RELATIVE HOMOLOGICAL ALGEBRA FOR THE PROPER CLASS ω_{f}

GRIGORY GARKUSHA

The classes of fp-injective and fp-flat modules have been introduced in [8] to study F P -injective and weakly quasi-Frobenius rings. Both classes are definable (=elementary) in the first order language of modules and naturally generalize the corresponding classes of F P -injective and flat modules.

Homological algebra based on F P -injective and flat modules is well studied and was developed in the 70-s (see e.g. [14, 23, 24]). One would like to construct homological algebra based on fp-injective and fp-flat modules. In this paper we shall show that such homological algebra can be realized as relative homological algebra corresponding to the proper class (in the sense of [11]) of monomorphisms

ω_{f} = {μ | R^{I} ⊗ μ is a monomorphism for any power R^{I} of a ring R}.

If we define the corresponding relative injective and relative flat modules in the natural way, we call them f-injective and f-flat modules respectively, then they will correspond to fp-injective pure-injective modules and fp-flat modules.

There are many results of a homological nature which may be generalized from Noetherian rings to coherent rings. In this direction, finitely generated modules are replaced by finitely presented modules and injective modules should be replaced by F P -injective modules (see [23, 24]). In our situation analogous results, which are related to the relative homological algebra we construct in this paper, are valid for all rings. This phenomenon can be explained as follows. The classical Chase theorem (see [25]) states that a ring R is left coherent iff each power R^{I} of I copies of the ring R is a flat right R-module. But if we consider only those exact sequences under which the functor R^{I} ⊗_{R} − is exact, we shall arrive at the proper class ω_{f}.

Every time we deal with definable subcategories the category of generalized modules

R𝒞 = (mod R, Ab)

is of great utility in this context as well as torsion theories of finite type of _{R}𝒞 (consult e.g. the works [13, 18]). For this reason the majority of statements of the paper is proved by using the technique of torsion/localizing functors in _{R}𝒞.

The paper is organized as follows. The first two sections are preliminary. There we present the necessary category-theoretic background and introduce the proper class ω_{f} as a class of monomorphisms of a certain quotient category in _{R}𝒞. In section 3 the classes of f-injective and f-flat modules are investigated. In section 4 we study basic properties for the functors Ext_{f} and Tor^{f} which are related to f-injective and f-flat resolutions. Also, certain homological dimensions are discussed in this section. In section 5 we describe the relative derived category D_{f}(R) and the functor Ext_{f} as Hom-sets in D_{f}(R). In the remaining section we study the relationship of the G-theory of a ring with the K-theory of some abelian subcategories of _{R}𝒞.

2000 Mathematics Subject Classification. 18G25, 16E10, 16E30.

Throughout the paper R denotes a ring with identity. The category of left (respectively right) R-modules is denoted by R Mod (respectively Mod R) and the category of finitely presented left (respectively right) R-modules by R mod (respectively mod R).

1. The category of generalized modules

The category of generalized left R-modules

R𝒞 = (mod R, Ab)

consists of additive covariant functors defined on the category of finitely presented right Rmodules mod R with values in the category of abelian groups Ab. In this section we collect basic facts about the category _{R}𝒞. For details and proofs we refer the reader to [13, 18], for example. All subcategories considered are assumed to be full.

We say that a subcategory 𝒮 of an abelian category 𝒞 is a Serre subcategory if for any short exact sequence

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0

in 𝒞 an object Y ∈ 𝒮 if and only if X, Z ∈ 𝒮. A Serre subcategory 𝒮 of a Grothendieck category 𝒞 is localizing if it is closed under taking direct limits. Equivalently, the inclusion functor i : 𝒮 → 𝒞 admits the right adjoint t = t_{𝒮} : 𝒞 → 𝒮 which takes every object X ∈ 𝒞 to the maximal subobject t(X) of X belonging to 𝒮 [16]. The functor t we call the torsion functor. An object C of _{R}𝒞 is said to be 𝒮-torsionfree if t(C) = 0. Given a localizing subcategory 𝒮 of _{R}𝒞 the quotient category _{R}𝒞/𝒮 consist of C ∈ _{R}𝒞 such that t(C) = t^{1}(C) = 0. The objects from _{R}𝒞/𝒮 we call 𝒮-closed objects. Given C ∈ _{R}𝒞 there exists a canonical exact sequence

0 ⟶ A′ ⟶ C ⟶^{λ_{C}} C_{𝒮} ⟶ A′′ ⟶ 0

with A′ = t(C), A′′ ∈ 𝒮, and where C_{𝒮} ∈ _{R}𝒞/𝒮 is the maximal essential extension of C̃ = C/t(C) such that C_{𝒮}/C̃ ∈ 𝒮. The object C_{𝒮} is uniquely defined and is called the 𝒮-envelope of C. Moreover, the inclusion functor i : _{R}𝒞/𝒮 → _{R}𝒞 has the left adjoint localizing functor (−)_{𝒮} : _{R}𝒞 → _{R}𝒞/𝒮, which is also exact. It takes each C ∈ _{R}𝒞 to C_{𝒮} ∈ _{R}𝒞/𝒮. Then,

Hom_{_{R}𝒞} (X, Y) ≃ Hom_{_{R}𝒞/𝒮} (X_{𝒮}, Y)

for all X ∈ _{R}𝒞 and Y ∈ _{R}𝒞/𝒮.

An object X of a Grothendieck category 𝒞 is finitely generated if whenever there are subobjects X_{i} ⊆ X with i ∈ I satisfying X = ∑_{i∈I} X_{i}, then there is a finite subset J ⊂ I such that X = ∑_{i∈J} X_{i}. The subcategory of finitely generated objects is denoted by fg 𝒞. A finitely generated object X is said to be finitely presented if every epimorphism γ : Y → X with Y ∈ fg 𝒞 has the finitely generated kernel Ker γ. By fp 𝒞 we denote the subcategory consisting of finitely presented objects. Finally, we refer to a finitely presented object X ∈ 𝒞 as coherent if every finitely generated subobject of X is finitely presented. The corresponding subcategory of coherent objects will be denoted by coh 𝒞.

The category_{R}𝒞 is a locally coherent Grothendieck category, i.e. every object C ∈ _{R}𝒞 is a direct limit C = \underrightarrow{lim}_{I} C_{i} of coherent objects C_{i} ∈ coh_{R}𝒞. Equivalently, the category coh_{R}𝒞 is abelian. Moreover, _{R}𝒞 has enough coherent projective generators {(M, −)}_{M∈mod R}. Thus, every coherent object C ∈ coh_{R}𝒞 has a projective presentation

(N, −) ⟶ (M, −) ⟶ C ⟶ 0,

where M, N ∈ mod R.

A localizing subcategory 𝒮 of _{R}𝒞 is of finite type if the inclusion functor i : _{R}𝒞/𝒮 → _{R}𝒞 commutes with direct limits. Then the quotient category _{R}𝒞/𝒮 is locally coherent and coh_{R}𝒞/𝒮 = {C_{𝒮} | C ∈ coh_{R}𝒞}.

Proposition 1.1. For a localizing subcategory 𝒮 of _{R}𝒞 the following are equivalent: (1) 𝒮 is of finite type;

(2) the inclusion functor i : _{R}𝒞/𝒮 → _{R}𝒞 commutes with direct unions;

(3) the torsion functor t_{𝒮} commutes with direct limits.

Proof. The equivalence (1) ⇔ (2) is a consequence of [6, Theorem 5.14]; (1) ⇔ (3) follows from [16, Lemma 2.4]. □

By a result of Herzog and Krause [13, 16] a localizing subcategory in _{R}𝒞 is of finite type iff it has the form 𝒮⃗ with 𝒮 some Serre subcategory of coh_{R}𝒞 and

𝒮⃗ = \{\underrightarrow{lim}_{i∈I} C_{i} | C_{i} ∈ 𝒮\}.

We say that M ∈ _{R}𝒞 is a coh-injective object if Ext_{_{R}𝒞}^{1} (C, M) = 0 for any C ∈ coh_{R}𝒞. The fully faithful functor −⊗_{R}? : R Mod → _{R}𝒞, M ↦ − ⊗_{R} M, identifies R Mod with the subcategory of coh-injective objects of _{R}𝒞. Moreover, the functor − ⊗_{R} M ∈ coh_{R}𝒞 iff M ∈ R mod. Given C ∈ coh_{R}𝒞 there is an exact sequence

0 ⟶ C ⟶ − ⊗_{R} M ⟶ − ⊗_{R} N

in coh_{R}𝒞 with M, N ∈ R mod.

A monomorphism μ : M → N in R Mod is a pure monomorphim if for any K ∈ Mod R the morphism K ⊗ μ is a monomorphism. Equivalently, the _{R}𝒞-morphism − ⊗ μ is a monomorphism. A module Q is pure-injective if the functor Hom_{R}(−, Q) takes the pure monomorphisms to epimorphisms. The injective objects of _{R}𝒞 are precisely the objects of the form − ⊗_{R} Q with Q a pure-injective module.

Let M be a left module; then

𝒮_{M} = {C ∈ coh_{R}𝒞 | (C, − ⊗_{R} M) = 0}

is Serre. Furthermore, every Serre subcategory of coh_{R}𝒞 arises in this fashion.

