COVERS AND ENVELOPES IN FUNCTOR CATEGORIES

MANUEL SAORÍN AND ALBERTO DEL VALLE

Abstract. Let R be a ring and let C be a skeletally small full subcategory of Mod_{R} containing the regular module R_{R}. In the category (C^{op}, Ab) of additive contravariant functors from C to the category Ab of abelian groups, we call pseudorepresentable a functor of the form Hom_{R} (−, M) |_{C} with M ∈ Mod_{R} not necessarily in C. We prove that every functor in (C^{op}, Ab) has a pseudorepresentable envelope that completes diagrams in a unique way, give conditions under which every functor has a pseudorepresentable (pre)cover and relate the existence of such covers to the existence of almost split sequences ending in a prescribed object of C with local endomorphism ring.

In particular, when C = mod_{R} is the category of all finitely presented right R-modules, our results imply the fact that every functor of (mod_{R}^{op}, Ab) has a pseudorepresentable (= flat, in this case) cover and the result of Auslander that every finitely presented module with local endomorphism ring is the third term in an almost split sequence in Mod_{R}.

1. Introduction

Throughout this paper R is a fixed arbitrary associative ring with identity. The notation M_{R} means that M is a right R-module, and Mod_{R} (resp. FMod_{R}, mod_{R} and proj_{R}) is the category of all (resp. all finitely generated, all finitely presented and all finitely generated projective) right R-modules. All maps between sets will act on the left of their arguments and the composition of maps will be written accordingly.

For a skeletally small preadditive category C, we denote by (C^{op}, Ab) the category of all additive contravariant functors from C to the category Ab of abelian groups. This category, which has proved to be a very valuable tool for the study of the category C (see, e.g., [1] or [4]) has small Hom sets because C is skeletally small, and is an abelian category with direct limits, inheriting kernels, cokernels, images, direct sums and direct limits from Ab. The representable functors Hom_{C} (−, C) with C ∈ C form a generating set of finitely generated projective objects in (C^{op}, Ab), and Yoneda’s lemma implies that the functor C → (C^{op}, Ab) (called the Yoneda embedding) given by C ↦ (−, C) is full and faithful.

In this paper, we take C to be a skeletally small full subcategory of Mod_{R} containing the regular module R_{R}. For any B ∈ Mod_{R}, we denote by (−, B) the

1991 Mathematics Subject Classification. 16D90, 16G70, 18A25, 18A40. Key words and phrases. Covers and envelopes, almost split sequences, pseudorepresentable

functors.

Both authors are supported by the DGES of Spain (PB96-0961-C02-02) and by the Fundación “Séneca”, Comunidad Autónoma de la Región de Murcia (00260/CV/97).

restriction to C of the functor Hom_{R} (−, B), and call a functor in (C^{op}, Ab) pseudorepresentable if it is naturally isomorphic to (−, B) for some B ∈ Mod_{R}. In Section 2, we prove that pseudorepresentable and flat envelopes exist in (C^{op}, Ab) and study the existence of pseudorepresentable covers in (C^{op}, Ab). We find that this study is worthy for two reasons: first, it implies the existence of flat covers in (mod_{R}^{op}, Ab), an interesting fact on its own right; and second, it is related to the existence of almost split sequences in Mod_{R}. This relation is made precise in Section 3, where we prove that, for a non-projective module C ∈ C with local endomorphism ring, the existence of a pseudorepresentable cover for the radical r (−, C) of the functor (−, C) is equivalent to the existence of an almost split sequence in Mod_{R} with third term C.

Next, we state the main definitions used in the paper.

Let A be any category and let B be any class of objects in A. Following [5], a morphism f : B → A in A is a B-precover for A if B ∈ B and, for every B′ ∈ B, the canonical map Hom_{A} (B′, f) : Hom_{A} (B′, B) → Hom_{A} (B′, A) is surjective. The morphism f is right minimal if the preimages of f by Hom_{A} (B, f) are all automorphisms of B. A right minimal B-precover is called a B-cover. The B-cover f completes the diagrams in a unique way if Hom_{A} (B′, f) is bijective for every B′ ∈ B. The dual concepts are B-(pre)envelope and left minimal.

