On the Relative Homology of Cleft Extensions of Rings and Abelian Categories

Apostolos Beligiannis

Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany Email Address: abelig@mathematik.uni-bielefeld.de

Current Address: 7^{th} Sintagmatos 21, 34100 Chalkis, Evia, Greece Email Address: abeligia@mail.otenet.gr

Abstract

We study the relative homological behaviour of the omnipresent class of cleft extensions of abelian categories. This class of extensions is a natural generalization of the trivial extensions studied in detail by Fossum, Griffith and Reiten and by Palmer and Roos. We apply our results to the relative homology of cleft extensions of rings.

Key Words: Relative Homology; Proper Classes; Derived Functors; Homological Dimension; Extensions of Rings; Extensions of Abelian Categories.

AMS Subj. Class: Primary 18G25, 18G20, 16E10, 16E70; Secondary 18G10, 18G05, 16D40, 16S70.

1 Introduction

Cleft extensions of abelian categories were introduced by the author in an earlier paper [6] as a generalization of the same concept of rings (Definitions and examples are recalled in Section 2). They provide a natural setting in which to study relative homology which is the topic of the present paper. Indeed, for simplicity and generality reasons, for the homological study of cleft ring extensions, it is natural to work instead in the context of cleft extensions of abelian categories. This leads to general theorems which can be applied back to the germinating case of rings, where they provide new information on global dimension, answer a question of Auslander-Reiten, give generalizations of some familiar results of Reiten and others, and yield new proofs of some well-known theorems of Mitchell.

The (homological) study of cleft extensions of rings is quite essential and has interesting applications to classical ring theory, since this class of ring extensions is very large: it includes free rings, polynomial rings, exterior rings, positively graded rings, basic semiperfect rings, group rings, supplemented rings, universal enveloping algebras of Lie algebras, trivial extension rings and in particular triangular matrix rings, truncated extension rings, quotients of quiver algebras, and all rings satisfying some version of the Principal Theorem of Wedderburn. From the point of view of representation theory, a cleft extension of an abelian category is a natural generalization of the module category of the ring kQ/<ρ>, where Q is a quiver, k is a (semisimple) ring and ρ is a set of relations.

The paper is organized as follows. Each time we prove a result about general cleft extensions of abelian categories, e.g. “Theorem n.m”, we usually include its ring-theoretic interpretation. The ring-theoretic results are labeled as “Corollary For Rings n.m+1”. Now we describe the contents of the paper section by section.

In Section 2, we review the basic facts from [6] about cleft extensions of abelian categories and cleft extensions of rings, we present a large list of constructions and examples of cleft extensions and we fix some notation.

The natural setting in which we study the homological behaviour of a cleft extension is that of relative homological algebra as covered in Mac Lane’s book [25]. In Section 3 we lift proper classes of short exact sequences ℜ in an abelian category D, to proper classes of short exact sequences 𝔉 in a cleft extension C of D; we lift always the minimal proper class of split short exact sequences in D to obtain the proper class Σ in C. We construct relative projective resolutions in C with respect to the proper classes 𝔉, Σ and we analyze the 𝔉−projective resolutions of the important class of primitive objects of C, which includes the “induced” from D objects of C. These objects are the Σ−proper extensions of non-zero objects of D. Our main aim in the remaining Sections is to study the 𝔉−, Σ−relative homological structure of C, using information directly available from D.

In Section 4 we study relative derived functors with respect to 𝔉, Σ and we obtain preliminary formulas for the relative extension functors, and for the relative derived functors of a particular functor C : C → D, which is part of the structure of a cleft extension. This functor plays an important role in the homological behaviour of C. We construct a natural morphism between the 𝔉−, Σ−relative derived functors of C and we study its properties. As an application we study when a relative (co-)tilting object in D can be lifted to a relative (co-)tilting object in C. As a consequence of the lifting of (co-)tilting objects, we prove that the trivial extension of a Cohen-Macaulay Artin algebra by a dualizing bimodule is Gorenstein, giving a positive answer to a question of Auslander-Reiten [2].

In Section 5, we characterize under various conditions, the finiteness of the relative (finitistic) global dimension of a cleft extension, giving a variety of bounds, and we prove generalized versions of the Hilbert Syzygy Theorem and the Hilbert Basis Theorem. Then we apply these results to some specific examples from Section 2. In this respect we compute the relative global dimension and the relative extension functors in free, polynomial, symmetric and exterior categories over an abelian category, equipped with a family of right exact endofunctors, generalizing most of the results of Mitchell [29], Chapter IX.

In Section 6 we study cleft extensions of small relative global dimension and relative Frobenius cleft extensions. More precisely we give necessary and sufficient conditions for the relative global dimension of C to be (less than or equal to) 1 or 2 and for C to be a (relative) Frobenius category.

In Section 7, we introduce some vanishing conditions which permit more precise formulas for the relative extension functors of C and the relative derived functors of the functor C mentioned above. These formulas become exact in the case of trivial extensions, comma-categories, categories of morphisms and in particular in the case of generalized triangular matrix rings. Finally we analyze the relations between the relative derived functors of C with respect to 𝔉, Σ, by studying the Butler-Horrocks spectral sequence for C, induced by the inclusion of proper classes Σ ⊆ 𝔉 in C. The results of this and the previous Sections generalize to the non-trivial case all the analogous results of Palmer-Roos [32] and Fossum-Griffith-Reiten [13].

Section 8 is devoted to the homological study of truncated extensions. We apply our previous results to a truncated extension C of an abelian category D, and under some mild vanishing condition we obtain exact formulas for the relative global dimension, the relative derived functors of C, and the relative extension functors of C, generalizing all the results of Marmaridis-Papistas [28]. In particular we prove the generalized form of the Strong No Loops Conjecture for truncated extensions [28], under much more general conditions and using different and more simple methods. As a final application we prove the Cartan Determinant Conjecture for finite-dimensional truncated algebras over a field.

A general convention used in this paper is that the composition of morphisms in a given category is meant in the diagrammatic order: if f, g are composable, then f ∘g, means first f then g. There are two exceptions: we use the usual antidiagrammatic order when we compose functors and when we apply elements to (compositions of) morphisms in concrete categories. If f_{1} : X → A, f_{2} : Y → A and g_{1} : A → X, g_{2} : A → Y are morphisms in an additive category, then we denote by ^{t}(f_{1}, f_{2}) : X ⊕Y → A and (g_{1}, g_{2}) : A → X ⊕Y, the uniquely induced morphisms.

2 Cleft Extensions of Rings and of Abelian Categories

In this Section we review from [6], some basic facts concerning cleft extensions of rings and abelian categories, we present constructions and examples of such extensions (mainly from ring theory), and we fix the notation.

2.1 Cleft and ϑ−extensions of Rings

Let Γ be an associative ring. A cleft extension of Γ is a triple (Λ, ε, μ) consisting of a ring Λ and ring morphisms ε : Λ → Γ, μ : Γ → Λ, such that μ ∘ ε = Id_{Γ}. We call ε the projection and μ the cleaving of (Λ, ε, μ). Cleft extensions of Γ are the objects of a category CE(Γ). A morphism in CE(Γ) between (Λ, ε_{1}, μ_{1}) and (Ξ, ε_{2}, μ_{2}), is a ring morphism ρ : Λ → Ξ such that ρ ∘ ε_{2} = ε_{1} and μ_{1} ∘ ρ = μ_{2}. For a better description of a cleft extension Λ of a ring Γ we need the notion of a ϑ−extension of Γ by a Γ − Γ−bimodule M, [27], [6]. Let _{Γ}M_{Γ} be a Γ − Γ−bimodule and let ϑ : M ⊗_{Γ} M → M be an associative Γ − Γ−bimodule morphism, i.e. the diagram below commutes:

M ⊗_{Γ} M ⊗_{Γ} M ⟶^{1_{M} ⊗ϑ} M ⊗_{Γ} M
ϑ⊗1_{M} ↓  ^{ϑ}↓
M ⊗_{Γ} M  ⟶^{ϑ} M.

We call the pair (M, ϑ) a multiplicative Γ − Γ−bimodule. The ϑ−extension ring Λ = Γ ⋉_{ϑ}M of Γ by M, is defined as follows: Λ = Γ ⊕ M in the category of abelian groups Ab. The multiplication of Λ 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 ϑ = 0, we obtain the familiar trivial extension of Γ by M. It is clear that Γ ⋉_{ϑ} M is a cleft extension of Γ with projection defined by π(γ, m) = γ and cleaving defined by ι(γ) = (γ, 0). We define the category E_{ϑ}(Γ) of ϑ−extensions of Γ as follows. The objects are triples (Γ ⋉_{ϑ} M, π, ι) as above. A morphism α : Γ ⋉_{ϑ} M → Γ ⋉_{φ} N is a ring morphism α which commutes with π and ι. Let (Λ, ε, μ) be a cleft extension of Γ and set M := Ker(ε). The ideal M is a multiplicative Γ − Γ−bimodule with actions Γ × M → M : (γ, m) ↦ μ(γ) · m and M × Γ → M : (m, γ) ↦ m · μ(γ) and multiplication ϑ(m_{1} ⊗ m_{2}) = m_{1}m_{2} (multiplication inside Λ). Hence we can form the ϑ−extension Γ ⋉_{ϑ}M and in this way we obtain a functor 𝔽 : CE(Γ) → E_{ϑ}(Γ). Now consider the category MB(Γ) with objects multiplicative Γ − Γ−bimodules (M, ϑ). 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. Then we define a functor 𝔾 : MB(Γ) → E_{ϑ}(Γ) by setting 𝔾(M, ϑ) = Γ ⋉_{ϑ}M and 𝔾(α)(γ, m) = (γ, α(m)).

The following result [6], which is a direct consequence of the definitions, allows us to reduce the study of cleft extensions to the more concrete class of ϑ−extensions.

Proposition 2.1. The functors 𝔽 : CE(Γ) → E_{ϑ}(Γ) and 𝔾 : MB(Γ) → E_{ϑ}(Γ) defined above are equivalences of categories. The equivalence 𝔾 restricts to an equivalence 𝔾 : Mod(Γ^{e}) ⟶^{≈} E_{ϑ=0}(Γ), where Γ^{e} is the enveloping ring of Γ.

If Λ = Γ ⋉_{ϑ} M is a cleft ϑ−extension of Γ, then the Γ − Γ−bimodule M is an ideal of Λ. The morphism ϑ is called nilpotent if the ideal M of Λ is nilpotent. This happens iff ∃i ∈ ℕ: (1_{M} ⊗1_{M} ⊗⋯⊗1_{M} ⊗ϑ)∘⋯∘(1_{M} ⊗ϑ)∘ϑ = 0 : M^{⊗_{i+1}} → M. Also ϑ is called right T-nilpotent if the ideal M is right T-nilpotent [5] and this happens iff ∀(m_{i})_{i∈ℕ} ⊆ M, ∃j ∈ ℕ : ϑ(m_{j} ⊗ ϑ(m_{j−1} ⊗ ϑ(m_{j−2} ⊗ ⋯ ⊗ ϑ(m_{3} ⊗ ϑ(m_{2} ⊗ m_{1})) ⋯) = 0. Left T-nilpotency of ϑ is defined similarly.

2.2 Cleft and η−Extensions of Abelian Categories

Throughout this Section we fix an abelian category D.

Definition 2.2. [6] A cleft extension of D is an abelian category C together with additive functors U : C → D (projection) and Z : D → C (cleaving) such that:

(i) U is faithful exact and admits a left adjoint T : D → C.

(ii) UZ = Id_{D}.

Example 2.3. The motivating example of a cleft extension is the following. Let (Λ, ε, μ) be a cleft extension of the ring Γ and let Mod(Λ), resp. Mod(Γ), be the category of right Λ−, resp. Γ−, modules. Set U = Hom_{Λ}[_{Γ}Λ_{Λ}, −] = − ⊗_{Λ} Λ_{Γ} : Mod(Λ) → Mod(Γ), T = − ⊗_{Γ} Λ_{Λ} : Mod(Γ) → Mod(Λ) and Z = Hom_{Γ}[_{Λ}Γ_{Γ}, −] : Mod(Γ) → Mod(Λ). It is easy to see that Mod(Λ) is a cleft extension of Mod(Γ).

For an internal description of cleft extensions of D, we need the notion of an η−extension D_{F} (η) of D by a right exact endofunctor F : D → D, as introduced in [27]. Here η : F^{2} → F is an associative natural morphism, i.e. ηF ∘ η = F η ∘ η, and D_{F} (η) has as objects pairs (X, f) where f : F X → X is a morphism of D such that: 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 such that f ∘ a = F a ∘ g. The above commutativities are shown in the following diagrams:

\begin{matrix}F^{3} & ⟶^{F η} & F^{2} \\ ^{ηF} ↓ & & ^{η}↓ \\ F^{2} & ⟶^{η} & F \end{matrix}  \begin{matrix}F^{2}(X) & ⟶^{F f} & F X \\ ^{η_{X}} ↓ & & ^{f}↓ \\ F X & ⟶^{f} & X \end{matrix}  \begin{matrix}F X & ⟶^{F a} & F Y \\ ^{f}↓ & & ^{g}↓ \\ X & ⟶^{a} & Y. \end{matrix}

Then D_{F} (η) is abelian and there are adjoint pairs of functors (T, U), (C, Z):

T : D → D_{F} (η), U : D_{F} (η) → D, C : D_{F} (η) → D, Z : D → D_{F} (η)

defined as follows. If a : X → Y is a morphism in D, and α : (X, f) → (Y, g) is a morphism in D_{F} (η), then:

U(X, f) = X, U(α) = α and Z(X) = (X, 0), Z(a) = a,

T(X) = (X ⊕ F X, t_{X}), where 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} ),

C(X, f) = Coker(f), C(α) = the unique morphism : C(X, f) → C(Y, g) in D, such that: α ∘ coker(g) = coker(f) ∘ C(α). Obviously U is faithful exact, Z is fully faithful exact, the following relations hold: CZ = Id_{D}, CT = Id_{D}, UZ = Id_{D} and D_{F} (η) is a cleft extension of D. We denote by ε : TU → Id_{D_{F} (η)} the counit and by δ : Id_{D} → UT = Id_{D} ⊕ F the unit of the adjoint pair (T, U). ε is defined by ε_{(X,f)} = ^{t}(1_{X}, f), and δ = (1_{Id_{D}}, 0). The unit λ : Id_{D_{F} (η)} → ZC of the adjoint pair (C, Z) is defined by λ_{(X,f)} = coker(f) and the counit is the identification CZ = Id_{D}. By [6] we have the following classification:

Theorem 2.4. For an abelian category C, the following are equivalent.

(i) C is a cleft extension of D.

(ii) There exists an equivalence C ⟶^{≈} D_{F} (η), where η : F^{2} → F is an associative natural morphism and F : D → D is a right exact functor.

In this case the functor Z has a left adjoint C : C → D and the following relations hold: CZ = Id_{D}, CT = Id_{D}, UZ = Id_{D}.

Remark 2.5. (1) η−extensions are the analogues of ϑ−extensions in the class of cleft extensions. One can define the category of cleft extensions of D and the category of η−extensions of D and prove a version of Proposition 2.1.

(2) By Theorem 2.4, the study of cleft extensions is equivalent to the study of η−extensions. Although one can describe the homological properties of a cleft extension intrinsically, we prefer to work with η−extensions for the following reasons. First the defining data of an η−extension depends only on information directly available from D, so our aim to reduce the study of D_{F} (η) to the study of the triad {D, F, η}, is more reasonable: we think of F as an “insertion of arrows” in D and η as “relations”. Our spirit is close to the spirit of Gabriel [15] who first described finite-dimensional algebras by quivers and relations. Second working with general cleft extensions is rather impractical for our purposes and offers no real advantages; for example the notation becomes more complicated. Finally most familiar cleft extensions of D are constructed using specific F : D → D and η : F^{2} → F.

2.3 The Module-Theoretic Intrepretation

The link between cleft extensions of abelian categories and cleft extensions of rings (equivalently ϑ−extensions of rings), is as nice as possible [6]: D_{F} (η) ≈ Mod(Λ) iff D ≈ Mod(Γ) and U (equivalently F), preserves coproducts. In this case by Watt’s Theorem, F ≅ − ⊗_{Γ} M for a Γ − Γ−bimodule _{Γ}M_{Γ} and by Theorem 2.4, there exists an associative natural morphism η : − ⊗_{Γ} M ⊗_{Γ} M → − ⊗_{Γ} M. Setting ϑ := η_{Γ}, we obtain an associative Γ − Γ−bimodule morphism ϑ : M ⊗_{Γ} M → M and an isomorphism of rings Λ ≅ Γ ⋉_{ϑ} M. Conversely if Λ = Γ ⋉_{ϑ} M is a cleft extension of Γ, then Mod(Λ) = Mod(Γ)_{F} (η). Here F = − ⊗_{Γ} M and η_{X} = 1_{X} ⊗ ϑ, where ϑ : M ⊗_{Γ} M → M is defined by ϑ(m_{1} ⊗ m_{2}) = (0, m_{1})(0, m_{2}). The above identification sends a right Λ−module X_{Λ} to the object (X_{Γ}, f) ∈ Mod(Γ)_{F} (η), where f : X ⊗_{Γ} M → X_{Γ} is defined as follows: f(x ⊗ m) = x · (0, m). Conversely we identify the object (X, f) of Mod(Γ)_{F} (η) with the Λ−module X_{Γ}, with right Λ−action: x · (γ, m) = x · γ + f(x ⊗ m); e.g. the ideal M = (M, ϑ) as an object of Mod(Γ)_{F} (η).

2.4 Cleft Coextensions

All the above definitions and constructions can be dualized in the following way. A cleft coextension of D is an abelian category C together with additive functors U : C → D and Z : D → C, such that U is faithful exact and admits a right adjoint H and UZ = Id_{D}. The primal examples of cleft coextensions of the abelian category D are the ζ−coextensions D^{G}(ζ) of D by a left exact endofunctor G : D → D. Here ζ : G → G^{2} is a coassociative natural morphism, i.e. ζ ∘ ζG = ζ ∘ Gζ, and D^{G}(ζ) has as objects pairs (X, f) where f : X → GX is a morphism of D such that: f ∘ Gf = f ∘ ζ_{X}. A morphism a : (X, f) → (Y, g) in D^{G}(ζ) is a morphism a : X → Y in D, such that a∘g = f ∘Ga. The category D^{G}(ζ) is abelian and there are adjoint pairs of functors (U, H), (Z, K):

H : D → D^{G}(ζ), U : D^{G}(ζ) → D, K : D^{G}(ζ) → D, Z : D → D^{G}(ζ)

defined as follows. If a : X → Y is a morphism in D and α : (X, f) → (Y, g) is a morphism in D^{G}(ζ), then:

H(X) = (GX ⊕ X, h_{X}), where h_{X} = ( \begin{matrix}ζ_{X} & 1_{GX} \\ 0 & 0 \end{matrix} ), and H(a) = ( \begin{matrix}Ga & 0 \\ 0 & a \end{matrix} ).

K(X, f) = Ker(f), and K(α) = the unique morphism : K(X, f) → K(Y, g) in D such that: K(α) ∘ ker(g) = ker(f) ∘ α. The functors U, Z are defined as in the case of cleft extensions. Trivially U is faithful exact, Z is fully faithful exact and the following relations hold: KZ = Id_{D}, KH = Id_{D}, UZ = Id_{D}. In particular D^{G}(ζ) is a cleft coextension of D. The dual of Theorem 2.4 is also true: the category C is a cleft coextension of D iff C ≈ D^{G}(ζ) for a coassociative natural morphism ζ : G → G^{2}, where G : D → D is a left exact functor. In this case Z has a right adjoint K and the following are true: KZ = Id_{D}, KH = Id_{D}, UZ = Id_{D}.

Since the concepts of cleft extension and cleft coextension are dual, we shall study only cleft extensions, noting that when it is necessary we shall state and use the dual results concerning cleft coextensions, leaving their proofs to the reader. We’ll see that the cleft extensions are suitable for the study of projective dimension, and the cleft coextensions are suitable for the study of injective dimension.

A nice situation occurs when a cleft extension D_{F} (η) of D is also a cleft coextension. This happens if (and only if) the functor F has a right adjoint G. In this case fixing counit ρ : F G → Id_{D} and unit σ : Id_{D} → GF of the adjoint pair (F, G), and defining ζ := σG ∘ GσF G ∘ G^{2}ηG ∘ G^{2}ρ, we obtain a coassociative natural morphism ζ : G → G^{2} and an isomorphism of categories 𝔻 : D_{F} (η) ⟶^{≅} D^{G}(ζ), defined by 𝔻(X, f) = (X, σ_{X} ∘ Gf), and 𝔻(a) = a. By the above remarks and Watt’s Theorem, a cleft extension of a module or Grothendieck category by a colimit preserving endofuctor F, is also a cleft coextension. Note that cleft coextensions can be used to study cleft extensions of coalgebras, using the cotensor product functor. Finally observe that if η = 0 (ζ = 0) then D_{F} (0) (D^{G}(0)) is the trivial extension (coextension) D ⋉ F (G ⋉ D) of D studied in detail in [13].

If the cleft extension D_{F} (η) of D is induced by a cleft extension of rings Λ = Γ ⋉_{ϑ} M, then by [6] the morphism ϑ is right, resp. left, T −nilpotent iff C(X, f) = 0 implies that (X, f) = 0, resp. K(X, f) = 0 implies that (X, f) = 0. The functors F, G, T, U, C, Z, H, K introduced above, can be realized as follows: F = −⊗_{Γ}M_{Γ}, G = Hom_{Γ}[_{Γ}M_{Γ}, −], T = −⊗_{Γ}Λ_{Λ}, U = Hom_{Λ}[_{Γ}Λ_{Λ}, −] ≅ −⊗_{Λ} Λ_{Γ}, C = − ⊗_{Λ} Γ_{Γ}, Z = Hom_{Γ}[_{Λ}Γ_{Γ}, −] ≅ − ⊗_{Γ} Γ_{Λ}, H = Hom_{Γ}[_{Λ}Λ_{Γ}, −], K = Hom_{Λ}[_{Γ}Γ_{Λ}, −].

2.5 Constructions and Examples of Cleft Extensions

In this subsection we present some constructions of cleft extensions and we indicate how familiar categories can be viewed as cleft η−extensions. The description of the functors F, T, U, Z, C and of the morphism η : F^{2} → F, will be clear from the context.

2.5.1 Free Categories

The free category D<F_{i}, i ∈ I> over D with respect to a family {F_{i}}_{i∈I} of right exact endofunctors of D has as objects, pairs (X, f_{i})_{i∈I} where f_{i} : F_{i}X → X are morphisms in D indexed by I. A morphism α : (X, f_{i})_{i∈I} → (Y, g_{i})_{i∈I} in D<F_{i}, i ∈ I>, is a morphism α : X → Y in D, such that f_{i} ∘ α = F α ∘ g_{i}, ∀i ∈ I. If J = max(ℵ_{0}, |I|), and if D has coproducts indexed by J which are preserved by all the functors F_{i}, we get from [6] that D<F_{i}, i ∈ I> = D(F) is a cleft extension of D, where F = ⊕_{i∈I}F_{i}. The definition of the functors U, Z is the obvious one and T(X) = (⊕_{i∈ℕ}F^{i}X, t^{X}), where t^{X} has components t_{i,j}^{X} = 0, ∀j ≠ i + 1 and t_{i,i+1}^{X} = 1_{F^{i}(X)}; for α : X → Y a morphism in D, T(α) = ⊕_{i∈ℕ}F^{i}(α). If F_{i} = Id_{D}, ∀i ∈ I, then D<F_{i}, i ∈ I> := D<I> is the free category over D in |I| noncommuting variables [29]. In case D ≈ Mod(Γ), we have that D<F_{i}, i ∈ I> ≈ Mod(T_{Γ}(M)), where T_{Γ}(M) is the tensor ring over Γ of 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| noncommuting variables. Note that categories of representations of quivers [15], [3], and more generally categories of diagrams, are free cleft extensions. In general any cleft extension of D can be realized as 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 considered as generators, and the relations are of the form η_{X} ∘f −F f ∘f. In the case of a cleft extension Λ of a ring Γ, it is very useful to view the Λ−modules as Γ−morphisms, satisfying relations given by the multiplication.

2.5.2 Polynomial and Symmetric Categories

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, then the symmetric category D[F_{1}, F_{2}, ..., F_{n}] of D with respect to the set {F_{i}}_{i=1}^{n}, is defined inductively as follows: D[F_{1}, F_{2}] = D(F_{1})(F_{2}^{∗}), where F_{2}^{∗} : D(F_{1}) → D(F_{1}) is defined by F_{2}^{∗}(X, f) = (F_{2}X, F_{2}f), F_{2}^{∗}(a) = F_{2}a, and so on. If 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 and of D[F_{1}, F_{2}, ..., F_{i−1}]. If F_{i} = Id_{D}, ∀i, then D[F_{1}, F_{2}, ..., F_{n}] := D[n] is the polynomial category of D in n−variables [29], and 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}]).

2.5.3 Exterior Categories

Let {F_{i}}_{i=1}^{n} be a set of commuting right exact endofunctors of D: F_{i}F_{j} = F_{j}F_{i}, ∀i, j. The exterior category ⋀(D)(F_{1}, F_{2}, ..., F_{n}) of D with respect to the set {F_{i}}_{i=1}^{n} is constructed inductively as follows. First we set ⋀(D)(F_{1}) = D_{F_{1}}(0). Then ⋀(D)(F_{1}, F_{2}) = [⋀(D)(F_{1})]_{F_{2}^{!}}(0), where F_{2}^{!} : ⋀(D)(F_{1}) → ⋀(D)(F_{1}) is defined by F_{2}^{!}(X, f) = (F_{2}X, −F_{2}f), F_{2}^{!}(a) = F_{2}a, and so on. Each category ⋀(D)(F_{1}, F_{2}, ..., F_{i}) is a cleft (trivial) extension of D and of ⋀(D)(F_{1}, F_{2}, ..., F_{i−1}). If F_{i} = Id_{D}, ∀i, then ⋀(D)(F_{1}, F_{2}, ..., F_{n}) := ⋀^{n}(D) is the exterior category of D in n−variables [29], 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)), where ⋀_{Γ}^{n}(Γ) is the exterior ring of Γ.

2.5.4 Trivial and Truncated Extensions

Let F : D → D be a right exact endofunctor. We view the trivial extension D_{F} (0) of D as the full subcategory of the free category D(F) consisting of objects f : F X → X such that F f ∘ f = 0. Special cases of the trivial extension construction are the comma-categories, the category of composable morphisms of any finite length and categories of complexes.

Let G : D → E be a right exact functor between abelian categories, and let C = (D, G, E) be the induced comma-category. Recall that an object of C is a triple (A, f, B) where f : GA → B is a morphism in E. A morphism γ : (A, f, B) → (A′, f′, B′) in C consists of two morphisms α : A → A′ in D and β : B → B′ in E such that f ∘ β = Gα ∘ f′. The category C can be considered as a trivial extension of D × E by the right exact functor F : D × E → D × E given by F (A, B) = (0, GA). This construction gives a nice description of the module category of a triangular matrix ring ( \begin{matrix}R & _{R}M_{S} \\ 0 & S \end{matrix} ). Here D = Mod(R),

E = Mod(S) and G = − ⊗_{R} M_{S}.

Let D^{n} be the category of composable morphisms of length n − 1, over D, in particular D^{2} is the category of morphisms of D. Obviously D^{2} is a trivial extension of D×D since it is the comma-category (D, Id_{D}, D). Inductively D^{n+1} is a trivial extension of D^{n} × D. Similarly the category of (bounded, bounded above or below, unbounded or of any given finite length) complexes over D is a trivial extension of the underlying graded category by the shift functor.

A natural generalization of the trivial extension is the t−truncated extension D ⋉_{t}F of D by the right exact endofunctor F : D → D, t ≥ 0. We recall from [28] that 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. There are adjoint pairs (T, U) : D → D ⋉_{t}F, (C, Z) : D ⋉_{t}F → D, such that U is faithful exact, and UZ = Id_{D}, [28]; hence D ⋉_{t} F is a cleft extension of D. We need only the description of T from [28]. If X ∈ D, then T(X) = (⊕_{i=0}^{t+1}F^{i}X, τ_{X}), where τ_{X} : ⊕_{i=1}^{t+2}F^{i}X → ⊕_{i=0}^{t+1}F^{i}X is represented by a (t + 1) × (t + 1) matrix, with all its entries zero, except the entries above the main diagonal, which are the identities 1_{F^{i}X}, i = 1, ..., t + 1. If a : X → Y is a morphism in D, then T(a) = ⊕_{i=0}^{t+1}F^{i}a. Setting F̃ := ⊕_{i=1}^{t+1}F^{i}, it is easy to see that there exists an associative natural morphism η̃ : F̃^{2} → F̃, and an isomorphism of categories:

𝔸 : D ⋉_{t} F ⟶^{≅} DF̃(η̃), defined by 𝔸(X, f) = (X,^{t}(f_{0}, f_{1}, ⋯, f_{t−1}, f_{t})), where:

f_{i} := F^{i}f ∘ F^{i−1}f ∘ ⋯ ∘ F^{2}f ∘ F f ∘ f : F^{i+1}X → X, and 𝔸(a) = a. If (T̃, Ũ), (C̃, Z̃) are the standard adjoint pairs defined on DF̃(η̃), then: Ũ𝔸 = U, 𝔸Z = Z̃, and moreover 𝔸T = T̃, C̃𝔸 = C; so 𝔸 is an isomorphism over D. Note that D ⋉_{t} Id_{D} is the category of (t + 1)−nilpotent endomorphisms of D, with objects D−morphisms f : X → X such that f^{t+1} = 0. The dual notion of an s−truncated coextension G ⋉_{s} D of D by a left exact functor G is defined dually. If D = Mod(Γ) and F = − ⊗_{Γ} M, then D ⋉_{t}F = Mod(Γ ⋉_{t}M), where Γ ⋉_{t} M is the t−truncated extension of Γ by M, i.e. the factor ring T_{Γ}(M)/< ⊗_{Γ}^{t+2} M>, where < ⊗_{Γ}^{t+2} M> is the ideal of the tensor ring T_{Γ}(M) generated by ⊗_{Γ}^{t+2}M.

