (1_{X}, f)) ∈ D^{T ^{∗}} ⇔ (X, f) ∈ D_{F} (η). Similarly we see that a : (X,^{t} (1_{X}, f)) → (Y,^{t} (1_{Y} , g)) is a morphism in D^{T ^{∗}} ⇔ a : (X, f) → (Y, g) is a morphism in D_{F} (η). Hence we can define a functor

S : D^{T ^{∗}} ⟶ D_{F} (η)

by S(X,^{t} (1_{X}, f)) = (X, f), S(a) = a, which obviously is an isomorphism of categories. Since the triples T^{∗}, T are isomorphic there exists an isomorphism R : D^{T ^{∗}} → D^{T} of the corresponding Eilenberg-Moore categories. Now we obtain the desired equivalence Ω : C → D_{F} (η) as the composition

Ω : C ⟶^{≈^{Θ}} D^{T ^{∗}} ⟶^{≅^{R}} D^{T} ⟶^{≅^{S}} D_{F} (η).

where Θ is the Eilenberg-Moore comparison functor and R, S are the isomorphisms constructed above. The direction (β) ⇒ (α) follows from lemma 2.5.

Suppose that C is a cleft extension of D via the functors T, U, Z. Then it is easy to see that the equivalence Ω is defined as follows:

Ω(A) = (U(A), μ_{UA} ∘ Uε_{A}), Ω(a) = a .

If U, Z, T are the standard functors connecting the categories D_{F} (η) and D we have by the definition of Ω:

∀A ∈ C : UΩ(A) = U(A), UΩ(a) = U(a) = a.

∀X ∈ D : ΩZ(X) = (UZX, μ_{UZ(X)} ∘ Uε_{Z(X)}) = (X, μ_{X} ∘ γ_{X}) = (X, 0) = Z(X),

ΩZ(a) = Z(a).

Let X ∈ D. Then ΘT(X) = (UT(X), μ_{UTX} ∘Uε_{TX}). Hence the underlying object of ΩT(X) is the object X ⊕ F X, and the morphism μ_{UTX} ∘ Uε_{TX} corresponds to the morphism^{t}(F δ_{X}, F μ_{X})∘μ_{UTX} ∘Uε_{TX} ∘(γ_{X}, ν_{X}). Calculating this composition we see that it goes via Ω to the morphism

( \begin{matrix}F δ_{X} ∘ μ_{UTX} ∘ Uε_{TX} ∘ γ_{X} & F δ_{X} ∘ μ_{UTX} ∘ Uε_{TX} ∘ ν_{X} \\ μ_{F X} ∘ UTμ_{X} ∘ Uε_{TX} ∘ γ_{X} & μ_{F X} ∘ UTμ_{X} ∘ Uε_{TX} ∘ ν_{X} \end{matrix} ) = t_{X} = ( \begin{matrix}0 & 1_{F X} \\ 0 & η_{X} \end{matrix} )

because μ_{F X} ∘ UTμ_{X} ∘ Uε_{TX} ∘ γ_{X} = (μF ∘ Uφ ∘ γ)_{X} = 0 by the claim, F δ_{X} ∘ μ_{UTX} ∘ Uε_{TX} ∘ γ_{X} = μ_{F X} ∘ UTδ_{X} ∘ Uε_{TX} ∘ γ_{X} = (μ_{F} ∘ UTδ ∘ UεT ∘ γ)_{X} = 0 by the claim, and finally F δ_{X} ∘ μ_{UTX} ∘ Uε_{TX} ∘ ν_{X} = μ_{F X} ∘ UTδ_{X} ∘ Uε_{TX} ∘ ν_{X} = (μ_{F} ∘ UTδ ∘ UεT ∘ ν)_{X} = 1_{F X}, using again the claim. Hence ∀X ∈ D : ΩT(X) =

(X ⊕ F X, t_{X}), ΩT(a) = ( \begin{matrix}a & 0 \\ 0 & F a \end{matrix} ) and we infer that: UΩ = U, ΩZ = Z, ΩT = T.

To prove (ii) suppose first that ∀(X, f) ∈ D_{F} (η) the cokernel coker(f) exists. We set C(X, f) = coker(f) and if a : (X, f) → (Y, g) is a morphism in D_{F} (η), then C(a) = the unique morphism in D with a ∘ coker(g) = coker(f) ∘ C(a). It is trivial to see that (C, Z) is an adjoint pair with counit the natural isomorphism CZ ≅ Id_{D} and unit the morphism coker(f) : (X, f) → ZC(X, f). Conversely if λ : Id_{D_{F} (η)} → ZC is the unit of the adjoint pair (C, Z), then obviously ∀(X, f) ∈ D_{F} (η), U(λ_{(X,f)}) = coker(f). The equivalence (b) ⇔ (c) follows from Proposition 2.3. Finally we show that: CΩ = C. From the definition of C in the Proposition 2.3 it suffices to show that: coker(μU ∘ Uε) = coker(Uε − UεZU). Since ^{t}(δU, μU) is a natural isomorphism we have: coker(Uε − UεZU) = coker[^{t}(δU, μU) ∘(Uε − UεZU)] = coker[^{t}(δU∘Uε−δU∘γU, μU∘Uε−μU∘γU)] = coker[^{t}(0, μU∘Uε)] = coker(μU∘Uε). Thus CΩ = C and then by proposition 2.3 we have CT = Id_{D}.

Remark 2.7. (1) Suppose that C is a cleft extension of the additive category D. By Proposition 2.3 it follows that Z is tripleable if D or C has cokernels. In this case we have an equivalence: C^{Z} ≈ D where C^{Z} is the Eilenberg-Moore category of the (idempotent) triple Z generated by the adjoint pair (C, Z).

(2) Let D be an additive category and G : D → D an additive functor. If ζ : G → G^{2} is a coassociative natural morphism, in the sense: ζ ∘ ζG = ζ ∘ Gζ, then we can define the ζ-coextension category D^{G}(ζ) [29] of D by the endofunctor G, as the category with objects (X, f) where f : X → GX is a D-morphism with f ∘ Gf = f ∘ ζ_{X}; a morphism a : (X, f) → (Y, g) in D^{G}(ζ) is a D-morphism a : X → Y with a ∘ g = f ∘ Ga. By the dual of Theorem 2.6 we have:

• An additive category C with kernels of split epics is a cleft coextension of D

iff there exists an equivalence of categories Σ : C ⟶^{≈} D^{G}(ζ). If C is a cleft extension of D through the functors T, U, Z, then:

• C is a cleft coextension of D via the functors U, Z iff the functor U has a right

adjoint H iff the functor F constructed in Theorem 2.6 has a right adjoint G. This follows from Theorem 2.6 and the fact that the tripleable functor U is cotripleable iff U has a right adjoint, see [7].

2.2. Preservation Properties. From now on assume that C is a cleft extension of the additive category D with cleaving data {T, U, Z, δ, ε} and we suppose always that D is closed under kernels of split epics. We denote by F the endofunctor of D constructed in Theorem 2.6 and we recall that by construction UT = Id_{D} ⊕ F.

Proposition 2.8. Let C be a cleft extension of the additive category D. (i) C is (finitely) complete ⇔ D is (finitely) complete. (ii) If C is (finitely) cocomplete then so is D. If D is (finitely) cocomplete and the

functor U, resp. F, preserves cokernels, then C is (finitely) cocomplete. (iii) The functor Z reflects limits, colimits and preserves limits. Also Z preserves

colimits of finite diagrams, and if C is a C_{2}-category [33] and the functor U, resp. F, preserves cokernels, then Z preserves colimits.

Proof. (i) If D is complete then so is C, since by [7] any tripleable functor reflects completeness. If C is complete, then let a : X → Y be a morphism in D and let k : A → Z(X) be the kernel of Z(a). Since U preserves kernels as a right adjoint and since UZ ≅ Id_{D}, we have U(k) = ker(a), so D has kernels. In the same way D has products and then D is complete.

(ii) Suppose that C is cocomplete. Since C has cokernels by Proposition 2.3 we have that Z is tripleable. Then D is cocomplete since by remark 2.7, D is the Eilenberg-Moore category of an idempotent triple defined on the cocomplete category C, see [39]. If D is cocomplete then by a well known theorem of Linton [39], it suffices to show that C has cokernels. By Theorem 2.6 we can assume that C ≈ D_{F} (η) for an endofunctor F of D with UT = Id_{D} ⊕ F, and we note that F is cokernel preserving by the hypothesis. Let a : (X, f) → (Y, g) be a morphism in D_{F} (η), and let m : Y → W be the cokernel of a = U(a). Since F m = coker(F a), there exists a unique morphism h : F W → W with: g ∘ m = F m ∘ h. One can check easily that (W, h) ∈ D_{F} (η) and that m : (Y, g) → (W, h) is the cokernel of m in D_{F} (η). Hence C ≈ D_{F} (η) has cokernels. We construct explicitly the coproduct in D_{F} (η): if (X_{i}, f_{i})_{i∈I} is a family of objects in D_{F} (η), then consider the coproduct ⊕X_{i} in D with injections ı_{i}, and the coproduct ⊕F X_{i} in D with injections ȷ_{i}. Next let ω : ⊕F X_{i} → F ⊕ X_{i} be the unique morphism with ȷ_{i} ∘ ω = F ı_{i}. One can check

easily that the morphism ω̂ = ( \begin{matrix}⊕f_{i} & −ω \\ 0 & F ⊕ f_{i} − F ω ∘ η_{⊕X_{i}} \end{matrix} ) : T(⊕F X_{i}) → T(⊕X_{i})

is a morphism in D_{F} (η). Suppose that ^{t}(a, b) : T(⊕X_{i}) → (X, f) is the cokernel of ω̂ in D_{F} (η). We leave to the reader to check that (X, f) = ⊕(X_{i}, f_{i}), with injections the morphisms ı_{i} ∘ a : (X_{i}, f_{i}) → (X, f).

(iii) Z reflects limits and colimits as a fully faithful functor. Using the relation UZ ≅ Id_{D} and the fact that U is faithful and preserves limits as a right adjoint, it follows that Z preserves limits. Also Z preserves finite colimits, since UZ ≅ Id_{D}. If C is a C_{2}-category and U (or F) preserves cokernels, then setting f_{i} = 0, ∀i ∈ I in (ii) we see that the unique morphism ξ : ⊕Z(X_{i}) → Z(⊕X_{i}) with the property ı_{i} ∘ a ∘ ξ = Z(ı_{i}) satisfies coker(ω̂) ∘ ξ = ε_{Z(⊕X_{i})}. Hence ξ is an epimorphism. Since C is a C_{2}-category, ξ is monic. Thus ξ is invertible and Z preserves colimits.

Corollary 2.9. Let C be a cleft extension of the category D.

(i) The functor F preserves cokernels (colimits) iff the functor U preserves cokernels (colimits).

(ii) If the category D is abelian (Grothendieck), and one of the functors U, F

preserves cokernels (colimits), then the category C is abelian (Grothendieck). (iii) If D is Grothendieck and U or F preserves colimits, then U and F have right

adjoints, U is cotripleable and C is a cleft coextension of D. (iv) If C = D_{F} (η) is abelian, then D is abelian ⇔ ∀(X, f) ∈ D_{F} (η) the cokernel

coker(f) exists. If C is a Grothendieck category and the functor F or the functor U preserves cokernels, then D is Grothendieck ⇔ ∀(X, f) ∈ D_{F} (η) the cokernel coker(f) exists.

Proof. (i) If the functor U preserves cokernels (colimits) then since UT = Id_{D} ⊕ F the same is true for the functor F. If F preserves cokernels, then U preserves cokernels by Proposition 2.8. If F preserves colimits, then the morphism ω in Proposition 2.8 is invertible. Then we see easily that (⊕X_{i}, ω^{−1}∘⊕f_{i}) = ⊕(X_{i}, f_{i}) is the cokernel of ω̂, via the morphism ^{t}(1_{⊕X_{i}}, ω^{−1} ∘ ⊕f_{i}). Thus U preserves coproducts.

(ii) Since C ≈ D^{T}, if one of the functors U, F is cokernel preserving, then the functor part UT of the triple T is cokernel preserving. Hence if D is abelian, then by [12] so is C. Suppose now that D is a Grothendieck category and U or F preserves colimits. By Proposition 2.8, C is a cocomplete abelian category. Since T admits a faithful right adjoint, it follows trivially that T preserves generators. Hence C admits a generator. Now the direct limit functor is exact on C since it is exact on D, and U preserves direct limits by hypothesis, and preserves and reflects short exact sequences as a faithful exact functor. Hence C is Grothendieck category.

(iii) Since by (ii), C is Grothendieck, it follows by [33] that U has a right adjoint and then U is cotripleable by [7]. So C is a cleft coextension of D.

(iv) Suppose that C = D_{F} (η) is abelian. If D is abelian then there is nothing to prove. If ∀(X, f) ∈ D_{F} (η) the cokernel coker(f) exists, then by Theorem 2.6 we have an adjoint pair (C, Z), and an equivalence C^{C} ≈ D where C^{C} is the Eilenberg-Moore category of the (idempotent) triple C. Since the functor part ZC of the triple C preserves cokernels we have that D is abelian by [12]. Finally assume that C is a Grothendieck category and F or U preserves cokernels. If ∀(X, f) ∈ D_{F} (η) the cokernel coker(f) exists, then as above D is abelian and by Proposition 2.8, D is cocomplete. Since C admits a faithful right afjoint, it follows trivially that C preserves generators. Hence D admits a generator. Finally the direct limit functor is exact in D since it is exact in C and Z reflects short exact sequences. We conclude that D is a Grothendieck category.

Combining the above results, we have the following characterization of abelian categories occuring as cleft extensions of a given abelian category.

Theorem 2.10. If C, D are abelian categories, then the following are equivalent:

(i) There exists an adjoint pair (T, U) of functors, where T : D → C, U : C → D, and a functor Z : D → C such that: (α) U is faithful exact. (β) UZ ≅ Id_{D}.

(ii) There exists an equivalence Ω : C ⟶^{≈} D_{F} (η) for an η-extension of D by a right

exact endofunctor F of D, such that: ΩZ ≅ Z, UΩ ≅ U. (iii) C is a cleft extension of D and the projection functor U is exact. If one of the above statements is true, then Z is a full exact imbedding which admits a left adjoint C with CZ = Id_{D}, CT = Id_{D}, CΩ ≅ C and it holds: ΩT ≅ T.

Proof. Follows directly from our previous results and the fact that, by the Weak Tripleability Theorem [28], a functor U with a left adjoint is tripleable, if it is exact and faithful.

Remark 2.11. Suppose that D is an exact category [37], and that the functor F is right exact with respect to the given exact structure F of D. Consider in C = D_{F} (η) the class of diagrams E = U^{−1}(F), i.e. a diagram (X, f) → (Y, g) → (W, h) belongs to E iff X → Y → W belongs to F. Trivially the class of diagrams E defines in D_{F} (η) an exact structure, such that the functors U, Z are exact. In particular if F is the minimal exact structure of split short exact sequences in D, then D_{F} (η) is always equipped with the exact structure of all short exact sequences which are split when considered, via U, as sequences in D.

2.3. Triples without Unit and Cleft Extensions. We close this section by studying the connection between the cleft extensions of a given category D and the triples defined on D. The information data defining an η-extension D_{F} (η) of a category D is the defining data of a triple on D except for the existence of a unit. So we’re led to the following definition:

Definition 2.12. A triple without unit (D, η, F) on a category D consists of an endofunctor F : D → D and an associative natural morphism η : F^{2} → F. A morphism ω : (D, η, F) → (D, θ, G) between triples without unit on D, is a natural morphism ω : F → G such that: η ∘ ω = F ω ∘ ωG ∘ θ.

The triples without unit on D constitute a category T_{−1}(D). If T (D) is the category of triples on D, then T (D) is a subcategory T_{−1}(D) of course not full. We denote by I : T (D) ↪ T_{−1}(D) the inclusion functor. If (D, η, μ, F) is a triple on D, then η is the multiplication and μ is the unit. On the other hand for an additive category D we define the category CE(D) of cleft extensions of D as follows: an object of CE(D) is a cleft extension C(T, U, Z) of D with cleaving data {T, U, Z}. A morphism 𝔸 : C_{1}(T_{1}, U_{1}, Z_{1}) → C_{2}(T_{2}, U_{2}, Z_{2}) is a functor 𝔸 : C_{1} → C_{2} such that U_{2}𝔸 = U_{1} and 𝔸Z_{1} = Z_{2}. Finally let EM(D) be the category of Eilenberg-Moore categories of triples defined on D, with morphisms functors A : D^{T_{1}} → D^{T_{2}} such that: AU_{2} = U_{1}, where U_{2}, U_{1} are the forgetful functors, see [7].

Proposition 2.13. Let D be an additive category. Then T (D) is a reflective subcategory of T_{−1}(D), and denoting by L the left adjoint of the inclusion functor I, we have: L(D, η, μ, F) ≅ (D, 1_{Id_{D}}, Id_{D}, 1_{Id_{D}}) ∏ (D, η, μ, F).

(i) If (D, η, F) ∈ T_{−1}(D), then: D_{F} (η) ≅ D^{L(D,η,F)}, where the latter is the Eilenberg-Moore category of the triple L(D, η, F).

(ii) If D has cokernels and F preserves them or if D_{F} (η) has cokernels, then for any triple T = (D, η, μ, F) ∈ T (D), the Eilenberg-Moore category D^{T} is a full reflective subcategory of D_{F} (η).

(iii) There are equivalences: EM(D)^{op} ≈ T (D), CE(D)^{op} ≈ T_{−1}(D) embedded in a commutative diagram:

T (D) ⟶^{≈} EM(D)^{op}
↓  ↓
T_{−1}(D) ⟶^{≈} CE(D)^{op} .

Proof. Define a functor L : T_{−1}(D) → T (D) as follows: if (D, η, F) ∈ T_{−1}(D), then L(D, η, F) = (D, η^{∗}, μ^{∗}, Id_{D} ⊕ F) where μ^{∗} = (1_{Id_{D}}, 0) : Id_{D} → Id_{D} ⊕ F,

η^{∗} = ( \begin{matrix}1_{Id_{D}} & 0 \\ 0 & 1_{F} \\ 0 & 1_{F} \\ 0 & η \end{matrix} ) : Id_{D} ⊕ F ⊕ F ⊕ F^{2} ⟶ Id_{D} ⊕ F .

We define L(ω) = ( \begin{matrix}1_{Id_{D}} & 0 \\ 0 & ω \end{matrix} ), for a morphism ω : (D, η_{1}, F_{1}) → (D, η_{2}, F_{2}) in

T_{−1}(D). It easy to check that L is a well defined functor. Let α : Id_{T_{−1}(D)} → IL be the natural morphism defined by α_{(D,η,F)} = (0, 1_{F}). Let β : LI → Id_{T (D)} be the natural morphism defined by β_{(D,η,μ,F)} = ^{t}(μ, 1_{F}). Trivially α is the unit and β is the counit of an adjunction (L, I). Hence T (D) is a reflective subcategory of T_{−1}(D). Let now (D, η, μ, F) ∈ T (D), and denote by I_{D} the identity triple on D. Consider the triple morphisms π_{1} = ^{t}(1_{Id_{D}}, 0) : LI(D, η, μ, F) → I_{D} and π_{2} = ^{t}(μ, 1_{F}) : LI(D, η, μ, F) → (D, η, μ, F). If χ : (D, η^{!}, μ^{!}, F^{!}) → I_{D}, ψ : (D, η^{!}, μ^{!}, F^{!}) → (D, η, μ, F) are triple morphisms, consider the natural morphism ϕ := (χ, ψ − χ ∘ μ) : F^{!} → Id_{D} ⊕ F. A boring calculation shows that ϕ := (χ, ψ − χ ∘ μ) : (D, η^{!}, μ^{!}, F^{!}) → LI(D, η, μ, F) is a morphism of triples and in addition we have: ϕ∘π_{1} = (χ, ψ −χ∘μ)∘^{t}(1_{Id_{D}}, 0) = χ, ϕ∘π_{2} = (χ, ψ −χ∘ μ)∘^{t}(μ, 1_{F}) = ψ. Trivially ϕ is the unique triple-morphism with: ϕ ∘ π_{1} = χ and ϕ ∘ π_{2} = ψ, so L(D, η, μ, F) is the product in T (D) of the identity triple I_{D} and the triple (D, η, μ, F).