Proposition 1.2. Let 𝒮_{M} be the Serre subcategory of coh_{R}𝒞 cogenerated by a left R-module M. Denote by 𝒳 the subcategory of R Mod consisting of the modules of the form \underrightarrow{lim}_{k∈K} M^{I_{k}} with some sets of indices K and I_{k}, k ∈ K, and M^{I_{k}} the product of I_{k} copies of M. Then an object F of _{R}𝒞 is 𝒮_{M}-torsionfree iff it is a subobject of − ⊗_{R} L with L ∈ 𝒳. Moreover,

𝒮⃗_{M} = {F ∈ _{R}𝒞 | (F, − ⊗_{R} L) = 0 for all L ∈ 𝒳}.

Proof. The proof is like that of [18, Theorem 2.1]. Let C ∈ coh_{R}𝒞 and I_{C} denote the set (C, − ⊗_{R} M). We have an exact sequence

(1.1)  0 ⟶ X ⟶ C ⟶^{ρ} − ⊗_{R} M^{I_{C}} .

Here we use the relation (− ⊗_{R} M)^{I_{C}} = − ⊗_{R} M^{I_{C}}. We claim that X = t(C). Indeed, if we apply the left exact 𝒮_{M}-torsion functor to (1.1), we shall obtain

0 ⟶ t(X) ⟶ t(C) ⟶ t(− ⊗_{R} M^{I_{C}}) = 0.

Therefore t(C) = t(X) ⊆ X. Let Y be a finitely generated subobject of X such that there exists a non-zero morphism φ : Y → − ⊗_{R} M. Since − ⊗_{R} M is coh-injective, φ can be extended to a non-zero morphism ψ : C → − ⊗_{R} M. But ψ factors through ρ. Hence φ = 0, a contrudiction to our assumption. Thus X = t(C).

Consider a direct limit of coherent objects C = \underrightarrow{lim}_{k∈K} C_{k}. Then the sequence

0 ⟶ \underrightarrow{lim} t(C_{k}) = t(C) ⟶ C ⟶ \underrightarrow{lim} C_{k}/t(C_{k}) = C/t(C) ⟶ 0

is exact. Given k ∈ K let J_{k} = {j ∈ K | k ⩽ j} and I_{k} = I_{C_{k}} × J_{k}. The canonically defined monomorphisms C_{k}/t(C_{k}) → − ⊗_{R} M^{I_{k}}, k ∈ K, induce a monomorphism C/t(C) → − ⊗_{R} L with L = \underrightarrow{lim}_{k∈K} M^{I_{k}} ∈ 𝒳. Thus the 𝒮_{M}-torsionfree objects of _{R}𝒞 are precisely the subobjects of the functors − ⊗_{R} L with L ∈ 𝒳.

Next, it is easy to see that (C, − ⊗_{R} L) = 0 for C ∈ 𝒮⃗_{M} and L ∈ 𝒳. Given an object F ∈ R𝒞 the relation (F, − ⊗_{R} L) = 0, for all L ∈ 𝒳, implies the relation (F/t(F), − ⊗_{R} L) = 0. Therefore F = t(F), as claimed. □

We refer to the subcategory

𝒵 = {N ∈ R Mod | − ⊗_{R} N is 𝒮_{M}-torsionfree}

as a definable subcategory.

Corollary 1.3. A left R-module belongs to 𝒵 iff it is a pure submodule of a module L from 𝒳.

In the sequel, we use the following Serre subcategories of coh_{R}𝒞:

𝒮^{R} = {C ∈ coh_{R}𝒞 | C(R) = 0}
𝒮_{R} = {C ∈ coh_{R}𝒞 | (C, − ⊗_{R} R) = 0},

as well as the localizing subcategories of finite type 𝒮⃗^{R} and 𝒮⃗_{R}

𝒮⃗^{R} = \{C ∈ _{R}𝒞 | C = \underrightarrow{lim} C_{i}, C_{i} ∈ 𝒮^{R}\}
𝒮⃗_{R} = \{C ∈ _{R}𝒞 | C = \underrightarrow{lim} C_{i}, C_{i} ∈ 𝒮_{R}\}.

The corresponding 𝒮⃗^{R}-torsion and 𝒮⃗_{R}-torsion functors will be denoted by t_{𝒮^{R}} and t_{𝒮_{R}} respectively.

2. Proper classes

Let T denote the functor that takes a left R-module M to the object (− ⊗_{R} M)_{𝒮^{R}} in R𝒞/𝒮⃗^{R}.

Lemma 2.1. The functor T is fully faithful, right exact and preserves direct limits.

Proof. Let 𝒫^{R} denote the localizing subcategory {F ∈ _{R}𝒞 | F (R) = 0} in _{R}𝒞. Then the functor M ↦ P (M) = (− ⊗_{R} M)_{𝒫^{R}} induces an equivalence between R Mod and _{R}𝒞/𝒫^{R} [9]. A quasi-inverse functor to P is induced by the functor _{R}𝒞 → Mod R taking an object F of R𝒞 to F (R). Since 𝒮⃗^{R} ⊆ 𝒫^{R} the functor P factors as P = L ∘ T, where L : _{R}𝒞/𝒮⃗^{R} → _{R}𝒞/𝒫^{R} is the localization functor with respect to the localizing subcategory 𝒫^{R}/𝒮⃗^{R}. It follows that T is faithful.

To show that T is full, it suffices to prove that the map L(μ) : LT (M) → LT (N) with 0 ≠ μ : T (M) → T (N) is non-zero. Assume the converse. Then μ(R) = 0 and, hence, D = Coker μ belongs to 𝒫^{R}.

Let λ : T (N) → LT (N) denote the 𝒫^{R}-envelope of T (N) in _{R}𝒞. Then λμ = 0 and therefore λ factors through D. Since LT (N) is 𝒫^{R}-torsionfree, wee see that λ = 0. This implies that T (N) belongs to 𝒫^{R} and thus N = 0. It follows that T (N) = 0, a contradiction.

Since both the tensor functor and the localizing functor preserve direct limits (right exact), we see that T preserves direct limits (is right exact) as well. □

A class of monomorphisms ω of R Mod is proper if it satisfies the following axioms: P 0. ω contains all split monomorphisms; P 1. the composition of two monomorphisms in ω, if defined, is also in ω; P 2. if a pushout diagram

(2.1)  \begin{matrix}A & ⟶^{σ} & B \\ ^{μ}↓ & & ↓^{μ′} \\ A′ & ⟶_{σ′} & B′ \end{matrix}

has σ ∈ ω, then σ′ ∈ ω;

P 3. if στ ∈ ω, then τ ∈ ω.

The axioms P 0′ − P 3′ for a proper class of epimorphisms are dually defined.

Proposition 2.2. The class of R-monomorphisms

ω_{f} = {μ | T μ is a monomorphism}

satisfies the axioms P 0 − P 3.

Proof. The axioms P 0 − P 1 and P 3 are obvious. If the square (2.1) is pushout, then the sequence

A ⟶^{(−μ,σ)^{T}} A′ ⨁ B ⟶^{(σ′,μ′)} B′ ⟶ 0

with (σ′, μ′) = coker(−μ, σ)^{T} is exact. By Lemma 2.1 the sequence

T A ⟶^{(−T μ,T σ)^{T}} T A′ ⨁ T B ⟶^{(T σ′,T μ′)} T B′ ⟶ 0

is exact. Thus T B′ = T A′ ∐_{T A} T B. Since T σ is a monomorphism, it follows that T σ′ is a monomorphism. This implies the claim. □

By Lemma 2.1 the proper class ω_{f} is inductively closed, i.e. given any direct system \{μ_{i}\}_{i∈I} of monomorphisms from ω_{f} the morphism \underrightarrow{lim} μ_{i} is in ω_{f}. Below we shall show that

ω_{f} = {μ | R^{I} ⊗ μ is a monomorphism for any power R^{I} of R}.

The most obvious example of a monomorphism in ω_{f} is a pure monomorphism, because it is a direct limit of split monomorphisms [22, Theorem 6.2]. We say that a module M is coinjective if any extension of M belongs to ω_{f}. As an example, every extension μ : M → E of a finitely presented module M ∈ R mod belongs to ω_{f}. Indeed, if X is a finitely generated subobject of Ker(− ⊗ μ), it is coherent and X(R) = 0. Hence Ker(− ⊗ μ) ∈ 𝒮⃗^{R}. Therefore the morphism T μ is a monomorphism, i.e. μ ∈ ω_{f}.

Proposition 2.3. A ring R is left coherent iff every left R-module is coinjective.

Proof. Let R be left coherent. Then any monomorphism μ : M → N is a direct limit μ = \underrightarrow{lim}_{I} μ_{i} of monomorphisms μ_{i} in R mod [17, Lemma 5.9]. Since each μ_{i} ∈ ω_{f}, it follows that μ = \underrightarrow{lim} μ_{i} is also in ω_{f}. Therefore M is coinjective.

Assume the converse. Since μ ∈ ω_{f} iff Ker(− ⊗ μ) ∈ 𝒮⃗^{R} and since any monomorphism belongs to ω_{f}, our assertion follows from [9, Theorem 2.5]. □

A module M ∈ R Mod is a f-submodule of N ∈ R Mod (respectively f-quotient module) if there is a monomorphism μ : M → N with μ ∈ ω_{f} (respectively if there is an epimorphism μ : N → M with ker μ ∈ ω_{f}). A short exact sequence

(2.2)  0 ⟶ M ⟶ N ⟶ L ⟶ 0

is f-exact if M is a f-submodule of N. Equivalently, if L is a f-quotient module of N. Clearly, the exact sequence (2.2) is f-exact iff the sequence

0 ⟶ T M ⟶ T N ⟶ T L ⟶ 0

is exact in _{R}𝒞/𝒮⃗^{R}.