Example 2.6. Consider the quiver Q : • ⇄_{β_{1}}^{α_{1}} • ⇄_{β_{2}}^{α_{2}} • ⋯ • ⇄_{β_{m}}^{α_{m}} •, with relations

R_{t,s} = {α_{1}β_{1}, α_{k+1}α_{k+2} ⋯ α_{k+t+1}, β_{l}β_{l−1} ⋯ β_{l−s}, β_{i}α_{i} − α_{i+1}β_{i+1}}

where: 0 ≤ k ≤ m − t − 1, s + 1 ≤ l ≤ m, i = 1, 2, ..., m − 1, 0 ≤ t, s ≤ m − 1.

Let k be a field and consider the family of algebras Λ_{t,s}^{m} = kQ/<R_{t,s}>. Set D = Mod(k) and let D^{(m+1)} be the product category of m + 1 copies of D. Define a functor F : D^{(m+1)} → D^{(m+1)} as follows. F (X_{1}, X_{2}, ..., X_{m+1}) = (0, X_{1}, X_{2}, ..., X_{m}). Let C = D^{(m+1)} ⋉_{t} F be the t−truncated extension of D^{(m+1)} by F. A typical object of C consists of k−linear maps f_{i} : X_{i} ⟶ X_{i+1}, 1 ≤ i ≤ m, X_{i} ∈ Mod(k), such that f_{k+1} ∘ ⋯ ∘ f_{k+t+1} = 0, 0 ≤ k ≤ m − t − 1. Define a functor G : C → C as follows. F (X_{1} ⟶^{f_{1}} X_{2} → ⋯ → X_{m} ⟶^{f_{m}} X_{m+1}) =

(0 → X_{1} ⟶^{f_{1}} X_{2} ⟶^{f_{2}} X → ⋯ → X_{m−1} ⟶^{f_{m−1}} X_{m}). Let E = G ⋉_{s} C be the

s−truncated coextension of C by G. It is easy to see that E is isomorphic to the module category Mod(Λ_{t,s}^{m}). Note that if char(k) = p > 0, then the algebras Λ_{0,0}^{m} obtained above for various m, are exactly the non-simple components of the basic algebra of the Schur algebra S_{k}(p, p) [41].

A host of (rather complicated) examples is obtained, by combining the free, symmetric, polynomial, exterior, truncated constructions and their duals.

2.6 Other Examples from Ring Theory

The previous examples of cleft extensions are of constructive nature. In ring theory there are also examples for structural reasons. The main source of such examples comes from the Hochschild version and its generalizations, of the classical Theorem of Wedderburn: let R be a commutative ring, let Λ be an R−algebra and let J ac(Λ) be the Jacobson radical of Λ. Then Λ is a cleft extension of Λ/J ac(Λ) in the following cases: (i) Λ is a basic semiperfect ring or Λ/J ac(Λ) is R−projective, J ac(Λ) is nilpotent and the Hochschild dimension Dim(Λ/J ac(Λ)) ≤ 1, see [36], [33]. (ii) R is a field, dim_{R}Λ < ∞ and either R is a perfect field or gl.dimΛ < ∞ [11], [25]. (iii) R is a von Neumann regular or Henselian ring, Λ_{R} is f. g. and Λ/J ac(Λ) is separable [39], [4]. In particular if Q is a finite quiver, k is a field and I is an admissible ideal of the path-algebra kQ, then the quotient algebra kQ/I is a cleft extension of the semisimple subalgebra k^{|Q_{0}|}, where Q_{0} is the set of vertices of Q. Also the repetitive algebra of an algebra [17] is a (trivial) cleft extension. If U(L) is the universal enveloping algebra of a finite-dimensional Lie algebra L over a field k, then by the Levi-Malcev Theorem [1], U(L) is a cleft extension of U(L/J) where J is the radical of L . Note that the class of cleft extensions of a ring Γ is closed under free products. For more examples see [6], [16], [22], [23], [24], [27], [39], [42].

The above plethora of constructions and examples of cleft extensions indicates also the difficulty of their study. The main dificulty is of course the complexity of the multiplication η : F^{2} → F in D_{F} (η), or the multiplication ϑ : M ⊗_{Γ} M → M in Λ = Γ ⋉_{ϑ} M. In case η or ϑ has a simple form, for example in the case of free, symmetric, exterior, trivial or truncated extensions, we obtain satisfactory results.

The notion of a cleft extension of rings is quite classical. Perhaps its first appearence is in the construction of a unital ring Λ from a ring Γ without unit. Here Λ is just the cleft ϑ−extension ℤ ⋉_{ϑ} Γ of ℤ, where ϑ : Γ ⊗_{ℤ} Γ → Γ is the multiplication. To the author’s best knowledge the concept goes back at least to Wedderburn and his celebrated Principal Theorem (stating that any finite-dimensional algebra over a field of characteristic zero is cleft). The terminology “cleft” it seems to be due to Vinograde [37] (in 1944). In the literature cleft extensions are also known as Everett extensions. Hochschild in his seminal paper [19] relates cleft extensions of algebras with their cohomology and he uses the term segregated extension. Cartan-Eilenberg in their book [8] use the term inessential extension. In Mac Lane’s book [25] the term cleft is used and Pierce in his book [33] uses the term split extension. Recently Cuntz-Quillen [10] studied ℂ−algebras Γ such that any nilpotent extension of Γ is cleft, under the name of quasi-free algebras [40]. They show that quasi-free algebras are the noncommutative analogues of smooth algebras and they provide a natural setting for noncommutative versions of certain aspects of manifolds, in connection with cyclic homology.

3 Proper Classes, Resolutions and Primitive Objects

3.1 Proper Classes

In this subsection we recall some definitions and concepts from relative homology. The setting in which we study the homological structure of D_{F} (η) is that of proper exact sequences as presented in Mac Lane’s book [25], to which we refer for details. In this subsection D can be an arbitrary abelian category. Let ℜ be a class of short exact sequences in D. The corresponding class of monics, resp. epics, is denoted by ℜ_{m}, resp. ℜ_{e}. A morphism in ℜ_{m}, resp. ℜ_{e} is called an ℜ−proper monic, resp. ℜ−proper epic. We denote by ℜ_{0} the class of split short exact sequences. We recall from [25] that ℜ is called a proper class if: (α) ℜ is closed under isomorphisms and ℜ_{0} ⊆ ℜ. (β) The classes ℜ_{e} and ℜ_{m} are closed under composition. (γ) If f, g are monics and f ∘ g ∈ ℜ_{m}, then f ∈ ℜ_{m}; if f, g are epics and f ∘ g ∈ ℜ_{e}, then g ∈ ℜ_{e}. We recall that an object P ∈ D is called ℜ−projective if 0 → D(P, X) → D(P, Y) → D(P, W) → 0 is exact for any 0 → X → Y → W → 0 in ℜ. The full subcategory of ℜ−projectives is denoted by P(ℜ). The full subcategory I(ℜ) of ℜ−injective objects is defined dually. We say that D has enough ℜ−projectives if any object X ∈ D is included in a sequence 0 → K → P → X → 0 in ℜ with P ∈ P(ℜ); such a sequence is called an ℜ−projective presentation of X. If D has enough ℜ−projectives, then the class of epics ℜ_{e} is a projective class in D in the sense of [12], see also [29], [18]. Obviously D has enough ℜ_{0}−projectives for the proper class ℜ_{0} and P(ℜ_{0}) = D. The class ℜ_{1} of all short exact sequences is always proper and D has enough ℜ_{1}−projectives iff D has enough projectives in the usual sense. For any proper class ℜ we have the relations: ℜ_{0} ⊆ ℜ ⊆ ℜ_{1} and P(ℜ_{1}) ⊆ P(ℜ) ⊆ P(ℜ_{0}) = D.

A morphism α : X ⟶ Y is called ℜ−proper if coker(α) : Y → Coker(α) is in ℜ_{e} and ker(α) : Ker(α) → X is in ℜ_{m}, or equivalently if im(α) : Im(α) → Y is in ℜ_{m} and coim(α) : X → Im(α) is in ℜ_{e}. A complex in D is called ℜ−proper if each of its differentials is an ℜ−proper morphism. An ℜ−proper exact sequence is called ℜ−exact. Clearly any ℜ−exact sequence is a Yoneda composition of members of ℜ. An additive functor 𝕂 : D → A with abelian range, is called ℜ−exact if 𝕂 sends ℜ−exact sequences to exact sequences in A. An ℜ−projective resolution of the object X ∈ D is an ℜ−exact sequence ⋯ → P_{1} → P_{0} → X → 0 such that P_{i} ∈ P(ℜ), ∀i ≥ 0. Such a resolution with X deleted, is called a deleted ℜ−projective resolution of X. If D has enough ℜ−projectives and if 𝕂 : D → A, 𝕃 : D^{op} ⟶ A are additive functors to an abelian category A, then we denote by L_{∗}^{ℜ}𝕂 the left ℜ−derived functors of 𝕂 and by R_{ℜ}^{∗}𝕃 the right ℜ−derived functors of 𝕃. In particular we denote by Ext_{ℜ}^{∗}[−, −] the ℜ−proper extension functors. These derived functors are computed in the usual way, by means of ℜ−projective resolutions. We denote by ℜ−p.dX the ℜ−projective dimension of X ∈ D and by ℜ−gl.dimD the ℜ−global dimension of D.

Finally we recall that a resolvent pair for ℜ in D is a pair (ℝ, ρ), where ℝ : D → D is an additive functor and ρ : ℝ → Id_{D} is a natural epimorphism such that ∀X ∈ D: ℝ(X) ∈ P(ℜ) and 0 → 𝕂(X) ⟶^{κ_{X}} ℝ(X) ⟶^{ρ_{X}} X → 0 is in ℜ, where κ : 𝕂 → ℝ is the kernel of ρ. The endofunctor 𝕂 is called the first ℜ−syzygy functor of D with respect to (ℝ, ρ). Clearly if a proper class admits a resolvent pair, then D admits functorial ℜ−projective resolutions.

3.2 Proper Resolutions in D_{F} (η)

We fix throughout the paper a cleft η− extension D_{F} (η) of the abelian category D by the right exact functor F : D → D, where η : F^{2} → F is an associative natural morphism. We follow always the notation introduced in the previous section.

Fix a proper class ℜ of short exact sequences in D and assume throughout that D has enough ℜ−projectives. We define a class of sequences 𝔉 in D_{F} (η) as follows:

𝔉 := U^{−1}(ℜ) = { (E) : 0 → (X, f) ⟶^{α} (Y, g) ⟶^{β} (W, h) → 0 is a sequence

in D_{F} (η) such that U(E) : 0 → X ⟶^{α} Y ⟶^{β} W → 0 ∈ ℜ }.

Using the adjoint pair (T, U), the Adjoint Theorem [12], [29], and the fact that U is faithful we have that 𝔉 is a proper class of short exact sequences in D_{F} (η). Moreover D_{F} (η) has enough 𝔉−projectives and an object (X, f) ∈ P(𝔉) iff (X, f) is a direct summand of T(P) for an object P ∈ P(ℜ).

Choosing ℜ = ℜ_{0} to be the proper class of split short exact sequences in D we obtain the proper class Σ = U^{−1}(ℜ_{0}) in D_{F} (η). If ℜ_{1} is the proper class of all short exact sequences in D then 𝔉_{1} = U^{−1}(ℜ_{1}) is the proper class of all short exact sequences in D_{F} (η). Obviously we have the relations: ℜ_{0} ⊆ ℜ ⊆ ℜ_{1}, and 𝔉_{0} ⊆ Σ ⊆ 𝔉 ⊆ 𝔉_{1}, where 𝔉_{0} is the proper class of all split short exact sequences in D_{F} (η). Then from the above inclusions we have: P(ℜ_{1}) ⊆ P(ℜ) ⊆ P(ℜ_{0}) = D and P(𝔉_{1}) ⊆ P(𝔉) ⊆ P(Σ) ⊆ P(𝔉_{0}) = D_{F} (η). Denoting as usual by ε : TU → Id_{D} the counit of (T, U), it is easy to see [6] that (TU, ε) is a resolvent pair for the proper class Σ in D_{F} (η) with corresponding Σ−syzygy functor Φ = ker(ε) : D_{F} (η) → D_{F} (η) given by Φ(X, f) = (F X, η_{X} − F f) and Φ(a) = F a. In particular D_{F} (η) has enough Σ−projectives and an object (X, f) ∈ P(Σ) iff the epic ε_{(X,f)} : T(X) → (X, f) splits. Fixing the above notation we’ll construct 𝔉−, Σ−projective resolutions of objects of D_{F} (η).

Let (X, f) ∈ D_{F} (η) and choose an ℜ−proper epic a_{0} : P_{0} → X with P_{0} ∈ P(ℜ), in such a way that if X ∈ P(ℜ) then P_{0} = X and a_{0} = 1_{X}. We form the pull-back diagram (0) below, and we consider the morphism F ν_{0} ∘ η_{P_{0}} − F μ_{0} : F Q_{0} → F P_{0}.

\begin{matrix}Q_{0} & ⟶^{ν_{0}} & F P_{0} \\ ^{μ_{0}}↓ & ^{F a_{0}∘f}↓ \\ P_{0} & ⟶^{a_{0}} & X \end{matrix} (0)

Then (F ν_{0}∘η_{P_{0}}−F μ_{0})∘F a_{0}∘f = F ν_{0}∘F^{2}a_{0}∘η_{X}∘f −F ν_{0}∘F^{2}a_{0}∘F f ∘f = 0, thus since (0) is a pullback diagram, there exists a unique morphism g_{0} : F Q_{0} → Q_{0} with: g_{0} ∘ μ_{0} = 0 and g_{0} ∘ ν_{0} = F ν_{0} ∘ η_{P_{0}} − F μ_{0}.

Now we proceed by induction on the following data:

(i) ∀i ≥ 0, a_{i} : P_{i} → Q_{i−1} is an ℜ-proper epic with P_{i} ∈ P(ℜ), such that if Q_{i−1} ∈ P(ℜ) then P_{i} = Q_{i−1} and a_{i} = 1_{Q_{i−1}}.

(ii) Every square (i) below is a pullback diagram in D.

\begin{matrix}Q_{i} & ⟶^{ν_{i}} & F P_{i} \\ ^{μ_{i}}↓ & ^{F a_{i}∘g_{i−1}}↓ \\ P_{i} & ⟶^{a_{i}} & Q_{i−1}. \end{matrix} (i)

(iii) ∀i ≥ 0, g_{i} : F Q_{i} → Q_{i} is the unique morphism in D such that: g_{i} ∘ μ_{i} = 0 , g_{i} ∘ ν_{i} = F ν_{i} ∘ η_{P_{i}} − F μ_{i} (∗)

By the first step of the above construction it follows that the following sequence in D_{F} (η) is in 𝔉, where α_{0} := ^{t}(a_{0}, F a_{0} ∘ f), and κ_{0} := (−μ_{0}, ν_{0}):

0 → (Q_{0}, g_{0}) ⟶^{κ_{0}} T(P_{0}) ⟶^{α_{0}} (X, f) → 0.

Since by construction T(P_{0}) ∈ P(𝔉), the above exact sequence is an 𝔉-projective presentation of (X, f). Similarly setting α_{i} := ^{t}(a_{i}, F a_{i} ∘ g_{i−1}) and κ_{i} := (−μ_{i}, ν_{i}) we obtain short exact sequences in 𝔉:

0 → (Q_{i}, g_{i}) ⟶^{κ_{i}} T(P_{i}) ⟶^{α_{i}} (Q_{i−1}, g_{i−1}) → 0

which are 𝔉-projective presentations of the objects (Q_{i}, g_{i}), ∀i ≥ 0, where (Q_{−1}, g_{−1}) = (X, f). Hence the Yoneda composition of the above short exact sequences is an 𝔉-projective resolution of (X, f), which we denote by P_{𝔉}^{(X,f)}:

⋯ → T(P_{i}) ⟶^{ϕ_{i}^{(X,f)}} T(P_{i−1}) → ⋯ → T(P_{1}) ⟶^{ϕ_{1}^{(X,f)}} T(P_{0}) ⟶^{α_{0}} (X, f) → 0.

Usually we denote by Ω_{𝔉}^{i}(X, f) the 𝔉−syzygy object (Q_{i−1}, g_{i−1}) of (X, f). Also denoting by ω_{i}^{(X,f)} = −a_{i+1} ∘ μ_{i} : P_{i+1} → P_{i}, we compute easily that

ϕ_{i}^{(X,f)} = ( \begin{matrix}ω_{i−1}^{(X,f)} & a_{i} ∘ ν_{i−1} \\ 0 & F ω_{i−1}^{(X,f)} + F (a_{i} ∘ ν_{i−1}) ∘ η_{P_{i−1}} \end{matrix} ) , i ≥ 1.

Applying the functor C to the resolution P_{𝔉}^{(X,f)}, we obtain the following complex of ℜ-projective objects in D, which we denote by 𝔉_{(X,f)}^{•}:

⋯ → P_{i+1} ⟶^{ω_{i}^{(X,f)}} P_{i} ⟶^{ω_{i−1}^{(X,f)}} P_{i−1} → ⋯ → P_{2} ⟶^{ω_{1}^{(X,f)}} P_{1} ⟶^{ω_{0}^{(X,f)}} P_{0} → 0.

We call 𝔉_{(X,f)}^{•} the 𝔉-associated complex of (X, f) with respect to the proper class 𝔉. The above construction applied to the proper class Σ, gives the following Σ−projective resolution of (X, f), which we denote by P_{Σ}^{(X,f)}:

⋯ → T(F^{i}X) ⟶^{θ_{i}^{(X,f)}} T(F^{i−1}X) → ⋯ → T(X) ⟶^{ε_{(X,f)}} (X, f) → 0.

Then θ_{i}^{(X,f)} = ( \begin{matrix}−f_{i−1}^{(X,f)} & 1_{F^{i}X} \\ 0 & f_{i}^{(X,f)} \end{matrix} ), f_{i}^{(X,f)} : F^{i+1}X → F^{i}X and the morphism

f_{i}^{(X,f)} is defined by the formula

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

The Σ-associated complex Σ_{(X,f)}^{•} of (X, f) is of the form:

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

In case η = 0, Σ_{(X,f)}^{•} is the complex constructed in an adhoc manner in [13].

Observe that the resolution P_{Σ}^{(X,f)} is the functorial normalized bar resolution [25], induced by the resolvent pair (TU, ε) in D_{F} (η). The corresponding Σsyzygy functors are given by Φ^{i}(X, f) = (F^{i}X, f_{i}^{(X,f)}), Φ^{i}(a) = F^{i}a, and the morphisms f_{i}^{(X,f)} satisfy the recursive relation: f_{i}^{(X,f)} = η_{F^{i−1}X} − F f_{i−1}^{(X,f)}, i ≥ 1, f_{0} = f.

If (Y, g) ∈ D_{F} (η), then define two complexes \overline{𝔉}_{(X,f),(Y,g)}^{•} and \overline{Σ}_{(X,f),(Y,g)}^{•} in Ab, as follows:

\overline{𝔉}_{(X,f),(Y,g)}^{•} = D_{F} (η)[P_{𝔉}^{(X,f)}, (Y, g)], \overline{Σ}_{(X,f),(Y,g)}^{•} = D_{F} (η)[P_{Σ}^{(X,f)}, (Y, g)].

Using the adjoint pair (T, U), a direct calculation shows that: \overline{𝔉}_{(X,f),(Y,g)}^{i} = D(P_{i}, Y) and the differential is given by : D(P_{i−1}, Y) → D(P_{i}, Y), m ↦ ω_{i−1} ∘ m+a_{i} ∘ν_{i−1} ∘F m∘g. Similarly \overline{Σ}_{(X,f),(Y,g)}^{i} = D(F^{i}X, Y) with differential given

by : D(F^{i−1}X, Y) → D(F^{i}X, Y), m ↦ −f_{i−1}^{(X,f)} ∘ m + F m ∘ g. Observe that if g = 0, i.e. if (Y, g) = Z(Y), then: \overline{𝔉}_{(X,f),Z(Y)}^{•} = D(𝔉_{(X,f)}^{•}, Y) and \overline{Σ}_{(X,f),Z(Y)}^{•} =

D(Σ_{(X,f)}^{•}, Y). Using the above resolutions and the formulas for \overline{𝔉}_{(X,f),(Y,g)}^{•} and \overline{Σ}_{(X,f),(Y,g)}^{•}, one can compute and compare 𝔉−, Σ−relative derived functors in the standard way.

Convention 3.1. From now on we assume that F (ℜ_{e}) ⊆ ℜ_{e}, i.e. the functor F preserves ℜ-proper epics. Also we assume that for any object (X, f) ∈ D_{F} (η), the morphism f : F X → X is ℜ−proper. These assumptions are made in order to simplify the exposition and they are not essential for all parts of the paper. Also if there is no confusion we write ϕ_{i} = ϕ_{i}^{(X,f)} and ω_{i} = ω_{i}^{(X,f)}.

Remark 3.2. (1) Consider the unit λ : Id_{D_{F} (η)} → ZC of the adjoint pair (C, Z). Since λ_{(X,f)} = coker(f), it follows that λ is an ℜ−proper epic, i.e. λ_{(X,f)} is an ℜ−proper epic, ∀(X, f) ∈ D_{F} (η). The morphism λ is always 𝔉−proper if 𝔉 = 𝔉_{1} and λ is always Σ−proper if D is semisimple.

(2) The functor T sends ℜ−proper epics to 𝔉−proper epics, i.e. T(ℜ_{e}) ⊆ 𝔉_{e}.The functor C sends 𝔉−proper epics to ℜ−proper epics, i.e. C(𝔉_{e}) ⊆ ℜ_{e}.

(3) The 𝔉−associated complex 𝔉_{(X,f)}^{•} of (X, f) is an ℜ−proper complex. This follows without difficulty from the construction of 𝔉_{(X,f)}^{•} and our conventions. Observe that from the pull-back diagram (i) above, we have: Coker(ω_{i}) ≅ Coker(a_{i+1} ∘ μ_{i}) ≅ Coker(μ_{i}) ≅ Coker(F a_{i} ∘ g_{i−1}) ≅ Coker(g_{i−1}) which by definition of the functor C is isomorphic to C(Q_{i−1}, g_{i−1}).

The next proposition is a kind of Universal Coefficient Theorem for D_{F} (η).

Proposition 3.3. If ℜ−gl.dimD ≤ 1, then ∀(X, f) ∈ D_{F} (η), ∀Y ∈ D:

(i) Ext_{𝔉}^{n}[(X, f), Z(Y)] ≅ Ext_{ℜ}^{1}[L_{n−1}^{𝔉}C(X, f), Y] ⊕ D(L_{n}^{𝔉}C(X, f), Y), ∀n ≥ 1.

(ii) If Θ : D → C is an additive functor to an abelian category C, then:

L_{n}^{𝔉}ΘC(X, f) ≅ L_{0}^{ℜ}Θ(L_{n}^{𝔉}C(X, f)) ⊕ L_{1}^{ℜ}Θ(L_{n−1}^{𝔉}C(X, f)), ∀n ≥ 1.

Proof. By Remark 3.2, the 𝔉-associated complex 𝔉_{(X,f)}^{•} of (X, f) is an ℜ-proper complex consisting of ℜ−projective objects. It is easy to see that the Universal Coefficient Theorems for homology and cohomology of complexes [36] are applied in our situation. Using these tools the proof is left to the reader.

3.3 Resolutions of Primitive Objects

We close this section studying the 𝔉−associated complex of an important class of objects of D_{F} (η). We call an object (X, f) ∈ D_{F} (η) primitive if (X, f) is a Σ−proper extension of a non-zero object of the form Z(Y), i.e. there exists a Σ−proper exact sequence 0 → (W, h) → (X, f) → Z(Y) → 0. If (X, f) is an

object of D_{F} (η) of the form (X, f) = (Y ⊕ W, ( \begin{matrix}0 & g \\ 0 & h \end{matrix} )) where Y ≠ 0, then it

is easy to see that we have: C(X, f) = Y ⊕ Coker(^{t}(g, h)), g : ΦZ(Y) → (W, h) is a morphism in D_{F} (η), and there is a push-out diagram:

0 → ΦZ(Y) ⟶^{(0,1_{F Y} )} T(Y) ⟶^{^{t}(1_{Y} ,0)} Z(Y) → 0
g↓  g^{!}↓  ∥
0 → (W, h) ⟶^{(0,1_{W})} (X, f) ⟶^{^{t}(1_{Y} ,0)} Z(Y) → 0

where g^{!} = ( \begin{matrix}1_{Y} & 0 \\ 0 & g \end{matrix} ). Hence any object of the above form is primitive and

conversely it is easy to see that any primitive object is of this form. Observe that if D is semisimple, then any object of D_{F} (η) is primitive, as follows from

the short exact sequence 0 → Ker(λ_{(X,f)}) → (X, f) ⟶^{λ_{(X,f)}} ZC(X, f) → 0.

Any primitive object (X, f) as above, defines two new objects (X_{∗}, f_{∗}) and (X^{∗}, f^{∗}) by the following exact sequence in D_{F} (η):

0 → (X_{∗}, f_{∗}) ⟶^{k_{g}} ΦZ(Y) ⟶^{g} (W, h) ⟶^{π_{g}} (X^{∗}, f^{∗}) → 0.

The most important primitive objects of D_{F} (η), are of course the objects of the form Z(X), T(X). For (X, f) = Z(X), we have (X_{∗}, f_{∗}) = ΦZ(X), (X^{∗}, f^{∗}) = 0, and for (X, f) = T(X), we have (X_{∗}, f_{∗}) = (X^{∗}, f^{∗}) = 0.

Theorem 3.4. Let (X, f) = (Y ⊕W, ( \begin{matrix}0 & g \\ 0 & h \end{matrix} )) be a primitive object in D_{F} (η).

(i) The 𝔉−associated complex of (X, f), is isomorphic to the direct sum of a deleted ℜ−projective resolution of Y and of a complex

P_{(X,f)}^{•} : ⋯ → P_{t} → P_{t−1} → ⋯ → P_{2} → P_{1} → P_{0} → 0

consisting of ℜ−projective objects. Further: ℜ−p.dY ≤ 𝔉−p.d(X, f).

(ii) If g is monic and L_{i}^{ℜ}F (Y) = 0, 1 ≤ i ≤ t, then the 𝔉−associated complex of the object (X^{∗}, f^{∗}), is of the form:

⋯ → P′_{t+2} → P_{t+1} → P_{t} → ⋯ → P_{2} → P_{1} → P_{0} → 0.

(iii) If g is epic and L_{i}^{ℜ}F (Y) = 0, 1 ≤ i ≤ t, then the 𝔉−associated complex of the object (X_{∗}, f_{∗}), is homotopy equivalent to a complex of the form:

⋯ → P′_{t+2} → P_{t+1} → P_{t} → ⋯ → P_{2} → P_{1} → 0.

(iv) If g is monic and L_{i}^{ℜ}F (Y) = 0, ∀i ≥ 1, then P_{(X,f)}^{•} is the 𝔉−associated complex of the object (X^{∗}, f^{∗}).

(v) If g is epic and L_{i}^{ℜ}F (Y) = 0, ∀i ≥ 1, then the complex

P_{(X,f)}^{•}[1] : ⋯ → P_{t} → P_{t−1} → ⋯ → P_{2} → P_{1} → 0

is homotopy equivalent to the 𝔉−associated complex of the object (X_{∗}, f_{∗}).

