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The Scientific World Journal
Volume 2013 (2013), Article ID 451091, 4 pages
http://dx.doi.org/10.1155/2013/451091
Research Article

On Flat Objects of Finitely Accessible Categories

Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, Street Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania

Received 6 August 2013; Accepted 8 September 2013

Academic Editors: A. Mimouni and J. Rada

Copyright © 2013 Septimiu Crivei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Flat objects of a finitely accessible additive category are described in terms of some objects of the associated functor category of , called strongly flat functors. We study closure properties of the class of strongly flat functors, and we use them to deduce the known result that every object of a finitely accessible abelian category has a flat cover.

1. Introduction

The famous Enochs's Flat Cover Conjecture played a key part in the development of the theory of module approximations, which has the root in the work of Auslander, Smalø, and Enochs [1, 2]. The conjecture stated that every module has a flat cover, and it was proved by Bican et al. [3, Theorem 3]. Afterwards, the problem was considered in various more general categories. For instance, Crivei et al. [4] and Rump [5] showed in two different ways that every object of a finitely accessible abelian category has a flat cover. Nevertheless, the knowledge about flat objects in such categories is rather limited. The present paper is intended to make a further step towards a better understanding of flat objects in finitely accessible additive categories.

It is well known that every finitely accessible additive category has an associated (Grothendieck) functor category consisting of all contravariant additive functors from the full subcategory of finitely presented objects of to the category of abelian groups. Moreover, Yoneda functor , defined on objects by the assignment , induces equivalence between and the full subcategory of flat objects of . We are interested in determining the objects of the functor category which correspond to flat objects in the original category via the above equivalence. These will be the so-called strongly flat objects of . We study some closure properties of the class of strongly flat objects, among which the closure under direct limits and pure epimorphic images. As an application, we use them to deduce the known result that every object of a finitely accessible abelian category has a flat cover. Note that every finitely accessible abelian category is already Grothendieck [6, Theorem 3.15].

2. Preliminaries

We recall some further terminology on finitely accessible additive categories, mainly following [6, 7]. Throughout the paper all categories and functors will be additive. An additive category is called finitely accessible if it has direct limits, the class of finitely presented objects is skeletally small, and every object is a direct limit of finitely presented objects. Let be a finitely accessible additive category. A sequence in is a pair of composable morphisms with . The above sequence in is called pure exact if it induces an exact sequence of abelian groups for every finitely presented object of . This implies that and form a kernel-cokernel pair, in which is called a pure monomorphism and a pure epimorphism. The pure exact sequences in are those which become exact sequences in through Yoneda embedding functor , defined on objects by and correspondingly on morphisms. The functor preserves and reflects purity [6, Corollary 5.11] and commutes with direct limits. An object of is called pure projective if it is projective with respect to every pure exact sequence and flat if every epimorphism is pure (e.g., see [6, 8]). If is a pure exact sequence in with flat, then and are flat (e.g., see [6, Proposition 5.9] and [9, Proposition 36.1]).

By a class of objects in an additive category we mean a class of objects closed under isomorphisms. Let be an object in and a class of objects in . Recall from [10] that a morphism in , with , is an -precover of if the induced abelian group homomorphism is an epimorphism for every . An -precover of is an -cover if every endomorphism with is an automorphism. The class is called (pre)covering if every object of has an -cover. Dually one defines the notions of relative (pre)envelope and (pre)enveloping class. For instance, every class of modules closed under direct products and pure submodules is preenveloping [11], whereas every class of modules closed under direct limits and pure epimorphic images is covering [4, 12].

3. Strongly Flat Objects in Functor Categories

We are interested in identifying certain objects of a finitely accessible additive category in terms of corresponding objects of its associated functor category through Yoneda functor . To this end, we introduce and study a specialization of flatness in , which is different from a strongly flat functor in the sense of [13]. Recall that every flat object of is of the form for some object of .

Definition 1. Let be a finitely accessible additive category. A flat object of is called strongly flat if for every morphism in such that is an epimorphism in , and for every finitely presented object of , the induced abelian group homomorphism is an epimorphism.

Theorem 2. Let be a finitely accessible abelian category. Then the class of strongly flat objects of is closed under pure epimorphic images, extensions, direct sums, and direct limits.