We refer to a morphism μ : M → N as an f-epimorphism if N is a f-quotient of M. The class of f-epimorphisms will be denoted by ω^{f}. A morphism μ : M → N is an f-homomorphism if μ = στ with σ ∈ ω_{f} and τ ∈ ω^{f}.

3. f-injective and f-flat modules

In this section we study relative injective and relative flat modules for the proper class ω_{f}.

Definitions. (1) A left R-module M is said to be F P -injective (or absolutely pure) if every monomorphism μ : M → N is pure. Equivalently, for all F ∈ R mod we have: Ext_{R}^{1}(F, M) = 0 [24]. A ring R is left F P -injective if the module _{R}R is F P -injective.

(2) M is an fp-injective module if for every monomorphism μ : K → L in R mod the morphism (μ, M) is an epimorphism. Clearly, F P -injective modules are fp-injective and every finitely presented fp-injective module is F P -injective.

(3) M is said to be fp-flat if for every monomorphism μ : K → L in mod R the morphism μ ⊗ M is a monomorphism. Clearly, every flat left R-module is fp-flat. The converse holds iff the ring R is right coherent [8, Theorem 2.4].

Let the functor S take a module M ∈ R Mod to the functor SM = (− ⊗_{R} M)_{𝒮_{R}}.

Proposition 3.1. The following statements are true:

(1) a module M is fp-injective iff − ⊗_{R} M = T M;

(2) a module M is fp-flat iff − ⊗_{R} M = SM;

(3) the subcategory of fp-injective (fp-flat) modules is definible.

Moreover, every object T M (SM) with M fp-injective (fp-flat) is coh-injective in _{R}𝒞/𝒮⃗^{R} (_{R}𝒞/𝒮⃗_{R}).

Proof. By [8, Proposition 2.2] M is fp-injective (fp-flat) iff − ⊗_{R} M is 𝒮^{R}-torsionfree (𝒮_{R}- torsionfree). This implies (3). The rest of the proof is a consequence of [13, Proposition 3.10].

□

The Auslander-Gruson-Jensen duality D (see [1, 12]) takes each object C ∈ coh_{R}𝒞 to the object DC ∈ coh 𝒞_{R} defined as follows:

(DC)(_{R}N) = Hom_{_{R}𝒞} (C, − ⊗_{R} N)

and a morphism α of coh_{R}𝒞 to the morphism Dα of coh 𝒞_{R} defined by the rule:

(Dα)(_{R}N) = Hom_{_{R}𝒞} (α, − ⊗_{R} N).

Proposition 3.2. [29, Lemma 2], [13, Proposition 5.6] Let _{S}M_{R} be an (S, R)-bimodule and let _{S}E be an injective S-module. Then for each C ∈ coh 𝒞_{R} there is an isomorphism

Hom_{S}(_{S}(C, M ⊗_{R} −),_{S}E) ≃ Hom_{_{R}𝒞} (DC, − ⊗_{R} (_{S}M_{R},_{S}E))

natural in C.

Throughout the paper the character module Hom_{Z}(M, Q/Z) of a module M is denoted by M̂.

Since

D𝒮^{R} = _{R}𝒮 = {C ∈ coh 𝒞_{R} | (C, R ⊗_{R} −) = 0}

and

D𝒮_{R} = ^{R}𝒮 = {C ∈ coh 𝒞_{R} | C(R) = 0}

it follows from [8, Proposition 2.2] the following.

Corollary 3.3. A left R-module M is fp-injective (fp-flat) iff its character module M̂ is fp-flat (fp-injective).

We observe that the Serre subcategory 𝒮^{R} of coh_{R}𝒞 is cogenerated by any injective cogenerator, say R̂, of R Mod.

Proposition 3.4. A left R-module is fp-injective (fp-flat) iff it is a pure submodule of \underrightarrow{lim}_{k∈K} R̂^{I_{k}} (\underrightarrow{lim}_{k∈K} R^{I_{k}}) for some sets of indices K and I_{k}, k ∈ K.

Proof. This follows from Proposition 1.2 and Corollary 1.3. □

A left R-module Q is said to be f-injective (respectively f-projective) if the functor Hom_{R}(−, Q) (respectively Hom_{R}(Q, −)) takes the monomorphisms of ω_{f} (respectively the epimorphisms of ω^{f}) to epimorphisms.

Proposition 3.5. An R-module Q is f-injective iff it is a fp-injective pure-injective module.

Proof. Since an injective object − ⊗_{R} Q of _{R}𝒞 is injective in _{R}𝒞/𝒮⃗^{R} iff it is 𝒮^{R}-closed, Proposition 3.1 implies that the _{R}𝒞/𝒮⃗^{R}-injective objects are precisely the fp-injective pureinjective modules.

Suppose that M is f-injective; then the _{R}𝒞/𝒮⃗^{R}-monomorphism T M → − ⊗_{R} Q = T Q with − ⊗_{R} Q = E(T M) splits. Therefore, M is a pure-injective fp-injective module. The converse is easy. □

A f-monomorphism μ : M → E = E_{f}(M) with E a f-injective module is called a finjective envelope of M if any endomorphism φ of E such that φμ = μ is an isomorphism.

Lemma 3.6. A f-monomorphism μ : N → E is a f-injective envelope iff T μ : T N → T E is an injective envelope of T N in _{R}𝒞/𝒮⃗^{R}. If μ′ : N → E′ is another f-injective envelope, then there exists an isomorphism ψ : E → E′ such that ψμ = μ′.

Proof. Straightforward. □

The existence of injective envelopes in _{R}𝒞/𝒮⃗^{R} implies the following.

Corollary 3.7. A f-injective envelope of a module always exists.

So we obtain that the proper class ω_{f} is injective, that is, every module is a f-submodule of a f-injective module. The projective proper classes are dually difined.

We refer to a right R-module M as a f-flat module if the tensor functor M ⊗_{R} − preserves f-exact sequences.

Theorem 3.8. A right R-module M is f-flat iff it is fp-flat. Moreover, the class of f-flat R-modules is closed under taking products and direct limits.

Proof. Since every monomorphism μ in R mod belongs to ω_{f}, it follows that the f-flat R-modules are fp-flat R-modules.

Consider a fp-flat right R-module M and a f-monomorphism μ : K → L. We want to show that 1 ⊗ μ : M ⊗_{R} K → M ⊗_{R} L is a monomorphism. From [13, Proposition 4.3] it follows that the character module M̂ is pure-injective. On the other hand, M̂ is fp-injective by Corollary 3.3. Proposition 3.5 implies that M̂ is f-injective. We see that the map

(3.1)  Hom_{R}(L, M̂) ⟶ Hom_{R}(K, M̂)

is an epimorphism. This map is isomorphic to the map

Hom_{Z}(M ⊗_{R} L, Q/Z) ⟶ Hom_{Z}(M ⊗_{R} K, Q/Z).

Since Q/Z is an injective cogenerator in Ab, it follows that 1 ⊗ μ : M ⊗_{R} K → M ⊗_{R} L is a monomorphism.

The fact that f-flat modules are closed under products and direct limits follows from [8, Proposition 2.3]. □

Corollary 3.9. A module M is fp-flat iff its character module M̂ is an fp-injective pureinjective module. In other words, a module is f-flat iff its character module is a f-injective module.

As for the class of flat R-modules by the Chase theorem [25, Proposition I.13.3] every product of flat right R-modules is flat iff the ring R is left coherent.

The class ω_{f} is flatly generated by fp-flat modules, that is, μ ∈ ω_{f} iff M ⊗ μ is a monomorphism for any fp-flat module M. Indeed, if μ ∈ ω_{f} and M is fp-flat, Theorem 3.8 implies that M ⊗ μ is a monomorphism.

On the other hand, if M ⊗ μ with μ : K → L is a monomorphism for each fp-flat module M, the map (3.1) is an epimorphism. In particular, it is an epimorphism for M = Q̂ with Q a f-injective module and, hence, Hom_{_{R}𝒞/𝒮⃗^{R}} (Ker T μ, − ⊗_{R} Q ) = 0. Since Q is a pure submodule of Q , it follows that Hom_{_{R}𝒞/𝒮⃗^{R}} (Ker T μ, − ⊗_{R} Q) = 0. As f-injectives cogenerate _{R}𝒞/𝒮⃗^{R} this implies Ker T μ = 0, and, hence, μ ∈ ω_{f}.

Proposition 3.10. ω_{f} is flatly generated by modules R^{I} with I a set of indices, R^{I} the product of I copies of R.

Proof. Let F be a fp-flat module. By Proposition 3.4 it is a pure submodule of \underrightarrow{lim}_{k∈K} R^{I_{k}} with K, I_{k}, k ∈ K, some sets of indices. Let 0 → M →^{μ} N → L → 0 be an exact sequence and the map R^{I} ⊗ μ is a monomorphism for any set I. We have then the following commutative diagram with exact rows:

F ⊗_{R} M  ⟶^{F ⊗μ} F ⊗_{R} N  ⟶ F ⊗_{R} L ⟶ 0
↓  ↓  ↓
0 ⟶ \underrightarrow{lim}_{k} R^{I_{k}} ⊗_{R} M ⟶ \underrightarrow{lim}_{k} R^{I_{k}} ⊗_{R} N ⟶ \underrightarrow{lim}_{k} R^{I_{k}} ⊗_{R} L ⟶ 0.