Proof. (i) We’ll analyze the 𝔉−projective resolution P_{𝔉}^{(X,f)} of (X, f) constructed before. We begin by considering an ℜ−projective resolution P_{ℜ}^{•}(Y) of Y:

⋯ → R_{n+1} ⟶^{a_{n+1}} R_{n} → ⋯ → R_{1} ⟶^{a_{1}} R_{0} ⟶^{a_{0}} Y → 0,

let a_{n+1} = l_{n+1} ∘ k_{n} be the canonical factorization of a_{n+1}, and set K_{n} = Im(a_{n+1}), ∀n ≥ 0. Let b_{0} : P_{0} → W be an ℜ−proper epic with P_{0} ∈ P(ℜ). By the construction of P_{𝔉}^{(X,f)}, we have the following 𝔉−projective presentation of (X, f): 0 → (Q_{0}, g_{0}) ⟶^{κ_{0}} T(R_{0}) ⊕ T(P_{0}) ⟶^{α_{0}} (X, f) → 0, where

Q_{0} = K_{0} ⊕ L_{0}, g_{0} = ( \begin{matrix}0 & x_{0} \\ 0 & y_{0} \end{matrix} ), κ_{0} = ( \begin{matrix}k_{0} & 0 \\ 0 & s_{0} \end{matrix} ), α_{0} = ( \begin{matrix}a_{0} & 0 \\ 0 & p_{0} \end{matrix} ), and

the object L_{0} is defined by the short exact sequence in D_{F} (η), (1): 0 →

(L_{0}, y_{0}) ⟶^{s_{0}} ΦZ(R_{0}) ⊕ T(P_{0}) ⟶^{p_{0}} (W, h) → 0, where s_{0} = (c_{0}, d_{0}, e_{0}), and

p_{0} = ^{t}(F a_{0} ∘ g, b_{0}, F b_{0} ∘ h). From the above construction it follows that the first 𝔉−syzygy (Q_{0}, g_{0}) of (X, f) is a primitive object. Since 𝔉−syzygies of primitive objects are primitive, we can proceed inductively. In the i^{th}−step of this construction we have the following 𝔉−projective presentation of (X, f):

0 → (Q_{i−1}, g_{i−1}) ⟶^{κ_{i−1}} T(R_{i−1}) ⊕ T(P_{i−1}) ⟶^{α_{i−1}} (Q_{i−2}, g_{i−2}) → 0, where

Q_{i−1} = K_{i−1} ⊕ L_{i−1}, g_{i−1} = ( \begin{matrix}0 & x_{i−1} \\ 0 & y_{i−1} \end{matrix} ), κ_{i−1} = ( \begin{matrix}k_{i−1} & 0 \\ 0 & s_{i−1} \end{matrix} ), α_{i−1} =

( \begin{matrix}l_{i−1} & 0 \\ 0 & p_{i−1} \end{matrix} ), and the object L_{i−1} is defined by the short exact sequence in

𝔉: 0 → (L_{i−1}, y_{i−1}) ⟶^{s_{i−1}} ΦZ(R_{i−1}) ⊕ T(P_{i−1}) ⟶^{p_{i−1}} (L_{i−2}, y_{i−2}) → 0, where

s_{i−1} = (c_{i−1}, d_{i−1}, e_{i−1}), and p_{i−1} = ^{t}(F l_{i−1} ∘ x_{i−2}, b_{i−1}, F b_{i−1} ∘ y_{i−2}), and b_{i−1} : P_{i−1} → L_{i−2} is an ℜ−proper epic with P_{i−1} ∈ P(ℜ). Composing the above 𝔉−projective presentations we get an 𝔉−projective resolution P_{𝔉}^{(X,f)} of (X, f):

⋯ → T(R_{i+1}) ⊕ T(P_{i+1}) ⟶^{ϕ_{i+1}} T(R_{i}) ⊕ T(P_{i}) ⟶^{ϕ_{i}} T(R_{i−1}) ⊕ T(P_{i−1}) → ⋯

⋯ → T(R_{1}) ⊕ T(P_{1}) ⟶^{ϕ_{1}} T(R_{0}) ⊕ T(P_{0}) ⟶^{α_{0}} (X, f) → 0.

We see easily that C(ϕ_{i}) := ω_{i} = ( \begin{matrix}a_{i} & 0 \\ 0 & b_{i} ∘ d_{i−1} \end{matrix} ) : R_{i} ⊕ P_{i} → R_{i−1} ⊕ P_{i−1}.

Hence the 𝔉−associated complex 𝔉_{(X,f)}^{•} of (X, f) is the following:

⋯ → R_{i} ⊕ P_{i} ⟶^{ω_{i}} R_{i−1} ⊕ P_{i−1} → ⋯ → R_{2} ⊕ P_{2} ⟶^{ω_{2}} R_{1} ⊕ P_{1} ⟶^{ω_{1}} R_{0} ⊕ P_{0} → 0.

Thus 𝔉_{(X,f)}^{•} is the direct sum of the complex P_{ℜ}^{•}(Y) : ⋯ → R_{n} ⟶^{a_{n}} R_{n−1} →

⋯ → R_{1} ⟶^{a_{1}} R_{0} → 0, and of the complex P_{(X,f)}^{•}: ⋯ → P_{i+1} ⟶^{δi+1} P_{i} → ⋯ →

P_{1} ⟶^{δ_{1}} P_{0} → 0, where δ_{i} = b_{i} ∘ d_{i−1}, ∀i ≥ 1. Observe that P_{ℜ}^{•}(Y) is a deleted ℜ−projective resolution of Y and P_{(X,f)}^{•} is a complex of ℜ−projective objects. The above proof also shows that: ℜ−p.dY ≤ 𝔉−p.d(X, f).

(ii) Suppose that g is monic and (+): L_{i}^{ℜ}F (Y) = 0, 1 ≤ i ≤ t. Keep the notation of part (i) and consider the morphism ζ_{0} := ^{t}(b_{0} ∘ π_{g}, F b_{0} ∘ h ∘ π_{g}) : T(P_{0}) → (X^{∗}, f^{∗}) which is an 𝔉−proper epic. Let ξ_{0} := ker(ζ_{0}) : (N_{0}, n_{0}) → T(P_{0}) be the kernel of ζ_{0} in D_{F} (η); then ξ_{0} is of the form (z_{0}, w_{0}), where z_{0} : N_{0} → P_{0} and w_{0} : N_{0} → F P_{0}. Using condition (+) and the 𝔉−exact sequence: 0 → (N_{0}, n_{0}) ⟶^{ξ_{0}} T(P_{0}) ⟶^{ζ_{0}} (X^{∗}, f^{∗}) → 0, we have the following ℜ−exact commutative diagram, except the lower left square:

0 → ΦZ(K_{0}) ⟶^{F k_{0}}  ΦZ(R_{0})  ⟶^{F a_{0}} ΦZ(Y) → 0
x_{0}↓  t_{0}↓  g↓
0 → (L_{0}, y_{0}) ⟶^{s_{0}} ΦZ(R_{0}) ⊕ T(P_{0}) ⟶^{p_{0}}  (W, h) → 0
∃!ρ_{0}↓  τ_{0}↓  π_{g}↓
0 → (N_{0}, n_{0}) ⟶^{ξ_{0}}  T(P_{0})  ⟶^{ζ_{0}}  (X^{∗}, f^{∗}) → 0

where t_{0} = (1_{ΦZ(R_{0})}, 0), τ_{0} = ^{t}(0, 1_{T(P_{0})}). Since the lower right square of the above diagram commutes, ∃!ρ_{0} : (L_{0}, y_{0}) → (N_{0}, n_{0}), such that the above diagram is commutative and has 𝔉−exact rows and columns, and then ρ_{0} = coker(x_{0}). Observe that x_{0} is monic ⇔ L_{1}^{ℜ}F (Y) = 0. Now we proceed inductively, noting that this inductive procedure continues for t ≥ 0, as long as the morphisms x_{i} are monics for 0 ≤ i ≤ t−1 or equivalently L_{i}^{ℜ}F (Y) = 0, 1 ≤ i ≤ t.

Then we get for 1 ≤ i ≤ t, an 𝔉−exact sequence 0 → (N_{i}, n_{i}) ⟶^{ξ_{i}} T(P_{i}) ⟶^{ζ_{i}} (N_{i−1}, n_{i−1}) → 0, where ξ_{i} = (z_{i}, w_{i}) = ker(ζ_{i}), and there exists a unique morphism ρ_{i} : (L_{i}, y_{i}) → (N_{i}, n_{i}), such that (∗): ρ_{i} = coker(x_{i}) and ρ_{i} ∘ z_{i} = d_{i}, ρ_{i} ∘ w_{i} = e_{i}. Observing also that there exists an 𝔉−epic T(P_{t+1}) → (N_{t}, n_{t}), and forming the Yoneda composition of the above 𝔉−exact sequences, we get an 𝔉−projective resolution P_{𝔉}^{(X^{∗},f^{∗})} of (X^{∗}, f^{∗}):

⋯ → T(P′_{t+2}) → T(P_{t+1}) ⟶^{ϕ̃_{t+1}} T(P_{t}) → ⋯ → T(P_{0}) ⟶^{ζ_{0}} (X^{∗}, f^{∗}) → 0.

Using the relations (∗), we compute easily that

ϕ̃_{i} = ( \begin{matrix}b_{i} ∘ d_{i−1} & b_{i} ∘ e_{i−1} \\ 0 & F b_{i} ∘ y_{i−1} ∘ e_{i−1} \end{matrix} ) , C(ϕ̃_{i}) = b_{i} ∘ d_{i−1} : P_{i} → P_{i−1}.

Hence the 𝔉−associated complex of (X^{∗}, f^{∗}), has the desired form.

(iii) Suppose that g is epic and (+): L_{i}^{ℜ}F (Y) = 0, 1 ≤ i ≤ t. Then using (+), we have the following exact commutative diagram:

0 → ΦZ(K_{0}) ⟶^{F k_{0}}  ΦZ(R_{0})  ⟶^{F a_{0}} ΦZ(Y) → 0
x_{0}↓  t_{0}↓  g↓
0 → (L_{0}, y_{0}) ⟶^{s_{0}} ΦZ(R_{0}) ⊕ T(P_{0}) ⟶^{p_{0}} (W, h) → 0

From the above diagram, using the snake lemma, we get a short exact sequence: 0 → (X_{∗}, f_{∗}) → (N_{0}, n_{0}) → T(P_{0}) → 0 in 𝔉 which of course splits. Using the same arguments as in (ii), we get an 𝔉−projective resolution P_{𝔉}^{(N_{0},n_{0})} of (N_{0}, n_{0}):

⋯ → T(P′_{t+2}) → T(P_{t+1}) ⟶^{ϕ̃_{t+1}} T(P_{t}) → ⋯ → T(P_{1}) ⟶^{ζ_{1}} (N_{0}, n_{0}) → 0.

Since (N_{0}, n_{0}) ≅ (X_{∗}, f_{∗}) ⊕ T(P_{0}), the above resolution is homotopy equivalent to the 𝔉−projective resolution of (X_{∗}, f_{∗}). Hence the 𝔉−associated complex of (X_{∗}, f_{∗}) is homotopy equivalent to the complex of the desired form.

(iv), (v) Direct consequences of (ii), (iii).

4 𝔉−Derived Functors And Σ−Derived Functors

4.1 Homology of Primitive Objects

We’re interested in the homology of primitive objects of the form T(X), Z(X).

Theorem 4.1. Suppose that Θ : D → E, Ξ : D^{op} → E, are additive functors to an abelian category E. Then we have the following:

(α) The following are true ∀i ≥ 0 :

(i) L_{i}^{ℜ}Θ is a direct summand of the functors L_{i}^{𝔉}(ΘC)Z, L_{i}^{𝔉}(ΘC)T.

(ii) R_{ℜ}^{i}Ξ is a direct summand of the functors R_{𝔉}^{i}(ΞC)Z, R_{𝔉}^{i}(ΞC)T.

(β) There are natural isomorphisms:

L_{1}^{𝔉}(ΘC)Z ≅ L_{1}^{ℜ}Θ⊕L_{0}^{ℜ}Θ(coker(η)), R_{𝔉}^{1}(ΞC)Z ≅ R_{ℜ}^{1}Ξ⊕R_{ℜ}^{0}Ξ(coker(η)).

(γ) There are natural isomorphisms:

L_{1}^{𝔉}(ΘC)T ≅ L_{1}^{ℜ}Θ , R_{𝔉}^{1}(ΞC)T ≅ R_{ℜ}^{1}Ξ,

and exact sequences, ∀X ∈ D:

L_{2}^{ℜ}Θ(X) → L_{2}^{𝔉}(ΘC)(T(X)) → L_{0}^{𝔉}(ΘC)(L_{1}^{ℜ}T(X)) → 0
0 → R_{𝔉}^{0}(ΞC)(L_{1}^{ℜ}T(X)) → R_{𝔉}^{2}(ΞC)(T(X)) → R_{ℜ}^{2}Ξ(X).

(δ) Suppose that L_{i}^{ℜ}F (X) = 0, 1 ≤ i ≤ t. Then ∀i ∈ ℕ, with: 1 ≤ i ≤ t + 1:

L_{i}^{𝔉}(ΘC)Z(X) ≅ L_{i}^{ℜ}Θ(X) ⊕ L_{i−1}^{𝔉}(ΘC)ΦZ(X),
R_{𝔉}^{i}(ΞC)Z(X) ≅ R_{ℜ}^{i}Ξ(X) ⊕ R_{𝔉}^{i−1}(ΞC)ΦZ(X),
L_{i}^{𝔉}(ΘC)T(X) ≅ L_{i}^{ℜ}Θ(X) , R_{𝔉}^{i}(ΞC)T(X) ≅ R_{ℜ}^{i}Ξ(X).

(ϵ) If the functor F is ℜ-exact, there are natural isomorphisms, ∀i ≥ 1:

L_{i}^{𝔉}(ΘC)Z ≅ L_{i}^{ℜ}Θ ⊕ L_{i−1}^{𝔉}(ΘC)ΦZ , R_{𝔉}^{i}(ΞC)Z ≅ R_{ℜ}^{i}Ξ ⊕ R_{𝔉}^{i−1}(ΞC)ΦZ,

L_{i}^{𝔉}(ΘC)T ≅ L_{i}^{ℜ}Θ , R_{𝔉}^{i}(ΞC)T ≅ R_{ℜ}^{i}Ξ.

Proof. Since ∀X ∈ D, Z(X) and T(X) are primitive objects we can apply Theorem 3.4. We consider only the assertions involving the functor Θ since the other parts are similar. Now part (α) follows easily from Theorem 3.4 by standard arguments.

(β) Using Theorem 3.4 it is not difficult to see that: L_{1}^{𝔉}(ΘC)Z(X) ≅ L_{1}^{ℜ}Θ(X) ⊕ H_{1}[ΘP_{Z(X)}^{•}]. Using the notation of (the proof of) Theorem 3.4, we have P_{0} = 0, and H_{1}[P_{Z(X)}^{•}] = Coker(δ_{1}) = Coker(b_{2} ∘ d_{1}) ≅ Coker(d_{1}), since b_{2} is epic. A simple calculation shows that Coker(d_{1}) ≅ Coker(η_{X}), via the epimorphism b_{1} ∘ F a_{0} ∘ coker(η_{X}) : P_{1} → Coker(η_{X}). This shows that H_{1}[ΘP_{Z(X)}^{•}] ≅ L_{0}^{ℜ}Θ(Coker(η_{X})).

(γ) Let (1) 0 → K_{0} ⟶^{k_{0}} R_{0} ⟶^{a_{0}} X → 0 be an ℜ−projective presentation of X in D. Then we have the 𝔉−projective presentation of T(X) in D_{F} (η):

0 → (Y, g) ⟶^{κ_{0}} T(R_{0}) ⟶^{T(a_{0})} T(X) → 0. The object (Y, g) is primitive of the

form (Y, g) = (K_{0} ⊕ W, ( \begin{matrix}0 & x \\ 0 & y \end{matrix} )), where F k_{0} = x ∘ ι is the canonical factor-

ization of F k_{0} and ι : W ↪ F R_{0} is the kernel of F a_{0}. Then L_{1}^{𝔉}(ΘC)T(X) ≅ Ker(L_{0}^{𝔉}(ΘC)(κ_{0})). But L_{0}^{𝔉}(ΘC)(Y, g) ≅ L_{0}^{ℜ}Θ(K_{0}), L_{0}^{𝔉}(ΘC)(κ_{0}) = L_{0}^{ℜ}Θ(k_{0}) and L_{0}^{𝔉}(ΘC)(T(R_{0})) ≅ L_{0}^{ℜ}Θ(R_{0}). Hence: L_{1}^{𝔉}(ΘC)T(X) ≅ L_{1}^{ℜ}Θ(X). Now from (1) we have a short exact sequence (2): 0 → L_{1}^{ℜ}T(X) ⟶^{μ} T(K_{0}) ⟶^{ν} (Y, g) → 0. Applying the functor L_{0}^{𝔉}(ΘC) to (2), we have the exact sequence:

⋯ → L_{1}^{𝔉}(ΘC)(T(K_{0})) → L_{1}^{𝔉}(ΘC)(Y, g) → L_{0}^{𝔉}(ΘC)(L_{1}^{ℜ}T(X)) →

→ L_{0}^{𝔉}(ΘC)(TK_{0}) ⟶^{L_{0}^{𝔉}(ΘC)(ν)} L_{0}^{𝔉}(ΘC)(Y, g)) → 0.

Since L_{0}^{𝔉}(ΘC)(ν) is (isomorphic to) the identity of L_{0}^{ℜ}Θ(K_{0}), L_{0}^{𝔉}(ΘC)(μ) = 0. Now the result follows from the isomorphisms: L_{1}^{𝔉}(ΘC)(T(K_{0})) ≅ L_{1}^{ℜ}Θ(K_{0}) ≅ L_{2}^{ℜ}Θ(X) and L_{1}^{𝔉}(ΘC)(Y, g) ≅ L_{2}^{𝔉}(ΘC)(T(X)).

(δ)−(ϵ) If (X, f) = Z(X), then (X_{∗}, f_{∗}) = ΦZ(X), and if (X, f) = T(X), then (X^{∗}, f^{∗}) = 0. Setting (X, f) = Z(X) and (X, f) = T(X) in Theorem 3.4, and using induction we get the result, noting that the case i = 1 is always true by part (β) and the first two isomorphisms in (γ).

The next corollary of the previous Theorem, contains a lot of information.

Corollary 4.2. (1) The functors Ext_{𝔉}^{i}[Z(−), Z(−)], Ext_{𝔉}^{i}[T(−), Z(−)] contain as a direct summand the functor Ext_{ℜ}^{i}[−, −], ∀i ≥ 0.

(2) There are natural isomorphisms:

UL_{i}^{ℜ}T ≅ L_{i}^{ℜ}F, ∀i ≥ 1,
Ext_{𝔉}^{1}[T(−), Z(−)] ≅ Ext_{ℜ}^{1}[−, −],
Ext_{𝔉}^{2}[T(−), Z(−)] ≅ Ext_{ℜ}^{2}[−, −] ⊕ D[CL_{1}^{ℜ}T−, −],
Ext_{𝔉}^{1}[Z(−), Z(−)] ≅ Ext_{ℜ}^{1}[−, −] ⊕ D[Coker(η_{−}), −],

L_{1}^{𝔉}CZ = Coker(η), L_{1}^{𝔉}CT = 0, L_{1}^{𝔉}C ≅ L_{1}^{Σ}C, L_{2}^{𝔉}CT ≅ CL_{1}^{ℜ}T.

(3) If L_{i}^{ℜ}F (X) = 0, 1 ≤ i ≤ n, then: ∀(Y, g) ∈ D_{F} (η):

Ext_{𝔉}^{i}[T(X), (Y, g)] ≅ Ext_{ℜ}^{i}[X, Y], 1 ≤ i ≤ n

and for all i ∈ ℕ, with: 1 ≤ i ≤ n + 1 :

L_{i}^{𝔉}CT(X) = 0, Ext_{𝔉}^{i}[Z(X), Z(Y)] ≅ Ext_{ℜ}^{i}[X, Y] ⊕ Ext_{𝔉}^{i−1}[ΦZ(X), Z(Y)].

(4) If L_{i}^{ℜ}F (X) = 0, ∀i ≥ 1, then:

(i) 𝔉−p.dT(X) = ℜ−p.dX and ∀(Y, g) ∈ D_{F} (η):

Ext_{𝔉}^{n}[T(X), (Y, g)] ≅ Ext_{ℜ}^{n}[X, Y].

(ii) There are isomorphisms ∀i ≥ 1:

L_{i}^{𝔉}CT(X) = 0, Ext_{𝔉}^{i}[Z(X), Z(−)] ≅ Ext_{ℜ}^{i}[X, −]⊕Ext_{𝔉}^{i−1}[ΦZ(X), Z(−)].

(5) If the functor F is ℜ-exact, there are natural isomorphisms, ∀i ≥ 1:

L_{i}^{𝔉}CT = 0, Ext_{𝔉}^{i}[T(−), −] ≅ Ext_{ℜ}^{i}[−, U(−)],

Ext_{𝔉}^{i}[Z(−), Z(−)] ≅ Ext_{ℜ}^{i}[−, −] ⊕ Ext_{𝔉}^{i−1}[ΦZ(−), Z(−)].

Proof. (1) In Theorem 4.1, choose ∀Y ∈ D: Ξ_{Y} = D[−, Y], and use that Ξ_{Y} C = D[−, Y]C = D[C−, Y] ≅ D_{F} (η)[−, Z(Y)].

(2) For the first isomorphism, let 0 → K ⟶^{k} P ⟶^{a} X → 0 be an ℜ−projective presentation of X ∈ D. Then L_{1}^{ℜ}F (X) = Ker(F k), L_{1}^{ℜ}T(X) = Ker(T(a)), and UL_{1}^{ℜ}T(X) = U(Ker(T(a))) ≅ Ker(U(T(a))) ≅ Ker(F k) = L_{1}^{ℜ}F (X). Hence: UL_{1}^{ℜ}T(X) ≅ L_{1}^{ℜ}F (X), ∀X ∈ D. This implies trivially that: UL_{i}^{ℜ}T ≅ L_{i}^{ℜ}F, ∀i ≥ 1. Consider now the functorial Σ−projective presentation of (X, f): (∗) 0 → Φ(X, f) → T(X) → (X, f) → 0. By definition L_{1}^{Σ}C(X, f) = Ker[C(Φ(X, f) → X)]. Applying C to (∗) and using that L_{1}^{𝔉}CT = 0, we get L_{1}^{𝔉}C(X, f) ≅ Ker[C(Φ (X, f) → X)] = L_{1}^{Σ}C(X, f), hence: L_{1}^{𝔉}C ≅ L_{1}^{Σ}C. The remaining isomorphisms are derived from Theorem 4.1, choosing ∀Y ∈ D: Ξ_{Y} = D[−, Y] and Θ = Id_{D}.

(3) The last two isomorphisms follow from Theorem 4.1. For the first, choose an ℜ−projective resolution ⋯ → P_{i} → P_{i−1} → ⋯ → P_{0} → X → 0 of X. Then since L_{i}^{ℜ}F (X) = 0, 1 ≤ i ≤ n, by the first isomorphism in (ii) we have L_{i}^{ℜ}T(X) = 0, 1 ≤ i ≤ n, because U is faithful. Hence ⋯ → T(P_{n+1}) → T(P_{n}) → ⋯ → T(P_{0}) → T(X) → 0 is part of an 𝔉−projective resolution of T(X). Then ∀(Y, g) ∈ D_{F} (η), applying the functor D_{F} (η)[−, (Y, g)] to the above resolution and using the adjoint pair (T, U), we see that the resulting complex is of the form 0 → D[X, Y] → D[P_{0}, Y] → ⋯ → D[P_{n}, Y] → D[P_{n+1}, Y] → ⋯, and the second isomorphism follows. Parts (4), (5) are consequences of (3).

From the above corollary it follows that ∀X, Y ∈ D we have an isomorphism: Ext_{𝔉}^{1}[Z(X), Z(Y)] ≅ Ext_{ℜ}^{1}[X, Y] ⊕ Ext_{Σ}^{1}[Z(X), Z(Y)], since trivially there is an isomorphism: Ext_{Σ}^{1}[Z(X), Z(Y)] ≅ D[Coker(η_{X}), Y]. So it is natural to ask, if the above formula extends to higher dimensions:

Corollary 4.3. Suppose that L_{i}^{ℜ}F (X) = 0, 1 ≤ i ≤ n − 1, and let X, Y ∈ D satisfy: Ext_{ℜ}^{p}[F^{q}X, Y] = 0, ∀p + q = n, 0 < p < n. Then:

Ext_{𝔉}^{i}[Z(X), Z(Y)] ≅ Ext_{ℜ}^{i}[X, Y] ⊕ Ext_{Σ}^{i}[Z(X), Z(Y)], 1 ≤ i ≤ n.

Proof. Consider the natural Σ−projective presentations:

0 → Φ^{i+1}Z(X) → T(F^{i}X) → Φ^{i}Z(X) → 0, 0 < i < n.

Then ∀Y ∈ D, applying to these presentations the functor D_{F} (η)[−, Z(Y)], the induced long exact sequences and the hypothesis produce isomorphisms:

Ext_{𝔉}^{1}[Φ^{i}Z(X), Z(Y)] ≅ Ext_{Σ}^{1}[Φ^{i}Z(X), Z(Y)] ≅ Ext_{Σ}^{i+1}[Z(X), Z(Y)],
Ext_{𝔉}^{1}[Φ^{i}Z(X), Z(Y)] ≅ Ext_{𝔉}^{i}[ΦZ(X), Z(Y)], 0 < i < n.

The assertion now follows from the third formula of Corollary 4.2(3).

We continue with other consequences of Corollary 4.2.

Corollary 4.4. If η = 0, the following are true for the cleft extension D_{F} (0):

L_{i}^{ℜ}T ≅ ZL_{i}^{ℜ}F, ∀i ≥ 1,
Ext_{𝔉}^{1}[Z(−), Z(−)] ≅ Ext_{ℜ}^{1}[−, −] ⊕ D[F −, −],
Ext_{𝔉}^{2}[T(−), Z(−)] ≅ Ext_{ℜ}^{2}[−, −] ⊕ D[L_{1}^{ℜ}F −, −],
L_{1}^{𝔉}CZ ≅ F, L_{2}^{𝔉}CZ ≅ F^{2} ⊕ L_{1}^{ℜ}F, L_{2}^{𝔉}CT ≅ L_{1}^{ℜ}F.

If in addition the functor F is ℜ-exact, then:

Ext_{𝔉}^{i}[Z(−), Z(−)] ≅ ⨁_{t=0}^{i} Ext_{ℜ}^{t}[F^{i−t}−, −], ∀i ≥ 0.

Proof. Since η = 0 implies ΦZ = ZF, all assertions are consequences of Corollary 4.2, except for the formula: L_{2}^{𝔉}CZ ≅ F^{2} ⊕ L_{1}^{ℜ}F. But L_{2}^{𝔉}CZ(X) ≅ L_{1}^{𝔉}CΩ_{𝔉}Z(X) ≅ L_{1}^{Σ}CΩ_{𝔉}Z(X). The Σ−associated complex of Ω_{𝔉}Z(X) is of the form:

⋯ → F^{2}K ⊕ F^{3}P ⟶^{z} F K ⊕ F^{2}P ⟶^{w} K ⊕ F P → 0

where z = ( \begin{matrix}0 & F^{2}k \\ 0 & 0 \end{matrix} ), w = ( \begin{matrix}0 & −F k \\ 0 & 0 \end{matrix} ), and 0 → K ⟶^{k} P ⟶^{a} X → 0 is an

ℜ−projective presentation of X. A direct calculation of the first homology of this complex shows that: L_{2}^{𝔉}CZ(X) ≅ L_{1}^{Σ}CΩ_{𝔉}Z(X) ≅ F^{2}X ⊕ L_{1}^{ℜ}F (X).

Corollary 4.5. (i) ℜ−p.dX ≤ min{𝔉−p.dZ(X), 𝔉−p.dT(X)}.

(ii) ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (η).

Corollary For Rings 4.6. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings.

(i) The functor T or_{n}^{Γ}[−, −] is a direct summand of the functors T or_{n}^{Λ}[Z−, Z−] , T or_{n}^{Λ}[T−, Z−], the functor Ext_{Γ}^{n}[−, −] is a direct summand of the functors Ext_{Λ}^{n}[Z−, Z−], Ext_{Λ}^{n}[T−, Z−], ∀n ≥ 0, and there are isomorphisms:

T or_{1}^{Λ}[T−, Z−] ≅ T or_{1}^{Γ}[−, −], Ext_{Λ}^{1}[T−, Z−] ≅ Ext_{Γ}^{1}[−, −],
T or_{1}^{Λ}[Z−, Z−] ≅ T or_{1}^{Γ}[−, −] ⨁ − ⊗_{Γ} M/M^{2} ⊗_{Γ} −,
Ext_{Λ}^{1}[Z−, Z−] ≅ Ext_{Γ}^{1}[−, −] ⨁ Hom_{Γ}[− ⊗_{Γ} M/M^{2}, −],
T or_{2}^{Λ}[T−, Z−] ≅ T or_{2}^{Γ}[−, −] ⨁ T or_{1}^{Γ}[−, Λ] ⊗_{Λ} Γ ⊗_{Γ} −,
Ext_{Λ}^{2}[T−, Z−] ≅ Ext_{Γ}^{2}[−, −] ⨁ Hom_{Γ}[T or_{1}^{Γ}[−, Λ] ⊗_{Λ} Γ, −].

(ii) w.gl.dimΓ ≤ w.gl.dimΛ and r.gl.dimΓ ≤ r.gl.dimΛ.

(iii) If M is left Γ−flat then there are isomorphisms, ∀n ≥ 1:

T or_{n}^{Λ}[T−, −] ≅ T or_{n}^{Γ}[−, U−], Ext_{Λ}^{n}[T−, −] ≅ Ext_{Γ}^{n}[−, U−],
T or_{n}^{Λ}[Z−, Z−] ≅ T or_{n}^{Γ}[−, −] ⨁ T or_{n−1}^{Λ}[ΦZ−, Z−],
Ext_{Λ}^{n}[Z−, Z−] ≅ Ext_{Γ}^{n}[−, −] ⨁ Ext_{Λ}^{n−1}[ΦZ−, Z−].

(iv) If M is left Γ−flat and ϑ = 0, then there are isomorphisms, ∀n ≥ 0:

T or_{n}^{Λ}[Z−, Z−] ≅ ⨁_{i=0}^{n} T or_{n}^{Γ}[− ⊗_{Γ} (⊗_{Γ}^{n−i}M), −],

Ext_{Λ}^{n}[Z−, Z−] ≅ ⨁_{i=0}^{n} Ext_{Γ}^{n}[− ⊗_{Γ} (⊗_{Γ}^{n−i}M), −].

