FUNCTORS ON LOCALLY FINITELY PRESENTED ADDITIVE CATEGORIES

HENNING KRAUSE

An additive category with direct limits is said to be locally finitely presented provided that the full subcategory of finitely presented objects is skeletally small and every object is a direct limit of finitely presented objects. Our aim in this paper is to study functors which are defined on locally finitely presented categories. Roughly, most of our results are of the following form. Given a category C and some property (∗), there exists a category D satisfying (∗) and a fully faithful functor d: C → D such that any functor f: C → D′ into a category satisfying (∗) can be extended uniquely to a functor f^{∗}: D → D′ which respects (∗) and satisfies f = f^{∗}∘d. The following properties (∗) for a category D and combinations thereof are relevant: D has cokernels; D has kernels; D is abelian; D has direct limits.

These “universal property” type results are used to study functors between locally finitely presented categories. For instance, we assign to each locally finitely presented category with products A a skeletally small abelian category A′ such that A is equivalent to the category of exact functors A′ → Ab into the category Ab of abelian groups. For any pair A and B of such categories there is a bijective correspondence between

(1) functors A → B commuting with direct limits and products, and

(2) functors B′ → A′ which are exact. In particular, we show that a functor f: A → Ab commutes with direct limits and products if and only if there exists a presentation Hom(Y, −) → Hom(X, −) → f → 0 such that X and Y are finitely presented.

Functors commuting with direct limits and products occur frequently, for instance in representation theory of artin algebras. In fact, a tensor functor − ⊗_{Γ} B: Mod(Γ) → Mod(Λ) has this property provided that B is finitely presented over Γ. In this case, −⊗_{Γ}B induces an exact functor fp(mod(Λ^{op}), Ab) → fp(mod(Γ^{op}), Ab) between categories of finitely presented functors, and this alternative description of a tensor functor has been used to establish a close relationship between endofinite modules in Mod(Γ) and Mod(Λ) [13].

If one studies functors, it is often useful to have the existence of left or right adjoints. We develop various criteria for their existence; of particular interest is the following application: A locally finitely presented category is complete if and only if it is cocomplete.

Another section is devoted to a discussion of contravariantly finite subcategories. We present a new criterion for a subcategory to be contravariantly finite which has already been used in studying the Krull-Gabriel dimension of an artin algebra [11] or the representation type of a stable module category [14].

A substantial part of this paper is devoted to locally coherent categories. A locally finitely presented category is locally coherent provided that the full subcategory formed by the finitely presented objects is abelian. Our motivation to study such categories is the following: Given any ring Λ, there is a locally coherent category D(Λ) which is an essential tool for studying pure-injective Λ-modules [10].

1991 Mathematics Subject Classification. 18A35, 18B15, 16B50.

In fact, there is a fully faithful functor Mod(Λ) → D(Λ) which identifies the pureinjective Λ-modules with the injective objects in D(Λ). However, it is often useful to pass from D(Λ) either to a localizing subcategory T or to a quotient category D(Λ)/T which are again locally coherent. These constructions have been used by various authors and it is therefore important to have available a general theory of locally coherent categories.

This article is of expository nature. Much of the material can be found in the literature and I have tried to include precise references. However, the coherent approach presented here as well as some of the results seem to be new.

1. Preliminaries

We introduce some terminology. Throughout we are working in a fixed universe 𝔘 containing an infinite set (e.g. see [8, Numéro 0]). All categories C are assumed to be 𝔘-categories in the sense that for each pair of objects X, Y ∈ C the set Hom(X, Y) is small, i.e. bijective with a set in 𝔘. A category C is skeletally small provided that the isomorphism classes of objects in C form a small set. A non-empty category I is said to be filtered provided that for each pair of objects λ_{1}, λ_{2} ∈ I there are morphisms φ_{i}: λ_{i} → μ for some μ ∈ I, and for each pair of morphisms φ_{1}, φ_{2} : λ → μ there is a morphism ψ: μ → ν with ψ∘φ_{1} = ψ∘φ_{2}. We call the colimit ⟶ lim_{λ∈I} X_{λ} of a functor X: I → A, λ ↦ X_{λ} a direct limit if I is a skeletally small filtered category.

Given a category A with direct limits the finitely presented and finitely generated objects of A play an important role. Recall that an object X ∈ A is finitely presented (finitely generated) provided that for every direct limit ⟶ lim Y_{λ} in A the natural morphism ⟶ lim Hom(X, Y_{λ}) → Hom(X, ⟶ lim Y_{λ}) is an isomorphism (a monomorphism). The full subcategory of finitely presented objects of A is denoted by fp(A). Following [4], an additive category A is called locally finitely presented if fp(A) is skeletally small and every object in A is a direct limit of objects in fp(A).

Recall that an abelian category A is a Grothendieck category provided that A has coproducts, exact direct limits and a small generating set of objects. An abelian category A is locally finitely presented if and only if it is a Grothendieck category with a generating set of finitely presented objects [3, Satz 1.5], [4, 2.4]. Now suppose that A is locally finitely presented and abelian. An object in A is finitely generated if and only if it is a quotient of some finitely presented object. The category A is said to be locally coherent provided that finitely generated subobjects of finitely presented objects are finitely presented, equivalently if fp(A) is abelian [18, Proposition 2.2].

Throughout the paper all functors between pre-additive categories are assumed to be additive. Given two pre-additive categories C and D, the class of functors F: C → D is denoted by (C, D) and Hom(F, G) denotes the class of natural transformations between two functors F and G in (C, D). If C is skeletally small, then (C, D) forms actually a category since the Hom sets are small. The category of abelian groups is denoted by Ab. Given any category C, one defines limits, colimits etc. in (C, Ab) pointwise and they coincide with the categorical notions if (C, Ab) is a category.

2. Finitely presented functors and abelian categories

In this section we collect some elementary facts about categories which have cokernels, kernels or which are abelian. Let C be a pre-additive category. A functor F: C^{op} → Ab is said to be finitely presented provided that there exists an exact sequence

∐_{i=1}^{n} Hom(−, X_{i}) ⟶ ∐_{j=1}^{m} Hom(−, Y_{j}) ⟶ F ⟶ 0

of functors in (C^{op}, Ab) such that n and m are finite. Note that the presentation of F could be replaced by Hom(−, X) → Hom(−, Y) → F → 0 if C is additive. Also F is said to be finitely generated provided that there exists an exact sequence ∐_{i=1}^{n} Hom(−, X_{i}) → F → 0 in (C^{op}, Ab) such that n is finite. The finitely presented functors form an additive category with cokernels which we denote by mod(C). Recall that Yoneda’s lemma gives a fully faithful functor h_{C}: C → mod(C), X ↦ Hom(−, X). We shall frequently use the category mod(C^{op})^{op} which we denote by mop(C), that is, we set

mop(C) = mod(C^{op})^{op}.

In this case Yoneda’s lemma gives a fully faithful functor C → mop(C), X ↦ Hom(X, −).

Universal Property 2.1. Let f: C → A be a functor into an additive category with cokernels. Then there exists, up to a natural isomorphism, a unique right exact functor f^{∗}: mod(C) → A such that f = f^{∗}∘h_{C}.

Proof. For any functor F ∈ mod(C) choose a presentation

∐_{i=1}^{n} Hom(−, X_{i}) ⟶^{(Hom(−,φ_{ij}))_{i,j}} ∐_{j=1}^{m} Hom(−, Y_{j}) ⟶ F ⟶ 0

and define f^{∗}(F) to be the cokernel of the morphism

∐_{i=1}^{n} f(X_{i}) ⟶^{(f(φ_{ij}))_{i,j}} ∐_{j=1}^{m} f(Y_{j}).

Any morphism ψ: F → G in mod(C) lifts to a morphism of the presentations, so it induces a unique morphism f^{∗}(ψ): f^{∗}(F) → f^{∗}(G). It is easily checked that this is well-defined and that f^{∗}: mod(C) → A is the unique right exact functor extending f.

The preceding result can be used to define a tensor product mod(C)×mod(C^{op}) → Ab, (F, G) ↦ F ⊗_{C} G which is characterized, up to a natural isomorphism, by the following properties.

(1) There are functorial isomorphism F ⊗_{C}Hom(X, −) ≅ F (X) and Hom(−, X)⊗_{C} G ≅ G(X) for X ∈ C.

(2) F ⊗_{C} − and − ⊗_{C} G are right exact.