Since the vertical arrows are monomorphisms, F ⊗ μ is a monomorphism. Thus μ ∈ ω_{f}, as claimed. □

Below we discuss some properties of fp-injective and fp-flat modules we shall use later.

Proposition 3.11. For a left R-module M the following statements are equivalent: (1) M is fp-injective;

(2) any f-exact sequence 0 → M → M′ → M′′ → 0 is pure;

(3) there is a pure-exact sequence 0 → M → M′ → M′′ → 0 with M′ a fp-injective module;

(4) Ext_{_{R}𝒞/𝒮⃗^{R}}^{1} (T F, T M) = 0 for all F ∈ R mod.

Moreover, any product and any direct limit of fp-injective modules is also a fp-injective module.

Proof. (1) ⟹ (2). Let F ∈ R mod then T F ∈ coh_{R}𝒞/𝒮⃗^{R}. Since T M is coh-injective in R𝒞/𝒮⃗^{R} by Proposition 3.1, it follows that the sequence

0 ⟶ (T F, T M) ⟶ (T F, T M′) ⟶ (T F, T M′′) ⟶ 0

is exact. By Lemma 2.1 the sequence

0 ⟶ (F, M) ⟶ (F, M′) ⟶ (F, M′′) ⟶ 0

is exact. So, the sequence 0 → M → M′ → M′′ → 0 is pure-exact.

(2) ⟹ (3). We may take M′ to be a f-injective envelope of M.

(3) ⟹ (4). Since − ⊗_{R} M is a subobject of the 𝒮^{R}-torsionfree object − ⊗_{R} M′, it is also 𝒮^{R}-torsionfree. From [13, Proposition 3.10] it follows that T M = − ⊗_{R} M is a coh-injective object for _{R}𝒞/𝒮⃗^{R}.

(4) ⟹ (1). Let μ be a monomorphism in R mod; then T μ is a monomorphism. By assumption, (T μ, T M) is an epimorphism. Since T is full and faithful by Lemma 2.1, the morphism (μ, M) is a monomorphism.

The fact that the class of fp-injective modules is closed under taking products and direct limits follows from [8, Proposition 2.3]. □

It is well-known that a direct limit of F P -injective left R-modules is F P -injective iff the ring R is left coherent (see, e.g., [24]).

Corollary 3.12. If in a short f-exact sequence 0 → M → M′ → M′′ → 0 modules M and M′ are fp-injective, then also M′′ is fp-injective.

Proof. By the preceding proposition the sequence of the corollary is pure-exact. If we apply the right exact 𝒮^{R}-torsion functor t = t_{𝒮^{R}} to the _{R}𝒞-exact sequence

0 ⟶ − ⊗_{R} M ⟶ − ⊗_{R} M′ ⟶ − ⊗_{R} M′′ ⟶ 0

we shall get an exact sequence

0 = t(− ⊗_{R} M′) ⟶ t(− ⊗_{R} M′′) ⟶ t^{1}(− ⊗_{R} M) = 0.

Hence M′′ is fp-injective by [8, Proposition 2.2]. □

The following two propositions extend the list of properties characterizing the coherent and Noetherian rings respectively (cf. [23]).

Proposition 3.13. The following assertions are equivalent for a ring R:

(1) R is left coherent;

(2) the character module of a fp-injective left module is a flat module;

(3) a left module is fp-injective iff its character module is a flat module;

(4) a pure-injective envelope of a right f-flat module is a flat module.

Proof. (1) ⟹ (2). Over a left coherent ring every fp-flat right module is flat [8, Theorem 2.4]. Now our implication follows from Corollary 3.3.

(2) ⟹ (3). This follows from Corollary 3.3.

(3) ⟹ (1). Suppose that the character module of an fp-injective left module M is flat. Then M is a pure submodule of the injective module M . Hence M is a F P -injective module. Now our assertion follows from [8, Theorem 2.4].

(1) ⟹ (4). A pure-injective envelope Q of a f-flat module M is always a f-flat module. Indeed, the 𝒞_{R}-injective functor Q ⊗_{R} − is an injective invelope of M ⊗_{R} − in 𝒞_{R}. Since M ⊗_{R} − is 𝒮_{R}-torsionfree, it follows that Q ⊗_{R} − is 𝒮_{R}-torsionfree, too. Proposition 3.1 implies that Q is f-flat. By assumption, R is left coherent and therefore Q is flat by [8, Theorem 2.4].

(4) ⟹ (1). It is directly verified that given a pure-exact sequence

0 ⟶ M ⟶ Q ⟶ N ⟶ 0

with Q flat, the module M is also flat.

Suppose that a right R-module M is f-flat. Then its pure-injective envelope Q is flat by assumption. Hence M is flat by the above. Therefore R is left coherent by [8, Theorem 2.4].

□

Remark. By a recent result of Rothmaler [21] there are non-coherent rings over which pure-injective envelopes of flat modules are flat.

Proposition 3.14. The following assertions are equivalent for a ring R:

(1) R is left Noetherian;

(2) every left fp-injective module is an injective module;

(3) a left module is f-injective iff its character module is a flat module.

Proof. (1) ⟹ (2). Since every left ideal I of R is finitely presented, then the representable functor Hom_{R}(−, M) takes the inclusion I ⊂ R to an epimorphism whenever M is fpinjective. Therefore M is an injective module by the Baer criterion.

(2) ⟹ (1). It suffices to observe that every direct limit of fp-injective modules is a fp-injective module.

(1), (2) ⟹ (3). This implication follows from Proposition 3.13.

(3) ⟹ (2). Let E be a f-injective envelope of a fp-injective module M. By Proposition 3.11 M is a pure submodule of E. Therefore M̂ is a direct summand of the flat module Ê. Then M is f-injective by assumption. By the preceding proposition the ring R is left coherent. Now our assertion follows from the fact that over the left coherent rings the classes of f-injective and injective left modules coincide (see [8, Theorem 2.4]). □

We conclude the section by characterizing the class of F P -injective rings. We give the most interesting criteria for this class of rings in terms of f-flat and f-injective modules. For details and proofs we refer the reader to [5, 8].

We recall that M ∈ R Mod is a F P -cogenerator if for any non-zero homomorphism ρ : K → L with K a finitely generated R-module and L a finitely presented module there exists a homomorphism μ : L → M such that μρ ≠ 0. Equivalently, every finitely presented left R-module is a submodule of a product M^{I} of some copies of the module M [5].

Lemma 3.15. M is a F P -cogenerator iff for any non-zero morphism ρ : K → L in R mod there exists a morphism μ : L → M such that μρ ≠ 0.

Proof. The necessary condition is trivial. Assume the converse. Let ρ : K → L be a non-zero homomorphism from a finitely generated R-module K to a finitely presented module L. Then there exists an epimorphism π : R^{n} → K. By assumption, there exists a homomorphism μ : L → M such that μ(ρπ) = (μρ)π ≠ 0. Hence μρ ≠ 0. □

Theorem 3.16. [5, 8] For a ring R the following conditions are equivalent:

(1) the module R_{R} is F P -injective;

(2) the module _{R}R is a F P -cogenerator;

(3) there is a f-flat cogenerator in R Mod;

(4) every left R-module is a (f-)submodule of a f-flat module;

(5) every fp-injective left R-module is a f-flat module;

(6) every f-injective left R-module is a f-flat module;

(7) every f-flat right R-module is a fp-injective module;

(8) every pure-injective f-flat right R-module is an f-injective module.

Remark. The most difficult statements for F P -injective rings are proved inside of the category _{R}𝒞 and use localization theory in _{R}𝒞. It would be interesting to have a proof (especially of the equivalence (1) ⟺ (4)) by using usual module-theoretic technique. The author does not know such a proof.

Rings over which any module is embedded into a projective module are quasi-Frobenius. This is the classical Faith-Walker theorem. Rings over which any module is embedded into a flat module, the IF -rings, were investigated in the early 70-s (see, e.g. [4, 14, 24]). The equivalent statements of the preceding theorem are, in a certain sense, similar to properties both for QF-rings and for IF -rings. We have thus the following proper inclusions:

\{ \begin{matrix}QF-rings \\ modules are submodules \\ of projective modules \end{matrix} \} ⊊ \{ \begin{matrix}IF-rings \\ modules are submodules \\ of flat modules \end{matrix} \} ⊊ \{ \begin{matrix}FP-injective rings \\ modules are submodules \\ of f-flat modules \end{matrix} \} .

The class of F P -injective rings is big enough. For example, given an arbitrary left selfinjective ring R and a locally finite group G, the group ring R(G) is left F P -injective [5, Theorem 3.2]. Moreover, if |G| = ∞, the group ring R(G) is left F P -injective but not left self-injective [5, Corollary 3.3].

4. The functors Ext_{f} and Tor^{f}

Definitions. Let N be a left R-module.

(1) The f-injective dimension inj. dim_{f} N is the minimum integer n (if it exists) such that there is a f-exact sequence

0 → N → E^{0} → E^{1} → ⋯ → E^{n} → 0

with E^{0}, ⋯, E^{n} being f-injective modules. We call such a sequence a f-resolution of N by f-injective modules.

(2) The left global f-dimension l. gl. dim_{f} R of a ring R is sup{inj. dim_{f} N | N ∈ R Mod}.

(3) The fp-injective dimension fp-inj. dim M is the minimum integer n (if it exists) such that there is a f-resolution

0 → N → E^{0} → E^{1} → ⋯ → E^{n} → 0

of N by fp-injective modules E^{0}, ⋯, E^{n}.