4.2 The Natural morphism : L_{∗}^{𝔉}C → L_{∗}^{Σ}C.

As we’ll see in the next section, the behaviour of 𝔉−global dimension of D_{F} (η) depends essentially on the behaviour of the sequence of functors L_{i}^{𝔉}C, i ≥ 0. It follows from the Σ−associated complex Σ_{(X,f)}^{•} of (X, f), especially when f = 0, that the study of the sequence of functors L_{i}^{Σ}C, i ≥ 0, requires information which is more close to the triad {D, F, η}. Hence it is useful to reduce the study of the sequence L_{i}^{𝔉}C, i ≥ 0, to the study of the sequence L_{i}^{Σ}C, i ≥ 0. In this respect we have the following result, in which we denote by KerC the full subcategory of D_{F} (η) with objects (X, f), such that C(X, f) = 0.

Theorem 4.7. There are natural morphisms:

ξ^{i} : L_{i}^{𝔉}C → L_{i}^{Σ}C, ∀i ≥ 0, such that :

(i) ξ^{0}, ξ^{1} are isomorphisms and ξ^{2} is an epimorphism.

(ii) Consider the following statements, ∀n ≥ 2:

(a) The morphisms ξ_{(X,f)}^{i} are isomorphisms, 1 ≤ i ≤ n, and ξ_{(X,f)}^{n+1} is an epimorphism.

(b) L_{i}^{𝔉}CT(X) = 0, 2 ≤ i ≤ n.

(c) L_{i}^{ℜ}F (X) = 0, 1 ≤ i ≤ n − 1.

Then (a) ⇔ (b) ⇐ (c). If KerC = 0, then all these statements are equivalent.

(iii) Consider the following statements:

(a) The morphisms ξ_{(X,f)}^{i} are isomorphisms, ∀i ≥ 0.

(b) L_{i}^{𝔉}CT(X) = 0, ∀i ≥ 2.

(c) L_{i}^{ℜ}F (X) = 0, ∀i ≥ 1, i.e. F is ℜ−exact.

Then (a) ⇔ (b) ⇐ (c). If KerC = 0, then all these statements are equivalent.

Proof. (i) By Corollary 4.2: L_{1}^{𝔉}CT = 0. Let now (X, f) ∈ D_{F} (η), and consider the natural Σ−projective presentation (1): 0 → Φ(X, f) ⟶^{m} T(X) ⟶^{ε_{(X,f)}} (X, f) → 0 of (X, f). The sequence (1) is 𝔉−exact since it is Σ−exact and Σ ⊆ 𝔉. Applying the functor C to (1) we get isomorphisms: ρ_{(X,f)}^{i} : L_{i+1}^{Σ}C(X, f) → L_{i}^{Σ}CΦ(X, f), ∀i ≥ 1, and a long exact sequence (2):

⋯ → L_{i+1}^{𝔉}CT(X) → L_{i+1}^{𝔉}C(X, f) ⟶^{σ_{(X,f)}^{i}} L_{i}^{𝔉}CΦ(X, f) → L_{i}^{𝔉}CT(X) → ⋯

⋯ → L_{2}^{𝔉}C(X, f) ⟶^{σ_{(X,f)}^{1}} L_{1}^{𝔉}CΦ(X, f) → L_{1}^{𝔉}CT(X) → L_{1}^{𝔉}C(X, f) →

→ CΦ(X, f) → X → C(X, f) → 0.

Since L_{1}^{𝔉}CT(X) = 0 we have L_{1}^{𝔉}C(X, f) ≅ Ker(CΦ(X, f) → X) = L_{1}^{Σ}C(X, f) by an isomorphism ξ_{(X,f)}^{1} and the morphism σ_{(X,f)}^{1} : L_{2}^{𝔉}C(X, f) → L_{1}^{𝔉}CΦ(X, f) is an epimorphism. Since (1) is functorial, the morphisms ρ_{(X,f)}^{i}, ξ_{(X,f)}^{1}, σ_{(X,f)}^{1} are natural, thus we get natural isomorphisms ρ^{i} : L_{i+1}^{Σ}C → L_{i}^{Σ}CΦ, ξ^{1} : L_{1}^{𝔉}C → L_{1}^{Σ}C, and a natural epimorphism σ^{1} : L_{2}^{𝔉}C → L_{1}^{𝔉}CΦ. We set: ξ^{2} = σ^{1} ∘(ξ^{1}Φ)∘ (ρ^{1})^{−1} : L_{2}^{𝔉}C → L_{2}^{Σ}C. Then ξ^{2} is an epimorphism. Now we proceed inductively setting ξ^{i+1} = σ^{i} ∘ ξ^{i}Φ ∘ (ρ^{i})^{−1} : L_{i+1}^{𝔉}C → L_{i+1}^{Σ}C, ∀i ≥ 1. Then the natural morphisms ξ^{i} : L_{i}^{𝔉}C → L_{i}^{Σ}C , ∀i ≥ 0, together with the obvious isomorphism ξ^{0} : L_{0}^{𝔉}C ≅ L_{0}^{Σ}C = C, is the desired sequence of natural morphisms.

(ii), (iii) (a) ⇔ (b) Follows directly from the long exact sequence (2), and the definition of the natural morphisms ξ^{i}. (c) ⇒ (b) follows from Corollary 4.2(3). Suppose now that KerC = 0, and (b) is true. Since L_{2}^{𝔉}CT(X) = 0, by the last isomorphism in Corollary 4.2(2) we have: CL_{1}^{ℜ}T(X) = 0. Since KerC = 0, L_{1}^{ℜ}T(X) = 0 ⇒ L_{1}^{ℜ}F (X) = 0. Let 0 → K_{0} → P_{0} → X → 0 be an ℜ−projective presentation of X. Since L_{1}^{ℜ}T(X) = 0, we have the 𝔉−projective presentation 0 → T(K_{0}) → T(P_{0}) → T(X) → 0 of T(X). Then 0 = L_{3}^{𝔉}CT(X) = L_{2}^{𝔉}CT(K_{0}) = CL_{1}^{ℜ}T(K_{0}) ⇒ L_{1}^{ℜ}F (K_{0}) = L_{2}^{ℜ}F (X) = 0. A simple induction and dimension-shifting argument completes the proof.

Remark 4.8. The above morphism ξ^{∗} : L_{∗}^{𝔉}C → L_{∗}^{Σ}C is a morphism of Σ− connected sequence of functors in the sense of [25], [18]. For more information concerning the morphism ξ^{∗}, see Section 7.

Corollary 4.9. If F is ℜ−exact, then for any proper class 𝔉 in D_{F} (η) of the form 𝔉 = U^{−1}(ℜ), where ℜ is a proper class in D, we have:

L_{i}^{𝔉}C ≅ L_{i}^{Σ}C, ∀i ≥ 0.

In particular if η = 0, then: L_{i}^{𝔉}CZ ≅ L_{i}^{Σ}CZ ≅ F^{i}, ∀i ≥ 0.

If Λ = Γ ⋉_{ϑ} M is a cleft extension of rings, then the proper class Σ is the class of short exact sequences of Λ−modules, which are split when considered as short exact sequences of Γ−modules. In this concrete case we use the notation Σ := (Λ, Γ).

Corollary For Rings 4.10. Let Λ = Γ ⋉_{ϑ} M be a cleft extension of rings. Then for the following statements:

(i) M is left Γ−flat.

(ii) T or_{i}^{Λ}[− ⊗_{Γ} Λ, Γ] = 0, ∀i ≥ 1.

(iii) T or_{i}^{Λ}[−, Γ] ≅ T or_{i}^{(Λ,Γ)}[−, Γ], ∀i ≥ 1.

we have (i) ⇒ (ii) ⇔ (iii); if ϑ is right T-nilpotent, then they are equivalent.

4.3 Lifting Tilting Objects

We close this section with an application to tilting theory. Let C be an abelian category and let ℜ be a proper class of short exact sequences in C. We recall that if X ∈ C, then add(X) is the full subcategory of C with objects the direct summands of all finite direct sums of copies of X. Define a full subcategory add(X)_{ℜ}^{t} of C as follows: add(X)_{ℜ}^{t} = {A ∈ C : there exists an ℜ−exact sequence : 0 → A → X_{0} → ⋯ → X_{t} → 0, with X_{i} ∈ add(X), ∀i = 0, 1, ..., t}, where t ≥ 0.

We call an object X ∈ C, a t−ℜ−tilting object iff (α) ℜ−p.dX = t < ∞, (β) Ext_{ℜ}^{i}[X, X] = 0, ∀i ≥ 1, and (γ) P(ℜ) ⊆ add(X)_{ℜ}^{t}.

Theorem 4.11. Suppose that KerC = 0.

(1) Assume that X ∈ D is an object with: L_{t}^{ℜ}F (X) = 0, 1 ≤ t ≤ n. Then for any t with: 1 ≤ t ≤ n, the following are equivalent:

(i) T(X) is a t−𝔉−tilting object.

(ii) (a) X is a t−ℜ−tilting object.

(b) Ext_{ℜ}^{i}[X, F X] = 0, 1 ≤ i ≤ t.

(2) The following are equivalent:

(i) T(X) is a 1−𝔉−tilting object.

(ii) (a) X is a 1−ℜ−tilting object.

(b) Ext_{ℜ}^{1}[X, F X] = L_{1}^{ℜ}F (X) = 0.

The ring End_{D_{F} (η)}[T(X)] is a cleft ϑ−extension of the ring End_{D}[X], by the bimodule D[X, F X], where ϑ is defined by ϑ(a ⊗ b) = a ∘ F b ∘ η_{X}, for a, b ∈ D[X, F X] and the bimodule structure of D[X, F X] is given by f ·a = f ∘a and b · g = b ∘ F g, for f, g ∈ End_{D}[X] and a, b ∈ D[X, F X].

Proof. (1) The hypothesis L_{t}^{ℜ}F (X) = 0, 1 ≤ t ≤ n, and Corollary 4.2(3), implies that (∗): Ext_{𝔉}^{t}[T(X), T(X)] ≅ Ext_{ℜ}^{t}[X, X ⊕ F X], 1 ≤ t ≤ n.

(a) ⇒ (b) Let t ∈ ℕ with 1 ≤ t ≤ n, and suppose that T(X) is a t − 𝔉−tilting object. Then since 𝔉−p.dT(X) = t and ℜ−p.dX ≤ 𝔉−p.dT(X), by (∗) we have ℜ−p.dX = 𝔉−p.dT(X) = t, and 0 = Ext_{𝔉}^{i}[T(X), T(X)] ≅ Ext_{ℜ}^{i}[X, X ⊕ F X] ≅ Ext_{ℜ}^{i}[X, X] ⊕ Ext_{ℜ}^{i}[X, F X], ∀i ≥ 1. Thus it remains to show that P(ℜ) ⊆ add(X)_{ℜ}^{t}. If P ∈ P(ℜ), then T(P) ∈ P(𝔉) and since T(X) is an 𝔉−tilting object, there exists an 𝔉−exact sequence (∗∗): 0 → T(P) → (X_{0}, f_{0}) → ⋯ → (X_{t}, f_{t}) → 0, with (X_{i}, f_{i}) ∈ addT(X), ∀i = 0, 1, ..., t. But then obviously C(X_{i}, f_{i}) ∈ add(X), ∀i = 0, 1, ..., t. We claim that the sequence C(∗∗): 0 → P → C(X_{0}, f_{0}) → ⋯ → C(X_{t}, f_{t}) → 0 is ℜ−exact. Indeed since by Corollary 4.2, L_{1}^{𝔉}C ≅ L_{1}^{Σ}C and since all the objects in (∗) are direct summands of objects in ImT ⊆ P(Σ), it follows that any object in (∗) is in the kernel of the functor L_{1}^{𝔉}C. Hence C(∗∗) is exact and it is ℜ−exact, since C sends 𝔉−proper epics in D_{F} (η) to ℜ−proper epics in D. Thus P(ℜ) ⊆ add(X)_{ℜ}^{t}.

(b) ⇒ (a) If X is a t−ℜ−tilting object, then ℜ−p.dX = 𝔉−p.dT(X) = t and Ext_{𝔉}^{i}[T(X), T(X)] ≅ Ext_{ℜ}^{i}[X, X ⊕ F X] ≅ Ext_{ℜ}^{i}[X, X] ⊕ Ext_{ℜ}^{i}[X, F X] = 0, ∀i ≥ 1, by hypothesis. So it remains to show that P(𝔉) ⊆ addT(X)_{𝔉}^{t}. Let (X, f) ∈ P(𝔉); then by [6] or Lemma 5.3 below, (X, f) ≅ T(P) with P ∈ P(ℜ). Since X is an ℜ−tilting object, there exists an ℜ−exact sequence (!): 0 → P → X_{0} → ⋯ → X_{t} → 0, with X_{i} ∈ add(X), ∀i = 0, 1, ..., t. Applying T we obtain the 𝔉−proper sequence T(!): 0 → T(P) → T(X_{0}) → ⋯ → T(X_{t}) → 0. Since X_{i} ∈ add(X), ∀i = 0, 1, ..., t and L_{n}^{ℜ}F X = 0, ∀n ≥ 1, and since L_{1}^{ℜ}F ≅ UL_{1}^{ℜ}T, we see easily that T(!) is 𝔉−exact. Thus P(𝔉) ⊆ addT(X)_{𝔉}^{t}.

(2) From the proof of (1), it is enough to prove that if T(X) is 1−𝔉−tilting object, then L_{1}^{ℜ}F (X) = 0. Since 𝔉−p.dT(X) = 1, we have L_{2}^{𝔉}CT(X) = 0. But by Corollary 4.2(2): L_{2}^{𝔉}CT(X) ≅ CL_{1}^{ℜ}T(X). Hence CL_{1}^{ℜ}T(X) = 0, and since KerC = 0, we have L_{1}^{ℜ}T(X) = 0. But then UL_{1}^{ℜ}T(X) = L_{1}^{ℜ}F (X) = 0.

The last assertion follows from [6].

Now consider a cleft ϑ−extension Λ = Γ ⋉_{ϑ} M of rings. In the following result which is a consequence of the above Theorem and its dual, an n−tilting (n−cotilting) module is a (co-)tilting module of projective (injective) dimension n, in the sense of [30]. We note that the case (1) below is a generalization of the main result of [26].

Corollary For Rings 4.12. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings and assume that ϑ is right T −nilpotent. Then:

(1) The following are equivalent:

(i) X ⊗_{Γ} Λ is a 1−tilting module.

(ii) X is a 1−tilting module, and Ext_{Γ}^{1}[X, X ⊗_{Γ} M] = T or_{1}^{Γ}[X, M] = 0.

(2) Suppose that T or_{i}^{Γ}[X, M] = 0, ∀i ≥ 1. Then the following are equivalent:

(i) X ⊗_{Γ} Λ is an n−tilting module.

(ii) X is an n−tilting module, and Ext_{Γ}^{i}[X, X ⊗_{Γ} M] = 0, ∀i ≥ 1.

If ϑ is left T −nilpotent, then:

(3) The following are equivalent:

(i) Hom_{Γ}[Λ, X] is a 1−cotilting module.

(ii) X is a 1−cotilting module, and Ext_{Γ}^{1}[Hom_{Γ}[M, X], X] = Ext_{Γ}^{1}[M, X] = 0.

(4) Suppose that Ext_{Γ}^{i}[M, X] = 0, ∀i ≥ 1. Then the following are equivalent:

(i) Hom_{Γ}[Λ, X] is an n−cotilting module.

(ii) X is an n−cotilting module, and Ext_{Γ}^{i}[Hom_{Γ}[M, X], X] = 0, ∀i ≥ 1.

Our next result is a corollary of part (4) of the above result.

Corollary 4.13. Let Λ = Γ ⋉_{0}M be a trivial cleft extension of rings. Suppose that Ext_{Γ}^{i}[M_{Γ}, M_{Γ}] = 0, ∀i ≥ 1, and the natural morphism ω_{r} : Γ → End_{Γ}(M_{Γ}) is an isomorphism. Then: Hom_{Γ}[Λ, M_{Γ}]_{Λ} is an n−cotilting module ⇔ M_{Γ} is an n−cotilting module; in this case: Λ ≅ End_{Λ}[Hom_{Γ}[Λ, M_{Γ}]], and: i.dΛ_{Λ} = n.

Auslander-Reiten in [2] ask if the trivial extension of a Cohen-Macaulay Artin algebra is Gorenstein (see [2] for the definition of the involved concepts). Our final result in this subsection is a direct consequence of the previous corollary and answers the question of Auslander-Reiten in the affirmative.

Corollary 4.14. Let Γ be a Cohen-Macaulay Artin algebra with dualizing bimodule _{Γ}M_{Γ} and consider the trivial extension Λ = Γ ⋉_{0}M. Then:

i.dΛ_{Λ} = i.d_{Λ}Λ = i.dM_{Γ} = i.d_{Γ}M < ∞

and Λ is a Gorenstein algebra.

5 𝔉−, Σ−Relative Global Dimension of Cleft Extensions

We continue to use the notation introduced in the previous section.

Definition 5.1. The category D_{F} (η) is called admissible if any object (X, f) ∈ D_{F} (η) has a finite filtration

0 = (X_{n}, f_{n}) ⊆ (X_{n−1}, f_{n−1}) ⊆ ⋯ ⊆ (X_{1}, f_{1}) ⊆ (X_{0}, f_{0}) = (X, f)

such that all quotient objects (X_{j}, f_{j})/(X_{j+1}, f_{j+1}) are in Z(D).

Hence D_{F} (η) is admissible iff the full subcategory Z(D) is admissible in the sense of [34]. By our conventions D_{F} (η) is admissible iff it is 𝔉−admissible in the sense that, in the filtration above all short exact sequences: 0 → (X_{j+1}, f_{j+1}) → (X_{j}, f_{j}) → (X_{j}, f_{j})/(X_{j+1}, f_{j+1}) → 0 are in 𝔉, ∀j = 0, 2, ..., n − 1.

Remark 5.2. (1) By a result of [6], D_{F} (η) is admissible ⇔ the functor R := Ker(λ) is locally nilpotent, i.e. ∀(X, f) ∈ D_{F} (η), ∃n ≥ 0 : R^{n}(X, f) = 0. Note that in this case by [34], the categories D and D_{F} (η) have isomorphic higher K−theory.

(2) If D_{F} (η) is admissible, then KerC = 0 (see [6]).

(3) The morphism η : F^{2} → F is called locally nilpotent, resp. nilpotent, if ∀X ∈ D, ∃t ∈ ℕ : η_{F^{t}X} ∘ η_{F^{t−1}X} ∘ ⋯ ∘ η_{F X} ∘ η_{X} = 0, resp. ∃t ∈ ℕ : η_{F^{t}} ∘ η_{F^{t−1}} ∘ ⋯ ∘ η_{F} ∘ η = 0. In the last case η is called t−nilpotent. It is not difficult to see that if η is locally nilpotent, then D_{F} (η) is admissible, in particular KerC = 0. It follows that a trivial extension or more generally a truncated extension is admissible.

(3) If η is t−nilpotent, then D_{F} (η) is a full exact subcategory of the truncated extension D ⋉_{t+1}F. This follows form the fact that ∀t ≥ 0, ∀(X, f) ∈ D_{F} (η) : η_{F^{t}X} ∘ η_{F^{t−1}X} ∘ ⋯ ∘ η_{F X} ∘ η_{X} ∘ f = F^{t+1}f ∘ F^{t}f ∘ ⋯ ∘ F^{2}f ∘ F f ∘ f.

(4) If D_{F} (η) is admissible, then it is 𝔉−admissible, hence the vanishing of the extension functors Ext_{𝔉}^{n}[−, −] is controlled by the vanishing of Ext_{𝔉}^{n}[Z(−), −].

(5) If the cleft extension is induced by a cleft extension of rings Λ = Γ ⋉_{ϑ} M, then [6]: ϑ is nilpotent iff η = 1 ⊗_{Γ} ϑ is (locally) nilpotent iff Mod(Λ) is admissible. As in Subsection 2.1, ϑ is right, resp. left, T −nilpotent ⇔ KerC = 0, resp. KerK = 0.

We recall that an epic α : X → Y in an abelian category A is called minimal if for any morphism β : Z → X, β ∘ α is epic implies that β is epic. If the category A has enough 𝔉−projectives with respect to a proper class of short exact sequences 𝔉, then a morphism α : P → X in 𝔉 is called an 𝔉−projective cover, if α is a minimal epic and P is 𝔉−projective. Then A is called 𝔉−perfect if any object X ∈ A has an 𝔉−projective cover. Consider now the full subcategory KerC = {(X, f) ∈ D_{F} (η) : C(X, f) = 0}. In [6] we have shown that KerC = 0 ⇔ λ is a minimal epic, i.e. ∀(X, f) ∈ D_{F} (η), λ_{(X,f)} is a minimal epic, and that the condition KerC = 0 has many pleasant consequences for the structure of the simple and projective objects of D_{F} (η). We restate Lemma 5.4 of [6] in the present setting:

Lemma 5.3. If KerC = 0, then we have the following:

(i) D_{F} (η) is 𝔉-perfect ⇔ D is ℜ-perfect.

(ii) (X, f) is simple ⇔ C(X, f) is simple and (X, f) ≅ ZC(X, f).

(iii) The following are equivalent:

(a) (X, f) ∈ P(𝔉).

(b) C(X, f) ∈ P(ℜ) and (X, f) ≅ TC(X, f).

(c) C(X, f) ∈ P(ℜ) and L_{1}^{Σ}C(X, f) = L_{1}^{𝔉}C(X, f) = 0.

(d) Ext_{𝔉}^{1}[(X, f), Z(−)] = 0.

Proof. We only prove that (a) ⇔ (d) in (iii). The other parts follows directly from Lemma 5.4 of [6]. Obviously (a) ⇒ (d). Suppose that (d) is true. Consider an 𝔉−projective presentation of (X, f): 0 → (Q_{0}, g_{0}) → T(P_{0}) → (X, f) → 0. Then we have the exact sequence: 0 → D_{F} (η)[(X, f), Z(−)] → D_{F} (η)[T(P_{0}), Z(−)] → D_{F} (η)[(Q_{0}, g_{0}), Z(−)] → 0 which is isomorphic to the short exact sequence: 0 → D[C(X, f), −] → D[P_{0}, −] → D[C(Q_{0}, g_{0}), −] → 0. This implies that the exact sequence: C(Q_{0}, g_{0}) → P_{0} → C(X, f) → 0 is indeed a split short exact sequence and C(X, f) ∈ P(ℜ) as a direct summand of P_{0}. The above argument shows also that L_{1}^{𝔉}C(X, f) = Ker(C(Q_{0}, g_{0}) → P_{0}) = 0, thus by (c), (X, f) ∈ P(𝔉).

If Λ = Γ ⋉_{ϑ} M is a cleft ring extension, then the next result follows from Lemma 5.3, using the realization of the functors F, G, C, T, H, K, Z, U from Section 2.

Corollary For Rings 5.4. (1) If ϑ is right T −nilpotent, then:

(i) Λ is right perfect ⇔ Γ is right perfect.

(ii) (X, f) is simple in Mod(Λ) ⇔ X is simple in Mod(Γ) and f = 0.

(iii) The following are equivalent:

(a) (X, f) is a projective, (resp. flat, resp. free), right Λ−module.

(b) (X, f) ⊗_{Λ} Γ_{Γ} is a projective, (resp. flat, resp. free), right Γ−module and Ker(f) = Im(1_{X} ⊗ ϑ − 1_{M} ⊗ f).

(c) (X, f) ⊗_{Λ} Γ_{Γ} is a projective, (resp. flat, resp. free), right Γ−module and (X, f) ⊗_{Λ} Γ ⊗_{Γ} Λ ≅ (X, f).

(d) Ext_{Λ}^{1}[(X, f), − ⊗_{Γ} Γ_{Λ}] = 0, (resp. T or_{1}^{Λ}[(X, f),_{Λ}Γ ⊗_{Γ} −] = 0).

(2) If ϑ is left T −nilpotent, then the following are equivalent:

(a) (X, f) is an injective right Λ−module.

(b) Hom_{Λ}[_{Γ}Γ_{Λ}, (X, f)] is an injective right Γ−module and Ker(f^{∗}) = Im(ζ_{X} −Hom_{Γ}[M, f^{∗}]), where f^{∗} : X → Hom_{Γ}[M, X] is defined by f^{∗}(x)(m) = f(x ⊗ m) and ζ_{X} : Hom_{Γ}[M, X] → Hom_{Γ}[M, Hom_{Γ}[M, X]] is defined by ζ_{X}(ϕ)(m)(n) = ϕ(ϑ(m ⊗ n)), ∀m, n ∈ M, ∀x ∈ X and ∀ϕ ∈ Hom_{Γ}[M, X].

(c) Hom_{Λ}[_{Γ}Γ_{Λ}, (X, f)] is an injective right Γ−module and (X, f) ≅ Hom_{Γ}[_{Λ}Λ_{Γ}, Hom_{Λ}[_{Γ}Γ_{Λ}, (X, f)]].

(d) Ext_{Λ}^{1}[Hom_{Γ}[_{Λ}Γ_{Γ}, −], (X, f)] = 0. Remark 5.5. Suppose that F ≠ 0. If KerC = 0, then C is never exact, and always Σ, 𝔉−gl.dimD_{F} (η) ≥ 1. Indeed otherwise applying C to the short exact sequence 0 → ΦZ(X) → T(X) → Z(X) → 0, we get directly that CΦZ(X) = 0, thus ΦZ(X) = 0 since KerC = 0. But then UΦZ(X) = F X = 0. Since this happens for any X ∈ D, we get that: F = 0.

5.1 Cleft Extensions of Finite Global Dimension

The next result gives necessary and sufficient conditions for the 𝔉−global dimension of D_{F} (η) to be finite. Theorem 5.6. (1) If KerC = 0, then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (η) < ∞. (ii) ℜ−gl.dimD < ∞ and ∃i ≥ 0 : L_{i+1}^{𝔉}C = 0. In any case we have the following bounds:

ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (η) ≤ min{n ≥ 0 : L_{n+1}^{𝔉}C = 0} + ℜ−gl.dimD.

(2) Suppose that D_{F} (η) is admissible. Then the following are equivalent: (i) 𝔉−gl.dimD_{F} (η) < ∞. (ii) ℜ−gl.dimD < ∞ and ∃i ≥ 0 : L_{i+1}^{𝔉}CZ = 0. In any case we have the following bounds: ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (η) ≤ min{n ≥ 0 : L_{n+1}^{𝔉}CZ = 0} + ℜ−gl.dimD. Proof. (1) (⇒) If 𝔉−gl.dimD_{F} (η) = m < ∞, then L_{m+1}^{𝔉}C = 0, and by Corollary 4.5 we have: ℜ−gl.dimD ≤ m < ∞.

(⇐) Suppose that ℜ−gl.dimD = k < ∞ and let n = min{i ≥ 0 : L_{i+1}^{𝔉}C = 0}. Let (X, f) ∈ D_{F} (η) be an arbitrary object and let 𝔉_{(X,f)}^{•} be the 𝔉−associated complex of (X, f). From the construction of 𝔉_{(X,f)}^{•}, the following complex

⋯ → P_{n+t+1} ⟶^{ω_{n+t}} P_{n+t} → ⋯ → P_{n+1} ⟶^{ω_{n}} P_{n} ⟶^{coker(ω_{n})} Coker(ω_{n}) → 0.

is an acyclic complex consisting of ℜ−projective objects. Since by Remark 3.2 the above complex is ℜ−proper, it is an ℜ−projective resolution of Coker(ω_{n}) ≅ C(Q_{n−1}, g_{n−1}). Since ℜ−gl.dimD = k, Coker(ω_{n+k}) is in P(ℜ) as a k^{th}−syzygy of Coker(ω_{n}) with respect to the proper class ℜ. Since by construction the object (Q_{j}, g_{j}) is a (j +1)^{th}−syzygy of (X, f) with respect to the proper class 𝔉, ∀j ≥ 0, the hypothesis implies that L_{1}^{𝔉}C(Q_{n+k−1}, g_{n+k−1}) = L_{n+k+1}^{𝔉}C(X, f) = 0. Hence by Lemma 5.3 we have (Q_{n+k−1}, g_{n+k−1}) ∈ P(𝔉) and consequently 𝔉−p.d(X, f) ≤ n + k. We conclude: 𝔉−gl.dimD_{F} (η) ≤ n + k < ∞.

(2) Since D_{F} (η) is admissible, by Remark 5.2(2) we have that KerC = 0, and the hypotheses of (1) are satisfied. Hence it suffices to show that: L_{n+1}^{𝔉}C = 0 ⇔ L_{n+1}^{𝔉}CZ = 0. The (⇒) direction is trivial and the (⇐) direction follows directly from the fact that D_{F} (η) is admissible.

Theorem 5.7. (1) If the functor F is ℜ−exact, then ∀(X, f) ∈ D_{F} (η):

𝔉−p.d(X, f) ≤ Σ−p.d(X, f) + ℜ−gl.dimD (∗)

(2) If the functor F is ℜ−exact, and F (P(ℜ)) ⊆ P(ℜ), then ∀(X, f) ∈ D_{F} (η):

𝔉−p.d(X, f) ≤ Σ−p.d(X, f) + ℜ−p.dX (∗∗)

In any of the above cases we have the bounds:

ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (η) ≤ ℜ−gl.dimD + Σ−gl.dimD_{F} (η).

