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GROTHENDIECK CATEGORIES

G. A. GARKUSHA

Abstract. This survey is devoted to the general theory of Grothendieck categories. The Popescu–Gabriel generalized theorem and theorems for projective generating families are given with proofs. These results make it possible to represent an arbitrary Grothendieck category as a quotient category of C_{A} = (mod A^{op}, Ab), where mod A^{op} is the category of finitely presented objects of some functor category. Also, various finiteness conditions for localizing subcategories are studied in detail, as well as their applications to the theory of rings and modules.

Contents

Introduction §1. Preliminaries

1.1. Ab-conditions

1.2. Localization in Grothendieck categories 1.3. Lattices of localizing subcategories 1.4. Locally finitely presented Grothendieck categories

§2. Module categories 2.1. The Gabriel topology 2.2. Localization in module categories

§3. Negligible objects and covering morphisms §4. The Popescu–Gabriel generalized theorem

4.1. The Gabriel formula

4.2. Projective generating families §5. Finiteness conditions for localizing subcategories

5.1. Left exact functors 5.2. The Ziegler topology

§6. Categories of generalized modules 6.1. Tensor products 6.2. The Auslander–Gruson–Jensen duality

§7. Grothendieck categories as quotient categories of (mod A^{op},

Ab) §8. FP-injective and flat modules Concluding remarks References

2000 Mathematics Subject Classification. Primary 18E15, 16D90. Key words and phrases. Grothendieck category, localizing subcategory, quotient category, Gabriel

topology, Ziegler spectrum.

\begin{matrix}c \\ ◯ \end{matrix}2002 American Mathematical Society

Introduction

In [1], in the course of investigation of the abstract properties of categories of sheaves, Grothendieck introduced the concept of an Ab5-category with a family of generators. Later, the categories of this class acquired the name Grothendieck categories. It was not clear from the outset whether the Grothendieck categories may have interesting applications. This question was resolved in Gabriel’s paper [2] in which the basic tool for the study of Grothendieck categories, namely, localization theory was proposed. In particular, this theory works perfectly in applications to the theory of rings and modules, resulting in deep and strong generalizations of classical facts pertaining to the theory of fractions. In this context, the language of torsion/localizing functors became a standard description language. Over long time since the moment of its creation by Gabriel, localization theory used to be applied mainly in this direction. Among the publications where the general theory of fractions is presented rather fully, at the first place we mention Stenström’s book [3]; we also refer the reader to the books [4, 5].

Starting with Ziegler’s paper [6], applications of Grothendieck categories were enriched due to interaction with model theory. This theory (see also the book [7] by Prest) revealed essentially new principles and problems concerning purely algebraic objects. For this reason, it gave birth to numerous investigations aimed at translating model-theoretic idioms to algebraic language.

It turned out that the principal model-theoretic concepts are realized in the category of generalized modules

C_{A} = (mod A^{op}, Ab).

This category consists of additive covariant functors that are defined on the category mod A^{op} of finitely presented left modules and take values in the category Ab of Abelian groups (see [8]). The category C_{A} has many remarkable properties. First, it is locally coherent [9], and every Grothendieck category is a quotient category of C_{A} [10]. In particular, this yields interesting applications to some problems of the theory of rings and modules, because the majority of statements for the classes of FP-injective and weakly quasi-Frobenius rings are of great utility in the context of torsion functors of the category C_{A} (or even are meaningless outside of it; see [11, 12]). Second, various problems pertaining to the representation theory of Artinian algebras admit natural reformulations in terms of the category C_{A}. This approach to problems of representation theory was proposed in the 70s by Auslander and is used at present by many mathematicians. Here we recommend the reader to consult Krause’s dissertation [13] in which further references and results on this topic can be found. Third, in C_{A} a certain fundamental modeltheoretic concept is realized, namely, the Ziegler spectrum of a ring. This topological space was constructed by Ziegler [6] in model-theoretic terms; recently, Herzog [14] and Krause [15] proposed an algebraic definition of the Ziegler topology for arbitrary locally coherent Grothendieck categories. By and large, remarkably, the category C_{A} plus the pair (t_{S}, (−)_{S}) of torsion/localizing functors turn out to be a good tool every time when something new about the category of modules arises in model theory. This is a good heuristic method in the context indicated.

The reasons listed above became the main stimulus for writing this paper; our main purpose is to give a self-contained and maximally general exposition of the basic results of the theory. Actually, the central facts are the Popescu–Gabriel generalized theorem (Theorem 4.1) and the theorems on projective generating sets (Subsection 4.2). These facts allow us to present any Grothendieck category C as a quotient category of C_{A} = (mod A^{op}, Ab); here A denotes the preadditive category formed by a family of generators U of the category C. We pay much attention to localizing subcategories of finite type, because many interesting applications are related to them. We only mention some applications to the theory of rings and modules, not dwelling on other applications such as the model theory of locally finitely presented Grothendieck categories, or representation theory (this would enlarge our exposition considerably). However, in the text below we give the relevant references.

The paper is organized as follows. The first section is preliminary; there we present the necessary category-theoretic background. In §2 we discuss the Gabriel topology and localization in the functor categories. §3 contains some technical material required for the proof of the Popescu–Gabriel generalized theorem. §4 is the main section; there we prove the Popescu–Gabriel generalized theorem and study projective generating sets. In §5 we show how various finiteness conditions for a family of generators U of a Grothendieck category C reflect some finiteness conditions for localizing subcategories. In particular, in that section we prove the Breitsprecher theorem. The basic properties of the categories of generalized modules, including the Auslander–Gruson–Jensen duality and the Herzog theorem are presented in §6. In the remaining sections we describe the Grothendieck categories as quotient categories of C_{A} and illustrate how localizing subcategories of C_{A} can be used for the study of rings and modules.

Acknowledgements. I would like to thank A. I. Generalov for a number of useful discussions. I am grateful to M. Prest and A. Jeremias who has kindly given me reprints of their papers. Also, I want to express my gratitude to Vladimir Kalinin for his help in preparing the manuscript and for his permanent interest to my work.

Notation. If no confusion concerning the ring A or the category B is possible, we usually abbreviate Hom_{A}(M, N) or Hom_{B}(M, N) to (M, N). We shall freely use the fact that every object X ∈ C of a Grothendieck category C has an injective envelope E(X) ∈ C (for the details, see [16]). This fact is also discussed in §4. If B is a category, then by a subcategory A of B we shall always mean a full subcategory of B. For such concepts as a subobject, epimorphism, injectivity, etc., we shall use a prefix (like A-subobject or B-subobject) to indicate the context. This prefix can be omitted if the concept in question is absolute with respect to the inclusion A ⊆ B. To indicate the context of an operation, such as Ker μ, E(X), or \underrightarrow{lim} X_{i}, we shall use a subscript (e.g.: Ker_{A} μ, E_{A}(X), etc.), which may also be omitted in the absolute case.

§1. Preliminaries

In this section we collect the basic definitions and facts about Grothendieck categories. Some of them can be treated for more general categories. For the details and proofs we refer the reader to [2, 5, 17].

1.1. Ab-conditions. Recall that an Abelian category C is cocomplete (or is an Ab3category) if it has arbitrary direct sums. A cocomplete Abelian category C is called an Ab5-category if for any directed family {A_{i}}_{i∈I} of subobjects of A and any subobject B of A we have

( ∑_{i∈I} A_{i}) ∩ B = ∑_{i∈I}(A_{i} ∩ B).

The Ab3-condition is equivalent to the existence of arbitrary inductive limits in C, and the Ab5-condition is equivalent to the fact that the inductive limit functor indexed by a directed set is exact. Namely, if I is a directed set, and

ε_{i}: 0 ⟶ A_{i} ⟶ B_{i} ⟶ C_{i} ⟶ 0

is an exact sequence in C, then \underrightarrow{lim} ε_{i} is also an exact sequence.

Let C be a category, and let U = {U_{i}}_{i∈I} be a family of objects of C. The family U is called a family of generators of C if for any object A of C and any subobject B of A distinct from A there exists an index i ∈ I and a morphism u: U_{i} → A that cannot be factored through the canonical injection i: B → A of B into A. An object U of C is called a generator of the category C if the family {U} is a family of generators for C.

If C is an Ab3-category, then U = {U_{i}}_{i∈I} is a family of generators for C if and only if ⨁_{i∈I} U_{i} is a generator (see [18, 5.33]). In accordance with [18, 5.35], any Ab3-category C possessing a family of generators U is locally finite. Furthermore, every object of C is isomorphic to a quotient object of ⨁_{j∈J} U_{j}, where J is some set of indices and U_{j} ∈ U for any j ∈ J (see [18, 5.34]).

The Ab5-categories possessing a family of generators are called Grothendieck categories.

Examples. (1) The category Mod A of right A-modules, where A is a ring with identity, and the category of (pre)sheaves of A-modules defined on an arbitrary topological space are Grothendieck categories.

(2) Let B be a preadditive small category. We denote by (B, Ab) the category whose objects are the additive functors F: B → Ab from B to the category of Abelian groups Ab; the morphisms of (B, Ab) are the natural transformations between functors. The fact that this category is Grothendieck follows from [3, V.2.2]. Moreover, the family of representable functors {h^{B} = (B, −)}_{B∈B} is a family of projective generators for (B, Ab) (see [3, IV.7.5]).

Let C be an Abelian category and U = {U_{i}}_{i∈I} some set of objects of C. Viewing U as a small preadditive category, we denote by (U^{op}, Ab) the category of additive contravariant functors from U to Ab. Let T: C → (U^{op}, Ab) be the functor defined as follows:

T X = _{C}(−, X), T f = (−, f),

where X ∈ C and f is a morphism of C.

Proposition 1.1. The functor T : C → (U^{op}, Ab) defined as above is faithful if and only if U is a family of generators for C.

Proof. Assume T is faithful. Let i: X′ → X be a monomorphism of C that is not an isomorphism, and let j: X → X/X′ be the cokernel of i. If T i is an isomorphism, then T j = 0 because T is left exact. It follows that j = 0, a contradiction.

Conversely, suppose U is a family of generators for C and f: X → Y is a morphism in C such that T f = 0. If f = pj is the canonical decomposition of f where p is a monomorphism and j is an epimorphism, then T j = 0, and if i: Ker f → X is the kernel of j, we see that T i is an isomorphism. Since U is a family of generators, i is an isomorphism, so that j = 0, whence f = 0. □

1.2. Localization in Grothendieck categories. A subcategory S of a Grothendieck category C is said to be closed under extensions if for any short exact sequence

0 ⟶ X′ ⟶ X ⟶ X′′ ⟶ 0

in C with X′, X′′ ∈ S the object X belongs to S. The subcategory S is a Serre subcategory provided that it is closed under passing to extensions, subobjects, and quotient objects. The corresponding quotient category C/S is constructed as follows. The objects of C/S are those of C, and

C/S(X, Y) = \underrightarrow{lim}_{C}(X′, Y/Y′),

where X′ ⊆ X, Y′ ⊆ Y, and X/X′, Y′ ∈ S. The set of such pairs is a partially ordered directed set with respect to the relation (X′, Y/Y′) ≤ (X′′, Y/Y′′) that is fulfilled if and only if X′′ ⊆ X′ and Y′ ⊆ Y′′. The direct limit is indexed by this partial order. Again C/S is Abelian, and there is a canonical quotient functor q: C → C/S such that q(X) = X; this functor is exact with Ker q = S (see [2, 17]). Here, by definition, the kernel Ker f of a functor f: C → D is the subcategory of all objects X such that f(X) = 0.

A Serre subcategory S of C is said to be localizing provided that the corresponding quotient functor admits a right adjoint s: C/S → C. Note that S is localizing if and only if S is closed under taking coproducts (see [17, 15.11]). In this case, S and C/S are again Grothendieck categories, by [2, III.4.9]. Moreover, the inclusion functor S → C admits a right adjoint t = t_{S}: C → S such that to X ∈ C it assigns the largest subobject t(X) of X belonging to S (see [15, 2.1]). The functor t_{S} is called the torsion functor generated by S. An object X is said to be S-periodic or simply periodic if t_{S}(X) = X. Furthermore, for any object X ∈ C there is a natural morphism λ_{X}: X → sq(X) for which Ker λ_{X}, Coker λ_{X} ∈ S and Ker λ_{X} = t_{S}(X) (see [17]).

We say that an object X ∈ C is S-closed (respectively, S-torsionfree) if λ_{X} is an isomorphism (respectively, a monomorphism). Thus, the section functor s induces an equivalence between C/S and the subcategory of S-closed objects in C (see [17, 15.19B]). Furthermore, by [15, 2.2] the subcategory of S-closed objects coincides with the perpendicular category S^{⊥} consisting of all X ∈ C such that _{C}(S, X) = 0 and Ext_{C}^{1}(S, X) = 0 for any S ∈ S. In what follows, the object sq(X) is denoted by X_{S}, and the morphism sq(α) is denoted by α_{S} for every X ∈ C and α ∈ Mor C. The morphism λ_{X} will be called the S-envelope of the object X. Thus, for any object X of C there is an exact sequence

(1.1)  0 ⟶ A′ ⟶ X ⟶^{λ_{X}} X_{S} ⟶ A′′ ⟶ 0,

where A′, A′′ ∈ S, and λ_{X} is the S-envelope for X. Observe that any two S-envelopes λ_{X}^{i}: X → X_{S}, i = 1, 2, of X ∈ C are isomorphic, and X_{S} ≈ (X_{S})_{S}. Also, we note that X_{S} = 0 if and only if the object X belongs to S.

Proposition 1.2 [17, 15.19C]. Let X be S-torsionfree. A monomorphism μ: X → Y is an S-envelope if and only if Y is S-closed and X/Y ∈ S. In this case, the following conditions are satisfied:

(1) μ is an essential monomorphism;

(2) if E is an essential extension of Y, then both E and E/Y are S-torsionfree.

Conversely, if conditions (1) and (2) are fulfilled and Y/X ∈ S, then μ is an Senvelope. Moreover, if E(X) is an injective envelope of X and X is S-torsionfree, then the S-envelope of X is the largest subobject D of E(X) containing X and such that D/X ∈ S. Thus, an S-torsionfree object X is S-closed if and only if E(X)/X is S-torsionfree.

Consider the localizing functor (−)_{S}: C → S^{⊥}, (−)_{S} = sq, where S^{⊥} is the subcategory of S-closed objects. Obviously, the inclusion functor i: S^{⊥} → C is fully faithful, and the localizing functor (−)_{S} is exact, because q is exact and, as was already mentioned, the functor s induces an equivalence of S^{⊥} and C/S. Suppose X, Y are some objects of C and α ∈ _{C}(X, Y). Then (−)_{S}(α) = α_{S} = (λ_{Y} α)_{S}, where λ_{Y} is the S-envelope of Y. Clearly, α_{S} = 0 if and only if Im α ⊆ t_{S}(Y). This implies that for any X ∈ C, Y ∈ S^{⊥} there is an isomorphism _{C}(X, Y) ≈ _{S^{⊥}} (X_{S}, Y), so that i is right adjoint to the localizing functor (−)_{S}. On the other hand, if C and D are Grothendieck categories, q′: C → D is some exact functor, and a functor s′: D → C is fully faithful and right adjoint to q′, then Ker q′ is a localizing subcategory, and there exists an equivalence C/ Ker q′ ≈^{H} D such that Hq′ = q, where q is the canonical functor (see [17, 15.18]).

In the sequel, by the quotient category C/S we always mean the subcategory of Sclosed objects S^{⊥} with the pair of functors (i, (−)_{S}), where i: C/S → C is the inclusion functor and (−)_{S}: C → C/S is the localizing functor.

Lemma 1.3. (1) An object E ∈ C/S is a C/S-injective if and only if it is C-injective. (2) Every S-torsionfree C-injective object E is S-closed.

Proof. (1). The inclusion functor i: C/S → C preserves injectivity since it is right adjoint to the exact functor (−)_{S}. If E ∈ C/S is C-injective, then any C/S-monomorphism μ: E → X is also a C-monomorphism, and, consequently, splits.

The second assertion follows from Proposition 1.2. □

Lemma 1.4. Let α: X → Y be some morphism of C/S. Then:

(1) the C-kernel of α is an S-closed object;

(2) α is a C/S-epimorphism if and only if Y/ Im_{C} α ∈ S.

Proof. (1) It suffices to observe that the inclusion functor i: C/S → C is left exact, because it is right adjoint to the localizing functor (−)_{S}.

(2) Localizing the exact sequence

X ⟶^{α} Y ⟶^{β} Y/ Im_{C} α ⟶ 0

with β = Coker α, we see that β_{S} = 0, which implies the relation (Y/ Im_{C} α)_{S} = 0. □

In particular, a C/S-morphism is a monomorphism if and only if it is a C-monomor-phism. Thus, for A, B ∈ C/S the relation A ≤ B is fulfilled in C/S if and only if so it is in C. Also, it is easy to show that for a C-morphism α: X → Y the C/S-morphism α_{S} is a C/S-monomorphism if and only if Ker α ∈ S, and that α_{S} is a C/S-epimorphism if and only if Y/ Im α ∈ S. Finally, α_{S} is a C/S-isomorphism if and only if Ker α ∈ S and Y/ Im α ∈ S.

1.3. Lattices of localizing subcategories. Let C be a Grothendieck category with a family of generators U = {U_{i}}_{i∈I}. We denote by L(C) the lattice consisting of the localizing subcategories of C ordered by inclusion.

We say that X ∈ C is a U-finitely generated object if there is an epimorphism ⨁_{i=1}^{n} U_{i} → X with U_{i} ∈ U. The subcategory of U-finitely generated objects is denoted by fg_{U} C. The fact that L(C) is a set follows from the fact that any localizing subcategory S is generated by the intersection fg_{U} S = S ∩ fg_{U} C. This means that every object X ∈ S can be written as a direct union ∑ X_{i} of objects in fg_{U} S. Since the category fg_{U} C is skeletally small, L(C) is indeed a set.

Proposition 1.5 [10, 2.7]. Suppose P and S are localizing subcategories of C; then P ⊆ S if and only if C/S is the quotient category of C/P with respect to the localizing subcategory S/P = { X ∈ C/P | X_{S} = 0 }.

Proposition 1.6. Let P be a localizing subcategory of C, and let A be a localizing subcategory of C/P. Then there exists a localizing subcategory S of C containing P and such that S/P = A.

Proof. Suppose the pair

i_{P}: C/P ⟶ C, (−)_{P}: C ⟶ C/P

determines C/P as a quotient category of C. Also, suppose that

i_{A}: (C/P)/A ⟶ C/P, (−)_{A}: C/P ⟶ (C/P)/A

determines (C/P)/A as a quotient category of C/P.

We introduce the functors

Q = (−)_{A} ∘ (−)_{P}: C ⟶ (C/P)/A,
I = i_{P} ∘ i_{A}: (C/P)/A ⟶ C.

The functor Q is exact because it is a composition of exact functors. Similarly, I is fully faithful because it is a composition of fully faithful functors. Furthermore, for X ∈ C and Y ∈ (C/P)/A we have

C(X, Y) ≈ _{C/P}(X_{P}, Y) ≈ _{(C/P)/A}(Q(X), Y),

which shows that Q is left adjoint to I. Thus, the pair (I, Q) determines (C/P)/A as a quotient category of C with respect to the localizing subcategory S = Ker Q. Clearly, P ⊆ S and S/P = A. □

Given a localizing subcategory P of C, consider the following sublattice of L(C):

L_{P}(C) = { S ∈ L(C) | S ⊇ P }.

Corollary 1.7. If P is a localizing subcategory of C, then the map

L: L_{P}(C) ⟶ L(C/P), S ⟼ S/P

is a lattice isomorphism.

We also note that

S/P = S_{P} = { S_{P} | S ∈ S }

for any S ∈ L_{P}(C). Indeed, obviously, S/P ⊂ S_{P}. Now, for S ∈ S we consider the exact sequence (1.1)

0 ⟶ A′ ⟶ S ⟶ S_{P} ⟶ A′′ ⟶ 0

with A′, A′′ ∈ P. Since P ⊆ S, it follows that A′, A′′ ∈ S. Consequently, S_{P} ∈ S, and since S_{P} is P-closed, we see that S_{P} ∈ S/P.

1.4. Locally finitely presented Grothendieck categories. In this subsection we fix a Grothendieck category C and define the most important subcategories of C to be used essentially in what follows. Namely, we describe the subcategories consisting of finitely generated, finitely presented, and coherent objects. These categories are ordered by inclusion as follows:

C ⊇ fg C ⊇ fp C ⊇ coh C.

Recall that an object A ∈ C is finitely generated if for any subobjects A_{i} ⊆ A, i ∈ I, satisfying A = ∑_{i∈I} A_{i}, there is a finite subset J ⊂ I such that A = ∑_{i∈J} A_{i}. The category of finitely generated subobjects of C is denoted by fg C. The category is locally finitely generated if every object X ∈ C is a directed sum X = ∑_{i∈I} X_{i} of finitely generated subobjects X_{i} or, equivalently, C possesses a family of finitely generated generators.

Theorem 1.8 [3, V.3.2]. An object C ∈ C is finitely generated if and only if the canonical homomorphism Φ: \underrightarrow{lim}_{C}(C, D_{i}) → _{C}(C, ∑ D_{i}) is an isomorphism for every object D ∈ C and every directed family {D_{i}}_{I} of subobjects of D.