(i) Follows from the proof of Theorem 2.6.

(ii) If T = (D, η, μ, F) ∈ T (D), then by construction the category D^{T} is a full subcategory of D_{F} (η). If (X, f) ∈ D_{F} (η) then obviously (F X, η_{X}), (F^{2}X, η_{F X}) ∈ D_{F} (η), and η_{X} −F f : (F^{2}X, η_{F X}) → (F X, η_{X}) is a morphism in D_{F} (η). Now by hypothesis D_{F} (η) has cokernels, hence we can set R(X, f) = (Y, g) := Coker(η_{X} − F f), c_{X} := coker(η_{X} −F f) : (F X, η_{X}) → (Y, g). Then c_{X} ∘μ_{Y} ∘g = μ_{F X} ∘F c_{X} ∘g = μ_{F X} ∘ η_{X} ∘ c_{X} = c_{X} and we see easily that μ_{Y} ∘ g is an endomorphism of (Y, g), with μ_{Y} ∘ g = 1_{(Y,g)}. This shows that 1_{Y} = μ_{Y} ∘ g and R(X, f) ∈ D^{T}. Consider now the morphism r_{(X,f)} := μ_{X} ∘ c_{X} : (X, f) → R(X, f) which is a morphism in D_{F} (η), since: f ∘ μ_{X} ∘ c_{X} = μ_{F X} ∘ F f ∘ c_{X} = μ_{F X} ∘ η_{X} ∘ c_{X} = c_{X}, and F μ_{X} ∘ F c_{X} ∘ g = F μ_{X} ∘ η_{X} ∘ c_{X} = c_{X}. We claim that r_{(X,f)} is the reflection of (X, f) on D^{T}. Indeed let (B, ξ) ∈ D^{T} and a : (X, f) → (B, ξ) be a morphism of D^{T}. Since (η_{X} − F f) ∘ f = 0 there exists a unique morphism ω : R(X, f) → (X, f) with c_{X} ∘ ω = f. Then r_{(X,f)} ∘ ω ∘ a = μ_{X} ∘ c_{X} ∘ ω ∘ a = μ_{X} ∘ f ∘ a = μ_{X} ∘ F a ∘ ξ = a ∘ μ_{B} ∘ ξ = a. If β : R(X, f) → (B, ξ) is another morphism with r_{(X,f)} ∘ β = a, then : f ∘ μ_{X} ∘ c_{X} ∘ β = f ∘ a = F a ∘ ξ. But: f ∘ μ_{X} ∘ c_{X} ∘ β = μ_{F X} ∘ F f ∘ c_{X} ∘ β = μ_{F X} ∘ η_{X} ∘ c_{X} ∘ β = c_{X} ∘ β, hence F a ∘ ξ = c_{X} ∘ β. Then: c_{X} ∘ (β − ω ∘ a) = c_{X} ∘ β − c_{X} ∘ ω ∘ a = c_{X} ∘ β − f ∘ a = F a ∘ ξ − f ∘ a = 0 and since c_{X} is epic we have β = ω ∘ a. This shows that r_{(X,f)} is the reflection of (X, f) in D^{T}, and D^{T} is a reflective subcategory of D_{F} (η).

(iii) The functor Ξ : T_{−1}(D) → CE(D)^{op} defined by Ξ(D, η, F) = D_{F} (η), and for a morphism α : (D, η, F) → (D, \overline{η}, \overline{F}) in T_{−1}(D), Ξ(α) := the functor A : D\overline{F} (\overline{η}) → D_{F} (η) defined by A(X, f) = (X, α_{X} ∘ f), and A(a) = a for a morphism a in D\overline{F} (\overline{η}) is a well-defined functor from T_{−1}(D) to CE(D)^{op}. Suppose now that A : D\overline{F} (\overline{η}) → D_{F} (η) is a morphism in CE(D). Since UA = \overline{U}, ∀X ∈ D : A\overline{T} (X) = (X ⊕ \overline{F} X, ( \begin{matrix}γ_{X} & α_{X} \\ β_{X} & δ_{X} \end{matrix} )) ∈ D_{F} (η). Consider the morphism \overline{ε}_{\overline{Z}(X)} : \overline{T} (X) → \overline{Z}(X).

Then since A\overline{Z} = Z, A(\overline{ε}_{\overline{Z}(X)}) = ^{t}(1_{X}, 0) : A\overline{T} (X) → Z(X) is a morphism in

D_{F} (η), hence ( \begin{matrix}γ_{X} & α_{X} \\ β_{X} & δ_{X} \end{matrix} ) ∘^{t} (1_{X}, 0) = 0 ⇒ γ_{X}, β_{X} = 0, and η_{X} ∘ α_{X} = F α_{X} ∘

δ_{X}, η_{\overline{F} X} ∘ δ_{X} = F δ_{X} ∘ δ_{X}. Trivially α : F → \overline{F} and δ : F \overline{F} → \overline{F} are natural with η ∘ α = F α ∘ δ, η\overline{F} ∘ δ = F δ ∘ δ. Next ∀(X, f) ∈ D\overline{F} (\overline{η}), consider the morphism

\overline{ϑ}_{(X,f)}^{1} = ( \begin{matrix}−f & 1_{\overline{F} X} \\ 0 & \overline{η}_{X} − F f \end{matrix} ) : \overline{T} \overline{F} (X) ⟶ \overline{T} (X) and the induced exact sequence

\overline{T} \overline{F} (X) ⟶^{\overline{ϑ}_{(X,f)}^{1}} \overline{T} (X) ⟶^{\overline{ε}_{(X,f)}} (X, f) ⟶ 0. One can check easily that the sequence

A \overline{T} \overline{F} (X) ⟶^{A(\overline{ϑ}_{(X,f)}^{1})} A \overline{T} (X) ⟶^{A(\overline{ε}_{(X,f)})} A(X, f) ⟶ 0 is exact in D_{F} (η). This

means that α_{\overline{F} X} ∘ \overline{η}_{X} = δ_{X} and A(X, f) = (X, α_{X} ∘ f). We deduce that: η ∘ α = F α ∘ δ = F α ∘ α\overline{F} ∘ \overline{η}, thus α : (D, η, F) → (D, \overline{η}, \overline{F}) is a morphism of triples without unit. Define a functor Θ : CE(D)^{op} → T_{−1}(D) as follows: if C = D_{F} (η), then Θ(D_{F} (η)) = (D, η, F), and if A : D\overline{F} (\overline{η}) → D_{F} (η) is a morphism in CE(D) then Θ(A):= the (unique) natural morphism α : F → \overline{F}. It is easy to see that Θ is an inverse of Ξ. The restriction of Ξ to the subcategory T (D) induces a functor Ξ : T (D) → EM(D)^{op}, which is an equivalence of categories, with proof as above (see also [7]). The last assertion is left to the reader.

3. Cleft Extensions of Rings and Categories of Modules

Throughout we fix an associative ring Γ and let _{Γ}M_{Γ} be a Γ-Γ-bimodule. A Γ-Γ-bimodule morphism ϑ : M ⊗_{Γ} M → M is called associative iff the following diagram commutes:

\begin{matrix}M ⊗_{Γ} M ⊗_{Γ} M & ⟶^{ϑ⊗1_{M}} & M ⊗_{Γ} M \\ ^{1_{M} ⊗ϑ}↓ & & ^{ϑ}↓ \\ M ⊗_{Γ} M & ⟶^{ϑ} & M \end{matrix} i.e. (ϑ ⊗ 1_{M}) ∘ ϑ = (1_{M} ⊗ ϑ) ∘ ϑ.

If ϑ is an associative Γ-Γ-bimodule homomorphism, Marmaridis in [29] introduced the notion of the ϑ−extension ring Γ ⋉_{ϑ}M of the ring Γ by the Γ-Γ-bimodule M as follows: as an abelian group Γ ⋉_{ϑ}M = Γ ⊕ M. If (r_{1}, m_{1}), (r_{2}, m_{2}) ∈ Γ ⋉_{ϑ}M, then the multiplication of the ring is defined by the formula:

(r_{1}, m_{1})(r_{2}, m_{2}) = (r_{1}r_{2}, r_{1}m_{2} + m_{1}r_{2} + ϑ(m_{1} ⊗ m_{2})).

If Γ ⋉_{ϑ} M is a ϑ−extension ring of Γ by M, then defining ε : Γ ⋉_{ϑ} M → Γ by ε(r, m) = r, we get a ring epimorphism which is split by the ring morphism μ : Γ → Γ ⋉_{ϑ}M defined by μ(r) = (r, 0). Hence the ring Γ ⋉_{ϑ}M is a cleft extension of Γ in the sense of the following definition, see also [27].

Definition 3.1. A cleft extension of Γ is a triple (Λ, ε, μ) consisting of ring morphisms ε : Λ ⟶ Γ (projection) and μ : Γ ⟶ Λ (cleaving), such that μ ∘ ε = Id_{Γ}.

The cleft extensions of Γ are the objects of a category CE(Γ). A morphism (Λ_{1}, ε_{1}, μ_{1}) ⟶ (Λ_{2}, ε_{2}, μ_{2}) in CE(Γ) consists of a ring morphism ρ : Λ_{1} ⟶ Λ_{2} such that ρ ∘ ε_{2} = ε_{1} and μ_{1} ∘ ρ = μ_{2}. Note that if (Λ, ε, μ) is a cleft extension of Γ then we have a split short exact sequence:

0 ⟶ Ker(ε) ⟶ Λ ⟶^{ε} Γ ⟶ 0 (1)

which defines a Γ-structure in the sense of Gerstenhaber [16], on the ideal Ker(ε) of Λ. Conversely any Γ-structure arises from a cleft extension of rings.

More presicely let ST R(Γ) be the category with objects the Γ-structures, i.e. the objects of ST R(Γ) are diagrams 0 ⟶ Ker(ϖ) ⟶ Λ ⟶^{ϖ} Γ ⟶ 0 where ϖ is a ring morphism, together with a fixed cleaving s : Γ → Λ of ϖ. A morphism from the Γ-structure 0 ⟶ Ker(ϖ) ⟶ Λ ⟶^{ϖ} Γ ⟶ 0 to the Γ−structure 0 ⟶ Ker(p) ⟶ Ξ ⟶^{p} Γ ⟶ 0 is a ring morphism f : Λ → Ξ such that f ∘ p = ϖ, and f also commutes with the (fixed) cleavings of p and ϖ. Let 𝕄od(Γ) be the full subcategory of ST R(Γ) consisting of all Γ-modules, i.e. Γ-structures as above with Ker(ε)^{2} = 0. Consider now the category MB(Γ) with objects the multiplicative Γ-Γ-bimodules, i.e. an object of MB(Γ) is of the form (M, ϑ) where M is a Γ-Γ-bimodule, and ϑ : M ⊗_{Γ} M → M is an associative Γ-Γ-bimodule morphism, see [36]. A morphism α : (M, ϑ) → (N, φ) in MB(Γ) is a Γ-Γ-bimodule morphism α : M → N such that ϑ ∘ α = α ⊗ α ∘ φ. We consider also the full subcategory T E(Γ) of CE(Γ) consisting of the trivial extensions of Γ, i.e. cleft ϑ−extensions Γ ⋉_{ϑ}M with ϑ = 0.

Proposition 3.2. The categories MB(Γ), CE(Γ), ST R(Γ) are equivalent. The categories Mod(Γ^{e}), T E(Γ), 𝕄od(Γ) are equivalent, where Γ^{e} = Γ ⊗_{ℤ} Γ^{op} is the enveloping ring of Γ. Further T E(Γ) is a reflective subcategory of CE(Γ).

Proof. Define a functor T : ST R(Γ) → MB(Γ) as follows : T takes the Γ-structure (1) to the Γ-Γ-bimodule M := Ker(ε). The Γ-Γ-bimodule structure of M is given by the rules: (r, m) ↦ μ(r)m, and (m, r) ↦ mμ(r), where μ : Γ → Λ is the fixed cleaving of ε. The bimodule M has an associative multiplication ϑ : M ⊗_{Γ} M → M given by: ∀m, n ∈ M : ϑ(m ⊗ n) = mn (multiplication inside Λ). Now define a functor S : MB(Γ) → ST R(Γ) as follows: S(M, ϑ) := the Γ-structure defined by the ϑ-extension ring Γ ⋉_{ϑ} M. It is easily seen that the functors T , S are inverse equivalences. On the other hand it is clear that the category MB(Γ) is equivalent to the category CE(Γ) of cleft extensions of Γ. Hence we have equivalences MB(Γ) ≈ CE(Γ) ≈ ST R(Γ). Obviously the above equivalences induce equivalences Mod(Γ^{e}) ≈ T E(Γ) ≈ 𝕄od(Γ). Finally we observe that T E(Γ) is a reflective subcategory of CE(Γ), with reflection f : Γ ⋉_{ϑ}M → Γ ⋉\overline{M}, where \overline{M} = coker(ϑ), and f(r, m) = (r, \overline{m}).

Now fix a cleft extension (Λ, ε, μ) of Γ. It is useful to have a canonical representation of Λ as a ϑ−extension ring. We denote by M the ideal Ker(ε) of Λ, and define actions of Γ on M by: (r, m) ↦ μ(r)m, and (m, r) ↦ mμ(r). These actions induce on M a Γ-Γ-bimodule structure. From the direct sum decomposition Λ = Γ ⊕ M in Mod(Γ^{e}), induced by μ, ϑ, we can define a morphism ϑ : M ⊗_{Γ} M → M as follows: ∀m, n ∈ M : ϑ(m ⊗ n) = (0, m)(0, n) and then extend Γ-Γ-bilinearly. Hence we can define the ϑ-extension ring Γ ⋉_{ϑ}M of Γ by M. It easy to see that we have a ring isomorphism Λ ≅ Γ ⋉_{ϑ}M, which from now on we view as an identification.

If Λ = Γ ⋉_{ϑ} M is a cleft extension of rings, then we can identify the module category Mod(Λ) as a cleft extension of Mod(Γ) as follows, see [29]: set F = − ⊗_{Γ} M, η : F^{2} → F where η_{X} = 1_{X} ⊗ ϑ. Then η is an associative natural morphism, so we can consider the category Mod(Γ)_{F} (η). Define a functor A : Mod(Λ) → Mod(Γ)_{F} (η) by A(A_{Λ}) = (A_{Γ}, f), where f : A ⊗_{Γ} M → A_{Γ} is defined by f(a ⊗ m) = a(0, m). Then A is an isomorphism of categories with inverse the functor B : Mod(Γ)_{F} (η) → Mod(Λ), where B(X_{Γ}, f) = X_{Λ} with action x(r, m) = xr + f(x ⊗ m). From now on we view the isomorphisms A, B as identifications.

The following proposition is a generalization of a result of [30].

Proposition 3.3. Suppose that C is a cleft extension of the additive category D, through the functors T, U, Z. For X ∈ D, set Γ := End_{D}(X), Λ := End_{C}(T(X)). Then Λ is a cleft extension of Γ: there exists a Γ-Γ-bimodule M, an associative Γ-Γ-bimodule morphism ϑ : M ⊗_{Γ}M → M, and an isomorphism of rings Λ ≅ Γ⋉_{ϑ}M. If F_{X} := − ⊗_{Γ} M : Mod(Γ) → Mod(Γ), then defining η^{X} : F_{X}^{2} → F_{X} by η_{Y}^{X} := 1_{Y} ⊗ ϑ we get an associative natural morphism η^{X} and isomorphisms of categories:

Mod(Λ) ≅ Mod(Γ ⋉_{ϑ} M) ≅ Mod(Γ)_{F_{X}} (η^{X}).

Proof. By Theorem 2.6 we can suppose that C = D_{F} (η). It is easy to see that

any element of Λ := End_{D_{F} (η)}(T(X)) is of the form ( \begin{matrix}f & m \\ 0 & F f + F m ∘ η_{X} \end{matrix} ), where

f ∈ Γ = End_{D}(X), m ∈ D(X, F X). Using the adjoint pair (T, U) we have an isomorphism in Ab : Λ ≅ Γ ⊕ D(X, F X). Set M := D(X, F X) and define actions: ∀f ∈ Γ, ∀a, b ∈ M : (f, a) ↦ f ∘ a, (b, f) ↦ b ∘ F f. With these actions M is a Γ-Γ-bimodule. Consider the following morphism

ϑ : M ⊗_{Γ} M → M, ϑ(a ⊗ b) = a ∘ F b ∘ η_{X}.

Since η is an associative natural morphism it follows easily that ϑ is an associative Γ-Γ-bimodule morphism. Consider the cleft extension ring Γ ⋉_{ϑ} M, and define an isomorphism ω : Γ ⋉_{ϑ} M → Λ in Ab as follows:

ω(f, m) = ( \begin{matrix}f & m \\ 0 & F f + F m ∘ η_{X} \end{matrix} ), where f ∈ Γ, m ∈ M.

Using the associativity of η and the Γ-Γ-bimodule structure of M as defined above we have, ∀(f, m), (g, n) ∈ Γ ⋉_{ϑ} M:

ω[(f, m)(g, n)] =

= ( \begin{matrix}f ∘ g & f ∘ n + m ∘ F g + ϑ(m ⊗ n) \\ 0 & F f ∘ f ∘ g + F f ∘ F n ∘ η_{X} + F m ∘ F^{2}g ∘ η_{X} + F ϑ(m ⊗ n) ∘ η_{X} \end{matrix} ) =

= ( \begin{matrix}f ∘ g & f ∘ n + m ∘ F g + m ∘ F n ∘ η_{X} \\ 0 & F f ∘ f ∘ g + F f ∘ F n ∘ η_{X} + F m ∘ η_{X} ∘ g + F m ∘ η_{X} ∘ F n ∘ η_{X} \end{matrix} ) =

= ω(f, m)ω(g, n).

Hence ω is an isomorphism of rings, and Λ is a cleft extension of Γ. The rest follows from the remarks preceding the proposition.

Now we can characterize the cleft extensions of module categories.

Proposition 3.4. Suppose that the abelian category C is a cleft extension of the abelian category D through the functors T, U, Z, and suppose that the functor U preserves cokernels. Then the following are equivalent:

(i) C is a module category. (ii) D is a module category and the functor U preserves coproducts. Moreover if one of the above equivalent statements hold and D ≈ Mod(Γ), then C ≈ Mod(Λ), where Λ is a cleft (ϑ−)extension of Γ.