Proof. If Σ−p.d(X, f) = ∞ or ℜ−gl.dimD = ∞ then the relation (∗) is true trivially. Suppose that ℜ−gl.dimD = t < ∞, Σ−p.d(X, f) < ∞ and we apply induction on Σ−p.d(X, f). If Σ−p.d(X, f) = 0, then (X, f) ∈ P(Σ) and the Σ−exact sequence: (1) : 0 → Φ(X, f) → T(X) ⟶^{ε_{(X,f)}} (X, f) → 0 splits. Hence we have: T(X) ≅ (X, f) ⊕ Φ(X, f), and

𝔉−p.dT(X) = max{𝔉−p.d(X, f), 𝔉−p.dΦ(X, f)}.

Since F is ℜ−exact, by Corollary 4.2 we have 𝔉−p.dT(X) = ℜ−p.dX, thus 𝔉−p.d(X, f) ≤ ℜ−p.dX ≤ ℜ−gl.dimD and (∗) is true. Let Σ−p.d(X, f) > 0 and assume (∗) is true for all objects (Y, g) with: Σ−p.d(Y, g) < Σ−p.d(X, f). Since Σ ⊆ 𝔉, the Σ−exact sequence (1) is also 𝔉−exact, hence for any (W, h) ∈ D_{F} (η) we have a long exact sequence:

⋯ → Ext_{𝔉}^{n−1}[T(X), (W, h)] → Ext_{𝔉}^{n−1}[Φ(X, f), (W, h)] → Ext_{𝔉}^{n}[(X, f), (W, h)]

→ Ext_{𝔉}^{n}[T(X), (W, h)] → Ext_{𝔉}^{n}[Φ(X, f), (W, h)] → ⋯

We set s := Σ−p.d(X, f) + t + 1. Since F is ℜ−exact from Corollary 4.2, we have an isomorphism: Ext_{𝔉}^{s−1}[T(X), (W, h)] ≅ Ext_{ℜ}^{s−1}[X, W] = 0, since s − 1 = Σ−p.d(X, f) + t > 0 and t = ℜ−gl.dimD. Hence the above long exact sequence induces an isomorphism Ext_{𝔉}^{s−1}[Φ(X, f), (W, h)] ≅ Ext_{𝔉}^{s}[(X, f), (W, h)]. By induction hypothesis, since Σ−p.d(X, f) − 1 = Σ−p.dΦ(X, f) < Σ−p.d(X, f), we have: 𝔉−p.dΦ(X, f) ≤ Σ−p.dΦ(X, f) + t = s − 2. But then: Ext_{𝔉}^{s−1}[Φ(X, f), (W, h)] ≅ Ext_{𝔉}^{s}[(X, f), (W, h)] = 0, and consequently 𝔉−p.d(X, f) ≤ s − 1 = Σ−p.d(X, f) + t because (W, h) was arbitrary. We conclude that (∗) is true.

If we have further F (P(ℜ)) ⊆ P(ℜ), then since F is ℜ−exact, obviously ∀X ∈ D: ℜ−p.dF X ≤ ℜ−p.dX. Applying the above induction hypothesis with t = ℜ−p.dX, we’ll arrive at the relation 𝔉−p.dΦ(X, f) ≤ ℜ−p.dF X + t, because Φ(X, f) = (F X, η_{X} − F f). Then 𝔉−p.dΦ(X, f) ≤ ℜ−p.dX + t and we can continue the induction as above, concluding that (∗∗) is true. The last relation follows from (∗) and Corollary 4.5.

It is interesting to note that the conditions in (2) of Theorem 5.7, forces Σ to be central in 𝔉, in the sense of [7]. Combining the results of Section 4 and Proposition 5.8(iv) of [6] (which also can be deduced easily from our previous results), we have the following Corollary, in which we denote by c(F) = ∞ or c(F) = min{n ≥ 0 : F^{n+1} = 0} the nilpotency class of F.

Corollary 5.8. (1) Σ−gl.dimD_{F} (η) ≤ c(F). If η = 0 then Σ−gl.dimD_{F} (0) < ∞ ⇔ the functor F is nilpotent; in the last case we have: Σ−gl.dimD_{F} (0) = c(F).

(2) If F is ℜ−exact, then we have the following:

(i) If the functor F is nilpotent, then:

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

(ii) If Σ−gl.dimD_{F} (η) < ∞: 𝔉−gl.dimD_{F} (η) < ∞ ⇔ ℜ−gl.dimD < ∞.

(iii) If Σ−gl.dimD_{F} (η) = 0 then 𝔉−gl.dimD_{F} (η) = c(F) + ℜ−gl.dimD.

Corollary For Rings 5.9. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings.

(i) (Λ, Γ)−r.gl.dimΛ ≤ min{m ∈ ℕ : ⊗_{Γ}^{n+1}M = 0}, with equality if ϑ = 0.

(ii) If M is left Γ−flat then:

r.gl.dimΓ ≤ r.gl.dimΛ ≤ (Λ, Γ)−r.gl.dimΛ + r.gl.dimΓ.

In particular: r.gl.dimΓ ≤ r.gl.dimΛ ≤ n + r.gl.dimΓ, if ⊗_{Γ}^{n+1}M = 0.

(iii) If M is left Γ−flat and right Γ−projective, then ∀(X, f) ∈ Mod(Λ):

p.d(X, f) ≤ (Λ, Γ)−p.d(X, f) + p.dX.

(iv) If ϑ is right T-nilpotent, then: r.gl.dimΛ < ∞ ⇔ r.gl.dimΓ < ∞ and w.d_{Λ}Γ < ∞. In any case we have the bounds:

r.gl.dimΓ ≤ r.gl.dimΛ ≤ w.d_{Λ}Γ + r.gl.dimΓ.

In particular if Γ is semisimple, then: r.gl.dimΛ = w.d_{Λ}Γ.

For a proper class ℜ in an abelian category C, we denote by P^{∞}(ℜ) the full subcategory of C consisting of objects with finite ℜ−projective dimension. We recall that the finitistic ℜ−global dimension ℜ−f.gl.dimC of C is defined as ℜ−f.gl.dimC = sup{ℜ−p.dX : X ∈ P^{∞}(ℜ)}. The next result generalizes Lemma 5.3(iii).

Lemma 5.10. Suppose that KerC = 0.

(1) If (X, f) ∈ P^{∞}(𝔉) and L_{n}^{𝔉}C(X, f) = 0, ∀n ≥ 1, then:

𝔉−p.d(X, f) = ℜ−p.dC(X, f).

(2) The following are equivalent for an object (X, f) ∈ D_{F} (η) :

(i) (X, f) ∈ P^{∞}(𝔉).

(ii) ∃n ≥ 0 : Ext_{𝔉}^{n+1}[(X, f), Z(−)] = 0.

(iii) ∃n ≥ 0 : Coker(ω_{n}^{(X,f)}) ∈ P^{∞}(ℜ), and L_{n+k}^{𝔉}C(X, f) = 0, ∀k ≥ 1, where ω_{n}^{(X,f)} is the n^{th}−differential of the 𝔉-associated complex 𝔉_{(X,f)}^{•} of (X, f).

If one of the above equivalent statements is true, then setting n = min{m ≥ 0 : L_{m+k}^{𝔉}C(X, f) = 0, ∀k ≥ 1}, we have:

𝔉−p.d(X, f) ≤ n + ℜ−p.dCoker(ω_{n}^{(X,f)}).

Proof. (1) By induction on 𝔉−p.d(X, f). If 𝔉−p.d(X, f) = 0, obviously ℜ−p.d C(X, f) = 0. Suppose that the result is true for all (Y, g) with 𝔉−p.d(Y, g) < 𝔉−p.d(X, f) and L_{t}^{𝔉}C(Y, g) = 0, ∀t ≥ 1. If 𝔉−p.d(X, f) > 0, consider the 𝔉− projective presentation 0 → (Q_{0}, g_{0}) ⟶^{κ_{0}} T(P_{0}) ⟶^{α_{0}} (X, f) → 0. Trivially (Q_{0}, g_{0}) satisfies the induction hypothesis, so we have: 𝔉−p.d(Q_{0}, g_{0}) = ℜ−p.dC(Q_{0}, g_{0}). Since L_{t}^{𝔉}C(X, f) = 0, ∀t ≥ 1, the sequence 0 → C(Q_{0}, g_{0})

⟶^{C(κ_{0})} P_{0} ⟶^{C(α_{0})} C(X, f) → 0 is an ℜ−projective presentation of C(X, f). So: 𝔉−p.d(X, f) = 1 − 𝔉−p.d(Q_{0}, g_{0}) = 1 − ℜ−p.dC(Q_{0}, g_{0}) = ℜ−p.dC(X, f).

(2) (i) ⇔ (ii) Follows from Lemma 5.3(iii), and an easy induction argument.

(i) ⇒ (iii) If 𝔉−p.d(X, f) = m < ∞, then the m^{th}−syzygy (Q_{m−1}, g_{m−1}) of (X, f) is 𝔉−projective. By Lemma 5.3(iii), we have: C(Q_{m−1}, g_{m−1}) ≅ Coker(ω_{m}) ∈ P(ℜ) and L_{1}^{𝔉}C(Q_{m−1}, g_{m−1}) = L_{m+1}^{𝔉}C(X, f) = 0.

(iii) ⇒ (i) If ℜ−p.dCoker(ω_{n}) = t < ∞, and L_{n+k}^{𝔉}C(X, f) = 0, ∀k ≥ 1, then the t^{th}−syzygy of Coker(ω_{n}), (which is Coker(ω_{n+t}) because L_{n+k}^{𝔉}C(X, f) = 0, ∀k ≥ 1), is ℜ−projective. But Coker(ω_{n+t}) ≅ C(Q_{n+t−1}, g_{n+t−1}). Now since L_{1}^{𝔉}C(Q_{n+t−1}, g_{n+t−1}) = L_{n+t+1}^{𝔉}C(X, f) = 0, we have by Lemma 5.3(iii), that (Q_{n+t−1}, g_{n+t−1}) ∈ P(𝔉). Since (Q_{n+t−1}, g_{n+t−1}) is a (n + t)^{th}−syzygy of (X, f), we have: 𝔉−p.d(X, f) ≤ n + t, and (X, f) ∈ P^{∞}(𝔉). The last assertion follows from the above proof.

Theorem 5.11. (1) If KerC = 0, then:

𝔉−f.gl.dimD_{F} (η) ≤ ℵ + ℜ−f.gl.dimD.

where: ℵ = sup{m ≥ 0 : L_{m+k}^{𝔉}C(X, f) = 0, ∀k ≥ 1, ∀(X, f) ∈ P^{∞}(𝔉)}.

(2) If the functor L_{n+1}^{𝔉}C vanishes on P^{∞}(𝔉), in particular if L_{n+1}^{𝔉}C = 0, then: 𝔉−f.gl.dimD_{F} (η) ≤ n + ℜ−f.gl.dimD. In particular if L_{t}^{𝔉}C|_{P^{∞}(𝔉)} = 0, ∀t ≥ 1, then: 𝔉−f.gl.dimD_{F} (η) ≤ ℜ−f.gl.dimD.

(3) If the functor F is ℜ−exact, then: ℜ−f.gl.dimD ≤ 𝔉−f.gl.dimD_{F} (η), and the following are equivalent:

(i) 𝔉−f.gl.dimD_{F} (η) < ∞.

(ii) (a) ℵ = sup{m ≥ 0 : L_{m+k}^{𝔉}C(X, f) = 0, ∀k ≥ 1, ∀(X, f) ∈ P^{∞}(𝔉)} < ∞.

(b) ℜ−f.gl.dimD < ∞.

If one of the above equivalent statements is true then:

ℜ−f.gl.dimD ≤ 𝔉−f.gl.dimD_{F} (η) ≤ ℵ + ℜ−f.gl.dimD.

Finally if in addition the functor L_{t}^{𝔉}C vanishes on P^{∞}(𝔉), ∀t ≥ 1, then:

𝔉−f.gl.dimD_{F} (η) = ℜ−f.gl.dimD.

(4) If F is ℜ−exact, then:

ℜ−f.gl.dimD ≤ 𝔉−f.gl.dimD_{F} (η) ≤ Σ−f.gl.dimD_{F} (η) + ℜ−gl.dimD.

If F is ℜ−exact and F (P(ℜ)) ⊆ P(ℜ), then:

ℜ−f.gl.dimD ≤ 𝔉−f.gl.dimD_{F} (η) ≤ Σ−f.gl.dimD_{F} (η) + ℜ−f.gl.dimD.

Proof. (1), (2) are consequences of Lemma 5.10. If F is exact, then by Corollary 4.2, we have ∀X ∈ D : ℜ−p.dX = 𝔉−p.dT(X). Hence ℜ−f.gl.dimD ≤ 𝔉−f.gl.dimD_{F} (η), and (3) follows from (2). Part (4) follows from Theorem 5.7.

Let Λ = Γ⋉_{ϑ}M be a cleft extension of rings. In the following consequence of Theorem 5.11, we denote by P^{∞}(Λ), the full subcategory of Mod(Λ) consisting of all modules of finite projective dimension.

Corollary For Rings 5.12. (1) If ϑ is right T −nilpotent, then:

f.gl.dimΛ ≤ ℵ + f.gl.dimΓ, where

ℵ = sup{m ≥ 0 : T or_{m+k}^{Λ}[(X, f), Γ] = 0, ∀k ≥ 1, ∀(X, f) ∈ P^{∞}(Λ)}.

(2) If the functor T or_{n+1}^{Λ}[−,_{Λ}Γ] vanishes on P^{∞}(Λ), in particular if w.d_{Λ}Γ ≤ n, then: r.f.gl.dimΛ ≤ n + r.f.gl.dimΓ.

(3) If M is left Γ−flat, then: r.f.gl.dimΓ ≤ r.f.gl.dimΛ, and the following are equivalent:

(i) f.gl.dimΛ < ∞.

(ii) (a) The number ℵ in (1) is finite.

(b) r.f.gl.dimΓ < ∞.

Finally if in addition the functor T or_{t}^{Λ}[−,_{Λ}Γ] vanishes on P^{∞}(Λ), ∀t ≥ 1, then:

r.f.gl.dimΛ = r.f.gl.dimΓ.

(4) If M is left Γ−flat, then:

r.f.gl.dimΓ ≤ r.f.gl.dimΛ ≤ (Λ, Γ)−r.f.gl.dimΛ + r.gl.dimΓ.

If M is left Γ−flat and right Γ−projective, then:

r.f.gl.dimΓ ≤ r.f.gl.dimΛ ≤ (Λ, Γ)−r.f.gl.dimΛ + r.f.gl.dimΓ.

5.2 Applications to Free, Symmetric, Polynomial and Exterior Categories

We apply the results obtained so far to the above mentioned categories. We follow the notation introduced in subsection 2.5.

Let D<F_{i}, i ∈ I> be the free category D<F_{i}, i ∈ I> of the abelian category D, with respect to the family of right exact endofunctors {F_{i}}_{i∈I} of D, F_{i} ≠ 0, ∀i. Let J := max{ℵ_{0}, |I|} and assume that D is an AB4^{(J)}−category, i.e. D has exact coproducts indexed by J. The following generalizes a result of Mitchell [29].

Theorem 5.13. If F_{i} is ℜ−exact ∀i ∈ I, then Σ−gl.dimD<F_{i}, i ∈ I> = 1 and

ℜ−gl.dimD ≤ 𝔉−gl.dimD<F_{i}, i ∈ I> ≤ 1 + ℜ−gl.dimD.

Moreover the following are equivalent:

(a) 𝔉−gl.dimD<F_{i}, i ∈ I> = 1 + ℜ−gl.dimD.

(b) ∃X ∈ D : sup_{i∈I}{ℜ−p.dF_{i}X} = ℜ−gl.dimD.

In particular 𝔉−gl.dimD<I> = 1 + ℜ−gl.dimD and ∀n ≥ 0 :

Ext_{𝔉}^{n}[Z(−), Z(−)] ≅ Ext_{ℜ}^{n}[−, −] ⨁ ∏_{i∈I} Ext_{ℜ}^{n−1}[F_{i}−, −].

Proof. (i) We view D<F_{i}, i ∈ I> as D(F), which is a cleft extension of D. Then F is ℜ−exact and moreover from the shape of the left adjoint T of U (see subsection 2.5.1 or [6]), it follows that ∀(X, f) ∈ D(F): Φ(X, f) ≅ T(F X) (∗). From (∗) we see that Σ−gl.dimD(F) ≤ 1. In addition if F_{i} ≠ 0 for some i ∈ I, then Σ−gl.dimD(F) = 1, since otherwise ∀X ∈ D, the Σ−epic ε_{Z(X)} splits, and this implies trivially that F (X) = 0. Then by Theorem 5.7: ℜ−gl.dimD ≤ 𝔉−gl.dimD(F) ≤ 1 + ℜ−gl.dimD. From the functorial Σ−projective presentation of (X, f) we deduce the following long exact sequence, ∀(Y, g) ∈ D(F) :

⋯ → Ext_{𝔉}^{n−1}[Φ(X, f), (Y, g)] → Ext_{𝔉}^{n}[(X, f), (Y, g)] → Ext_{𝔉}^{n}[T(X), (Y, g)] →

→ Ext_{𝔉}^{n}[Φ(X, f), (Y, g)] → ⋯

(a) ⇒ (b) Suppose ℜ−gl.dimD = m < ∞ and 𝔉−gl.dimD(F) = 1 + m. Since F is ℜ−exact by (∗) and Corollary 4.2(3), the above long exact sequence induces an epic: Ext_{ℜ}^{m}[F X, Y] → Ext_{𝔉}^{m+1}[(X, f), (Y, g)] → 0. Hence ℜ−p.dF X = m, since otherwise 𝔉−gl.dimD(F) < 1 + m contrary to our assumption. (b) ⇒ (a) Since F is ℜ−exact, by Corollary 4.2(3) we have:

Ext_{𝔉}^{m+1}[Z(X), Z(Y)] ≅ Ext_{𝔉}^{m}[ΦZ(X), Z(Y)] ≅ Ext_{𝔉}^{m}[T(F X), Z(Y)] ≅
≅ Ext_{ℜ}^{m}[F X, Y].

Since by hypothesis the last group is ≠ 0, we have: 𝔉−gl.dimD(F) = 1 + m. The proof of the last assertion follows from Corollary 4.2(3).

As a corollary of the above Theorem and its dual we have the following result (part (1) is classical and part (2) is a result of Roganov, see [9]).

Corollary For Rings 5.14. (1) r.gl.dimΓ<S> = 1 + r.gl.dimΓ.

(2) Let M be a Γ−Γ−bimodule, flat as a left Γ−module. Then:

r.gl.dimΓ ≤ r.gl.dimT_{Γ}(M) ≤ 1 + r.gl.dimΓ

and the following are equivalent:

(i) r.gl.dimT_{Γ}(M) = 1 + r.gl.dimΓ.

(ii) ∃X ∈ Mod(Γ) : p.dX ⊗_{Γ} M = r.gl.dimΓ.

(iii) ∃X ∈ Mod(Γ) : i.dHom_{Γ}[M, X] = r.gl.dimΓ.

The following are equivalent:

(i) r.gl.dimT_{Γ}(M) ≤ 1.

(ii) r.gl.dimΓ ≤ 1 and ∀X ∈ Mod(Γ), the Γ−module X ⊗_{Γ} M is projective.

(iii) r.gl.dimΓ ≤ 1 and ∀X ∈ Mod(Γ), the Γ−module Hom_{Γ}[M, X] is injective.

Corollary 5.15. If Q is a quiver with finitely many vertices and Γ is semisimple algebra, then the quiver-algebra ΓQ is hereditary.

Consider the symmetric category D[F_{1}, F_{2}, ..., F_{m}] of D with respect to a set {F_{i}}_{i=1}^{n} of commuting right exact endofunctors of D and let D[m] be the polynomial category of D in m−variables. Assume that D is an AB4^{(ℵ_{0})}−category. The following includes a result of Mitchell [29].

Theorem 5.16. Assume that F_{i} is ℜ−exact, ∀i = 1, ..., m. Then:

ℜ−gl.dimD ≤ 𝔉−gl.dimD[F_{1}, F_{2}, ..., F_{m}] ≤ m + ℜ−gl.dimD,
Σ−gl.dimD[F_{1}, F_{2}, ..., F_{m}] = m, 𝔉−gl.dimD[m] = m + ℜ−gl.dimD,

Ext_{𝔉}^{n}[Z(−), Z(−)] ≅ Ext_{ℜ}^{n}[−, −] ⊕ ⨁_{i=1}^{m} Ext_{ℜ}^{n−1}[F_{i}−, −] ⊕ ⨁_{i<j} Ext_{ℜ}^{n−2}[F_{i}F_{j}−, −]⊕

⊕ ⨁_{i<j<k} Ext_{ℜ}^{n−3}[F_{i}F_{j}F_{k}−, −] ⊕ ⋯ ⊕ Ext_{ℜ}^{n−m}[F_{1}F_{2} ⋯ F_{m}−, −].

∀X ∈ D, the 𝔉−cohomology ring Ext_{𝔉}^{∗}[Z(X), Z(X)] of Z(X) in the polynomial category D[m], is the exterior ring of End_{D}(X) in m−variables, and is a cleft extension of the ℜ−cohomology ring Ext_{ℜ}^{∗}[X, X] of X in D.

Proof. Follows directly from Theorem 5.13 and induction.

Corollary For Rings 5.17. r.gl.dimΓ[x_{1}, x_{2}, ⋯, x_{m}] = m + r.gl.dimΓ, and

∀n ≥ 0 : Ext_{Γ[x_{1},x_{2},··· ,x_{m}]}^{n}[Γ, Γ] ≅ Γ^{( \frac{m!}{n!(m−n)!} )}.

Finally consider the exterior category ⋀(D)(F_{1}, F_{2}, ..., F_{m}) of D with respect to a set of commuting right exact endofunctors {F_{i}}_{i=1}^{m} of D and the exterior category ⋀^{m}(D) of D in m−variables. The next Theorem which includes a result of Mitchell [29], follows by induction directly from the results of this section.

Theorem 5.18. Assume that each F_{i} is ℜ−exact.

(1) The following are equivalent:

(i) 𝔉−gl.dim ⋀(D)(F_{1}, F_{2}, ..., F_{m}) < ∞.

(ii) ℜ−gl.dimD < ∞ and F_{i} is nilpotent, ∀i = 1, 2, ..., m.

(2) 𝔉−gl.dim ⋀^{m}(D) = ∞, and ∀n ≥ 0:

Ext_{𝔉}^{n}[Z(−), Z(−)] ≅ ⨁_{k_{m+1}+k_{m}+···+k_{1}=n} Ext_{ℜ}^{k_{m+1}}[F_{1}^{k_{m}}F_{2}^{k_{m−1}} ⋯ F_{m−1}^{k_{2}}F_{m}^{k_{1}}−, −],

ℜ−gl.dimD ≤ 𝔉−gl.dim ⋀(D)(F_{1}, F_{2}, ..., F_{m}) ≤ ∑_{i=1}^{m} c(F_{i}) + ℜ−gl.dimD.

Corollary For Rings 5.19. r.gl.dim ⋀_{Γ}^{n}(Γ) = ∞.

5.3 The Hilbert Basis Theorem

We seek conditions for the cleft extension D_{F} (η) to be (locally) Noetherian. The results of this subsection are independent of the results of the previous Sections.

Lemma 5.20. If X, F X are Noetherian objects in D, then so is T(X) in D_{F} (η). Hence D_{F} (η) is Noetherian iff D is Noetherian.

Proof. Let (X_{0}, f_{0}) ⊆ ⋯ ⊆ (X_{n}, f_{n}) ⊆ ⋯ ⊆ T(X) be a chain of subobjects. Applying U we have a chain of subobjects X_{0} ⊆ ⋯ ⊆ X_{n} ⊆ ⋯ ⊆ X ⊕ F X. Since X ⊕ F X is Noetherian, there exists n_{0} such that X_{n} = X_{n_{0}}, ∀n ≥ n_{0}. Since U is faithful, we have (X_{n}, f_{n}) = (X_{n_{0}}, f_{n_{0}}), ∀n ≥ n_{0}. Hence T(X) is Noetherian. If D is Noetherian then D_{F} (η) is Noetherian, since any (X, f) is a quotient of T(X). The converse is easy since D = Z(D) is closed under subobjects in D_{F} (η).

From now on we assume that D has injectives and exact coproducts and the functor F has a right adjoint G. Then it is easy to see that D_{F} (η) has coproducts and D_{F} (η) is also a cleft coextension. We view the isomorphism D^{G}(ζ) ≅ D_{F} (η) of Section 2, as an identification. Then we have the adjoint pairs (T, U), (U, H), (C, Z) and (Z, K). The adjoint pair (U, H) induces an exact sequence of functors

0 → Id_{D_{F} (η)} ⟶^{μ} HU ⟶^{ν} Ψ → 0

where μ is the unit of the adjoint pair (U, H) defined by μ_{(X,f)} = (f, 1_{X}), ν_{(X,f)} = ^{t}(1_{GX}, −f) and the functor Ψ is defined in D^{G}(ζ) by Ψ(X, f) = (GX, ζ_{X} − Gf). Obviously the functor Ψ is a right adjoint of the functor Φ introduced in Section 3.2. Recall that Φ is defined in D_{F} (η) by Φ(X, f) = (F X, η_{X} − F f).

Theorem 5.21. Assume that F is exact and its right adjoint G preserves coproducts. Any coproduct of injectives in D is injective iff the same is true in D_{F} (η).

Proof. Since F is exact it follows trivially that the functor Φ is exact. Hence its right adjoint Ψ preserves injectives. Similarly since U is exact, its right adjoint H preserves injectives. Let (X_{i}, f_{i})_{i∈I} be a set of injective objects in D^{G}(ζ) and consider the object (X, f) := ⊕_{i∈I}(X_{i}, f_{i}). Then we have short exact sequences

(∗) : 0 → (X_{i}, f_{i}) ⟶^{μ_{(X_{i},f_{i})}} H(X_{i}) ⟶^{ν_{(X_{i},f_{i})}} Ψ(X_{i}, f_{i}) → 0, ∀i ∈ I.

Since (X_{i}, f_{i}) is an injective object, ∀i ∈ I, we have that H(X_{i}) ≅ (X_{i}, f_{i}) ⊕ Ψ(X_{i}, f_{i}). Since (X_{i}, f_{i}) is injective and Ψ preseves injectives, it follows that H(X_{i}) is an injective object ∀i ∈ I. Since Z is exact, it follows that K preserves injectives. Since KH = Id_{D^{G}(ζ)}, we have that X_{i} is injective, ∀i ∈ I. Since any coproduct of injectives D is injective, we have that ⊕_{i∈I}X_{i} is injective in D. Since H preserves injectives, we have that H(⊕_{i∈I}X_{i}) is injective in D^{G}(ζ). But since G preserves coproducts, it follows easily that the functors H, Ψ preserve coproducts. Hence ⊕_{i∈I}H(X_{i}) is injective. The coproduct of the split short exact sequences (∗), is the split short exact sequence:

0 → ⊕_{i∈I}(X_{i}, f_{i}) → ⊕_{i∈I}H(X_{i}) → ⊕_{i∈I}Ψ(X_{i}, f_{i}) → 0.

Since ⊕_{i∈I}(X_{i}, f_{i}) is a direct summand of ⊕_{i∈I}H(X_{i}), it is injective. The converse is easy and is left to the reader.

From now on we assume that D is a Grothendieck category and F : D → D preserves colimits. By [6], D_{F} (η) is a Grothendieck category. Since F preserves colimits and D is Grothendieck, by the Special Adjoint Theorem it follows that F has a right adjoint G : D → D. We recall that an object X ∈ D is called finitely presented, if the functor D(X, −) preserves filtered colimits. We recall that D is called locally finitely presented if D has a set of finitely presented generators. The category D is called locally Noetherian if D has a set of Noetherian generators. Note that in this case by Gabriel’s theory [14], the category of finitely presented objects of D is identified with the category of Noetherian objects of D.

Theorem 5.22. Assume that D is a locally finitely presented Grothendieck category and the functor F preserves colimits. If F preserves finitely presented objects (equivalently its right adjoint G preserves filtered colimits), then: D_{F} (η) is locally Noetherian iff so is D.

Proof. From the adjoint pair (T, U) and the fact that U is faithful and preserves colimits, it follows directly that T preserves generators and finitely presented objects. Hence D_{F} (η) is locally finitely presented. If D is locally Noetherian by Lemma 5.20, it suffices to show that F preserves Noetherian or equivalently finitely presented objects, and this is true by hypothesis. The converse and the parenthetical assertion are easy and are left to the reader.

Corollary For Rings 5.23. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings. If Γ is right Noetherian and M_{Γ} is finitely generated, then Λ is right Noetherian.

Applying Theorems 5.21, 5.22 to the free cleft extension D(F) and using induction we have the following.

Corollary 5.24. Let {F_{1}, F_{2}, ..., F_{m}} be a set of commuting endofunctors of D. Assume that D has exact coproducts and each F_{i} has a right adjoint G_{i}.

(i) Suppose that each F_{i} is exact and its right adjoint G_{i}, preserves coproducts. Then any coproduct of injectives objects in D is an injective object iff the same is true for the symmetric category D[F_{1}, F_{2}, ..., F_{m}].

(ii) If D is locally finitely presented and if each G_{i} preserves filtered colimits, then D is locally Noetherian iff so is D[F_{1}, F_{2}, ..., F_{m}].

As a Corollary of the above result, we obtain the classical Hilbert Basis Theorem:

Corollary For Rings 5.25. If the ring Γ is right Noetherian, then so is the polynomial ring Γ[x_{1}, x_{2}, ⋯, x_{m}].

6 Cleft Extensions of Small Global Dimension and Frobenius Cleft Extensions

6.1 Cleft Extensions of Global Dimension ≤ 2

We characterize when D_{F} (η) has relative 𝔉−global dimension (smaller or equal to) 1 or 2.