Let A, B, C ∈ Mod_{R}; a R-homomorphism f : B → C is right almost split if it is not a retraction and any morphism C′ → C in Mod_{R} which is not a retraction factors through f; dually, g : A → B is left almost split if it is not a section and, for any A′ ∈ Mod_{R}, each non-section A → A′ factors through g. A non-split exact sequence 0 → A →^{g} B →^{f} C → 0 in Mod_{R} is said to be an almost split sequence if g is left almost split and f is right almost split; this is equivalent to the fact that f is right almost split and right minimal (see [1, (II.4.4)]).

2. Pseudorepresentable Functors

For the rest of the paper we let C be a skeletally small full subcategory of Mod_{R} such that R_{R} ∈ C. Given two objects F, G ∈ (C^{op}, Ab), we write [F, G] for the set of all morphisms (i.e., natural transformations) between the functors F and G. Given any A ∈ Mod_{R}, we write (−, A) for the restriction to C of the contravariant functor Hom_{R} (−, A) : Mod_{R} → Ab. Also, given a morphism f : A → B in Mod_{R}, we shall write (−, f) : (−, A) → (−, B) for the morphism in (C^{op}, Ab) which sends X ∈ C to the map (X, f) : (X, A) → (X, B) given by (X, f) (h) = f ∘ h.

We recall that, for any F ∈ (C^{op}, Ab) and any X ∈ C, the abelian group F X has a natural structure of right End_{R} (X)-module. Moreover, for any τ ∈ [F, G], the homomorphism τ_{X} : F X → GX is End_{R} (X)-linear [1, (I.1)]. When X = R_{R}, this gives a right R-module F R for any F ∈ (C^{op}, Ab) and a R-homomorphism τ_{R} : F R → GR for any τ ∈ [F, G]. We shall write σ : 1_{Mod_{R}} → Hom_{R} (R, −) for the standard natural isomorphism. Using this, it is clear that:

Lemma 2.1. For a functor F ∈ (C^{op}, Ab), the following conditions are equivalent: (a) F is naturally isomorphic to (−, A) for some A ∈ Mod_{R}.

(b) F is naturally isomorphic to (−, F R).

A functor satisfying the above conditions will be called pseudorepresentable, and we shall denote by B the class of all pseudorepresentable functors in (C^{op}, Ab).

Clearly, a pseudorepresentable functor F is representable if and only if F R ∈ C. Pseudorepresentable functors are closely related to flat functors, i.e. functors isomorphic to direct limits of representable functors. These have been studied in [10] and [7] (see also [4]), where they are shown to have most of the properties of flat modules. By [11, (24.10) and (25.4)] we have:

Lemma 2.2. (i) If C = FMod_{R} then each pseudorepresentable functor is flat. (ii) If C = mod_{R} then a functor is pseudorepresentable if and only if it is flat.

We start by proving the existence of pseudorepresentable envelopes in (C^{op}, Ab), and for this it is important to describe the morphisms between pseudorepresentable functors. These are easy to deal with thanks to the following result. It is easily verified and we shall use it throughout the paper without further reference.

Lemma 2.3. Given A, B ∈ Mod_{R}, the assignations τ ↦ σ_{B}^{−1} ∘ τ_{R} ∘ σ_{A} and f ↦ (−, f) define inverse isomorphisms of abelian groups between [(−, A) , (−, B)] and Hom_{R} (A, B).

Thus we have a sort of Yoneda embedding Mod_{R} → (C^{op}, Ab), i.e. a full and faithful functor given by A ↦ (−, A) which establishes an equivalence of categories Mod_{R} → B.

Now we prove that pseudorepresentable envelopes always exist in (C^{op}, Ab); these complete the diagrams in a unique way and admit a clear description.