A finitely generated object B ∈ C is finitely presented if the kernel Ker η of every epimorphism η: A → B with A finitely generated is also finitely generated. The subcategory of finitely presented objects of C is denoted by fp C. The corresponding categories of finitely presented left and right A-modules over the ring A are denoted by mod A^{op} = fp(Mod A^{op}) and mod A = fp(Mod A), respectively. Observe that the subcategory fp C of C is closed under extensions. Moreover, if

0 ⟶ A ⟶ B ⟶ C ⟶ 0

is a short exact sequence in C, and B is finitely presented, then C is finitely presented if and only if A is finitely generated.

The most obvious example of a finitely presented object of C is a finitely generated projective object P. We say that C has enough finitely generated projectives if every finitely generated object A ∈ C admits an epimorphism η: P → A, where P is a finitely generated projective object. If C has enough finitely generated projectives, then, by the above remarks, every finitely presented object B ∈ C is isomorphic to the cokernel of a morphism between finitely generated projective objects. This is expressed by an exact sequence of the form

P_{1} ⟶ P_{0} ⟶ B ⟶ 0,

called a projective presentation of B ∈ fp C.

Examples. The category Mod A of left A-modules has enough finitely generated projectives.

Another example is furnished by the category of functors (B, Ab) from a small preadditive category B to the category Ab of Abelian groups. In this category, every finitely generated projective object is a coproduct factor of a finite coproduct of representable objects ⨁_{i=1}^{n}(B_{i}, −) (see [14, 1.2]). Moreover, if B is an additive category, i.e., B is preadditive and has finite products/coproducts, and the idempotents split in B, then every finitely generated projective object in (B, Ab) is representable [14, 2.1].

The category C is said to be locally finitely presented provided that every object B ∈ C is a direct limit B = \underrightarrow{lim} B_{i} of finitely presented objects B_{i} or, equivalently, C possesses a family of finitely presented generators. For example, every locally finitely generated Grothendieck category having enough finitely generated projectives {P_{i}}_{i∈I} is locally finitely presented. In this case, the {P_{i}}_{i∈I} are generators for C. For instance, the set of representable functors {h^{B}}_{B∈B} of the functor category (B, Ab), where B is a small preadditive category, forms a family of finitely generated projective generators for (B, Ab). Thus, (B, Ab) is a locally finitely presented Grothendieck category (see [14, 1.3]).

Theorem 1.9 [3, V.3.4]. Let C be a locally finitely generated Grothendieck category. An object B ∈ C is finitely presented if and only if the functor _{C}(B, −): C → Ab commutes with the direct limits.

We say that a finitely presented object C ∈ C is coherent provided that every finitely generated subobject B ⊆ C is finitely presented. Obviously, any finitely generated subobject of a coherent object is also coherent. The subcategory of coherent objects of C is denoted by coh C. The category C is locally coherent if every object of C is a direct limit of coherent objects. This is equivalent to saying that fp C is Abelian [19, §2], or that C possesses a family of coherent generators. For instance, the category of right A-modules is locally coherent if and only if the ring A is right coherent.

In order to describe the case where, for an additive category B, the category (B, Ab) is locally coherent (i.e., fp C = coh C; see [19, §2]), we recall that a morphism ψ: Y → Z

is called a pseudo-cokernel for φ: X → Y in B if the sequence h^{Z} →^{(ψ,−)} h^{Y} →^{(φ,−)} h^{X} is exact, i.e., every morphism δ: Y → Z′ with δφ = 0 factors through ψ.

Lemma 1.10 [13, C.3]. The following statements are equivalent for C:

(1) fp C is Abelian;

(2) every morphism in B has a pseudo-cokernel.

A classical example of a locally coherent Grothendieck category is the category of right (left) generalized A-modules C_{A} = (mod A^{op}, Ab) (_{A}C = (mod A, Ab)) that consist of the covariant additive functors from the category mod A^{op} (mod A) of finitely presented left (right) A-modules to Ab. By the above lemma, the category fp C_{A}, which will be called the category of coherent functors in the sequel, is Abelian. We have already mentioned that the finitely generated projective objects of C_{A} are the representable functors (M, −) = Hom_{A}(M, −) for some M ∈ mod A^{op}, and they are generators for C_{A}.

There is a natural right exact and fully faithful functor

(1.2)  ? ⊗_{A} −: Mod A ⟶ C_{A}

that takes every module M_{A} to the tensor functor M ⊗_{A} −. We recall that a short exact sequence

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0

of right A-modules is said to be pure if for any M ∈ mod A the sequence of Abelian groups

0 ⟶ Hom_{A}(M, X) ⟶ Hom_{A}(M, Y) ⟶ Hom_{A}(M, Z) ⟶ 0

is exact. Equivalently, the C_{A}-sequence

0 ⟶ X ⊗_{A} − ⟶ Y ⊗_{A} − ⟶ Z ⊗_{A} − ⟶ 0

is exact. The module Q ∈ Mod A is pure-injective if the functor Hom_{A}(−, Q) takes the pure-monomorphisms to epimorphisms.

The functor (1.2) identifies the pure-injective A-modules with the injective objects of C_{A} (see [20, 1.2] and also [14, 4.1]). Furthermore, the functor M ⊗_{A} − belongs to coh C_{A} if and only if M ∈ mod A (see [9, 14]).

The category coh C_{A} has enough injectives and, by [14, 5.2], they are precisely the objects of the form M ⊗_{A} − with M ∈ mod A. Thus, every coherent object C ∈ coh C_{A} has both a projective presentation in C_{A}

(K, −) ⟶ (L, −) ⟶ C ⟶ 0

and an injective presentation in coh C_{A}

0 ⟶ C ⟶ M ⊗_{A} − ⟶ N ⊗_{A} −.

Here K, L ∈ mod A^{op} and M, N ∈ mod A.

It should be noted that the concept of purity is extremely important for applications to the representation theory of finite-dimensional algebras, because the pure-injective modules play a prominent role among the non-finitely-generated modules. For this reason, many concepts and problems are naturally formulated and solved in the category C_{A}. For this subject we recommend the reader to consult [13].

Another important application of the category C_{A} comes from the model theory of modules since its basic conceptions are realized in C_{A} (see [8, 14]). One of such concepts (“the Ziegler spectrum”) will be discussed in §5.

§2. Module categories

The terminology listed below is borrowed from the classical theory of rings and modules. By analogy with Mod A, the Grothendieck categories C possessing a family of finitely generated projective generators A = {P_{i}}_{i∈I} will be denoted by Mod A, and we refer to A as a ring of projective generators {P_{i}}_{i∈I}, or simply a ring. The category Mod A is called the category of right A-modules. Finally, every subobject 𝔞 of the object P_{i} ∈ A is called an ideal of the ring A corresponding to P_{i}.

2.1. The Gabriel topology. Consider the family 𝔉 = ⋃_{i∈I} 𝔉^{i} of ideals, where 𝔉^{i} is some family of ideals corresponding to the object P_{i}. We refer to 𝔉 as a Gabriel topology for A provided that the following axioms are fulfilled: T1) P_{i} ∈ 𝔉^{i} for every i ∈ I; T2) if 𝔞 ∈ 𝔉^{i} and μ ∈ Hom_{A}(P_{j}, P_{i}), P_{j} ∈ A, then { 𝔞 : μ } = μ^{−1}(𝔞) ∈ 𝔉^{j}; T3) if 𝔞 and 𝔟 are ideals of A corresponding to P_{i} and such that 𝔞 ∈ 𝔉^{i} and { 𝔟 : μ } ∈ 𝔉^{j}

for any μ ∈ Hom_{A}(P_{j}, P_{i}) with Im μ ⊂ 𝔞, P_{j} ∈ A, then 𝔟 ∈ 𝔉^{i}. If A = {A} is a ring and 𝔞 is a right ideal of A, then for every endomorphism μ: A → A

of the module A_{A} we have

μ^{−1}(𝔞) = { 𝔞 : μ(1) } = { a ∈ A | μ(1)a ∈ 𝔞 }.

On the other hand, if x ∈ A, then { 𝔞 : x } = μ^{−1}(𝔞), where μ ∈ End A is such that μ(1) = x.

Remark. The following properties of the Gabriel topologies 𝔉 = ⋃_{i∈I} 𝔉^{i} of a ring A = {P_{i}}_{i∈I} will be useful.

(1) If 𝔞, 𝔟 ∈ 𝔉^{i} and 𝔞 ⊆ 𝔟, then 𝔟 ∈ 𝔉^{i}. Indeed, if μ ∈ (P_{j}, P_{i}) is such that Im μ ⊆ 𝔞, then { 𝔟 : μ } = P_{j} ∈ 𝔉^{j}.

(2) If 𝔞, 𝔟 ∈ 𝔉^{i}, then 𝔞 ∩ 𝔟 ∈ 𝔉^{i}. Indeed, since { 𝔞 ∩ 𝔟 : μ } = { 𝔞 : μ } ∩ { 𝔟 : μ } for any μ ∈ (P_{j}, P_{i}), we see that { 𝔞 ∩ 𝔟 : μ } = { 𝔞 : μ } ∈ 𝔉^{j} for μ ∈ (P_{j}, P_{i}) with Im μ ⊂ 𝔟.

Thus the system 𝔉^{i}, i ∈ I, is a downward directed system of ideals.

Theorem 2.1 (Gabriel). The map

S ⟼ 𝔉(S) = { 𝔞 ⊆ P_{i} | i ∈ I, P_{i}/𝔞 ∈ S }

establishes a bijection between the Gabriel topologies for A and the localizing subcategories of Mod A.

Proof. Suppose 𝔉 = ⋃_{i∈I} 𝔉^{i} is a Gabriel topology for A. We introduce the following subcategory S of Mod A: A ∈ S if and only if for every δ: P_{i} → A the kernel Ker δ belongs to 𝔉^{i}. We claim that the subcategory S is localizing. Indeed, let A ∈ S, let i: A′ → A be a monomorphism, and let δ: P_{i} → A′. Then Ker δ = Ker(iδ) ∈ 𝔉^{i}. Now suppose that p: A → A′′ is an epimorphism, δ: P_{i} → A′′. Since P_{i} is projective, there exists γ: P_{i} → A such that pγ = δ. Then Ker γ ⊂ Ker δ, implying Ker δ ∈ 𝔉^{i}.

We show that S is closed under extensions. To see this, consider an exact sequence

0 ⟶ A′ ⟶^{i} A ⟶^{p} A′′ ⟶ 0

with A′, A′′ ∈ S. Let δ: P_{i} → A; then 𝔞 = Ker(pδ) ∈ 𝔉^{i}. Let γ: P_{j} → P_{i} be such that Im γ ⊂ 𝔞. Since pδγ = 0, there exists α: P_{j} → A′ such that δγ = iα. Therefore, { Ker δ : γ } = γ^{−1}(Ker δ) = Ker(δγ) = Ker α ∈ 𝔉^{j}. From T3 it follows that Ker δ ∈ 𝔉^{i}. So, S is a Serre subcategory. If δ: P_{i} → ⨁ A_{k}, A_{k} ∈ S, then there exist k_{1}, …, k_{s} such that Im δ ⊂ ⨁_{j=1}^{s} A_{k_{j}} (because P_{i} is finitely generated). Since S is a Serre subcategory, any finite product of objects in S belongs to S, whence Ker δ ∈ 𝔉^{i}. Thus, S is a localizing subcategory.

Conversely, assume that S is a localizing subcategory of Mod A. Let 𝔉^{i} = { 𝔞 ⊂ P_{i} | P_{i}/𝔞 ∈ S }. Obviously, P_{i} ∈ 𝔉^{i}. If 𝔞 ∈ 𝔉^{i} and δ: P_{j} → P_{i}, then { 𝔞 : δ } = Ker(pδ), where p: P_{i} → P_{i}/𝔞 is the canonical epimorphism. Since P_{i}/𝔞 ∈ S, we conclude that { 𝔞 : δ } ∈ 𝔉^{j}. It remains to check T3. Let 𝔞 ∈ 𝔉^{i} and 𝔟 ⊂ P_{i} be such that for any μ: P_{j} → P_{i} with Im μ ⊂ 𝔞 we have μ^{−1}(𝔟) ∈ 𝔉^{j}. Consider an exact sequence

0 ⟶ 𝔞 + 𝔟/𝔟 ⟶ P_{i}/𝔟 ⟶ P_{i}/𝔞 + 𝔟 ⟶ 0.

Since 𝔞 ⊂ 𝔞+𝔟, we have 𝔞+𝔟 ∈ 𝔉^{i}. Let p: P_{i} → P_{i}/𝔟 be the canonical epimorphism, γ_{μ} = pμ for μ ∈ (P_{j}, P_{i}). Since μ^{−1}(𝔟) = Ker γ_{μ}, we get P_{j}/μ^{−1}(𝔟) = Im γ_{μ} = p(μ(P_{j}) + 𝔟). In particular, if Im μ ⊂ 𝔞, then p(μ(P_{j}) + 𝔟) = P_{j}/μ^{−1}(𝔟) ∈ S, whence we see that

𝔞 + 𝔟/𝔟 = ∑_{\begin{matrix}μ∈(P_{j},P_{i}):Im μ⊂𝔞, \\ P_{j}∈A \end{matrix}} p(μ(P_{j}) + 𝔟)

belongs to S. Since S is closed under extensions, P_{i}/𝔟 ∈ S. □

Proposition 2.2 (Freyd [16]). The functor M ↦^{T} (−, M) establishes an equivalence between an arbitrary module category Mod A, where A = {P_{i}}_{i∈I} is the ring of finitely generated projective generators, and the functor category (A^{op}, Ab).

Proposition 2.2 shows that, in order to define a right A-module M (respectively, an A-homomorphism), it suffices to define M as a functor from A^{op} to Ab (respectively, as a natural transformation of functors). Conversely, every functor F: A^{op} → Ab (respectively, a natural transformation of functors from (A^{op}, Ab)) can be viewed as a right A-module (respectively, an A-homomorphism). In what follows, we do not distinguish between the A-modules and the functors in (A^{op}, Ab), freely using this without explicit references.

2.2. Localization in module categories. We fix a localizing subcategory S in Mod A, A = {P_{i}}_{i∈I}. Let 𝔉 = ⋃_{i∈I} 𝔉^{i} be the corresponding Gabriel topology for A and t = t_{S} the corresponding S-torsion functor. As was observed in Subsection 2.1, 𝔉^{i} is a downward directed system. Let X be some right A-module. Every pair 𝔞, 𝔟 ∈ 𝔉^{i} such that 𝔟 ⊂ 𝔞 gives rise to a homomorphism

Hom_{A}(𝔞, X/t(X)) ⟶ Hom_{A}(𝔟, X/t(X))

induced by the inclusion of 𝔟 into 𝔞. Clearly, the Abelian groups Hom_{A}(𝔞, X/t(X)) equipped with these homomorphisms form an inductive system over 𝔉^{i}.

Consider the functor H: Mod A → Mod A defined as follows. For every M ∈ Mod A and every P_{i} ∈ A we put

(2.1)  H(M)(P_{i}) = \underrightarrow{lim}_{𝔞∈𝔉^{i}} Hom_{A}(𝔞, M/t(M))

(we also call relation (2.1) the Gabriel formula).

We show that the Abelian groups (2.1) determine H(M) as a functor from A^{op} to Ab. To see this, consider a morphism μ: P_{j} → P_{i} and an element m ∈ H(M)(P_{i}). Let u: 𝔞 → M/t(M) be a morphism representing the element m of the direct limit. We define H(M)(μ)(m) ∈ H(M)(P_{j}) as the map represented by the composition

(2.2)  μ^{−1}(𝔞) ⟶^{μ̄} 𝔞 ⟶^{u} M/t(M).

It is easy to check that H(M)(μ) is well defined, i.e., is independent of the choice of a representing morphism u. Thus, H(M) becomes a right A-module.

Now, let f: M → N be a morphism of Mod A. Obviously, f(t(M)) is contained in t(N), so that f induces a unique morphism f′: M/t(M) → N/t(N). In turn, f′ induces a unique morphism H(f): H(M) → H(N), which is a homomorphism of right A-modules. This completes the construction of the functor H.

Ordering the Gabriel topology 𝔉 by inclusion, we obtain a directed set (see the remark in Subsection 2.1). The Ab5-condition for Ab and the construction of H imply that H is a left exact functor. Moreover, if M ∈ S, then H(M) = 0 because t(M) = M.

Let ζ_{i} be the canonical morphism from Hom_{A}(P_{i}, M/t(M)) into the direct limit H(M)(P_{i}), and let p: M → M/t(M) be the canonical epimorphism. Consider the map of A-modules Φ_{M}: M → H(M) defined as follows. For P_{i} ∈ A, α ∈ Hom_{A}(P_{i}, M) we put Φ_{M}(P_{i})(α) = ζ_{i}(pα). Let μ: P_{j} → P_{i} be a morphism. From the construction of the map H(M)(μ) it easily follows that the diagram

Hom_{A}(P_{i}, M) ⟶^{Φ_{M} (P_{i})(μ)} H(M)(P_{i})
(μ,M)↓  ↓^{H(M)(μ)}
Hom_{A}(P_{j}, M) ⟶_{Φ_{M} (P_{j})(μ)} H(M)(P_{j})

commutes; hence, Φ_{M} is an A-homomorphism. Direct verification shows that Φ_{M} is functorial in M. Thus, we obtain a functorial morphism Φ: 1_{Mod A} → H.

Proposition 2.3. Ker Φ_{M} and Coker Φ_{M} belong to S for every M ∈ Mod A.

Proof. As above, we can construct a morphism Ψ_{M}: M/t(M) → H(M), Ψ_{M}(P_{i})(μ) = ζ_{i}(μ) for μ ∈ Hom_{A}(P_{i}, M/t(M)). In a similar way we check that Ψ_{M} is an A-homomor-phism. Now, from the definitions of Φ_{M} and Ψ_{M} it easily follows that the diagram with exact rows

0 ⟶ t(M) ⟶ M ⟶ M/t(M)
∥  ↓^{Ψ_{M}}
0 ⟶ Ker Φ_{M} ⟶ M ⟶_{Φ_{M}} H(M)

is commutative, whence t(M) ⊂ Ker Φ_{M}. We show that Ψ_{M} is a monomorphism or, equivalently, Φ_{M} is a monomorphism for any S-torsionfree object M. Indeed, assume that M is S-torsionfree and let μ: P_{i} → M be such that Φ_{M}(P_{i})(μ) = 0. Then there exists an element 𝔞 ∈ 𝔉^{i} such that the restriction μ|_{𝔞} is zero. This implies that 𝔞 is contained in Ker μ, whence Ker μ ∈ 𝔉^{i} (see the remark in Subsection 2.1). Consequently, Im μ ∈ S. Since M is S-torsionfree, it follows that Im μ = 0, i.e., μ = 0. Thus, Ker Φ_{M} = t(M). Therefore, if H(M) = 0, then M ∈ S. Indeed, then we have Ker Φ_{M} = M, so that M ∈ S. Therefore, H(M) = 0 if and only if M ∈ S.

It remains to check that Coker Φ_{M} ∈ S. Let μ: P_{i} → H(M). It suffices to show that the ideal μ^{−1}(Im Φ_{M}) is an element of 𝔉^{i}. Indeed, if p: P_{i} → Coker Φ_{M}, then there exists μ: P_{i} → H(M) such that p = Coker Φ_{M} ∘ μ (because P_{i} is projective). Next, since the sequence

0 ⟶ μ^{−1}(Im Φ_{M}) ⟶ P_{i} ⟶^{p} Coker Φ_{M}

is exact and μ^{−1}(Im Φ_{M}) belongs to 𝔉^{i} by assumption, we see that Coker Φ_{M} belongs to S. Without loss of generality we may assume that Φ_{M} is a monomorphism and identify M with Im Φ_{M}. We put 𝔟 = μ^{−1}(M).

Let u: 𝔞 → M be an A-homomorphism representing μ in the direct limit H(M)(P_{i}), and let ξ: P_{j} → P_{i} be such that Im ξ ⊆ 𝔞. Consider the following commutative diagram:

ξ^{−1}(𝔟) ⟶ 𝔟 ⟶ M
ι↓  ↓  ↓^{Φ_{M}}
P_{j}  ⟶_{ξ} P_{i} ⟶_{μ} H(M).

We recall that the element μξ is represented by the composed morphism uξ̄: { 𝔞 : ξ } → M (see (2.2)). Since Im ξ ⊆ 𝔞, we have { 𝔞 : ξ } = P_{j}, whence Φ_{M}(P_{j})(uξ̄) = μξ. Since both squares of the diagram are pullback, so is the outer square; hence, there exists a morphism κ: P_{j} → ξ^{−1}(𝔟) such that ικ = 1_{P_{j}}. Thus, ξ^{−1}(𝔟) = P_{j} ∈ 𝔉^{j}. Recalling T3, we deduce that 𝔟 ∈ 𝔉^{i}. □

Theorem 2.4. For an arbitrary right A-module M, the module H(M) is S-closed. Moreover, the homomorphism Φ_{M} is an S-envelope for M.