Recall that given a morphism ψ: Y → Z in C, then a morphism φ: X → Y is a pseudo-kernel for ψ provided that the induced sequence of functors Hom(−, X) → Hom(−, Y) → Hom(−, Z) is exact. This concept was introduced by Freyd under the name “weak kernel” and he proved the following classical result [5, Theorem 1.4].

Lemma 2.2. The category mod(C) is abelian iff C has pseudo-kernels.

Let A be an abelian category. We denote by proj(A) the full subcategory of projective objects in A and we say that A has sufficently many projectives provided that for every object X ∈ A there exists an epimorphism Y → X with Y ∈ proj(A). Analogous terminology is used for injective objects in A.

Proposition 2.3. There is, up to equivalence, a bijective correspondence between (skeletally small) additive categories with split idempotents and pseudo-kernels and (skeletally small) abelian categories with sufficently many projectives. The correspondence is given by

C ↦ mod(C)  and  A ↦ proj(A).

Proof. If C is an additive categories with split idempotents and pseudo-kernels, then mod(C) is abelian and the representable functors Hom(−, X) are precisely the projective objects in mod(C) by Yoneda’s lemma. Conversely, given an abelian category A with sufficently many projectives, then proj(A) has pseudo-kernels and the inclusion proj(A) → A extends to a functor mod(proj(A)) → A which is an equivalence.

Lemma 2.4. Let A be an abelian category with sufficently many projectives and suppose that f: proj(A) → B is a functor into a category with cokernels. Then there exists, up to a natural isomorphism, a unique right exact functor f^{∗}: A → B such that f^{∗}|_{proj(A)} = f.

Proof. Combine Property 2.1 and Proposition 2.3.

Lemma 2.5. Let f: A → B be a right exact functor between abelian categories and suppose that A has sufficently many projectives. Then f is exact iff f(X) → f(Y) → f(Z) is exact for each exact sequence X → Y → Z of projective objects in A.

Proof. Taking projective presentations of an arbitrary exact sequence in A and applying f the assertion is an immediate consequence of the snake lemma.

Let f: C → D be a functor between additive categories. We denote by f^{∗}: mod(C) → mod(D) the unique right exact functor which extends f. We list some properties of f^{∗}.

Lemma 2.6. (1) f is fully faithful iff f^{∗} is fully faithful.

(2) Suppose that C and D have pseudo-kernels. Then f: C → D preserves pseudokernels iff f^{∗} is exact.

(3) Suppose that C has kernels and that D has pseudo-kernels. Then f is left exact iff f preserves pseudo-kernels.

Proof. (1) Straightforward.

(2) Use Lemma 2.5.

(3) If C has kernels, then a morphism φ: X → Y is a pseudo-kernel for ψ iff φ decomposes as [φ_{1} φ_{2}]: X_{1} ∐ X_{2} → Y where φ_{1} is a kernel and ψ∘φ_{2} = 0. Thus a left exact functor preserves pseudo-kernels. Conversely, suppose that f preserves pseudo-kernels. It follows from (2) that f^{∗}: mod(C) → mod(D) is exact. Therefore f^{∗}∘h_{C} = h_{D}∘f is left exact and we conclude the left exactness of f.

Lemma 2.7. Let f: C → D be a fully faithful functor between additive categories and suppose that D has cokernels. Then the following are equivalent.

(1) f admits a left adjoint.

(2) f admits a right exact functor g: D → C satisfying g∘f = id_{C}. In this case g is a left adjoint and C has cokernels.

Proof. Straightforward.

Lemma 2.8. The Yoneda functor h_{C}: C → mod(C) has the following properties.

(1) If C has split idempotents and pseudo-kernels, then h_{C} preserves pseudokernels iff C has kernels.

(2) h_{C} has a left adjoint iff C has cokernels. A left adjoint is right exact.

Proof. Use the above lemmata.

We include the following well-known criterion for a right adjoint functor to be exact.

Lemma 2.9. Let f: A → B be a functor between abelian categories and let g: B → A be a left adjoint.

(1) If f is exact, then g preserves projectives.

(2) If B has sufficently many projectives and g preserves projectives, then f is exact.

Let C be a pre-additive category and define A(C) = mod(mop(C)). Further let

a_{C}: C → A(C), X ↦ Hom(−, Hom(X, −))

be the composition of the corresponding Yoneda functors. The category A(C) has the following property.

Universal Property 2.10. The category A(C) is abelian and given any functor f: C → A into an abelian category, there exists, up to a natural isomorphism, a unique exact functor f^{∗}: A(C) → A such that f = f^{∗}∘a_{C}.

Proof. The category A(C) is abelian by Lemma 2.2 since mop(C) has kernels. A functor f: C → A extends uniquely to a left exact functor f′: mop(C) → A and this extends uniquely to a right exact functor f^{∗}: A(C) → A. This follows from Property 2.1 and we deduce from Lemma 2.5 and Lemma 2.6 the exactness of f^{∗} since f′ is left exact. Any other exact functor A(C) → A extending f needs to be isomorphic to f^{∗} since the restriction to mop(C) is left exact and therefore isomorphic to f′.

Corollary 2.11. There exists, up to a natural isomorphism, a unique duality d: A(C) → A(C^{op}) which extends the canonical duality C → C^{op}. This duality F ↦ d(F) is given by

d(F)(X) = Hom(F, X ⊗_{C} −).

Proof. The functor d is the unique exact functor which makes the following diagram commutative.

C ⟶ C^{op}
↓a_{C}  ↓a_{Cop}
A(C) ⟶^{d} A(C^{op})

Corollary 2.12. There exists, up to a natural isomorphism, a unique equivalence

mod(mop(C)) ⟶ mop(mod(C))

which sends Hom(−, Hom(X, −)) to Hom(Hom(−, X), −) for every X ∈ C.

Corollary 2.13. The category A(C) has sufficently many projectives which are the functors Hom(−, X), X ∈ mod(C^{op}), and sufficently many injectives which are the functors X ⊗_{C} −, X ∈ mod(C). Finally, the projective-injective objects are precisely the direct summands of functors ∐_{i=1}^{n} Hom(−, Hom(X, −)), X_{i} ∈ C.

Corollary 2.14. The functor d: mod(C) → A(C), F ↦ F ⊗_{C} − induces an equivalence between mod(C) and inj(A(C)). Moreover, the Yoneda functor h: C → mop(C) induces a functor h^{∗} which is isomorphic to d.

Remark 2.15. If C has precisely one object, in other words that it is a ring, then the universal property of the category A(C) has been studied by Gruson [9], see also [5].

3. Contravariantly finite subcategories

Let A be an additive category. Following [2], an additive subcategory C is said to be contravariantly finite provided that every object X ∈ A has a right C-approximation, i.e. a morphism Y → X with Y ∈ C such that the induced sequence of functors Hom(−, Y)|_{C} → Hom(−, X)|_{C} → 0 from C^{op} to Ab is exact. Of course there is the dual notion of covariant finiteness.

Let f: C → D be a functor between additive categories and denote by f^{∗}: mod(C) → mod(D) the unique right exact functor extending f. Also we have for each functor F: D^{op} → Ab the restriction f_{∗}(F) = F ∘f. We start with some preliminary results.

Lemma 3.1. The following are equivalent.

(1) f_{∗}(F) is finitely presented for all F ∈ mod(D).

(2) f^{∗} has a right adjoint.

Proof. (1) ⇒ (2) Take f_{∗} as a right adjoint.

(2) ⇒ (1) Let g be a right adjoint of f^{∗}. The isomorphism

g(F)(X) ≅ Hom(Hom(−, X), g(F)) ≅ Hom(f^{∗}(Hom(−, X)), F)
≅ Hom(Hom(−, f(X)), F) ≅ F (f(X)) = f_{∗}(F)(X)

shows that g takes the same values as f_{∗}.

Lemma 3.2. Suppose that D has pseudo-kernels and that f is fully faithful. Then the following are equivalent.

(1) The image Im(f) of f is contravariantly finite in D.

(2) f_{∗}(Hom(−, X)) is finitely generated for all X ∈ D.

(3) f_{∗}(F) is finitely presented for all F ∈ mod(D).

Proof. (1) ⇔ (2) ⇐ (3) This is a direct consequence of the definitions involved.

(2) ⇒ (3) It suffices to show that f_{∗}(Hom(−, X)) is finitely presented for all X ∈ D since f_{∗} is exact and mod(C) is closed under cokernels. To this end let f(Y) → X be a right Im(f)-approximation of X and let f(Y′) → X′ be a left Im(f)-approximation of a pseudo-kernel X′ of f(Y) → X. Choose a morphism φ: Y′ → Y such that f(φ) equals the composition f(Y′) → X′ → f(Y). It is easily seen that φ induces a presentation Hom(−, Y′) → Hom(−, Y) → Hom(−, X)∘f → 0 of f_{∗}(Hom(−, X)) = Hom(−, X)∘f. This finishes the proof.

