© Hindawi Publishing Corp. FLAT COVERS OF REPRESENTATIONS OF THE QUIVER A∞

Rooted quivers are quivers that do not contain 
A∞≡⋯→•→• as a subquiver. The existence of flat covers and cotorsion envelopes 
for representations of these quivers have been studied by Enochs et al. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for 
representations of A∞. We first characterize finitely generated projective representations of A∞. We also see that there are no 
projective covers for representations of A∞, which adds more 
interest to the problem of the existence of flat covers.


Introduction.
It is known that, for an arbitrary quiver Q, the category of representations by modules of Q is a Grothendieck category with a generating system of projective representations.An explicit construction of such a system was given in [6], where, furthermore, flat representations were characterized, not for all quivers, but for a large class of them called rooted quivers.This characterization of flat representations of rooted quivers was very useful in proving that every representation of a rooted quiver admits a flat cover and a cotorsion envelope, see [6,Theorem 4.3].Rooted quivers were also characterized in [6] as those quivers with no path of the form ••• → • → •.In this paper, we study flat representations and flat covers of representations of the quiver A ∞ ≡ ••• → • → • as a first step in the treatment of nonrooted quivers.
In Section 2, we characterize finitely generated and projective representations of A ∞ , and this will allow us to define flat representations as direct limits of them.To do this, we give necessary and sufficient conditions for a representation P of A ∞ to be projective.Finally, we give an example which shows that projective covers of representations of A ∞ do not exist in general.This makes Theorem 5.3, the main result of Section 5, more interesting: we prove that any representation by modules of A ∞ has a flat cover.This will be done using the techniques developed by Eklof and Trlifaj in [4] concerning cotorsion theories cogenerated by sets in the categories of modules (see [7] for a more detailed explanation about cotorsion theories) and the generalizations of these techniques to Grothendieck categories with projective generators given in [1].

Preliminaries.
All rings considered in this paper will be associative with identity and, unless otherwise specified, they are not necessarily commutative.The letter R will usually denote a ring.
A quiver Q is a directed graph whose edges are called arrows.An arrow of a quiver from a vertex v 1 to a vertex v 2 is denoted by a : A quiver Q may be thought of as a category in which the objects are the vertices of Q and the morphisms are the paths (a path is a sequence of arrows) of Q.
A representation by modules X of a given quiver Q is a functor X : Q → R-Mod.Such a representation is determined by giving a module X(v) (or X v ) for each vertex v of Q and a homomorphism X(a) : Thus, the representation of a quiver Q by modules over a ring R is a category denoted by (Q, R-Mod), which is a Grothendieck category with enough projectives.
As usual, A ∞ will be used to denote the quiver where v i , i = 1,...,n, are the vertices of the quiver; representation will mean representation by modules of A ∞ .For a representation M of A ∞ , and for an arrow a : v n → v n−1 , we will often use the notation A morphism ψ between two representations will be a family {ψ k : k ≥ 1} satisfying the conditions above.
Since we will prove the existence of flat covers and cotorsion envelopes making use of the techniques developed by Eklof and Trlifaj (see [4]) over cotorsion theories, we introduce some general definitions on covers and envelopes and recall what is understood by a pair of classes cogenerated by a set.
Recall from [5] that, given a class Ᏺ of objects in an abelian category Ꮽ, an Ᏺ-precover (resp., an Ᏺ-pre-envelope) of an object C ∈ Ob(Ꮽ) is a morphism is an automorphism, then ϕ is said to be an Ᏺ-cover (resp., an Ᏺ-envelope).For the same class Ᏺ, Ᏺ ⊥ will denote the class of all objects C of Ꮽ such that Ext 1 (F , C) = 0 for every F ∈ Ᏺ.Then, the pair of classes (Ᏺ, Ᏺ ⊥ ) is said to be cogenerated by a set if there exists a set of objects of Ꮽ, say Z, such that C ∈ Ᏺ ⊥ if and only if Ext 1 (F , C) = 0 for every F ∈ Z.
We also have to recall the definition of a finitely generated representation of a quiver.Definition 2.1.Let Q be a quiver, D a representation of Q, and Z a set of elements of D. The subrepresentation of D generated by the set Z is defined as the intersection of all representations of Q containing Z.The representation D is said to be finitely generated provided that D is generated by a finite subset of elements, or equivalently, if it is finitely generated as an object of the category of representations of Q.

It follows immediately from Definition 2.1 that a representation S of the quiver A ∞ is finitely generated if and only if it is of the form
for some natural number n ≥ 1 and with S i finitely generated as an R-module for all i ≥ 1.