Proof. First, we show that H(M) is S-torsionfree. Let S be a subobject of H(M) belonging to S, let μ: P_{i} → S, and let μ be represented by u: 𝔞 → M/t(M) in the direct limit H(M)(P_{i}). Also, suppose ξ: P_{j} → P_{i} is such that Im ξ ⊆ Ker μ; then the relation μξ = 0 implies that the image of the composed map uξ̄: { 𝔞 : ξ } → M/t(M) in H(M)(P_{i}) equals zero. So, by the properties of direct limits, there is an ideal 𝔟 ∈ 𝔉^{j} such that the restriction of uξ̄ to 𝔟 that represents μξ equals zero; hence, 𝔟 ⊆ Ker(uξ̄). Then Ker(uξ̄) ∈ 𝔉^{j}, i.e., Im(uξ̄) ∈ S. But M/t(M) is S-torsionfree, whence Im(uξ̄) = 0 and uξ̄ = 0. Since this is true for every ξ: P_{j} → P_{i} such that Im ξ ⊆ Ker μ, we infer that Ker μ ⊆ Ker u. Since Ker μ ∈ 𝔉^{i}, the kernel Ker u is also an element of 𝔉^{i}. In that case, μ equals zero, because μ is the image of the zero homomorphism u ∘ Ker u from Hom_{A}(Ker u, M/t(M)). Since this is true for any μ ∈ Hom_{A}(P_{i}, S), we conclude that S = 0.

Now we prove that a module M is S-closed if and only if Φ_{M} is an isomorphism. Indeed, if M is S-closed, then, by the preceding proposition, Ker Φ_{M} = t(M) = 0, i.e., Φ_{M} is a monomorphism. Since Ext^{1}(Coker Φ_{M}, M) = 0, there exists a morphism α: H(M) → M such that αΦ_{M} = 1_{M}, i.e., Coker Φ_{M} is a direct summand of H(M). Since H(M) is S-torsionfree, we see that Coker Φ_{M} = 0, which implies that Φ_{M} is an isomorphism.

On the other hand, if the morphism Φ_{M} is an isomorphism, then t(M) = Ker Φ_{M} = 0, whence M is S-torsionfree. If we show that every short exact sequence

0 ⟶ M ⟶^{i} N ⟶ S ⟶ 0

with S ∈ S splits, it will follow that Ext^{1}(S, M) = 0 for any S ∈ S, implying that M is S-closed. For this, consider the commutative diagram

0 ⟶ M ⟶^{i} N ⟶ S ⟶ 0
Φ_{M} ↓  ↓^{Φ_{N}}
0 ⟶ H(M) ⟶_{H(i)} H(N) ⟶ H(S),

where the bottom row is exact and H(S) = 0. We deduce that H(i) is an isomorphism. Hence, Φ_{M}^{−1}H(i)^{−1}Φ_{N}i = 1_{M}, and we see that i splits.

Thus, in order to prove that H(M) is S-closed, it suffices to show that Φ_{H(M)} : H(M) → H(H(M)) is an isomorphism. First, we prove that H(Φ_{M}) is an isomorphism. By the construction of H(M) we have H(M) = H(M/t(M)). Let p: M → M/t(M) be the canonical epimorphism. Applying the functor H to the commutative diagram

M ⟶^{p} M/t(M)
Φ_{M} ↓  ↓^{Φ_{M/t(M)}}
H(M)  H(M/t(M)),

we obtain H(Φ_{M}) = H(Φ_{M/t(M)}). Since H(Coker Φ_{M/t(M)}) = 0, it follows that the morphism H(Φ_{M/t(M)}) is an isomorphism; hence, so is H(Φ_{M}).

Next, since Φ is a functorial morphism, the following relations are true:

(2.3)  Φ_{H(M)}Φ_{M} = H(Φ_{M})Φ_{M}

and

(2.4)  \begin{aligned}Φ_{H^{2}(M)}Φ_{H(M)} & = H(Φ_{H(M)})Φ_{H(M)}, \\ Φ_{H^{2}(M)}H(Φ_{M}) & = H^{2}(Φ_{M})Φ_{H(M)}. \end{aligned}

Applying the functor H to (2.3), we get H(Φ_{H(M)})H(Φ_{M}) = H^{2}(Φ_{M})H(Φ_{M}). Since H(Φ_{M}) is an isomorphism, H(Φ_{H(M)}) = H^{2}(Φ_{M}). Relations (2.4) imply that

(2.5)  Φ_{H^{2}(M)}Φ_{H(M)} = Φ_{H^{2}(M)}H(Φ_{M}).

Since H^{2}(M) is S-torsionfree, the first part of the proof of the preceding proposition shows that Φ_{H^{2}(M)} is a monomorphism. Therefore, from (2.5) it follows that Φ_{H(M)} = H(Φ_{M}). So, Φ_{H(M)} is an isomorphism, and, consequently, H(M) is S-closed.

In particular, if we consider the exact sequence

0 ⟶ Ker Φ_{M} = t(M) ⟶ M ⟶^{Φ_{M}} H(M) ⟶ Coker Φ_{M} ⟶ 0

and apply the exact localizing functor (−)_{S}, we obtain

M_{S} ≈ (H(M))_{S} ≈ H(M).

This implies immediately that Φ_{M} is an S-envelope for M. □

Now, suppose i: Mod A/S → Mod A is the inclusion functor. The functor H is left adjoint to i. The unit of adjointness between i and H is given by the functor morphism Φ: 1_{Mod A} → H = i ∘ H constructed above, and the counit of adjointness is the functor morphism Ψ: H ∘ i → 1_{Mod A/S} defined by the rule Ψ_{M} = (Φ_{M})^{−1} for all M ∈ Mod A/S. The fact that Ψ_{H(M)}H(Φ_{M}) = 1_{H(M)} follows from the relation Φ_{H(M)} = H(Φ_{M}) proved in the preceding theorem. The identity i(Ψ_{M})Φ_{i(M)} = 1_{i(M)} is trivial. Thus, H is indeed left adjoint to i. Since the localizing functor (−)_{S} is also left adjoint to i, it follows that the functors (−)_{S} and H are equivalent.

§3. Negligible objects and covering morphisms

Let C be an Abelian category and M some class of objects in C. We put S = { C ∈ C | _{C}(C, M) = 0, Ext_{C}^{1}(C, M) = 0 for every M ∈ M }.

Proposition 3.1. (1) S is closed under extensions. An object S belongs to S if and only if for every M ∈ M, X ∈ C, and every epimorphism f: X → S the canonical homomorphism (X, M) → (Ker f, M) is an isomorphism. Furthermore, if C is cocomplete, then S is closed under coproducts.

(2) The following conditions on S are equivalent: (a) S is a Serre subcategory;

(b) S is closed under subobjects;

(c) S ∈ S if and only if for every M ∈ M and every morphism f: X → S, X ∈ C, the canonical homomorphism (X, M) → (Ker f, M) is an isomorphism.

Moreover, if C is a Grothendieck category, then S is a localizing subcategory.

Proof. (1) A short exact sequence

(3.1)  0 ⟶ S′ ⟶^{i} S ⟶^{j} S′′ ⟶ 0

in C induces the exact sequence

0 → (S′′, M) → (S, M) → (S′, M)
→ Ext^{1}(S′′, M) → Ext^{1}(S, M) → Ext^{1}(S′, M).

(3.2)

If M ∈ M and S′, S′′ ∈ S, then, obviously, S ∈ S. Thus, S is closed under extensions.

Let S ∈ S, M ∈ M, X ∈ C, and let f: X → S be an arbitrary epimorphism. Looking at the exact sequence

(3.3)  0 → (S, M) = 0 → (X, M) → (Ker f, M) → Ext^{1}(S, M) = 0,

we obtain (X, M) ≈ (Ker f, M).

Conversely, consider the identity morphism 1_{S}: S → S. Then

0 = (Ker 1_{S}, M) ≈ (S, M).

It remains to check that Ext^{1}(S, M) = 0. For this, we show that any short exact sequence

0 ⟶ M ⟶^{h} P ⟶^{f} S ⟶ 0

splits. By assumption, for 1_{M}: M → M there exists g: P → M such that gh = 1_{M}, i.e., h splits.

In turn, if C is cocomplete, then S is closed under coproducts, because the functor Ext_{C}^{1}(−, M) commutes with the coproducts.

(2) The implication (a) ⟹ (b) is obvious.

(b) ⟹ (c). Suppose S ∈ S and f: X → S is an arbitrary morphism. Then, by assumption, Im f belongs to S. Replacing S by Im f in (3.3), we see that for any M ∈ M the canonical morphism (X, M) → (Ker f, M) is an isomorphism. The proof of the converse is similar to the proof of (1).

(c) ⟹ (a). By the first statement, the subcategory S is closed under extensions. Suppose now that, in the exact sequence (3.1), the object S belongs to S and f: X → S′, X ∈ C; then Ker f = Ker(if). Hence, S′ ∈ S. Next, considering the exact sequence (3.2) with S′, S ∈ S, we obtain S′′ ∈ S.

If C is a Grothendieck category, then S is closed under coproducts, and, consequently, it is a localizing subcategory by [17, 15.11]. □

Let C be a Grothendieck category, and let M be some class of objects of C. An object S ∈ C is said to be M-negligible if for any M ∈ M, X ∈ C, and f: X → S the canonical homomorphism (X, M) → (Ker f, M) is an isomorphism.

Example. Every localizing subcategory S of C is precisely the subcategory of C/Snegligible objects, where C/S is the quotient category of C with respect to S, because S = (C/S)^{⊥}.

Lemma 3.2. The subcategory S of C consisting of M-negligible objects is the largest localizing subcategory for which all M ∈ M are S-closed.

Proof. Indeed, if P is a localizing subcategory such that every object M ∈ M is P-closed, then for any f: X → S, S ∈ P, the object Im f belongs to P. Hence, looking at the exact sequence (3.3) with S = Im f, we conclude that the homomorphism (X, M) → (Ker f, M) is an isomorphism. Therefore, S ∈ S by Proposition 3.1, so that P ⊂ S. □

Let S be a localizing subcategory of the category C. We say that an object M ∈ C cogenerates S if S = { C ∈ C | _{C}(C, M) = 0 }.

Lemma 3.3. The localizing subcategory S consisting of M-negligible objects is cogenerated by the objects E(M) ⊕ E(E(M)/M), where M ∈ M and E(M) is the injective envelope for M.

Proof. Let P denote the localizing subcategory cogenerated by E(M) ⊕ E(E(M)/M). We must show that S = P.

Let M ∈ M. Consider the short exact sequence

0 ⟶ M ⟶ E(M) ⟶ E(M)/M ⟶ 0.

For every S ∈ C, it induces the exact sequence

0 ⟶ (S, M) ⟶ (S, E(M)) ⟶ (S, E(M)/M) ⟶ Ext^{1}(S, M) ⟶ 0.

If S ∈ S, then Ext^{1}(S, M) = 0, and also (S, E(M)) = 0 because M is an S-torsionfree object. Therefore, (S, E(M)/M) = 0, whence (S, E(E(M)/M)) = 0, which implies the relation S ⊂ P.

On the other hand, suppose that S ∈ P; then (S, E(M)/M) = 0 because E(E(M)/M) is a P-torsionfree object. Hence, (S, M) = 0 and Ext^{1}(S, M) = 0. Thus, M is P-closed. But, by assumption, S is the largest localizing subcategory such that every M is S-closed. Consequently, P ⊂ S. □

In the proof of the lemma we used the fact that an object M ∈ C is S-torsionfree if and only if its injective envelope E(M) is S-torsionfree.

We say that an object C of the category C is cyclic if there is an epimorphism f: U_{i} → C for some U_{i} ∈ U.

Lemma 3.4. Suppose C, D are objects of C; then _{C}(C, E(D)) = 0 if and only if _{C}(C′, D) = 0 for every cyclic subobject C′ of C.

Proof. This is obvious. □

Proposition 3.5. Let Mod A be the category of right A-modules, where A is the ring of finitely generated projective generators {P_{i}}_{i∈I}, and let M be some class of A-modules. Then an object S is M-negligible if and only if for an arbitrary morphism f: P_{i} → S with P_{i} ∈ A the canonical homomorphism (P_{i}, M) → (Ker f, M) with M ∈ M is an isomorphism.

Proof. The “only if” part is obvious. Let S′ be an arbitrary cyclic subobject of the object S; then (S′, M) = 0. Indeed, there exists an epimorphism f: P_{i} → S′ for some P_{i} ∈ A. The exact sequence

0 ⟶ Ker f ⟶^{i} P_{i} ⟶^{f} S′ ⟶ 0

shows that (S′, M) = 0, because, by assumption, the homomorphism (i, M) is an isomorphism. Taking Lemma 3.4 into account, we get (S, E(M)) = 0.

Now we consider the exact sequence

0 ⟶ M ⟶ E(M) ⟶ E(M)/M ⟶ 0.

It induces the exact sequence

(S′, E(M)) ⟶ (S′, E(M)/M) ⟶ Ext^{1}(S′, M) ⟶ 0.

Since (S, E(M)) = 0, we have (S′, E(M)) = 0, whence (S′, E(M)/M) ≈ Ext^{1}(S′, M). Our proof will be finished if we show that any short exact sequence of the form

0 ⟶ M ⟶^{g} N ⟶^{h} S′ ⟶ 0

splits. For this, consider the commutative diagram

Ker f  Ker f
i′↓  ↓^{i}
0 ⟶ M ⟶^{ν} NΠ_{S′}P_{i} ⟶^{π} P_{i}
∥  ^{f′}↓  ↓^{f}
0 ⟶ M ⟶_{g} N ⟶_{h} S′,

where the lower right square is pullback. Since f and h are epimorphisms, so are f′ and π. Since P_{i} is a projective A-module, ν is a splitting monomorphism. Let π′ be the canonical projection onto M, i.e., π′ν = 1_{M}. By assumption, there exists a morphism β: P_{i} → M such that π′i′ = βi. Since βi = βπi′, we have (π′ − βπ)i′ = 0. Then there exists q: N → M such that π′ − βπ = qf′, whence 1_{M} = π′ν = qf′ν = qg. Thus, g is a splitting monomorphism. This completes the proof. □

Let M be some class of objects of a Grothendieck category C. A morphism u: X → Y is said to be M-covering if Coker u is an M-negligible object.

Let Mod A be the category of right A-modules, where A = {P_{i}}_{i∈I} is a ring. We consider morphisms u: X → Y and f: P_{i} → Y, and also a commutative diagram

Ker(pf)
↓^{h}
0 ⟶ Ker u ⟶^{i′} XΠ_{Y} P_{i} ⟶^{u′} P_{i}
∥  ^{f′}↓  ↓^{f}
0 ⟶ Ker u ⟶_{i} X ⟶_{u} Y  ⟶_{p} Coker u ⟶ 0,

in which the rows are exact, and the couple (Ker(pf), h) is the kernel of pf.

Lemma 3.6. The morphism u: X → Y is M-covering if and only if for every P_{i} ∈ A, M ∈ M and every morphism f: P_{i} → Y the sequence

0 ⟶ Hom_{A}(P_{i}, M) ⟶^{φ} Hom_{A}(XΠ_{Y} P_{i}, M) ⟶^{ψ} Hom_{A}(Ker u, M)

induced by u′ and i′ is exact.

Proof. Since pfu′ = puf′ = 0, there exists a unique morphism q: XΠ_{Y} P_{i} → Ker(pf) such that u′ = hq. We claim that q is an epimorphism. Indeed, consider the following diagram:

XΠ_{Y} P_{i} ⟶^{q} Ker(pf) ⟶^{h} P_{i}
f′↓  ↓  ↓^{f}
X ⟶_{n} Ker p ⟶_{k} Y ⟶_{p} Coker u,

where (Ker p, k) is the kernel of p, kn = u, and the middle arrow is a unique morphism that makes the diagram commute. Since the outer and right squares are pullback, it follows that so is the left square. Then the epimorphism n: M → Im u = Ker p shows q is an epimorphism.

Suppose u is M-covering and α: P_{i} → M is a morphism satisfying αu′ = 0. Then αhq = 0, and so αh = 0. Since Coker u is M-negligible, it follows that α = 0 . Thus, the homomorphism φ induced by u′ is a monomorphism. Now, suppose that a morphism α: XΠ_{Y} P_{i} → M satisfies αi′ = 0. Since (Ker(pf), q) is the cokernel of i′, there exists a morphism t: Ker(pf) → M such that tq = α. Since Coker u is M-negligible, there exists l: P_{i} → M such that lh = t. Then lu′ = lhq = tq = α.

Conversely, suppose the sequence occurring in the lemma is exact, and g: P_{i} → Coker u is an arbitrary morphism. Since P_{i} is projective, there exists a morphism f: P_{i} → Y satisfying pf = g, so that (Ker(pf), h) is the kernel of g. If a morphism α: P_{i} → M satisfies αh = 0, then αu′ = αhq = 0. It follows that α = 0, because φ is a monomorphism. Now, let β ∈ Hom_{A}(Ker g, M). Since βqi′ = 0 and Ker ψ = Im φ, there exists α: P_{i} → M such that βq = αu′, i.e., βq = αhq. Therefore, β = αh. Thus, the canonical homomorphism Hom_{A}(P_{i}, M) → Hom_{A}(Ker g, M) is an isomorphism. Proposition 3.5 implies that Coker u is M-negligible. □

§4. The Popescu–Gabriel generalized theorem

Consider a Grothendieck category C. Let U = {U_{i}}_{i∈I} be some family of generators for C. In Subsection 1.1 we introduced the functor T: C → Mod A = (U^{op}, Ab) that takes every object X ∈ C to T X = (−, X) (here A denotes the ring of projective generators h_{U_{i}} = (−, U_{i}), i ∈ I, of the functor category (U^{op}, Ab)). The next theorem was obtained also by Prest in [21].

Theorem 4.1 (Popescu, Gabriel). Let C be a Grothendieck category with a family of generators U = {U_{i}}_{i∈I}, and let T be the functor defined above. Then:

(1) T is full and faithful;

(2) T induces an equivalence between C and the quotient category Mod A/S, where S denotes the largest localizing subcategory in Mod A for which all modules T X = (−, X) are S-closed.

Proof. (1) By Proposition 1.1, the functor T is faithful. To see that it is full, we must show that if X and Y are objects in C and Φ: (−, X) → (−, Y) is a functor, then Φ is of the form Φ(f) = φf for some φ: X → Y. We denote by Λ_{i} = Hom_{A}(U_{i}, X) the set of all morphisms U_{i} → X and put Λ = ⋃_{i∈I} Λ_{i}. For α ∈ Λ_{i}, let i_{α}: U_{i} → U_{Λ} = ⨁_{β∈Λ} U_{β}, denote the corresponding injection; here U_{β} = U_{i} if and only if β ∈ Λ_{i}. There exists a unique morphism λ: U_{Λ} → X such that λi_{α} = α for all α ∈ Λ, and λ is an epimorphism since U is a family of generators. Similarly, there exists a unique morphism μ: U_{Λ} → Y such that μi_{α} = Φ(U_{i})(α) for every α ∈ Λ. Let κ: K → U_{Λ} be the kernel of λ. We claim that μκ = 0; then μ factors as μ = φλ with some φ: X → Y, and for every i ∈ I, α: U_{i} → X we get Φ(U_{i})(α) = μi_{α} = φλi_{α} = φα, so that our assertion will be proved.

Thus, we need to check that μκ = 0. For any finite subset J of Λ and any α ∈ J there are canonical morphisms π′_{α}: U_{J} = ⨁_{β∈J} U_{β} → U_{α}, i′_{α}: U_{α} → U_{J}, and i_{J}: U_{J} → U_{Λ}; here U_{β} = U_{i} (respectively, U_{α} = U_{i}) if β ∈ Λ_{i} (respectively, if α ∈ Λ_{i}). Let κ_{J}: K_{J} → U_{J} be the kernel of the composed morphism λi_{J}: U_{J} → X. Since K is the direct limit of the kernels K_{J} for all finite subsets J of Λ, it suffices to show that μi_{J}κ_{J} = 0. Since Φ is a functor, for every U_{i} ∈ U, β: U_{i} → K_{J} we have

\begin{aligned}μi_{J}κ_{J}β & = μi_{J}( ∑_{α∈J} i′_{α}π′_{α})κ_{J}β = ∑_{α∈J} μi_{α}π′_{α}κ_{J}β = ∑_{α∈J} Φ(U_{α})(α)π′_{α}κ_{J}β \\ & = ∑_{α∈J} Φ(U_{i})(απ′_{α}κ_{J}β) = Φ(U_{i})( ∑_{α∈J} λi_{J}i′_{α}π′_{α}κ_{J}β) = Φ(U_{i})(λi_{J}κ_{J}β) = 0, \end{aligned}

because λi_{J}κ_{J} = 0. Being true for arbitrary β: U_{i} → K_{J}, this relation implies that μi_{J}κ_{J} = 0.

(2) Let S be the largest localizing subcategory in Mod A for which all modules of the form T X = (−, X) are S-closed. This subcategory exists by Lemmas 3.2 and 3.3, and it is cogenerated by the class of injective modules of the form E(T X) ⊕ E(E(T X)/T X). Let T = {T X}_{X∈C}; the corresponding T -negligible objects and T -covering morphisms will be referred to as negligible and covering (respectively), without the prefix T . Since every module T X is S-closed, there is a functor T′: C → Mod A/S such that T = iT′, where i: Mod A/S → Mod A is the inclusion functor. We must show that T′ is an equivalence. Since iT′ = T is full and faithful by (1), so is T′; thus, it suffices to show that every S-closed module is isomorphic to a module of the form T X.