It is well-known that a subcategory C of an additive category A is contravariantly finite if the inclusion functor i: C → A has a right adjoint. Consider the following example.

Example 3.3. Let C be an additive category and h: C → mop(C) be the Yoneda functor. The image of h is contravariantly finite iff C has pseudo-kernels. The functor h has a right adjoint iff C has kernels.

Replacing i: C → A by i^{∗}: mod(C) → mod(A) we obtain the following criterion.

Theorem 3.4. Let A be an additive category with pseudo-kernels. If i: C → A denotes the inclusion of a full additive subcategory, then the following are equivalent.

(1) C is contravariantly finite.

(2) The right exact functor i^{∗}: mod(C) → mod(A) which extends i has a right adjoint.

(3) Restriction induces a functor i_{∗}: mod(A) → mod(C). In this case the category C has pseudo-kernels and i_{∗} is exact.

Proof. Combine Lemma 3.1 and Lemma 3.2 to prove the equivalence of (1) – (3). To obtain a pseudo-cokernel for a morphism in C take a right C-approximation of a pseudo-cokernel in A. The exactness of i_{∗} is clear since kernels and cokernels in mod(C) and mod(A) are computed pointwise.

Remark 3.5. If A⌋⌈C denotes the additive quotient of A modulo all morphisms which factor through objects in C, then mod(A⌋⌈C) and the kernel of i_{∗} are identified via the canonical functor A → A⌋⌈C, and i_{∗} induces an equivalence between the abelian quotient mod(A)/ mod(A⌋⌈C) and mod(C).

4. Module categories and tensor products

Module categories and tensor products are the most important tools for studying locally finitely presented categories and their functors. Let C be a skeletally small pre-additive category. We denote by Mod(C) the category of functors from C^{op} to the category Ab of abelian groups. The objects in Mod(C) are called C-modules [16]. It is well-known that Mod(C) is a locally finitely presented abelian category and that fp(Mod(C)) and mod(C) coincide. We denote by h_{C}: C → Mod(C) the Yoneda functor X ↦ Hom(−, X). This functor has the following property.

Universal Property 4.1. Let f: C → A be a functor into an additive category with colimits. Then there exists, up to a natural isomorphism, a unique functor f^{∗}: Mod(C) → A such that f = f^{∗}∘h_{C} and f^{∗} commutes with colimits. Moreover, f^{∗} has a right adjoint.

Proof. The category Mod(C) has sufficently many projectives and proj(Mod(C)) consists precisely of the direct summands of coproducts ∐_{i∈I} Hom(−, X_{i}). First we define

f^{∗}(∐_{i} Hom(−, X_{i})) = ∐_{i} f(X_{i})

and for a direct summand M we define f^{∗}(M) to be the cokernel of f^{∗}(id − ε) where ε ∈ End(∐_{i} Hom(−, X_{i})) is the idempotent corresponding to M. Similarly, we define f^{∗} on morphisms between projectives. Thus we obtain a unique functor proj(Mod(C)) → A extending f and commuting with colimits. Now one uses Property 2.1 to extend this functor uniquely to a right exact functor f^{∗}: Mod(C) → A. The right adjoint f_{∗}: A → Mod(C) is obtained by f_{∗}(M)(X) = Hom(f(X), M) for M ∈ A. It is routine to check that there is a functorial isomorphism Hom(f^{∗}(N), M) ≅ Hom(N, f_{∗}(M)) for N ∈ Mod(C). The uniqueness of f^{∗} follows from the fact that f^{∗} commutes with colimits.

The preceding result can be used to define a tensor product Mod(C)×Mod(C^{op}) → Ab, (M, N) ↦ M ⊗_{C} N which extends the tensor product defined on mod(C) × mod(C^{op}) [12]. It is characterized, up to a natural isomorphism, by the following properties.

(1) There are functorial isomorphism M⊗_{C}Hom(X, −) ≅ M(X) and Hom(−, X)⊗_{C} N ≅ N(X) for X ∈ C.

(2) M ⊗_{C} − and − ⊗_{C} N have right adjoints.

Let f: C → D be a functor between skeletally small additive categories. The restriction functor f_{∗}: Mod(D) → Mod(C), M ↦ M ∘f has a left adjoint

f^{∗}: Mod(C) → Mod(D), M ↦ N with N(X) = M ⊗_{C} (Hom(X, −)∘f)

and a right adjoint

∗f: Mod(C) → Mod(D), M ↦ N with N(X) = Hom(Hom(−, X)∘f, M).

We discuss some properties of these functors.

Lemma 4.2. (1) h_{D}∘f = f^{∗}∘h_{C} where h_{C}: C → Mod(C) and h_{D}: D → Mod(D) are given by X ↦ Hom(−, X).

(2) f^{∗} is fully faithful iff f is fully faithful.

Proof. (1) Clear.

(2) It follows from (1) that f is fully faithful if f^{∗} is fully faithful since h_{C} and h_{D} are fully faithful. Conversely, suppose that f is fully faithful. The isomorphism

Hom(∐_{i} Hom(−, X_{i}), ∐_{j} Hom(−, Y_{j})) ≅ ∏_{i}∐_{j} Hom(X_{i}, Y_{j})

shows that f^{∗} induces a fully faithful functor proj(Mod(C)) → proj(Mod(D)) since f^{∗}(Hom(−, X)) = Hom(−, f(X)) for every X ∈ C and f^{∗} commutes with coproducts. It follows from Lemma 2.6 that the induced right exact functor mod(proj(Mod(C))) → mod(proj(Mod(D))) is fully faithful. But this functor is isomorphic to f^{∗}.

5. Locally finitely presented categories

Let C be a skeletally small pre-additive category. A C-module M is said to be flat provided that M ⊗_{C} − is exact. The full subcategory of Mod(C) formed by the flat modules is denoted by Flat(C). It is well-known that this is a locally finitely presented category [4, Theorem 1.4]. In fact the category Flat(C) is closed under direct limits in Mod(C), so has direct limits itself and a C-module M is flat if and only if it is a direct limit of modules of the form Hom(−, X) with X ∈ C.

The following description of an arbitrary locally finitely presented category is due to Crawley-Boevey [4, Theorem 1.4]. The result goes back to work of Grothendieck and Verdier [8, Numéro 8]; see also [1, Theorem 2.26]. It generalizes results for locally noetherian categories [6, II.4, Théorème 1], locally coherent categories [18, Proposition 2.2], locally finitely presented abelian categories [3, Satz 2.4] and locally finitely presented categories in the sense of Gabriel and Ulmer [7, Korollar 7.9].

Proposition 5.1. If A is a locally finitely presented category, then the functor A → Flat(fp(A)) given by X ↦ Hom(−, X)|_{fp(A)} is an equivalence. There is, up to equivalence, a bijective correspondence between skeletally small additive categories with split idempotents and locally finitely presented categories. The correspondence is given by

C ↦ Flat(C)  and  A ↦ fp(A).

This result has some obvious consequences which we now discuss. Let A be a locally finitely presented category. Denote by m_{A}: A → Mod(fp(A)) the fully faithful functor given by X ↦ Hom(−, X)|_{fp(A)}.

Universal Property 5.2. Let f: A → B be a functor into an additive category with cokernels and direct limits. Suppose also that f commutes with direct limits. Then there exists, up to a natural isomorphism, a unique right exact functor f^{∗}: Mod(fp(A)) → B such that f = f^{∗}∘m_{A} and f^{∗} commutes with direct limits.

Proof. The assumption on B means that B has colimits. Thus f|_{fp(A)} extends uniquely to a functor f^{∗}: Mod(fp(A)) → B which commutes with colimits by Property 4.1. Using the identification A = Flat(fp(A)) the assertion follows since f commutes with direct limits.

Following Roos [18] we call A^{co} = Flat(fp(A)^{op}) the conjugate category of A. The contravariant functor fp(A) → A^{co}, X ↦ Hom(X, −)|_{fp(A)} induces a duality between fp(A) and fp(A^{co}) which we denote by d.

Proposition 5.3. Let A be a locally finitely presented category. There exists a tensor product functor A×A^{co} → Ab, (M, N) ↦ M ⊗_{fp(A)}N which is characterized, up to a natural isomorphism, by the following properties.