If no finite resolution exists, we set inj. dim_{f} N, fp-inj. dim N equal to ∞.

(4) The left global fp-dimension l. gl. fp-dim R of a ring R is sup{fp-inj. dim N | N ∈ R Mod}.

(5) If M ∈ R Mod, let Ext_{f}^{n}(M, N) denote Ext_{_{R}𝒞/𝒮⃗^{R}}^{n} (T M, T N).

The abelian groups Ext_{f}^{n}(M, N) can be computed with the use of f-injective f-resolutions of N. Namely, let

0 → N → E^{0} → E^{1} → ⋯

be a f-injective f-resolution for N. Then

0 → T N → T E^{0} → T E^{1} → ⋯

is an injective resolution in _{R}𝒞/𝒮⃗^{R} for T N by T E^{i} = − ⊗_{R} E^{i} and the cohomology groups for the complex

0 → Hom_{R}(M, E^{0}) → Hom_{R}(M, E^{1}) → ⋯

are those of the complex

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

Any injective resolution for T N in _{R}𝒞/𝒮⃗^{R} gives an f-injective f-resolution for N. In a similar way, any f-resolution for N by fp-injective modules determines a resolution for T N by coh-injective objects in _{R}𝒞/𝒮⃗^{R}, and vica versa. We also note that any f-exact sequence

0 ⟶ L ⟶ M ⟶ N ⟶ 0

yields a long exact sequence for any F ∈ R Mod ⋯ → Ext_{f}^{n−1}(F, N) → Ext_{f}^{n}(F, L) → Ext_{f}^{n}(F, M) → Ext_{f}^{n}(F, N) → ⋯

Lemma 4.1. The following are equivalent for a left R-module N:

(1) inj. dim_{f} N ⩽ n;

(2) Ext_{f}^{p}(M, N) = 0 for all p > n and all R-modules M;

(3) Ext_{f}^{n+1}(M, N) = 0 for all R-modules M;

(4) if 0 → N → E^{0} → ⋯ → E^{n−1} → L^{n} → 0 is a f-resolution of N by f-injective modules E^{i}, then L^{n} is also f-injective.

Proof. Easy. □

Corollary 4.2. l. gl. dim_{f} R = sup{n | Ext_{f}^{n}(M, N) ≠ 0 for some R-modules M and N}.

An acyclic complex M^{∗} = (M^{n}, d^{n}) is said to be f-acyclic if each d^{n}, n ∈ Z, is an f-homomorphism. Similarly to the absolute case, the elements of Ext_{f}^{n}(M, N), n ⩾ 1, are represented by f-acyclic complexes of the form:

0 → N → M^{n−1} → M^{n−2} → ⋯ → M^{0} → M → 0.

So, the groups Ext_{f}^{n}(M, N) can be defined alternatively using “Baer addition” on the set of equivalence classes of such complexes (see [11]).

The proof of the following lemma is straightforward (cf. [24, Lemma 3.1]).

Lemma 4.3. The following are equivalent for a left R-module N:

(1) fp-inj. dim N ⩽ n;

(2) Ext_{f}^{p}(M, N) = 0 for all p > n and all finitely presented R-modules M;

(3) Ext_{f}^{n+1}(M, N) = 0 for all finitely presented R-modules M;

(4) if 0 → N → E^{0} → ⋯ → E^{n−1} → L^{n} → 0 is a f-resolution of N by fp-injective modules E^{i}, then L^{n} is also fp-injective.

It is well-known that a ring R is left Noetherian iff every direct limit of left modules of injective dimension ⩽ n has injective dimension ⩽ n. In turn, let FP-inj. dim N denote the smallest integer n ⩾ 0 such that there is a resolution

0 → N → E^{0} → E^{1} → ⋯ → E^{n} → 0

of N by F P -injective modules E^{i}. Then R is left coherent iff every direct limit of left modules of FP-injective dimension ⩽ n has FP-injective dimension ⩽ n [24, Theorem 3.2].

Corollary 4.4. Let R be an arbitrary ring. Then every direct limit of left modules of fp-injective dimension ⩽ n has fp-injective dimension ⩽ n.

Proof. Let {M_{i}}_{I} be a direct system of left modules of fp-injective dimension ⩽ n and let

0 → M_{i} → E_{i}^{0} → E_{i}^{1} → ⋯

be the f-injective f-resolution for M_{i}, i ∈ I, constructed below. Choose the direct system of f-injective modules E_{i}^{0} ⊇ M_{i} in the following way. Let Q be an f-injective cogenerator. Such a module exists: we can take an injective cogenerator − ⊗_{R} Q in _{R}𝒞/𝒮⃗^{R} and then Q is a f-injective cogenerator. We set E_{i}^{0} equal to Q^{Hom(M_{i},Q)}. The direct system (M_{i})_{I} is then extended to a direct system (E_{i}^{0})_{I}. The modules E_{i}^{k⩾1} are similarly constructed.

By Lemma 4.3

0 → M_{i} → E_{i}^{0} → ⋯ → E_{i}^{n−1} → L_{i}^{n} → 0

is a f-resolution of M_{i} by fp-injective modules. Since every direct limit of fp-injective modules is fp-injective, M has a f-resolution

0 → M → \underrightarrow{lim} E_{i}^{0} → ⋯ → \underrightarrow{lim} E_{i}^{n−1} → \underrightarrow{lim} L_{i}^{n} → 0

by fp-injective modules. Hence fp-inj. dim M ⩽ n. □

For a left finitely presented module F and a direct system {M_{i}}_{I} of left modules, we consider the canonical homomorphism

ξ_{n} : \underrightarrow{lim}_{I} Ext_{f}^{n}(F, M_{i}) ⟶ Ext_{f}^{n}(F, \underrightarrow{lim}_{I} M_{i}).

Recall that finite presentation of F is equivalent to ξ_{0} being an isomorphism for every {M_{i}}_{I} [25, Proposition V.3.4].

Theorem 4.5. ξ_{n} : \underrightarrow{lim}_{I} Ext_{f}^{n}(F, M_{i}) ⟶ Ext_{f}^{n}(F, \underrightarrow{lim}_{I} M_{i}) are isomorphisms for all n ⩾ 0, for every finitely presented module F and direct system {M_{i}}_{I}.

Proof. ξ_{0} is an isomorphim. First we shall show that ξ_{1} is an isomorphism. Let us consider the f-injective f-resolution

0 → M_{i} → E_{i}^{0} →^{d_{i}^{0}} E_{i}^{1} →^{d_{i}^{1}} ⋯

for each M_{i} constructed above.

Let L_{i}^{0} = E_{i}^{0}/M_{i}. Since the module \underrightarrow{lim}_{I} E_{i}^{0} is fp-injective, it follows that Ext_{f}^{1}(F, \underrightarrow{lim}_{I} E_{i}^{0}) = 0. We get then the following commutative diagram with exact rows:

\underrightarrow{lim}(F, E_{i}^{0}) ⟶ \underrightarrow{lim}(F, L_{i}^{0}) ⟶ \underrightarrow{lim} Ext_{f}^{1}(F, M_{i}) ⟶ 0
ξ_{0}↓  ↓^{ξ_{0}}  ↓^{ξ_{1}}
(F, \underrightarrow{lim} E_{i}^{0}) ⟶ (F, \underrightarrow{lim} L_{i}^{0}) ⟶ Ext_{f}^{1}(F, \underrightarrow{lim} M_{i}) ⟶ 0.

Since the first two arrows of the diagram are isomorphisms, ξ_{1} is an isomorphism as well.

Let L_{i}^{n} = Im d_{i}^{n}. Then Ext_{f}^{n}(F, M_{i}) = Ext_{f}^{1}(F, L_{i}^{n−2}), n ⩾ 2. The sequence

0 → \underrightarrow{lim} M_{i} → \underrightarrow{lim} E_{i}^{0} → \underrightarrow{lim} E_{i}^{1} → ⋯

is a f-resolution of \underrightarrow{lim} M_{i} by fp-injective modules. Then

\underrightarrow{lim} Ext_{f}^{n}(F, M_{i}) = \underrightarrow{lim} Ext_{f}^{1}(F, L_{i}^{n−2}) →^{ξ_{1}} Ext_{f}^{1}(F, \underrightarrow{lim} L_{i}^{n−2}) = Ext_{f}^{n}(F, \underrightarrow{lim} M_{i}).

Since ξ_{1} is an isomorphism, it follows that ξ_{n} is an isomorphism. □

Stenström [24] showed that ξ_{n} : \underrightarrow{lim}_{I} Ext_{R}^{n}(F, M_{i}) ⟶ Ext_{R}^{n}(F, \underrightarrow{lim}_{I} M_{i}) are isomorphisms for all n ⩾ 0, for every finitely presented module F and direct system {M_{i}}_{I} iff the ring R is left coherent.

Corollary 4.6. Let M be a left module and M = \underrightarrow{lim}_{I} M_{i} with M_{i} ∈ R mod. Then Ext_{f}^{n}(F, M) = \underrightarrow{lim}_{I} Ext_{f}^{n}(F, M_{i}) for any finitely presented left module F.

A f-resolution of a left module M by fp-injective modules can be constructed as follows. Let M = \underrightarrow{lim}_{I} M_{i} be written as a direct limit of finitely presented modules M_{i}. Choose for each i ∈ I the resolution M_{i} → E_{i}^{∗} of M_{i} by the f-injective modules E_{i}^{∗} we have constructed above. Then M → \underrightarrow{lim}_{I} E_{i}^{∗} is a f-resolution of M by fp-injective modules.