For an S-closed A-module M we can choose an exact sequence

⨁_{k∈K} h_{V_{k}} ⟶^{α} ⨁_{j∈J} h_{V_{j}} ⟶ M ⟶ 0

in Mod A/S; here V_{k}, V_{j} ∈ U, and K, J are some sets of indices.

If we show that T′ preserves colimits, we shall have isomorphisms

\begin{matrix}( ⨁_{k∈K} h_{V_{k}}, ⨁_{j∈J} h_{V_{j}}) ≈ ( ⨁_{k∈K} T′V_{k}, ⨁_{j∈J} T′V_{j}) \\ ≈ (T′( ⨁_{k∈K} V_{k}), T′( ⨁_{j∈J} V_{j})) ≈ _{C}( ⨁_{k∈K} V_{k}, ⨁_{j∈J} V_{j}). \end{matrix}

Then there exists β such that T′β = α.

Let M̃ be defined by the exact sequence

(4.1)  ⨁_{k∈K} V_{k} ⟶^{β} ⨁_{j∈J} V_{j} ⟶ M̃ ⟶ 0.

If we prove that the functor T′ is exact, then, applying T′ to (4.1), we shall obtain the following commutative diagram in Mod A/S with exact rows:

⨁_{k∈K} h_{V_{k}} ⟶^{α} ⨁_{j∈J} h_{V_{j}} ⟶ M ⟶ 0
∥  ∥
⨁_{k∈K} h_{V_{k}} ⟶_{T′β=α} ⨁_{j∈J} h_{V_{j}} ⟶ T′(M̃) ⟶ 0.

Then M = T′(M̃). Thus, to complete the proof it remains to verify the following statement.

Lemma 4.2. The functor T′: C → Mod A/S is exact and preserves direct sums.

Proof. First, we prove the exactness of T′. Obviously, the functor T′ = (−)_{S}T, where (−)_{S} is the corresponding localizing functor, is left exact, so that it suffices to check that T′ preserves epimorphisms. This means that if u: X → Y is an epimorphism in C, then the morphism T u of Mod A is covering, i.e., in accordance with Lemma 3.6, that for any object Z of C and U_{i} ∈ U we have the exact sequence

(4.2)  0 → Hom_{A}(h_{U_{i}}, T Z) → Hom_{A}(h_{U_{i}}Π_{T Y} T X, T Z) → Hom_{A}(Ker T u, T Z)

induced by an arbitrary A-homomorphism f: h_{U_{i}} → T Y. Since T is full and faithful and h_{U_{i}} = T U_{i}, there exists a morphism g: U_{i} → Y such that f = T g. Therefore, we have the commutative diagram

0 ⟶ Ker u ⟶^{i′} U_{i}Π_{Y} X ⟶^{u′} U_{i} ⟶ 0
↓  ↓^{g}
X ⟶_{u} Y ⟶ 0

with exact rows. The short exact sequence

0 ⟶ Ker u ⟶ U_{i}Π_{Y} X ⟶ U_{i} ⟶ 0

induces the exact sequence

0 ⟶ _{C}(U_{i}, Z) ⟶ _{C}(U_{i}Π_{Y} X, Z) ⟶ _{C}(Ker u, Z).

Since T is fully faithful, we have the exact sequence

0 ⟶ Hom_{A}(h_{U_{i}}, T Z) ⟶ Hom_{A}(T (U_{i}Π_{Y} X), T Z) ⟶ Hom_{A}(T (Ker u), T Z).

Since T is left exact, T (Ker u) is isomorphic to Ker(T u), T (U_{i}Π_{Y} X) is obviously isomorphic to h_{U_{i}}Π_{T Y} T X, and these isomorphisms are functorial. Thus, the sequence (4.2) is exact for all Z ∈ C, U_{i} ∈ U, and now it suffices to apply Lemma 3.6.

It remains to prove that T′ preserves direct sums. Actually, we prove a little more, namely, that it preserves direct unions. Suppose {X_{γ}}_{γ∈Γ} is a directed family of subobjects of X ∈ C such that X = ∑_{γ∈Γ} X_{γ}, where Γ is some set of indices. We need to show that the canonical monomorphism

u: ∑_{γ∈Γ} T X_{γ} ⟶ T ( ∑_{γ∈Γ} X_{γ})

is covering. Let q be an arbitrary morphism T U_{i} → Coker u. Since T U_{i} = h_{U_{i}} is a projective object of Mod A, there exists a morphism g: T U_{i} → T X such that q = vg, where v = Coker u. Since T is fully faithful, there exists a morphism f: U_{i} → X such that g = T f. Let X_{γ}Π_{X}U_{i} be the fibered product associated with the diagram

U_{i}
↓^{f}
X_{γ} ⟶_{u_{γ}} X,

where u_{γ} is the canonical monomorphism. Since T is left exact, T (X_{γ}Π_{X}U_{i}) is the fibered product associated with the diagram

T U_{i}
↓^{T f}
T X_{γ} ⟶_{T u_{γ}} T X.

Thus, we see that ∑_{γ∈Γ} T (X_{γ}Π_{X}U_{i}) is the fibered product associated with the diagram

T U_{i}
↓^{T f}
∑_{γ∈Γ} T X_{γ} ⟶_{T u} T X,

which implies that ∑_{γ∈Γ} T (X_{γ}Π_{X}U_{i}) ≈ ( ∑_{γ∈Γ} T X_{γ})Π_{T X}T U_{i}. Since

∑_{C}(X_{γ}Π_{X}U_{i}) ≈ ( ∑_{C} X_{γ})Π_{X}U_{i} = XΠ_{X}U_{i} ≈ U_{i}

and T is fully faithful, we obtain

Hom_{A} ( ∑_{γ∈Γ} T (X_{γ}Π_{X}U_{i}), T Z) ≈ \underleftarrow{lim} Hom_{A}(T (X_{γ}Π_{X}U_{i}), T Z)

≈ \underleftarrow{lim}_{C}((X_{γ}Π_{X}U_{i}), Z) ≈ _{C}(( ∑_{γ∈Γ} X_{γ}Π_{X}U_{i}), Z) ≈ _{C}(U_{i}, Z)

≈ Hom_{A}(h_{U_{i}}, T Z)

for all Z ∈ C. Hence, by Lemma 3.6, Coker u is negligible. □

So, the functor T′ is an equivalence. This completes the proof of the Popescu–Gabriel generalized theorem. □

Corollary 4.3. Let C be a Grothendieck category with a family of generators U = \{U_{i}\}_{i∈I}, and let M = \{M_{j}\}_{j∈J} an arbitrary family of objects of C. Put \overline{U} = U ∪ M and A = \{h_{U} = (−, U)\}_{U∈\overline{U}}. Then the functor T : C → Mod A, T X = (−, X), establishes equivalence between C and the quotient category Mod A/S, where S denotes the largest localizing subcategory in Mod A for which all modules T X are S-closed.

Proof. It suffices to observe that \overline{U} is again a family of generators for C. □

Corollary 4.4 (Popescu and Gabriel [22]). If C is a Grothendieck category, and U is a generator of C, then the functor _{C}(U, −) establishes equivalence between the categories C and Mod A/S, where A = End U is the endomorphism ring of U, and S is some localizing subcategory of Mod A.

Proof. By Theorem 4.1 the category C is equivalent to the quotient category of (U^{op}, Ab) with U = {U}. Since the representable functor h_{U} is a finitely generated projective generator for (U^{op}, Ab), from the Mitchell theorem it follows that (U^{op}, Ab) is equivalent to the category Mod A of right A-modules. □

Remark. In the proof of Lemma 4.2 we used Lemma 3.6 substantially; in its turn, the proof of the latter leans upon the fact that every A-module has an injective envelope, and this fact can be proved in different ways, independently of the Popescu–Gabriel theorem [22] (see, e.g., [16] or [5, §III.3.10]).

4.1. The Gabriel formula. Let P be a localizing subcategory of C. Via the functor T: C → Mod A, A = {h_{U_{i}}}_{i∈I}, we identify the category C and the quotient category Mod A/S. By Proposition 1.6, there exists a localizing subcategory T of Mod A such that T ⊇ S and T /S = P. Moreover, the quotient category C/P is equivalent to the quotient category Mod A/T .

Let 𝔉 be the Gabriel topology of A that corresponds to T . Like in the case of module categories, the family 𝔊 = ⋃_{i∈I} 𝔊^{i} with

𝔊^{i} = { X | X ⊆ U_{i}, U_{i}/X ∈ P }

will be called a Gabriel topology for U. It is easy to verify that 𝔊 ⊆ 𝔉 and

𝔊 = 𝔉_{S} = { 𝔞_{S} | 𝔞 ∈ 𝔉 }.

Proposition 4.5. For a Gabriel topology 𝔊 of U, the following conditions are satisfied: T1′) U_{i} ∈ 𝔊^{i} for each i ∈ I; T2′) if 𝔞 ∈ 𝔊^{i} and μ ∈ _{C}(U_{j}, U_{i}), U_{j} ∈ U, then { 𝔞 : μ } = μ^{−1}(𝔞) ∈ 𝔊^{j}; T3′) if 𝔞 and 𝔟 are subobjects of U_{i} such that 𝔞 ∈ 𝔊^{i} and { 𝔟 : μ } ∈ 𝔊^{j} for every

μ ∈ _{C}(U_{j}, U_{i}) with Im μ ⊂ 𝔞, U_{j} ∈ U, then 𝔟 ∈ 𝔊^{i}.

Proof. T1′ is obvious.

T2′. Since the left square of the commutative diagram

0 ⟶ μ^{−1}(𝔞) ⟶ U_{j} ⟶ U_{j}/μ^{−1}(𝔞) ⟶ 0
↓  ↓^{μ}  ↓^{δ}
0 ⟶ 𝔞 ⟶ U_{i} ⟶ U_{i}/𝔞 ⟶ 0

is pullback, δ is a monomorphism. Since P is closed under taking subobjects, we have U_{j}/μ^{−1}(𝔞) ∈ P.

T3′. Let 𝔞 ∈ 𝔊^{i}, and let μ: h_{U_{j}} → h_{U_{i}} be an A-homomorphism such that Im μ ⊆ 𝔞; then 𝔞 ∈ 𝔉. Since (−)_{T} = (−)_{P}(−)_{S}, we get

(h_{U_{j}}/μ^{−1}(𝔟))_{T} = (U_{j}/μ_{S}^{−1}(𝔟))_{P} = 0,

because U_{j}/μ_{S}^{−1}(𝔟) ∈ P. Therefore μ^{−1}(𝔟) ∈ 𝔉, and T3 implies that 𝔟 ∈ 𝔉. But 𝔟 is S-closed, and, consequently, 𝔟 ∈ 𝔊. □

Remark. In contrast to the Gabriel topology defined for a ring A = {P_{i}}_{i∈I}, conditions T1′–T3′ fail to imply that any family 𝔊 = ⋃_{i∈I} 𝔊^{i} satisfying these conditions determines some localizing subcategory in C. The sharp conditions (namely, axioms T1′–T2′ and an axiom somewhat stronger than T3′) ensuring that 𝔊 determines localization in C can be found in [23].

Further, the construction of the P-envelope X_{P} of an object X in C is similar to that in the case of Mod A (see also [24]). Namely, since the localizing functor (−)_{T} factors as (−)_{T} = (−)_{P}(−)_{S}, for the localization X_{P} of X we have the Gabriel formula

X_{P}(U_{i}) = X_{T} (U_{i}) = \underrightarrow{lim}_{𝔞∈𝔉^{i}}(𝔞, X/t_{T} (X)) = \underrightarrow{lim}_{𝔞∈𝔉^{i}}(𝔞, X/t_{P}(X))
≈ \underrightarrow{lim}_{𝔞_{S}∈𝔊^{i}}(𝔞_{S}, X/t_{P}(X)).

This isomorphism is functorial in both arguments. Here we have used the fact that t_{T} (X) = t_{P}(X) for every X ∈ C.

Using Proposition 3.5 and the properties of localizing subcategories, we can easily prove the following statement.

Proposition 4.6. Let M be some class of objects of C. Then an object S is M-negligible if and only if for an arbitrary morphism f: U_{i} → S, U_{i} ∈ U, the canonical homomorphism (U_{i}, M) → (Ker f, M) with M ∈ M is an isomorphism.

4.2. Projective generating families. Suppose U = {U_{i}}_{i∈I} is some family of objects in C. We denote by G_{U} the subcategory of the category C that consists of the objects generated by the family U. Namely, C ∈ G_{U} if and only if C admits a representation

⨁_{K} V_{k} ⟶ ⨁_{J} V_{j} ⟶ C ⟶ 0

with V_{k}, V_{j} ∈ U. We refer to U as a generating family for G_{U}. When U = {U}, we write G_{U} = G_{U}. Clearly, G_{U} = C if and only if U is the family of generators for C.

We consider the categories G_{U} generated by projective families U. So, let P be a projective object of C. For every M ∈ G_{P} there is a projective presentation

(4.3)  ⨁_{I} P_{i} ⟶ ⨁_{J} P_{j} ⟶ M ⟶ 0,

where I, J are some sets of indices, P_{i} = P_{j} = P for any i, j.

Theorem 4.7. Let C be a Grothendieck category, and let P be some projective object of C. Then the subcategory S = { C ∈ C | _{C}(P, C) = 0 } is localizing and the localized object P_{S} is a C/S-projective generator. Moreover, the localizing functor (−)_{S} induces an equivalence of the categories C/S and G_{P}.

Proof. First, we show that S is localizing. Let

0 ⟶ A′ ⟶ A ⟶ A′′ ⟶ 0

be a short exact sequence in C. Applying the exact functor _{C}(P, −), we obtain the short exact sequence

0 ⟶ _{C}(P, A′) ⟶ _{C}(P, A) ⟶ _{C}(P, A′′) ⟶ 0,

whence it follows that _{C}(P, A) = 0 if and only if _{C}(P, A′) = 0 and _{C}(P, A′′) = 0, i.e., S is a Serre subcategory. Next, the map

C(P, ⨁ A_{i}) ⟶^{φ} _{C}(P, ∏ A_{i}) ≈ ∏_{C}(P, A_{i}) = 0,

where A_{i} ∈ S and φ is the monomorphism induced by the canonical monomorphism ⨁ A_{i} → ∏ A_{i}, shows that S is closed under coproducts. So, S is a localizing subcategory.

In the remaining part of the proof we show that P_{S} is a projective generator for C/S. First, we consider a short exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0

in C/S. In C, it induces an exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ S ⟶ 0

with S ∈ S. Applying the exact functor _{C}(P, −), we get the following commutative diagram of Abelian groups:

0 ⟶ _{C}(P, A) ⟶ _{C}(P, B) ⟶ _{C}(P, C) ⟶ 0
↓  ↓  ↓
0 ⟶ _{C/S}(P_{S}, A) ⟶ _{C/S}(P_{S}, B) ⟶ _{C/S}(P_{S}, C) ⟶ 0,

where the vertical arrows are isomorphisms. We see that P_{S} is a C/S-projective object.

It remains to check that P_{S} is a generator for C/S. Let A be an arbitrary object of C/S, and let I = _{C}(P, A); then there exists a morphism u: ⨁_{i∈I} P_{i} → A with P_{i} = P, u_{i} = i for all i ∈ I. In C we have the exact sequence

⨁ P_{i} ⟶^{u} A ⟶ Coker u ⟶ 0.

Let w: P → A be an arbitrary morphism. By the construction of u, we have Im w ⊆ Im u. Therefore, there exists a morphism p: P → ⨁ P_{i} such that w = up (since P is projective), which implies the relation _{C}(P, Coker u) = 0, i.e., Coker u ∈ S.

Thus,

u_{S}: ( ⨁ P_{i})_{S} ≈ ⨁ (P_{i})_{S} ⟶ A_{S}

is a C/S-epimorphism. This shows that any object A ∈ C/S is a quotient object of ⨁(P_{i})_{S}, and so P_{S} is a generator for C/S.

Finally, we check that the restriction of (−)_{S} to G_{P} determines an equivalence of C/S and G_{P}. If M ∈ C, we have the exact sequence

0 ⟶ A′ ⟶ M ⟶^{λ} M_{S} ⟶ A′′ ⟶ 0

of the form (1.1) with A′, A′′ ∈ S. Hence, λ induces the isomorphisms

(4.4)  _{C}(P, M) ≈ _{C}(P, M_{S}) ≈ _{C/S}(P_{S}, M_{S}).

Then

C( ⨁ P_{i}, M) ≈ ∏_{C}(P_{i}, M) ≈ _{C}( ⨁ P_{i}, M_{S}) ≈ _{C/S}( ⨁(P_{i})_{S}, M_{S})

with P_{i} = P for any i ∈ I. Now, if we consider the projective presentation (4.3) for M ∈ G_{P}, we obtain the commutative diagram

0 ⟶ _{C}(M, N) ⟶ _{C}(⨁ P_{j}, N) ⟶ _{C}(⨁ P_{i}, N)
↓  ↓
0 ⟶ _{C}(M, N_{S}) ⟶ _{C}(⨁ P_{j}, N_{S}) ⟶ _{C}(⨁ P_{i}, N_{S})

in which the rows are exact and the vertical arrows are isomorphisms. Thus,

C(M, N) ≈ _{C}(M, N_{S}) ≈ _{C/S}(M_{S}, N_{S}),

which implies that (−)_{S}|_{G_{P}} is a fully faithful functor. Now, let N ∈ C/S. Consider a C/S-projective presentation of N of the form

⨁(P_{i})_{S} ⟶^{α} ⨁(P_{j})_{S} ⟶ N ⟶ 0,

where P_{i} = P_{j} = P. Then there exists β: ⨁ P_{i} → ⨁ P_{j} such that α = β_{S}; consequently, N ≈ (Coker β)_{S}, as required. □

Corollary 4.8. Under the conditions of Theorem 4.7, the quotient category C/S is equivalent to the quotient category Mod A/P, where A = End P is the endomorphism ring of P, and P is some localizing subcategory of Mod A.

Proof. It suffices to observe that the isomorphisms (4.4) ensure ring isomorphism

A = _{C}(P, P) ≈ _{C}(P, P_{S}) ≈ _{C/S}(P_{S}, P_{S}).

Since P_{S} is a generator for C/S, our assertion follows from Corollary 4.4. □

Now, we consider an arbitrary family of projective objects U = {P_{i}}_{i∈I}. Let G_{U} be the category of C generated by U.

Corollary 4.9. The subcategory S = { C ∈ C | _{C}(P, C) = 0 for all P ∈ U } is localizing, and C/S is equivalent to the quotient category Mod A/P, where A = {h_{P} }_{P ∈U}, and P is some localizing subcategory of Mod A. Moreover, the functor (−)_{S} induces an equivalence of the categories C/S and G_{U}.

Proof. We denote Q = ⨁_{P ∈U} P; then S = { C ∈ C | _{C}(Q, C) = 0 }. The preceding theorem implies that S is localizing, and Q_{S} = ⨁_{P ∈U} P_{S} is a C/S-projective generator. Therefore, U_{S} = { P_{S} | P ∈ U } is a family of projective generators for C/S.

The isomorphisms (4.4) show that U and U_{S} are equivalent. Now our assertion follows from Theorem 4.1. □

Remark. Under the conditions of Corollary 4.9, the localizing functor (−)_{S} factors through G_{U},

\xymatrix{C \ar[rr]^{(−)_{S}} \ar[dr]_{F} &  & C/S \\  & G_{U}; \ar[ur]_{G}}

here G is the restriction of (−)_{S} to G_{U}, and F is constructed as follows. If C ∈ C and I = ⋃_{P ∈U} I_{P} with I_{P} = _{C}(P, C), then there is a morphism φ: ⨁_{μ∈I} P_{μ} → C such that φ_{μ} = μ. In a similar way we can construct a morphism ψ: ⨁_{ν∈J} P_{ν} → Ker φ with J = ⋃_{P ∈U} J_{P}, J_{P} = _{C}(P, Ker φ). This yields the following commutative diagram:

\xymatrix{⨁_{ν∈J} P_{ν} \ar[rr]^{ζ} \ar[dr]_{ψ} &  & ⨁_{μ∈I} P_{μ} \ar[r]^{φ} & C \\  & Ker φ, \ar[ur]_{i}}

where i is the kernel of φ. By definition, we put F (C) = Coker ζ.

Let A = {P_{i}}_{i∈I} be a family of finitely generated projective objects of C. By the preceding arguments, we have isomorphism G_{A} ≈ (A^{op}, Ab).

Theorem 4.10. Suppose C is a Grothendieck category and A = {P_{i}}_{i∈I} is some family of finitely generated projective objects for C. Then the subcategory S = { C ∈ C | _{C}(P_{i}, C) = 0 for all P_{i} ∈ A } is localizing, and A_{S} = {(P_{i})_{S}}_{i∈I} is a family of finitely generated projective generators for the quotient category C/S. Moreover, the localizing functor (−)_{S} induces an equivalence of C/S and Mod A.

Proof. By Corollary 4.9, A_{S} is a family of projective generators for C/S. It remains to show that every (P_{i})_{S} ∈ A_{S} is C/S-finitely generated.