Proposition 2.4. Every functor in (C^{op}, Ab) has a pseudorepresentable envelope that completes the diagrams in a unique way.

Proof. Let F ∈ (C^{op}, Ab), X ∈ C and a ∈ F X. We define a map ϕ_{X,a} : X → F R as follows: for every x ∈ X, let λ_{x} : R → X be the R-homomorphism given by left multiplication by x. Then we have F λ_{x} : F X → F R in Ab, and we define ϕ_{X,a} (x) = F λ_{x} (a). This map ϕ_{X,a} is easily seen to be in Hom_{R} (X, F R), so that the assignation a ↦ ϕ_{X,a} gives a map ϕ_{X} : F X → (X, F R). Now, it is easy to check that this amounts to a natural transformation ϕ : F → (−, F R), which is the desired envelope since the map [(−, F R) , (−, A)] ⟶^{−∘ϕ} [F, (−, A)] has an inverse given by τ ↦ (−, σ_{A}^{−1} ∘ τ_{R}). □

Remark 2.1. (1) The existence of a B-envelope completing the diagrams in a unique way for every object of a category A is equivalent to the fact that B is a reflective subcategory of A (see, e.g., [6] or [9]). By [6, (36.9) and (36.13)], the class B of all pseudorepresentable functors of (C^{op}, Ab) is closed for direct summands and projective limits.

(2) If the category C has cokernels (e.g., if it is closed for cokernels in Mod_{R}) then one can check that every finitely presented functor in (C^{op}, Ab) has a representable envelope completing the diagrams in a unique way. Therefore, by [9, Theorem 4.1], the flat functors also form a reflective subcategory of (C^{op}, Ab).

In the rest of this section we study pseudorepresentable (pre)covers in (C^{op}, Ab). The starting point for this investigation is a theorem of Belshoff, Enochs and Xu [2] asserting that every left module of finite flat dimension over a right coherent ring has a flat cover. We shall give a generalized version of it that fits our purposes, after introducing some notation.

If B is any class of objects of an abelian category A then we shall write B^{⊥} for the class of all objects K in A for which there are no non-split exact sequences 0 → K → X → B → 0 in A with B ∈ B. By Wakamatsu’s lemma [12], the kernel of any B-cover in A lies in B^{⊥}. On the other hand, if 0 → K → B → A → 0 is an exact sequence in A with B ∈ B and K ∈ B^{⊥} then B → A is clearly a B-precover; following [12], an epimorphism f : B → A with B ∈ B and Ker f ∈ B^{⊥} will be called a special B-precover for A. Finally, we shall say that an object A ∈ A has B-dimension ≤ n if there is an exact sequence in A of the form

0 → B_{n} → B_{n−1} → ⋯ → B_{1} → B_{0} → A → 0

with each B_{i} ∈ B. Now we can state:

Theorem 2.5. Let A be an abelian category and let B be a class of objects in A closed for extensions and satisfying the following conditions:

(a) If the sequence 0 → X → Y → Z → 0 is exact in A with X, Y ∈ B^{⊥} then Z ∈ B^{⊥}.

(b) For every B ∈ B there exists an exact sequence 0 → B → B_{1} → B_{2} → 0 in A with B_{1} ∈ B^{⊥} and B_{2} ∈ B.

Then every object of finite B-dimension of A has a special B-precover.

Proof. The proof follows the lines of that of [2, Theorem 3.4]. It is clearly enough to show that, if 0 → A′ → B → A → 0 is an exact sequence in A such that A′ has a special B-precover and B ∈ B, then A has a special B-precover. Let then 0 → K → B′ → A′ → 0 be an exact sequence in A with K ∈ B^{⊥} and B′ ∈ B, and let 0 → B′ → B′_{1} → B′_{2} → 0 the sequence guaranteed by (b); taking the pushout of B′ → A′ and B′ → B′_{1} we get a commutative diagram with exact rows in A