Projective representations.
Projective representations of A ∞ by vector spaces over a field K were characterized in [2, Example, page 102] as those representations P such that the homomorphisms P n → P n−1 are always injections and that lim ← P n = 0.The condition is indeed necessary; however, we will now give an example of a representation of A ∞ by vector spaces satisfying these two conditions, but which is not a projective representation.Notice first that it is immediate to see that any representation of A ∞ of the form where where n is the position where U appears for the first time in T ) for every representation M of A ∞ .
Example 3.1.Let K be a field and L the following representation of A ∞ : For every n ∈ N, we consider the representation P (n) of A ∞ given by where the first K appears in nth place.The direct sum ⊕ ∞ n=0 P (n) is a projective representation of A ∞ since each P (n) is a projective representation.Furthermore, it is easy to see that ⊕ ∞ n=0 P (n) is a projective generator for the category of representations by K-modules of the quiver A ∞ , so if L was a projective representation of A ∞ , then L should be a direct summand of (⊕ ∞ n=0 P (n)) (X) for some set X. We prove that L cannot be contained in (⊕ ∞ n=0 P (n)) (X) , so we will have a contradiction.
It is immediate to observe that the kernel of any morphism L → P (n) contains the subrepresentation where the K-module ∞ n+2 K is the corresponding module to the vertex v n+2 .Then, any morphism L → P (n) factors through the quotient L/T (which is clearly a finitely generated representation by the comments made in Section 2), and then we have Hom L, P (n) (X)  Hom L T , P (n) (X) . (3.5) Suppose we have for some set X.Then, for any natural number n, we have a morphism and by (3.5), we see that there exists a finite subset X n ⊆ X such that (3.7) factors through ) , and this is impossible in general since (⊕ ∞ n=0 P (n) (X ) )(v 0 ) has a countable base and L(v 0 ) does not in general (take, e.g., Q).
In Propositions 3.2 and 3.3, we give, respectively, necessary and sufficient conditions for a representation of A ∞ to be projective in the general case where R is an arbitrary ring.These will lead to a characterization of finitely generated projective representations of A ∞ (Proposition 3.4).

Proposition 3.2. Let
Then the following statements hold: (1) P n is a projective R-module for every n ∈ N; (2) f n is a splitting monomorphism for every n ∈ N; (3) lim Proof.The first statement is easy: take n ∈ N and let M h → N be an epimorphism of R-modules.Now, if P n gn → N is a morphism of R-modules, we may extend h and g n to morphisms of representations by putting M and id M (resp., N and id N ) for k ≥ n and 0 on the rest.We finally apply that P is a projective representation to get the desired extension P n → M of g n .
Consider now the family of representations of A ∞ (first P n is in nth position) for all n ∈ N.Then, the direct sum ⊕ n≥0 K n can be considered as the representation where each λ j is the canonical injection.
It is clear that the map ϕ : ⊕ n≥0 K n → P given by is a morphism of representations and that it is in fact an epimorphism of representations.But P is projective by hypothesis, so there exists φ : P → ⊕ k≥0 K n with ϕ • φ = id P , which means that ϕ n • φ n = id Pn , for all n ∈ N.
If we now look at the canonical projections π n : , where the last equality holds since φ is a morphism of representations (so, . Therefore, we immediately obtain that each f n is a splitting monomorphism. It only remains to prove (3).We have already seen that P is a direct summand of ⊕ n≥0 K n , that is, P ⊕ T = ⊕ n≥0 K n for some representation T of A ∞ , and it is clear that ⊕ n≥0 K n satisfies (3), so we are done since inverse limits commute with finite direct sums.