Proof. First note that from Proposition 2.8, C is cocomplete iff D is cocomplete. Next since the right adjoints U, Z of the functors T, C are exact and faithful, we have that the functors T, C preserve projective objects and generators.

(ii) ⇒ (i) If D is a module category then D admits a small projective generator P. Then T(P) is a projective generator, and since U preserves coproducts from the adjointness (T, U) we infer that T(P) is small. It follows that C is the module category Mod(End_{D}(T(P))).

(i) ⇒ (ii) If C is a module category then C admits a small projective generator Q. Then C(Q) is a projective generator. Since by Proposition 2.8 the functor Z preserves coproducts, and since C(Q) is small, we have :

D(C(Q), ⊕_{i∈I}X_{i}) ≅ C(Q, Z(⊕_{i∈I}X_{i})) ≅ C(Q, ⊕_{i∈I}Z(X_{i})) ≅

≅ ⊕_{i∈I}C(Q, Z(X_{i})) ≅ ⊕_{i∈I}D(C(Q), X_{i}).

Hence D is the module category Mod(Γ), where Γ = End_{D}(C(Q)). Suppose now that C is a module category, and let C ≈ Mod(Λ), where Λ = End_{C}(Q) and Q is a small projective generator of C. Obviously we can suppose that Q is of the form T(P) for a small projective generator P in D. Then it is easy to see that U is isomorphic to the functor Hom_{Λ}(_{Γ}Λ_{Λ}, −), where Γ = End_{D}(P), so trivially U preserves coproducts. The rest follows from Proposition 3.3.

We now indicate briefly how the preceding theory applies to the dual situation. Let K be an additive monoidal category with kernels and a multiplication ⊗ : K × K → K which is a kernel preserving functor [28]. Suppose that Λ, Γ are coassociative coalgebras with unit over K, and let α : Λ → Γ be a coalgebra morphism which is split monic, see [24]. Since K has kernels and ⊗ is kernel preserving, we can construct the cotensor product functor − ⊙_{Λ} − : Comod(Λ) × Comod(Λ^{op}) → K. Then denoting by M = Coker(α), we see easily as in the case of rings that M ∈ Λ−Comod−Λ, and there exists a ⊙ − coassociative morphism φ : M → M ⊙_{Λ} M in Λ−Comod−Λ, i.e. φ ∘ φ ⊙ 1_{M} = φ ∘ 1_{M} ⊙ φ. Consider the functor G = − ⊙_{Λ} M : Comod−Λ → Comod−Λ, and define a coassociative natural morphism ζ : G → G^{2}, by ζ_{X} = 1_{X} ⊙ φ. Hence we can form the ζ-coextension category (Comod−Λ)^{ζ}(− ⊙_{Λ} M), and then with the above notation we have the following result, observing that by construction the functor G is kernel preserving.

Proposition 3.5. (Comod(Λ))^{ζ}(− ⊙_{Λ} M) ≅ Comod(Γ). The category Comod(Γ) is abelian iff the category Comod(Λ) is abelian. The last condition holds if K is abelian and the functor − ⊗ Λ is exact.

It follows that any split monomorphism of coalgebras over K, gives rise to a cleft coextension of the corresponding categories of comodules. Instances of this situation are the following:

(i) Suppose that K is abelian, ⊗ is kernel preserving, Λ is a coalgebra over K which is an injective K-object, and H^{2}(−, Λ) = 0 where the H^{∗}(−, Λ) is Jonah’s-Cohomology functor, see [24]. Then any coalgebra-extension Γ of Λ with conilpotent cokernel is a cleft extension of Λ, see [24].

(ii) By the splitting Theorem of Sullivan [1], any coalgebra Γ over a field k with the property that the subcoalgebra corad(Γ) is separable, (this happens for example if k is perfect), is a cleft extension of corad(Γ).

One can dualize the theory developed so far, as well as the theory of the remaining sections, to obtain the analogous results connecting the cleft coextensions and the category of comodules and/or contramodules over a cleft coalgebra extension. In a similar vein one can study cleft extensions of bialgebras, Hopf algebras, and so on.

4. Constructions and Examples of Cleft Extensions

In this section we present methods for constructing cleft extensions and we give a plethora of illustrating examples.

Fix a commutative ring k and let C be a k-algebroid in the sense of Mitchell [34], i.e. C is a small k-linear category. A morphism of k-algebroids is a k-linear functor. An extension over the k-algebroid D is a full k-linear functor ϖ : C → D which is the identity on objects. The above extension splits iff there exists a k-linear functor σ : D → C such that ϖσ = Id_{D}. For a k-algebroid C we denote by Mod(C^{op}) the category of additive covariant functors from C to the category of abelian groups Ab, and by Dim(D) we denote the Hochschild-dimension of D [10].

Proposition 4.1. (i) Let C, D be two k-algebroids. If C is a split extension of D, then Mod(C^{op}) is a cleft extension and coextension of Mod(D^{op}).

(ii) Suppose that D is a k-algebroid with Dim(D) ≤ 1. Then any extension ϖ : C → D of k-projective k-algebroids with a nilpotent kernel splits. Consequently Mod(C^{op}) is a cleft extension and coextension of Mod(D^{op}).

Proof. (i) Suppose that ϖ : C → D, σ : D → C are functors with the property that ϖσ = Id_{D}. Consider the restriction functors ϖ^{∗} : Mod(D^{op}) → Mod(C^{op}), σ^{∗} : Mod(C^{op}) → Mod(D^{op}). It is well known that the functors σ^{∗}, ϖ^{∗} are limit and colimit preserving. Note also that we have σ^{∗}ϖ^{∗} = Id_{Mod(D^{op})}. Since σ^{∗} has a left and a right adjoint given by the Kan construction, and since σ^{∗} is obviously faithful because σ is dense, see [39], we conclude that Mod(C^{op}) is a cleft extension and coextension of Mod(D^{op}), by Theorem 2.10.

(ii) Adapt the proof of Whitehead-Hochschild Theorem in [27] using [10].

As follows from the above Proposition if D is a k-projective algebroid, such that the Jacobson radical J (C) of C is nilpotent and the algebroid C/J (C) is k-separable [10], then Mod(C^{op}) is a cleft extension and coextension of Mod((C/J (C))^{op})). This observation can be regarded as a categorical version of the Principal Theorem of Wedderburn.

From now on D will denote an abelian category.

4.1. Free Categories and Representations of Quivers. Suppose F : D → D is a right exact functor. We define the free category D(F) over D with respect to the functor F, as follows: the objects of D(F) are of the form (X, f) where f : F X → X is a morphism of D, and a morphism a : (X, f) → (Y, g) in D(F) is a morphism a : X → Y of D, with: f ∘ a = F a ∘ g. We have the obvious functors U : D(F) → D, Z : D → D(F), defined by U(X, f) = X, U(a) = a , and Z(X) = (X, 0), Z(a) = a, and trivially it holds: UZ = Id_{D}. It is easy to see that D(F) is an abelian category, the functor Z is a full exact embedding and the functor U is faithful exact.

Lemma 4.2. Suppose that D has countable coproducts and the functor F preserves them. Then D(F) is a cleft extension of D. Any cleft extension D_{F} (η) of D appears as a full reflective subcategory of the free cleft extension D(F).

Proof. For the first part it suffices to show that the functor U has a left adjoint. This is a well known consequence of the fact that F preserves colimits and the easy proof is left to the reader. Obviously D_{F} (η) is a full subcategory of D(F). The reflection of D(F) in D_{F} (η) is obtained as follows: if (X, f) ∈ D(F) then one can check easily that setting X^{!} = Coker(η_{X} ∘ f − F f ∘ f), there exists a unique morphism f^{!} : F X^{!} → X^{!} such that (X^{!}, f^{!}) ∈ D_{F} (η). Then the morphism coker(η_{X} ∘f −F f ∘f) : (X, f) → (X^{!}, f^{!}) is a morphism in D(F) and is the reflection of D(F) in D_{F} (η).

By Proposition 3.4 and Watt’s Theorem we have that if D ≈ Mod(Γ) then F ≅ − ⊗_{Γ} M, and D(F) ≈ Mod(T_{Γ}(M)), where T_{Γ}(M) is the tensor ring of Γ by the Γ-Γ-bimodule M.

Generalizing the situation above we define the free category D<F_{i}, i ∈ I> over the abelian category D with respect to a family {F_{i}}_{i∈I} of right exact endofunctors of D, as follows: the objects of D<F_{i}, i ∈ I> are families (X, f_{i})_{I}, i ∈ I, where f_{i} : F_{i}X → X is a morphism in D, ∀i ∈ I, and a morphism a : (X, f_{i})_{I} → (Y, g_{i})_{I} in D<F_{i}, i ∈ I> is a morphism a : X → Y in D with : f_{i} ∘ a = F a ∘ g_{i}, ∀i ∈ I. Again D<F_{i}, i ∈ I> is an abelian category, and we have the faithful exact functor U : D<F_{i}, i ∈ I> → D, with U((X, f_{i})_{I}) = X, U(a) = a, as well as the full exact embedding Z : D ⟶ D<F_{i}, i ∈ I> where Z(X) = (X, f_{i})_{I}, f_{i} = 0, ∀i ∈ I, and Z(a) = a. Obviously UZ = Id_{D}. Letting J = max(ℕ, I), and supposing that D has coproducts indexed by J which are preserved by all the functors F_{i}, we get as in the preceding lemma that D<F_{i}, i ∈ I> is a cleft extension of D. In particular if F_{i} = Id_{D}, ∀i ∈ I, then D<F_{i}, i ∈ I> := D<I> is the free category over D in |I| variables. In case D ≈ Mod(Γ), we have D<F_{i}, i ∈ I> ≈ Mod(T_{Γ}(M)), where T_{Γ}(M) is the tensor ring of Γ by the unique Γ-Γ-bimodule M, with F ≅ − ⊗_{Γ} M, where F = ⊕_{i∈I}F_{i}. In particular if F_{i} = Id_{D}, ∀i ∈ I, then D<I> ≈ Mod(Γ<I>) is the module category of the free Γ-ring in |I| variables.

A special case of free categories are the representations of quivers [4]. Let Q be a quiver with |Q| = I, and let D be a cocomplete additive category. Define the category Rep_{D}(Q) of representations of Q in D to be the free cleft extension D(F), where F : D^{(I)} → D^{(I)} is the functor F (X_{i})_{i∈I} = (Y_{j})_{j∈I}, with (Y_{j})_{j∈I} = ⊕_{i∈J_{j}}X_{i} and J_{j} is the set of direct predecessors of the vertex j. F is defined in an obvious way on morphisms. Since D is cocomplete, by the above lemma, Rep_{D}(Q) is a cleft extension of D. This definition includes as a special case the category of representations of a finite quiver over a field. For example if Q is the quiver

E_{6} : •1 → •2 →•6_{↓} ; •3← •4 ← •5

and kQ is the quiver-algebra of Q over a field k then Mod(kE_{6}) ≈ Mod(k)^{6}(F), where F (X_{1}, ..., X_{6}) = (0, X_{1}, X_{2} ⊕ X_{6} ⊕ X_{4}, X_{5}, 0).

4.2. Symmetric and Polynomial Categories. Suppose now that {F_{i}}_{i=1,...,n} is a set of commuting right exact endofunctors of D: F_{i}F_{j} = F_{j}F_{i}, ∀i, j = 1, ..., n. Define the symmetric category D[F_{1}, F_{2}, ..., F_{n}] of D with respect to the set {F_{i}}_{i=1,...,n}, inductively as follows: first set D[F_{1}] = D(F_{1}). Since F_{1}F_{2} = F_{2}F_{1}, the functor F_{2} induces a right exact endofunctor F_{2}^{!} of D[F_{1}] by: F_{2}^{!}(X, f_{1}) = (F_{2}(X), F_{2}(f_{1})), F_{2}^{!}(a) = F_{2}(a). Define D[F_{1}, F_{2}] := D[F_{1}](F_{2}), and inductively define D[F_{1}, F_{2}, ..., F_{n}] = D[F_{1}, F_{2}, ..., F_{n−1}](F_{n}). It is obvious that the definition of D[F_{1}, F_{2}, ..., F_{n}] is independent of the ordering of the construction. Suppose that D has countable coproducts and each F_{i} preserves them. Then each D[F_{1}, F_{2}, ..., F_{i}] is a cleft extension of D[F_{1}, F_{2}, ..., F_{i−1}], say by the functors T_{i}, U_{i}, Z_{i}. Since D[F_{1}] = D(F_{1}) is a cleft extension of D say by the functors T_{1}, U_{1}, Z_{1}, setting T := T_{i}T_{i−1} ⋯ T_{1}, U := U_{1}U_{2} ⋯ U_{i}, Z := Z_{i}Z_{i−1} ⋯ Z_{1} we get that each D[F_{1}, F_{2}, ..., F_{i}] is a cleft extension of D.

In particular if F_{i} = Id_{D}, ∀i = 1, ..., n then D[F_{1}, F_{2}, ..., F_{n}] := D[n] is the polynomial category of D in n variables, and it is easy to see that a typical object of D[n] consists of n-endomorphisms f_{i} : X → X, X ∈ D with f_{i} ∘ f_{j} = f_{j} ∘ f_{i}. Of course if D = Mod(Γ), then D[n] = Mod(Γ[x_{1}, x_{2}, ..., x_{n}]).

4.3. Exterior Categories. If F : D → D is a right exact endofunctor then we can always form the trivial (cleft) extension D ⋉ F = D_{F} (0). This class of cleft extensions includes the category of morphisms of a given category, or more generally the various comma-categories defined by an additive functor, see [15].

If {F_{i}}_{i=1}^{n} is a set of commuting right exact endofunctors of D: F_{i}F_{j} = F_{j}F_{i}, ∀i, j = 1, ..., n, then define the exterior category ⋀(D)(F_{1}, F_{2}, ..., F_{n}) of D with respect to the set of endofunctors {F_{i}}_{i=1}^{n}, inductively as follows: ⋀(D)(F_{1}) = D ⋉ F_{1}, ⋀(D)(F_{1}, F_{2}) := ⋀(D)(F_{1}) ⋉ F_{2}^{⋆}, where the endofunctor F_{2}^{⋆} of ⋀(D)(F_{1}) is defined by: F_{2}^{⋆}(X, f_{1}) = (F_{2}(X), −F_{2}(f_{1})), F_{2}^{⋆}(a) = F_{2}(a), and so on. Then each ⋀(D)(F_{1}, F_{2}, ..., F_{i}) is a cleft (trivial) extension of ⋀(D)(F_{1}, F_{2}, ..., F_{i−1}), i = 1, ..., n. As in the symmetric case this sequence of cleft extensions allows us to consider the category ⋀(D)(F_{1}, F_{2}, ..., F_{n}) as a cleft extension of D. In case F_{i} = Id_{D}, ∀i = 1, ..., n, the category ⋀(D)(F_{1}, F_{2}, ..., F_{n}) := ⋀^{n}(D) is the exterior category of D in n variables, and a typical object of ⋀^{n}(D) consists of n endomorphisms f_{i} : X → X, X ∈ D, with f_{i}^{2} = 0, f_{i} ∘ f_{j} + f_{j} ∘ f_{i} = 0. If D ≈ Mod(Γ), then ⋀^{n}(D) ≈ Mod(⋀_{Γ}^{n}(Γ)) is the module category of the exterior Γ−ring of Γ. Note that the categories D<I>, D[n], ⋀^{n}(D) were first considered in [33].

4.4. Truncated and Idempotent Categories. We recall from [31] the definition of the t−truncated extension D ⋉_{t} F of the abelian category D, where t ≥ 0. The category D ⋉_{t}F is the full subcategory of the free cleft extension D(F) with objects (X, f) where: F^{t+1}f ∘ F^{t}f ∘ ⋯ ∘ F^{2}f ∘ F f ∘ f = 0. Denoting by U_{t} the restriction of the functor U : D(F) → D to D ⋉_{t}F, we have from [31] that U_{t} has a left adjoint T_{t} : D → D ⋉_{t} F. Viewing the functor Z : D ⟶ D(F) as a functor Z_{t} : D ⟶ D ⋉_{t}F and noting that U_{t}Z_{t} = Id_{D} we see by Theorem 2.6 that D ⋉_{t}F is a cleft extension of D by the functor F̃ := ⊕_{i=1}^{t+1}F^{i}. In case D = Mod(Γ) then D ⋉_{t}F = Mod(Γ ⋉_{t}M) where Γ ⋉_{t}M is the factor ring T_{Γ}(M)/< ⊗_{Γ}^{t+2} M>, where < ⊗_{Γ}^{t+2} M> is the ideal generated by ⊗_{Γ}^{t+2}M.

The ∞-truncated extension D ⋉_{∞} F of D is defined by D ⋉_{∞} F = lim_{t→∞}D ⋉_{t} F, observing that ∀t, s ∈ ℕ with t ≤ s, D ⋉_{t}F is a full exact subcategory of D ⋉_{s}F. Of course again D ⋉_{∞} F is an abelian category. In particular if F = Id_{D}, then D ⋉_{t} Id_{D} := D(t) is the category of of t-nilpotent endomorphisms of D, and D ⋉_{∞}Id_{D} := D(∞) is the category of nilpotent endomorphisms of D. If Γ is a ring and P_{Γ} is the category of the finitely generated projective right Γ-modules, then the direct limit P_{Γ}(∞) = lim_{t→∞}P_{Γ}(i) of cleft extensions P_{Γ}(i) is the category N il(Γ) which plays an important role in Algebraic K-theory.

In a somewhat opposite direction define the idempotent extension D^{!} of an arbitrary category D as the cleft extension D_{Id_{D}}(1_{Id_{D}}). Trivially D^{!} is the category of idempotent endomorphisms of D; in fact D^{!} is the idempotent or Karoubian completion of D via the functor ΦZ : D → D^{!}, where Φ is the endofunctor of D_{F} (η) defined by Φ(X, f) = (F X, η_{X} − F f). In particular the category of projective modules over a ring is a cleft (idempotent) extension of the category of free modules.

4.5. Repetitive Categories and Categories of Complexes. Let F : D → D be a right exact endofunctor. Denote by D^{ℤ} the associated graded category, and let S_{+}, S_{−} : D^{ℤ} → D^{ℤ} be the shift functors: S_{+}(X^{•})^{i} = X^{i+1}, S_{−}(X^{•})^{i} = X^{i−1}. Then F extends to a functor F^{∘} : D^{ℤ} → D^{ℤ} in an obvious way. Then set: F̂ := F^{∘}S_{+} and define the F -repetitive category D̂(F) of D to be the trivial cleft extension (D^{ℤ})⋉F̂. The bounded F -repetitive category D̂^{b}(F) of D is defined as the trivial cleft extension D^{b} ⋉F̂ of the full subcategory D^{b} of bounded objects in D^{ℤ}. Similarly we define the left bounded, right bounded, positive or negative F -repetitive category of D. Especially if F = Id_{D} then the bounded, left bounded, right bounded, positive or negative F -repetitive category of D is the category of bounded, left bounded, right bounded, positive or negative complexes of D respectively. In case D = mod(Λ), is the category of finitely presented right Λ−modules over an artin algebra Λ, then: D̂^{b}(F) = mod(Λ̂), where F := − ⊗_{Λ} D(Λ) is the Nakayama functor and Λ̂ is the repetitive algebra of Λ [20].

Finally note the easily checked observation that given a cleft extension D_{F} (η) of D, the category C(D_{F} (η)) of complexes of D_{F} (η) is a cleft extension of the corresponding category C(D) of complexes of D.