0 → K ⟶ B′ ⟶ A′ → 0
∥ ↓ ↓
0 → K ⟶ B′_{1} ⟶ A′_{1} → 0

and an exact sequence 0 → A′ → A′_{1} → B′_{2} → 0 where A′_{1} ∈ B^{⊥} by (a). Taking now the pushout of A′ → B and A′ → A′_{1} we get a commutative diagram with exact rows in A

0 → A′ ⟶ B ⟶ A → 0
↓ ↓ ∥
0 → A′_{1} ⟶ B_{1} ⟶ A → 0

and an exact sequence 0 → B → B_{1} → B′_{2} → 0 which shows that B_{1} ∈ B and therefore the lower row of the latter diagram gives us the desired special B-precover of A. □

Remark 2.2. (1) If A is locally small and B is closed for direct limits then one can adapt the lemmas preceding Theorem 3.1 in [5] to get the existence of a B-cover of A from any B-precover of any object A ∈ A, and therefore, in the conclusion of the theorem, we can write B-cover instead of special B-precover.

(2) When the category A has enough projective objects then condition (a) above can be replaced by: (a’) For every projective object P of A and every epimorphism P → B with B ∈ B, the kernel of P → B is in B. Indeed, in this case the fact that X ∈ B^{⊥} implies that Ext_{A}^{i} (B, X) = 0 for every B ∈ B and every i ≥ 1, and hence we get (a).

Next we look for conditions on C under which the class B of pseudorepresentable functors of (C^{op}, Ab) fulfils the requirements in Theorem 2.5, and we start by relating the exactness of sequences in Mod_{R} to the exactness of the corresponding sequences of pseudorepresentable functors.

We shall say that a sequence 0 → A → B → C → 0 in Mod_{R} is C-pure (or that A → B is a C-pure monomorphism, or that B → C is a C-pure epimorphism) if it is exact and any X ∈ C is projective relative to it (see, e.g., [11]). When C = proj_{R} (resp. C = mod_{R}) these are just the exact (resp. pure-exact) sequences. The following properties are easily checked and will be used freely in what follows.

Proposition 2.6. Let f : A → B and g : B → C be morphisms in Mod_{R}. Then:

(1) (−, f) is a monomorphism (resp. epimorphism, section or retraction) in (C^{op}, Ab) if and only if f is a monomorphism (resp. C-pure epimorphism, section or retraction) in Mod_{R}.

(2) The sequence 0 → (−, A) ⟶^{(−,f)} (−, B) ⟶^{(−,g)} (−, C) is exact in (C^{op}, Ab) if and only if the sequence 0 → A →^{f} B →^{g} C is exact in Mod_{R}.

(3) The sequence 0 → (−, A) ⟶^{(−,f)} (−, B) ⟶^{(−,g)} (−, C) → 0 is exact in (C^{op}, Ab) if and only if the sequence 0 → A →^{f} B →^{g} C → 0 is C-pure in Mod_{R}.

(4) The sequence 0 → (−, A) ⟶^{(−,f)} (−, B) ⟶^{(−,g)} (−, C) → 0 splits in (C^{op}, Ab) if and only if the sequence 0 → A →^{f} B →^{g} C → 0 splits in Mod_{R}.

Proposition 2.7. The class of pseudorepresentable functors is closed under extensions.

Proof. Let 0 → (−, A) →^{α} F →^{β} (−, B) → 0 be an exact sequence in (C^{op}, Ab) with A, B ∈ Mod_{R}, and set f = α_{R} ∘ σ_{A} and g = σ_{B}^{−1} ∘ β_{R}. We then get an exact

sequence 0 → A →^{f} F R →^{g} B → 0 and, if ϕ : F → (−, F R) is the B-envelope of F, then (−, g) ∘ ϕ = β. Thus we get a diagram with exact rows