(1) There are functorial isomorphism M ⊗_{fp(A)} Y ≅ Hom(d(Y), M) for Y ∈ fp(A^{co}) and X ⊗_{fp(A)} N ≅ Hom(d(X), N) for X ∈ fp(A).

(2) M ⊗_{fp(A)} − and − ⊗_{fp(A)} N commute with direct limits.

Proof. Using the identification A = Flat(fp(A)) and A^{co} = Flat(fp(A)^{op}) the assertion follows if one restricts the tensor product which is defined on Mod(fp(A)) × Mod(fp(A)^{op}).

The next few results show how certain properties of a locally finitely presented category A are reflected by properties of the category C = fp(A). We shall need a canonical presentation of any object X and any morphism φ: X → Y in A as a direct limit of objects and morphisms, respectively, in C. To this end define categories C/X and C/φ as follows. An object λ ∈ C/X is given by a morphism α_{λ}: X_{λ} → X with X_{λ} ∈ C and a morphism θ: λ → μ is given by a morphism α_{θ}: X_{λ} → X_{μ} satisfying α_{λ} = α_{μ}∘α_{θ}. An object λ ∈ C/φ is given by a commuting diagram of the form

X_{λ} ⟶^{φ_{λ}} Y_{λ}
↓α_{λ}  ↓β_{λ}
X ⟶^{φ} Y

with X_{λ}, Y_{λ} ∈ C and a morphism θ: λ → μ is given by a commuting diagram of the form

X_{λ} ⟶^{φ_{λ}} Y_{λ}
↓α_{θ}  ↓β_{θ}
X_{μ} ⟶^{φ_{μ}} Y_{μ}

with α_{λ} = α_{μ}∘α_{θ} and β_{λ} = β_{μ}∘β_{θ}.

Lemma 5.4. (1) The category C/X is filtered and ⟶ lim_{λ∈C/X} X_{λ} = X. (2) The category C/φ is filtered and ⟶ lim_{λ∈C/φ} φ_{λ} = φ.

(3) φ induces a natural functor C/X → C/Y.

Proof. Straightforward.

Lemma 5.5. Let F_{λ}: I_{λ} → I, λ ∈ J, be a filtered family of functors between filtered categories and suppose that the family is cofinal, i.e.

(i) for any ν ∈ I there is an object μ ∈ I_{λ} for some λ ∈ J and a morphism ν → F_{λ}(μ);

(ii) for any pair of morphisms φ_{1}, φ_{2} : ν → F_{λ}(μ) there is a morphism ψ: μ → χ such that F_{λ}(ψ)∘φ_{1} = F_{λ}(ψ)∘φ_{2}. Then for any functor X from I to a category with direct limits, the natural morphism

φ: ⟶ lim_{λ∈J} (⟶ lim_{μ∈I_{λ}} X_{F_{λ}(μ)}) ⟶ ⟶ lim_{ν∈I} X_{ν}

is an isomorphism.

Proof. Using (i) – (ii) one constructs for any object Y and any morphism

σ: ⟶ lim_{λ∈J} (⟶ lim_{μ∈I_{λ}} X_{F_{λ}(μ)}) ⟶ Y

a unique morphism τ: ⟶ lim_{ν∈I} X_{ν} → Y satisfying σ = τ ∘φ. Thus φ induces an isomorphism

Hom(⟶ lim_{ν∈I} X_{ν}, Y) ⟶ Hom(⟶ lim_{λ∈J} (⟶ lim_{μ∈I_{λ}} X_{F_{λ}(μ)}), Y)

which is functorial in Y, and the assertion follows.

We are now in position to prove a universal property of a locally finitely presented category. In fact the property characterizes locally finitely presented categories, and in this generality the result can be found in the exposition of Adámek and Rosický [1, Theorem 2.26].

Universal Property 5.6. Let f: fp(A) → B be a functor into an additive category with direct limits. Then there exists, up to a natural isomorphism, a unique functor f^{∗}: A → B such that f^{∗}|_{fp(A)} = f and f^{∗} commutes with direct limits.

Proof. Define f^{∗}(X) = ⟶ lim_{λ∈C/X} f(X_{λ}) and f^{∗}(φ) = ⟶ lim_{λ∈C/φ} f(φ_{λ}) for every object X and every morphism φ in A. We show that f^{∗} commutes with direct limits. To this end let X = ⟶ lim X_{λ} be a direct limit in A. The corresponding family of functors C/X_{λ} → C/X satisfies the assumption of Lemma 5.5 and we obtain therefore

⟶ lim_{λ} f^{∗}(X_{λ}) = ⟶ lim_{λ}(⟶ lim_{μ∈C/X_{λ}} f(X_{μ})) ≅ ⟶ lim_{ν∈C/X} f(X_{ν}) = f^{∗}(X).

The uniqueness follows easily from the requirement that the functor A → B extending f commutes with direct limits.

We proceed with some further properties of A which are related to properties of fp(A).

Lemma 5.7. A locally finitely presented category A has cokernels iff fp(A) has cokernels. In that case the embedding fp(A) → A is right exact, direct limits in A are right exact, and any exact sequence L → M → N → 0 in A is a direct limit of exact sequences L_{λ} → M_{λ} → N_{λ} → 0 in fp(A).

Proof. If A has cokernels, then fp(A) is closed under cokernels, so has cokernels itself and the embedding fp(A) → A is right exact. Conversely, suppose that fp(A) has cokernels. It is not hard to check that each cokernel in fp(A) is also a cokernel in A. The fact that direct limits preserve colimits implies that direct limits are right exact. Now let φ: L → M be a morphism in A. Write φ as a direct limit of morphisms φ_{λ}: L_{λ} → M_{λ} in fp(A) and consider for each λ a cokernel M_{λ} → N_{λ} in fp(A). Again, the direct limit M → ⟶ lim N_{λ} is a cokernel for φ since direct limits preserve colimits. Thus A has cokernels and the lemma is proven.

Lemma 5.8. A locally finitely presented category A has kernels, if fp(A) has kernels. In that case the embedding fp(A) → A is left exact, direct limits in A are left exact, and any exact sequence 0 → L → M → N in A is a direct limit of exact sequences 0 → L_{λ} → M_{λ} → N_{λ} in fp(A).

Proof. The fully faithful functor A → Mod(fp(A)), X ↦ Hom(−, X)|_{fp(A)} shows immediately that the embedding fp(A) → A is left exact and that direct limits in A are left exact. Now let φ: M → N be a morphism in A. Write φ as a direct limit of morphisms φ_{λ}: M_{λ} → N_{λ} in fp(A) and consider for each λ a kernel L_{λ} → M_{λ} in fp(A). We claim that ⟶ lim L_{λ} → M is a kernel for φ. Using again the functor A → Mod(fp(A)) this follows since 0 → Hom(X, ⟶ lim L_{λ}) → Hom(X, ⟶ lim M_{λ}) → Hom(X, ⟶ lim N_{λ}) is exact for each X ∈ fp(A). Thus A has kernels and the lemma is proven.

Lemma 5.9. If A is a locally coherent category, then the embedding fp(A) → A is exact, direct limits in A are exact, and any exact sequence 0 → L → M → N → 0 in A is a direct limit of exact sequences 0 → L_{λ} → M_{λ} → N_{λ} → 0 in fp(A).

Proof. Combine the previous lemmata.

Proposition 5.10. Let A be a locally finitely presented category and let f: A → B be a functor into an additive category which preserves direct limits.

(1) If fp(A) and B have cokernels, then f is right exact if and only if f|_{fp(A)} is right exact.

(2) If fp(A) and B have kernels and direct limits in B are left exact, then f is left exact if and only if f|_{fp(A)} is left exact.

(3) If fp(A) and B are abelian and direct limits in B are exact, then f is exact if and only if f|_{fp(A)} is exact.

Proof. Combine the previous lemmata.

Let A and B be a pair of locally finitely presented categories. Any functor f: fp(A) → fp(B) extends uniquely to a functor A → B which commutes with direct limits by Property 5.6. Alternatively, one could use the induced functor f^{∗}: Mod(fp(A)) → Mod(fp(B)) which restricts to a functor Flat(fp(A)) → Flat(fp(B)). This observation has some important implication.

Given a full additive subcategory C of fp(A) denote by C^{→} the full subcategory of A formed by the objects which are direct limits of objects in C. This subcategory has the following description which is due to Crawley-Boevey [4, Theorem 4.1].

Proposition 5.11. C^{→} is a locally finitely presented category which is closed under direct limits in A, and fp(C^{→}) consists of the direct summands of objects in C. An object X ∈ A belongs to C^{→} if and only if any morphism from an object in fp(A) to X factors through some object in C.