For this, we consider an object X of C/S. Let {X_{j}}_{j∈J} be a directed family of C/S-subobjects of X such that X = ∑_{C/S} X_{j}. Since the X_{j} are also C-subobjects, it follows that the C-direct union ∑_{C} X_{j} is a subobject of X and that the quotient object A = X/ ∑_{C} X_{j} belongs to S. Indeed, if we apply the exact localizing functor (−)_{S}, commuting with the direct limits, to the short exact sequence

0 ⟶ ∑_{C} X_{j} ⟶ X ⟶ A ⟶ 0,

we obtain the short exact sequence

0 ⟶ ( ∑_{C} X_{j})_{S} = ∑_{C/S} X_{j} ⟶ X ⟶ A_{S} ⟶ 0.

This shows that A_{S} = 0, i.e., A ∈ S. Consequently, for any P_{i} ∈ A we have

C(P_{i}, ∑_{C} X_{j}) ≈ _{C}(P_{i}, X) ≈ _{C/S}((P_{i})_{S}, ∑_{C/S} X_{j}).

Thus,

C/S((P_{i})_{S}, ∑_{C/S} X_{j}) ≈ _{C}(P_{i}, ∑_{C} X_{j}) ≈ \underrightarrow{lim}_{C}(P_{i}, X_{j}) ≈ \underrightarrow{lim}_{C/S}((P_{i})_{S}, X_{j}).

By Theorem 1.8, (P_{i})_{S} ∈ fg C/S.

The proof of the fact that the localizing functor (−)_{S} induces an equivalence between C/S and Mod A is similar to that of Theorem 4.7. □

Corollary 4.11 [10, 2.1]. Let C be a Grothendieck category and P some finitely generated projective object of C. Then the subcategory S = { C ∈ C | _{C}(P, C) = 0 } is localizing, and the functor _{C/S}(P_{S}, −) determines an equivalence of the categories C/S and Mod A, where A = End P the endomorphism ring of the object P.

Proof. By the preceding theorem, the localized object P_{S} is a finitely generated projective generator. Applying the Mitchell theorem and Corollary 4.8, we see that the quotient category C/S is equivalent to the category of modules over the ring A = End_{C} P. □

Example. Consider the category C_{A} = (mod A^{op}, Ab) of generalized right A-modules. For M ∈ mod A^{op}, we put R = End_{A} M and S_{M} = { F ∈ C_{A} | F (M) = 0 }. By Corollary 4.11, there exists an equivalence of categories C_{A}/S_{M} →^{h} Mod R, where h(F_{S_{M}}) = F_{S_{M}} (M) = F (M) for any F ∈ C_{A}. The functor quasiinverse to h is constructed as follows: g: Mod R → C_{A}/S_{M}, E ↦^{g} ((M, E) ⊗_{A} −)_{S_{M}}. In particular, for any F ∈ C_{A} we have isomorphism F_{S_{M}} ≈ ((M, F (M)) ⊗_{A} −)_{S_{M}}.

Suppose Mod A, A = {P_{i}}_{i∈I}, is the category of right A-modules. For any P_{i} ∈ A we put

S_{i} = { M ∈ Mod A | Hom_{A}(P_{i}, M) = 0 }.

By Corollary 4.11, we have Mod A/S_{i} ≈ Mod A_{i} with A_{i} = End P_{i}. We regard this equivalence as identification. Then

⋃_{P_{i}∈A} Mod A_{i} ⊆ Mod A.

As the next example shows, generally speaking, the categories Mod A_{i} of A_{i}-modules do not cover Mod A entirely.

Example. Consider the category_{ℤ}C = (mod ℤ, Ab) of generalized Abelian groups. In §1 we mentioned that the functor M ↦ − ⊗ M identifies the pure-injective Abelian groups and the injective objects of_{ℤ}C. By the Kaplansky theorem (see [25]), the indecomposable pure-injective Abelian groups Zg_{ℤ}C are precisely the Abelian groups of the following type:

1) the injective modules ℚ and, for every prime p, the Prüfer groups ℤ_{p^{∞}};

2) the cyclic groups ℤ_{p^{n}} of order p^{n}, where p is a prime;

3) for every prime p, the p-adic completion \overline{ℤ}_{p} = \underleftarrow{lim} ℤ_{p^{n}} of the integers. In [6] Ziegler showed that a similar argument applies to Dedekind domains.

Let S_{M} = { C ∈ _{ℤ}C | C(M) = 0 } with M ∈ mod ℤ. Since any finitely generated Abelian group is isomorphic to ℤ^{⊕m} ⊕ ℤ_{p_{i}^{k_{i}}}^{⊕n} with p_{1}, …, p_{n} prime, it suffices to show

that

Mod ℤ ⋃[ ⋃_{p is prime} Mod ℤ_{p^{n}}] ⊊ _{ℤ}C.

In accordance with [10, 2.4], − ⊗ Q ∈ Zg_{ℤ}C belongs to _{ℤ}C/S_{M} ≈ Mod A_{M} with A_{M} = End M if and only if Q ≈ Hom_{A_{M}} (M, E), where E is an indecomposable injective A_{M}- module. In this case, E ≈ M ⊗ Q. Clearly, ℚ, ℤ_{p^{∞}} ∈ _{ℤ}C/S_{ℤ} and ℤ_{p^{n}} ∈ _{ℤ}C/S_{ℤ_{p^{n}}}. However \overline{ℤ}_{p} does not satisfy the condition indicated. Indeed, it fails to be an injective Abelian group, so that − ⊗ \overline{ℤ}_{p} ∉ _{ℤ}C/S_{ℤ}. Since for prime p ≠ q the object (ℤ_{p}, −) belongs to S_{ℤ_{q^{n}}}, and ℤ_{p} ⊗ \overline{ℤ}_{p} ≈ ℤ_{p} ≠ 0, it follows that t_{S_{ℤ_{q^{n}}}} (− ⊗ Q) ≠ 0, where t_{S_{ℤ_{q^{n}}}} is the S_{ℤ_{q^{n}}} - torsion functor. Hence, by Lemma 1.3, − ⊗ \overline{ℤ}_{p} ∉ _{ℤ}C/S_{ℤ_{q^{n}}}. Finally, − ⊗ \overline{ℤ}_{p} ∉ _{ℤ}C/S_{ℤ_{p^{n}}}, because (ℤ_{p^{n}}, ℤ_{p^{n}} ⊗ \overline{ℤ}_{p}) ≈ ℤ_{p^{n}} ≉ \overline{ℤ}_{p}.

§5. Finiteness conditions for localizing subcategories

Let S be a localizing subcategory of a Grothendieck category C. In this section we show how some properties for the family of generators U = {U_{i}}_{i∈I} of the category C describe various finiteness conditions for S.

Consider the inclusion functor i: C/S → C. We say that S is of prefinite type (respectively of finite type) if i commutes with the direct unions (respectively, with the direct limits), i.e., for any S-closed object C and any directed family {C_{i}}_{I} of S-closed subobjects of C we have

∑_{C/S} C_{i} = ∑_{C} C_{i}

(respectively, \underrightarrow{lim}_{C/S} C_{i} = \underrightarrow{lim}_{C} C_{i}).

It should be noted that if S is of prefinite type, then, in particular, every direct sum of S-closed objects is S-closed. We also recall that a subcategory B ⊆ A of an Abelian category A is said to be exact if it is Abelian and the inclusion functor of B into A is exact.

Proposition 5.1 [16, 3.41]. A subcategory B of A is an exact subcategory if and only if the following two conditions are fulfilled:

(1) if B_{1}, B_{2} ∈ B, then the coproduct B_{1} ⊕ B_{2} is an object of B;

(2) if β: B_{1} → B_{2} is a morphism in B, then both the A-kernel and A-cokernel of β are objects of B.

Finally, a subcategory B of an Abelian category A is said to be coexact if the perpendicular subcategory

B^{⊥} = { A ∈ A | (B, A) = 0 , Ext^{1}(B, A) = 0 for all B ∈ B }

is exact. For instance, if C is a Grothendieck category, and S ⊆ C is localizing, then S is coexact if and only if the quotient category C/S is exact, because C/S = S^{⊥}.

Proposition 5.2. If S ⊆ C is a coexact localizing subcategory, then the localization P_{S} of any projective object P of C is C/S-projective. If C has a family of projective generators A = {P_{i}}_{i∈I} and every (P_{i})_{S} is C/S-projective, then S is coexact.

Proof. If S is coexact, then any short exact sequence

(5.1)  0 ⟶ A ⟶ B ⟶^{β} C ⟶ 0

in C/S is also exact in C. Suppose P ∈ C is projective; then we have the commutative diagram

0 ⟶ _{C}(P, A) ⟶ _{C}(P, B) ⟶^{(P,β)} _{C}(P, C) ⟶ 0
↓  ↓  ↓
0 ⟶ _{C/S}(P_{S}, A) ⟶ _{C/S}(P_{S}, B) ⟶_{(P_{S},β)} _{C/S}(P_{S}, C) ⟶ 0,

where the rows are exact and the vertical arrows are isomorphisms. Hence, P_{S} is C/Sprojective.

Conversely, we must show that any C/S-epimorphism is a C-epimorphism. Consider the exact sequence (5.1). By Lemma 1.4, C/ Im_{C} β ∈ S. By assumption, the morphism (P_{S}, β) with P ∈ A is an epimorphism; hence, (P, β) is an epimorphism. Therefore, (P, C/ Im_{C} β) = 0 for every P ∈ A. Since A is a family of generators, we conclude that C/ Im_{C} β = 0. □

The following statement characterizes the coexact localizing subcategories of finite or prefinite type.

Proposition 5.3. For a coexact localizing subcategory S of a Grothendieck category C the following statements are equivalent:

(1) S is of finite type;

(2) S is of prefinite type;

(3) the inclusion functor i: C/S → C commutes with the coproducts.

Proof. (1) ⟹ (2) ⟹ (3). Obvious.

(3) ⟹ (1). Let \underrightarrow{lim}_{C/S} C_{i} be the C/S-direct limit of C_{i} ∈ C/S, i ∈ I, and let Λ denote the subset of I × I that consists of the pairs (i, j) with i ≤ j. For any λ ∈ Λ we put s(λ) = i, t(λ) = j. By [3, IV.8.4], we have

\underrightarrow{lim}_{C/S}C_{i} = Coker_{C/S}[⨁_{λ∈Λ} C_{s(λ)} ⟶^{φ} ⨁_{i∈I} C_{i}],

where φ is induced by φ_{λ} = u_{j}γ_{ij} −u_{i}: C_{s(λ)} → ⨁_{i∈I} C_{i}, λ = (i, j), and the γ_{ij}: C_{s(λ)} → C_{t(λ)} are the canonical morphisms. Since, by assumption, the inclusion functor C/S → C is exact and commutes with the coproducts, we obtain

\underrightarrow{lim}_{C/S}C_{i} = Coker_{C}[⨁_{λ∈Λ} C_{s(λ)} ⟶^{φ} ⨁_{i∈I} C_{i}] = \underrightarrow{lim}_{C}C_{i},

which implies the claim. □

We shall say that a localizing subcategory S of a Grothendieck category C is strongly coexact if the S-torsion functor t_{S} is exact.

Proposition 5.4. For a localizing subcategory S of a Grothendieck category C the following statements are equivalent:

(1) S is strongly coexact;

(2) every quotient object of an S-torsionfree object is S-torsionfree;

(3) every S-torsionfree object is S-closed;

(4) every subobject of an S-closed object is S-closed.

Proof. The implications (1) ⟹ (2) and (3) ⟺ (4) are obvious.

(2) ⟹ (3). Let t_{S}(X) = 0. Consider the short exact sequence

0 ⟶ X ⟶ X_{S} ⟶ A ⟶ 0

with A ∈ S. Since t_{S}(X_{S}) = 0, we get A = 0.

(3), (4) ⟹ (1). Any short exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0

in C can be embedded in the following commutative diagram:

0  0  0
↓  ↓  ↓
0 ⟶ t(A) ⟶ t(B) ⟶^{α} t(C)
↓  ↓  ↓
0 ⟶ A ⟶ B ⟶ C ⟶ 0
φ↓  ^{ψ}↓  ↓^{γ}
0 ⟶ A_{S} ⟶ B_{S} ⟶_{δ} C_{S}.

Since, by assumption, every S-torsionfree object is S-closed, the morphisms φ, ψ, γ, δ are epimorphisms. By the (3 × 3)-lemma, α is an epimorphism. □

Corollary 5.5. S is strongly coexact if and only if C/S = { X/t_{S}(X) | X ∈ C }.

Corollary 5.6. Every strongly coexact localizing subcategory is a coexact subcategory of finite type.

Proof. Since a C/S-morphism α: X → Y is a C/S-epimorphism if and only if Y/ Im_{C} α ∈ S, it is easy to see that the strongly coexact subcategories are coexact.

Now, if X = ∑_{C/S} X_{i}, X_{i} ∈ C/S, then, obviously, ∑_{C} X_{i} is a subobject of X. By Proposition 5.4, ∑_{C} X_{i} is an S-closed object, and by Proposition 5.3(2), S is of finite type. □

Lemma 5.7. Let S be a localizing subcategory of C. The S-torsion functor t = t_{S} preserves the direct limits if and only if every C-direct limit of S-closed objects is Storsionfree.

Proof. Suppose that t preserves the direct limits. If {C_{i}}_{I} is a directed system of S-closed objects, then

t(\underrightarrow{lim}_{C}C_{i}) = \underrightarrow{lim} t(C_{i}) = 0

because t(C_{i}) = 0. Consequently, \underrightarrow{lim}_{C} C_{i} is S-torsionfree.

Conversely, we have the exact sequence

0 ⟶ t(C_{i}) ⟶ C_{i} ⟶^{λ_{C_{i}}} (C_{i})_{S}.

Since the direct limit functor is exact, we get the exact sequence

(5.2)  0 ⟶ \underrightarrow{lim} t(C_{i}) ⟶ \underrightarrow{lim} C_{i} ⟶ \underrightarrow{lim}(C_{i})_{S}.

Observe that \underrightarrow{lim} t(C_{i}) ∈ S because S is closed under taking direct limits. Now, applying t to (5.2), we obtain

\underrightarrow{lim} t(C_{i}) = t(\underrightarrow{lim} t(C_{i})) = t(\underrightarrow{lim} C_{i})

because t(\underrightarrow{lim}(C_{i})_{S}) = 0. □

Let 𝔊 be the Gabriel topology for U that corresponds to S. By a basis for the topology 𝔊 we mean a subset 𝔅 of 𝔊 such that every object in 𝔊 contains some 𝔟 ∈ 𝔅.

Theorem 5.8. Suppose C is a locally finitely generated Grothendieck category with a family of finitely generated generators U ⊆ fg C, and S is a localizing subcategory of C. The following conditions are equivalent:

(1) S is of prefinite type;

(2) for any U ∈ U the natural morphism \underrightarrow{lim}_{C}(U, C_{i}) → _{C}(U, ∑_{C/S} C_{i}) induced by the S-envelope λ of ∑_{C} C_{i} is an isomorphism;

(3) U_{S} = {U_{S}}_{U∈U} is the family of C/S-finitely generated generators for C/S;

(4) if C is C-finitely generated, then C_{S} is C/S-finitely generated;

(5) the torsion functor t commutes with the direct limits;

(6) any C-direct limit of S-closed objects is S-torsionfree;

(7) 𝔊 has a basis of finitely generated objects.

Thus, if conditions (1)–(7) are fulfilled, then C/S is a locally finitely generated Grothendieck category, and every C/S-finitely generated object D is the localization C_{S} of some C ∈ fg C.

Proof. The equivalence (5) ⟺ (6) follows from Lemma 5.7.

(1) ⟹ (2). Since ∑_{C/S} C_{i} = ∑_{C} C_{i} by assumption, our statement follows from Theorem 1.8.

(2) ⟹ (3). Let λ: ∑_{C} C_{i} → ∑_{C/S} C_{i} be the S-envelope of ∑_{C} C_{i}. By assumption, the composed map

\underrightarrow{lim}_{C}(U, C_{i}) ⟶^{Φ} _{C}(U, ∑_{C} C_{i}) ⟶^{(U,λ)} _{C}(U, ∑_{C/S} C_{i}),

where Φ is the canonical morphism, is an isomorphism. Hence,

\underrightarrow{lim}_{C/S}(U_{S}, C_{i}) ≈ \underrightarrow{lim}_{C}(U, C_{i}) ≈ _{C}(U, ∑_{C/S} C_{i}) ≈ _{C/S}(U_{S}, ∑_{C/S} C_{i}).

Applying Theorem 1.8, we see that U_{S} ∈ fg C/S.

(3) ⟹ (4). For C ∈ fg C there is an epimorphism η: ⨁_{i=1}^{n} U_{i} → C with some U_{1}, …, U_{n} ∈ U. Since ⨁_{i=1}^{n}(U_{i})_{S} ∈ fg C/S, we get C_{S} ∈ fg C/S.

(4) ⟹ (7). Suppose 𝔞 ∈ 𝔊, i.e., 𝔞_{S} = U_{S} for some U ∈ U. We write 𝔞 = ∑_{C} 𝔞_{i} as a directed sum of C-finitely generated subobjects 𝔞_{i}. Then U_{S} = ∑_{C/S}(𝔞_{i})_{S}. By assumption, there exists i_{0} such that U_{S} = (𝔞_{i_{0}})_{S}, whence 𝔞_{i_{0}} ∈ 𝔊.

(7) ⟹ (1). First, we show that U_{S} ∈ fg C/S for U in U. Indeed, let U_{S} = ∑_{C/S} 𝔞_{i}. Then 𝔞 = λ_{U}^{−1}(∑_{C} 𝔞_{i}) = ∑_{C} λ_{U}^{−1}(𝔞_{i}) ∈ 𝔊, where λ_{U} is the S-envelope of U. By assumption, there is a finitely generated subobject 𝔟 of 𝔞 such that 𝔟 ∈ 𝔊. Then there exists i_{0} such that 𝔟 ⊆ λ_{U}^{−1}(𝔞_{i_{0}}). Since

U_{S} = 𝔟_{S} ⊆ (λ_{U}^{−1}(𝔞_{i_{0}}))_{S} ⊆ 𝔞_{i_{0}} ⊆ U_{S},

we have 𝔞_{i_{0}} = U_{S}, whence U_{S} ∈ fg C/S.

Next, since the isomorphism

\begin{aligned}_{C}(U, ∑_{C/S} C_{i}) & ≈ _{C/S}(U_{S}, ∑_{C/S} C_{i}) ≈ \underrightarrow{lim}_{C/S}(U_{S}, C_{i}) ≈ \underrightarrow{lim}_{C}(U, C_{i}) \\ & ≈ _{C}(U, ∑_{C} C_{i}) \end{aligned}

is functorial in U, we obtain ∑_{C/S} C_{i} = ∑_{C} C_{i}.

(1) ⟹ (6). The direct limit \underrightarrow{lim}_{I} C_{i} can be written as a quotient object of the coproduct ⨁_{I} C_{i}. More precisely, let R be the subset of I × I that consists of all pairs (i, j) with i ≤ j. For every S ⊆ R we put

C_{S} = ∑_{(i,j)∈S} Im(u_{i} − u_{j}γ_{ij}) ⊆ ⨁ C_{i},

where u_{i}: C_{i} → ⨁ C_{i} is the canonical monomorphism for i ∈ I, and γ_{ij}: C_{i} → C_{j} is the canonical monomorphism for i ≤ j. Then \underrightarrow{lim} C_{i} = ⨁ C_{i}/C_{R} = ⨁ C_{i}/ ∑ C_{S}, where S runs over all finite subsets of R. By assumption, both ⨁ C_{i} and ∑ C_{i} are S-closed; hence, being a quotient object of S-closed objects, \underrightarrow{lim} C_{i} is S-torsionfree.

(5) ⟹ (1). Suppose X is S-closed. We write X = ∑_{C/S} X_{i} as a direct union of Sclosed subobjects. Then X/ ∑_{C} X_{i} = t(X/ ∑_{C} X_{i}) = t(\underrightarrow{lim}_{C} X/X_{i}) = \underrightarrow{lim} t(X/X_{i}) = 0, because t(X/X_{i}) = 0. Consequently, ∑_{C/S} X_{i} = ∑_{C} X_{i}.

In turn, let D ∈ fg C/S. Writing D = ∑_{C} D_{i} as a direct union of D_{i} ∈ fg C, we see that D = D_{S} = ∑_{C/S}(D_{i})_{S}, whence D = (D_{i_{0}})_{S} for some i_{0}. □

Proposition 5.9. Suppose C is a locally finitely presented Grothendieck category with a family of finitely presented generators U ⊆ fp C, and S is a localizing subcategory of C. The following conditions are equivalent:

(1) S is of finite type;

(2) for every U ∈ U the canonical morphism \underrightarrow{lim}_{C}(U, C_{i}) → _{C}(U, \underrightarrow{lim}_{C/S} C_{i}) induced by the S-envelope λ of the object \underrightarrow{lim}_{C} C_{i} is an isomorphism;

(3) U_{S} = {U_{S}}_{U∈U} is the family of C/S-finitely presented generators for C/S;

(4) if C ∈ fp C, then C_{S} ∈ fp C/S.

Thus, if these conditions are fulfilled, then C/S is a locally finitely presented Grothendieck category, and every C/S-finitely presented object D is the localization C_{S} of some C ∈ fp C.