0 → (−, A) ⟶^{α}  F  ⟶^{β} (−, B) → 0
∥  ↓^{ϕ}  ∥
0 → (−, A) ⟶^{(−,f)} (−, F R) ⟶^{(−,g)} (−, B)

in which the right square commutes. Next we see that so does the second. For, take X ∈ C and let us see that ϕ_{X} ∘ α_{X} = (X, f), i.e. that, for any h : X → A, we have f ∘ h = ϕ_{X,α_{X}(h)} with the notation of Proposition 2.4. Now, for each x ∈ X, the naturality of α applied to λ_{x} : R → X gives F λ_{x} ∘ α_{X} = α_{R} (− ∘ λ_{x}), so that

ϕ_{X,α_{X}(h)} (x) = F λ_{x} (α_{X} (h)) = α_{R} (h ∘ λ_{x}) = α_{R} (λ_{hx}) = α_{R} (σ_{A} (hx)) = f (hx) .

Thus the middle column in the diagram is an isomorphism and hence F is pseudorepresentable. □

Now we can identify the pseudorepresentable functors that are in B^{⊥}. A module A ∈ Mod_{R} is C-pure-injective if Hom (−, A) carries C-pure sequences to exact sequences, or equivalently if any C-pure sequence starting on A splits [11, (33.7)].

Proposition 2.8. If A ∈ Mod_{R}, then (−, A) ∈ B^{⊥} if and only if A is C-pure-injective.

Proof. If A is C-pure-injective and 0 → (−, A) → F → G → 0 is exact with G ∈ B then also F ∈ B, so that the sequence is of the form 0 → (−, A) → (−, B) → (−, C) → 0 and therefore it comes from a C-pure (hence split) sequence 0 → A → B → C → 0 in Mod_{R}, whence the first sequence splits, thus showing that (−, A) ∈ B^{⊥}. Conversely, if (−, A) ∈ B^{⊥} and the sequence 0 → A → B → C → 0 is pure then the exact sequence 0 → (−, A) → (−, B) → (−, C) → 0 splits and so does the first one, so that A is C-pure-injective. □

Next we see that if C ⊆ FMod_{R}, then B satisfies condition (a) in Theorem 2.5 and all objects of (C^{op}, Ab) have finite B-dimension.

Proposition 2.9. If each module of C is finitely generated (i.e., C ⊆ FMod_{R}) then: (1) The class B of pseudorepresentable functors is closed under direct sums.

(2) Each functor of (C^{op}, Ab) is a quotient of a pseudorepresentable functor.

(3) Each functor of (C^{op}, Ab) has B-dimension at most 2.

(4) Each projective functor of (C^{op}, Ab) is pseudorepresentable.

(5) The class B satisfies condition (a) in Theorem 2.5.

Proof. (1) follows from [11, 24.10]. If F ∈ (C^{op}, Ab) and C_{0} is a skeleton of C, then the canonical morphism ⊕_{X∈C_{0}} (−, X)^{([(−,X),F])} = ⊕_{X∈C_{0}} (−, X)^{(F X)} → F is an epimorphism in (C^{op}, Ab), so that (2) follows from (1). Now (3) and (4) are clear and (5) follows from Remark 2.2.(2). □

Finally, we see what is needed for B to satisfy all conditions in Theorem 2.5.

Proposition 2.10. Assume that C ⊆ FMod_{R} and that, for every A ∈ Mod_{R}, there exists a C-pure-injective module Ā and a C-pure monomorphism A → Ā. Then every functor of (C^{op}, Ab) has a special pseudorepresentable precover.