4.6. Examples of Cleft Extensions of Module Categories. The constructions of the previous paragraphs cover the module categories of the free, symmetric, polynomial, exterior, truncated rings (in particular trivial extension rings). More generally since by Proposition 3.4 the only module categories which are cleft extensions of a given abelian category are the module categories of the cleft extensions of rings, we conclude that the following module categories Mod(Λ) are cleft extensions, where Λ is: a supplemented ring, a positively graded ring, a group ring, a universal enveloping algebra, a symmetric algebra, a tensor algebra, a path-algebra, etc. Further let k be a commutative ring, G a group, and H a subgroup of G with finite index. Let Γ[G−H] be the Γ−submodule of ΓG with free basis G−H. Then Γ[G−H] is a ΓH-subbimodule of ΓG, and ΓG is a cleft extension of ΓH.

All these examples are of constructive nature. There are module categories which are cleft extensions for structural reasons. The main examples are provided by the module categories of algebras satisfying satisfy some version of the Principal Theorem of Wedderburn. For instance let Q be a finite quiver [4], and let k be an arbitrary field. If I is an admissible ideal in the path-algebra kQ, then the quotient algebra kQ/I is a cleft extension of the semisimple subalgebra k^{|Q_{0}|} of kQ/I.

In what follows k is a commutative ring, and Λ is a k-algebra. We denote by J (Λ) the Jacobson radical of Λ, and Dim the Hochschild dimension. The algebra Λ is a cleft extension of Λ/J (Λ) in the following cases:

(i) p.d_{k}Λ/J (Λ) = 0, J (Λ) is nilpotent, and Dim(Λ/J (Λ)) ≤ 1, see [36]. (ii) k is a field, dim_{k}Λ < ∞ and either k is a perfect field, or gl.dimΛ < ∞, see

[27], [11]. (iii) k is a Henselian ring, Λ_{k} is finitely generated, and Λ/J (Λ) is a separable

k-algebra, see [5]. (iv) k is a von Neumann regular ring, Λ_{k} is finitely generated, and Λ/J (Λ) is a

separable k-algebra, see [9].

(v) Λ is a ring of generalized triangular type (T-ring in the sense of [41]), or a triangular algebra in the sense of [18].

If Λ is commutative and I ⊆ J (Λ) is an ideal with Λ/I a separable k-algebra, then Λ is a cleft extension of Λ/I, if I is nilpotent or Λ is I-complete [22], [23]. If Λ is any commutative ring and L(Λ) is the lower Baer radical then Λ is a cleft extension of Λ/L(Λ), if Λ is finitely generated projective k-module and Λ/L(Λ) is separable [42]. Another example of cleft extension comes from Kleiner’s generalization of the Principal Theorem of Wedderburn in the category of R-rings, see [26].

Let L be a finite-dimensional Lie algebra over a field k. Then from the Levi-Malcev theorem [1] we have a decomposition in mod(k): L ≅ R ⊕ L/R, where R is the radical of L. If U(−) is the universal enveloping algebra functor, then U(L) is a cleft extension of U(L/R) by the twosided ideal U(R) of U(L) generated by i(R), where i : L → U(L) is the canonical inclusion [1]. Generally if Ring is the category of rings with unit and unit preserving ring homomorphisms, and C → Ring is any functor from an arbitrary category C, then any split epic in C produces a cleft extension of rings. Proposition 2.12 includes the following well known situation. Let Rng be the category of rings not necessarilly with unit and ring homomorphisms which need not necessarily preserve units. Then the (non full) inclusion functor: Ring ⊆ Rng admits a left adjoint Ω : Rng → Ring which can be described as the cleft extension: Ω(R) = ℤ ⋉_{ϑ} R, where ϑ : R ⊗_{ℤ} R → R, is given by the multiplication of R. Hence the classical procedure of adjoining a unit to a ring without unit is an example of cleft extension of rings.

We close this section by noting the following observation: if Λ is finite dimensional algebra over a field then: Λ is hereditary iff any extension of Λ is cleft. This follows trivially from the Theorem of Whitehead-Hochschild [27].

5. Resolutions and Filtrations

Throughout this section we fix a cleft extension C of an abelian category D and assume that the projection U : C ⟶ D preserves cokernels. By Theorem 2.6 we can assume that C is an η−extension D_{F} (η) of D, by a right exact endofunctor F of D, where η : F^{2} → F is an associative natural morphism.

Consider the standard functors T : D → D_{F} (η), U : D_{F} (η) → D, Z : D → D_{F} (η), C : D_{F} (η) → D connecting D, D_{F} (η). We recall that T(X) = (X⊕F X, t_{X}),

t_{X} = ( \begin{matrix}0 & 1_{F X} \\ 0 & η_{X} \end{matrix} ) and T(a) = ( \begin{matrix}a & 0 \\ 0 & F a \end{matrix} ), U(X, f) = X and U(a) = a, Z(X) =

(X, 0) and Z(a) = a, C(X, f) = Coker(f) and C(a) = the unique morphism : C(X, f) → C(Y, g) with coker(f) ∘ C(a) = a ∘ coker(g), where a : (X, f) → (Y, g).

Then we have the adjoint pairs (T, U), (C, Z). The counit ε : TU → Id_{D_{F} (η)} of the adjunction (T, U), is defined by ε_{(X,f)} = ^{t}(1_{X}, f) and the unit δ : Id_{D} → UT = Id_{D} ⊕ F by δ_{X} = (1_{X}, 0). The unit λ : Id_{D_{F} (η)} → ZC of the adjunction (C, Z), is defined by λ_{(X,f)} = coker(f), and the counit is the natural identification CZ = Id_{D}. The functors U, Z are faithful exact, with Z in addition full, the functor T is faithful, and we have the identifications: CT = Id_{D}, UZ = Id_{D}, CZ = Id_{D}.

5.1. Filtrations, Simple and Projective Objects. We shall study the cleft extension D_{F} (η) through the adjoint pairs (T, U), (C, Z) and the endofunctors Φ : D_{F} (η) → D_{F} (η), R : D_{F} (η) → D_{F} (η) which are defined by the exact sequences:

0 ⟶ Φ ⟶^{μ} TU ⟶^{ε} Id_{D_{F} (η)} ⟶ 0 and 0 ⟶ R ⟶^{ι} Id_{D_{F} (η)} ⟶^{λ} ZC ⟶ 0 (1)

i.e. Φ := Ker(ε), R := Ker(λ). It is easy to see that ∀(X, f) ∈ D_{F} (η) : Φ(X, f) = (F X, η_{X} − F f), Φ(a) = F a, μ_{(X,f)} = (−f, 1_{X}) and R(X, f) = (Imf, f^{1}) where f^{1} : F Im(f) → Im(f) is the unique morphism with: f^{1} ∘im(f) = F im(f)∘f. The natural morphism ι is defined by ι_{(X,f)} = ker(λ_{(X,f)}), and if a : (X, f) → (Y, g) is a morphism in D_{F} (η), then R(a) = is the unique morphism with the property: R(a) ∘ ι_{(Y,g)} = ι_{(X,f)} ∘ a. Inductively we define the endofunctors Φ^{n} : D_{F} (η) → D_{F} (η), R^{n} : D_{F} (η) → D_{F} (η), ∀n ∈ ℕ, by Φ^{n} = Φ(Φ^{n−1}), R^{n} = R(R^{n−1}). It is not difficult to see that: Φ^{n}(X, f) = (F^{n}X, f_{n}^{(X,f)}), where f_{0}^{(X,f)} := f and

f_{n}^{(X,f)} = ∑_{i=0}^{n−1}(−1)^{i}F^{i}η_{F^{n−i−1}X} + (−1)^{n}F^{n}f : F^{n+1}X ⟶ F^{n}X, ∀n ≥ 1.

By construction the endofunctors Φ^{n} are right exact, ∀n ∈ ℕ. The second short exact sequence in (1) induces a filtration of the identity functor Id_{D_{F} (η)}:

⋯ ⊆ R^{n} ⊆ R^{n−1} ⊆ · · · · · · ⊆ R^{2} ⊆ R ⊆ Id_{D_{F} (η)} (2)

with successive inclusions ι_{R^{n−1}} : R^{n} ↪ R^{n−1}. Since trivially RZ = 0, it follows that R is a radical in D_{F} (η).

Remark 5.1. R^{2} = 0 iff η = 0 i.e. D_{F} (η) = D ⋉ F is a trivial extension.

Indeed: if R^{2} = 0 then R^{2}T = 0. But ∀X ∈ D : RT(X) = ΦZ(X) = (F X, η_{X}), hence if R^{2}T(X) = 0 we have that λ_{(F X,η_{X})} is an isomorphism ⇒ coker(η_{X}) is an isomorphism ⇒ η_{X} = 0, ∀X ∈ D and then η = 0. If η = 0, then ∀(X, f) ∈ D_{F} (η) : F f ∘ f = 0 ⇒ F im(f) ∘ f = 0 ⇒ f^{1} ∘ ι_{(X,f)} = 0 ⇒ f^{1} = 0. Hence R(X, f) = Z(Im(f)) and then R^{2}(X, f) = RZ(Im(f)) = 0 ⇒ R^{2} = 0.

The filtration (2) shows that any object of D_{F} (η) has a (infinite in general) filtration of subobjects, with quotients in the full subcategory Z(D) ≈ D, and suggests the following definition. The cleft extension D_{F} (η) is called admissible, if any object (X, f) ∈ D_{F} (η) admits a finite filtration by subobjects with all the quotients in the full subcategory Z(D). We call the functor R locally nilpotent, if ∀(X, f) ∈ D_{F} (η), ∃n ∈ ℕ : R^{n}(X, f) = 0.

Proposition 5.2. The following are equivalent:

(i) D_{F} (η) is admissible. (ii) The functor R is locally nilpotent.

Proof. (i) ⇒ (ii) Let D_{F} (η) be admissible, let (X, f) ∈ D_{F} (η) and suppose that: 0 = (X_{k+1}, g_{k+1}) ⊆ (X_{k}, g_{k}) ⊆ ⋯ ⊆ (X_{1}, g_{1}) ⊆ (X_{0}, g_{0}) := (X, f) is a filtration with quotients in Z(D). Then we have short exact sequences:

0 ⟶ (X_{t+1}, g_{k+1}) ⟶^{m_{t}} (X_{t}, g_{t}) ⟶^{p_{t}} Z(Y_{t}) ⟶ 0

∀t = 0, 1, ..., k, g_{0} = f, X_{0} = X. We show inductively that R^{n}(X, f) ⊆ (X_{n}, g_{n}). Since f ∘ p_{0} = 0, ∃ ν_{0} : ZC(X, f) → Z(Y_{0}), with λ_{(X,f)} ∘ ν_{0} = p_{0}, hence we have the exact commutative diagram:

0 ⟶ R(X, f) ⟶^{ι_{(X,f)}} (X, f) ⟶^{λ_{(X,f)}} ZC(X, f) ⟶ 0
∃ γ_{1}↓  ^{||}↓  ^{ν_{0}}↓
0 ⟶ (X_{1}, g_{1}) ⟶^{m_{0}} (X, f) ⟶^{p_{0}} Z(Y_{0}) ⟶ 0

from which we get that γ_{1} is monic. Then f^{1} ∘ γ_{1} = F γ_{1} ∘ g_{1} ⇒ f^{1} ∘ γ_{1} ∘ p_{1} = F γ_{1} ∘ g_{1} ∘ p_{1} = 0, since p_{1} is a morphism in D_{F} (η). Since λ_{R(X,f)} = coker(f^{1}), ∃ ν_{1} : ZCR(X, f) → Z(Y_{1}), with: λ_{R(X,f)} ∘ ν_{1} = γ_{1} ∘ p_{1}, and we get an exact commutative diagram:

0 ⟶ R^{2}(X, f) ⟶^{ι_{R(X,f)}} R(X, f) ⟶^{λ_{R(X,f)}} ZCR(X, f) ⟶ 0
∃ γ_{2}↓  ^{γ_{1}}↓  ^{ν_{1}}↓
0 ⟶ (X_{2}, g_{2}) ⟶^{m_{1}} (X_{1}, f_{1}) ⟶^{p_{1}}  Z(Y_{1}) ⟶ 0

from which we deduce that γ_{2} is monic, since ι_{R(X,f)}, γ_{1} are monics. Continuing in this way we get a sequence of monics γ_{n} : R^{n}(X, f) ↪ (X_{n}, g_{n}), ∀n ∈ ℕ. Since (X_{k+1}, g_{k+1}) = 0 we have R^{k+1}(X, f) = 0. Hence R is locally nilpotent.

(ii) ⇒ (i) If R is locally nilpotent then the result follows from the filtration (2), since the quotients of the filtration (2) are the functors ZCR^{k}, k ∈ ℕ.

We are interested in the full subcategory KerC := {(X, f) ∈ D_{F} (η) | C(X, f) = 0} of D_{F} (η). We recall from Proposition 2.4 that: KerC = 0 iff λ : Id_{D_{F} (η)} → ZC is a minimal epimorphism. We call the natural morphism η locally nilpotent, if ∀X ∈ D, ∃i = i_{X} ∈ ℕ : η_{F^{i}X} ∘ η_{F^{i−1}X} ∘ ⋯ ∘ η_{F X} ∘ η_{X} = 0. The natural morphism η is called nilpotent, if ∃i ∈ ℕ : η_{F^{i}} ∘ η_{F^{i−1}} ∘ ⋯ ∘ η_{F} ∘ η = 0. Finally we call a right exact sequence A_{3} ⟶^{β} A_{2} ⟶^{α} A_{1} ⟶ 0 in a given category, a minimal presentation (of A_{1}), if the morphisms α, coim(β) are minimal epimorphisms.

Lemma 5.3. (i) If D_{F} (η) is admissible, or if η is (locally) nilpotent, then:

KerC = 0. (ii) Suppose that KerC = 0. Then the functor T preserves and reflects minimal

presentations. The functor C preserves minimal epimorphisms, and reflects minimal presentations.

Proof. (i) Let D_{F} (η) be admissible and let (X, f) ∈ D_{F} (η) with C(X, f) = 0. Then from the finite filtration 0 = R^{n}(X, f) ⊆ R^{n−1}(X, f) ⊆ ⋯ ⊆ R^{2}(X, f) ⊆ R(X, f) ⊆ (X, f), we have (X, f) ≅ R(X, f) and ZCR(X, f) = 0. But then R^{2}(X, f) ≅ R(X, f) ≅ (X, f) and inductively 0 = R^{n}(X, f) ≅ ⋯ ≅ R(X, f) ≅ (X, f). Hence KerC = 0. If η is (locally) nilpotent, the result follows from [29].

(ii) We show first that C preserves and reflects minimal epimorphisms. Suppose that α : (X_{2}, f_{2}) → (X_{1}, f_{1}) is a minimal epimorphism and let γ : Y → C(X_{2}, f_{2}) be a morphism with γ ∘C(α) an epimorphism. Then Z(γ)∘ZC(α) is an epimorphism. Consider the following pullback diagram:

0 ⟶ R(X_{2}, f_{2}) ⟶ (W, h) ⟶^{m}  Z(Y) ⟶ 0
||↓  ^{k}↓  ^{Z(γ)}↓
0 ⟶ R(X_{2}, f_{2}) ⟶ (X_{2}, f_{2}) ⟶^{λ_{(X_{2},f_{2})}} ZC(X_{2}, f_{2}) ⟶ 0

from which we have that: m ∘ Z(γ) ∘ ZC(α) = k ∘ λ_{(X_{2},f_{2})} ∘ ZC(α) = k ∘ α ∘ λ_{(X_{1},f_{1})} is an epimorphism. But then k is an epimorphism, since α, λ_{(X_{1},f_{1})} are minimal epimorphisms. This implies trivially that γ is an epimorphism, and hence that C(α) is a minimal epimorphism. If C(α) : C(X_{2}, f_{2}) → C(X_{1}, f_{1}) is a minimal epimorphism, let γ : (Y, g) → (X_{2}, f_{2}) be a morphism with γ ∘ α an epimorphism. Then C(γ) ∘ C(α) is an epimorphism, and this implies that C(γ) and ZC(γ) are epimorphisms. Since γ ∘ λ_{(X_{2},f_{2})} = λ_{(Y,g)} ∘ ZC(γ), we have directly that γ is an epimorphism and then α is a minimal epimorphism.

Now let X_{3} ⟶^{β} X_{2} ⟶^{α} X_{1} ⟶ 0 be a minimal presentation of X_{1}. Since CT = Id_{D} and C reflects minimal epimorphisms, trivially T(α) is minimal. A simple calculation shows that C(Im(T(β))) = Im(β) and C(coim(T(β))) = coim(β), from which we get as above that coim(T(β)) is a minimal epimorphism. Thus T

preserves minimal presentations. Next let T(X_{3}) ⟶^{T(β)} T(X_{2}) ⟶^{T(α)} T(X_{1}) ⟶ 0 be a minimal presentation of T(X_{1}), and let γ : Y → X_{2} be a morphism with γ∘α an epimorphism. Then T(γ) ∘ T(α) is an epimorphism, thus T(γ) is an epimorphism because T(α) is a minimal one. But then γ = CT(γ) is an epimorphism, so α minimal. Since C(coim(T(β))) = coim(β) and C preserves minimal epimorphisms, coim(β) is minimal. Hence T reflects minimal presentations.

To show that C reflects minimal presentations it suffices to show that if β : (X_{3}, f_{3}) → (X_{2}, f_{2}) is a morphism with coim(C(β)) a minimal epimorphism, then coim(β) is a minimal epimorphism. Let d : (Y, g) → (X_{3}, f_{3}) be a morphism with d ∘ coim(β) an epimorphism, and let ω : C(Im(β)) → Im(C(β)) be the unique epimorphism with C(coim(β)) ∘ ω = coim(C(β)). Then C(d) ∘ C(coim(β)) ∘ ω = C(d)∘coim(C(β)) is an epimorphism, thus C(d) is an epimorphism. Then as above we conclude that d is an epimorphism and C reflects minimal presentations.

Suppose now that the category D has enough projectives, and let (X, f) ∈ D_{F} (η). Let a : P → X be an epimorphism with P projective in D. Then T(a) ∘ ε_{(X,f)} = t(a, F a ∘ f) : T(P) → (X, f) is obviously an epimorphism, and T(P) is a projective object of D_{F} (η), since T preserves projectives as a left adjoint of the exact functor U. Hence D_{F} (η) has enough projectives. Conversely if D_{F} (η) has enough projectives, let X ∈ D, and choose an epimorphism a : (P, f) → Z(X) with (P, f) projective in D_{F} (η). Then C(a) : C(P, f) → CZ(X) = X is an epimorphism and C(P, f) is projective in D, since C preserve projectives as a left adjoint of the exact functor Z. Hence D has enough projectives.

The condition KerC = 0, is sufficient for the complete determination of the simple and projective objects of D_{F} (η), in terms of the simple and projective objects of D. Such a determination was given in [29], under the condition that the natural morphism η is nilpotent. We recall that an abelian category A is called perfect, if any object A of A admits a projective cover, i.e. a minimal epimorphism P ⟶ A where P is a projective object. Since nilpotency of η implies KerC = 0, combining the above remarks and Lemma 5.3, we get the following generalization of the corresponding results of [29].

Proposition 5.4. (1) D_{F} (η) has enough projectives iff D has enough projectives. Moreover an object (X, f) is projective in D_{F} (η) iff (X, f) is a direct summand of an object of the form T(P), where P is projective in D.