Proof. It suffices to observe that for every C ∈ fp C the representable functor (C, −) commutes with the direct limits (see Theorem 1.9), and there is a presentation

⨁_{i=1}^{n} U_{i} ⟶ ⨁_{j=1}^{m} U_{j} ⟶ C ⟶ 0

of C by objects in U. Now, the theorem can be proved by word for word repetition of the proof of Theorem 5.8.

In turn, if D ∈ fp C/S, there is an epimorphism η: B_{S} → D with B ∈ fp C. Then Ker η ∈ fg C/S. Referring to [14, 2.13], we can choose C ⊆ B such that C ∈ fg C and C_{S} = Ker η. Hence, D = (B/C)_{S}. □

Now, we consider a localizing subcategory S of the module category Mod A, where A = {P_{i}}_{i∈I} is some ring. Then the family A_{S} = {P_{S}}_{P ∈A} generates Mod A/S, and we call this family the ring of quotients of A with respect to S.

Consider the following (natural) functor j: Mod A/S → Mod A_{S} = (A_{S}^{op}, Ab):

Hom_{A_{S}} (P_{S}, j(M)) := Hom_{A}(P, M)

for every P ∈ A and M ∈ Mod A/S. The next statement is an analog of the Walker and Walker theorem (see [3, XI.3.4]).

Proposition 5.10. The functor j: Mod A/S → Mod A_{S} is an equivalence if and only if the localizing subcategory S is coexact and of finite type.

Proof. Suppose j is an equivalence. Then every P_{S} is finitely generated and projective in Mod A/S. Proposition 5.2 implies that S is coexact, and it is of finite type by Proposition 5.9.

Conversely, let S be coexact and of finite type. By Proposition 5.2, every P_{S} is projective in Mod A/S; by Proposition 5.9, P_{S} is finitely generated in Mod A/S. Now, our statement follows from Proposition 2.2. □

5.1. Left exact functors. Let C be a locally finitely presented Grothendieck category. By Theorem 4.1, the functor T: C → Mod A with A = {h_{X}}_{X∈fp C} induces an equivalence of C and Mod A/S, where S is some localizing subcategory of Mod A.

We denote by L the subcategory of Mod A consisting of all L ∈ Mod A such that for every x ∈ L(X) with X ∈ fp C there exists an epimorphism f: Y → X satisfying L(f)(x) = 0. We leave to the reader to check that L is a localizing subcategory. Let 𝔉 = ⋃_{X∈fp C} 𝔉^{X} denote the corresponding Gabriel topology for A.

Lemma 5.11. If X′ →^{p} X →^{f} X′′ → 0 is an exact sequence in fp C, then Coker(T f) ∈ L.

Proof. Let Y ∈ fp C and y ∈ Coker(T f)(Y). There is a morphism g ∈ _{C}(Y, X′′) such that u_{y} = r∘T g, where r: h_{X′′} → Coker(T f) is the canonical epimorphism and u_{y,Y} (1_{Y}) = y. In C, consider the commutative diagram

XΠ_{X′′}Y ⟶^{f′} Y
g′↓  ↓^{g}
X ⟶_{f} X′′

in which f′ is an epimorphism. Since Y ∈ fp C, there is a finitely generated subobject Z′ of XΠ_{X′′}Y such that f′(Z′) = Y. There exists an epimorphism h: Z → Z′ with Z ∈ fp C. So, f′h is an epimorphism. It is easy to show that Coker(T f)(f′h)(y) = 0, whence Coker(T f) ∈ L. □

We recall that a functor M ∈ Mod A, A = {h_{X}}_{X∈fp C}, is left exact provided that for any exact sequence

X′ ⟶ X ⟶ X′′ ⟶ 0

in fp C the sequence of Abelian groups

0 ⟶ M(X′′) ⟶ M(X) ⟶ M(X′)

is exact.

Proposition 5.12. Every left exact functor M ∈ Mod A is L-closed.

Proof. Let M: fp C → Ab be a contravariant left exact functor, and let X ∈ fp C and x ∈ M(X) be such that Ker u_{x} ∈ 𝔉^{X}. Here, u_{x} denotes a unique morphism such that u_{x,X}(1_{X}) = x. Then there exists an epimorphism f: Y → X in fp C satisfying M(f)(x) = 0. Since M(f) is a monomorphism of Abelian groups, we have x = 0. So, M is L-torsionfree.

Now, let 𝔞 ∈ 𝔉^{X} and let g: 𝔞 → M be a morphism in Mod A. There exists an epimorphism f: Y → X such that Im(T f) ⊆ 𝔞. By Lemma 5.11, Im(T f) ∈ 𝔉^{X}. Consider the morphism g ∘ T f = t: h_{Y} → M. Since Ker f ∈ fg C, there exists an epimorphism Z → Ker f, and we arrive at the exact sequence

Z ⟶^{p} Y ⟶^{f} X ⟶ 0.

We have the following commutative diagram of Abelian groups:

0 ⟶ h_{Y} (X) ⟶^{h_{Y} (f)} h_{Y} (Y) ⟶^{h_{Y} (p)} h_{Y} (Z)
t_{X} ↓  ↓^{t_{Y}}  ↓^{t_{Z}}
0 ⟶ M(X) ⟶_{M(f)} M(Y) ⟶_{M(p)} M(Z).

Thus,

(M(p)t_{Y})(1_{Y}) = (t_{Z}h_{Y} (p))(1_{Y}) = (g_{Z}(T f)_{Z}h_{Y} (p))(1_{Y}) = g_{Z}(fp) = 0.

Then there exists an element x ∈ M(X) such that M(f)(x) = t_{Y} (1_{Y}) = (gT f)_{Y} (1_{Y}) = g_{Y} (f). Hence, u_{x}|_{Im(T f)} = g|_{Im(T f)}. Since Im(T f) ∈ 𝔉^{X} and M is L-torsionfree, it follows that u_{x}|_{𝔞} = g. Thus, u_{x}: h_{X} → M is a morphism extending g. The uniqueness of u_{x} follows from the fact that M is L-torsionfree. So, M is L-closed. □

An object C of a Grothendieck category C with a family of generators U is said to be U-finitely presented if there is an exact sequence

⨁_{k=1}^{n} Y_{k} ⟶ ⨁_{j=1}^{m} X_{j} ⟶ C ⟶ 0

with Y_{k}, X_{j} ∈ U. The corresponding subcategory of U-finitely presented objects will be denoted by fp_{U} C.

We define Lex((fp C)^{op}, Ab) to be the category of contravariant left exact functors from fp C to Ab. Now, we are in a position to prove the following result.

Theorem 5.13 (Breitsprecher [26]). Let C be a Grothendieck category with a family of generators U = {U_{i}}_{i∈I}. Then the representation functor T = (−,?): C → ((fp_{U} C)^{op}, Ab) determines an equivalence between C and ((fp_{U} C)^{op}, Ab)/S, where S is some localizing subcategory of ((fp_{U} C)^{op}, Ab). Moreover, S is of finite type if and only if fp_{U} C = fp C. In this case, C is equivalent to the category Lex((fp C)^{op}, Ab).

Proof. The first part of the theorem easily follows from Corollary 4.3 and Proposition 5.9. Now, let fp_{U} C = fp C. Our statement will be proved if we show that S = L. Since for every X ∈ C the functor T (X) is left exact, Proposition 5.12 implies that T (X) is Lclosed, and, consequently, L ⊆ S. Conversely, let M ∈ S. We can choose a projective presentation for M

⨁ h_{Y_{j}} ⟶^{g} ⨁ h_{X_{i}} ⟶ M ⟶ 0.

Proposition 5.9 implies that S is of finite type. Therefore, any coproduct of S-closed objects is an S-closed object. Thus, there exists a morphism f ∈ Mor C such that T f = g. Furthermore, f is an epimorphism in C because M_{S} = 0. So, there is no loss of generality in assuming that for every M ∈ S there exists an exact sequence

T Y ⟶^{T f} T X ⟶ M ⟶ 0,

where f is a C-epimorphism.

In accordance with [27, 5.7], f is a direct limit f_{α}: Y_{α} → X_{α} of epimorphisms f_{α} in fp C, so that M ≈ \underrightarrow{lim} Coker(T f_{α}). By Lemma 5.11, every Coker(T f_{α}) belongs to L, whence M ∈ L. □

The next result characterizes the localizing subcategories of finite type for locally coherent Grothendieck categories.

Theorem 5.14. Let C be a locally coherent Grothendieck category with a family of coherent generators U ⊆ coh C. The following conditions are equivalent:

(1) S is of finite type;

(2) S is of prefinite type;

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

(4) any C-direct limit of S-closed objects is S-torsionfree;

(5) for any U ∈ U the natural morphism \underrightarrow{lim}_{C}(U, C_{i}) → _{C}(U, \underrightarrow{lim}_{C/S} C_{i}) induced by the S-envelope λ of \underrightarrow{lim}_{C} C_{i} is an isomorphism;

(6) U_{S} = {U_{S}}_{U∈U} is the family of C/S-coherent generators for C/S;

(7) if C is C-coherent, then C_{S} is C/S-coherent;

(8) 𝔊 has a basis of coherent objects;

Thus, if these conditions are fulfilled, then the category C/S is locally coherent, and every C/S-coherent object D is the localization C_{S} of some C ∈ coh C.

Proof. (1) ⟹ (5). This follows from Proposition 5.9.

(5) ⟹ (6). Proposition 5.9 shows that U_{S} ⊆ fp C/S. Now, we check that for any U ∈ U the object U_{S} is coherent. Let C ⊆ U_{S} be a C/S-finitely generated subobject of U_{S}. By [14, 2.13], there exists a C-finitely generated subobject A ⊆ U such that A_{S} = C. Since U is coherent, the object A belongs to coh C. By Proposition 5.9, A_{S} ∈ fp C/S.

(6) ⟹ (7). Obvious.

(7) ⟹ (1). S is of finite type by Proposition 5.9.

(2) ⟺ (3) ⟺ (4). This follows from Theorem 5.8.

(2) ⟺ (8). Since every finitely generated subobject of a coherent object is coherent, the equivalence in question follows from Theorem 5.8.

(1) ⟹ (2). Obvious.

(3) ⟹ (1). This is a consequence of [15, 2.4]. □

As in [14], given an arbitrary subcategory X of C, we denote by X⃗ the subcategory of C consisting of the direct limits of objects in X. Herzog [14] and Krause [15] observed that the localizing subcategories of finite type in the locally coherent categories are determined by the Serre subcategories of the Abelian subcategory coh C. Namely, the following statement is true.

Theorem 5.15 (Herzog, Krause). Let C be a locally coherent Grothendieck category. There is a bijective correspondence between the Serre subcategories P of coh C and the localizing subcategories S of C of finite type. This correspondence is given by the functions

P ⟼ P⃗
S ⟼ coh S = S ∩ coh C,

which are mutual inverses.

In what follows, for any Serre subcategory P of coh C, the P⃗-torsion functor will be denoted by t_{P}.

We recall that an object C ∈ C of a Grothendieck category C is said to be Noetherian if any subobject of C is finitely generated. The category C is locally Noetherian if it has a family of Noetherian generators. In this case,

fg C = fp C = coh C.

Moreover, every localizing subcategory S of C is of finite type. Indeed, every X ∈ S is a direct union ∑_{i∈I} X_{i} of objects X_{i} ∈ fg C ∩ S = coh C ∩ S = coh S. Consequently, any quotient category C/S is locally Noetherian, with the family of Noetherian generators {C_{S}}_{C∈coh C}.

5.2. The Ziegler topology. The study of pure-injective (= algebraically compact) modules over various classes of rings plays an important role in the theory of rings and modules. Since the pure-injective modules can be defined by using some solvability condition for systems of linear equations (in this module), many problems concerning the (algebraic!) structure of pure-injective modules admit reformulations in terms of model theory. Precisely this approach led Ziegler (see [6]) to the construction of a topological space (“Ziegler spectrum”) the points of which are the indecomposable pure-injective modules. Not long ago, Herzog [14] and Krause [15] proposed an algebraic definition of the Ziegler spectrum.

Let C be a Grothendieck category with a family of generators U. We denote by Zg C the set of isomorphism classes of indecomposable injective objects in C and call Zg C the Ziegler spectrum of C. The fact that Zg C forms a set follows from the fact that any indecomposable injective object in C occurs as the injective envelope of some Ufinitely generated object X ∈ fg_{U} C, and fg_{U} C is skeletally small. It will be convenient to identify each isomorphism class with a representative belonging to it. If S is a localizing subcategory in C, the rule X ↦ E(X) induces injective maps Zg S → Zg C and Zg C/S → Zg C. We view both maps as embeddings. They satisfy Zg S ∪ Zg C/S = Zg C and Zg S ∩ Zg C/S = ∅ (see [2, III.3.2]).

Now, suppose C is a locally coherent category, so that we may assume U ⊆ coh C. With an arbitrary subcategory X ⊂ coh C we associate the following subset of Zg C:

O(X) = { E ∈ Zg C | _{C}(C, E) ≠ 0 for some C ∈ X }.

If X = {C} is singleton, we abbreviate O(X) to O(C); thus, O(X) = ⋃_{C∈X} O(C).

We shall restrict our discussion to the subcategories of the form O(S), where S ⊆ coh C is a Serre subcategory. In this case, we have

O(S) = { E ∈ Zg C | t_{S}(E) ≠ 0 }.

Theorem 5.16 (Herzog [14], Krause [15]). The following statements are true for any locally coherent Grothendieck category C.

(1) The collection { O(S) | S is a Serre subcategory }

of subsets of Zg C satisfies the axioms for the open sets of a topological space on Zg C. This topological space will also be called the Ziegler spectrum of C.

(2) There is a bijective, inclusion preserving correspondence between the Serre subcategories S of coh C and the open subsets O of Zg C. This correspondence is given by the mappings

S ⟼ O(S)
O ⟼ S_{O} = { C ∈ coh C | O(C) ⊆ O },

which are mutual inverses.

We recall that a topological space X is quasicompact if for every family {O_{i}}_{i∈I} of open subsets satisfying X = ⋃_{i∈I} O_{i} we have X = ⋃_{i∈J} O_{i} for some finite subset J of I. A subset of X is quasicompact if it is quasicompact with respect to the induced topology.

By [14, 3.9] and [15, 4.6], an open subset O of Zg C is quasicompact if and only if it is one of the basic open subsets O(C) with C ∈ coh C.

The Serre subcategories S of coh C arise in the following natural way. An object M ∈ C is said to be coh-injective if Ext_{C}^{1}(C, M) = 0 for every C ∈ coh C. Then the subcategory

S_{M} = { C ∈ coh C | _{C}(C, M) = 0 }

cogenerated by M is Serre. Moreover, every Serre subcategory of coh C arises in this way (see [14, 3.11]).

Examples. Here are some examples of Ziegler-closed subsets.

(1) Let C be a locally coherent Grothendieck category; then the functor category ((coh C)^{op}, Ab) is locally coherent. By Theorem 5.13, C is equivalent to Lex((coh C)^{op}, Ab) = ((coh C)^{op}, Ab)/L⃗, where L is the Serre subcategory of coh((coh C)^{op}, Ab) consisting of the objects isomorphic to Coker(−, μ) for some epimorphism μ: A → B in coh C. Thus, Zg C is closed in Zg((coh C)^{op}, Ab).

(2) If C = C_{A}, then for the Ziegler spectrum of C_{A} we have

Zg C_{A} = { Q ⊗_{A} − | Q_{A} is an indecomposable pure-injective module }.

We note that Zg C_{A} is quasicompact because Zg C_{A} = O(A ⊗_{A} −). Prest, Rothmaler, and Ziegler proved (see [28, 4.4] and also [10, 2.5]) that a ring A is right coherent if and only if the set

I_{inj} = { Q ⊗_{A} − ∈ Zg C_{A} | Q is an injective module }

is closed in Zg C_{A}.

(3) If A is left coherent, then the result of [11, 2.4] implies that

I_{flat} = { Q ⊗_{A} − ∈ Zg C_{A} | Q is a flat module }

is closed in Zg C_{A}.

A ring A is said to be weakly quasi-Frobenius (see [11, 12]) if the functor Hom_{A}(−, A) puts the categories mod A^{op} and mod A of finitely presented modules in duality. The weakly quasi-Frobenius rings can be characterized as the (two-sided) coherent FP-injec-tive rings (see [11, 2.11]). Over the weakly quasi-Frobenius rings we have I_{inj} = I_{flat}. Conversely, if A is a (two-sided) coherent ring and I_{inj} = I_{flat}, then A is weakly quasi-Frobenius [11, 2.12].

(4) Let ρ: A → B be a ring homomorphism; then ρ induces an exact functor ρ^{∗}: coh C_{A} → coh C_{B}. If ρ is an epimorphism of rings, then the map M_{B} ↦ M_{A} induces a homeomorphism Zg C_{B} → Zg C_{A}\O(S) with S = Ker ρ^{∗}. Thus, Zg C_{B} is closed in Zg C_{A} [29].

§6. Categories of generalized modules

In this section, by analogy with the category C_{A} of generalized A-modules, we construct the category C_{A}, where A = {P_{i}}_{i∈I} is the ring of finitely generated projective objects of the category Mod A. The relevant model-theoretic constructions can be found in [30, 31]. Mainly, here we shall follow the lines of the paper [14] by Herzog.

6.1. Tensor products. Let Mod A be the category of right A-modules with the family of finitely generated projective generators A = {P_{i}}_{i∈I}. By Proposition 2.2, Mod A ≈ (A^{op}, Ab). We refer to the functor category (A, Ab) as the category of left A-modules and denote it by Mod A^{op}. This is a locally finitely presented Grothendieck category with the family of finitely generated projective generators A^{op} = {h^{P} = Hom_{A}(P, −)}_{P ∈A}.

Proposition 6.1 [5, III.6.3]. Let M: A^{op} → Ab (N: A → Ab) be a right (left) Amodule. Then there exists a unique functor

M ⊗_{A} −: Mod A^{op} → Ab (− ⊗_{A} N: Mod A → Ab)

with the following properties:

(1) there are functorial isomorphisms M ⊗_{A} h^{P} ≈ M(P) and h_{P} ⊗_{A} N ≈ N(P) for P ∈ A;

(2) the functors M ⊗_{A} − and − ⊗_{A} N have right adjoints.

We also note that the tensor product functor M ⊗_{A} − is right exact and commutes with the direct limits.

Let Mod A = (A^{op}, Ab) (Mod A^{op} = (A, Ab)) be the category of right (left) A-modules with A = {P_{i}}_{i∈I}, and let mod A^{op} (mod A) be the category of finitely presented left (right) A-modules. By definition, every M ∈ mod A admits a projective presentation in Mod A:

(6.1)  ⨁_{j=1}^{n} h_{P_{j}} ⟶ ⨁_{k=1}^{m} h_{P_{k}} ⟶ M ⟶ 0.

Similarly, every M ∈ mod A^{op} admits a presentation

⨁_{j=1}^{n} h^{P_{j}} ⟶ ⨁_{k=1}^{m} h^{P_{k}} ⟶ M ⟶ 0

in Mod A^{op}. Let C_{A} = (mod A^{op}, Ab) (_{A}C = (mod A, Ab)) be the category of additive covariant functors defined on mod A^{op} (mod A); we call C_{A} (_{A}C) the category of generalized right (left) A-modules. These categories naturally generalize the corresponding categories of generalized A-modules with A = {A}. Since mod A^{op} is closed under cokernels, C_{A} is a locally coherent Grothendieck category. The finitely generated projective objects of C_{A} are precisely the objects of the form {(M, −)}_{M∈mod A^{op}}, and this family generates C_{A}. The tensor product functor ? ⊗_{A} −: Mod A → C_{A} defined by the rule M_{A} ↦ M ⊗_{A} − is fully faithful and right exact.

Every finitely presented (= coherent) generalized module C ∈ C_{A} has a projective presentation

(K, −) ⟶^{(f,−)} (L, −) ⟶ C ⟶ 0

with K, L ∈ mod A^{op}. Moreover, M ∈ mod A if and only if M ⊗_{A} − ∈ coh C_{A}. Indeed, observing that the functor (M ⊗_{A} −, −) preserves the direct limits if and only if M ⊗_{A} − ∈ coh C_{A}, it suffices to use Theorem 1.9.

Lemma 6.2. An object E ∈ C_{A} is coh-injective if and only if it is isomorphic to one of the functors M ⊗_{A} −, where M is a right A-module.

Proof. In [14, 2.2], Herzog proved that E is coh-injective if and only if it is right exact. Thus, the functor M ⊗_{A} − is coh-injective.

Suppose E is coh-injective. We define M_{A} by the rule Hom_{A}(P, M) = E(h^{P}) for P ∈ A. Now, the proof of our assertion is similar to that of [3, IV.10.1]. □

We see that the category Mod A of right A-modules can be viewed as the subcategory of coh-injective objects of the category C_{A}.

In order to describe the points of the Ziegler spectrum Zg C_{A} of C_{A}, we recall that a short exact sequence

0 ⟶ L ⟶^{μ} M ⟶^{δ} N ⟶ 0

in a locally finitely presented Grothendieck category C is said to be pure-exact if the sequence

0 ⟶ _{C}(X, L) ⟶ _{C}(X, M) ⟶ _{C}(X, N) ⟶ 0

is exact for all X ∈ fp C. In this case, μ is called a pure-monomorphism. An object Q ∈ C is said to be pure-injective if every pure-exact sequence with the first term Q splits.