Corollary 4.7. The following numbers are the same for any ring R:

(1) l. gl. fp-dim R;

(2) sup{n | Ext_{f}^{n}(F, M) ≠ 0 for some F ∈ R mod and M ∈ R Mod};

(3) sup{fp-inj. dim M | M ∈ R mod}.

Proof. Obviously, l. gl. fp-dim R = sup(2) and sup(2) ⩾ sup(3). The preceding corollary implies sup(3) ⩾ sup(2). □

A ring R is said to be almost regular if every (both left and right) module is fp-injective; equivalently, f-flat (see [8]). Since every (both left and right) module is fp-injective, every f-exact sequence is pure by Proposition 3.11. An almost regular ring is (von Neumann) regular iff it is left or right coherent [8]. Let l. gl. FP-dim R denote sup{FP-inj. dim M | M ∈ R Mod}.

Obviously, l. gl. FP-dim R = l. gl. fp-dim R when R is left coherent. Also, l. gl. fp-dim R = l. gl. dim R when R is left Noetherian. By [24, Proposition 3.6] l. gl. FP-dim R = 0 iff R is (von Neumann) regular. In turn, it follows that l. gl. fp-dim R = r. gl. fp-dim R = 0 iff R is almost regular. The class of non-regular almost regular rings is big enough. Simple almost regular rings we call indiscrete rings [19, 8]. Given a non-regular finite dimensional algebra of finite representation type one can construct a non-regular indiscrete ring as a twisted limit of matrix rings [19, §2.4]. In turn, if R is a non-regular almost regular ring, G a non-trivial locally finite group and the order |H| of every finite subgroup H of G is invertible in R, then the group ring R(G) is a non-regular non-indiscrete almost regular ring [7].

There is an interesting problem in our context. A f-flat precover of a right module M is a f-epimorphism φ : F → M with F a f-flat module such that the induced map φ^{∗} : Hom_{R}(F′, F) → Hom_{R}(F′, M) is an epimorphism for any f-flat module F′. We refer to φ as a f-cover if for every endomorphism ψ : F → F the relation φψ = φ implies ψ is an isomorphism.

Question. Every right R-module has a f-flat (pre-)cover.

Rings over which every module has a f-flat cover exist. For example, over an almost regular ring every module is f-flat. This is the easiest case. In the absolute case, every module has a flat cover. This has recently been proved by Bican, El Bashir and Enochs in [2].

Proposition 4.8. For a ring R the following conditions are equivalent:

(1) l. gl. fp-dim R ⩽ 1;

(2) every f-quotient module of a f-injective module is fp-injective;

(3) every f-quotient module of a fp-injective module is fp-injective;

(4) if in a short exact sequence 0 → M → E → N → 0 with M a finitely presented module a module E is (F P -)injective, then N is fp-injective.

Proof. Apply Corollary 4.7. □

A ring R is left f-semihereditary if it satisfies the equivalent conditions of the preceding proposition.

Corollary 4.9. A left f-semihereditary ring R is left semihereditary iff it is left coherent.

Proof. Any left semihereditary ring is left coherent. Conversely, over a left coherent ring the functors Ext_{R}^{∗} and Ext_{f}^{∗} coincide. Now our assertion follows from [23, Proposition 1.22]. □

If M = \underrightarrow{lim}_{α∈I} M_{α} with M_{α} finitely presented modules, then there exists a spectral sequence for each module N

E_{2}^{pq} = \underleftarrow{lim}_{I}^{p} Ext_{R}^{q}(M_{α}, N) ⟹ Ext_{R}^{n}(M, N).

In [15, Théorème 4.2], the spectral sequence is constructed from a double complex. Precisely, we construct the pure-exact resolution of M

⋯ ⟶^{∂_{3}} R_{2} ⟶^{∂_{2}} R_{1} ⟶^{∂_{1}} R_{0} ⟶^{∂_{0}} M,

where

R_{n} = ⨁_{α_{0}⩽α_{1}⩽···⩽α_{n}} M_{α_{0},α_{1},··· ,α_{n}}

and M_{α_{0},α_{1},··· ,α_{n}} is a copy of M_{α_{0}}.

Let

N ⟶ E^{0} ⟶^{∂^{0}} E^{1} ⟶^{∂^{1}} E^{2} ⟶^{∂^{2}} ⋯

be an injective resolution of N. Then the bicomplex is

E_{0}^{pq} = Hom_{R}(R_{p}, E^{q}),

where the vertical and horizontal differentials are

d_{0}^{pq} = (−1)^{p} Hom_{R}(R_{p}, ∂^{q}) : E_{0}^{pq} ⟶ E_{0}^{p,q+1}

and

d_{1}^{pq} = (−1)^{p} Hom_{R}(∂_{p+1}, E^{q}) : E_{0}^{pq} ⟶ E_{0}^{p+1,q},

respectively.

If we replace the injective resolution N → E^{∗} by an f-injective f-resolution of N and also apply Corollary 4.6, we shall get the following result.

Theorem 4.10. Let M and N be two modules, M = \underrightarrow{lim}_{α} M_{α} and N = \underrightarrow{lim}_{β} N_{β} with M_{α} and N_{β} finitely presented modules. Then the following relation is valid:

\underleftarrow{lim}_{I}^{p} \underrightarrow{lim}_{J} Ext_{f}^{q}(M_{α}, N_{β}) = \underleftarrow{lim}_{I}^{p} Ext_{f}^{q}(M_{α}, N) ⟹ Ext_{f}^{n}(M, N).

Suppose M has a f-resolution by f-flat modules F_{∗} → M. Given a left module N we put

Tor_{∗}^{f}(M, N) = H_{∗}(F_{∗} ⊗_{R} N).

Any f-exact sequence of left R-modules

0 ⟶ L ⟶ M ⟶ N ⟶ 0

yields a long exact sequence for any G ∈ Mod R having a f-flat f-resolution ⋯ → Tor_{n+1}^{f}(G, N) → Tor_{n}^{f}(G, L) → Tor_{n}^{f}(G, M) → Tor_{n}^{f}(G, N) → ⋯

Remark. The definition of the functor Tor^{f} depends on the choice of a f-flat f-resolution. In the absolute case, the groups Tor^{R} do not depend on the choice of a flat resolution (see e.g. [28]) since every module admits a projective resolution. It is not clear, however, whether the class ω_{f} is projective. The case of almost regular rings is trivial, because Tor_{n}^{f} ≡ 0 for all n ⩾ 1.

We consider the duality homomorphisms

ρ : Ext_{R}^{n}(M, N̂) ⟶ Tor_{n}^{R}(M, N)

and

σ : Tor_{n}^{R}(N̂, F) ⟶ Ext_{R}^{n}(F, N)

where M ∈ Mod R, F, N ∈ R Mod. The first homomorphism is always an isomorphism [3, Proposition VI.5.1] whereas σ is an isomorphism for every finitely generated module F whenever R is a left Noetherian [3, Proposition VI.5.3] or for every finitely presented module F whenever R is a left coherent [23, Lemma 1.12].

It is not clear, however, whether a homomorphism μ̂ is a f-homomorphism if μ is a fhomomorphism. Therefore we specify some classes of modules to construct the analogous duality homomorphisms corresponding to the functors Ext_{f} and Tor^{f}. Precisely, let ℱ(R) (ℱ(R^{op})) denote the subcategory of right (left) modules M that have a f-flat f-resolution F_{∗} → M such that M̂ → F̂^{∗} is a f-injective f-resolution for the character module M̂. The class ℱ(R) is non-empty, because every fp-flat module belongs to ℱ(R). In a similar way, ℐ(R^{op}) (ℐ(R)) is the subcategory of left (right) modules M that have f-injective f-resolution M → E^{∗} such that Ê_{∗} → M̂ is a f-flat f-resolution for the character module M̂. Obviously the fp-injective left modules belong to ℐ(R^{op}).

Proposition 4.11. For a right F P -injective ring R and M ∈ ℐ(R^{op}) the following are equivalent:

(1) fp-inj. dim M = 0;

(2) fp-inj. dim M ⩽ n.

Proof. The implication (1) ⟹ (2) is trivial. Let us show (2) ⟹ (1). Let M → E^{∗} be a f-injective f-resolution for M such that Ê_{∗} → M̂ is a f-flat f-resolution for M̂. Then the module K^{n} = Im d^{n−1} in the exact sequence

0 → M → E^{0} → ⋯ → E^{n−1} →^{d^{n−1}} K^{n} → 0

is fp-injective. It is enough to show that L^{n−1} = Ker d^{n−1} is fp-injective. By assumption, the sequence

0 → K̂^{n} → Ê^{n−1} → ⋯ → Ê^{0} → M̂ → 0

is a f-flat f-resolution of M̂. By Theorem 3.16 all the E^{i}, i < n, and the K^{n} are f-flat modules. Therefore K̂^{n} is a direct summand of Ê^{n−1} by Corollary 3.9. Hence L̂^{n−1} is a direct summand of Ê^{n−1} as well. This implies L̂^{n−1} is f-flat, and L^{n−1} is fp-injective. □

Given modules M ∈ Mod R and N ∈ ℱ(R^{op}) we consider the duality homomorphism

ρ : Ext_{f}^{n}(M, N̂) ⟶ Tor_{n}^{f}(M, N) .