(2) If KerC = 0, then we have the following:

(i) The functor T preserves and reflects projectives and minimal projective presentations and the functor C preserves projectives and projective covers and reflects minimal projective presentations.

(ii) D_{F} (η) is perfect iff D is perfect. (iii) (X, f) is projective iff C(X, f) is projective and (X, f) ≅ TC(X, f). (iv) (X, f) is simple iff C(X, f) is simple and (X, f) ≅ ZC(X, f).

Proof. (1) Follows from the above discussion.

(2) Part (i) follows from Lemma 5.3 and the above remarks. If D_{F} (η) is perfect, then since C preserves projective covers and CZ = Id_{D}, it follows that D is perfect. Conversely assume that D is perfect and let (X, f) ∈ D_{F} (η). Let β : P ⟶ C(X, f) be a projective cover. Since U(λ_{(X,f)}) = coker(f) : X ⟶ C(X, f) is an epimorphism, there exists a morphism α : P ⟶ X such that α ∘ coker(f) = β. Consider the morphism ρ := T(α) ∘ ε_{(X,f)} : T(P) ⟶ (X, f). Then T(P) is projective and it is easy to see that C(ρ) = β and the composition ρ ∘ λ_{(X,f)} equals ^{t}(β, 0) which is an epimorphism, since β is so. Since λ_{(X,f)} is minimal, ρ is an epimorphism. If γ : (Y, g) ⟶ T(P) is a morphism such that γ ∘ ρ is an epimorphim, then so is C(γ) ∘ C(ρ) = C(γ) ∘ β. Since β is minimal, C(γ), hence ZC(γ), is an epimorphism. Since γ ∘ λ_{T(P)} = λ_{(X,f)} ∘ ZC(γ) and λ_{T(P)} is minimal, we infer that γ is an epimorphism. Hence ρ is a minimal epimorphism, i.e. ρ : T(P) ⟶ (X, f) is a projective cover and D_{F} (η) is perfect. Part (iii) is a direct consequence of (ii).

It remains to show part (iv). If C(X, f) is simple in D and (X, f) ≅ ZC(X, f), then obviously (X, f) is simple in D_{F} (η), since Z is a full exact embedding. Conversely if (X, f) is simple in D_{F} (η), then λ_{(X,f)} : (X, f) ⟶ ZC(X, f) is invertible, since oherwise ZC(X, f) = 0 ⇒ C(X, f) = 0 ⇒ (X, f) = 0 and this is not the case. Now let α : Y ⟶ C(X, f) is a non-zero morphism and consider the morphism Z(α) ∘ λ_{(X,f)}^{−1} : Z(Y) ⟶ (X, f), which is non-zero since otherwise, α ∘ C(λ_{(X,f)}^{−1}) = 0 ⇒ α = 0 and this is not the case. Hence Z(α) ∘ λ_{(X,f)}^{−1} is a monomorphism and then so is UZ(α) ∘ U(λ_{(X,f)}^{−1}) = α ∘ U(λ_{(X,f)}^{−1}). It follows that α is a monomorphism. Hence C(X, f) is a simple object in D.

Remark 5.5. (1) If KerC = 0 and the category D_{F} (η) has arbitrary intersections, then the functor R is a subfunctor of the Jacobson radical Rad [40] of D_{F} (η). Indeed if (X, f) ∈ D_{F} (η) and (M, m) is a maximal subobject of (X, f), then the quotient (X, f)/(M, m) is a simple object of D_{F} (η); hence by Proposition 5.4 it is of the form Z(S) for a simple object S of D. Then as in the proof of Proposition 5.2, we have R(X, f) ⊆ (M, m). Hence R(X, f) is a subobject of the intersection of all maximal subobjects of (X, f), and this shows that : R ⊆ Rad. In particular if D is semisimple, then R = Rad is the Jacobson radical of D_{F} (η).

(2) Suppose that in D_{F} (η) any non-zero object has a maximal subobject. Then KerC = 0 ⇔ any simple object of D_{F} (η) is of the form Z(S) where S is a simple object of D. Indeed the (⇒) direction follows from Proposition 5.4. For the (⇐) direction let C(X, f) = 0 with (X, f) ≠ 0, and let (M, m) be a maximal subobject of (X, f). Then (X, f)/(M, m) is a simple object of D_{F} (η), hence it is of the form Z(S) for a simple object S of D. Then C(X, f) = 0 ⇒ CZ(S) = S = 0 and this is impossible. Hence (X, f) = 0, and KerC = 0.

Denote by P(D_{F} (η)), P(D) the full subcategories of projective objects of the categories D_{F} (η), D and let K_{i}(P(D_{F} (η))), K_{i}(P(D)), K_{i}(D_{F} (η)), K_{i}(D) be the Kgroups of Quillen [37] of the corresponding exact categories.

Corollary 5.6. (1) The abelian groups K_{i}(P(D)), K_{i}(D) are direct summands of the abelian groups K_{i}(P(D_{F} (η))), K_{i}(D_{F} (η)), ∀i ≥ 0. If KerC = 0, then:

K_{i}(P(D_{F} (η))) ≅ K_{i}(P(D)), ∀i ≥ 0.

(2) Suppose that one of the following conditions holds:

(i) The functor R is locally nilpotent. (ii) The category D is Artinian and KerC = 0. (iii) The category D is Artinian and η is (locally) nilpotent. (iv) Any object of D has finite projective dimension.

(v) Any object of D has finite length. Then: K_{i}(D_{F} (η)) ≅ K_{i}(D), ∀i ≥ 0.

Proof. (1) Since the functors U, Z are exact and UZ = Id_{D}, it follows that K_{i}(D) is a direct summand of K_{i}(D_{F} (η)). Similarly since the functors T, C preserve projective objects and CT = Id_{P(D)}, we have that T induces a split monomorphism K_{i}(T) : K_{i}(P(D)) → K_{i}(P(D_{F} (η))). If KerC = 0, it follows by Proposition 5.4 that the exact functor T : P(D) → P(D_{F} (η)) induces a bijection on the isomorphism classes of objects and the claim follows from Theorem C′ of [19].

(2) In case (i) we have by Proposition 5.2 that the full subcategory Z(D) which is equivalent to D is admissible. Then the result follows from the Devissage Theorem, see [37]. In cases (ii), (iii) the conditions insure that the category D_{F} (η) is admissible. Indeed calculating the filtration (2) on an arbitrary object (X, f) ∈ D_{F} (η), and then applying the exact functor U, we have a filtration of subobjects of X in D. Since D is artinian we have that for some n ∈ ℕ : UR^{n}(X, f) ≅ UR^{n−1}(X, f) and then R^{n}(X, f) ≅ R^{n−1}(X, f) since U is faithful. Hence ZCR^{n−1}(X, f) = 0 ⇒ CR^{n−1}(X, f) = 0, from which we get R^{n−1}(X, f) = 0, since KerC = 0, in either case. In cases (iv), (v) the result follows from [2], since the exact functor U is faithful and UZ = Id_{D}.

5.2. Resolutions. We now use the adjoint pair (T, U) to construct a resolution of the identity functor Id_{D_{F} (η)}. Consider the short exact sequence of endofunctors

0 ⟶ Φ ⟶^{μ} TU ⟶^{ε} Id_{D_{F} (η)} ⟶ 0

Since TUΦ^{n} = TF^{n}U, we get short exact sequences, ∀n ≥ 1:

0 ⟶ Φ^{n} ⟶^{μΦ^{n−1}} TF^{n−1}U ⟶^{εΦ^{n−1}} Φ^{n−1} ⟶ 0.

Considering the Yoneda composition of these short exact sequences we have the following long exact sequence of endofunctors of D_{F} (η):

Σ^{•} ⋯ ⟶ TF^{n}U ⟶^{θ^{n}} TF^{n−1}U ⟶ · · · · · · ⟶ TF U ⟶^{θ^{1}} TU ⟶^{ε} Id_{D_{F} (η)} ⟶ 0.

By definition we have:

μΦ_{(X,f)}^{n} = (−f_{n}^{(X,f)}, 1_{F^{n}X}), εΦ_{(X,f)}^{n} = ^{t}(1_{F^{n}X}, −f_{n}^{(X,f)})

θ_{(X,f)}^{n} = ( \begin{matrix}−f_{n−1}^{(X,f)} & 1_{F^{n}X} \\ 0 & f_{n}^{(X,f)} \end{matrix} )

f_{n}^{(X,f)} = η_{F^{n−1}X} − F (f_{n−1}^{(X,f)}), η_{F^{n}X} ∘ f_{n}^{(X,f)} = F (f_{n}^{(X,f)}) ∘ η_{F^{n−1}X}.

The last relation shows that f_{n}^{(X,f)} : (F^{n+1}X, η_{F^{n}X}) → (F^{n}X, η_{F^{n−1}X}) are morphisms in D_{F} (η). Applying the functor C to the exact sequence (Σ^{•}) we get the following complex in D:

C(Σ^{•}) ⋯ ⟶ F^{n}U ⟶^{C(θ^{n})} F^{n−1}U ⟶ ⋯ ⟶ F U ⟶^{C(θ^{1})} U ⟶ 0 .

For any (X, f) ∈ D_{F} (η) we have directly C(θ_{(X,f)}^{n}) = −f_{n−1}^{(X,f)}, ∀n ≥ 1 and f_{0}^{(X,f)} = f. Hence we get the following complex C(Σ_{(X,f)}^{•}) in D:

⋯ ⟶ F^{n}X ⟶^{−f_{n−1}^{(X,f)}} F^{n−1}X ⟶ ⋯ ⟶ F^{2}X ⟶^{−f_{1}^{(X,f)}} F X ⟶^{f_{0}^{(X,f)}} X ⟶ 0.

We call the complex C(Σ_{(X,f)}^{•}) in D the standard complex of the object (X, f). In this way we obtain a faithful right exact additive functor S : D_{F} (η) → C^{≤0}(D) from D_{F} (η) to the abelian category C^{≤0}(D) of negative complexes over D, defined by S(X, f) = C(Σ_{(X,f)}^{•}) and S(a)^{−n} = F^{n}. Note that S is full iff F is so.

Now it is easy to see that the morphisms θ_{n}^{0} = ( \begin{matrix}0 & 1_{F^{n}X} \\ 0 & η_{F^{n−1}X} \end{matrix} ), θ_{n}^{n} = T(F^{n−1}f),

θ_{n}^{i} = T(F^{i−1}η_{F^{n−i−1}X}) : T(F^{n}X) → T(F^{n−1}X) for 0 < i < n define a simplicial object without degeneracies augmented over (X, f), whose associated complex is Σ_{(X,f)}^{•}. Hence the standard complex C(Σ_{(X,f)}^{•}) is the associated complex of the simplicial object without degeneracies augmented over C(X, f), with face operators C(θ_{n}^{0}) = 0, C(θ_{n}^{n}) = F^{n−1}f, C(θ_{n}^{i}) = F^{i−1}η_{F^{n−i−1}X} : F^{n}X → F^{n−1}X for 0 < i < n. Note that the standard complex was constructed in the case of the trivial extension (η = 0) in [15], in an adhoc manner. Consider now the projective class ℜ_{0} of split epics in D. By the Adjoint Theorem [13], we have the projective class Σ = U^{−1}(ℜ_{0}) on D_{F} (η) which consists of all morphisms a : (X, f) → (Y, g) such that a : X → Y is split epic. The full subcategory P(Σ) of Σ-projective objects consists of all objects of the form T(X) and their retracts. Moreover (TU, ε) is a resolvent pair for Σ, i.e. ∀(X, f) ∈ D_{F} (η), TU(X, f) ∈ P(Σ) and ε_{(X,f)} ∈ Σ; consequently (X, f) ∈ P(Σ) ⇔ ε_{(X,f)} splits. Hence the endofunctors Φ^{n} : D_{F} (η) → D_{F} (η) are the n^{th}−syzygy functors for the projective class Σ in D, and the resolution Σ^{•} is a Σ-exact Σ-projective resolution of Id_{D_{F} (η)} with corresponding Σ-syzygy functors Φ^{n}, n ∈ ℕ. We conclude that the homology of the complex C(Σ^{•}) computes the Σ-left derived functors of C : H_{n}[C(Σ^{•})] = L_{n}^{Σ}C : D_{F} (η) → D, ∀n ∈ ℕ.

On the other hand consider the cotriple V [7] induced in D_{F} (η) by the adjoint pair (T, U). Then it is trivial that (X, f) ∈ D_{F} (η) is V-projective in the sense of [7] iff (X, f) ∈ P(Σ). Since the Barr-Beck-resolution [6] of Id_{D_{F} (η)} is obviously Σ-exact we have that the cotriple derived functors H_{n}(−, C)_{V} of C and H^{n}[−, (X, f)]_{V} of D_{F} (η)(−, (X, f)) are naturally isomorphic to the Σ-left derived functors of C and to the Σ-right derived functors of D_{F} (η)(−, (X, f)) respectively:

H_{n}(−, C)_{V} ≅ L_{n}^{Σ}C ≅ H_{n}[C(Σ^{•})] , ∀n ∈ ℕ.
H^{n}[−, (X, f)]_{V} ≅ Ext_{Σ}^{n}[−, (X, f)], ∀n ∈ ℕ, ∀(X, f) ∈ D_{F} (η).

Hence the Σ-left derived functors of C is a cotriple (simplicial) homology, and the Σ-extension functors of D_{F} (η) is a cotriple (cosimplicial) cohomology. For a detailed study of the relative homological behaviour of D_{F} (η) we refer to [8], where the standard complex introduced above plays an important role.

Remark 5.7. (1) It follows trivially from the construction of C(Σ^{∗}), that if η = 0, then L_{n}^{Σ}CZ = F^{n} and Ext_{Σ}^{n}[Z(−), Z(−)] = D[F^{n}(−), −].

(2) Ext_{Σ}^{n}[−, Z(X)] = H^{n}D[C(Σ^{∗})(−), X]. Hence for any injective object X ∈ D there exists an isomorphism Ext_{Σ}^{n}[−, Z(X)] = D[L_{n}^{Σ}C(−), X].

(3) Ext_{Σ}^{1}[Z(−), Z(−)] ≅ D_{F} (η)[ΦZ(−), Z(−)] ≅ D[Coker(η_{−}), −]. Indeed from the natural Σ-projective presentation of Id_{D_{F} (η)}: 0 ⟶ Φ ⟶^{μ} TU ⟶^{ε} Id_{D_{F} (η)} ⟶ 0 evaluated at Z(X) we get the Σ-projective presentation 0 ⟶ ΦZ(X) ⟶ T(X) ⟶ Z(X) ⟶ 0 of Z(X). Then we have an exact sequence:

0 ⟶ D_{F} (η)[Z(X), Z(Y)] ⟶ D_{F} (η)[T(X), Z(Y)] ⟶ D_{F} (η)[ΦZ(X), Z(Y)] ⟶

⟶ Ext_{Σ}^{1}[Z(X), Z(Y)] ⟶ 0

Since the first morphism in the above exact sequence is an isomorphism we have

Ext_{Σ}^{1}[Z(X), Z(Y)] ≅ D_{F} (η)[ΦZ(X), Z(Y)] ≅ D[C(ΦZ(X)), Y] ≅ D[Coker(η_{X}), Y].

We define the nilpotency class of F by c(F) = min{n ∈ ℕ : F^{n+1} = 0}, if the functor F is nilpotent, otherwise c(F) = ∞. The next result shows in particular that if the functor F is nilpotent, then the finiteness of the global dimension gl.dimD_{F} (η) of D_{F} (η) is controlled by the finiteness of the global dimension gl.dimD of D. Note that the isomorphism in part (1) of the following Proposition generalizes a result of [29].

Proposition 5.8. (1) ∀X, Y ∈ D, ∀n ∈ ℕ : the abelian group Ext_{D}^{n}[X, Y] is a direct summand of Ext_{D_{F} (η)}^{n}[Z(X), Z(Y)], and there is a natural isomorphism

Ext_{D_{F} (η)}^{1}[Z(X), Z(Y)] ≅ Ext_{D}^{1}[X, Y] ⊕ Ext_{Σ}^{1}[Z(X), Z(Y)].

(2) ∀X, Y ∈ D, ∀n ∈ ℕ : the abelian group Ext_{D}^{n}[X, Y] is a direct summand of Ext_{D_{F} (η)}^{n}[T(X), Z(Y)], and there is a natural isomorphism

Ext_{D_{F} (η)}^{1}[T(X), Z(Y)] ≅ Ext_{D}^{1}[X, Y].

(3) ∀X ∈ D : p.d_{D}X ≤ min{p.d_{D_{F} (η)}Z(X), p.d_{D_{F} (η)}T(X)} and

gl.dimD ≤ gl.dimD_{F} (η).

(4) If F is exact then p.d_{D}X = p.d_{D_{F} (η)}T(X) and there exists a natural isomorphism of bifunctors, induced from the adjoint pair (T, U):

Ext_{D_{F} (η)}^{n}[T−, −] ≅ Ext_{D}^{n}[−, U−], ∀n ≥ 0.

Moreover:

gl.dimD ≤ gl.dimD_{F} (η) ≤ c(F) + gl.dimD.

(5) If D_{F} (η), D are Krull-Schmidt categories and Σ−gl.dimD_{F} (η) = 0, then: D_{F} (η) has finite representation type ⇔ D has finite representation type.

(6) Σ−gl.dimD_{F} (η) ≤ c(F). If KerC = 0 then: Σ−gl.dimD_{F} (η) = 0 ⇔ F = 0. If η = 0 then: Σ−gl.dimD_{F} (0) < ∞ ⇔ the functor F is nilpotent. In the last case: Σ−gl.dimD_{F} (0) = c(F).

Proof. (1) Since U, Z are exact functors, we have induced homomorphisms

U_{∗}^{n} : Ext_{D_{F} (η)}^{n}[Z(X), Z(Y)] → Ext_{D}^{n}[X, Y]
Z_{∗}^{n} : Ext_{D}^{n}[X, Y] → Ext_{D_{F} (η)}^{n}[Z(X), Z(Y)].

Since UZ = Id_{D}, we have U_{∗}^{n} ∘ Z_{∗}^{n} = Id_{Ext_{D}^{n}[X,Y]}. In case n = 1, obviously we have that the kernel of U_{∗}^{1} consists of all equivalence classes of short exact sequences in Ext_{D_{F} (η)}^{1}[Z(X), Z(Y)] which split as short exact sequences in D, thus KerU_{∗}^{1} = Ext_{Σ}^{1}[Z(X), Z(Y)].

(2), (3) The proof of (2) is the same as in (1), using the fact that T(X) ∈ P(Σ) and (3) follows from (2).

(4) Since F is exact iff T is exact, the first part follows from [34], since then (T, U) is an adjoint pair of exact functors between abelian categories. If F is in addition nilpotent, then for any object (X, f) ∈ D_{F} (η) we have a finite resolution:

0 ⟶ T(F^{n}X) ⟶ T(F^{n−1}X) ⟶ ⋯ ⟶ T(F X) ⟶ T(X) ⟶ (X, f) ⟶ 0

where n = c(F). Since F is exact, we have p.d_{D_{F} (η)}T(F^{i}X) = p.d_{D}F^{i}X ≤ gl.dimD, ∀i = 0, ..., n. Hence p.d_{D_{F} (η)}(X, f) ≤ c(F) + gl.dimD.

