On Flat Objects of Finitely Accessible Categories

Abstract

Flat objects of a finitely accessible additive category 𝒞 are described in terms of some objects of the associated functor category of 𝒞, 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 𝒞 has an associated (Grothendieck) functor category (fp(𝒞)^op,Ab) consisting of all contravariant additive functors from the full subcategory fp(𝒞) of finitely presented objects of 𝒞 to the category Ab of abelian groups. Moreover, Yoneda functor 𝐻∶𝒞→(fp(𝒞)^op,Ab), defined on objects by the assignment 𝑋↦𝐻_𝑋=Hom_𝒞(−,𝑋)|_fp(𝒞), induces equivalence between 𝒞 and the full subcategory of flat objects of (fp(𝒞)^op,Ab). We are interested in determining the objects of the functor category (fp(𝒞)^op,Ab) which correspond to flat objects in the original category 𝒞 via the above equivalence. These will be the so-called strongly flat objects of (fp(𝒞)^op,Ab). 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 𝒞 is called finitely accessible if it has direct limits, the class fp(𝒞) of finitely presented objects is skeletally small, and every object is a direct limit of finitely presented objects. Let 𝒞 be a finitely accessible additive category. A sequence 0→𝑋−→^𝑓𝑌→−^𝑔𝑍→0 in 𝒞 is a pair of composable morphisms with 𝑔𝑓=0. The above sequence in 𝒞 is called pure exact if it induces an exact sequence of abelian groups 0→Hom_𝒞(𝑃,𝑋)→Hom_𝒞(𝑃,𝑌)→Hom_𝒞(𝑃,𝑍)→0 for every finitely presented object 𝑃 of 𝒞. This implies that 𝑓 and 𝑔 form a kernel-cokernel pair, in which 𝑓 is called a pure monomorphism and 𝑔 a pure epimorphism. The pure exact sequences in 𝒞 are those which become exact sequences in (fp(𝒞)^op,Ab) through Yoneda embedding functor 𝐻∶𝒞→(fp(𝒞)^op,Ab), defined on objects by 𝑋↦𝐻_𝑋=Hom_𝒞(−,𝑋)|_fp(𝒞) and correspondingly on morphisms. The functor 𝐻 preserves and reflects purity [6, Corollary 5.11] and commutes with direct limits. An object 𝑍 of 𝒞 is called pure projective if it is projective with respect to every pure exact sequence and flat if every epimorphism 𝑌→𝑍 is pure (e.g., see [6, 8]). If 0→𝐴→𝐵→𝐶→0 is a pure exact sequence in (fp(𝒞)^op,Ab) with 𝐵 flat, then 𝐴 and 𝐶 are flat (e.g., see [6, Proposition 5.9] and [9, Proposition 36.1]).

By a class of objects in an additive category 𝒞 we mean a class of objects closed under isomorphisms. Let 𝑀 be an object in 𝒞 and 𝒳 a class of objects in 𝒞. Recall from [10] that a morphism 𝑓∶𝑋→𝑀 in 𝒞, with 𝑋∈𝒳, is an 𝒳-precover of 𝑀 if the induced abelian group homomorphism Hom(𝑋^′,𝑓)∶Hom(𝑋^′,𝑋)→Hom(𝑋^′,𝑀) is an epimorphism for every 𝑋^′∈𝒳. An 𝒳-precover 𝑓∶𝑋→𝑀 of 𝑀 is an 𝒳-cover if every endomorphism 𝑔∶𝑋→𝑋 with 𝑓𝑔=𝑓 is an automorphism. The class 𝒳 is called (pre)covering if every object of 𝒞 has an 𝒳-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 𝒞 in terms of corresponding objects of its associated functor category through Yoneda functor 𝐻∶𝒞→(fp(𝒞)^op,Ab). To this end, we introduce and study a specialization of flatness in (fp(𝒞)^op,Ab), which is different from a strongly flat functor in the sense of [13]. Recall that every flat object of (fp(𝒞)^op,Ab) is of the form 𝐻_𝑍 for some object 𝑍 of 𝒞.

Definition 1. Let 𝒞 be a finitely accessible additive category. A flat object 𝐻_𝑍 of (fp(𝒞)^op,Ab) is called strongly flat if for every morphism 𝐻_𝑔∶𝐻_𝑌→𝐻_𝑍 in (fp(𝒞)^op,Ab) such that 𝑔∶𝑌→𝑍 is an epimorphism in 𝒞, and for every finitely presented object 𝑃 of (fp(𝒞)^op,Ab), the induced abelian group homomorphism Hom(𝑃,𝐻_𝑔)∶Hom(𝑃,𝐻_𝑌)→Hom(𝑃,𝐻_𝑍) is an epimorphism.