suppose that
(1) P n is a projective R-module for every n; (2) f n is a splitting monomorphism for every n; (3) there exists a set of epimorphisms {α n : P n−1 → P n | n ≥ 1} such that α n • f n = id Pn for every n ∈ N and that for any x ∈ P n there is a natural Proof.Let K n and ϕ : ⊕ n≥0 K n → P be the representations of A ∞ and the epimorphism of representations given in Proposition 3.2.By the previous comments, each K n is projective and so is ⊕ n≥0 K n .Therefore, if we prove that P is a direct summand of ⊕ n≥0 K n , we will have that P is also projective.
We want to define a morphism w : P → ⊕ n≥0 K n so that ϕ • w = id P .For any x n ∈ P n , we define for each j ≥ 0. Notice that w n is well defined by condition (3).
With this definition, it is an easy computation to check that λ n •w n = w n−1 • f n and that ϕ n • w n (x n ) = x n for all x n ∈ P n , for all n ∈ N, so we are done.Using Propositions 3.2 and 3.3 and the comments given in Section 2, we immediately obtain the following result.Proposition 3.4.Let P be a representation of A ∞ .Then P is finitely generated projective if and only if there exists an n ∈ N such that (1) P k = 0 for every k > n; (2) P k is a finitely generated projective module for every k with 0 ≤ k ≤ n; (3) f k : P k → P k−1 is a split monomorphism for every k ∈ N.
We finish this section with an example showing that, in general, not every representation of A ∞ has a projective cover.This will raise the interest in the problem of the existence of flat covers for all representations of A ∞ (which will be treated in Section 5) as well as the characterization of rings for which every representation by modules of A ∞ has a projective cover.
Example 3.5.Let R be a ring and consider the representation of A ∞ , and for n ∈ N, the subrepresentation with first R in nth position.It is clear that, for any n 0 ∈ N, there exists an epimorphism ⊕ n≥n 0 T n → T which of course is a projective precover of T (each T n is a projective representation of A ∞ ).Suppose T has a projective cover Then S is a direct summand of ⊕ n≥n 0 T n , and so g n is an isomorphism for all n, 0 ≤ n ≤ n 0 , and all n 0 ∈ N, which contradicts the hypotheses of Proposition 3.2.Therefore, T does not have a projective cover.

Flat representations.
Since the category of representations by modules of a quiver has enough projectives, we can define flat representations of A ∞ as direct limits of projective representations.Furthermore, it is easy to see that we can assume that a flat representation is a direct limit of finitely generated and projective representations which have been characterized in Proposition 3.4.This section is therefore devoted to characterizing flat representations of A ∞ .This turns out to be very useful in proving that the pair (Ᏺ, Ᏺ ⊥ ) (where Ᏺ is the class of all flat representations of A ∞ ) is cogenerated by a set, as was noticed in [6], where the same result was proved for rooted quivers.
In [6,Proposition 3.4] it is stated that, for any quiver Q, if a representation F is flat, then F(v) is a flat module for every vertex v of Q and that the homomorphism ⊕ t(a)=v F(i(a)) → F(v) is a pure monomorphism for all vertices v of Q, where t(a) and i(a) denote the terminal and initial vertices of the arrow a.Furthermore, it was also proved in [6,Theorem 3.7] that these conditions are sufficient for a representation to be flat provided that Q is rooted.Now we prove that this characterization also holds for a nonrooted quiver A ∞ .

Proposition 4.1. A representation F of the quiver A ∞ is flat if and only if the following statements hold:
(1) F v is a flat module for every vertex v of A ∞ ; (2) the homomorphism f j+1 : F j+1 → F j is a pure injection for every j ∈ N.
Proof.As we have seen above, the conditions are necessary.We prove that they are also sufficient.For every n ∈ N, we define the subrepresentation F n of F given by It is clear that F = lim →n∈N F n , so if F n is a flat representation for any n ∈ N, then F is also a flat representation of A ∞ .But it is easy to see that F n is a flat representation of A ∞ if and only if is a flat representation of the rooted quiver v n → v n−1 → ••• → v 1 → v 0 .By hypothesis, F j is a flat module for every j ≤ n and the homomorphisms f j are pure injections for every j ∈ {1,...,n}, so [6,Theorem 3.7] gives us that (4.2) is a flat representation for all n ∈ N, and we are done.
Throughout the rest of this paper, the class of all flat representations of A ∞ will be denoted by the symbol Ᏺ.
As an immediate consequence of the previous results, we have the following proposition.Proposition 4.2.Let F be a flat representation of A ∞ and G a subrepresentation of F in such a way that F/G is flat.Then G is also a flat representation of A ∞ .

Flat covers and cotorsion envelopes.
We will now prove the existence of a flat cover and a cotorsion envelope for every representation of A ∞ .This will be deduced from the fact that the pair of classes of flat representations and cotorsion representations of A ∞ , (Ᏺ, Ᏺ ⊥ ) is cogenerated by a set.
Proof.The class Ᏺ is closed under extensions and arbitrary direct limits, so every representation of A ∞ has a cotorsion envelope by [1,Corollary 2.11].Furthermore, Ᏺ contains all projective representations of A ∞ , so by [1, Corollary 2.12], every representation of A ∞ has a flat cover.