(5) Always D is of finite representation type if D_{F} (η) is. If Σ−gl.dimD_{F} (η) = 0 and D is of representation finite, then ∀(X, f) ∈ D_{F} (η), the natural epic ε_{(X,f)} : T(X) → (X, f) splits, hence T(X) ≅ (X, f) ⊕ Φ(X, f). Let {X_{1}, X_{2}, ..., X_{m}} be a complete list of the isomorphism classes of the indecomposable in D. Then any indecomposable object (X, f) ∈ D_{F} (η) is a direct summand of ⊕_{i=1}^{m}T(X_{i})^{k_{i}}. This implies that D_{F} (η) is representation finite.

(6) Letting n = c(F) < ∞, we have directly that Φ^{n+1} = 0. Since the functors Φ^{n} by construction are the n^{th}−syzygy functors for the projective class Σ, the first part follows. If F is nilpotent then from (1) we have that Σ−gl.dimD_{F} (0) < ∞. If Σ−gl.dimD_{F} (0) = n then ∀X ∈ D, the n^{th} Σ-syzygy object Φ^{n}Z(X) = Z(F^{n}X) is Σ-projective. Hence the natural Σ-epic ε_{Z(F^{n}X)} : T(F^{n}X) → Z(F^{n}X) splits. Obviously this is equivalent to F^{n+1}X = 0. Since this happens for any X ∈ D, we have F^{n+1} = 0. Trivially then Σ−gl.dimD_{F} (0) = c(F).

Lemma 5.9. The functor C is exact on exact sequences with right end term from the subcategory P(Σ).

Proof. Let 0 ⟶ (W, h) ⟶^{b} (Y, g) ⟶^{a} (X, f) ⟶ 0 be an exact sequence in D_{F} (η) and suppose that L_{1}^{Σ}C(X, f) = 0. Consider the exact commutative diagram:

Φ(W, h) ⟶^{Φ(b)} Φ(Y, g) ⟶^{Φ(a)} Φ(X, f) ⟶ 0
↓^{μ_{(W,h)}}  ↓^{μ_{(Y,g)}}  ↓^{μ_{(X,f)}}
T(W) ⟶^{T(b)} T(Y) ⟶^{T(a)} T(X) ⟶ 0
↓^{ε_{(W,h)}}  ↓^{ε_{(Y,g)}}  ↓^{ε_{(X,f)}}
0 ⟶ (W, h) ⟶^{b} (Y, g) ⟶^{a} (X, f) ⟶ 0.

Applying the functor C we get the following exact commutative diagram in D:

⋯ ⟶ CΦ(W, h) ⟶^{CΦ(b)} CΦ(Y, g) ⟶^{CΦ(a)} CΦ(X, f) ⟶ 0
↓  ↓^{C(μ_{(W,h)})}  ↓^{C(μ_{(Y,g)})}  ↓^{C(μ_{(X,f)})}
0 ⟶  W  ⟶^{b}  Y  ⟶^{a}  X  ⟶ 0

By the snake Lemma we get the following exact sequence: L_{1}^{Σ}C(W, h) ⟶ L_{1}^{Σ}C(Y, g)

⟶ L_{1}^{Σ}C(X, f) ⟶ C(W, h) ⟶^{C(b)} C(Y, g) ⟶^{C(a)} C(X, f) ⟶ 0 in D. Since (X, f) is

Σ−projective, we have L_{1}^{Σ}C(X, f) = 0, so C(b) is a monomorphism.

Corollary 5.10. (i) If D has enough projectives, then the functors L_{1}C, L_{1}^{Σ}C

are isomorphic and ∀(X, f) ∈ D_{F} (η): L_{1}C ≅ L_{1}^{Σ}C = Ker(f)/Im(η_{X} − F f). (ii) If D has enough projectives and KerC = 0, then an object (X, f) is projective

⇔ C(X, f) is projective and L_{1}^{Σ}C(X, f) = Ker(f)/Im(η_{X} − F f) = 0. (iii) Suppose that KerC = 0. If L_{1}^{Σ}C(X, f) = 0, then a morphism a : (Y, g) →

(X, f) is an isomorphism ⇔ C(a) is an isomorphism. (iv) If KerC = 0, then the functor C preserves minimal (projective) presentations

of objects (X, f) with L_{1}^{Σ}C(X, f) = Ker(f)/Im(η_{X} − F f) = 0.

(v) Suppose that KerC = 0 and let (X, f) ∈ D_{F} (η) with L_{1}^{Σ}C(X, f) = 0. Then the ring End_{D_{F} (η)}((X, f)) is local if the ring End_{D}(C(X, f)) is local. In particular ∀X ∈ D : End_{D_{F} (η)}(T(X)) is local ⇔ End_{D}(X) is local.

Proof. (i) First we note that L_{1}C(T(X)) = 0, ∀X ∈ D, as follows trivially from Lemma 5.9. Let now (X, f) ∈ D_{F} (η), and consider the standard Σ-exact se-

quence (1) 0 ⟶ Φ(X, f) ⟶^{μ_{(X,f)}} TX ⟶^{ε_{(X,f)}} (X, f) ⟶ 0. Then L_{1}^{Σ}C(X, f) =

Ker(C(μ_{(X,f)})). Applying C to (1) we obtain the long exact sequence

⋯ ⟶ L_{1}C(T(X)) ⟶ L_{1}C(X, f) ⟶ C(Φ(X, f)) ⟶ X ⟶ C(X, f) ⟶ 0

from which we get L_{1}C(X, f) ≅ L_{1}^{Σ}C(X, f) = Ker(C(μ_{(X,f)})), since L_{1}C(T(X)) = 0. Since the Σ−exact sequence (1) is natural, the above isomorphism is natural.

(ii) Follows from Proposition 5.4 and (i).

(iii) Consider a morphism a : (Y, g) → (X, f) in D_{F} (η), such that C(a) is invertible, and let b : (W, h) → (Y, g) be the kernel of a. Then ZC(a) is invertible and since λ_{(Y,g)} ∘ ZC(a) = a ∘ λ_{(X,f)}, we have that a is an epimorphism, since

λ_{(X,f)} is a minimal epimorphism. Hence we have an exact sequence 0 ⟶ (W, h) ⟶^{b} (Y, g) ⟶^{a} (X, f) ⟶ 0 in D_{F} (η). By our assumptions and Lemma 5.9 we have C(W, h) = 0. Since KerC = 0, (W, h) = 0 and a is invertible.

(iv) If (W, h) ⟶^{β} (Y, g) ⟶^{α} (X, f) ⟶ 0 is a minimal (projective) presentation of (X, f) and L_{1}^{Σ}C(X, f) = 0, then from Lemma 5.9 we see that C(im(β)) is a

monomorphism. Hence C(W, h) ⟶^{C(β)} C(Y, g) ⟶^{C(α)} C(X, f) ⟶ 0 is a minimal (projective) presentation of C(X, f) by Lemma 5.3.

(v) Consider the ring morphism ϕ : End_{D_{F} (η)}((X, f)) → End_{D}(C(X, f)) where ϕ(r) = C(r). Suppose that r is a non-unit in End_{D_{F} (η)}((X, f)). Then C(r) is a nonunit by (iii), hence 1_{C(X,f)} − C(r) is a unit, since the ring End_{D}(C(X, f)) is local. Then by (iv), 1_{(X,f)} − r is a unit, thus End_{D_{F} (η)}((X, f)) is a local ring. The last part follows from the fact that End_{D}(X) is a factor ring of End_{D_{F} (η)}(T(X)).

6. Cleft Coextensions

Throughout this section we fix an η−extension D_{F} (η) of the abelian category D by the right exact endofunctor F of D, and suppose further that F has a right adjoint G. We fix counit ρ : F G → Id_{D} and unit σ : Id_{D} → GF of the adjoint pair (F, G). Consider the natural morphism ξ : F G → G where ξ := σF G ∘ GηG ∘ Gρ and define a functor H : D → D_{F} (η) as follows:

H(X) := (GX ⊕ X, ω_{X}) where ω_{X} = ( \begin{matrix}ξ_{X} & ρ_{X} \\ 0 & 0 \end{matrix} ), H(a) = ( \begin{matrix}Ga & 0 \\ 0 & a \end{matrix} ).

It is easy to see that H is indeed a functor from D to D_{F} (η). Consider the natural morphisms: ρ̃ : UH → Id_{D} , σ̃ : Id_{D_{F} (η)} → HU, given by ρ̃_{X} = ^{t}(0, 1_{X}), and σ̃_{(X,f)} = (σ_{X} ∘ Gf, 1_{X}), and define an endofunctor Ψ : D_{F} (η) → D_{F} (η), as follows : Ψ(X, f) = (GX, ξ_{X} − ρ_{X} ∘ σ_{X} ∘ Gf), and Ψ(a) = Ga. Further define a functor K : D_{F} (η) → D, as follows : K(X, f) := Ker(σ_{X} ∘ Gf) and if a : (X, f) → (Y, g) then K(a) := the unique D−morphism with K(a) ∘ ker(σ_{Y} ∘ Gg) = a ∘ (σ_{X} ∘ Gf). If (X, f) ∈ D_{F} (η), then usually we denote f^{∗} the adjoint morphism σ_{X} ∘ Gf.

Lemma 6.1. (1) (U, H) is an adjoint pair with unit σ̃ and counit ρ̃.

(2) D_{F} (η) is a cleft extension and coextension of D. More precisely setting ζ := ξ^{∗} = σ_{G} ∘ Gξ : G → G^{2} we obtain an associative natural morphism and an isomorphism of categories D_{F} (η) ≅ D^{G}(ζ).

(3) (Φ, Ψ) is an adjoint pair with counit ρ and unit σ and there is a short exact sequence of functors, where ȷ_{(X,f)} = ^{t}(1_{GX}, −σ_{X} ∘ Gf):

0 ⟶ Id_{D_{F} (η)} ⟶^{σ̃} UH ⟶^{ȷ} Ψ ⟶ 0

(4) (Z, K) is an adjoint pair of functors with counit κ : ZK → Id_{D_{F} (η)} defined by κ_{(X,f)} = ker(f^{∗}), unit the natural identification KZ = Id_{D}, and there is an exact sequence of functors:

0 ⟶ ZK ⟶^{κ} Id_{D_{F} (η)} ⟶^{ν} S ⟶ 0 .

Proof. We use the fact that D_{F} (η) is isomorphic to the cleft coextension D^{G}(ζ), where ζ = σ_{G} ∘ Gξ : G → G^{2} as in section 2. Details are left to the reader.

Since by the Lemma D_{F} (η) is also a cleft coextension of D, one can dualize the theory of the previous sections to obtain the dual results. For example we have the following: D_{F} (η) has enough injectives iff D has enough injectives. If KerK = 0, then D_{F} (η) has injective envelopes iff D has enough injective envelopes. If KerK = 0, then an object (X, f) ∈ D_{F} (η) is injective, resp. simple, iff HK(X, f) ≅ (X, f), resp. ZK(X, f) ≅ (X, f), and K(X, f) is injective, resp. simple, in D. The functor ZK is a radical in D_{F} (η) which is the socle functor in case D_{F} (η) has unions and D is semisimple. In this case we can construct easily the socle series of an arbitrary (X, f) ∈ D_{F} (η). Also the class of monomorphisms Ξ in D_{F} (η) which are split monomorphisms in D, is an injective class in D_{F} (η), (UH, σ̃) is a coresolving pair for Ξ, with corresponding Ξ-cosyzygy functor Ψ, and an object (X, f) ∈ D_{F} (η) is Ξ-injective iff σ̃_{(X,f)} splits. One has dually the costandard complex of any object of D_{F} (η) using the functorial Ξ-injective coresolution Ξ_{•} of Id_{D_{F} (η)}. Composing the exact complexes Σ^{•}, Ξ_{•} we get an infinite in both directions exact sequence which we call the complete resolution of Id_{D_{F} (η)}. We leave the details to the reader, observing that in case D_{F} (η) is a module category, then from section 2 we have that the functor F has a right adjoint.

Remark 6.2. By the dual of Lemma 5.9 and Corollary 5.10, we have:

(1) The functor K is exact on exact sequences with left end term from the subcategory I(Ξ) of Ξ-injective objects. (2) If D has enough injectives then the functors R_{Ξ}^{1}K, R^{1}K are isomorphic. (3) If KerK = 0, and if D has enough injectives, then an object (X, f) is injective ⇔ K(X, f) is injective and R_{Ξ}^{1}K(X, f) = Ker(f^{∗})/Im(ζ_{X} − G(f^{∗})) = 0. (4) If KerK = 0 and R_{Ξ}^{1}K(X, f) = 0, then a morphism a : (X, f) → (Y, g) is an isomorphism ⇔ K(a) is an isomorphism. (5) If KerK = 0, then the ring End_{D_{F} (η)}(H(X)) is local ⇔ the ring End_{D}(X) is local.

Now let (1) 0 ⟶ Y ⟶^{b} E ⟶^{a} X ⟶ 0 be a sequence in D, and consider the object

(M, m) in D_{F} (η), where M := GY ⊕E ⊕F X and m := ( \begin{matrix}ξ_{Y} & ρ_{Y} ∘ b & 0 \\ 0 & 0 & F a \\ 0 & 0 & η_{X} \end{matrix} ). Then

we have a sequence in D_{F} (η):

0 ⟶ H(Y) ⟶^{b̃} (M, m) ⟶^{ã} T(X) ⟶ 0 (2)

where ã = ( \begin{matrix}0 & 0 \\ a & 0 \\ 0 & 1_{F X} \end{matrix} ), and b̃ = ( \begin{matrix}1_{GY} & 0 & 0 \\ 0 & b & 0 \end{matrix} ).

The following generalizes an unpublished result of Marmaridis. For the concept of an almost split sequence we refer to [4]

Proposition 6.3. The sequence (1) is short exact in D_{F} (η) iff the sequence (2) is short exact in D. If further KerC = 0 and KerK = 0, then the following statements are equivalent:

(i) The sequence (1) is an almost split sequence in D. (ii) The sequence (2) is an almost split sequence in D_{F} (η).

Proof. It is easy to see that (1) is short exact iff (2) is short exact.

(i) ⇒ (ii) Suppose that (1) is an almost split sequence. Then End_{D}(X) is a local ring, hence End_{D_{F} (η)}[T(X)] is a local ring by Corrolary 5.10. If ã is split epic then U(ã) is split epic, thus trivially a is also split epic. Let c : (W, h) → T(X) be non split epic; then U(c) = c is also non split epic since otherwise c would be a split epic as a Σ-epic to the Σ-projective object T(X). Of course c = (c_{1}, c_{2}) : W → X ⊕ F X in D, and we have h ∘ c_{1} = 0 and h ∘ c_{2} = F c_{1} + F c_{2} ∘ η_{X}. Let d : X → W with d∘c_{1} = 1_{X}; then the morphism^{t}(d, F d∘h) : T(X) → (W, h) is a morphism in D_{F} (η)

with ^{t}(d, F d ∘ h) ∘ (c_{1}, c_{2}) = ( \begin{matrix}1_{X} & d ∘ c_{2} \\ 0 & 1_{F X} + F (d ∘ c_{2}) ∘ η_{X} \end{matrix} ). Since End_{D_{F} (η)}(T(X))

is local, ^{t}(d, F d∘h)∘(c_{1}, c_{2}) is unit or 1_{T(X)} −^{t}(d, F d∘h)∘(c_{1}, c_{2}) is a unit. An easy calculation shows that the second alternative is impossible, thus^{t}(d, F d∘h)∘(c_{1}, c_{2}) is a unit and (c_{1}, c_{2}) is split epic and this is not the case; hence c_{1} : W → X is non-split epic. Since a is right almost split we have that c_{1} = d ∘ a for a morphism d : W → E; since 0 = h ∘ c_{1} = h ∘ d ∘ a, there is a morphism e : F W → Y with : h ∘ d = e ∘ b. Consider now the morphism f : (W, h) → (M, m) given by f = (σ_{W} ∘ Ge, d, c_{2}). Taking into account the definition of the natural morphism ξ, we see easily that f is a morphism in D_{F} (η). Then f ∘ ã = c and this shows that ã is a right almost split morphism. Dual considerations show that b̃ is a left almost split morphism and (2) is an almost split sequence.

(ii) ⇒ (i) Suppose that (2) is an almost split sequence. Obviously End_{D}(X) is a local ring as a factor of the local ring End_{D_{F} (η)}[T(X)]. If a is split epic with

x ∘ a = 1_{X} then the D_{F} (η)−morphism p := ( \begin{matrix}0 & x & 0 \\ 0 & 0 & 1_{F X} \end{matrix} ) : T(X) → (M, m) has

the property that p ∘ ã = 1_{T(X)}, and this is impossible since ã is right almost split. Hence a is non-split epic. If c : W → X is non-split epic then trivially T(c) is non-

split epic, thus T(c) = f ∘ã for a morphism f = ( \begin{matrix}f_{1} & f_{2} & f_{3} \\ f_{4} & f_{5} & f_{6} \end{matrix} ) : T(W) → (M, m),

since ã is right almost split. Then c = f_{2} ∘ a and we infer that a is right almost split. By duality b is left almost split and (1) is an almost split sequence.

7. Applications to Cleft Extensions of Rings

Throughout this section we consider a cleft extension of rings: Λ = Γ ⋉_{ϑ}M, for a Γ-Γ-bimodule M and an associative Γ-Γ-bimodule morphism ϑ : M ⊗_{Γ} M → M. We identify the category of (right) Λ-modules Mod(Λ) with the η−extension category Mod(Γ)_{F} (η), where F := − ⊗_{Γ} M : Mod(Γ) → Mod(Γ), and η : F^{2} → F with η_{X} = 1_{X} ⊗ ϑ. Denoting by G := Hom_{Γ}(M, −) the right adjoint of F, we know that Mod(Λ) = Mod(Γ)_{F} (η) ≅ Mod(Γ)^{G}(ζ), where the associative natural morphism ζ : G → G^{2} of the previous section, is defined as follows: ζ_{X}(ϕ)(m)(n) = ϕ(ϑ(m ⊗ n)), where m, n ∈ M, and ϕ ∈ GX = Hom_{Γ}(M, X). The last isomorphism takes the right Λ−module (X, f) to the object (X, f^{∗}), where f^{∗} : X → GX is the adjoint morphism of f defined by f^{∗}(x)(m) = f(x ⊗ m) for x ∈ X, m ∈ M. It is easy to see that the functors T, U, C, Z, H, K of the previous sections, between Mod(Λ), Mod(Γ), have in our case the following realizations: C = − ⊗_{Λ} Γ_{Γ}, Z = Hom_{Γ}(_{Λ}Λ_{Γ}, −), T = − ⊗_{Γ} Λ_{Λ}, U = Hom_{Λ}(_{Γ}Λ_{Λ}, −), K = Hom_{Λ}(_{Γ}Γ_{Λ}, −), H = Hom_{Γ}(_{Λ}Λ_{Γ}, −). We describe now the functors Φ, Ψ, R, S.

Consider the exact sequence of endofunctors of Mod(Λ):

Σ^{•} : ⋯ ⟶ TF^{n}U ⟶ TF^{n−1}U ⟶ · · · · · · ⟶ TF U ⟶ TU ⟶ Id_{Mod(Λ)} ⟶ 0 .