Proof. Using the identification A = Flat(fp(A)) the inclusion C → fp(A) induces a fully faithful functor Mod(C) → Mod(fp(A)) by Lemma 4.2, which restricts to an equivalence between Flat(C) and C^{→}.

6. Adjoints

We develop various criteria for the existence of left and right adjoints for functors between locally finitely presented categories. This is strongly related to the question whether a functor preserves finitely presented objects. We shall use the following crucial fact.

Lemma 6.1. Let f: A → B be a functor between categories with direct limits. Suppose there exists a right adjoint which commutes with direct limits. Then f(X) is finitely presented (finitely generated), if X is a finitely presented (finitely generated) object in A.

Proof. Let X ∈ A and ⟶ lim Y_{λ} ∈ B. We have the following sequence of morphisms

⟶ lim Hom(f(X), Y_{λ}) ≅ ⟶ lim Hom(X, g(Y_{λ})) →^{φ} Hom(X, ⟶ lim g(Y_{λ})) ≅ Hom(X, g(⟶ lim Y_{λ}))

≅ Hom(f(X), ⟶ lim Y_{λ})

since f and g is a pair of adjoint functors and g commutes with direct limits. Now the assertion follows since φ is an isomorphism (a monomorphism), if X is finitely presented (finitely generated).

Let A and B be a pair of locally finitely presented categories. Denote for any functor f: fp(A) → fp(B) by f^{∗}: A → B the unique functor which extends f and commutes with direct limits.

Theorem 6.2. For a functor f: fp(A) → fp(B) the following are equivalent. (1) f^{∗}: A → B has a left adjoint.

(2) f: fp(A) → fp(B) has a left adjoint. Moreover, if g: fp(B) → fp(A) is a left adjoint of f, then g^{∗}: B → A is a left adjoint of f^{∗}: A → B.

Proof. (1) ⇒ (2) The left adjoint of f^{∗} maps finitely presented to finitely presented objects by Lemma 6.1 and gives therefore a left adjoint of f.

(2) ⇒ (1) We shall identify A = Flat(fp(A)) and B = Flat(fp(B)), respectively. Now let g: fp(B) → fp(A) be a left adjoint of f. It is easily verified that g_{∗}: Mod(fp(A)) → Mod(fp(B)) is a left adjoint of f_{∗}: Mod(fp(B)) → Mod(fp(A)). This fact implies that g_{∗}(M) ≅ f^{∗}(M) is flat for every M ∈ A. The assertion now follows since g^{∗}: B → A is a left adjoint of the functor A → B, M ↦ g_{∗}(M).

We have the following application of which one direction has been proved by Gabriel and Ulmer [7, Korollar 5.8] using some different arguments.

Corollary 6.3. Let C be a skeletally small additive category. Then the inclusion Flat(C) → Mod(C) has a left adjoint if and only if C has cokernels.

Proof. The inclusion C → mod(C) has a left adjoint iff C has cokernels by Lemma 2.8. The assertion now follows from the preceding theorem.

Lemma 6.4. (1) If C has pseudo-cokernels, then a C-module M is flat iff M maps any sequence X →^{φ} Y →^{ψ} Z in C where ψ is a pseudo-cokernel for φ, to an exact sequence M(Z) → M(Y) → M(X). (2) If C has cokernels, then a C-module M is flat iff the functor M: C^{op} → Ab is left exact.

Proof. The category mod(C^{op}) is abelian and therefore Proposition 5.10 applies, saying that M ⊗_{C} − is exact iff the restriction M ⊗_{C} −|_{mod(C^{op})} is exact. Now the assertions follow from Lemma 2.5 and Lemma 2.6.

Lemma 6.5. If C has pseudo-cokernels, then a functor f: C → D preserves pseudocokernels iff f_{∗}: Mod(D) → Mod(C) preserves flatness.

Proof. The first implication follows from the characterization of flatness in the previous lemma. Conversely, the exactness of f_{∗}(Hom(−, Y))⊗_{C} − for every Y ∈ D implies that f preserves pseudo-cokernels since f_{∗}(Hom(−, Y)) ⊗_{C} Hom(X, −) ≅ Hom(f(X), Y) for every X ∈ C.

The characterization of flat modules in the previous lemma allows one to deduce the following theorem of Crawley-Boevey [4, Theorem 2.1] which generalizes a wellknown result of Chase for modules over a ring.

Proposition 6.6. For a locally finitely presented category A the following are equivalent. (1) A has products.

(2) fp(A) has pseudo-cokernels.

(3) A product of flat fp(A)-modules is flat.

The next result gives a criterion for the existence of a right adjoint.

Theorem 6.7. Let f: A → B be a functor between locally finitely presented categories with products and suppose that f commutes with direct limits. Then the following are equivalent.

(1) f has a right adjoint which commutes with direct limits.

(2) f restricts to a functor fp(A) → fp(B) which preserves pseudo-cokernels.

Proof. (1) ⇒ (2) Let g: B → A be a right adjoint of f commuting with direct limits. It follows from Lemma 6.1 that f restricts to a functor fp(A) → fp(B). Let X →^{φ} Y →^{ψ} Z be a sequence in fp(A) where ψ is a pseudo-cokernel for φ in fp(A). It is easily checked that ψ is also a pseudo-cokernel in A. It follows that Hom(Z, g(M)) → Hom(Y, g(M)) → Hom(X, g(M)) is exact for every M ∈ B and therefore Hom(f(Z), M) → Hom(f(Y), M) → Hom(f(X), M) is exact which shows that f(ψ) is a pseudo-cokernel for f(φ).

(2) ⇒ (1) Denote by h: fp(A) → fp(B) the restriction of f. The functor h_{∗}: Mod(fp(B)) → Mod(fp(A)) preserves flatness by Lemma 6.5. Thus Flat(fp(B)) → Flat(fp(A)), X ↦ h_{∗}(X) is a right adjoint of f which commutes with direct limits, since f is isomorphic to Flat(fp(A)) → Flat(fp(B)), X ↦ h^{∗}(X).

We finish our discussion of adjoint functors with the following example.

Proposition 6.8. Let C and D be skeletally small additive categories with split idempotents. A functor f: Mod(C) → Mod(D) is isomorphic to g_{∗} for some additive functor g: D → C if and only if f commutes with limits and colimits.

Proof. One direction is clear. So suppose that f commutes with limits and colimits. Thus f has a left adjoint since f commutes with limits and this left adjoint preserves finitely presented objects by Lemma 6.1 and projectives since f is exact. Using the Yoneda embeddings we obtain g by restricting the left adjoint to a functor from D to C.

7. Products

In this section we discuss the property of certain functors to preserve products. It is well-known that for a ring Λ a right Λ-module M is finitely presented (finitely generated) iff the natural map M ⊗_{Λ} (∏_{i} N_{i}) → ∏_{i}(M ⊗_{Λ} N_{i}) is an isomorphism (epimorphism) for every family (N_{i})_{i∈I} of left Λ-modules. This generalizes to arbitrary module categories as follows.

Lemma 7.1. Let C be a skeletally small additive category. For M ∈ Mod(C) the following are equivalent.

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

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

(3) The natural morphism M ⊗_{C} (∏_{i} Hom(X_{i}, −)) → ∏_{i}(M ⊗_{C} Hom(X_{i}, −)) is an isomorphism (epimorphism) for every family (X_{i})_{i∈I} in C.

Proof. Adopt the proof for modules over a ring.

Let f: C → D be a functor between skeletally small additive categories. Recall that f induces the restriction functor f_{∗}: Mod(D) → Mod(C) which has a left adjoint f^{∗}: Mod(C) → Mod(D).

Proposition 7.2. The following are equivalent.

(1) (f^{op})_{∗} restricts to a functor mod(D^{op}) → mod(C^{op}).

(2) The natural morphism f^{∗}(∏_{i} M_{i}) → ∏_{i} f^{∗}(M_{i}) is an isomorphism for all families (M_{i})_{i∈I} in Mod(C).

(3) The natural morphism f^{∗}(∏_{i} Hom(−, X_{i})) → ∏_{i} f^{∗}(Hom(−, X_{i})) is an isomorphism for all families (X_{i})_{i∈I} in C.

Proof. The assertion follows from the definition of f_{∗} and f^{∗} using the lemma above.

Theorem 7.3. Let A and B be locally finitely presented categories with products. Suppose that f: fp(A) → fp(B) is a functor and denote by f^{∗}: A → B the induced functor which commutes with direct limits. Then the following are equivalent.