Proposition 6.1. (1) If KerC = 0, then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (η) ≤ 1.

(ii) (a) ℜ−gl.dimD ≤ 1.

(b) The functor F is ℜ−exact.

(c) ∀(X, f) ∈ D_{F} (η): Ker(f)/Im(η_{X} − F f) ∈ P(ℜ) and Ker(η_{X} − F f) = Im(η_{F X} − F η_{X} + F^{2}f).

(2) If D_{F} (η) is admissible, then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (η) ≤ 1.

(ii) (a) ℜ−gl.dimD ≤ 1.

(b) The functor F is ℜ−exact.

(c) ∀X ∈ D : Coker(η_{X}) ∈ P(ℜ) ( F X ∈ P(ℜ), if η = 0 ).

(d) Ker(η) = Im(ηF − F η) ( F^{2} = 0, if η = 0 ).

(3) In (1), (2) if F ≠ 0, then 𝔉−gl.dimD_{F} (η) = 1.

Proof. (1) Suppose that 𝔉−gl.dimD_{F} (η) ≤ 1. By Corollary 4.5, we have that ℜ−gl.dimD ≤ 1. Since L_{2}^{𝔉}C = 0, by Theorem 4.7 we have L_{2}^{Σ}C(X, f) = Ker(η_{X} − F f)/Im(η_{F X} − F η_{X} + F^{2}f) = 0, and by Corollary 4.2 we have L_{1}^{Σ}C(X, f) = Ker(f)/Im(η_{X} − F f) ∈ P(ℜ). Since L_{2}^{𝔉}CT = 0, by Corollary 4.2 we have CL_{1}^{ℜ}T = 0 and then L_{1}^{ℜ}T = 0, since KerC = 0. Hence L_{1}^{ℜ}F = 0 and F is ℜ−exact. Suppose now that (a), (b), (c) in (ii) are true. Since F is ℜ−exact, by Theorem 4.7 we have L_{i}^{𝔉}C(X, f) ≅ L_{i}^{Σ}C(X, f), ∀i ≥ 0. Hence by (c) we have L_{2}^{𝔉}C(X, f) = 0 and L_{1}^{𝔉}C(X, f) ∈ P(ℜ). If 0 → (Q_{0}, g_{0}) → T(P_{0}) → (X, f) → 0 is an 𝔉−projective presentation of (X, f) then we have the induced ℜ−exact sequence: 0 → L_{1}^{𝔉}C(X, f) → C(Q_{0}, g_{0}) → P_{0} → C(X, f) → 0. Since ℜ−gl.dimD ≤ 1 and L_{1}^{𝔉}C(X, f) ∈ P(ℜ), we see directly that C(Q_{0}, g_{0}) ∈ P(ℜ). But L_{1}^{𝔉}C(Q_{0}, g_{0}) ≅ L_{2}^{𝔉}C(X, f) = 0, so by Lemma 5.2, (Q_{0}, g_{0}) ∈ P(𝔉). Hence 𝔉−gl.dimD_{F} (η) ≤ 1. Parts (2), (3) follow from (1).

The case ϑ = 0 of the next result is due to I. Reiten [35]. Corollary For Rings 6.2. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings, and suppose that the morphism ϑ is nilpotent. Then the following are equivalent:

(i) Λ is right hereditary. (ii) (a) Γ is right hereditary, and _{Γ}M is flat.

(b) Ker(ϑ) = Im(ϑ ⊗ 1_{M} − 1_{M} ⊗ ϑ) ( M ⊗_{Γ} M = 0, if ϑ = 0 ).

(c) ∀X ∈ Mod(Γ) : X ⊗_{Γ} Coker(ϑ) ( X ⊗_{Γ} M, if ϑ = 0 ) is right

Γ−projective.

If M ≠ 0 and (ii) is true, then r.gl.dimΛ = 1.

The characterization of when 𝔉−gl.dimD_{F} (η) ≤ 2, is much more difficult, and is more precise if η = 0 or more generally if D_{F} (η) is a truncated extension, see Section 8. First we need to construct for any X ∈ D, a specific natural morphism

φ^{X} : D[L_{2}^{𝔉}CZ(X), −] ⟶ Ext_{ℜ}^{3}[X, −] ⊕ Ext_{ℜ}^{2}[Coker(η_{X}), −].

Let X ∈ D; since Z(X), Ω_{𝔉}Z(X) are primitive objects, we have as in Theorem 3.4, an 𝔉−projective presentation of Ω_{𝔉}Z(X) of the form: (1) : 0 → Ω_{𝔉}^{2}Z(X) ⟶^{κ_{1}} T(P_{1}) ⊕ T(Q) ⟶^{β_{1}} Ω_{𝔉}Z(X) → 0. It is easy to see that C(Ω_{𝔉}Z(X)) ≅ K_{0} ⊕ Coker(η_{X}). Hence applying to (1) the functor C we get the short exact sequence: 0 → L_{2}^{𝔉}CZ(X) → C(Ω_{𝔉}^{2}Z(X)) → Ω_{ℜ}^{2}(X) ⊕ Ω_{ℜ}^{1}(Coker(η_{X})) → 0. From this we get the following exact sequence and the desired morphism φ^{X}:

⋯ → D[C(Ω_{𝔉}^{2}Z(X)), −] → D[L_{2}^{𝔉}CZ(X), −] ⟶^{ϕ^{X}} Ext_{ℜ}^{3}[X, −]⊕

Ext_{ℜ}^{2}[Coker(η_{X}), −] → Ext_{ℜ}^{1}[C(Ω_{𝔉}^{2}Z(X)), −] → Ext_{ℜ}^{1}[L_{2}^{𝔉}C(Z(X)), −] → ⋯ Now we are ready to prove the following result. Theorem 6.3. If D_{F} (η) is admissible, then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (η) ≤ 2. (ii) (a) ∀X ∈ D : L_{2}^{Σ}C(Ω_{𝔉}Z(X)) = 0.

(b) ∀P ∈ P(ℜ) : L_{2}^{𝔉}C(ΦZ(P)) = 0.

(c) ∀X ∈ D : L_{2}^{𝔉}C(Z(X)) ∈ P(ℜ).

(d) L_{2}^{ℜ}F = 0.

(e) ∀X ∈ D, the natural morphism φ^{X} constructed above, is epic. If (ii) is true and F^{2} ≠ 0, then 𝔉−gl.dimD_{F} (η) = 2. Proof. (i) ⇒ (ii) By Corollary 4.5: ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (η) ≤ 2. Let X ∈ D; since 𝔉−gl.dimD_{F} (η) ≤ 2, Ω_{𝔉}^{2}Z(X) ∈ P(𝔉), thus CΩ_{𝔉}^{2}Z(X)) ∈ P(ℜ). Applying C to the 𝔉−projective presentation (1) : 0 → Ω_{𝔉}^{2}Z(X) ⟶^{κ_{1}} T(P_{1}) ⊕ T(Q) ⟶^{β_{1}} Ω_{𝔉}Z(X) → 0 above, we get the exact sequence 0 → L_{2}^{𝔉}CZ(X) → C(Ω_{𝔉}^{2}Z(X)) → P_{1} ⊕ Q → C(Ω_{𝔉}Z(X)) → 0, which is therefore an ℜ−projective resolution of C(Ω_{𝔉}Z(X)) ≅ Ω_{ℜ}(X) ⊕ Coker(η_{X}). Hence L_{2}^{𝔉}CZ(X) ∈ P(ℜ). Further since C(Ω_{𝔉}^{2}Z(X)) ∈ P(ℜ), the natural morphism φ^{X} is an epimorphism. Since 𝔉−gl.dimD_{F} (η) ≤ 2, L_{3}^{𝔉}CZ(X) = 0. But L_{3}^{𝔉}CZ(X) is isomorphic to L_{2}^{𝔉}C(Ω_{𝔉}Z(X)) and L_{2}^{Σ}C(Ω_{𝔉}Z(X)) is an epimorphic image of L_{2}^{𝔉}C(Ω_{𝔉}Z(X)), and this proves (a). If P ∈ P(ℜ), then obviously ΦZ(P) is a first 𝔉-syzygy of Z(P), and this implies that L_{2}^{𝔉}C(ΦZ(P)) = L_{3}^{𝔉}C(Z(P)) = 0. It remains to prove (d). Consider the primitive object Ω_{𝔉}^{2}Z(X). We know from (the proof of) Theorem 3.4 that if 0 → K_{1} → P_{1} → P_{0} → X → 0 is an ℜ−projective resolu-

tion of X, then Ω_{𝔉}^{2}Z(X) = (K_{1} ⊕ L_{1}, ( \begin{matrix}0 & x_{1} \\ 0 & y_{1} \end{matrix} )), and there exists a push-out

diagram:

0 → ΦZ(K_{1}) →  T(K_{1}) ⟶^{^{t}(1_{K_{1}} ,0)} Z(K_{1}) → 0
x_{1}↓  x_{1}^{!}↓  ∥
0 → (L_{1}, y_{1}) → Ω_{𝔉}^{2}Z(X) ⟶^{^{t}(1_{K_{1}} ,0)} Z(K_{1}) → 0

where x_{1}^{!} = ( \begin{matrix}1_{K_{1}} & 0 \\ 0 & x_{1} \end{matrix} ). Since K_{1} = Ω_{ℜ}^{2}(X), we have K_{1} ∈ P(ℜ) and

T(K_{1}) ∈ P(𝔉). Since D_{F} (η) is admissible, KerC = 0, hence the unit of the adjoint pair (C, Z) is a minimal epic. But λ_{T(K_{1})} = ^{t}(1_{K_{1}}, 0). Thus ^{t}(1_{K_{1}}, 0) : T(K_{1}) → Z(K_{1}) is an 𝔉−projective cover. This implies that x_{1}^{!} is split monic since Ω_{𝔉}^{2}Z(X) ∈ P(𝔉). But then x_{1} is split monic. From the proof of Theorem 3.4 we have that x_{1} is monic iff L_{1}^{ℜ}F K_{0} = L_{2}^{ℜ}F X = 0. This proves (d).

(ii) ⇒ (i) Since D_{F} (η) is admissible it suffices to show that Ω^{2}Z(X) ∈ P(𝔉), ∀X ∈ D. This happens iff C(Ω^{2}Z(X)) ∈ P(ℜ) and L_{1}^{𝔉}C(Ω^{2}Z(X)) = L_{3}^{𝔉}C(Z(X)) = 0. By (e), (c) and the long exact sequence above which defines the morphism ϕ^{X}, we have that Ext_{ℜ}^{1}[C(Ω^{2}Z(X)), −] = 0, thus C(Ω^{2}Z(X)) ∈ P(ℜ). Now consider the functorial Σ−projective presentation of Ω_{𝔉}Z(X): 0 → Φ(Ω_{𝔉}Z(X)) → TU(Ω_{𝔉}Z(X)) → Ω_{𝔉}Z(X) → 0, and apply to it the functor C. Then we get the following exact sequence:

⋯ → L_{2}^{𝔉}C(Φ(Ω_{𝔉}Z(X))) → L_{2}^{𝔉}C(TU(Ω_{𝔉}Z(X))) → L_{3}^{𝔉}C(Z(X)) →
→ L_{1}^{𝔉}C(Φ(Ω_{𝔉}Z(X))) → L_{1}^{𝔉}C(TU(Ω_{𝔉}Z(X))) → ⋯

Since L_{1}^{𝔉}C(Φ(Ω_{𝔉}Z(X))) ≅ L_{2}^{Σ}C(Ω_{𝔉}Z(X)) = 0 by (a), we have that L_{3}^{𝔉}C(Z(X)) is an epimorphic image of L_{2}^{𝔉}C(TU(Ω_{𝔉}Z(X))). Thus it suffices to show that L_{2}^{𝔉}C(TU(Ω_{𝔉}Z(X))) = 0. Now since L_{2}^{𝔉}CT ≅ CL_{1}^{ℜ}T and KerC = 0, we have that L_{2}^{𝔉}C(TU(Ω_{𝔉}Z(X))) = 0 ⇔ L_{1}^{ℜ}T(U(Ω_{𝔉}Z(X))) = 0. But U(Ω_{𝔉}Z(X)) ≅ Ω_{ℜ}X⊕F P_{0}. Hence L_{1}^{ℜ}T(U(Ω_{𝔉}Z(X))) = 0 ⇔ L_{1}^{ℜ}T(Ω_{ℜ}X⊕F P_{0}) = L_{2}^{ℜ}T(X)⊕ L_{1}^{ℜ}T(F P_{0}) = 0. Since U(L_{i}^{ℜ}T) = L_{i}^{ℜ}F, by (d) we have that L_{2}^{ℜ}T(X) = 0, and it remains to show that L_{1}^{ℜ}F (F P) = 0, ∀P ∈ P(ℜ). Consider the Σ−projective presentation of Z(P): 0 → Φ(Z(P)) → T(F P) → Z(P) → 0, and apply to it the functor C. Since L_{1}^{𝔉}C(T(F P)) = 0, in the induced long exact sequence:

⋯ → L_{2}^{𝔉}C(ΦZ(P)) → L_{2}^{𝔉}C(T(F P)) → L_{2}^{𝔉}C(Z(P)) ⟶^{m} L_{1}^{𝔉}C(ΦZ(P)) → ⋯

the morphism m is an epimorphism and by (b), L_{2}^{𝔉}C(ΦZ(P)) = 0. Since ΦZ(P) = Ω_{𝔉}Z(P) we have that L_{1}^{𝔉}C(ΦZ(P)) ≅ L_{2}^{𝔉}C(Z(P)) and the morphism m is an isomorphism. Hence L_{2}^{𝔉}C(T(F P)) = 0. But L_{2}^{𝔉}C(T(F P)) = CL_{1}^{ℜ}T(F P)) and this is zero iff L_{1}^{ℜ}T(F P)) = 0 iff L_{1}^{ℜ}F (F P)) = 0.

The last assertion follows from Proposition 6.1.

Corollary 6.4. The following are equivalent:

(i) 𝔉−gl.dimD_{F} (0) ≤ 2.

(ii) (a) F^{3} = L_{1}^{ℜ}F^{2} = L_{2}^{ℜ}F = 0.

(b) L_{1}^{ℜ}F (F P) = 0, ∀P ∈ P(ℜ).

(c) ∀X ∈ D : F^{2}X, L_{1}^{ℜ}F (X) ∈ P(ℜ).

(d) ∀X ∈ D, the natural morphism ϕ^{X} constructed above is epic.

Proof. We translate the conditions of Theorem 6.3, when necessary, to the case η = 0, and we use the formula L_{2}^{𝔉}CZ ≅ F^{2} ⊕ L_{1}^{ℜ}F of Corollary 4.4.

(i) ⇒ (ii) ∀P ∈ P(ℜ) : 0 = L_{3}^{𝔉}CZ(P) ≅ L_{2}^{𝔉}CΩ_{𝔉}Z(P) = L_{2}^{𝔉}CZ(F P) ≅ L_{1}^{ℜ}F (F P) ⊕ F^{3}P. Hence ∀P ∈ P(ℜ) : L_{1}^{ℜ}F (F P) = 0 and F^{3}P = 0, so F^{3} = 0. Since L_{2}^{Σ}CΩ_{𝔉}Z(X) = 0 and F^{3} = 0, the Σ−associated complex of Ω_{𝔉}Z(X) is

of the form: ⋯ → 0 → F^{2}K ⟶^{(0,F^{2}k)} F K ⊕ F^{2}P → K ⊕ F P → 0, see the proof

of Corollary 4.4. Hence 0 = L_{2}^{Σ}CΩ_{𝔉}Z(X) = Ker(F^{2}k) = L_{1}^{ℜ}F^{2}X.

(ii) ⇒ (i) By (a) we have: L_{2}^{Σ}CΩ_{𝔉}Z(X) ≅ L_{1}^{ℜ}F^{2}X = 0 and L_{2}^{𝔉}CΩ_{𝔉}ΦZ(P) = L_{2}^{𝔉}CΩ_{𝔉}Z(F P) = L_{1}^{ℜ}F (F P) = 0.

The next corollary of the above result, improves a Theorem of I. Palmer [31].

Corollary For Rings 6.5. If Λ = Γ ⋉_{0}M is trivial cleft extension of Γ, then the following are equivalent:

(i) r.gl.dimΛ ≤ 2.

(ii) (a) ⊗_{Γ}^{3}M = T or_{1}^{Γ}(M, M) = 0 and: w.d ⊗_{Γ}^{2} M = 0, w.d_{Γ}M ≤ 1. (b) ∀X ∈ Mod(Γ) : X⊗_{Γ}^{2}M, T or_{1}^{Γ}(X, M) are projective right Γ−modules.

(c) ∀X ∈ Mod(Γ), the naturally induced morphism:

Hom_{Γ}[T or_{1}^{Γ}(X, M) ⊕ X ⊗_{Γ} (⊗_{Γ}^{2}M), −] ⟶
⟶ Ext_{Γ}^{2}[X ⊗_{Γ} M, −] ⊕ Ext_{Γ}^{3}[X, −]

is an epimorphism.

If M ⊗_{Γ} M ≠ 0 and condition (ii) is true, then r.gl.dimΛ = 2.

6.2 Frobenius Cleft Extensions

We close this section by studing when a cleft extension D_{F} (η) is 𝔉-Frobenius. We recall that an abelian category C equipped with a proper class of short exact sequences 𝔉 is called 𝔉−Frobenius, if C has enough 𝔉−projectives, enough 𝔉−injectives and the class of 𝔉−projectives P(𝔉) coincides with the class I(𝔉) of 𝔉−injectives.

We suppose now that the functor F has a right adjoint G with counit ρ : F G → Id_{D} and unit σ : Id_{D} → GF. We denote by H : D → D_{F} (η) the right adjoint of U and by K : D_{F} (η) → D the right adjoint of Z, using the isomorphism 𝔻 : D_{F} (η) ≅ D^{G}(ζ) from Section 2. We assume that D, in addition to our default assumption that has enough ℜ−projectives, it has also enough ℜ−injectives. Then by the dual of the Adjoint Theorem [12], D^{G}(ζ) ≅ D_{F} (η) has also enough 𝔉−injectives.

Proposition 6.6. If KerC = 0, KerK = 0, then the following are equivalent for the cleft extension D_{F} (η) of D:

(i) D_{F} (η) is 𝔉−Frobenius.

(ii) (a) ∀P ∈ P(ℜ) : KT(P) ∈ I(ℜ) and R_{ℜ}^{1}KT(P) = 0.

(b) ∀I ∈ I(ℜ) : CH(I) ∈ P(ℜ) and L_{1}^{𝔉}CH(I) = 0.

Proof. If KerC = 0, then (X, f) ∈ P(𝔉) ⇔ C(X, f) ∈ P(ℜ) and L_{1}^{𝔉}C(X, f) = 0 ⇔ (X, f) ≅ T(P) for P ∈ P(ℜ). Dually if KerK = 0, then (X, f) ∈ I(𝔉) ⇔ K(X, f) ∈ I(ℜ) and R_{𝔉}^{1}K(X, f) = 0 ⇔ (X, f) ≅ H(I) for I ∈ I(ℜ). Hence P(𝔉) = I(𝔉) ⇔ ∀P ∈ P(ℜ), ∀I ∈ I(ℜ) : KT(P) ∈ I(ℜ), R_{𝔉}^{1}KT(P) = 0, CH(I) ∈ P(ℜ) and L_{1}^{𝔉}CH(I) = 0.

Corollary 6.7. The following are equivalent for the cleft extension D_{F} (0):

(i) D_{F} (0) is 𝔉−Frobenius.

(ii) The following are true, ∀I ∈ I(ℜ), ∀P ∈ P(ℜ):

(a) F P, Ker(σ_{P}) ∈ I(ℜ), σ_{P} is epic and GKer(σ_{P}) = 0.

(b) GI, Coker(ρ_{I}) ∈ P(ℜ), ρ_{I} is monic and F Coker(ρ_{I}) = 0.

Proof. Obviously KerC = 0 = KerK. We observe that: KT(P) ≅ Ker(σ_{P}) ⊕ F P, and CH(I) ≅ Coker(ρ_{I}) ⊕ GI. Also using Theorem 4.7(i) and its dual, we see that: R_{𝔉}^{1}KT(P) ≅ GKer(σ_{P})⊕Coker(σ_{P}), and L_{1}^{𝔉}CH(I) ≅ F Coker(ρ_{I})⊕ Ker(ρ_{I}). Then the result follows from Proposition 6.6.

In the following let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings. We denote by Lann_{Γ}(M) the left annihilator of M in Γ.

Corollary For Rings 6.8. (1) Let Λ = Γ ⋉_{ϑ} M be a cleft extension of rings and suppose that ϑ is right T-nilpotent. Then the following are equivalent:

(i) Λ is a right selfinjective ring (QF-ring).

(ii) Ext_{Λ}^{1}[Γ, Λ] = 0 and Hom_{Γ}[Γ, Λ] is an injective right Γ−module (and Γ is right Noetherian and M_{Γ} is finitely generated).

(2) If ϑ = 0, then the following are equivalent.

(i) Λ is a right selfinjective ring (QF-ring).

(ii) (a) _{Γ}M,_{Γ} Lann_{Γ}(M) are injective, and Hom_{Γ}[M, Lann_{Γ}(M)] = 0.

(b) The natural morphism Γ → End_{Γ}(M_{Γ}) is an epimorphism (and Γ is right Noetherian and M_{Γ} is finitely generated).

A direct consequence of the above results is the following.

Corollary 6.9. Let Γ be a ring and M be a Γ − Γ−bimodule such that the functor −⊗_{Γ}M : Mod(Γ) → Mod(Γ) induces an equivalence −⊗_{Γ}M : P_{Γ} ⟶^{≈} I_{Γ} between the category P_{Γ} of projective right Γ−modules and the category I_{Γ} of injective Γ−modules. Then the trivial extension Λ = Γ ⋉_{0}M is QF.

Examples of rings satisfying the assumptions of the above Corollary are rings with a (Morita) self-duality, for instance Artin algebras.

7 Formulas for L_{n}^{𝔉}CZ, Ext_{𝔉}^{n}[Z(−), Z(−)] and Spectral Sequences

In this section we use spectral sequences to study further the relative homological behaviour of D_{F} (η), with respect the proper class 𝔉 = U^{−1}(ℜ). We pose a condition on the functor F for a nice behaviour of the derived spectral sequences. In general we get better estimates for 𝔉−gl.dimD_{F} (η), and more precise formulas for the functors L_{n}^{𝔉}CZ, Ext_{𝔉}^{n}[Z(−), Z(−)], when we impose on the functor F the following vanishing condition (see also [13]):

(†) L_{i}^{ℜ}F^{j}(F P) = 0, ∀i, j ≥ 1, ∀P ∈ P(ℜ).

In case the cleft extension is induced by a cleft extension of rings Λ = Γ ⋉_{ϑ}M, the vanishing condition (for the absolute theory) is the following:

T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1.

These estimates and formulas become exact, when η = 0. The condition (†) will be very useful in the study of the Butler-Horrocks spectral sequence. We begin with the following preliminary result:

Proposition 7.1. There is a functorial spectral sequence:

E_{2}^{p,q} = Ext_{ℜ}^{p}[L_{q}^{𝔉}C−,?] ⟹ Ext_{𝔉}^{n}[−, Z(?)].

Proof. Let (X, f) ∈ D_{F} (η) and Y ∈ D, and consider the functor h_{Y} := D[−, Y] : D → Ab. Obviously the right h_{Y} −acyclic objects of h_{Y} with respect to ℜ are precisely the objects of P(ℜ). Since C(P(𝔉)) ⊆ P(ℜ), the functor C sends objects of P(𝔉) to h_{Y} −acyclic objects with respect to ℜ. From the adjoint pair (C, Z) we have an isomorphism h_{Y} C ≅ D_{F} (η)[−, Z(Y)]. Since R_{ℜ}^{n}h_{Y} = Ext_{ℜ}^{n}[−, Y], and R_{𝔉}^{n}h_{Y} C ≅ R_{𝔉}^{n}D_{F} (η)[−, Z(Y)] = Ext_{𝔉}^{n}[−, Z(Y)], we have a third quadrant spectral sequence of Grothendieck type E_{2}^{p,q} = Ext_{ℜ}^{p}[L_{q}^{𝔉}C(−), Y] ⟹ Ext_{𝔉}^{n}[−, Z(Y)], if the 𝔉−associated complex 𝔉_{(X,f)}^{•} of any (X, f) ∈ D_{F} (η) has an ℜ−proper ℜ−projective resolution in the sense of Cartan-Eilenberg. This happens in our case, as follows from our conventions and the classical construction of the Grothendieck’s spectral sequence, see [18], [36] or [40].

As a corollary of the above Proposition we have the following:

Corollary 7.2. Let (X, f) ∈ D_{F} (η) be an object with: L_{n}^{𝔉}C(X, f) = 0, ∀n ≥ 1. Then ∀Y ∈ D, ∀n ≥ 1, there is an isomorphism:

Ext_{𝔉}^{n}[(X, f), Z(Y)] ≅ Ext_{ℜ}^{n}[C(X, f), Y].

If in addition KerC = 0, then: 𝔉−p.d(X, f) = ℜ−p.dC(X, f).

Proof. Trivial since the spectral sequence of Proposition 7.1 collapses. If KerC = 0, the result follows from Lemma 5.3.

Corollary 7.3. The following are equivalent for the cleft extension D_{F} (η):

(i) The natural morphism Z^{i} : Ext_{ℜ}^{i}[−, −] → Ext_{𝔉}^{i}[Z(−), Z(−)] induced by the exact functor Z is an isomorphism, ∀i ≥ 1.

(ii) The natural morphism U^{i} : Ext_{𝔉}^{i}[Z(−), Z(−)] → Ext_{ℜ}^{i}[−, −] induced by the exact functor U is an isomorphism, ∀i ≥ 1.

(iii) L_{i}^{𝔉}CZ(P) = 0, ∀i ≥ 1, ∀P ∈ P(ℜ).

(iv) L_{i}^{𝔉}CZ = 0, ∀i ≥ 1.

(v) Z(D) is extension-closed in D_{F} (η) and L_{i}^{𝔉}CZ(P) = 0, ∀i ≥ 2, ∀P ∈ P(ℜ).

(vi) η is epic and L_{i}^{𝔉}CZ(P) = 0, ∀i ≥ 2, ∀P ∈ P(ℜ).

Proof. (i) ⇔ (ii) Trivial since U^{i} ∘ Z^{i} = Id_{Ext_{ℜ}^{i}[−,−]}, ∀i ≥ 1. (iii) ⇔ (iv) Assuming (iii), let X ∈ D, and P^{•} → X be an ℜ−projective resolution of X. Then Z(P^{•}) → Z(X) is an 𝔉−exact complex consisting of C−acyclic objects with respect to ℜ. Hence L_{i}^{𝔉}CZ(X) = H_{i}[CZ(P^{•})] = 0, ∀i ≥ 1, since CZ = Id_{D}. The other direction is trivial. (iv) ⇒ (v) It suffices to show that L_{1}^{𝔉}CZ = 0 iff η is epic. But ∀X ∈ D : L_{1}^{𝔉}CZ(X) = Coker(η_{X}). (v) ⇔ (vi) Obviously Z(D) is extension closed iff the morphism Z^{1} is an isomorphism. Since Z^{1} is always split monic and

Coker(Z^{1}) ≅ Ext_{Σ}^{1}[Z(X), Z(Y)] ≅ D[ΦZ(X), Z(Y)] ≅ D[Coker(η_{X}), Y],

we have that Z(D) is extension closed iff η is epic. (iv) ⇒ (i) Assuming (iv), the spectral sequence of Proposition 7.1 for f = 0, collapses, so the horizontal edge morphism e^{i} : Ext_{ℜ}^{i}[X, Y] → Ext_{𝔉}^{i}[Z(X), Z(Y)], ∀X, Y ∈ D, ∀i ≥ 1 is an isomorphism. It is easy to see that e^{i} coincides with the natural morphism Z^{i} induced by the exact functor Z. (i) ⇒ (iv) The proof consists of a simple induction argument and Theorem 5.12, page 328 of [8].

7.1 The vanishing condition (†) : L_{i}^{ℜ}F^{j}(F P) = 0, ∀i, j ≥ 1, ∀P ∈ P(ℜ) and its consequences

Throughout this subsection we assume that the condition (†) cited above holds. The next Lemma describes an equivalent formulation of (†) and indicates some useful consequences.

Lemma 7.4. (α) Consider the following conditions:

(i) L_{i}^{ℜ}F (F^{j}P) = 0, ∀i, j ≥ 1, ∀P ∈ P(ℜ).

(ii) L_{i}^{ℜ}F^{j}(F P) = 0, ∀i, j ≥ 1, ∀P ∈ P(ℜ).

(iii) L_{i}^{𝔉}CT(F^{j}P) = 0, ∀i, j ≥ 1, ∀P ∈ P(ℜ).

Then: (i) ⇔ (ii) ⇒ (iii) and if KerC = 0, then these conditions are equivalent. (β) If one of the above conditions is true, then we have natural isomorphisms:

L_{n}^{𝔉}CZ ≅ L_{n}^{Σ}CZ , L_{n+m}^{𝔉}CZ ≅ L_{n}^{𝔉}CΦ^{m}Z : P(ℜ) → D , ∀n, m ≥ 0.

Proof. (α) (i) ⇒ (ii) Assuming (i), the functor F^{r}, ∀r ≥ 1, sends ℜ−projectives to F −acyclic objects with respect to ℜ. Hence there exists a family of spectral sequences of Grothendieck type:

rE_{p,q}^{2} = L_{p}^{ℜ}F [L_{q}^{ℜ}F^{r}(X)] ⟹ L_{n}^{ℜ}F^{r+1}(X), ∀X ∈ D, ∀r ≥ 1.