Proposition 6.3. Let ε : 0 → L → M → N → 0 be a short exact sequence in C. Then the following statements are equivalent:

(1) ε is pure in C;

(2) ε is a direct limit of split exact sequences 0 → L_{i} → M_{i} → N_{i} → 0 in C.

Proof. We write N = \underrightarrow{lim}_{I} N_{i} as a direct limit of finitely presented objects N_{i}. For every i ∈ I, consider the commutative diagram

ε_{i}: 0 ⟶ L ⟶ M_{i} ⟶^{δ_{i}} N_{i} ⟶ 0
∥  ^{ψ_{i}}↓  ↓^{φ_{i}}
0 ⟶ L ⟶ M ⟶_{δ} N ⟶ 0

in which the right square is pullback and φ_{i} is the canonical morphism. Since for i ≤ j we have

δψ_{i} = φ_{i}δ_{i} = φ_{j}(φ_{ij}δ_{i}),

there exists a unique ψ_{ij}: M_{i} → M_{j} such that ψ_{i} = ψ_{j}ψ_{ij}. Clearly, the system {M_{i}, ψ_{ij}}_{I} is direct and \underrightarrow{lim} ε_{i} = ε. By assumption, there exists a morphism f: N_{i} → M such that δf = φ_{i}; hence, there exists g: N_{i} → M_{i} such that δ_{i}g = 1_{M_{i}}, i.e., each sequence ε_{i} splits.

Conversely, if each ε_{i} splits, then the sequence

0 ⟶ Hom_{A}(X, L_{i}) ⟶ Hom_{A}(X, M_{i}) ⟶ Hom_{A}(X, N_{i}) ⟶ 0

is exact. The fact that Hom_{A}(X, −) with X ∈ fp C commutes with the direct limits implies (1). □

Corollary 6.4. A sequence ε: 0 → L → M → N → 0 of right A-modules is pure-exact if and only if the C_{A}-sequence ε ⊗_{A} −: 0 → L ⊗_{A} − → M ⊗_{A} − → N ⊗_{A} − → 0 is exact.

Proof. Since tensoring commutes with the direct limits, the “only if” part follows from the preceding proposition. Conversely, if ε ⊗_{A} − is exact, then for any X ∈ mod A we have the exact sequence

\begin{aligned}0 & ⟶ (X ⊗_{A} −, L ⊗_{A} −) ⟶ (X ⊗_{A} −, M ⊗_{A} −) \\ & ⟶ (X ⊗_{A} −, N ⊗_{A} −) ⟶ Ext^{1}(X ⊗_{A} −, L ⊗_{A} −). \end{aligned}

Since X ⊗_{A} − ∈ coh C_{A} and the object L ⊗_{A} − is coh-injective, it follows that

Ext^{1}(X ⊗_{A} −, L ⊗_{A} −) = 0.□

Since tensoring commutes with the direct limits, a monomorphism μ: L → M is pure if and only if μ ⊗_{A} X is a monomorphism for any left A-module X.

The following result can be proved as in [14, 4.1].

Proposition 6.5 (Gruson, Jensen). An object E ∈ C_{A} is injective if and only if it is isomorphic to one of the functors Q ⊗_{A} −, where Q is a pure-injective right A-module.

Thus, the points of the Ziegler spectrum Zg C_{A} of C_{A} are represented by the pureinjective indecomposable right A-modules. We note that every indecomposable injective right A-module E_{A} is pure-injective; hence, E ⊗_{A} − is a point of Zg C_{A}.

It is easy to check that Zg C_{A} = ⋃_{P ∈A} O(P ⊗_{A} −) is a union of basic open quasicompact subsets O(P ⊗_{A} −). In [31] Prest showed that Zg C_{A} need not be compact, in contrast to the case where A = {A} and A is a ring.

6.2. The Auslander–Gruson–Jensen duality. Let A be an arbitrary ring. Gruson and Jensen [20] and Auslander [32] proved that there is a duality D: (coh_{A}C)^{op} ≈ coh C_{A} between the corresponding subcategories of coherent objects in _{A}C and in C_{A}. A similar duality can easily be constructed for the categories _{A}C and C_{A} with A = {P_{i}}_{i∈I}.

Namely, let D: (coh_{A}C)^{op} → coh C_{A} be the functor given by

(DC)(_{A}N) = _{_{A}C}(C, − ⊗_{A} N)

with C ∈ coh_{A}C and N ∈ mod A^{op}. If δ: B → C is a morphism in coh_{A}C, we define

D(δ)_{N}: (DC)(_{A}N) ⟶ (DB)(_{A}N)

to be equal to _{_{A}C}(δ, − ⊗_{A} N).

Theorem 6.6 (Auslander, Gruson, and Jensen). The functor D: (coh_{A}C)^{op} → coh C_{A} defined above puts the categories coh_{A}C and coh C_{A} in duality. Furthermore, for M_{A} ∈ mod A and _{A}N ∈ mod A^{op} we have

D(M_{A}, −) ≈ M ⊗_{A} − and D(− ⊗_{A} N) ≈ (_{A}N, −).

Proof. The proof is similar to the case where A = {A} (see [14]). □

Since the category coh_{A}C has enough projectives, the duality D gives the following.

Proposition 6.7 (Auslander). The category coh C_{A} has enough injectives, and these are precisely the objects of the form M ⊗_{A} −, where M_{A} ∈ mod A.

Thus, every coherent object C ∈ coh C_{A} has both a projective presentation in C_{A}

(6.2)  (_{A}K, −) ⟶ (_{A}L, −) ⟶ C ⟶ 0

and an injective presentation in coh C_{A}

(6.3)  0 ⟶ C ⟶ M ⊗_{A} − ⟶ N ⊗_{A} −,

where K, L ∈ mod A^{op} and M, N ∈ mod A.

We conclude this section with the Herzog theorem. Let S ⊆ coh_{A}C; then the subcategory

DS = { DC | C ∈ S }

is Serre in coh C_{A}, and the restriction to S of the duality D gives a duality D: S^{op} → DS. By Theorem 5.16, the map O(S) ↦ O(DS) induced on the open subsets of the Ziegler spectrum is an inclusion preserving bijection. Arguing as in [14], we easily show that the functor D induces an isomorphism

AC/S⃗(A, B) ≈ _{C_{A}/\overrightarrow{DS}}(DB, DA)

of Abelian groups, where A and B ∈ coh_{A}C and the map given by A_{S} ↦ (DA)_{DS} is functorial. Thus, we have the following statement.

Theorem 6.8 (Herzog). Let A be a ring. The correspondence given by

S ⟼ DS

is an inclusion preserving bijective correspondence between the Serre subcategories of coh_{A}C and those of coh C_{A}. The induced map O(S) ↦ O(DS) is an isomorphism between the topologies, that is, between the respective algebras of open sets of the Ziegler spectra of _{A}C and C_{A}. Furthermore, the duality D induces dualities between the corresponding subcategories D: S^{op} → DS and D: (coh_{A}C/S⃗)^{op} → coh C_{A}/\overrightarrow{DS}, and we have the following commutative diagram of Abelian categories:

0 ⟶ S ⟶ coh_{A}C ⟶ coh_{A}C/S⃗ ⟶ 0
D↓  ^{D}↓  ↓^{D}
0 ⟶ DS ⟶ coh C_{A} ⟶ coh C_{A}/\overrightarrow{DS} ⟶ 0.

§7. Grothendieck categories as quotient categories of (mod A^{op}, Ab)

In this section we give another representation of an arbitrary Grothendieck category C as a quotient category of C_{A}, where A = {P_{i}}_{i∈I} is a ring. Given an arbitrary functor F ∈ C_{A}, we denote by F (A) the right A-module defined by F (A)(P) = F (h^{P}) for P ∈ A. Direct verification shows that F (A) ∈ Mod A.

Proposition 7.1. Let Mod A be the category of right A-modules, where A = {P_{i}}_{i∈I} is a ring, and let C_{A} be the corresponding category of generalized right A-modules. Then the category Mod A is equivalent to the quotient category C_{A} with respect to the localizing subcategory P^{A} = { F ∈ C_{A} | F (h^{P}) = 0 for all P ∈ A }.

Proof. By Theorem 4.10, the functor Φ: C_{A} → Mod A, F ↦ F (A), provides an equivalence for the categories Mod A and C_{A}/ Ker Φ. Clearly, P^{A} = Ker Φ. □

Since the functors h_{P} ⊗_{A} − and (h^{P} , −) are isomorphic, Theorem 4.10 implies that the quotient category C_{A}/P^{A} is equivalent to the subcategory { M ⊗_{A} − | M ∈ Mod A } generated by the set {P ⊗_{A} −}_{P ∈A}.

Remark. Obviously,

P^{A} = { Ker(μ ⊗_{A} −) | μ is a monomorphism in Mod A }.

It should also be mentioned that, by Lemma 1.3, the C_{A}/P^{A}-injective objects are precisely the objects of the form E ⊗_{A} −, where E_{A} is an injective right A-module.

Now, consider a Grothendieck category C with a family of generators U. As usual, let Mod A be the category of right A-modules with A = {h_{U}}_{U∈U}.

Theorem 7.2. Every Grothendieck category C with a family of generators U is equivalent to a quotient category of C_{A} with respect to some localizing subcategory S of C_{A}.

Proof. By Theorem 4.1, there is a pair of functors (s, q), s: C → Mod A and q: Mod A → C, that determines C as a quotient category of Mod A. Next, by Proposition 7.1, there is a pair of functors (g, h), g: Mod A → C_{A} and h: C_{A} → Mod A, that determines Mod A as a quotient category of C_{A}. It suffices to show that the functor gs is a fully faithful functor, and that the functor qh is exact and left adjoint to gs. Indeed, the composition gs of the fully faithful functors g and s is fully faithful, and the composition qh of the exact functors q and h is exact. The fact that gs is right adjoint to qh follows from the isomorphisms

C_{A}(F, gs(C)) ≈ Hom_{A}(h(F), s(C)) ≈ _{C}(qh(F), C).

Hence, C is equivalent to the quotient category of C_{A} with respect to the localizing subcategory S = Ker(qh). □

Corollary 7.3 [10, 2.3]. Every Grothendieck category C with a generator U is equivalent to the quotient category of C_{A}, A = End_{C} U, with respect to some localizing subcategory S of C_{A}.

Proof. This is a consequence of Corollary 4.4 and Theorem 7.2. □

A ring A is said to be right coherent if every object P ∈ A is coherent. Now, let C be locally coherent, i.e., we may assume that U ⊆ coh C, and let Mod A be the corresponding category of modules with A = {h_{U}}_{U∈U}. It is easy to verify that A is right coherent.

Theorem 7.4. Let C be a Grothendieck category. Consider the following conditions: (1) C is locally coherent, i.e., there is a family of generators U ⊆ coh C;

(2) the localizing subcategory S of the preceding theorem is of finite type;

(3) S is of prefinite type;

(4) Zg C = { E ⊗_{A} − | E is C-injective } is closed in Zg C_{A}.

Conditions (1), (2), and (3) are equivalent, and (1), (2), (3) imply (4). If C is a locally finitely generated category, then (4) implies (1), (2), (3).

Proof. The equivalence (2) ⟺ (3) follows from Theorem 5.14.

(1) ⟹ (4). By assumption, the ring A is right coherent. Therefore, the category Mod A of right A-modules is locally coherent. By Theorem 5.14, Zg C is closed in Zg(Mod A). Our assertion will be proved if we show that Zg(Mod A) is a closed subset of Zg C_{A}. By Theorem 5.16 and Theorem 5.15, it suffices to check that the localizing subcategory P^{A} of C_{A} is of finite type. Thus, we must show that P^{A} = S⃗^{A}, where S^{A} = P^{A} ∩ coh C_{A} is a Serre subcategory of coh C_{A}. Clearly, S⃗^{A} ⊆ P^{A}. We verify the reverse inclusion.

Let F ∈ P^{A}; then we have the exact sequence

0 ⟶ F ⟶^{ι} M ⊗_{A} − ⟶^{μ⊗−} N ⊗_{A} − ⟶^{ν⊗−} L ⊗_{A} − ⟶ 0,

where M ⊗_{A} − = E(F), N ⊗_{A} − = E(coker ι), and μ: M → N is a monomorphism. By assumption, A is right coherent; hence, the exact sequence 0 → M →^{μ} N →^{ν} L → 0 is the direct limit of exact sequences 0 → M_{i} →^{μ_{i}} N_{i} →^{ν_{i}} L_{i} → 0 with M_{i}, N_{i}, L_{i} ∈ mod A (see [27, 5.7]). If C_{i} = Ker(μ_{i} ⊗ −), then C_{i} ∈ S^{A}. Consider the commutative diagram

(7.1)  \begin{aligned}0 & ⟶ & C_{i} & ⟶^{ρ_{i}} & M_{i} ⊗_{A} − & ⟶^{μ_{i}⊗−} & N_{i} ⊗ − \\ & & & & ↓^{α_{i}⊗−} & & ↓^{β_{i}⊗−} \\ 0 & ⟶ & F & ⟶^{ι} & M ⊗ − & ⟶^{μ⊗−} & N ⊗ − \end{aligned}

in which α_{i} and β_{i} are the canonical homomorphisms. Since (μα_{i} ⊗ −)ρ_{i} = (β_{i}μ_{i} ⊗ −)ρ_{i} = 0, there exists γ_{i}: C_{i} → F such that ιγ_{i} = (α_{i} ⊗ −)ρ_{i}. Similarly, given i ≤ j, we can construct the commutative diagrams

0 ⟶ C_{i} ⟶^{ρ_{i}} M_{i} ⊗_{A} − ⟶^{μ_{i}⊗−} N_{i} ⊗_{A} −
γ_{ij} ↓  ↓^{α_{ij}⊗−}  ↓^{β_{ij}⊗−}
0 ⟶ C_{j} ⟶^{ρ_{j}} M_{j} ⊗_{A} − ⟶^{μ_{j}⊗−} N_{j} ⊗_{A} −.

An immediate verification shows that the family {C_{i}, γ_{ij}} is direct. Now, taking the direct limit in the upper row in (7.1), we obtain F = \underrightarrow{lim} C_{i}. Thus, F ∈ S⃗^{A}.

(2) ⟹ (1). By Theorem 5.14, the S-localization (h_{U} ⊗_{A} −)_{S} ≈ U of the C_{A}- coherent object h_{U} ⊗_{A} − is C-coherent.

(4) ⟹ (2). Suppose C is a locally finitely generated category and Zg C = Zg C_{A}/S is a closed subset of Zg C_{A}. By Theorem 5.16, there is a Serre subcategory P of coh C_{A} such that Zg C = Zg C_{A}/P⃗. From [10, 2.7] it follows that S = P⃗, and our assertion becomes a consequence of Theorem 5.15. □

Corollary 7.5. Let Mod A be the module category with A = {P_{i}}_{i∈I}. The following statements are equivalent:

(1) the ring A is right coherent;

(2) the localizing subcategory P^{A} is of finite type;

(3) the localizing subcategory P^{A} is of prefinite type;

(4) Zg(Mod A) = { E ⊗_{A} − | E_{A} is an injective right A-module } is closed in Zg C_{A}.

The next statement extends the list of properties characterizing the coherent rings (see also [33]).

Proposition 7.6. For a ring A = {P_{i}}_{i∈I}, the following statements are equivalent: (1) A is right coherent;

(2) for any finitely presented left A-module M, the right A-module M^{∗} = Hom_{A}(M, A) is finitely presented;

(3) for any finitely presented left A-module M, the right A-module M^{∗} = Hom_{A}(M, A) is finitely generated;

(4) for any coherent object C ∈ coh C_{A}, the right A-module C(A) is finitely presented;

(5) for any coherent object C ∈ coh C_{A}, the right A-module C(A) is finitely generated.

Proof. By Proposition 7.1, the functor C_{A} → Mod A, F ↦ F (A), yields an equivalence of the categories C_{A}/P^{A} and Mod A. By the preceding corollary, A is right coherent if and only if P^{A} is of finite type (= of prefinite type). Since the family {(_{A}M, −)}_{M∈mod A^{op}} is a family of generators for C_{A}, our assertion follows immediately from Theorem 5.8 and Proposition 5.9. □

Corollary 7.7. For a ring A = {P_{i}}_{i∈I}, the following statements are equivalent: (1) P^{A} is coexact;

(2) for any finitely presented left A-module M, the right A-module M^{∗} = Hom_{A}(M, A) is projective.

Proof. This follows from Proposition 5.2. □

As an example, we describe the rings for which the subcategory P^{A} of the category C_{A} is coexact.

Proposition 7.8. For a right coherent ring A, the following statements are equivalent: (1) r. w. dim A ≤ 2;

(2) P^{A} is coexact;

(3) for every finitely presented left A-module M, the right A-module M^{∗} = Hom(M, A) is projective.

Proof. Since A is right coherent, it is easily seen that

r. w. dim A = sup{ pd M | M ∈ mod A }.

The preceding corollary shows that conditions (2) and (3) are equivalent.

(1) ⟹ (3). For M ∈ mod A^{op}, consider an exact sequence

P_{1} ⟶^{α} P_{0} ⟶ M ⟶ 0,

where P_{1} and P_{0} are finitely generated projective modules. Then there is an exact sequence

0 ⟶ M^{∗} ⟶ P_{0}^{∗} ⟶^{α^{∗}} P_{1}^{∗} ⟶ N ⟶ 0

with N = Coker α. By assumption, M^{∗} is projective.

(3) ⟹ (1). For N ∈ mod A, consider an exact sequence of the form

0 ⟶ K ⟶ P_{1} ⟶^{α} P_{0} ⟶ N ⟶ 0.

We must prove that the module K is projective. Indeed, let M = Coker α^{∗}; then K is isomorphic to M^{∗}. Since M^{∗} is projective by assumption, K is projective. □

Corollary 7.9 (Bass [34]). If A is a two-sided Noetherian ring, then r. dim A ≤ 2 if and only if for every finitely generated left A-module M the right A-module M^{∗} = Hom(M, A) is projective.

Proof. Over the Noetherian rings, the categories of finitely generated and of finitely presented modules coincide. □

Now, we want to describe the localizing subcategories of prefinite type in the locally finitely generated Grothendieck categories in terms of localizing subcategories of finite type of the category C_{A}. Let S^{A} be the Serre subcategory P^{A} ∩ coh C_{A} of C_{A}. If P is a subcategory of C_{A}, we denote by P(A) the subcategory of Mod A that consists of the modules F (A) with F ∈ P.

Proposition 7.10. For any localizing subcategory S of Mod A, A = {P_{i}}_{i∈I}, the following statements are equivalent:

(1) S is of prefinite type;

(2) there is a Serre subcategory T of coh C_{A} such that T ⊇ S^{A} and T⃗ (A) = S.

Proof. Let P denote the localizing subcategory of C_{A} such that P ⊇ P^{A} and P(A) = S (see Proposition 1.6). Let T = P ∩ coh C_{A}; then T is a Serre subcategory of coh C_{A}. Since, clearly, T ⊇ S^{A} and T⃗ ⊆ P, we have T⃗ (A) ⊆ S. We prove the reverse inclusion.

Let 𝔉 = ⋃_{P ∈A} 𝔉^{P} be the Gabriel topology that corresponds to S. Our statement will be proved if we show that 𝔉 has a basis consisting of ideals 𝔠 ⊆ P such that 𝔠 = 𝔟(A), where 𝔟 is a coherent object of P ⊗_{A} − satisfying (P ⊗_{A} −)/𝔟 ∈ T .

So, let 𝔞 ∈ 𝔉^{P}. Consider the following exact sequence:

0 ⟶ Ker(α ⊗ −) ⟶ 𝔞 ⊗_{A} − ⟶^{α⊗−} P ⊗_{A} −.

Since α is a monomorphism, Ker(α ⊗ −) ∈ P^{A}. Let 𝔞̃ = Im(α ⊗ −); then 𝔞̃(A) = 𝔞 and (P ⊗_{A} −)/𝔞̃ ∈ P. We write 𝔞̃ = ∑_{i∈I} 𝔞_{i} as a sum of finitely generated subobjects 𝔞_{i} ⊆ 𝔞̃. Since every 𝔞_{i} is a subobject of the C_{A}-coherent object P ⊗_{A} −, it follows that 𝔞_{i} is coherent. We have

P_{S} = 𝔞_{S} = (𝔞̃(A))_{S} = ∑_{i∈I}(𝔞_{i}(A))_{S}.

By Theorem 5.8, the object P_{S} belongs to fg(Mod A/S); hence, there is a finite subset J of I such that P_{S} = ∑_{i∈J}(𝔞_{i}(A))_{S}. Let 𝔟 = ∑_{i∈J} 𝔞_{i}; then (P ⊗_{A} −)/𝔟 ∈ P. Since (P ⊗_{A} −)/𝔟 is coherent, (P ⊗_{A} −)/𝔟 ∈ T . Consequently, 𝔟(A) ⊆ 𝔞 and (𝔟(A))_{S} = P_{S}. This implies the claim. □

Now, we consider a locally finitely generated Grothendieck category C. By Theorem 7.2, there is a localizing subcategory S of C_{A} such that C is equivalent to C_{A}/S with A = {h_{U}}_{U∈U}. If Q and P are localizing subcategories of C_{A}, we denote by Q_{P} the subcategory of C_{A}/P that consists of { Q_{P} | Q ∈ Q }. Also, we denote by L the Serre subcategory S ∩ coh C_{A} of coh C_{A}.