Proof. Let be a pure exact sequence in with strongly flat. Then is flat, hence and are also flat. It follows that , , and for some objects , , and of . Then the initial pure exact sequence has the form for some morphisms , in . Now let be a morphism in such that is an epimorphism in , and let be a finitely presented object of . Consider the pullback of and in in order to obtain the following commutative diagram with exact rows:
xy(2) Since and are flat, so is . Hence for some object of , and then for some morphism in . The full and faithful functor reflects pullbacks [14, Chapter II, Theorem 7.1]. Since is abelian, pullbacks preserve epimorphisms; hence is an epimorphism in . Since is strongly flat and is part of a pure exact sequence, and are epimorphisms. Then the commutative diagram xy(3) shows that is an epimorphism. Hence is strongly flat.
Now let be a short exact sequence in with and strongly flat. Then and are flat, and so is also flat. It follows that , , and for some objects , , and of . Then the initial short exact sequence has the form for some morphisms in , and it is pure by the flatness of . Now let be a morphism in such that is an epimorphism in , and let be a finitely presented object of . Consider the pullback of and in in order to obtain the following commutative diagram with exact rows: xy(5) Since is flat, the upper row of the diagram is pure. Since is flat, it follows that is also flat. Hence for some object of , and then for some morphism in . Using that is full and faithful and is abelian, one deduces as in the first part of the proof that is an epimorphism in . Since is strongly flat, is an epimorphism. Then the induced commutative diagram with exact rowsxy(6)implies that is an epimorphism. Hence is strongly flat.
The closure of the class of strongly flat objects of under extensions implies its closure under finite direct sums. Now let be a direct sum of strongly flat objects of . Let be a morphism in such that is an epimorphism in , and let be a finitely presented object of . Then there is a finite subset of such that is an epimorphism, where is the inclusion morphism. Then , where is the inclusion morphism. Consider the pullback of and in : xy(8) Since is abelian, is an epimorphism in . Since is strongly flat, it follows that is an epimorphism. Then the induced commutative diagram xy(9) implies that is an epimorphism. Hence is strongly flat.
Finally, let be a direct system of strongly flat objects of . Then there is a pure epimorphism in (e.g., see [9, Example 33.9]). We have already proved that the class of strongly flat objects of is closed under direct sums and pure epimorphic images. Hence the direct limit is strongly flat.

4. Flat Objects in Finitely Accessible Categories

Now let us relate flat objects of a finitely accessible additive category and strongly flat objects of its associated functor category .

Theorem 3. Let be a finitely accessible additive category. Then the equivalence induced by the Yoneda functor between and the full subcategory of flat objects of restricts to equivalences between the following full subcategories:(1)pure-projective objects of and projective objects of ,(2)flat objects of and strongly flat objects of ,(3)projective objects of and strongly flat projective objects of .

Proof. (1) By [7, Lemma 3.1].
(2) Assume first that is a flat object of . Let be a morphism in such that is an epimorphism in , and let be a morphism in with finitely presented. Since is flat in , is a pure epimorphism, and so there is a pure exact sequence in . Then the induced sequence is pure exact in . Now lifts to a morphism , showing that is strongly flat in .
Conversely, assume that is a strongly flat object of . Consider in an epimorphism , a finitely presented object , and a morphism . Then is finitely generated projective and so finitely presented in (e.g., see [15, Theorem 1.1]). Since is strongly flat in , there is a morphism such that . Now we have for some morphism in . Then , showing that is a pure epimorphism in , and so is flat in .
(3) This follows by (1) and (2).

Using the above theorems we may deduce the following known result on the existence of flat covers in finitely accessible abelian (Grothendieck) categories (see [4, Corollary 3.3] and [5, page 1604]).

Corollary 4. Let be a finitely accessible abelian category. Then the class of flat objects of is covering.

Proof. The class of strongly flat objects of the functor category is closed under direct limits and pure epimorphic images by Theorem 2. Then it is a covering class in by [4, Theorem 2.4] (also see [12, Theorem 2.5]). By Theorem 3 and [4, Lemma 2.5] it follows that the class of flat objects of is a covering class.

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