Proof. We just need to check condition (b) to apply Theorem 2.5, but if (−, A) ∈ B then the sequence 0 → (−, A) → (−, Ā) → (−, Ā/A) → 0 does the job. □

Remark 2.3. It is easy to see that the Yoneda embedding Mod_{R} → (C^{op}, Ab) takes pushout diagrams in Mod_{R} in which one of the source maps is a C-pure monomorphism to pushout diagrams in (C^{op}, Ab). So, in the setting of the above proposition, the proof of Theorem 2.5 give us a way to construct a special pseudorepresentable precover for any functor F ∈ (C^{op}, Ab) with pseudorepresentable

dimension 1. Indeed, given an exact sequence 0 → (−, A) ⟶^{(−,f)} (−, B) → F → 0 in (C^{op}, Ab) and a pushout diagram

\begin{matrix}A & ⟶^{f} & B \\ ↓ & & ↓ \\ Ā & ⟶ & C \end{matrix}

in Mod_{R}, where A → Ā is as above, we get a commutative diagram with exact rows in (C^{op}, Ab)

0 → (−, A) ⟶ (−, B) ⟶ F → 0
↓ ↓ ∥
0 → (−, Ā) ⟶ (−, C) ⟶ F → 0

and the morphism (−, C) → F is a special pseudorepresentable precover.

Corollary 2.11. Every functor of (mod_{R}^{op}, Ab) has a flat cover.

Proof. When C = mod_{R} then B is the class of flat functors (Lemma 2.2) and Cpurity is the usual concept of purity. Since, for every module A, the canonical map to its double character module A → A^{++} is a pure monomorphism to a pureinjective module, we can apply the above proposition and the fact that flat functors are closed under direct limits to get the result. □

In view of Proposition 2.10 it is interesting, for different choices of C, to study the existence of C-pure monomorphisms M → M̄ with M̄ C-pure-injective for a given module M. Of course these exist for every module when C = proj_{R} or C = mod_{R}. We present here a different instance of this situation.

Example 2.1. Let S ∈ Mod_{R} be simple and non-projective, and let C = add (R ⊕ S) be the class of all direct summands of finite direct sums of copies of R ⊕ S. Also, for each X ∈ Mod_{R}, let X [S] be the S-homogeneous component of Soc (X). We propose the reader to check the following assertions about modules in Mod_{R}:

(1) An epimorphism Y → X is C-pure if and only if the inclusion X [S] ↪ X factors through it.

From now on assume that S is Σ-injective; this means that the inclusion X [S] ↪ X splits for every module X.

(2) M is C-pure-injective if and only if Ext_{R}^{1} (X, M) = 0 when X [S] = 0.

(3) If R is right pure-semisimple then direct sums of C-pure-injective modules are again C-pure-injective.

Now let R be the algebra of finite representation type given by the Dynkin diagram 𝔸_{3} (· → · → ·). Its Auslander-Reiten quiver is well known to be the wing

P_{1} = I_{3}
↗  ↘
P_{2}  I_{2}
↗  ↘  ↗  ↘
S_{3} = P_{3}  S_{2}  I_{1} = S_{1}

where the “going up” arrows are monomorphisms, the “going down” arrows are epimorphisms and, for each i = 1, 2, 3, S_{i} is simple with projective cover P_{i} and injective envelope I_{i}. We set C = add (R ⊕ S_{1}). By using the above claims, one checks that the indecomposable C-pure-injective modules are I_{1}, I_{2}, I_{3} and S_{2}, and that the obvious sequences

0 → P_{3} → I_{3} → I_{2} → 0 0 → P_{2} → I_{3} ⊕ S_{2} → I_{2} → 0

are C-pure. Therefore we have the desired map M → M̄ for every indecomposable M and it is then clear how to construct one such for an arbitrary M.

3. Application to Almost Split Sequences

We still assume that C is a skeletally small subcategory of Mod_{R} containing R_{R}. For a functor F ∈ (C^{op}, Ab) we denote its radical (i.e., the intersection of all its maximal subfunctors) by rF.

Theorem 3.1. Assume that C ∈ C is not projective and has a local endomorphism ring. Then the following conditions are equivalent:

(a) There exists an almost split sequence in Mod_{R} with third term C.

(b) r (−, C) has a pseudorepresentable cover in (C^{op}, Ab).

Proof. We first note that, by [1, (II.1.9) and (II.1.5)], r (−, C) is a proper subfunctor of the representable functor (−, C) which contains every proper subfunctor of (−, C). We write μ : r (−, C) ↪ (−, C) for be the inclusion functor.