Proposition 7.11. Let Q be a localizing subcategory of a locally finitely generated Grothendieck category C with a family of generators U ⊆ fg C. If A = {h_{U}}_{U∈U} is the ring generated by U, then the following statements are equivalent:

(1) Q is of prefinite type;

(2) there is a Serre subcategory T of coh C_{A} such that T ⊇ L and T⃗_{S} = Q.

Proof. By Theorem 4.1, there is a localizing subcategory P of Mod A such that C is equivalent to Mod A/P and, by Proposition 1.6, a localizing subcategory V of Mod A such that V ⊇ P and V/P = Q. Since both P and Q are of prefinite type, so is V. The preceding proposition implies that V = T⃗ (A) for some localizing subcategory T⃗ of finite type in C_{A}. Then Q = V_{P} = (T⃗ (A))_{P} = T⃗_{S}. □

Question. Is it true that if T⃗ is a localizing subcategory of finite type in C_{A} containing the subcategory S⃗^{A}, then the subcategory T⃗ (A) is localizing and is of prefinite type in Mod A? Should this be true, we could construct the Ziegler topology for an arbitrary locally finitely generated Grothendieck category.

§8. FP-injective and flat modules

In this section we sketch how the classes of FP-injective and flat A-modules can be studied with the help of some torsion/localizing functors of the category C_{A}.

Let C be a locally finitely presented Grothendieck category. An object C ∈ C is said to be FP-injective (or absolutely pure) if Ext_{C}^{1}(X, C) = 0 for any X ∈ fp C. An object C ∈ C is said to be fp-injective if for any monomorphism μ: X → Y in fp C the morphism_{C}(μ, C) is an epimorphism. Obviously, every FP-injective object is fp-injective, and every fp-injective finitely presented object is FP-injective. The ring A = {P_{i}}_{i∈I} is right FP-injective if every right A-module P ∈ A is FP-injective.

Let Mod A, A = {P_{i}}_{i∈I}, be the category of right A-modules. A module M ∈ Mod A is flat if the tensor functor M ⊗_{A} − is exact. We refer to M as an fp-flat module if for any monomorphism μ: _{A}K → _{A}L in mod A^{op} the morphism M ⊗_{A} μ is a monomorphism. Clearly, every flat module is fp-flat.

The next lemma is verified directly.

Lemma 8.1. Let C be a locally finitely presented Grothendieck category, and let C be an object of C. The following statements are equivalent:

(1) C is FP-injective;

(2) every exact sequence 0 → C → C′ → C′′ → 0 is pure;

(3) there exists a pure-exact sequence 0 → C → C′ → C′′ → 0 such that C′ is FP-injective.

In the sequel, we use the following notation:

P^{A} = { F ∈ C_{A} | F (A) = 0 },
S^{A} = { C ∈ coh C_{A} | C(A) = 0 },
S_{A} = { C ∈ coh C_{A} | (C, P ⊗_{A} −) = 0 for all P ∈ A }.

The definitions of the subcategories ^{A}S and _{A}S of coh_{A}C are similar to those of S^{A} and S_{A}, respectively. By Theorem 5.15, S⃗^{A} and S⃗_{A} are localizing subcategories of finite type. By Corollary 7.5, P^{A} = S⃗^{A} if and only if the ring A is right coherent. From the presentation (6.3) it easily follows that

S^{A} = { Ker(μ ⊗ −) | μ: M → N is a monomorphism in mod A }.

In a similar way,

S_{A} = { Coker(μ, −) | μ: L → K is a monomorphism in mod A^{op} }.

Proposition 8.2. If K ∈ Mod A, then:

(1) K is FP-injective if and only if the functor K ⊗_{A} − is P^{A}-torsionfree;

(2) K is fp-injective if and only if the functor K ⊗_{A} − is S⃗^{A}-torsionfree;

(3) K is fp-flat if and only if the functor K ⊗_{A} − is S⃗_{A}-torsionfree.

Proof. It suffices to adapt the proof to the case where A = {A} (see [11, 2.2]). □

Corollary 8.3. The set of all pure-injective fp-injective (fp-flat) modules is closed in Zg C_{A}.

Proposition 8.4. For any ring A the following statements are equivalent: (1) A is right semihereditary;

(2) the subcategory P^{A} is strongly coexact;

(3) any quotient module of an FP-injective right A-module is FP-injective;

(4) any quotient module of an injective right A-module is FP-injective.

Proof. The proof of the equivalences (1) ⟺ (3) ⟺ (4) is similar to that presented in [3, I.9.5].

(2) ⟹ (3). Since, by assumption, the functor t_{P^{A}} is exact, our assertion follows from the preceding proposition.

(3) ⟹ (2). Clearly, every P^{A}-torsionfree object X has the form X ≈ (M ⊗_{A} −)/Y with M = X(A), Y = t_{P^{A}}(M ⊗_{A} −). If E ⊗_{A} − = E(X) is the injective envelope of X, we have the exact sequence

0 ⟶ X ⟶ E ⊗_{A} − ⟶ (E/M) ⊗_{A} − ⟶ 0.

Obviously, the module E is injective. By assumption, t_{P^{A}}((E/M) ⊗_{A} −) = 0. The exact sequence

0 = t_{P^{A}}((E/M) ⊗_{A} −) ⟶ t_{P^{A}}^{1}(X) ⟶ t_{P^{A}}^{1}(E ⊗_{A} −) = 0

shows that t_{P^{A}}^{1}(X) = 0, i.e., X is P^{A}-closed. Referring to Proposition 5.4(3), we complete the proof. □

Corollary 8.5. A ring A is right semihereditary if and only if the functor

M_{A} ⟼ M ⊗_{A} −/t_{P^{A}}(M ⊗_{A} −)

yields equivalence between the category Mod A and the category of P^{A}-torsionfree objects.

Proof. It suffices to apply Proposition 7.1 and Corollary 5.5. □

It is well known (see, e.g., [3]) that, for a ring A, a right A-module M is finitely presented (finitely generated) if and only if the natural map

M ⊗_{A} ( ∏_{i∈I} N_{i}) → ∏_{i∈I}(M ⊗_{A} N_{i})

is an isomorphism (epimorphism) for every family {N_{i}}_{i∈I} of left A-modules. This fact can be generalized to arbitrary module categories Mod A as follows.

Lemma 8.6 [27, 7.1]. Let A be a ring. For M ∈ Mod A, the following statements are equivalent:

(1) M is finitely presented (finitely generated);

(2) the natural morphism M ⊗_{A} (∏_{i∈I} N_{i}) → ∏_{i∈I}(M ⊗_{A} N_{i}) is an isomorphism (epimorphism) for every family {N_{i}}_{i∈I} in Mod A^{op};

(3) the natural morphism M ⊗_{A} (∏_{i∈I} P_{i}) → ∏_{i∈I}(M ⊗_{A} P_{i}) = ∏_{i∈I} M(P_{i}) is an isomorphism (epimorphism) for every family {P_{i}}_{i∈I} in A.

Now, the next result can be proved as in [11, 2.3].

Proposition 8.7. If A = {P_{i}}_{i∈I} is a ring, then:

(1) for every family of right A-modules {M_{i}}_{I}, the module ∏_{I} M_{i} is FP-injective (respectively, fp-injective, fp-flat) if and only if every M_{i} is FP-injective (respectively, fp-injective, fp-flat);

(2) the direct limit \underrightarrow{lim} M_{i} of fp-injective (respectively fp-flat) right A-modules M_{i} is fp-injective (respectively, fp-flat).

Now, let C be a locally finitely presented Grothendieck category with a family of generators U ⊆ fp C. As usual, let Mod A be the category of modules with A = {h_{U}}_{U∈U}. By Theorem 4.1, C is equivalent to a quotient category Mod A/S. Furthermore, S is of finite type by Proposition 5.9. By Theorem 7.2, there is a localizing subcategory P of C_{A} such that C is equivalent to the quotient category C_{A}/P. Like in the case of the category of modules, the FP-/fp-injective objects in C can be described in terms of torsion functors of C_{A}. To begin with, we prove the following assertion.

Proposition 8.8. If C is an object of C, then:

(1) C is FP-injective if and only if C is FP-injective as a right A-module.

(2) C is fp-injective if and only if C is fp-injective as a right A-module.

Proof. (1). Let C be an FP-injective object of C, and let M ∈ mod A. We must show that Ext_{A}^{1}(M, C) = 0. This is equivalent to saying that any short exact sequence

0 ⟶ C ⟶^{α} X ⟶ M ⟶ 0

of right A-modules splits. By Proposition 5.9, we have M_{S} ∈ fp C. By assumption, the morphism α_{S} splits, i.e., there exists β: X_{S} → C such that βα_{S} = 1_{C}. Then (βλ_{X})α = βα_{S} = 1_{C}, where λ_{X} is the S-envelope for X. So, α splits.

Conversely, let C be an FP-injective right A-module, and let

ε: 0 ⟶ C ⟶^{α} E ⟶^{β} X ⟶ 0

be the C-exact sequence with E = E(C) and X = E/C. By assumption, the short exact sequence

ε̄: 0 ⟶ C ⟶^{α} E ⟶^{β̄} Im β ⟶ 0

is pure-exact in Mod A. Clearly, ε̄_{S} = ε. Proposition 6.3 implies that ε̄ is a direct limit of split exact sequences

ε̄_{i}: 0 ⟶ C_{i} ⟶ E_{i} ⟶ M_{i} ⟶ 0

in Mod A. Then ε is the direct limit of the split exact sequences ε_{i} = (ε̄_{i})_{S}. Thus, C is an FP-injective object of C.

(2) Suppose C is an fp-injective object of C and μ: M → N is a monomorphism in mod A. Since S is of finite type, the morphism μ_{S} is a monomorphism in fp C. Consider the commutative diagram

Hom_{A}(N, C) ⟶^{(μ,C)} Hom_{A}(M, C)
↓  ↓
C(N_{S}, C) ⟶_{(μ_{S},C)} _{C}(M_{S}, C)

in which the vertical arrows are isomorphisms. Since (μ_{S}, C) is an epimorphism, it follows that (μ, C) is an epimorphism.

Conversely, suppose μ: X → Y is a monomorphism in fp C. Then there is a monomorphism γ: M → N in mod A such that γ_{S} = μ. Indeed, we can embed μ into the following commutative diagram in C with exact rows:

(−, ⨁_{i=1}^{n} U_{i}) ⟶^{ψ} (−, ⨁_{j=1}^{m} U_{j}) ⟶ X ⟶ 0
↓  ↓  ↓^{μ}
(−, ⨁_{k=1}^{s} U_{k}) ⟶_{φ} (−, ⨁_{l=1}^{t} U_{l}) ⟶ Y ⟶ 0.

Since every ⨁ h_{U_{i}} is S-closed and finitely generated projective in Mod A, both Coker ψ and Coker φ are finitely presented right A-modules. We put M = Coker ψ and N = Coker φ. There is a unique morphism γ: M → N. Since M_{S} = X and N_{S} = Y, we have γ_{S} = μ. Consider the commutative diagram

Hom_{A}(N, C) ⟶^{(γ,C)} Hom_{A}(M, C)
↓  ↓
C(Y, C) ⟶_{(μ,C)}  _{C}(X, C)

where the vertical arrows are isomorphisms. Since (γ, C) is an epimorphism, so is (μ, C). Thus, C is fp-injective in C. □

Corollary 8.9. The ring A = {h_{U}}_{U∈U} is FP-injective if and only if every U ∈ U is an FP-injective object of C.

Proof. It suffices to observe that every h_{U} is S-closed (see Theorem 4.1) and then apply the preceding proposition. □

We denote T = coh C ∩ P and let t_{P} and t_{T} be the torsion functors corresponding to the localizing subcategories P and T⃗ of C_{A}.

Proposition 8.10. If C ∈ C, then:

(1) C is an FP-injective object of C if and only if t_{P}(C ⊗_{A} −) = 0;

(2) C is an fp-injective object of C if and only if t_{T} (C ⊗_{A} −) = 0.

Proof. (1) Let C be FP-injective. By Proposition 8.8, C is an FP-injective right Amodule. Let E be an injective envelope for C. Then C ⊗_{A} − is a subobject of E ⊗_{A} −. Since E ⊗_{A} − is P-torsionfree, so is C ⊗_{A} −. Conversely, since P ⊇ P^{A}, our assertion follows from Propositions 8.2 and 8.8.

(2) Let C be fp-injective, and let T ∈ T . Consider the exact sequence (6.3)

0 ⟶ T ⟶ M ⊗_{A} − ⟶^{μ⊗−} N ⊗_{A} −

with M, N ∈ mod A. Since 0 = T_{P} = T (A)_{S}, the morphism μ_{S} is a monomorphism in fp C. Consequently, the morphism (μ_{S}, C) is an epimorphism, and, hence, so is (μ, C). Since C ⊗_{A} − is a coh-injective object, we have the exact sequence

(N ⊗_{A} −, C ⊗_{A} −) ⟶^{(μ⊗−,C⊗_{A}−)} (M ⊗_{A} −, C ⊗_{A} −) ⟶ (T, C ⊗_{A} −) ⟶ 0.

The fact that (μ, C) is an epimorphism implies the relation (T, C ⊗_{A} −) = 0, whence t_{T} (C ⊗_{A} −) = 0. Since T ⊇ S^{A}, the converse statement follows from Propositions 8.8 and 8.2. □

Let A be a ring; the Chase theorem states that A is left coherent if and only if any direct product ∏ M_{i} of flat right A-modules M_{i} is flat. This fact can be generalized to an arbitrary ring A = {P_{i}}_{i∈I} as follows.

Proposition 8.11 (Chase). Let A = {P_{i}}_{i∈I} be a ring. The following statements are equivalent:

(1) A is left coherent;

(2) every product of flat right A-modules is flat;

(3) for an arbitrary set J, every product ∏_{j∈J} P_{j} of P_{j} ∈ A is a flat right A-module.

Proof. (1) ⟹ (2). Let {M_{j}}_{j∈J} be a family of flat right A-modules. Proposition 8.7 shows that the module ∏_{j∈J} M_{j} is fp-flat. Let φ: K → L be a monomorphism in Mod A^{op}. Since A is left coherent, φ = \underrightarrow{lim} φ_{i} is the direct limit of some monomorphisms φ_{i} in mod A^{op} (see [27, 5.9]). Then the morphism ∏ M_{j} ⊗ φ = \underrightarrow{lim}(∏ M_{j} ⊗ φ_{i}) is the direct limit of the monomorphisms ∏ M_{j} ⊗ φ_{i}, and, consequently, is a monomorphism itself.

The implication (2) ⟹ (3) is trivial.

(3) ⟹ (1). Let _{A}K be a finitely generated submodule of a finitely presented module AL. For any index set J, we have a commutative diagram

∏ P_{j} ⊗_{A} K ⟶ ∏ P_{j} ⊗_{A} L
φ_{K} ↓  ↓^{φ_{L}}
∏ K(P_{j}) ⟶ ∏ L(P_{j}),

where the horizontal arrows are monomorphisms. Since φ_{L} is a monomorphism by Lemma 8.6, so is φ_{K}. Thus, K is finitely presented by Lemma 8.6. □

In contrast to the FP-injective right A-modules, the realization of the class of the flat right A-modules in C_{A} as the class of functors M ⊗_{A} − that satisfy t_{S}(M ⊗_{A} −) = 0 for some localizing subcategory S of C_{A} is possible if and only if A is left coherent.

Theorem 8.12. For any ring A, the following statements are equivalent:

(1) A is left coherent;

(2) there is a localizing subcategory S of C_{A} such that any right A-module M is flat if and only if the functor M ⊗_{A} − is S-torsionfree;

(3) every left fp-injective A-module is FP-injective;

(4) every right fp-flat A-module is flat;

(5) any direct limit of FP-injective left A-modules is FP-injective.

Proof. The equivalence (1) ⟺ (5) follows from [27, 9.3].

The remaining statements can be proved as in [11, 2.4]. □

Theorem 8.13. For any ring A, the following statements are equivalent:

(1) A is right FP-injective;

(2) S^{A} ⊆ S_{A};

(3) _{A}S ⊆ ^{A}S;

(4) every fp-flat right A-module is fp-injective;

(5) every indecomposable pure-injective fp-flat right A-module is fp-injective;

(6) every pure-injective fp-flat right A-module is fp-injective;

(7) every fp-injective left A-module is fp-flat;

(8) every indecomposable pure-injective fp-injective left A-module is fp-flat;

(9) every pure-injective fp-injective left A-module is fp-flat.

Proof. It suffices to adapt the proof to the case where A = {A} (see [11, 2.5]). □

Example. Let A = {A} be a ring, and let C_{A} be the category of generalized right Amodules. Then the ring B = {(M, −)}_{M∈mod A^{op}} is right FP-injective if and only if A is (von Neumann) regular.

Indeed, let B be right FP-injective; then each (K, −) with K ∈ mod A^{op} is coh-injec-tive. Therefore (K, −) is isomorphic to the object K^{∗} ⊗_{A} −, where K^{∗} = Hom_{A}(K, A). Since every coherent object C ∈ coh C_{A} is a cokernel

(K, −) ⟶^{(α,−)} (L, −) ⟶ C ⟶ 0

of (α, −) (see the sequence (6.2)), it follows that C is isomorphic to (Coker α^{∗}) ⊗_{A} −. So, every C ∈ coh C_{A} is coh-injective, and, consequently, A is a regular ring [14, 4.4]. The converse statement is also a consequence of [14, 4.4].

In conclusion, we give a duality criterion for the categories of finitely presented left and right A-modules. A ring A over which the functor Hom_{A}(−, A) puts these categories in duality is said to be weakly quasi-Frobenius. For the case where A = {A} we refer the reader to [11].

Theorem 8.14. For any ring A = {P_{i}}_{i∈I}, the following statements are equivalent: (1) A is weakly quasi-Frobenius;

(2) A is (left and right) FP-injective and (left and right) coherent;

(3) the classes of flat right A-modules and right FP-injective A-modules coincide;

(4) A is left FP-injective and left coherent, and any flat right A-module is FP-injective;

(5) A is right FP-injective and right coherent, and any FP-injective right A-module is flat.

Proof. (1) ⟹ (2). By assumption, the right A-module M^{∗} = Hom_{A}(M, A) is finitely presented for any finitely presented left A-module M. Proposition 7.6 implies that A is right coherent. By symmetry, A is left coherent. Since the functor Hom_{A}(−, P) with P ∈ A is exact both on mod A^{op} and on mod A, it follows that P is an FP-injective left and right A-module. So, A is a (two-sided) FP-injective ring.

(2) ⟹ (1). Since A is a (two-sided) coherent ring, from Proposition 7.5 it follows that P^{A} = S⃗^{A}; hence, there is an equivalence of the categories mod A and coh C_{A}/S⃗^{A}. In a similar way, there is an equivalence of the categories mod A^{op} and coh_{A}C/^{A}S⃗. Theorem 8.13 yields the relations

S^{A} = D(_{A}S) = D(^{A}S)
S_{A} = D(^{A}S) = D(_{A}S).

Now, our implication follows from Theorem 6.8.

(2) ⟹ (3), (2) ⟹ (4). It suffices to apply Theorems 8.12 and 8.13.

(3) ⟹ (5). It suffices to show that A is right coherent. For this, consider a direct system of FP-injective right A-modules {M_{i}}_{i∈I}. Since each M_{i} is flat by assumption, the module \underrightarrow{lim} M_{i} is flat, and, therefore, FP-injective. Consequently, A is right coherent by Theorem 8.12.

(4) ⟹ (3). By Theorem 8.13, any FP-injective right A-module is fp-flat; hence it is flat by Theorem 8.12.

(5) ⟹ (2). Since the ring A is right FP-injective, the module ∏_{J} P_{j} is FP-injective (here P_{j} ∈ A and J is some set of indices); the latter module is flat by assumption. By Proposition 8.11, A is left coherent. Theorem 8.12 says that any fp-injective right R-module is FP-injective, and, therefore, flat. From Theorem 8.13 it follows that A is left FP-injective. □

Concluding remarks

Talking about the further study of the classes of FP-injective and weakly quasi-Frobenius rings, here we have not touched upon yet another important concept, namely, that of an FP-cogenerator (see [12]). Actually, the condition of right FP-injectivity of a ring turns out to be equivalent to the fact that this ring is a left FP-cogenerator (see [12]).

In fact, when we carry the properties of the category of modules Mod A over to the category of finitely presented modules mod A, many concepts arise in couples of the type FPproperty/fp-property (for instance, FP-injectivity/fp-injectivity, or FP-cogenerator/fp-cogenerator; see [12]). It is here that the properties of the category _{A}C find full-scale application, and the torsion functors turn out to be an extremely convenient tool in this field. Moving in this direction, it is difficult to keep track of all consequences, and we invite the reader to the further study of the category mod A.

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St. Petersburg State University, Department of Mathematics and Mechanics, St. Petersburg, Bibliotechnaya pl. 2, 198504, Stary Peterhof, Russia

   E-mail address: ggarkusha@mail.ru

Received 10/MAY/2000

Translated by THE AUTHOR