(1) (f^{op})_{∗} restricts to a functor mod(fp(B)^{op}) → mod(fp(A)^{op}).

(2) The natural morphism f^{∗}(∏_{i} M_{i}) → ∏_{i} f^{∗}(M_{i}) is an isomorphism for all families (M_{i})_{i∈I} in A.

(3) The natural morphism f^{∗}(∏_{i} M_{i}) → ∏_{i} f^{∗}(M_{i}) is an isomorphism for all families (M_{i})_{i∈I} in fp(A).

Proof. The assertion follows immediately from the previous result if one identifies A and B with Flat(fp(A)) and Flat(fp(B)), respectively.

Specializing the preceding theorem to the case that f is an embedding and using Theorem 3.4 we obtain the following result which is due to Crawley-Boevey [4, Theorem 4.2].

Corollary 7.4. Let A be a locally finitely presented category with products and C a full additive subcategory of fp(A). Then C^{→} is closed under products in A if and only if C is covariantly finite in fp(A).

8. The category D(A)

Let A be a locally finitely presented category. In this section we construct a functor d_{A}: A → D(A) into a locally finitely presented category with kernels which is an analogue of the functor A → Mod(fp(A)), X ↦ Hom(−, X)|_{fp(A)}. This is based on a construction which was introduced by Gruson and Jensen [10] and further extended by Simson [19] and Crawley-Boevey [4].

Let C = fp(A) and denote by h: C → mop(C) the Yoneda functor. We define D(A) = Flat(mop(C)) and d_{A} = h^{∗} is the unique functor commuting with direct limits which extends h. Note that D(A) has kernels and left exact direct limits by Lemma 5.8.

Universal Property 8.1. Let f: A → B be a functor into an additive category with kernels and left exact direct limits. Supppose also that f commutes with direct limits. Then there exists, up to a natural isomorphism, a unique left exact functor f^{∗}: D(A) → B such that f = f^{∗}∘d_{A} and f^{∗} commutes with direct limits.

Proof. The restriction f|_{C}: C → B extends uniquely to a left exact functor f′: mop(C) → B by Property 2.1 and this extends uniquely to a functor f^{∗}: D(A) → B commuting with direct limits by Property 5.6. The left exactness of f^{∗} follows from Proposition 5.10.

Corollary 8.2. The category A has kernels if and only if d_{A} admits a right adjoint. A right adjoint commutes with direct limits, and it preserves finitely presented objects if and only if fp(A) has kernels.

Proof. Combine the above theorem with the statements of Lemma 2.7 and Lemma 2.8.

Corollary 8.3. The category A has products if and only if D(A) has products. In this case d_{A} commutes with products.

Proof. The category A has products iff C has pseudo-cokernels by Proposition 6.6. It follows that A has products iff D(A) has products since C has pseudo-cokernels iff mop(C) = fp(D(A)) has pseudo-cokernels by Lemma 2.2. It remains to check that the image of d_{A} is closed under products taken in D(A). This follows with Corollary 7.4 since the image of the Yoneda functor h is covariantly finite in mop(C).

Corollary 8.4. Let A be a locally finitely presented category with products. Then A has kernels if and only if A has cokernels.

Proof. By Theorem 6.7, the functor d_{A}: A → D(A) has a right adjoint commuting with direct limits iff the Yoneda functor C → mop(C) = fp(D(A)) preserves pseudocokernels. This happens iff C has cokernels, equivalently if A has cokernels, by Lemma 2.8 and Lemma 5.7. Now the assertion follows from Corollary 8.2.

Recall that an additive category is (co)complete if it has arbitrary (co)limits, equivalently if (co)kernels and (co)products exist. The following consequence of the preceding result seems to be well-known (e.g. see [17, Theorem 6.1.4]).

Corollary 8.5. A locally finitely presented category is complete if and only if it is cocomplete.

Our discussion suggests a further construction. We define a functor c_{A}: A → C(A) into a locally coherent category as follows. Let C(A) = Flat(A(C)) and c_{A} = a_{C}^{∗} is the unique functor commuting with direct limits which extends a_{C}: C → A(C), X ↦ Hom(−, Hom(X, −)). Note that C(A) = D(Mod(C)) and that c_{A} is isomorphic to the composition of A → Mod(C), X ↦ Hom(−, X)|_{C} with d_{Mod(C)}.

Universal Property 8.6. Let f: A → B be a functor into an abelian category with exact direct limits. Supppose also that f commutes with direct limts. Then there exists, up to a natural isomorphism, a unique exact functor f^{∗}: C(A) → B such that f = f^{∗}∘c_{A} and f^{∗} commutes with direct limits.

Proof. Analogous to the proof of Property 8.1.

9. Locally coherent categories

Recall that a locally finitely presented category A is locally coherent provided that fp(A) is abelian. Note that in this case the embedding of fp(A) into A is exact. In this section we discuss how certain properties of such a category A are reflected by properties of fp(A). We need some preparations.

Lemma 9.1. Let A be a Grothendieck category. An object X ∈ A is finitely presented (finitely generated) if and only if for every direct limit ⟶ lim Y_{λ} in A with Y_{λ} ∈ inj(A) for all λ the natural morphism ⟶ lim Hom(X, Y_{λ}) → Hom(X, ⟶ lim Y_{λ}) is an isomorphism (a monomorphism).

Proof. See [3, Korollar 1.8].

Recall that an object M of a Grothendieck category A is fp-injective provided that Ext^{1}(Z, M) = 0 for all Z ∈ fp(A). We denote the full subcategory of fpinjective objects in A by fpinj(A). Note that fpinj(A) is automatically closed under products. The following lemma is taken from [4].

Lemma 9.2. Let A be a locally finitely presented abelian category. For Z ∈ fp(A) and M ∈ A the following are equivalent.

(1) Ext^{1}(Z, M) = 0.

(2) 0 → Hom(Z, M) → Hom(Y, M) → Hom(X, M) → 0 is exact for all exact sequences 0 → X → Y → Z → 0 in A with Y finitely presented.

Proof. Straightforward.

Using the above lemmata we obtain the following.

Lemma 9.3. Let A be a locally finitely presented abelian category.

(1) If (M_{i})_{i∈I} is a filtered family of subobjects M_{i} ∈ fpinj(A) of some M ∈ A, then ∑_{i} M_{i} ∈ fpinj(A). In particular, fpinj(A) is closed under coproducts.

(2) A is locally coherent iff fpinj(A) is closed under direct limits.

Proof. Let 0 → X → Y → Z → 0 be an exact sequence in A with Y ∈ fp(A). Suppose also that Z ∈ fp(A), equivalently that X is finitely generated. Let ⟶ lim M_{λ} be a direct limit of fp-injective objects in A. Then we obtain the following commutative diagram

⟶ lim Hom(Y, M_{λ}) ⟶ ⟶ lim Hom(X, M_{λ}) ⟶ 0
↓ψ  ↓φ
Hom(Y, ⟶ lim M_{λ}) ⟶ Hom(X, ⟶ lim M_{λ}) ⟶ 0

where the upper row is exact and ψ is an isomorphism by our assumptions.

(1) If the M_{λ}’s form a filtered family of subobjects of some M ∈ A, then φ is an isomorphism since X is finitely generated. Thus the lower row is exact and ∑ M_{λ} = ⟶ lim M_{λ} is fp-injective.

(2) The map φ is an isomorphism iff the lower row is exact. Thus a finitely generated subobject X of Y is finitely presented iff ⟶ lim M_{λ} is fp-injective for any choice of the M_{λ}’s.

Remark 9.4. Let A be a locally coherent category. The equivalence A → Flat(fp(A)) given by X ↦ Hom(−, X)|_{fp(A)} induces an equivalence between fpinj(A) and the full subcategory of exact functors from fp(A)^{op} to Ab.

The following is another preliminary result.

Lemma 9.5. Let A be a locally coherent category. Then fpinj(A) has direct limits and

fp(fpinj(A)) = fpinj(A) ∩ fp(A) = inj(fp(A)).

Proof. The assertion follows from the definitions involved using the above lemmata.

We say that a locally coherent category has sufficently many fp-injectives provided that fpinj(A) is locally finitely presented. The following is a consequence of our discussion of fp-injective objects.

Theorem 9.6. A locally coherent category has sufficently many fp-injectives if and only if fp(A) has sufficently many injectives. The finitely presented objects in fpinj(A) are precisely the injective objects in fp(A).