We note that evaluating the sequence Σ^{•} at Λ, it is clear that the induced exact sequence is a resolution of Λ in Mod(Λ^{e}):

Σ^{•}(Λ) : ⋯ ⟶ Λ ⊗_{Γ} M ⊗_{Γ} ⋯ ⊗_{Γ} M ⊗_{Γ} Λ ⟶ ⋯
⋯ ⟶ Λ ⊗_{Γ} M ⊗_{Γ} M ⊗_{Γ} Λ ⟶ Λ ⊗_{Γ} M ⊗_{Γ} Λ ⟶ Λ ⊗_{Γ} Λ ⟶ Λ ⟶ 0.

Let Ω_{Λ/Γ}^{n} be the Λ−Λ−bimodule of n^{th}-Kahler differentials of Λ over Γ, i.e. the

image of the morphism: Λ ⊗_{Γ} M^{⊗^{n}} ⊗_{Γ} Λ ⟶ Λ ⊗_{Γ} M^{⊗^{n−1}} ⊗_{Γ} Λ, n ≥ 1. In particular Ω_{Λ/Γ}^{1} = the kernel of the multiplication : Λ ⊗_{Γ} Λ ⟶ Λ. The proof of the following proposition is an easy consequence of the results of the previous section and is left to the reader, noting that part (i) was first observed in [17].

Proposition 7.1. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of k−algebras over a commutative ring k and assume that k acts centrally on M.

(i) The exact sequence Σ^{∗}(Λ) is a Γ-Γ-relative resolution of Λ by Γ-Γ-relative projective bimodules in Mod(Λ^{e}). The Γ-relative Hochschild cohomology functors of Λ can be computed as:

H^{∗}(Λ, Γ, −) = H^{∗}Hom_{Λ^{e}}(Σ^{•}(Λ), −).

(ii) Φ ≅ − ⊗_{Λ} Ω_{Λ/Γ}^{1}, Ψ ≅ Hom_{Λ}(Ω_{Λ/Γ}^{1}, −) and Φ (resp. Ψ) is a lifting of the loop space (resp. suspension) functor of Heller : \underline{Mod}_{(Λ,Γ)}(Λ) → \underline{Mod}_{(Λ,Γ)}(Λ) (resp. : \overline{Mod}_{(Λ,Γ)}(Λ) → \overline{Mod}_{(Λ,Γ)}(Λ)) in the stable category of Mod(Λ) modulo the relative Γ-projectives (resp. injectives). In addition: Ω_{Λ/Γ}^{n} = ⊗_{Λ}^{n}Ω_{Λ/Γ}^{1}.

(iii) If Γ is k−separable, then Σ^{•}(Λ) is a k-relative projective resolution of Λ in Mod(Λ^{e}), thus H^{∗}(Λ, −) = H^{∗}(Σ^{•}(Λ), −). If further k is semisimple, then Φ ≅ − ⊗_{Λ} Ω_{Λ/Γ}^{1} (Ψ ≅ Hom_{Λ}(Ω_{Λ/Γ}^{1}, −)) is a lifting of the loop space (suspension) functor of Heller in the stable category \underline{Mod}(Λ) (\overline{Mod}(Λ)) of Mod(Λ) modulo projectives (injectives).

Consider now the kernel Ker(ε) of the canonical split epimorphism ε : Λ → Γ. Then as an object of Mod(Λ): Ker(ε) = (M, ϑ). Hence by the definition of the functor R : Mod(Λ) → Mod(Λ), we have easily that: R(X, f) = (X, f)(M, ϑ). Similarly ZK(X, f) = Ann_{(M,ϑ)}(X, f) = {x ∈ X : f(x ⊗ m) = 0, ∀m ∈ M}. Hence S(X, f) = (X, f)/Ann_{(M,ϑ)}(X, f). We are interested first for the relation between the ideals (M, ϑ), J ac(Λ).

Lemma 7.2. For the cleft extension Λ = Γ ⋉_{ϑ}M, the following are equivalent:

(i) (M, ϑ) ⊆ J ac(Λ). (ii) Z(J ac(Γ)) ⊕ (M, ϑ) = J ac(Λ). (iii) ∀n ∈ M, ∃m ∈ M : ϑ(m ⊗ n) = ϑ(n ⊗ m) = −m − n. (iv) If (X, f) ∈ Mod(Λ) is a simple module, then C(X, f) is a simple Γ-module

and (X, f) ≅ ZC(X, f).

(v) The epimorphism ϖ : Λ → Γ is minimal in Mod(Λ). (vi) For any finitely generated non-zero right Λ-module (X, f), the epimorphism

λ_{(X,f)} : (X, f) → ZC((X, f)) is minimal. (vii) For any finitely generated (X, f) ∈ Mod(Λ) : C(X, f) = 0 ⇒ (X, f) = 0.

Proof. (i) ⇔ (ii) Obviously 1 − r ∈ Γ is invertible iff (1 − r, 0) ∈ Λ is invertible. Hence Z(J ac(Γ)) = {(r, 0) : r ∈ J ac(Γ)} ⊆ J ac(Λ). Then by (i), I := ZJ ac(Γ) ⊕ (M, ϑ) ⊆ J ac(Λ), and consequently we can define a ring-homomorphism ε^{!} : Λ/I → Γ/J ac(Γ), by ε^{!}[(r, m) + I] = [r + J ac(Γ)]. We see easily that ε^{!} is an isomorphism of rings; but then J ac(Λ) ⊆ I. Hence: ZJ ac(Γ) ⊕ (M, ϑ) = J ac(Λ).

(i) ⇔ (iii) The proof is a simple calculation using the characterization of the elements of Jacobson radical of a ring.

(i) ⇔ (iv) If (M, ϑ) ⊆ J ac(Λ) and (X, f) ∈ Mod(Λ) is a simple module, then by Nakayama lemma, R(X, f) = (X, f)(M, ϑ) = 0. But then (X, f) ≅ ZC(X, f) and of course C(X, f) is a simple Γ-module. If (iv) holds then any simple Λ−module is of the form Z(S) where S is a simple Γ−module. Since RZ = 0 and R is multiplication with the ideal (M, ϑ), it follows that (M, ϑ) annihilates any simple Λ−module. Hence: (M, ϑ) ⊆ J ac(Λ).

It is well-known and easy to see that (i) ⇔ (v) [14]. The equivalences (v) ⇔ (vi) ⇔ (vii) follows from Lemma 5.3 and Proposition 5.4.

Definition 7.3. (α) The morphism ϑ : M ⊗_{Γ} M → M is called nilpotent iff there is an i ∈ ℕ :

(1_{M} ⊗ 1_{M} ⊗ ⋯ ⊗ 1_{M} ⊗ ϑ) ∘ ⋯ ∘ (1_{M} ⊗ ϑ) ∘ ϑ = 0 : M^{⊗_{i+1}} → M.

(β) The morphism ϑ : M ⊗_{Γ}M → M is called right T-nilpotent iff for any (m_{i})_{i∈ℕ} ⊆ M there is a j ∈ ℕ :

ϑ(m_{j} ⊗ ϑ(m_{j−1} ⊗ ϑ(m_{j−2} ⊗ ⋯ ⊗ ϑ(m_{3} ⊗ ϑ(m_{2} ⊗ m_{1})) ⋯) = 0.

We consider the category Mod(Λ^{op}) as the cleft extension D̃F̃(η̃), where D̃ := Mod(Γ^{op}), F̃ := M ⊗_{Γ} − : Mod(Γ^{op}) → Mod(Γ^{op}), η̃_{X} = ϑ ⊗ 1_{X}. Denote also by G̃ : Mod(Γ^{op}) → Mod(Γ^{op}) the right adjoint of F̃, and by ζ̃ : G̃ → G̃^{2} the corresponding associative natural morphism. By C̃, K̃, Z̃, T̃, Ũ, H̃, R̃ we denote the standard functors between the categories Mod(Λ^{op}) = D̃F̃(η̃) ≅ D̃G̃(ζ̃) and Mod(Γ^{op}) = D̃.

Proposition 7.4. If Λ = Γ ⋉_{ϑ}M is a cleft extension of rings then:

(i) The morphism ϑ is nilpotent iff the category Mod(Λ) is admissible iff the functor R is nilpotent iff Mod(Λ^{op}) is admissible iff the functor R̃ is nilpotent.

(ii) The morphism ϑ is right (left) T-nilpotent iff KerC = 0 (KerC̃ = 0). (iii) The morphism ϑ is left (right) T-nilpotent iff KerK = 0 (KerK̃ = 0).

Proof. (i) Assume that Mod(Λ) is admissible. Then by Proposition 5.2 the functor R is locally nilpotent. Since ∀(X, f) ∈ Mod(Λ) : R(X, f) = (X, f)(M, ϑ) we have that R(M, ϑ) = (M, ϑ)^{2} and thus inductively: R^{n}(M, ϑ) = (M, ϑ)^{n+1}. Hence ∃n ∈ ℕ : (M, ϑ)^{n+1} = 0. By the definition of the multiplication of the ring Λ, the ideal (M, ϑ) is nilpotent iff the morphism ϑ is nilpotent. Let now the morphism ϑ be nilpotent. Then the ideal (M, ϑ) of Λ is nilpotent. Hence (M, ϑ)^{n} = 0, for some n ∈ ℕ. If (X, f) ∈ Mod(Λ), then R^{n}(X, f) = (X, f)(M, ϑ)^{n} = 0. It follows that the functor R is nilpotent and Mod(Λ) is admissible, by Proposition 5.2. Obviously the morphism ϑ is nilpotent iff the functor R is nilpotent.

(ii) Of course the morphism ϑ is right T-nilpotent iff the ideal (M, ϑ) is right Tnilpotent. Hence ∀(X, f) ∈ Mod(Λ) : (X, f) ∈ KerC ⇒ (X, f) = 0 ⇔ C(X, f) = 0 ⇒ (X, f) = 0 ⇔ R(X, f) = (X, f) ⇒ (X, f) = 0 ⇔ (X, f)(M, ϑ) = (X, f) ⇒ (X, f) = 0 ⇔ the ideal (M, ϑ) is right T-nilpotent, where the last equivalence follows from the well known characterization of right T-nilpotent ideals [38].

(iii) We know that ZK(X, f) = Ann_{(M,ϑ)}(X, f), ∀(X, f) ∈ Mod(Λ). Since ϑ is left T-nilpotent ⇔ the ideal (M, ϑ) is left T-nilpotent, we have: ϑ is left T-nilpotent ⇔ ∀(X, f) ∈ Mod(Λ), Ann_{(M,ϑ)}(X, f) = 0 ⇒ (X, f) = 0, see [25]. Hence ϑ is left T-nilpotent ⇔ ∀(X, f) ∈ Mod(Λ), ZK(X, f) = 0 ⇒ (X, f) = 0 ⇔ KerK = 0.

Using the above observations we obtain easily the following:

Proposition 7.5. Suppose that Λ = Γ ⋉_{ϑ}M is a cleft extension of rings.

(i) If the ring Γ is semiprimary and the morphism ϑ is nilpotent, then Λ is semiprimary. If the ring Λ is semiprimary, then Γ is semiprimary.

(ii) If the ring Γ is right perfect and the morphism ϑ is right T-nilpotent, then Λ

is right perfect. If the ring Λ is right perfect, then Γ is right perfect. (iii) If the ring Γ is semiperfect and (M, ϑ) ⊆ J ac(Λ), then Λ is semiperfect. If

the ring Λ is semiperfect, then Γ is semiperfect. (iv) Suppose that ϑ is (left) right T-nilpotent. Then Λ is (left) right Noetherian iff

Γ is (left) right Noetherian and M is finitely generated (left) right Γ-module. Also Λ is (left) right Artinian iff Γ is (left) right Artinian and M is finitely generated (left) right Γ-module.

(v) If (M, ϑ) ⊆ Jac(Λ) then J ac(Λ) is nilpotent iff J ac(Γ) is nilpotent and ϑ is nilpotent. In this case we have :

LL(Γ) ≤ LL(Λ) ≤ LL(Γ) + c(ϑ)

where LL denotes Loewy length and c(ϑ) the nilpotency class of ϑ.

Proof. (i) If the morphism ϑ is nilpotent, then the ideal (M, ϑ) ⊆ J ac(Λ), and by Lemma 7.2 we have J ac(Λ) = Z(J ac(Γ)) ⊕ (M, ϑ) = J ac(Λ), and Λ/J ac(Λ) ≅ Γ/J ac(Γ). Hence if further Γ is semiprimary, the last isomorphism shows that Λ is semiprimary. If Λ is semiprimary then it follows easily that Γ is semiprimary.

(ii) If the morphism ϑ is right T-nilpotent, then KerC = 0 by Proposition 7.4. Hence Γ is right perfect iff Λ is right perfect as follows from Proposition 5.4. If Λ is right perfect, then Γ is right perfect as a factor ring of Λ.

(iii) The proof is similar as in (ii), using Lemma 7.2.

(iv) Suppose first that Λ is right Noetherian. Then obviously Γ is right Noetherian. Since (M, ϑ) is finitely generated nil ideal of Λ, by Levitzki Theorem we have that (M, ϑ) is a nilpotent ideal and thus the morphism ϑ is nilpotent. But then by Proposition 7.4, Mod(Λ) is admissible. Using the finite R-filtration of (M, ϑ), an easy induction argument shows that M is finitely generated in Mod(Γ). If Γ is right Noetherian then obviously Z(Γ) = Γ_{Λ} is right Noetherian. If M_{Γ} is finitely generated, then (M, ϑ) is Noetherian as a right Λ−module. Indeed take any ascending sequence of right ideals in (M, ϑ) and apply to this sequence the faithful functor U. Then we get an ascending sequence of right submodules of M; since M is finitely generated right Γ−module and Γ is right Noetherian, we have that M_{Γ} is Noetherian. Hence the last sequence eventually stops, and since U is faithful the original sequence stops implying that (M, ϑ)_{Λ} is Noetherian. Then Λ is right Noetherian, as follows from the short exact sequence 0 ⟶ (M, ϑ) ⟶ Λ ⟶ Γ ⟶ 0. The Artinian case follows similarly. Part (v) is easy and is left to the reader.

Remark 7.6. Suppose that ϑ is right T-nilpotent.

(1) P is an indecomposable (projective) right Γ−module ⇔ T(P) is an indecomposable (projective) right Λ−module. Indeed, since ϑ is right T-nilpotent by Proposition 7.4, KerC = 0, thus T preserves and reflects indecomposable (projective) modules by Proposition 5.4.

(2) If (X, f) ∈ Mod(Λ) is finitely generated, then Rad(X, f) = Ker(λ_{(X,f)} ∘ Zπ) where π : C(X, f) → C(X, f)/RadC(X, f) is the canonical projection. Indeed, since ϑ is right T-nilpotent the epimorphism λ_{(X,f)} : (X, f) → ZC(X, f) is minimal. Also since (X, f) is finitely generated, the canonical epimorphism π : C(X, f) → C(X, f)/RadC(X, f) is minimal. Thus the composition λ_{(X,f)} ∘ Z(π) : (X, f) → Z(C(X, f)/RadC(X, f)) is a minimal epimorphism and the assertion follows.

(3) For finitely generated X ∈ Mod(Γ), RadTX = (RadX ⊕ F X, ( \begin{matrix}0 & F μ \\ 0 & η_{X} \end{matrix} ))

where μ : RadX ↪ X is the natural inclusion. In particular: C(RadT(X)) = RadX ⊕ Coker(η_{X}). Consequently RadT(X) = 0 ⇔ RadX = 0 and F X = 0. Indeed, the first part follows by computation, using (2). The second part follows from the easy calculation: C(RadT(X)) = RadX ⊕ Coker(η_{X}).

We recall that a ring R is a Nakayama ring iff R is Artinian and any left and right indecomposable projective module is uniserial [14].

Proposition 7.7. Suppose that Λ = Γ ⋉_{ϑ}M is a cleft extension of rings, and the morphism ϑ is left and right T-nilpotent. Then we have the following:

(i) The ring Λ is Nakayama. (ii) The ring Γ is Nakayama, M is finitely generated as a left and right Γ-module,

and the following conditions hold:

(a) For any indecomposable projective right Γ-module P: P ⊗_{Γ} M = 0 or the right Γ-module P ⊗_{Γ} Coker(ϑ)/Rad(P ⊗_{Γ} Coker(ϑ)) is simple.

(b) For any indecomposable projective left Γ-module P: M ⊗_{Γ} P = 0 or the left Γ-module Coker(ϑ) ⊗_{Γ} P/Rad(Coker(ϑ) ⊗_{Γ} P) is simple.

Proof. (i) ⇒ (ii) Suppose that Λ is Nakayama. Then Λ is Artinian and thus by Proposition 7.5, Γ is Artinian and M is finitely generated as left and right Γ−module. Consider an indecomposable projective right Λ−module T(P); then P is projective Γ-module, and P is indecomposable since ϑ is right T-nilpotent. From [14] it follows that either RadT(P)/Rad^{2}T(P) is a simple Λ−module or RadT(P) = 0. Setting R_{P} := Coker(η_{P})/RadCoker(η_{P}) and using remark 7.6, we compute easily: RadT(P)/Rad^{2}T(P) = Z(RadP/Rad^{2}P) ⊕ Z(R_{P}). Hence Z(RadP/Rad^{2}P) ⊕ Z(R_{P}) is a simple Λ−module or RadT(P) = 0. From Remark 7.6, RadT(P) = 0 ⇔ RadP = 0 or F P = 0. This implies that either RadP/Rad^{2}P is a simple Γ-module and F P = 0 or RadP = 0 and R_{P} is a simple Γ-module. But then from [14], p.225, we have that in Γ every indecomposable right projective module P is uniserial and either R_{P} is a simple Γ-module or F P = 0. Using the left T-nilpotency of ϑ we get similarly the same conclusion for the left side.

(ii) ⇒ (i) By the hypothesis and Proposition 7.5(iv), we have that Λ is Artinian. Now using (a), (b), the fact that Γ is Nakayama and working as above we get similarly that Λ is Nakayama.

By Proposition 5.4 we know the projective and simple modules of Mod(Λ), where Λ = Γ⋉_{ϑ}M, in terms of the corresponding projective and simple modules of Mod(Γ), if KerC = 0 or equivalently if ϑ is right T-nilpotent. We turn our attention to flat modules. Consider the character module functor 𝔻 := Hom_{ℤ}(−, ℚ/ℤ) : Mod(Λ) → Mod(Λ^{op}). We denote also by 𝔻 the corresponding functors between Mod(Γ^{op}) and Mod(Γ). Using the adjoint associativity, we obtain easily the following natural isomorphisms: ω : 𝔻F ≅ G̃𝔻, H𝔻 ≅ 𝔻T̃. We recall that Σ, resp. Ξ, is the projective, resp. injective, class of morphisms in Mod(Λ) or Mod(Λ^{op}), which are split epimorphisms, resp. monomorphisms, in Mod(Γ) or Mod(Γ^{op}).

Lemma 7.8. There exists a natural isomorphism of functors, ∀n ≥ 0:

𝔻L_{n}^{Σ}C ≅ R_{Ξ}^{n}K̃𝔻 : Mod(Λ) → Mod(Γ^{op}).