We apply induction to j ≥ 0; the hypothesis ensures that (ii) is true for j = 1, and we assume that is true for j − 1. We set X = F P, for P ∈ P(ℜ) and consider the spectral sequence _{j−1}E_{p,q}^{2}. From the induction hypothesis this spectral sequence collapses, thus L_{p}^{ℜ}F [F^{j−1}(F P)] = L_{p}^{ℜ}F [F^{j}P] ≅ L_{p}^{ℜ}F^{j}(F P), ∀p ≥ 1. Hence since by our hypothesis L_{p}^{ℜ}F [F^{j}P] = 0, ∀j ≥ 1, we have that L_{p}^{ℜ}F^{j}(F P) = 0, ∀p ≥ 1, which completes the proof. Part (ii) ⇒ (i) is similar to (i) ⇒ (ii) and the rest follows from Theorem 4.7. Part (β) is easy and is left to the reader.

Theorem 7.5. If D_{F} (η) is admissible , then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (η) < ∞.

(ii) (a) ℜ−gl.dimD < ∞. (b) ∃m ≥ 0 : L_{n}^{Σ}CZ(P) = 0, ∀n ≥ m + 1, ∀P ∈ P(ℜ).

Setting m_{0} = min{m ≥ 0 : L_{n}^{Σ}CZ(P) = 0, ∀n ≥ m + 1, ∀P ∈ P(ℜ)}, we have:

ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (η) ≤ m_{0} + 2 · ℜ−gl.dimD.

Proof. (i) ⇒ (ii) Follows from Theorem 5.6, and Lemma 7.4(β). (ii) ⇒ (i) By Theorem 5.6(2), it is enough to show that if ℜ−gl.dimD < ∞, and m_{0} is as in the statement of the Theorem, then 𝔉−gl.dimD_{F} (η) ≤ m_{0} + 2 · ℜ−gl.dimD. Suppose that ℜ−gl.dimD = n < ∞, and for an arbitrary X ∈ D, choose an ℜ−projective resolution 0 → P_{n} → P_{n−1} → ⋯ → P_{0} → X → 0, with corresponding ℜ−syzygies K_{0}, K_{1}, ⋯, K_{n−1} = P_{n}. From the long exact sequence induced by the 𝔉−exact sequence 0 → Z(K_{0}) → Z(P_{0}) → Z(X) → 0, using our hypothesis we have: L_{m_{0}+k+1}^{𝔉}CZ(X) ≅ L_{m_{0}+k}^{𝔉}CZ(K_{0}), ∀k ≥ 1. In the same way from the long exact sequence induced by the 𝔉−exact sequence 0 → Z(K_{1}) → Z(P_{1}) → Z(K_{0}) → 0, using our hypothesis we have: L_{m_{0}+k+1}^{𝔉}CZ(K_{0}) ≅ L_{m_{0}+k}^{𝔉}CZ(K_{1}), ∀k ≥ 1. Similarly L_{m_{0}+k+1}^{𝔉}CZ(K_{i}) ≅ L_{m_{0}+k}^{𝔉}CZ(K_{i−1}), ∀k ≥ 1, ∀i = 0, 1, ⋯, n − 1. Hence:

L_{m_{0}+n+1}^{𝔉}CZ(X) ≅ L_{m_{0}+n}^{𝔉}CZ(K_{0}) ≅ L_{m_{0}+n−1}^{𝔉}CZ(K_{1}) ≅ ⋯
⋯ ≅ L_{m_{0}+1}^{𝔉}CZ(K_{n−1}) ≅ L_{m_{0}+1}^{𝔉}CZ(P_{n}) ≅ L_{m_{0}+1}^{Σ}CZ(P_{n}) = 0.

We conclude that L_{m_{0}+n+1}^{𝔉}CZ(X) = 0, ∀X ∈ D. But since D_{F} (η) is 𝔉− admissible, we have L_{m_{0}+n+1}^{𝔉}C = 0. Finally by Theorem 5.6, we have the formula: 𝔉−gl.dimD_{F} (η) ≤ m_{0} + n + ℜ−gl.dimD = m_{0} + 2 · ℜ−gl.dimD.

As a corollary we get a generalization of the main result of [13] about the global dimension of the trivial extension D_{F} (0).

Corollary 7.6. If η = 0, then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (0) < ∞.

(ii) (a) ℜ−gl.dimD < ∞. (b) The functor F is nilpotent.

If c(F) is the nilpotency class of F, we have the bounds:

ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (0) ≤ c(F) + 2 · ℜ−gl.dimD.

Proof. If η = 0 then D_{F} (0) is admissible and of course: L_{n}^{Σ}CZ = F^{n}. Hence L_{n}^{Σ}CZ(P) = 0, ∀P ∈ P(ℜ) ⇔ F^{n}(P) = 0, ∀P ∈ P(ℜ) ⇔ F^{n} = 0. The rest follows from the previous Theorem.

Remark 7.7. Fix the abelian category D and the functor F. Consider the class H of all nilpotent and associative “multiplications” η : F^{2} → F in D. Set

∀η ∈ H : 𝔪(η) := min{m ≥ 0 : L_{n}^{Σ}CZ(P) = 0, ∀n ≥ m + 1, ∀P ∈ P(ℜ)},

𝔪(D, F, η) := sup{𝔪(η); η ∈ H}, ℵ(D, F, η) := sup_{η:F^{2}→F ∈ H}{𝔉−gl.dimD_{F} (η)}.

It is clear that 𝔪(D, F, η) = 𝔪(0) = c(F), hence 𝔪(D, F, η) is independent of η. We conjecture that ℵ(D, F, η) is also independent of η and:

ℵ(D, F, η) = 𝔉−gl.dimD_{F} (0).

Clearly 𝔉−gl.dimD_{F} (0) ≤ ℵ(D, F, η). Assuming the vanishing condition (†), by the above results:

ℜ−gl.dimD ≤ ℵ(D, F, η) ≤ c(F) + 2 · ℜ−gl.dimD.

If D is semisimple, so the homological theory is the absolute one, then for any admissible D_{F} (η), we have that 𝔪(η) = gl.dimD_{F} (η), as follows from the isomorphism Ext_{D_{F} (η)}^{n}[Z(X), Z(Y)] ≅ D[L_{n}^{Σ}CZ(X), Z(Y)] ≅ D[H_{n}Σ_{Z(X)}^{•}, Y], where Σ_{Z(X)}^{•} is the Σ−associated complex of Z(X). It follows in this case that ℵ(D, F, η) = c(F) = gl.dimD_{F} (0) and the above conjecture is true. In particular if gl.dimD_{F} (0) < ∞, or equivalently if F is nilpotent, then

∀η : gl.dimD_{F} (η) ≤ gl.dimD_{F} (0) = c(F) < ∞.

Let Λ = Γ⋉_{ϑ}M be a cleft extension of rings. The tensor-nilpotency class c(M) of M is the nilpotency class of the functor − ⊗_{Γ} M, i.e. c(M) = ∞ or c(M) = min{k ∈ ℕ : ⊗_{Γ}^{k+1}M = 0}. We leave to the reader to interpret the above Remark in the case of rings, noting that by [6] the class of algebras Λ = Γ ⋉_{ϑ}M we obtain using tensor nilpotent Γ − Γ−bimodules M over semisimple k−algebras Γ over a field k, coincides with the class of triangular algebras, i.e. algebras without oriented cycles in their ordinary quiver [3].

Corollary For Rings 7.8. Let Λ = Γ ⋉_{ϑ}M be a cleft extension of rings and assume that ϑ is nilpotent and: T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1. Then:

(i) r.gl.dimΛ < ∞.

(ii) r.gl.dimΓ < ∞ and ∃m ≥ 0 : T or_{n}^{(Λ,Γ)}[Γ, Γ] = 0, ∀n ≥ m + 1.

If m_{0} = min{m ≥ 0 : T or_{n}^{(Λ,Γ)}[Γ, Γ] = 0, ∀n ≥ m + 1}, then:

r.gl.dimΓ ≤ r.gl.dimΛ ≤ m_{0} + 2 · r.gl.dimΓ.

In particular if ϑ = 0, then:

r.gl.dimΓ ≤ r.gl.dimΛ ≤ c(M) + 2 · r.gl.dimΓ.

Our purpose now is to construct, ∀X ∈ D, an 𝔉−exact resolution of Z(X) consisting of C−acyclic objects with respect to 𝔉. We’ll use this resolution to compute the derived functors L_{i}^{𝔉}CZ, Ext_{𝔉}^{i}[Z(−), Z(−)]. We begin by considering an arbitrary ℜ−projective resolution P_{ℜ}^{•}(X) of X:

⋯ → P_{n+1} ⟶^{a_{n+1}} P_{n} ⟶^{a_{n}} P_{n−1} → ⋯ → P_{1} ⟶^{a_{1}} P_{0} ⟶^{a_{0}} X → 0.

Let a_{n+1} = l_{n+1} ∘ k_{n} be the canonical factorization of a_{n+1}, and let K_{n} = Im(a_{n+1}), ∀n ≥ 0 be the corresponding ℜ−syzygies of X.

Lemma 7.9. There exists an 𝔉−exact resolution of Z(X) consisting of C−acyclic objects with respect to 𝔉, of the form 𝔛_{𝔉}^{•}Z(X) :

⋯ → T(⊕_{p+q=n+1}F^{p}P_{q}) ⟶^{ϖ_{n}} T(⊕_{p+q=n}F^{p}P_{q}) ⟶^{ϖ_{n−1}} T(⊕_{p+q=n−1}F^{p}P_{q}) →

⋯ → T(P_{2} ⊕ F P_{1} ⊕ F^{2}P_{0}) ⟶^{ϖ_{1}} T(P_{1} ⊕ F P_{0}) ⟶^{ϖ_{0}} T(P_{0}) ⟶^{α} Z(X) → 0.

Proof. As in the proof of Theorem 3.4, we have the following 𝔉−projective presentation of Z(X): 0 → (Q_{0}, g_{0}) ⟶^{κ_{0}} T(P_{0}) ⟶^{α_{0}} Z(X) → 0, where Q_{0} =

K_{0} ⊕ F P_{0}, g_{0} = ( \begin{matrix}0 & F k_{0} \\ 0 & η_{P_{0}} \end{matrix} ), κ_{0} = ( \begin{matrix}k_{0} & 0 \\ 0 & 1_{F P_{0}} \end{matrix} ), α_{0} = ^{t}(a_{0}, 0). It is easy

to see that we have an 𝔉−exact sequence 0 → (Q_{1}, g_{1}) ⟶^{κ_{1}} T(P_{1} ⊕ F P_{0}) ⟶^{α_{1}} (Q_{0}, g_{0}) → 0. By induction we have 𝔉−exact sequences

0 → (Q_{n−1}, g_{n−1}) ⟶^{κ_{n−1}} T(⊕_{p+q=n−1}F^{p}P_{q}) ⟶^{α_{n−1}} (Q_{n−2}, g_{n−2}) → 0

where Q_{n−1} = K_{n−1} ⊕ ⊕_{p+q=n−1}F^{p}P_{q}. The Yoneda composition of the above 𝔉−exact sequences, is the desired 𝔉−exact resolution of Z(X), which by condition (†), consists of C−acyclic objects with respect to 𝔉.

From the proof of the above Lemma, an easy calculation shows that:

C(ϖ_{n}) = (\begin{matrix}a_{n+1} & 0 & 0 & … & 0 & 0 \\ 0 & (−1)^{n}F a_{n} & 0 & … & 0 & 0 \\ 0 & η_{P_{n−1}}^{1} & (−1)^{n}F^{2}a_{n−1} & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & … & η_{P_{1}}^{n−1} & (−1)^{n}F^{n}a_{1} \\ 0 & 0 & 0 & … & 0 & η_{P_{0}}^{n} \end{matrix}),

where η_{P_{i}}^{j} = ∑_{k=0}^{j−1}(−1)^{k}F^{k}η_{F^{j−k−1}P_{j}} : F^{j+1}P_{i} → F^{j}P_{i}.

Theorem 7.10. The 𝔉−associated complex 𝔉_{Z(X)}^{•} of Z(X) is quasi-isomorphic to the complex C𝔛_{𝔉}^{•}Z(X), thus:

L_{n}^{𝔉}CZ(X) ≅ H_{n}[C𝔛_{𝔉}^{•}Z(X)], ∀n ≥ 0.

In particular if η = 0, then:

L_{n}^{𝔉}CZ ≅ ⨁_{p+q=n} L_{q}^{ℜ}F^{p}, ∀n ≥ 0.

Proof. By Lemma 7.9, the 𝔉−exact resolution 𝔛_{𝔉}^{•}Z(X) of Z(X), consists of C−acyclic objects. So L_{i}^{𝔉}CZ(X) ≅ H_{i}[C𝔛_{𝔉}^{•}Z(X)], ∀i ≥ 0. If η = 0, then η_{P_{i}}^{j} = 0, ∀i, j ≥ 0. Then the assertion follows from the shape of the morphism C(ϖ_{n}).

Corollary 7.11. If η = 0, then the following are equivalent:

(i) 𝔉−gl.dimD_{F} (0) < ∞.

(ii) (a) ℜ−gl.dimD < ∞. (b) ∃n ≥ 0 : ⨁_{p+q=n} L_{q}^{ℜ}F^{p} = 0.

If m = min{n ≥ 0 : ⨁_{p+q=n+1} L_{q}^{ℜ}F^{p} = 0}, we have the bounds:

ℜ−gl.dimD ≤ 𝔉−gl.dimD_{F} (0) ≤ m + ℜ−gl.dimD.

Corollary For Rings 7.12. Let Λ = Γ ⋉_{0} M be a trivial cleft extension of rings and assume that: T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1. Then:

∀X ∈ Mod(Γ) : T or_{n}^{Λ}[X_{Λ},_{Λ} Γ] ≅ ⨂_{p+q=n} T or_{q}^{Γ}[X, ⊗_{Γ}^{p}M], ∀n ≥ 0,

gl.dimΓ ≤ gl.dimΛ ≤ min{n ≥ 0 : T or_{q}^{Γ}[−, ⊗_{Γ}^{p}M] = 0, ∀p+q = n+1}+gl.dimΓ.

From now on we assume that D has enough ℜ−injectives.

Theorem 7.13. ∀X, Y ∈ D, there exists an isomorphism:

Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ H^{n}[TotM^{••}], ∀n ≥ 0

where M^{••} = D[C𝔛_{𝔉}^{•}Z(X), I_{ℜ}^{•}(Y)] is the double complex arising from an arbitrary ℜ−injective resolution I_{ℜ}^{•}(Y) of Y in D. In particular if η = 0, then:

Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ ⨁_{p+q=n} R_{ℜ}^{p}D[F^{q}X, Y], ∀n ≥ 0.

Proof. For any X ∈ D, consider the 𝔉−projective resolution P_{𝔉}^{Z(X)}, constructed in Section 3, and the 𝔉−exact C−acyclic resolution 𝔛_{𝔉}^{•}Z(X) constructed above. Obviously the identity morphism of Z(X) induces a morphism of resolutions ξ^{∘} : P_{𝔉}^{Z(X)} → 𝔛_{𝔉}^{•}Z(X). By Theorem 7.10, the morphism C(ξ^{∘}) is a quasiisomorphism, i.e. C(ξ^{∘}) induces an isomorphism H_{n}[C(ξ^{∘})] : H_{n}[C(P_{𝔉}^{Z(X)})] → H_{n}[C(𝔛_{𝔉}^{•}Z(X))] ≅ L_{n}^{𝔉}CZ(X), ∀n ≥ 0. For any Y ∈ D, the morphism C(ξ^{∘}), induces a morphism of complexes: ζ^{∘} = D[C(ξ^{∘}), Y] : D[C(𝔛_{𝔉}^{•}Z(X)), Y] → D[C(P_{𝔉}^{Z(X)}), Y], and in turn ζ^{∘} induces a morphism in cohomology H^{n}(ζ^{∘}) : H^{n}[D[C(𝔛_{𝔉}^{•}Z(X)), Y]] → H^{n}[D[C(P_{𝔉}^{Z(X)}), Y]]. By Section 3, we have an isomorphism: H^{n}[D[C(P_{𝔉}^{Z(X)}), Y]] ≅ Ext_{𝔉}^{n}[Z(X), Z(Y)]. Hence we have a well defined morphism

H^{n}(ζ^{∘}) : H^{n}[D[C(𝔛_{𝔉}^{•}Z(X)), Y]] → Ext_{𝔉}^{n}[Z(X), Z(Y)], ∀n ≥ 0.

Consider now an ℜ−injective resolution I_{ℜ}^{•}(Y) of Y in D, and denote by M^{••} = D[C𝔛_{𝔉}^{•}Z(X), I_{ℜ}^{•}(Y)], N^{••} = D[CP_{𝔉}^{Z(X)}, I_{ℜ}^{•}(Y)] the induced double complexes.

The morphism ζ^{∘}, extends to a morphism of double complexes ζ^{∘∘} : M^{••} → N^{••} and we have the following usual spectral sequences arising from M^{••}, N^{••}:

I^{1}E_{2}^{p,q} = _{I}H^{q}[_{II}H^{p}(M^{••})] ⟹ H^{p+q}[TotM^{••}] (i)
II^{1}E_{2}^{p,q} = _{II}H^{p}[_{I}H^{q}(M^{••})] ⟹ H^{p+q}[TotM^{••}] (ii)
I^{2}E_{2}^{p,q} = _{I}H^{q}[_{II}H^{p}(N^{••})] ⟹ H^{p+q}[TotN^{••}] (iii)
II^{2}E_{2}^{p,q} = _{II}H^{p}[_{I}H^{q}(N^{••})] ⟹ H^{p+q}[TotN^{••}] (iv)

Calculating the spectral sequence (ii), we have:

IH^{q}(M^{••}) ≅ H^{q}[D[C𝔛_{𝔉}^{•}Z(X), I^{p}]] ≅ D[L_{q}^{𝔉}CZ(X), I^{p}].

Hence: _{II}^{1}E_{2}^{p,q} = Ext_{ℜ}^{p}[L_{q}^{𝔉}CZ(X), Y]. Similarly calculating the spectral sequence (iii), we see easily that (iii) collapses giving isomorphisms:

I^{2}E_{2}^{n,0} ≅ Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ H^{n}[TotN^{••}], ∀n ≥ 0.

Finally calculating the spectral sequence (iv), we get as in the case of (ii) that:

II^{2}E_{2}^{p,q} ≅ Ext_{ℜ}^{p}[L_{q}^{𝔉}CZ(X), Y].

Now the morphism ζ^{∘∘} : M^{••} → N^{••}, induces a morphism of spectral sequences (ii) → (iv), such that ζ^{∘∘} induces isomorphisms between the terms II^{1}E_{2}^{p,q},_{II}^{2}E_{2}^{p,q}. From the Comparison Theorem of spectral sequences, see [8], the morphism ζ^{∘∘} : M^{••} → N^{••}, induces isomorphisms in cohomology: H^{n}(ζ^{∘∘}) : H^{n}(TotM^{••}) → H^{n}(TotN^{••}), ∀n ≥ 0. Since H^{n}(TotN^{••}) is isomorphic to Ext_{𝔉}^{n}[Z(X), Z(Y)], we get finally that ζ^{∘∘} induces isomorphisms:

Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ H^{n}(TotM^{••}), ∀X, Y ∈ D, ∀n ≥ 0.

Suppose now η = 0. Then from the shape of the complex C𝔛_{𝔉}^{•}Z(X), we have:

(TotM^{••})^{n} = ⨁_{p+q=n} D[C𝔛_{𝔉}^{•}Z(X)_{p}, I^{q}] = ⨁_{p+q=n} D[⊕_{r+s=p}F^{r}P_{s}, I^{q}] =

= ⨁_{p+q=n} ⨁_{r+s=p} D[F^{r}P_{s}, I^{q}].

Hence (TotM^{••})^{n} = ⊕_{p+q=n}(TotK_{q}^{••})^{p}, where TotK_{q}^{••} is the total complex of the double complex D[F^{q}(P_{ℜ}^{•}(X)), I_{ℜ}^{•}(Y)] and P_{ℜ}^{•}(X) is an ℜ−projective resolution of X. Consequently H^{n}[TotM^{••}] = ⊕_{p+q=n}H^{p}[TotK_{q}^{••}], ∀n ≥ 0. From the complex TotK_{q}^{••}, we see directly that H^{p}[TotK_{q}^{••}] = R_{ℜ}^{p}D[F^{q}X, Y] is the p − th right derived functor of the non-balanced functor D[F^{q}−, −] : D^{op} × D → Ab. We conclude that:

Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ ⨁_{p+q=n} H^{p}[TotK_{q}^{••}] ≅ ⨁_{p+q=n} R_{ℜ}^{p}D[F^{q}X, Y], ∀n ≥ 0.

The case η = 0 in the previous Theorem as well as in the following corollaries generalizes, simplifies and completes all the analogous results of [32].

Corollary 7.14. If η = 0, then:

𝔉−gl.dimD_{F} (0) = sup{n ≥ 0 : ⨁_{p+q=n+1} R_{ℜ}^{p}D[F^{q}−, −] = 0}.

Corollary For Rings 7.15. Λ = Γ ⋉_{0} M be a trivial cleft extension of rings and assume that: T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1. Then:

gl.dimΛ = sup{n ≥ 0 : ⨁_{p+q=n+1} R^{p}Hom_{Γ}[− ⊗_{Γ} (⊗_{Γ}^{q}M), −] = 0}.

7.2 Morphism and Comma Categories, Categories of Complexes

Corollary 7.16. If η = 0, and the following vanishing condition (††) below is true:

(††) L_{i}^{ℜ}F (F P) = 0, ∀i ≥ 0, ∀P ∈ P(ℜ)

then we have the following:

L_{n}^{𝔉}CZ = L_{n−1}^{ℜ}F, Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ Ext_{𝔉}^{n}[X, Y] ⨁ R_{ℜ}^{n−1}D[F X, Y], ∀n ≥ 1,

𝔉−gl.dimD_{F} (0) = max{ℜ−gl.dimD, sup{n ≥ 0 : R_{ℜ}^{n−1}D[F −, −] = 0}}.

Proof. The condition (††) ensures that F^{2} = 0, and that the vanishing condition (†) is true for F. The rest follows from Corollary 7.14.

Corollary For Rings 7.17. Let Λ = Γ⋉_{0}M be a trivial cleft extension of rings and assume that: T or_{i}^{Γ}[M, M] = 0, ∀i ≥ 0. Then ∀X, Y ∈ Mod(Γ), ∀n ≥ 1:

T or_{n}^{Λ}[X_{Λ},_{Λ} Γ] ≅ T or_{n−1}^{Γ}[X_{Γ},_{Γ} M],
Ext_{Λ}^{n}[X_{Λ}, Y_{Λ}] ≅ Ext_{Λ}^{n}[X_{Γ}, Y_{Γ}] ⊕ R^{n−1}Hom_{Γ}[X ⊗_{Γ} M, Y],

gl.dimΛ = max{gl.dimΓ, sup{n ≥ 0 : R^{n−1}Hom_{Γ}[− ⊗_{Γ} (⊗_{Γ}^{q}M), −] = 0}.

Let G : D → E be a right exact functor between abelian categories, and let C = (D, G, E) be the induced comma-category. As in Section 2, C can be considered as a trivial extension of D ×E by the right exact functor F : D ×E → D × E given by F (A, B) = (0, GA). It is clear that the functor F satisfies the condition (††) of Corollary 7.16. Hence we have the following.

Corollary 7.18. ∀(X_{1}, Y_{1}), (X_{2}, Y_{2}) ∈ C, ∀n ≥ 0 :

Ext_{C}^{n}[Z(X_{1}, Y_{1}), Z(X_{2}, Y_{2})] ≅ Ext_{D}^{n}[X_{1}, X_{2}]⊕Ext_{E}^{n}[Y_{1}, Y_{2}]⊕R^{n−1}D×E[GX_{1}, Y_{2}].

gl.dimC = max{gl.dimD, gl.dimE, sup{n : R^{n}Hom_{D×E}[G−, −] = 0}}.

Let _{R}M_{S} be an R−S−bimodule, and let ( \begin{matrix}R & _{R}M_{S} \\ 0 & S \end{matrix} ) be the induced

triangular matrix ring. A direct consequence of the above result is the following.

Corollary For Rings 7.19. r.gl.dim ( \begin{matrix}R & _{R}M_{S} \\ 0 & S \end{matrix} ) =

= max{r.gl.dim.R, r.gl.dimS, sup{n : R^{n−1}Hom_{R×S}[− ⊗_{R} M_{S}, −] = 0}}.

Let D^{n} be the category of composable morphisms of length n − 1, over the abelian category D. Then D^{n} is a trivial extension (of D^{n−1} × D), as is the category C^{[0,n]}(D) of complexes of length n over D. Let C^{∅}(D) be the category of complexes over D and let C^{b}(D), C^{−}(D) and C^{+}(D), be the full subcategories of bounded, bounded below and bounded above complexes respectively. The following is a trivial consequence of the above results.

Corollary 7.20. ∀n ≥ 0: gl.dimD^{n} = gl.dimC^{[0,n]}(D) = n + gl.dimD,

gl.dimC^{∗}(D) = ∞, for ∗ = b, −, +, ∅.

7.3 The Butler-Horrocks Spectral Sequence

For any object (X, f) ∈ D_{F} (η) we consider the 𝔉−projective resolution of (X, f):

P_{𝔉}^{(X,f)} : ⋯ → T(P_{p}) ⟶^{ϕ_{p}^{(X,f)}} T(P_{p−1}) → ⋯ → T(P_{0}) ⟶^{α_{0}^{(X,f)}} (X, f) → 0

and the functorial Σ−projective resolutions of the objects (X, f), T(P_{p}), ∀p ≥ 0:

P_{Σ}^{(X,f)} : ⋯ → T(F^{q}X) ⟶^{θ_{q}^{(X,f)}} T(F^{q−1}X) → ⋯ → T(X) ⟶^{ε_{(X,f)}} (X, f) → 0,

P_{Σ}^{T(P_{p})} : ⋯ → T(F^{q}P_{p} ⊕ F^{q+1}P_{p}) ⟶^{θ_{q}^{T(Pp)}} T(F^{q−1}P_{p} ⊕ F^{q}P_{p}) → ⋯

⋯ → T(F P_{p} ⊕ F^{2}P_{p}) ⟶^{θ_{1}^{T(Pp)}} T(P_{p} ⊕ F P_{p}) ⟶^{ε_{T(Pp)}} T(P_{p}) → 0

constructed in section 3. All these resolutions are imbedded in the following complex of Σ−projective resolutions:

⋯ → P_{Σ}^{T(P_{p})} → P_{Σ}^{T(P_{p−1})} → ⋯ → P_{Σ}^{T(P_{1})} → P_{Σ}^{T(P_{0})} → P_{Σ}^{(X,f)} → 0 (1)

The differentials in (1) are TF^{q}U(ϕ_{p}^{(X,f)}) : P_{Σ^{p}}^{T(P_{p})} → P_{Σ^{p}}^{T(P_{p−1})}. Now applying to this sequence the functor C, we obtain the following sequence of the Σ−associated complexes of the objects (X, f), T(P_{p}), ∀p ≥ 0:

⋯ → Σ_{T(P_{p})}^{•} → Σ_{T(P_{p−1})}^{•} → ⋯ → Σ_{T(P_{1})}^{•} → Σ_{T(P_{0})}^{•} → Σ_{(X,f)}^{•} → 0 (2)

The differentials in (2) are F^{q}U(ϕ_{p}^{(X,f)}) : Σ_{T(P_{p})}^{p} → Σ_{T(P_{p−1})}^{p}. Using the sign Lemma [36], we can view the sequences (1), (2) as first quadrant bicomplexes _{i}M_{••}^{(X,f)}, i = 1, 2, in which we have deleted the first row and column. Then _{2}M_{p,q}^{(X,f)} = F^{q}P_{p} ⊕ F^{q+1}P_{p}, ∀p, q ≥ 0 and the differentials are given by: δ_{p,q}^{1} = F^{q}U(ϕ_{p}^{(X,f)}) : M_{p,q}^{(X,f)} → M_{p−1,q}^{(X,f)}, δ_{p,q}^{2} = (−1)^{p}C(θ_{q}^{T(P_{p})}) : M_{p,q}^{(X,f)} → M_{p,q−1}^{(X,f)}. Since the columns are the Σ−associated complexes of the objects T(P_{p}), all columns are contractible and the rows are obtained from the deleted 𝔉−projective resolution of (X, f) by applying succesively the functors F^{q}U, ∀ q ≥ 0. We observe that we have an augmentation Tot_{1}M_{••}^{(X,f)} → (X, f) → 0.

Lemma 7.21. The total complex Tot_{1}M_{••}^{(X,f)} is an 𝔉-exact resolution of (X, f), which consists of C−acyclic objects with respect to 𝔉 if the functor F satisfies the vanishing condition (†), in which case:

L_{n}^{𝔉}C(X, f) ≅ H_{n}[Tot_{1}M_{••}^{(X,f)}], ∀n ≥ 0.

Proof. The first part follows from the spectral sequence induced by the bicomplex _{1}M_{••}^{(X,f)}, and the second follows from the first.