(a)⇒(b). Let f : B → C be right almost split and right minimal in Mod_{R}. Since f is not a retraction, (−, f) is not epic and therefore it factors as (−, f) = μ ∘ τ for a morphism τ : (−, B) → r (−, C), and this is a B-cover of r (−, C). Indeed, as f is right minimal, so is (−, f) and hence τ, and we just have to check that, given any D ∈ Mod_{R}, any morphism σ : (−, D) → r (−, C) factors through τ. For, take h : D → C in Mod_{R} with μ∘σ = (−, h). As h is not a retraction, it factors through f, i.e. there exists t : D → B with h = f ∘ t, whence

μ ∘ σ = (−, h) = (−, f) ∘ (−, t) = μ ∘ τ ∘ (−, t)

and therefore σ = τ ∘ (−, t), as desired.

(b)⇒(a). Let τ : (−, B) → r (−, C) be a pseudorepresentable cover in (C^{op}, Ab) and take f : B → C in Mod_{R} with μ∘τ = (−, f). Given a non-retraction g : D → C in Mod_{R}, we have Im (−, g) ⊆ r (−, C) and hence (−, g) = μ ∘ σ for a morphism σ : (−, D) → r (−, C). Then, by the preenveloping condition of τ, there exists h : D → B in Mod_{R} with σ = τ ∘ (−, h) and therefore

(−, g) = μ ∘ σ = μ ∘ τ ∘ (−, h) = (−, f) ∘ (−, h) = (−, f ∘ h) ,

so that g factors through f and hence f is right almost split. Since C is not projective, f is epic. On the other hand, since τ is right minimal and μ is monic, the composition μ ∘ τ = (−, f) is right minimal, and hence so is f. □

Remark 3.1. (1) If C_{R} is projective then it has a unique maximal submodule B and r (−, C) = (−, B) is a pseudorepresentable functor, so that the above arguments also give a right almost split right minimal morphism B ↪ C.

(2) By the first part of the above proof, a B-precover (−, B) → r (−, C) gives a right almost split morphism B → C. So, since r (−, C) has B-dimension at most one, Remark 2.3 gives us a way for constructing such a morphism.

Corollary 3.2. [1, (II.5.1)] Every nonprojective C ∈ mod_{R} with local endomorphism ring is the third term of an almost split sequence in Mod_{R}.

Proof. This follows from Theorem 3.1 and Corollary 2.11. □

Remark 3.2. Corollaries 2.11 and 3.2 admit the following partial dualization. If we consider the category D (R) = (_{R} mod, Ab) of additive covariant functors R mod → Ab then the assignation A ↦ (A ⊗ −) gives an embedding Mod_{R} → D (R) which induces an equivalence of categories between Mod_{R} and the full subcategory of D (R) consisting of the FP-injective objects. By essentially the same argument of [8, Corollary 3.5(c)], one can show that FP-injective preenvelopes exist in D (R). Moreover, if S ∈ D (R) is a simple functor and τ : S → (A ⊗ −) is an injective envelope (so that A is pure-injective with local endomorphism ring) then, taking a FP-injective preenvelope of Coker τ, we get a short exact sequence 0 → S → (A ⊗ −) ⟶^{−⊗f} (B ⊗ −) in D (R), and it can be proven that f : A → B is left almost split (although it is not left minimal unless the FP-injective preenvelope of Coker τ is indeed an envelope). In fact, by [3, Theorem 2.3], A is the source of a left almost split morphism A → B in Mod_{R} if and only if (A ⊗ −) is is the injective envelope of a simple functor.

We would like to thank Henning Krause for calling our attention to this remark.

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(M. Saorín and A. del Valle) Departamento de Matemáticas. Universidad de Murcia, Aptdo. 4021. Espinardo 30.100, Murcia, Spain

E-mail address: msaorinc@fcu.um.es and alberto@fcu.um.es