Similar to the absolute case [3, Proposition VI.5.1] ρ is an isomorphism. It follows that given two f-flat f-resolutions for N we obtain the same functors Tor_{n}^{f}(−, N) . This follows from the fact that the functors Ext_{f}^{n}(−, N̂) do not depend on the choice of a f-injective f-resolution for N̂ and that ρ is an isomorphism.

Let F ∈ R Mod and N ∈ ℐ(R^{op}). There is a f-injective f-resolution N → E^{∗} for N such that Ê_{∗} → N̂ is a f-flat f-resolution for the character module N̂. In this case we consider the groups Tor_{n}^{f}(N̂, F) relative to this resolution. There is a duality homomorphism

σ : Tor_{n}^{f}(N̂, F) ⟶ Ext_{f}^{n}(F, N) .

If F is finitely presented, then each natural homomorphism Ê^{i} ⊗_{R} F → Hom_{R}(F, E^{i}) with E^{i} the ith component of the complex E^{∗}, i ⩾ 0, is an isomorphism. We see that σ is an isomorphism whenever F is finitely presented.

Now let F be a finitely generated left module without finite presentations, 0 → H → P →^{φ} F → 0 an exact sequence with P a finitely generated free module. Then F = \underrightarrow{lim} F_{α} where F_{α} = P/H_{α}, H_{α} are finitely generated submodules of H. Moreover, φ = \underrightarrow{lim} φ_{α} where φ_{α} are epimorphisms P → F_{α}. Let G denote the kernel Ker T φ, G_{α} the kernel Ker T φ_{α} and D_{α} = G/G_{α}.

Lemma 4.12. If a left module F is finitely generated without finite presentations and N is a f-injective module, then the kernel of the natural homomorphism N̂ ⊗_{R} F → Hom_{R}(F, N) is the abelian group \underrightarrow{lim} Hom_{_{R}𝒞/𝒮⃗^{R}} (D_{α}, T N) = \underrightarrow{lim} Hom_{_{R}𝒞/𝒮⃗^{R}} (D_{α}, − ⊗_{R} N) .

Proof. Indeed, for each α we have an exact sequence 0 → D_{α} → T F_{α} → T F → 0. If N is a f-injective module, then the sequence

0 ⟶ (D_{α}, T N) ⟶ (T F_{α}, T N) ⟶ (T F, T N) ⟶ 0

is exact. Since the functor T is fully faithful and the functor of direct limit is exact, we get the exact sequence

0 ⟶ \underrightarrow{lim}(D_{α}, T N) ⟶ \underrightarrow{lim} Hom_{R}(F_{α}, N) ⟶ Hom_{R}(F, N) ⟶ 0.

All the modules F_{α} are finitely presented, and so all the groups Hom_{R}(F_{α}, N) are isomorphic to N̂ ⊗_{R} F_{α}. It remains to observe that \underrightarrow{lim} N̂ ⊗_{R} F_{α} = N̂ ⊗_{R} F. □

Theorem 4.13 (The Sklyarenko exact sequence). If F is a finitely generated left module, then for any module N ∈ ℐ(R^{op}) the duality homomorphism σ fits into the exact sequence

⋯ →^{σ} Ext_{f}^{n+1}(F, N) →^{δ} \underrightarrow{lim} Ext_{_{R}𝒞/𝒮⃗^{R}}^{n} (D_{α}, T N) → Tor_{n}^{f}(N̂, F) →

→^{σ} Ext_{f}^{n}(F, N) → ⋯ →^{δ} \underrightarrow{lim} Hom_{_{R}𝒞/𝒮⃗^{R}} (D_{α}, T N) → N̂ ⊗_{R} F →^{σ} Hom_{R}(F, N) → 0.

Proof. Let N → Y^{∗} be a f-injective f-resolution of N. By the preceding lemma the sequence of complexes

(4.1)  0 ⟶ \underrightarrow{lim}(D_{α}, T Y^{∗}) ⟶ Ŷ_{∗} ⊗_{R} F ⟶ Hom_{R}(F, Y^{∗}) ⟶ 0

is exact. The sequence of the theorem is the homological sequence that corresponds to (4.1). Indeed, the homology groups of the complex Hom_{R}(F, Y^{∗}) are obviously the groups Ext_{f}^{∗}(F, N) , the homology groups of the complex Ŷ_{∗} ⊗_{R} F are Tor_{∗}^{f}(N̂, F), because Ŷ_{∗} is a f-flat fresolution of N̂, by assumption. Finally,

H_{n}(\underrightarrow{lim}(D_{α}, T Y^{∗}) ) = \underrightarrow{lim} H_{n}((D_{α}, T Y^{∗}) ) =
= \underrightarrow{lim} H^{n}((D_{α}, T Y^{∗})) = \underrightarrow{lim} Ext_{_{R}𝒞/𝒮⃗^{R}}^{n} (D_{α}, T N)

because the homology functor H_{n} commutes with the exact functors \underrightarrow{lim} and X → X̂. □

Now we suppose that R is a left coherent ring, and let F be a finitely generated left module and 0 → G → P → F → 0 an exact sequence with P a finitely generated free module. Then F = \underrightarrow{lim} F_{α} where F_{α} = P/G_{α} and G_{α} are finitely presented submodules of P. As above, let D_{α} denote G/G_{α}.

Corollary 4.14. [23] If R is a left coherent ring, F is a finitely generated left module, then for any module N ∈ R Mod the duality homomorphism σ fits into an exact sequence

⋯ →^{σ} Ext_{R}^{n+1}(F, N) →^{δ} \underrightarrow{lim} Ext_{R}^{n}(D_{α}, N) → Tor_{n}^{R}(N̂, F) →

→^{σ} Ext_{R}^{n}(F, N) → ⋯ →^{δ} \underrightarrow{lim} Hom_{R}(D_{α}, N) → N̂ ⊗_{R} F →^{σ} Hom_{R}(F, N) → 0.

Proof. Since over a left coherent ring the character module of any injective module is flat, the proof is similar to that of Theorem 4.13. □

5. The derived categories D_{f}(R) and D(_{R}𝒞/𝒮⃗^{R})

In this section we use Generalov’s construction of the relative derived category (see [11]) to describe the derived category of the locally coherent Grothendieck category _{R}𝒞/𝒮⃗^{R}.

Let K(R) and K(_{R}𝒞/𝒮⃗^{R}) denote the corresponding homotopy categories of R Mod and R𝒞/𝒮⃗^{R}, i.e. the quotient categories of Kom(R) and Kom(_{R}𝒞/𝒮⃗^{R}) respectively modulo homotopy equivalence. The functor T induces the fully faithful embeddings

Kom(R) ⟶ Kom(_{R}𝒞/𝒮⃗^{R})

and

K(R) ⟶ K(_{R}𝒞/𝒮⃗^{R})

that take a complex M^{∗} = (M^{n}, d^{n}) to T M^{∗} = (T M^{n}, T d^{n}). We recall that a mapping cone of a morphism μ : M^{∗} → N^{∗} in Kom(R) is defined as the complex

C(μ) = (M^{n+1} ⨁ N^{n}, d_{C(μ)}), d_{C(μ)=} (\begin{matrix}−d_{M} & 0 \\ μ & d_{N} \end{matrix})

We have the following sequence of complexes

M ⟶^{μ} N ⟶^{ν} C(μ) ⟶^{ρ} M[1]

where (M[1])^{n} = M^{n+1}, ν^{n} : N^{n} → C(μ)^{n} and ρ^{n} : C(μ)^{n} → M^{n+1} are the canonical injection and projection respectively. The family of such sequences (up to isomorphism the so-called “distinguish triangles”) defines the structure of a triangulated category on the homotopy category K(R) (see [27, 10, 28]).

A morphism μ : X^{∗} → Y^{∗} in K(R) is called a f-quasi-isomorphism if its mapping cone C(μ) is f-acyclic. Obviously, if μ is a f-quasi-isomorphism then T μ is a quasi-isomorphism in K(_{R}𝒞/𝒮⃗^{R}). We denote by 𝒮_{f} the class of all f-quasi-isomorphisms in K(R).

Theorem 5.1 (Generalov [11]). The class of f-quasi-isomorphisms is localising in K(R).

We can thus construct the localization of K(R) with respect to 𝒮_{f}, and we define the relative derived category D_{f}(R) of R Mod as this localization: D_{f}(R) = K(R)[𝒮_{f}] (see [11]). The category D_{f}(R) inherits the structure of a triangulated category from K(R). It is easy to see that the functor T induces an embedding of D_{f}(R) into the derived category D(_{R}𝒞/𝒮⃗^{R}) of _{R}𝒞/𝒮⃗^{R}. If we start with the homotopy category K^{+}(R) (respectively K^{−}(R) or K^{b}(R)) of complexes bounded from below (respectively bounded from above or bounded complexes), then we get in a similar way the derived categories D_{f}^{+}(R) (respectively D_{f}^{−}(R) or D_{f}^{b}(R)).

Let K^{+}(ℐ_{f}) denote the homotopy category of complexes bounded from below over the full subcategory ℐ_{f} of R Mod consisting of f-injectives. Generalov has shown [11] that the natural functor Φ : K^{+}(ℐ_{f}) → D_{f}^{+}(R) that takes E^{∗} ∈ K^{+}(ℐ_{f}) ⊆ K^{+}(R) to its image in D_{f}^{+}(R) is an equivalence of categories. Obviously, the functor

T : K^{+}(ℐ_{f}) ⟶ K^{+}(ℐ), E^{∗} ↦ T E^{∗}

is an equivalence of the category K^{+}(ℐ_{f}) and the homotopy category of complexes bounded from below over the full subcategory ℐ of _{R}𝒞/𝒮⃗^{R} consisting of _{R}𝒞/𝒮⃗^{R}-injectives. By [10, Theorem 3.5.21] the natural functor Φ̃: K^{+}(ℐ) → D^{+}(_{R}𝒞/𝒮⃗^{R}) is an equivalence of categories. Let Ψ denote the equivalence of categories quasi-inverse to Φ.