Proof. We have C = inj(fp(A)) = fp(fpinj(A)) by the preceding lemma. Suppose first that fpinj(A) is locally finitely presented. Thus fpinj(A) = C^{→}. We need to show that for every X ∈ fp(A) there exists a monomorphism X → Y with Y ∈ C. To this end let φ: X → ⟶ lim Y_{λ} be an injective envelope in A with Y_{λ} ∈ C for all λ. The morphism φ factors through some Y_{λ} since X is finitely presented and the corresponding morphism X → Y_{λ} is a monomorphism. Now suppose that fp(A) has sufficently many injectives. We claim that C^{→} = fpinj(A). We have C^{→} ⊆ fpinj(A) since fpinj(A) is closed under direct limits by Lemma 9.3. Using Proposition 5.11 it remains to show that every morphism X → Y with X ∈ fp(A) and Y ∈ fpinj(A) factors through some object Z ∈ C. But this is clear since we can choose a monomorphism X → Z and therefore the proof is complete.

Having discussed the case that fp(A) has sufficently many injectives we turn now to the case that fp(A) has sufficently many projectives.

Theorem 9.7. Let A be a locally coherent category. Then A is equivalent to a module category Mod(C) if and only if fp(A) has sufficently many projectives. In this case A and Mod(proj(fp(A))) are equivalent.

Proof. Suppose first that A ≅ Mod(C). Any finitely presented object in Mod(C) is a quotient of some representable functor which is projective by Yoneda’s lemma. Thus fp(A) has sufficently many projectives. Conversely, suppose that fp(A) has sufficently many projectives and let C = proj(fp(A)). Using Proposition 2.3 we deduce that fp(A) ≅ mod(C) and this implies A ≅ Mod(C) by Proposition 5.1. This finishes the proof.

Let A be a locally finitely presented category with products. It has been shown by Crawley-Boevey that under this assumption the category D(A) is locally coherent and that d_{A}: A → D(A) identifies A with fpinj(D(A)) [4, Theorem 3.3]. The next result shows that Crawley-Boevey’s construction is essentially unique.

Corollary 9.8. There is, up to equivalence, a bijective correspondence between locally finitely presented categories with products and locally coherent categories with sufficently many fp-injectives. The correspondence is given by

A ↦ D(A)  and  A ↦ fpinj(A).

Proof. Combine the correspondence between locally finitely presented categories with products and skeletally small additive categories with split idempotents and pseudo-cokernels given by A ↦ fp(A), the correspondence between skeletally small additive categories with split idempotents and pseudo-cokernels and skeletally small abelian categories with sufficently many injectives given by fp(A) ↦ mop(fp(A)), and finally, the correspondence between skeletally small abelian categories with sufficently many injectives and locally coherent categories with sufficently many fp-injectives given by mop(fp(A)) ↦ Flat(mop(fp(A))).

10. Functors between locally coherent categories

To develop further the theory of locally coherent categories one needs to study their functors. The most natural type of functors, namely those which preserve direct limits, finitely presented objects and exactness, will be the main objective of this section.

Theorem 10.1. Let A and B be locally coherent categories. Suppose that f: fp(A) → fp(B) is a functor and denote by f^{∗}: A → B the induced functor which commutes with direct limits. Then f^{∗} is exact if and only if f is exact. Moreover, if f is exact, then there exists a right adjoint f_{∗}: B → A which commutes with direct limits and sends fp-injective objects to fp-injective objects.

Proof. The assertion is a consequence of Proposition 5.10 and Theorem 6.7, except the fact that f_{∗} preserves fp-injectivity. To this end let M ∈ fpinj(B) and 0 → X → Y → Z → 0 be an exact sequence in fp(A). It suffices to show that 0 → Hom(Z, f_{∗}(M)) → Hom(Y, f_{∗}(M)) → Hom(X, f_{∗}(M)) → 0 is exact by Lemma 9.2. Applying f^{∗} we get an exact sequence 0 → Hom(f^{∗}(Z), M) → Hom(f^{∗}(Y), M) → Hom(f^{∗}(X), M) → 0 since f^{∗} is exact and M is fp-injective. But this sequence is isomorphic to 0 → Hom(Z, f_{∗}(M)) → Hom(Y, f_{∗}(M)) → Hom(X, f_{∗}(M)) → 0 since f_{∗} is a right adjoint of f^{∗}. Thus f_{∗}(M) is fp-injective.

Remark 10.2. The right adjoint f_{∗} sends injective objects to injective objects since f^{∗} is exact. This follows from Lemma 2.9.

In view of Theorem 10.1 it is natural to ask whether given locally coherent categories A and B any functor fpinj(B) → fpinj(A) commuting with direct limits and products can be extended to a functor f_{∗}: B → A for some exact functor f: fp(A) → fp(B). We begin our discussion of this question with a preliminary result.

Lemma 10.3. Let A be an abelian category with products and sufficently many injectives. For a functor f: A → B into an additive category the following are equivalent.

(1) f preserves limits.

(2) f is left exact and f|_{inj(A)} preserves products.

Proof. (1) ⇒ (2) Clear.

(2) ⇒ (1) A limit can be computed as a kernel between a particular map between two products. Thus it suffices to show that f preserves products. To this end let (X_{i})_{i∈I} be a family of objects in A and choose injective copresentations 0 → X_{i} → Y_{i} → Z_{i} for each i. We obtain the following exact commutative diagram for every M ∈ B.

0 ⟶ Hom(M, f(∏_{i} X_{i})) ⟶ Hom(M, f(∏_{i} Y_{i})) ⟶ Hom(M, f(∏_{i} Z_{i}))
↓ ↓
Hom(M, ∏_{i} f(Y_{i})) ⟶ Hom(M, ∏_{i} f(Z_{i}))
↓ ↓
0 ⟶ ∏_{i} Hom(M, f(X_{i})) ⟶ ∏_{i} Hom(M, f(Y_{i})) ⟶ ∏_{i} Hom(M, f(Z_{i}))

All vertical maps are isomorphisms and therefore the canonical map

Hom(M, f(∏_{i} X_{i})) ⟶ ∏_{i} Hom(M, f(X_{i}))

is an isomorphism. This shows that f(∏_{i} X_{i}) is a product of the f(X_{i})’s.

Universal Property 10.4. Let A be a locally finitely presented category with products and let f: A → B be a functor into an additive category with kernels and left exact direct limits. Supppose also that f commutes with direct limits. Then there exists, up to a natural isomorphism, a unique left exact functor f^{∗}: D(A) → B such that f = f^{∗}∘d_{A} and f^{∗} commutes with direct limits. Moreover, the following are equivalent.

(1) f preserves products.

(2) f^{∗} preserves products.

(3) f^{∗} has a left adjoint.

Proof. The existence of f^{∗} is Property 8.1. It remais to discuss (1) – (3). If f preserves products, then f^{∗} preserves limits by the lemma above since d_{A} induces an equivalence between A and fpinj(D(A)). The existence of a left adjoint then follows from the adjoint functor theorem [15, V, Corollary 3.2]. The other implications are trivial.

We are now in position to answer the above question related to Theorem 10.1.

Corollary 10.5. Let A and B be locally coherent categories and suppose that A has sufficently many fp-injectives. Let f: fpinj(A) → fpinj(B) be a functor commuting with direct limits and denote by f^{∗}: A → B the unique left exact functor commuting with direct limits which extends f. Then f^{∗} has a left adjoint if and only if f commutes with products. In that case a left adjoint B → A is exact and preserves finitely presented objects.

Proof. The first part of the assertion follows from the previous theorem and it remains to verify the properties of a left adjoint. First observe that such a left adjoint restricts to a functor g: fp(B) → fp(A) since f^{∗} commutes with direct limits. Now it suffices to show that g is exact by Proposition 5.10. Let 0 → X →^{φ} Y →^{ψ} Z → 0 be an exact sequence in fp(B). We need to check that g(φ) is a mono since a left adjoint preserves cokernels. Choose a mono χ: g(X) → M into some fp-injective object. Using the adjointness, the exactness of Hom(Y, f(M)) → Hom(X, f(M)) → 0 implies that χ factors through g(φ). Thus g(φ) is a mono and the proof is complete.

11. Functors commuting with direct limits and products

Let A and B be a pair of locally finitely presented categories. The functors f: A → B which commute with direct limits form a category which can be identified with (fp(A), B) by Property 5.6. If A and B have products then we denote by Fp(A, B) the category of functors f: A → B which commute with direct limits and products. We shall give a description of this category using results of the previous section. The following well-known fact is needed [15, V, Proposition 2.1].