Proof. If (X, f) ∈ Mod(Λ), then using the natural isomorphism ω : 𝔻F ≅ G̃𝔻, we see easily that 𝔻(X, f) = (𝔻(X), 𝔻(f)∘ω_{X}) as object of Mod(Λ^{op}) = Mod(Γ^{op})G̃(ζ̃), and ω_{X}^{−1} ∘ 𝔻(η_{X}) = ζ̃_{𝔻(X)} ∘ G̃(ω_{X}^{−1}) ∘ ω_{F X}^{−1}. These relations show that the family of isomorphisms (1_{𝔻(X)}, ω_{X}, ω_{F X} ∘ G̃(ω_{X}), ω_{F^{2}X} ∘ G̃(ω_{F X}) ∘ G̃^{2}(ω_{X}), ⋯) is an isomorphism between the costandard complex K̃(Ξ_{•}^{(𝔻(X,f)}) of 𝔻(X, f) with respect to the injective class Ξ in Mod(Γ^{op})G̃(ζ̃), and the image of the standard complex C(Σ_{(X,f)}^{•}) under the functor 𝔻. Hence these two complexes have isomorphic coho-

mology. But the n^{th}−cohomology of K̃(Ξ_{•}^{(𝔻(X,f)}) is the object R_{Ξ}^{i}K̃𝔻(X, f), and the n^{th}−cohomology of 𝔻C(Σ_{(X,f)}^{•}) is obviously the object 𝔻L_{i}^{Σ}C(X, f).

Corollary 7.9. For a right Λ−module (X, f) ∈ Mod(Λ), we have the following.

(i) If the morphism ϑ is right T-nilpotent, then the following are equivalent: (a) (X, f) is projective (flat).

(b) C(X, f) is projective (flat) and (X, f) ≅ TC(X, f).

(c) C(X, f) is projective (flat) and Ker(f) = Im(1_{X} ⊗ ϑ − 1_{M} ⊗ f). (ii) If the morphism ϑ is left T-nilpotent, then the following are equivalent:

(a) (X, f) is injective.

(b) K(X, f) is injective and HK(X, f) ≅ (X, f).

(c) K(X, f) is injective and Ker(f^{∗}) = Im(ζ_{X} − Hom_{Γ}(M, f^{∗})).

Proof. (i) The case of projectives follows from Proposition 5.4. By the Bourbaki-Lambek Theorem, we have: (X, f) is flat ⇔ 𝔻(X, f) is injective. Since ϑ is right T-nilpotent, by Proposition 7.4 we have KerK̃ = 0, hence: 𝔻(X, f) is injective ⇔ K̃𝔻(X, f) is injective and H̃K̃𝔻(X, f) ≅ 𝔻(X, f). But K̃𝔻(X, f) = Ker(𝔻f ∘ ω_{X}) = 𝔻Coker(f) = 𝔻C(X, f), and H̃K̃𝔻(X, f) = H̃𝔻C(X, f) ≅ 𝔻TC(X, f). Now 𝔻Coker(f) is injective ⇔ Coker(f) is flat, and 𝔻TC(X, f) ≅ 𝔻(X, f) ⇔ TC(X, f) ≅ (X, f). Hence we have proved the equivalence (a) ⇔ (b). For the equivalence (b) ⇔ (c), we use the natural isomorphism H𝔻 ≅ 𝔻T̃, the above Lemma, and the fact that a module (Y, g) ∈ Mod(Λ^{op}) = Mod(Γ^{op})G̃(ζ̃) is injective ⇔ K̃(Y, g) is injective and H̃K̃(Y, g) ≅ (Y, g) ⇔ K̃(Y, g) is injective and R_{Ξ}^{1}K̃(Y, g) = 0, as follows from Section 6, since by Proposition 7.4, KerK̃ = 0. We have: C(X, f) is flat and Ker(f) = Im(1_{X} ⊗ ϑ − 1_{M} ⊗ f) ⇔ 𝔻C(X, f) is injective and L_{1}^{Σ}C(X, f) = 0 ⇔ K̃𝔻(X, f) is injective and 𝔻L_{1}^{Σ}C(X, f) = 0 ⇔ K̃𝔻(X, f) is injective and R_{Ξ}^{1}K̃𝔻(X, f) = 0 ⇔ K̃𝔻(X, f) is injective and H̃K̃𝔻(X, f) ≅ 𝔻(X, f) ⇔ 𝔻C(X, f) is injective and 𝔻TC(X, f) ≅ 𝔻(X, f) ⇔ C(X, f) is flat and TC(X, f) ≅ (X, f).

(ii) Since by Proposition 7.4, ϑ is left T-nilpotent ⇔ KerK = 0, the result follows from Section 6.

For the rest of this section we suppose that Λ is a finite-dimensional k-algebra over a perfect field k. By the Principal Theorem of Wedderburn, Λ is a cleft extension of the semisimple subalgebra Γ := Λ/J ac(Λ), and Λ = Γ ⊕ J ac(Λ). We set D = mod(Γ), M = J ac(Λ) as a Γ-Γ-bimodule, F = − ⊗_{Γ} M : mod(Γ) → mod(Γ), define ϑ : M ⊗_{Γ} M → M by ϑ(m_{1} ⊗ m_{2}) = (0, m_{1})(0, m_{2}) and η : F^{2} → F by η_{X} = 1_{X} ⊗ ϑ. We use also the previous notations concerning an η-extension and in particular since in our setting, F has a right adjoint G = Hom_{Γ}(M, −), we have that D_{F} (η) = mod(Λ) is a cleft coextension of D and the functors H, K of section 6 are defined. Of course ϑ, η and ζ are nilpotent. Let {S_{1}, S_{2}..., S_{n}} be a complete set of non-isomorphic simple right Γ-modules. Then {Z(S_{1}), Z(S_{2})..., Z(S_{n})} is a complete set of non-isomorphic simple right Λ-modules, {T(S_{1}), T(S_{2})..., T(S_{n})} is a complete set of non-isomorphic indecomposable projective right Λ−modules, and {H(S_{1}), H(S_{2})..., H(S_{n})} is a complete set of non-isomorphic indecomposable

injective right Λ−modules. The short exact sequences 0 ⟶ ΦZ(S_{i}) ⟶^{(0,1_{F S_{i}})}

T(S_{i}) ⟶^{^{t}(1_{S_{i}} ,0)} Z(S_{i}) ⟶ 0, i = 1, 2, ..., n, are projective covers, and the short

exact sequences 0 ⟶ Z(S_{i}) ⟶^{(0,1_{S_{i}})} H(S_{i}) ⟶^{^{t}(1_{GS_{i}} ,0)} ΨZ(S_{i}) ⟶ 0, i = 1, 2, ..., n,

are injective envelopes of the simple Λ−modules. If (X, f) ∈ mod(Λ), then (X, f) is projective (injective) iff (X, f) ≅ TC(X, f) ((X, f) ≅ HK(X, f)). The projective cover of (X, f) is the morphism ^{t}(a, F a ∘ f) : TC(X, f) → (X, f), where a : C(X, f) → X is an arbitrary morphism in mod(Γ) with a ∘ coker(f) = 1_{C(X,f)}. Dually the injective envelope of (X, f) is the morphism (f^{∗} ∘ Gb, b) : (X, f) → HK(X, f), where b : K(X, f^{∗}) → X is an arbitrary morphism in mod(Γ) with ker(f^{∗}) ∘ b = 1_{K(X,f^{∗})}, and f^{∗} : X → GX is the adjoint morphism of f : F X → X. Also Soc(X, f) = ZK(X, f) and (X, f)/Rad(X, f) = ZC(X, f).

Denoting by s(Λ) the number of isomorphic classes of the simple right Λ−modules, we have the following simple proof of a result of L.E.P. Huppert [21].

Corollary 7.10. Suppose that Λ is a finite dimensional k-algebra over an algebraically closed or a perfect field k. Then the following are equivalent :

(i) Λ is Nakayama algebra. (ii) At most an arrow begins from a vertex and at most an arrow ends to a vertex

in the ordinary quiver of Λ. (iii) ∀j = 1, ..., s(Λ) : ∑_{i=1}^{s(Λ)} dim_{k}Ext_{Λ}^{1}[S_{i}, S_{j}] ≤ 1, ∑_{i=1}^{s(Λ)} dim_{k}Ext_{Λ}^{1}[S_{j}, S_{i}] ≤ 1.

Proof. Trivially (ii) ⇔ (iii). The equivalence (i) ⇔ (iii) follows directly by Proposition 7.7, and the formula (3) of Remark 5.7.

Consider now the exact sequence of endofunctors of mod(Λ):

Σ^{•} : ⋯ ⟶ TF^{n}U ⟶ TF^{n−1}U ⟶ · · · · · · ⟶ TF U ⟶ TU ⟶ Id_{mod(Λ)} ⟶ 0

which associates to any right Λ−module (X, f) a projective resolution of (X, f), and consider also the exact sequence of endofunctors of mod(Λ):

Ξ_{•} : 0 ⟶ Id_{mod(Λ)} ⟶ HU ⟶ HGU ⟶ · · · · · · ⟶ HG^{n−1}U ⟶ HG^{n}U ⟶ ⋯

which associates to any right Λ−module (X, f) an injective resolution of (X, f). Finally consider the complete resolution of Id_{mod(Λ)}:

⋯ ⟶ TF^{2}U ⟶^{ϑ^{2}} TF U ⟶^{ϑ^{1}} TU ⟶^{ω} HU ⟶^{ϑ_{1}} HGU ⟶^{ϑ_{1}} HG^{2}U ⟶ ⋯

which is the Yoneda composition of Σ^{•} and Ξ_{•}. We observe that setting Γ = Λ/Jac(Λ), the projective resolution Σ^{•}(Λ) of Λ in mod(Λ^{e}) has the property that ∀(X, f) ∈ mod(Λ): (X, f) ⊗_{Λ} Σ^{•}(Λ), is the projective resolution Σ_{(X,f)}^{•} of (X, f),

and Hom_{Λ}[Σ^{•}(Λ), (X, f)] is the injective resolution Ξ_{•}^{(X,f)} of (X, f). Observe also that the bimodule of n^{th}−Kahler differentials Ω_{Λ/Γ}^{n} of Λ over Γ, is an n^{th}-syzygy of Λ in Mod(Λ^{e}). The next result gives a new answer to a problem of E. Green [3].

Corollary 7.11. Suppose that Λ is a finite dimensional k-algebra over an algebraically closed or a perfect field k. Let Γ = Λ/J ac(Λ) and M = J ac(Λ).

(1) There exists an adjoint pair of exact endofunctors (Φ, Ψ) of mod(Λ), with the following properties: Φ preserves projective modules and the induced functor \underline{Φ} : \underline{mod}(Λ) → \underline{mod}(Λ) is the loopspace functor Ω. Ψ preserves injective modules and the induced functor \overline{Ψ} : \overline{mod}(Λ) → \overline{mod}(Λ) is the suspension functor Σ. Further: Φ ≅ − ⊗_{Λ} Ω_{Λ/Γ}^{1} ≅ − ⊗_{Λ} M ⊗_{Γ} Λ, Ψ ≅ Hom_{Λ}(Ω_{Λ/Γ}^{1}, −) ≅ Hom_{Λ}(M ⊗_{Γ} Λ, −).

(2) There exists an adjoint pair (Δ, ∇) of endofunctors of mod(Λ), with the following properties: Δ sends injective modules to projective ones, and the induced functor \overline{Δ} : \overline{mod}(Λ) → \underline{mod}(Λ) is the functor TrD. ∇ sends projective modules to injective ones, and the induced functor \underline{∇} : \underline{mod}(Λ) → \overline{mod}(Λ) is the functor DTr. Further we have natural isomorphisms: N^{+}Δ ≅ Ψ^{2} and N^{−}∇ ≅ Φ^{2} where N^{+} is the Nakayama functor and N^{−} its right adjoint. If N := CokerHom_{Λ^{e}}(ϑ_{Λ}^{1}, Λ^{e}) ∈ mod(Λ^{e}), then: Δ ≅ − ⊗_{Λ} N and ∇ ≅ Hom_{Λ}(N, −).

Proof. The assumptions on Λ allows us to consider Λ as a cleft extension of the semisimple subalgebra Γ = Λ/J ac(Λ). Then part (1) follows from Proposition 7.1.

(2) Consider the Nakayama functor N^{+} := −⊗_{Λ}D(Λ) and its right adjoint N^{−} := Hom_{Λ}(D(Λ), −), where D is the ordinary duality with respect to k. Consider also the complete resolution of Id_{mod(Λ)} constructed above, and set: ∇ := Ker(N^{+}(ϑ^{1})) and Δ := Coker(N^{−}ϑ_{1})). If (X, f) is a projective module then obviously ∇(X, f) = N^{+}Φ^{2}(X, f), thus ∇(X, f) is injective module, since Φ preserves projectives and N^{+} sends projectives to injectives. Similarly if (X, f) is an injective module then Δ(X, f) = N^{−}Ψ^{2}(X, f), thus Δ(X, f) is a projective module. By construction we have \overline{Δ} = TrD and \underline{∇} = DTr in the stable categories. Now we show that (Δ, ∇) is an adjoint pair. Consider the following exact sequences

0 ⟶ ∇ ⟶ N^{+}TF U ⟶^{N^{+}ϑ^{1}} N^{+}TU ⟶ N^{+} ⟶ 0 (1)

0 ⟶ N^{−} ⟶ N^{−}HU ⟶^{N^{−}ϑ_{1}} N^{−}HGU ⟶ Δ ⟶ 0 (2)

0 ⟶ Φ^{2}Δ ⟶ TF UΔ ⟶^{ϑ_{1}Δ} TUΔ ⟶ Δ ⟶ 0 (3)

Since N^{−}HGU, N^{−}HU take projective values there exists morphisms α : N^{−}HGU ⟶ TUΔ and β : N^{−}HU ⟶ TF UΔ such that: N^{−}ϑ_{1} ∘ α = β ∘ ϑ^{1}Δ, hence N^{+}N^{−}ϑ_{1} ∘ N^{+}α = N^{+}β ∘ N^{+}ϑ^{1}Δ. But since (4): N^{+}N^{−}HGU ≅ HGU, N^{+}N^{−}HU ≅ HU and KerN^{+}ϑ^{1}Δ = ∇Δ, there exists a unique natural morphism γ : Id_{mod(Λ)} → ∇Δ, such that γ ∘ ker(N^{+}(ϑ^{1})Δ) = σ̃ ∘ μ_{HU}^{−1} ∘ N^{+}(β), where σ̃ is the unit of the adjoint pair (U, H), and μ is the counit of the adjoint pair (N^{+}, N^{−}). Dually there exists a natural morphism δ : Δ∇ → Id_{mod(Λ)}. The proof that γ is the unit and δ is the counit of the adjoint pair (Δ, ∇) is standard and is left to the reader. Finally since by construction Coker(N^{+}N^{−}(ϑ_{1}) = N^{+}Δ, using the relations (4), we have also that Coker(N^{+}N^{−}(ϑ_{1})) ≅ Ψ^{2}, hence N^{+}Δ ≅ Ψ^{2}, and dually N^{−}∇ ≅ Φ^{2}. To prove the last assertion, we observe first that we have isomorphisms:

HU(Λ_{Λ}) = Hom_{Γ}(_{Λ}Λ_{Γ}, Hom_{Λ}(_{Γ}Λ_{Λ}, Λ_{Λ})) ≅ Hom_{Λ}(Λ ⊗_{Γ} Λ, Λ_{Λ}),

N^{−}HU(Λ_{Λ}) ≅ Hom_{Λ}(D(Λ), Hom_{Λ}(Λ ⊗_{Γ} Λ, Λ_{Λ})) ≅ Hom_{Λ}(D(Λ) ⊗_{Λ} Λ ⊗_{Γ} Λ, Λ_{Λ})

≅ Hom_{Λ^{e}}(Λ ⊗_{Γ} Λ, Λ^{e})

where the last isomorphism follows from [3]. Similarly N^{−}HGU(Λ_{Λ}) ≅ Hom_{Λ^{e}}(Λ⊗_{Γ} J ac(Λ) ⊗_{Γ} Λ, Λ^{e}). Moreover the following diagram commutes:

Hom_{Λ^{e}}(Λ ⊗_{Γ} Λ, Λ^{e}) ⟶^{(ϑ_{Λ}^{1},Λ^{e})} Hom_{Λ^{e}}(Λ ⊗_{Γ} J ac(Λ) ⊗_{Γ} Λ, Λ^{e})
≅↓  ≅↓
N^{−}HU(Λ_{Λ})  ⟶^{N^{−}(ϑ_{Λ}^{1})}  N^{−}HGU(Λ_{Λ}).

By construction: Δ(Λ_{Λ}) = N and then Δ ≅ − ⊗_{Λ} N and ∇ ≅ Hom_{Λ}(N, −).

We have seen that Φ : mod(Λ) → mod(Λ) is a lifting of the loop space functor of Heller Ω : \underline{mod}(Λ) → \underline{mod}(Λ), if Γ is semisimple. Since gl.dimΛ < ∞ iff the functor Ω is nilpotent, it is interesting to know when the functor Φ is nilpotent.

Corollary 7.12. Suppose that Λ is a finite dimensional k-algebra over an algebraically closed or a perfect field k. Then the following are equivalent:

(i) The lifting Φ of the loop functor Ω in mod(Λ) is nilpotent. (ii) The functor F := − ⊗_{Γ} J ac(Λ) is nilpotent. (iii) The ordinary quiver of Λ has no oriented cycles, i.e. Λ is triangular. (iv) gl.dimΛ/J ac(Λ)^{2} < ∞. If one of these equivalent statements is true then: gl.dimΛ ≤ c(Φ) ≤ s(Λ) − 1.

Proof. The equivalence (i) ⇔ (ii) is obvious. Suppose that Φ is nilpotent and without loss of generality let: 1. → 2. → 3. → ⋯ → m. → 1. be an oriented cycle in the ordinary quiver of Λ. Using formula (3) of Remark 5.7, we have non-zero epics α_{i} : ΦZ(S_{i}) → Z(S_{i+1}), i = 1, 2, ..., m − 1, and α_{m} : ΦZ(S_{m}) → Z(S_{1}). Then we have a non-zero epic α := Φ^{m−1}(α_{1}) ∘ Φ^{m−2}(α_{2}) ⋯ Φ(α_{m−1}) ∘ α_{m} : Φ^{m}(Z(S_{1})) → Z(S_{1}), thus a non-zero epic α : F^{m}S_{1} → S_{1}. Hence S_{1} is a direct summand of F^{m}S_{1}, so F, Φ, cannot be nilpotent. Conversely if F^{m} ≠ 0, ∀m ≥ 1, then there is an oriented cycle of length at least m, as follows from the multiplication of Λ.

The equivalence (iii) ⇔ (iv) is well known. We include a proof in our setting. Consider the algebra Λ/J ac(Λ)^{2} as a trivial cleft extension of Γ = Λ/J ac(Λ) by J ac(Λ)/J ac(Λ)^{2}, and set F^{!} := − ⊗_{Γ} J ac(Λ)/J ac(Λ)^{2}. Since obviously Λ and Λ/J ac(Λ)^{2} have the same ordinary quiver we have: Λ is triangular ⇔ F^{!} is nilpotent ⇔ gl.dimΛ/J ac(Λ)^{2} < ∞. The last equivalence follows from (vi) of Proposition 5.8. Observe that gl.dimΛ/J ac(Λ)^{2} = c(F^{!}).

The last claim follows by Proposition 5.8 and the filtration by (bi-)localizing subcategories of mod(Λ), which is induced by the exact endofunctors Φ^{m}:

0 ⊆ KerΦ ⊆ KerΦ^{2} ⊆ ⋯ ⊆ KerΦ^{c(Φ)+1} = mod(Λ)

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