Theorem 5.2. The derived categories D_{f}^{+}(R) and D^{+}(_{R}𝒞/𝒮⃗^{R}) are naturally equivalent via the functor Φ̃ ∘ T ∘ Ψ.

Let the functor I_{f} send a module M ∈ R Mod to the complex ⋯ 0 → M → 0 ⋯ concentrated at zero degree. Then the following relation holds:

Ext_{f}^{n}(M, N) = Hom_{D_{f} (R)}(I_{f}(M), I_{f}(N)[n])

where M, N ∈ R Mod [11]. The groups Ext_{f}^{n}(M, N) can also be defined [11] by using “Baer addition” on the set of equivalence classes of f-acyclic complexes of the form:

0 → N → M^{n−1} → ⋯ → M^{0} → M → 0.

On the other hand, let the functor I take every M ∈ R Mod to the complex ⋯ 0 → T M → 0 ⋯ Then

Ext_{f}^{n}(M, N) = Hom_{D(_{R}𝒞/𝒮⃗^{R})} (I(M), I(N)[n]).
6. K-groups for the category coh_{R}𝒞/𝒮⃗^{R}

An exact category 𝒞 is a full subcategory of an abelian category 𝒜 which is closed under extensions and which contains a zero object of 𝒜. A sequence 0 → E′ → E → E′′ → 0 in 𝒞 is called exact if it is exact in 𝒜. We say that a map i : M → N in an exact category 𝒞 is an admissible monomorphism if there is a short exact sequence 0 → M → N → L → 0 in 𝒞. In other words, the cokernel of i in the abelian category 𝒜 is in 𝒞. Similarly j : M → N in an exact category 𝒞 is an admissible epimorphism if there is a short exact sequence 0 → L → M → N → 0 in 𝒞.

If ℳ is an exact category define Qℳ to be the category with the same objects as ℳ and with morphisms defined as follows. A morphism from M to N in Qℳ is an equivalence class of diagrams of the form

M ↞ X ↣ N

in ℳ. Here ↠ denotes an admissible epimorphism (with kernel in ℳ) and ↣ denotes an admissible monomorphism (with cokernel in ℳ). We say that M ↞ X ↣ N and M ↞ Y ↣ N are equivalent if there is an isomorphism X ≃ Y making

M ⟵ X ⟶ N
∥  ↓^{≃}  ∥
M ⟵ Y ⟶ N

commutative (see [20, 26]).

Theorem 6.1 (Resolution). Let 𝒞 be an exact category an let ℳ ⊆ 𝒞 be a full subcategory of 𝒞 closed under extensions. Assume

(1) if 0 → M′ → M → M′′ → 0 is a short exact sequence in 𝒞 with M′ and M in ℳ, then M′′ is in ℳ;

(2) for every object C in 𝒞 there is a finite resolution

0 ⟶ C ⟶ M_{0} ⟶ M_{1} ⟶ ⋯ ⟶ M_{n} ⟶ 0

with M_{i} in ℳ.

Then K_{n}(ℳ) ⟶^{≃} K_{n}(𝒞).

Lemma 6.2. For any C ∈ coh_{R}𝒞/𝒮⃗^{R} there is an exact sequence

0 ⟶ C ⟶^{i} T M ⟶^{T μ} T N ⟶ T L ⟶ 0

with M, N and L finitely presented modules.

Proof. By [13, Theorem 2.16] there exists a coherent object D of coh_{R}𝒞 such that C = D_{𝒮^{R}}. The object D fits into an exact sequence

0 ⟶ D ⟶ − ⊗_{R} M ⟶^{−⊗μ} − ⊗_{R} N ⟶ − ⊗_{R} L ⟶ 0

with M, N, L ∈ R mod. If we apply the exact functor of 𝒮^{R}-localization to this sequence, we shall obtain the exact sequence of lemma. □

Lemma 6.3. Let ℳ = {T M | M ∈ R mod}. Then ℳ is closed under extensions in coh_{R}𝒞/𝒮⃗^{R}.

Proof. Let

0 ⟶ T M ⟶ C ⟶ T N ⟶ 0

be an exact sequence in coh_{R}𝒞/𝒮⃗^{R} with M and N finitely presented. Then the module K = C(R) is finitely presented. By the preceding lemma there is an exact sequence

0 ⟶ C ⟶^{i} T L ⟶^{T μ} T E

with L and E finitely presented modules. Then K = Ker μ and there is a unique morphism φ : T K → C such that T ρ = iφ with ρ = ker μ. We claim that φ is an isomorphism.

The module F = L/K is finitely presented and the exact sequense

0 ⟶ K ⟶ L ⟶ F ⟶ 0

is f-exact. We have a commutative diagram with exact rows:

0 ⟶ T K ⟶^{T ρ} T L ⟶^{T ν} T F ⟶ 0
φ↓  ∥  ↓^{T α}
0 ⟶ C ⟶_{i} T L ⟶_{T μ} T E.

Since F is a finitely presented submodule of E, it follows that T α is a monomorphism. By the snake lemma φ is an isomorphism and, hence, C ∈ ℳ. □

G-theory of a ring R is, by definition, the K-theory of the exact category of finitely presented modules R mod. Preceding two lemmas imply the following.

Proposition 6.4. The subcategory ℳ = T (R mod) of 𝒞 satisfies all the hypotheses of the Resolution Theorem. In particular, the functor T induces an isomorphism of K-groups

G_{n}(R) ⟶^{≃} K_{n}(coh_{R}𝒞/𝒮⃗^{R}), n ⩾ 0.

Let ^{R}𝒮 = {C ∈ coh 𝒞_{R} | C(R) = 0}. The Auslander-Gruson-Jensen duality D takes the category ^{R}𝒮 to 𝒮_{R}. By [13, Theorem 5.5] D induces a duality

D : coh 𝒞_{R}/^{R}𝒮⃗ ⟶ coh_{R}𝒞/𝒮⃗_{R}.

Then the classifying spaces for the categories Q coh 𝒞_{R}/^{R}𝒮⃗ and Q coh_{R}𝒞/𝒮⃗_{R} are homeomorphic.

Proposition 6.5. The functors T and D induce an isomorphism of K-groups G_{n}(R^{op}) ⟶^{≃} K_{n}(coh_{R}𝒞/𝒮⃗_{R}), n ⩾ 0.

Corollary 6.6. If R is a left and right F P -injective ring, then the functors T and D induce an isomorphism of K-groups G_{n}(R^{op}) ⟶^{≃} G_{n}(R), n ⩾ 0.

Proof. A ring R is left and right F P -injective iff 𝒮^{R} = 𝒮_{R} [8, Theorem 2.5]. Therefore our assertion follows from the preceding two propositions. □

The duality D yields isomorophisms of K-groups K_{i}(coh_{R}𝒞) ⟶^{≃} K_{i}(coh 𝒞_{R}). Since every coherent object C ∈ coh_{R}𝒞 fits into an exact sequence

0 → C → − ⊗_{R} M → − ⊗_{R} N → − ⊗_{R} L → 0

the Resolution theorem implies that the K-theory of coh_{R}𝒞 is equivalent to the K-theory of R mod⨁. Here R mod⨁ denotes the exact category of finitely presented left R-modules where only short exact sequences are used. Let _{R}𝒮 denote the Serre subcategory D𝒮^{R} = {C ∈ coh 𝒞_{R} | (C, R ⊗_{R} −) = 0} of coh 𝒞_{R}.

Proposition 6.7. The functors T and D induce two isomorphic long exact sequences

⋯ G_{n+1}(R) →^{δ} K_{n}(𝒮^{R}) → K_{n}(R mod⨁) → G_{n}(R) →

⋯ →^{δ} K_{0}(𝒮^{R}) → K_{0}(R mod⨁) → G_{0}(R) → 0

and

⋯ G_{n+1}(R) →^{δ} K_{n}(_{R}𝒮) → K_{n}(mod⨁ R) → G_{n}(R) →

⋯ →^{δ} K_{0}(_{R}𝒮) → K_{0}(mod⨁ R) → G_{0}(R) → 0.

Proof. Since 𝒮^{R} is a Serre subcategory of coh_{R}𝒞, by [20, Theorem 5.5] we have a long exact sequence

⋯ ⟶ K_{1}(coh_{R}𝒞/𝒮⃗^{R}) ⟶^{δ} K_{0}(𝒮^{R}) ⟶ K_{0}(R mod⨁) ⟶ K_{0}(coh_{R}𝒞/𝒮⃗^{R}) ⟶ 0.

By Proposition 6.4 G_{n}(R) is isomorphic to K_{n}(coh_{R}𝒞/𝒮⃗^{R}) for all n ⩾ 0. We obtain then the first long exact sequence. Proposition 6.5 implies the second long exact sequence. □

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St. Petersburg State University, Department of Mathematics and Mechanics, Universitetsky pr. 28, 198504, Stary Peterhof, Russia

E-mail address: ggarkusha@mail.ru