Now we define the Σ−projective resolution functor S : D_{F} (η) → C^{0]}(D_{F} (η)) to the abelian category C^{0]}(D_{F} (η)) of positive complexes of D_{F} (η), as follows: S(X, f) = P_{Σ}^{(X,f)}, and if a : (X, f) → (Y, g) is a morphism in D_{F} (η), then S(a)^{i} = T(F^{i}a), ∀i ≥ 0. Let C^{0]} : C^{0]}(D_{F} (η)) → C^{0]}(D) be the natural extension of the functor C : D_{F} (η) → D, and define:

Σ^{•} = C^{0]}S : D_{F} (η) → C^{0]}(D).

Then Σ^{•}(X, f) = Σ_{(X,f)}^{•} = the Σ−associated complex of (X, f), and if a : (X, f) → (Y, g) is a morphism in D_{F} (η), then Σ^{•}(a)^{i} = F^{i}a, ∀i ≥ 0. The complexes (1), (2) are therefore the images of P_{𝔉}^{(X,f)} under the functors S, Σ^{•}. Obviously Σ^{•} is a right 𝔉−exact and Σ−exact embedding. So it is interesting to consider the functors L_{q}^{𝔉}Σ^{•}. The objects L_{q}^{𝔉}Σ^{•}(X, f) are by construction positive complexes in D. The following describes the Butler-Horrocks spectral sequence for the functor C, induced by the inclusion of the proper classes Σ ⊆ 𝔉, see [7], [12].

Theorem 7.22. ∀(X, f) ∈ D_{F} (η), there is a spectral sequence of the form:

(!) E_{p,q}^{2} = L_{p}^{Σ}[L_{q}^{𝔉}C](X, f) = H_{p}L_{q}^{𝔉}Σ^{•}(X, f) ⟹ L_{n}^{𝔉}C(X, f).

If the vanishing condition (†) is true for F, then the following are equivalent:

(i) ∀p, q ≥ 1 such that p + q ≥ n + 1 : L_{p}^{ℜ}F^{q} = 0.

(ii) L_{m}^{𝔉}C ≅ L_{m}^{Σ}C, ∀m ≥ n + 1.

In particular if η = 0, the vanishing condition (††) of Corollary 7.16 is true for F, and (i) above holds, then: L_{m}^{𝔉}C = 0, ∀m ≥ n + 1.

Proof. We compute the spectral sequences arising from the bicomplex _{2}M_{••}^{(X,f)}. For the first filtration, since the columns are contractible, we have: ^{I}E_{p,q}^{2} = 0, ∀q ≥ 1, and ^{I}E_{p,0}^{2} = H_{p}[𝔉_{(X,f)}^{•}]. Thus the spectral sequence arising from the

first filtration collapses, and ^{I}E_{n,0}^{2} ≅ L_{n}^{𝔉}C(X, f) ≅ H_{n}[Tot(_{2}M_{••}^{(X,f)})], ∀n ≥ 0. For the filtration II, we have: H_{p,q}^{1}[_{2}M_{••}^{(X,f)}] = Ker(F^{p}U(ϕ^{q}))/Im(F^{p}U(ϕ^{q+1})) = (L_{q}^{𝔉}Σ^{•}(X, f))^{p}. Hence ^{II}E_{p,q}^{2} = H_{p}(L_{q}^{𝔉}Σ^{•}(X, f)).

Suppose now that the condition (†) is true for F. Then the underline graded object of the complex L_{q}^{𝔉}Σ^{•}(X, f) is L_{p}^{ℜ}F^{q}X, ∀p, q ≥ 1. Indeed: as we have seen (L_{q}^{𝔉}Σ^{•}(X, f))^{p} = Ker(F^{p}U(ϕ^{q}))/Im(F^{p}U(ϕ^{q+1})). But this homology is obtained applying the 𝔉−exact functor U to the 𝔉−projective resolution P_{𝔉}^{(X,f)} of (X, f) and then applying to the resulting ℜ−exact sequence U[P_{𝔉}^{(X,f)}] the functor F^{p}. By (†), the ℜ−exact resolution U[P_{𝔉}^{(X,f)}] of X consists of F^{p}−acyclic objects with respect to ℜ. So Ker(F^{p}U(ϕ^{q}))/Im(F^{p}U(ϕ^{q+1})) = (L_{q}^{𝔉}Σ^{•}(X, f))^{p} ≅ L_{p}^{ℜ}F^{q}X, ∀p, q ≥ 1. Observe that ^{II}E_{p,0}^{2} = H_{p}(Σ^{•}(X, f)) = H_{p}[Σ_{(X,f)}^{•}] ≅ L_{p}^{Σ}C(X, f).

That (i) ⇒ (ii) follows easily from the theory of spectral sequences, and the proof of (ii) ⇒ (i), consists of a simple induction argument and is left to the reader.

Combining Theorems 5.11(2), 7.22, we have the following

Corollary 7.23. Suppose that the vanishing condition (†) is true for F, and ∀p ≥ 0, q ≥ 1 : p + q = n + 1, L_{p}^{ℜ}F^{q} = 0. Then:

𝔉−f.gl.dimD_{F} (η) ≤ n + ℜ−f.gl.dimD.

The sequence of complexes (2), viewed as a bicomplex, has been constructed in an adhoc manner in case η = 0 in [13]. The above Theorem and Corollary generalizes and improves all the corresponding results of [13].

Corollary For Rings 7.24. Let Λ = Γ ⋉_{ϑ} M be a cleft extension of rings. and suppose that: T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1. If T or_{q}^{Γ}[−, ⊗_{Γ}^{p}M] = 0, ∀p ≥ 1, q ≥ 1 : p + q = n + 1, then: T or_{i}^{Λ}[−, Γ] ≅ T or_{i}^{(Λ,Γ)}[−, Γ], ∀i ≥ n + 1. If T or_{q}^{Γ}[−, ⊗_{Γ}^{p}M] = 0, ∀p ≥ 0, q ≥ 1 : p + q = n + 1, then:

r.f.gl.dimΛ ≤ n + r.f.gl.dimΓ.

8 A Homological Quide to the Truncated Extensions of Rings and Abelian Categories

Throughout this section we consider a t−truncated extension D ⋉_{t} F of the abelian category D by the right exact functor F, where t ≥ 0. We recall that the objects of D⋉_{t}F are of the form (X, f) where f : F X → X is a morphism in D such that f_{t+1} = 0, where f_{i} := F^{i}f ∘F^{i−1}f ∘⋯∘F^{2}f ∘F f ∘f : F^{i+1}X → X. From Section 2 we have an isomorphism of categories 𝔸 : D ⋉_{t}F → DF̃(η̃), for an η̃−extension, where F̃ := ⊕_{i=1}^{t+1}F^{i} and η̃ : F̃^{2} → F̃ is a suitable associative morphism. Hence we can apply the results of the previous sections, using the isomorphism 𝔸. We point out that the absolute homology of D ⋉_{t}F in the case F is exact, is studied in [28] using triple complexes. We begin the study of the 𝔉, Σ−homology of D ⋉_{t}F by determining the Σ−associated complex of an arbitrary object (X, f) of D ⋉_{t}F. We denote by 𝔉, Σ the usual proper classes in D ⋉_{t}F and in DF̃(η̃). From the isomorphism 𝔸 above and the shape of the Σ−associated complex Σ_{(X,f)}^{•} of an object (X, f) ∈ DF̃(η̃), it follows that Σ_{(X,f)}^{•} is too big for calculations. However we have the following:

Lemma 8.1. ∀(X, f) ∈ D ⋉_{t}F ≅ DF̃(η̃), the Σ−associated complex Σ_{(X,f)}^{•} of (X, f) is homotopy equivalent to the following complex:

tΣ_{(X,f)}^{•} : ⋯ → F^{ϕ(2n+1)}X ⟶^{F^{ϕ(2n)}f} F^{ϕ(2n)}X ⟶^{−F^{ϕ(2n−1)}f_{t}} F^{ϕ(2n−1)}X → ⋯

⋯ → F^{ϕ(3)}X ⟶^{−F^{ϕ(2)}f} F^{ϕ(2)}X ⟶^{F^{ϕ(1)}f_{t}} F^{ϕ(1)}X ⟶^{−F^{ϕ(0)}f} F^{ϕ(0)}X → 0

where ϕ : ℕ → ℕ is the function defined by:

ϕ(n) = \{ \begin{matrix}m · (t + 2) & if n = 2 · m \\ m · (t + 2) + 1 & if n = 2 · m + 1. \end{matrix}

Proof. Let (X, f) ∈ D ⋉_{t}F. Then we have a short exact sequence:

0 → (⊕_{i=1}^{t+1}F^{i}X, σ_{0}) ⟶^{κ_{0}} T(X) ⟶^{ϖ_{(X,f)}} (X, f) → 0

where ϖ is the counit of the adjunction (T, U), defined by ϖ_{(X,f)} = ^{t}(1_{X}, f_{0}, f_{1}, , ⋯, f_{t}), where σ_{0} : ⊕_{i=2}^{t+2}F^{i}X → ⊕_{i=1}^{t+1}F^{i}X is represented by a (t+1)×(t+1) matrix (a_{i,j}), where: a_{i,1} = −F f_{i}, i = 0, 1, ⋯, t, a_{i,i+1} = 1_{F^{i+1}X}, i = 0, 1, ⋯, t, and all other entries are zero, and finally κ_{0} = ker(ϖ_{(X,f)}). Next we have the following exact sequence in D ⋉_{t}F:

0 → (F^{t+2}X, F^{t+2}f_{0}) ⟶^{κ_{1}} T(F X) ⟶^{π_{1}} (⊕_{i=1}^{t+1}F^{i}X, σ_{0}) → 0

where π_{1} is represented by a (t + 1) × (t + 2) matrix (a_{i,j}), where: a_{i,i−1} = −F^{i−1}f_{0}, i = 2, ⋯, t + 1, a_{i,i} = 1_{F^{i}X}, i = 1, ⋯, t + 1, and all other entries are zero, and finally κ_{1} = ker(π_{1}). The Yoneda composition of the above sequences gives the following exact sequence:

0 → (F^{t+2}X, F^{t+2}f_{0}) → T(F X) → T(X) → (X, f) → 0.

Continuing the above procedure starting now with (F^{t+2}X, F^{t+2}f_{0}), and using the function ϕ, we have a resolution of (X, f):

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

Obviously (1) is a functorial Σ−projective resolution of (X, f) in D⋉_{t}F. We denote (1) by^{t}P_{Σ}^{(X,f)}. Now using the isomorphism 𝔸, 𝔸(^{t}P_{Σ}^{(X,f)}) is a Σ−projective resolution of 𝔸(X, f) in DF̃(η̃). Hence 𝔸(^{t}P_{Σ}^{(X,f)}) is homotopy equivalent to P_{Σ}^{𝔸(X,f)}, and this implies that Σ_{𝔸(X,f)}^{•} is homotopy equivalent to ^{t}Σ_{(X,f)}^{•}.

We note the function ϕ is defined in [28]. From now on we view the isomorphism 𝔸 as an identification, and we call the complex ^{t}Σ_{(X,f)}^{•}, the t−truncated Σ−associated complex of (X, f). In case t = 0, then (and only then) D_{F} (0) ≅ D ⋉_{0} F is the trivial extension, ϕ = Id_{ℕ} and ^{0}Σ_{(X,f)}^{•} = Σ_{(X,f)}^{•}. Having established a convenient description of Σ_{(X,f)}^{•} as ^{t}Σ_{(X,f)}^{•}, we can apply the results of the previous sections, observing that in D ⋉_{t} F we have KerC = 0, since by Remark 5.2(3), D ⋉_{t} F is 𝔉−admissible. Many of these results apply directly to the truncated case. Namely the results that are independent of the multiplication η̃. This is true for example for Theorem 4.11 concerning lifting of tilting objects. This Theorem is true as stated replacing everywhere F with F̃ = ⊕_{i=1}^{t+1}F^{i}. Now we state some results which depend on the index t in D ⋉_{t}F.

Corollary 8.2. The following are equivalent for the cleft extension D ⋉_{t}F.

(i) 𝔉−gl.dimD ⋉_{t}F ≤ 1.

(ii) (a) ℜ−gl.dimD ≤ 1.

(b) F is ℜ−exact.

(c) F^{t+2} = 0.

(d) ∀X ∈ D : F X ∈ P(ℜ).

If condition (i) is true, then D ⋉_{t} F is isomorphic to the free cleft extension D(F).

Proof. Exactly as in Proposition 6.1.

Corollary For Rings 8.3. Let Λ = Γ ⋉_{t}M be the t−truncated cleft extension of Γ by M, t ≥ 0. Then the following are equivalent:

(i) Λ = Γ ⋉_{t}M is right hereditary. (ii) (a) Γ is right hereditary, and _{Γ}M is flat.

(b) ⊗_{Γ}^{t+2}M = 0.

(c) ∀X ∈ Mod(Γ) : X ⊗_{Γ} M is a projective right Γ−module. Corollary 8.4. The following are equivalent for the cleft extension D ⋉_{t}F.

(i) 𝔉−gl.dimD ⋉_{t}F ≤ 2. (ii) (a) F^{t+3} = L_{1}^{ℜ}F^{t+2} = L_{2}^{ℜ}F^{i} = 0, ∀i = 1, 2, ..., t + 1.

(b) ∀P ∈ P(ℜ) : L_{1}^{ℜ}F F (P) = 0.

(c) ∀X ∈ D : F^{t+2}X, L_{1}^{ℜ}F X ∈ P(ℜ).

(d) ∀X ∈ D the naturally induced morphism φ^{X} is an epimorphism:

φ^{X} : D[L_{1}^{ℜ}F (X) ⊕ F^{t+2}X, −] → Ext_{ℜ}^{2}[F X, −] ⊕ Ext_{ℜ}^{3}[X, −].

Proof. The proof is the same as in Corollary 6.4, using Theorem 6.3. Corollary For Rings 8.5. Let Λ = Γ ⋉_{t}M be the t−truncated cleft extension of Γ by M, t ≥ 0. Then the following are equivalent:

(i) r.gl.dimΛ ≤ 2. (ii) (a) ⊗_{Γ}^{t+3}M = T or_{1}^{Γ}(M, M) = 0 and w.d ⊗_{Γ}^{t+2} M = 0, w.d ⊗_{Γ}^{i} M ≤ 1,

∀i = 1, 2, ..., t + 1.

(b) ∀X ∈ Mod(Γ) : X ⊗_{Γ}^{t+2} M, T or_{1}^{Γ}(X, M) are projective right Γ−modules.

(c) ∀X ∈ Mod(Γ), the naturally induced morphism:

Hom_{Γ}[T or_{1}^{Γ}(X, M) ⊕ X ⊗_{Γ} (⊗_{Γ}^{t+2}M), −] ⟶

⟶ Ext_{Γ}^{2}[X ⊗_{Γ} M, −] ⊕ Ext_{Γ}^{3}[X, −]

is an epimorphism. With the notation preceding the Proposition 6.6 we have the following:

Corollary 8.6. The following are equivalent for the cleft extension D ⋉_{t}F. (i) D ⋉_{t}F is 𝔉−Frobenius. (ii) The following are true ∀P ∈ P(ℜ), ∀I ∈ I(ℜ), ∀i = 0, 1, ..., t:

(a) F^{t+1}P, Ker(σ_{F^{i}P}) ∈ I(ℜ).

(b) G^{t+1}I, Coker(ρ_{G^{i}I}) ∈ P(ℜ).

(c) σ_{F^{i}P} is epic and: GKer(σ_{P} ∘ Gσ_{F P} ∘ ⋯ ∘ G^{t}σ_{F^{t}P}) = 0.

(d) ρ_{G^{i}I} is monic and: F Coker(F^{t}ρ_{G^{t}I} ∘ ⋯ ∘ F ρ_{GI} ∘ ρ_{I}) = 0.

Hence if the category D is ℜ−Frobenius, then the category D ⋉_{t} Id_{D} of (t + 1)−nilpotent endomorphisms of D is 𝔉−Frobenius. Proof. The proof is as in Corollary 6.7, using Lemma 5.3 and the t−truncated Σ−associated complex of the objects T(P), H(I).

We recall now the vanishing condition

(†) L_{i}^{ℜ}F^{j}(F P) = 0, ∀i, j ≥ 1, ∀P ∈ P(ℜ)

of the last Section. We define the t−truncated nilpotency class c_{t}(F) of F to be ∞ if F is non-nilpotent, otherwise c_{t}(F) = min{m ≥ 0 : F^{ϕ(m+1)} = 0}. Theorem 8.7. Let D ⋉_{t}F be the t−truncated extension of D by F, t ≥ 0.

(1) The following are equivalent:

(i) The natural morphisms ξ^{i} : L_{i}^{𝔉}C → L_{i}^{Σ}C , ∀i ≥ 0 are isomorphisms. (ii) The functor F is ℜ−exact. If one of the above statements is true, then ∀(X, f) ∈ D ⋉_{t}F, ∀n ≥ 1:

L_{2n−1}^{𝔉}C(X, f) ≅ Ker(F^{ϕ(2n−2)}f)/Im(F^{ϕ(2n−1)}f_{t}),

L_{2n}^{𝔉}C(X, f) ≅ Ker(F^{ϕ(2n−1)}f_{t})/Im(F^{ϕ(2n)}f),
L_{n}^{𝔉}CZ ≅ L_{n}^{Σ}CZ ≅ F^{ϕ(n)}, ∀n ≥ 0.

(2) Suppose that the condition (†) is true for F. The following are equivalent: (i) 𝔉−gl.dimD ⋉_{t}F < ∞. (ii) ℜ−gl.dimD < ∞ and F is nilpotent. In any case we have the bounds:

ℜ−gl.dimD ≤ 𝔉−gl.dimD ⋉_{t}F ≤ c_{t}(F) + 2 · ℜ−gl.dimD.

(3) Suppose that the vanishing condition (†) is true for F. Then:

L_{n}^{𝔉}CZ ≅ ⨁_{p+q=n} L_{q}^{ℜ}F^{ϕ(p)}, ∀n ≥ 0,

Ext_{𝔉}^{n}[Z(X), Z(Y)] ≅ ⨁_{p+q=n} R_{ℜ}^{p}D[F^{ϕ(q)}X, Y], ∀n ≥ 0,

𝔉−gl.dimD ⋉_{t}F = sup{n ≥ 0 : ⨁_{p+q=n+1} R_{ℜ}^{p}D[F^{ϕ(q)}−, −] = 0}.

Proof. (1) Since the statement is independent of η̃, this is true by Theorem 4.7. The last assertion follows from the description of ^{t}Σ_{Z(X)}^{•}. Part (2) follows directly from Theorem 7.5, and the proof of part (3) is similar to the proof of Theorem 7.10 and Theorem 7.13. The necessary modifications are left to the reader.

Corollary For Rings 8.8. Let Λ = Γ ⋉_{t}M be a t−truncated cleft extension of Γ by M, t ≥ 0. Assume that: T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1.

(1)

T or_{n}^{Λ}[Z−, Γ] ≅ ⨁_{p+q=n} T or_{p}^{Γ}[−, ⊗_{Γ}^{ϕ(q)}M], ∀n ≥ 0.

In particular: T or_{n}^{Λ}[Γ, Γ] ≅ ⊗_{Γ}^{ϕ(n)}M. If in addition: T or_{q}^{Γ}[−, ⊗_{Γ}^{p}M] = 0, ∀p, q ≥ 1 : p + q = n, then: T or_{i}^{Λ}[−, Γ] ≅ T or_{i}^{(Λ,Γ)}[−, Γ], ∀i ≥ n.

(2)

Ext_{Λ}^{n}[Z−, Z−] ≅ ⨁_{p+q=n} R^{p}Hom_{Γ}[− ⊗_{Γ}^{ϕ(q)} M, −],

T or_{n}^{Λ}[Z−, Z−] ≅ ⨁_{p+q=n} L_{p}[− ⊗_{Γ}^{ϕ(q)} M−],

r.gl.dimΛ = sup{n ≥ 0 : R^{p}Hom_{Γ}[− ⊗_{Γ}^{ϕ(q)} M, −] = 0, ∀p + q = n + 1},

w.gl.dimΛ = sup{n ≥ 0 : L^{p}[− ⊗_{Γ}^{ϕ(q)} M−] = 0, ∀p + q = n + 1}.

(3) Suppose that _{Γ}M is flat. Then we have the following:

Ext_{Λ}^{n}[Z−, Z−] ≅ ⨁_{p+q=n} Ext_{Γ}^{p}[− ⊗_{Γ}^{ϕ(q)} M, −],

T or_{n}^{Λ}[Z−, Z−] ≅ ⨁_{p+q=n} T or_{p}^{Γ}[− ⊗_{Γ}^{ϕ(q)} M, −],

r.gl.dimΛ = sup{n ≥ 0 : Ext_{Γ}^{p}[− ⊗_{Γ}^{ϕ(q)} M, −] = 0, ∀p + q = n + 1},

w.gl.dimΛ = sup{n ≥ 0 : T or_{p}^{Γ}[− ⊗_{Γ}^{ϕ(q)} M, −] = 0, ∀p + q = n + 1}.

We leave to the reader to state the corresponding results about 𝔉−f.gl.dim D ⋉_{t} F and f.gl.dimΓ ⋉_{t} M, applying for example Theorem 5.11, or using the techniques developed so far. The isomorphisms of part (2) in Corollary 8.8 take a more familiar form under some mild conditions on Γ, X, Y:

Proposition 8.9. Suppose that Γ is a k−algebra over a commutative ring k, which acts centrally on M. Let Γ be k−projective, T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1, and let Λ = Γ⋉_{t}M be the t−truncated cleft extension. Let X, Y ∈ Mod(Γ), W ∈ Mod(Γ^{op}) and let Γ^{e} = Γ ⊗_{k} Γ^{op} be the enveloping ring of Γ over k.

(1) If Ext_{k}^{n}[X, Y] = 0, ∀n ≥ 1, then:

Ext_{Λ}^{n}[Z(X), Z(Y)] ≅ ⨁_{p+q=n} Ext_{Γ^{e}}^{p}[⊗_{Γ}^{ϕ(q)}M, Hom_{k}(X, Y)], ∀n ≥ 0.

(2) If T or_{n}^{k}[X, W] = 0, ∀n ≥ 1, then:

T or_{n}^{Λ}[Z(X), Z(W)] ≅ ⨁_{p+q=n} T or_{p}^{Γ^{e}}[⊗_{Γ}^{ϕ(q)}M, X ⊗_{k} W], ∀n ≥ 0.

Proof. It is easy to see that we have an natural isomorphism, ∀q ≥ 1:

Hom_{Γ^{e}}[⊗_{Γ}^{q}M, Hom_{k}(X, Y)] ≅ Hom_{Γ}[X ⊗_{Γ} (⊗_{Γ}^{q}M), Y].

From this isomorphism, the hypotheses and standard arguments [8], we have:

R^{p}Hom_{Γ}[X ⊗_{Γ}^{q} M, Y] ≅ Ext_{Γ^{e}}^{p}[⊗_{Γ}^{q}M, Hom_{k}(X, Y)].

Hence by Theorem 8.8, part (1) follows and (2) is similar.

The t−truncated tensor nilpotent class of the bimodule M is defined by

c_{t}(M) = min{n ≥ 0 : ⊗_{Γ}^{ϕ(n+1)}M = 0}.

Corollary 8.10. Let Γ be a finite-dimensional k−algebra over an algebraically closed field k, which acts centrally on M. Let T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1, and consider the t−truncated cleft extension Λ = Γ ⋉_{t} M of Γ by M, t ≥ 0. Then:

gl.dimΛ = max{c_{t}(M), p.d_{Γ^{e}}(⊗_{Γ}^{ϕ(i)}M) + i ; i = 0, 1, ..., c_{t}(M)}.

Proof. Follows from the previous proposition and the fact that in our case all simple Γ^{e}-modules are of the form Hom_{k}(S, T) for simple Γ−modules S, T.

The following is a version of the Strong No Loops Conjecture [21], [28].

Corollary 8.11. Let S be a simple object of D without selfextensions, i.e. we have: Ext_{ℜ}^{1}[S, S] = 0, and consider the t−truncated extension D ⋉_{t} F of D by F. If the vanishing condition (†) is true for F, for the simple object Z(S) of D ⋉_{t}F, consider the following statements:

(i) Ext_{𝔉}^{1}[Z(S), Z(S)] ≠ 0.

(ii) Ext_{Σ}^{1}[Z(S), Z(S)] ≠ 0.

(iii) Ext_{Σ}^{i}[Z(S), Z(S)] ≠ 0, ∀i ≥ 0.

(iv) Ext_{𝔉}^{i}[Z(S), Z(S)] ≠ 0, ∀i ≥ 0.

(v) 𝔉−p.dZ(S) = ∞.

Then: (i) ⇔ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v). In particular if D is semisimple, then for any simple object Z(S) of D ⋉_{t}F with Ext_{D⋉_{t}F}^{1} [Z(S), Z(S)] ≠ 0, we have:

Ext_{D⋉_{t}F}^{i} [Z(S), Z(S)] = D[F^{ϕ(i)}S, S] ≠ 0, ∀i ≥ 0

and p.d_{D⋉_{t}F} Z(S) = ∞.

Proof. Since Ext_{𝔉}^{1}[Z(S), Z(S)] ≅ Ext_{Σ}^{1}[Z(S), Z(S)] ⊕ Ext_{ℜ}^{1}[S, S], the hypothesis implies that (i) ⇔ (ii). From the t−truncated Σ−associated complex ^{t}Σ_{Z(S)}^{•}, we see that ∀i ≥ 0 : Ext_{Σ}^{i}[Z(S), Z(S)] = D[F^{ϕ(i)}S, S] and this is a direct summand of Ext_{𝔉}^{i}[Z(S), Z(S)] as follows from part (3) of the previous Theorem. Hence (iii) ⇒ (iv). Since S is simple and F is right exact, trivially (ii) ⇒ (iii). The implication (iv) ⇒ (v) is obvious.

Corollary For Rings 8.12. Let Λ = Γ ⋉_{t}M be a t−truncated cleft extension of Γ by M, t ≥ 0, and suppose that T or_{i}^{Γ}[M, ⊗_{Γ}^{j}M] = 0, ∀i, j ≥ 1. Let S a simple Γ−module with Ext_{Γ}^{1}[S, S] = 0. If for the simple Λ−module Z(S) we have Ext_{Λ}^{1}[Z(S), Z(S)] ≠ 0, then Ext_{Λ}^{i}[Z(S), Z(S)] ≠ 0, ∀i ≥ 0 and p.dZ(S) = ∞.

Let Λ be a finite dimensional k−algebra over the algebraically closed field k. Let {S_{1}, S_{2}, ..., S_{n}} be a complete set of isomorphism classes of simple right Λ−modules and let P_{i} be the projective cover of S_{i}. The Cartan matrix C(Λ) of Λ is the n×n matrix with entries C_{ij}(Λ) = dim_{k}Hom_{Λ}[P_{i}, P_{j}]. It is conjectured by Eilenberg that if Λ has finite global dimension, then DetC(Λ) = 1.

Proposition 8.13. Let Γ be a semisimple k−algebra over the algebraically closed field k and let Λ = Γ⋉_{t}M be a t−truncated cleft extension of Γ by M, t ≥ 0. If gl.dimΛ < ∞, then DetC(Λ) = 1.

Proof. First we show that C_{ii}(Λ) = 1, ∀i = 1, ..., n. From the adjoint pair (T, U) we have that Hom_{Λ}[T(S_{i}), T(S_{i})] = Hom_{Γ}[S_{i}, S_{i} ⊕ F S_{i} ⊕ F^{2}S_{i} ⊕ ⋯ ⊕ F^{t+1}S_{i}]. If Hom_{Λ}[S_{i}, F^{k}S_{i}] ≠ 0 for some k = 1, 2, ..., t+1, then we have a monomorphism S_{i} ↪ F^{k}S_{i} which splits since Γ is semisimple. Hence S_{i} is a direct summand of F^{k}S_{i}. Consider the trivial extension E := D ⋉_{0}F^{k}, where D = mod(Γ). Since Λ has finite global dimension, we have that F, hence F^{k} is nilpotent. This implies that also that E has finite global dimension. Now Ext_{E}^{1}[Z(S_{i}), Z(S_{i})] = D[F^{k}S_{i}, S_{i}] ≠ 0. Since Z(S_{i}) is a simple object in E, from Corollary 8.11, we have that p.dZ(S_{i}) = ∞ and this is impossible. Hence Hom_{Λ}[S_{i}, F^{k}S_{i}] = 0, ∀k = 1, ..., t + 1. This shows that Hom_{Λ}[T(S_{i}), T(S_{i})] = Hom_{Γ}[S_{i}, S_{i}], hence C_{ii}(Λ) = 1, ∀i = 1, ..., n.

Next we show that if i ≠ j and C_{ij}(Λ) ≠ 0, then C_{ji}(Λ) = 0. Indeed otherwise we have as above non zero split monics S_{i} ↪ F^{k}S_{j} and S_{j} ↪ F^{m}S_{i}, where 1 ≤ k, m ≤ t + 1. Then we have a non zero split monic S_{j} ↪ F^{m}S_{i} ↪ F^{k+m}S_{j}. But then Hom_{Γ}[F^{k+m}S_{j}, S_{j}] ≠ 0. By the above argument this is impossible. Since C_{ii}(Λ) = 1, ∀i = 1, ..., n and for i ≠ j, C_{ij}(Λ) ≠ 0 implies C_{ji}(Λ) = 0, we can arrange the simples in such a way that C(Λ) is an upper triangular matrix with all diagonal entries equal to 1. Then obviously we have DetC(Λ) = 1.

Acknowledgement. The author is grateful to the referee for his/her many helpful and very valuable comments, suggestions and corrections which resulted in a significant improvement of the paper.

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