Lemma 11.1. Let f_{i}: A → B and g_{i}: B → A, i = 1, 2, be pairs of functors such that g_{i} is a left adjoint for f_{i}, i = 1, 2. If φ: f_{1} → f_{2} is a natural transformation, then there is a unique natural transformation ψ: g_{2} → g_{1} making the following diagram commutative for all X ∈ A and Y ∈ B.

Hom(g_{1}(Y), X) ⟶^{∼} Hom(Y, f_{1}(X))
↓Hom(ψ_{Y} ,X)  ↓Hom(Y,φ_{X})
Hom(g_{2}(Y), X) ⟶^{∼} Hom(Y, f_{2}(X))

Given a pair of abelian categories C and D, we denote by Ex(C, D) the class of functors f: C → D which are exact, together with its natural transformations. Now fix a pair A and B of locally finitely presented categories with products. Any functor f ∈ Fp(A, B) extends by Corollary 10.5 uniquely to a functor f^{∗}: D(A) → D(B) which has an exact left adjoint. This functor restricts to an exact functor g: fp(D(B)) → fp(D(A)).

Theorem 11.2. The assignment f ↦ g is functorial and induces an anti-equivalence between Fp(A, B) and Ex(mop(fp(B)), mop(fp(A))).

Proof. The assignment is functorial and fully faithful by the preceding lemma since the relevant natural transformations are determined by their values on the finitely presented objects. The functor is dense by Theorem 10.1.

The following result is a reformulation of the previous one and makes the description of Fp(A, B) more explicit.

Corollary 11.3. Let f: A → B be a functor which commutes with direct limits and products. Denote for every X ∈ fp(B) by F_{X} the functor A → Ab, M ↦ Hom(X, f(M)). Then the assignment X ↦ F_{X} defines a functor F: fp(B)^{op} → Fp(A, Ab). The assignment f ↦ F is functorial and induces an equivalence between

Fp(A, B) and the category of functors F: fp(B)^{op} → Fp(A, Ab) such that F (Z) →^{F (ψ)}

F (Y) →^{F (φ)} F (X) is exact whenever ψ is a pseudo-cokernel for φ.

Proof. An inverse for f ↦ F is obtained as follows. Given a functor F, one defines a functor f: A → B = Flat(fp(B)) commuting with direct limits and products by

f(M): fp(B)^{op} → Ab, X ↦ F (X)(M)

for every M ∈ A.

We give now an interpretation of this result in case that B = Ab.

Theorem 11.4. Let A be a locally finitely presented category with products. A functor f: A → Ab commutes with direct limits and products if and only if there exists a presentation Hom(Y, −) → Hom(X, −) → f → 0 with X and Y in fp(A). The category Fp(A, Ab) of these functors is abelian.

Proof. We use the above notation with B = Mod(ℤ) = Ab. Moreover, we view A and B in the natural way as subcategories of D(A) and D(B), respectively. The adjointness gives an isomorphism f(M) = Hom(ℤ, f(M)) ≅ Hom(g(ℤ), M) with g(ℤ) ∈ fp(D(A)). Thus there is an exact sequence 0 → g(ℤ) → X → Y in D(A) with X, Y ∈ fp(A) and this gives a presentation Hom(Y, M) → Hom(X, M) → f(M) → 0 since M is fp-injective in D(A). Conversely, any functor having such a presentation commutes with direct limits and products since Hom(X, −) has this property for any finitely presented object X and direct limits and products are right exact. Identifying (fp(A), Ab) with those functors f: A → Ab which commute with direct limits, the above result says that Fp(A, Ab) = fp(fp(A), Ab). Thus Fp(A, Ab) is abelian since fp(A) has pseudo-cokernels.

Remark 11.5. In case that A is the category of modules over some ring the characterization of functors commuting with direct limits and product has also been observed by Crawley-Boevey.

12. Functors between module categories

To illustrate the results of the preceding section we give some examples from module theory.

Corollary 12.1. Let C and D be a pair of skeletally small pre-additive categories. Then there is a canonical duality

Fp(Mod(C), Mod(D)) ⟶ Fp(Mod(C^{op}), Mod(D^{op}))

between the categories of functors commuting with direct limits and products.

Proof. Recall from Section 2 that there are canonical dualities mop(mod(C)) → mop(mod(C^{op})) and mop(mod(D)) → mop(mod(D^{op})). Using this fact the assertion follows from Theorem 11.2.

Corollary 12.2. Let Λ be a ring and A be a locally finitely presented category with products. Suppose that f: A → Mod(Λ) is a functor which commutes with direct limits and products, and denote by F: A → Ab its composition with the forgetful functor. Then there is a presentation Hom(Y, −) → Hom(X, −) → F → 0 with X, Y ∈ fp(A) and a ring homomorphism φ: Λ → End(F)^{op} such that f(M) = φ_{∗}(F (M)) for all M ∈ A. Conversely, any functor F: A → Ab having such a presentation together with a ring homomorphism φ: Λ → End(F)^{op} gives a functor f: A → Mod(Λ) which commutes with direct limits and products.

Proof. Adopt the proof of Theorem 11.4 by replacing ℤ by Λ. The homomorphism φ is obtained by composing the canonical morphism Λ = End(Λ) → End(g(Λ)) with End(g(Λ)) ≅ End(F)^{op}.

Given a module M over some ring, we denote by ℓ_{end}(M) its length over End(M)^{op}.

Corollary 12.3. Let f: Mod(Λ) → Mod(Γ) be a functor commuting with direct limits and products. Then there exists c_{f} ∈ ℕ such that ℓ_{end}(f(M)) ≤ c_{f} · ℓ_{end}(M) for all M ∈ Mod(Λ).

Proof. There is a presentation Hom(Y, −) → Hom(X, −) → f → 0 of f with X, Y ∈ mod(Λ). Choosing an epimorphism Λ^{n} → X we put c_{f} = n.

Let R be a commutative artinian ring and suppose that Λ and Γ are R-algebras. Suppose also that Λ is finitely generated as an R-module.

Corollary 12.4. Let f: Mod(Λ) → Mod(Γ) be an R-linear functor commuting with direct limits and products. Then f(M) is finitely generated as an R-module for every M ∈ mod(Λ).

References

[1] J. Adámek and J. Rosický, Locally presentable categories and accessible categories, London Math. Soc. Lec. Note Series 189 (1994).

[2] M. Auslander and S.O. Smalø, Preprojective modules over artin algebras, J. Algebra 66 (1980) 61–122.

[3] S. Breitsprecher, Lokal endlich präsentierbare Grothendieck-Kategorien, Mitt. Math. Sem. Giessen, 85 (1970) 1–25.

[4] W. Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994) 1644–1674.

[5] P. Freyd, Representations in abelian categories, in: Proceedings of the conference on categorical algebra, La Jolla (1966) 95–120.

[6] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France, 90 (1962), 323–448.

[7] P. Gabriel and F. Ulmer, Lokal präsentierbare Kategorien, Springer Lecture Notes in Math. 221 (1971).

[8] A. Grothendieck and J.L. Verdier, Prefaisceaux, in: Théorie des Topos et Cohomologie Etale des Schémas, Springer Lecture Notes in Math. 269 (1972) 1–217.

[9] L. Gruson, Simple coherent functors, in: Representations of algebras, Springer Lecture Notes in Math. 488 (1975) 156–159.

[10] L. Gruson and C.U. Jensen, Dimensions cohomologique reliées aux functeurs ⟵ lim^{(i)}, in: Sém d’Algèbre P. Dubreil et M.-P. Malliavin, Springer Lecture Notes in Math. 867 (1981) 234–249.

[11] I. Herzog, The endomorphism ring of a localized coherent functor, preprint (1996).

[12] D.M. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958) 294–329.

[13] H. Krause, Generic modules over artin algebras, Proc. London Math. Soc., to appear.

[14] H. Krause, Stable equivalence preserves the representation type, Comment. Math. Helv., to appear.

[15] B. Mitchell, Theory of categories, Academic Press (1965).

[16] B. Mitchell, Rings with several objects, Adv. in Math. 8 (1972) 1–161.

[17] M. Makkai and R. Paré, Accessible categories: the foundations of categorical model theory, Contemp. Math. 104 (1989).

[18] J.-E. Roos, Locally noetherian categories, in: Category Theory, Homology Theory and their Applications II, Springer Lecture Notes in Math. 92 (1969) 197–277.

[19] D. Simson, On pure semi-simple Grothendieck categories I, Fund. Math. 100 (1978) 211–222.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

   E-mail address: henning@mathematik.uni-bielefeld.de