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

Proof. Let 0→𝐴→𝐵→𝐶→0 be a pure exact sequence in (fp(𝒞)^op,Ab) with 𝐵 strongly flat. Then 𝐵 is flat, hence 𝐴 and 𝐶 are also flat. It follows that 𝐴≅𝐻_𝑋, 𝐵≅𝐻_𝑌, and 𝐶≅𝐻_𝑍 for some objects 𝑋, 𝑌, and 𝑍 of 𝒞. Then the initial pure exact sequence has the form 0⟶𝐻⟶_𝑋^𝐻𝑓𝐻⟶_𝑌^𝐻𝑔𝐻_𝑍⟶0 (1) for some morphisms 𝑓, 𝑔 in 𝒞. Now let 𝐻_𝑤∶𝐻_𝑍′→𝐻_𝑍 be a morphism in (fp(𝒞)^op,Ab) such that 𝑤∶𝑍^′→𝑍 is an epimorphism in 𝒞, and let 𝑃 be a finitely presented object of (fp(𝒞)^op,Ab). Consider the pullback of 𝐻_𝑔 and 𝐻_𝑤 in (fp(𝒞)^op,Ab) in order to obtain the following commutative diagram with exact rows:
  (2) Since 𝐻_𝑋 and 𝐻_𝑍′ are flat, so is 𝐵^′. Hence 𝐵^′≅𝐻_𝑌′ for some object 𝑌^′ of 𝒞, and then 𝛽=𝐻_𝑣 for some morphism 𝑣∶𝑌^′→𝑌 in 𝒞. The full and faithful functor 𝐻 reflects pullbacks [14, Chapter II, Theorem 7.1]. Since 𝒞 is abelian, pullbacks preserve epimorphisms; hence 𝑣 is an epimorphism in 𝒞. Since 𝐻_𝑌 is strongly flat and is part of a pure exact sequence, Hom(𝑃,𝛽) and Hom(𝑃,𝐻_𝑔) are epimorphisms. Then the commutative diagram (3) shows that Hom(𝑃,𝐻_𝑤) is an epimorphism. Hence 𝐶≅𝐻_𝑍 is strongly flat.
Now let 0→𝐴→𝐵→𝐶→0 be a short exact sequence in (fp(𝒞)^op,Ab) with 𝐴 and 𝐶 strongly flat. Then 𝐴 and 𝐶 are flat, and so 𝐵 is also flat. It follows that 𝐴≅𝐻_𝑋, 𝐵≅𝐻_𝑌, and 𝐶≅𝐻_𝑍 for some objects 𝑋, 𝑌, and 𝑍 of 𝒞. Then the initial short exact sequence has the form 0⟶𝐻⟶_𝑋^𝐻𝑓𝐻⟶_𝑌^𝐻𝑔𝐻_𝑍⟶0 (4) for some morphisms 𝑓,𝑔 in 𝒞, and it is pure by the flatness of 𝐻_𝑍. Now let 𝐻_𝑣∶𝐻_𝑌′→𝐻_𝑌 be a morphism in (fp(𝒞)^op,Ab) such that 𝑣∶𝑌^′→𝑌 is an epimorphism in 𝒞, and let 𝑃 be a finitely presented object of (fp(𝒞)^op,Ab). Consider the pullback of 𝐻_𝑓 and 𝐻_𝑣 in (fp(𝒞)^op,Ab) in order to obtain the following commutative diagram with exact rows: (5) Since 𝐻_𝑍 is flat, the upper row of the diagram is pure. Since 𝐻_𝑌′ is flat, it follows that 𝐴^′ is also flat. Hence 𝐴^′≅𝐻_𝑋′ for some object 𝑋^′ of 𝒞, and then 𝛼=𝐻_𝑢 for some morphism 𝑢∶𝑋^′→𝑋 in 𝒞. Using that 𝐻 is full and faithful and 𝒞 is abelian, one deduces as in the first part of the proof that 𝑢 is an epimorphism in 𝒞. Since 𝐻_𝑋 is strongly flat, Hom(𝑃,𝛼) is an epimorphism. Then the induced commutative diagram with exact rows(6)implies that Hom(𝑃,𝐻_𝑣) is an epimorphism. Hence 𝐵≅𝐻_𝑌 is strongly flat.
The closure of the class of strongly flat objects of (fp(𝒞)^op,Ab) under extensions implies its closure under finite direct sums. Now let ⊕_𝑖∈𝐼𝐻_𝑍𝑖≅𝐻_{⊕𝑖∈𝐼𝑍𝑖} be a direct sum of strongly flat objects of (fp(𝒞)^op,Ab). Let 𝐻_𝑔∶𝐻_𝑌→𝐻_{⊕𝑖∈𝐼𝑍𝑖} be a morphism in (fp(𝒞)^op,Ab) such that 𝑔∶𝑌→⊕_𝑖∈𝐼𝑍_𝑖 is an epimorphism in 𝒞, and let 𝑃 be a finitely presented object of (fp(𝒞)^op,Ab). Then there is a finite subset 𝐹 of 𝐼 such that Hom𝑃,𝜑∶Hom𝑃,_𝑖∈𝐹𝐻_𝑍𝑖⟶Hom𝑃,_𝑖∈𝐼𝐻_𝑍𝑖 (7) is an epimorphism, where 𝜑∶⊕_𝑖∈𝐹𝐻_𝑍𝑖→⊕_𝑖∈𝐼𝐻_𝑍𝑖 is the inclusion morphism. Then 𝜑≅𝐻_𝑢∶𝐻_{⊕𝑖∈𝐹𝑍𝑖}→𝐻_{⊕𝑖∈𝐼𝑍𝑖}, where 𝑢∶⊕_𝑖∈𝐹𝑍_𝑖→⊕_𝑖∈𝐼𝑍_𝑖 is the inclusion morphism. Consider the pullback of 𝑢 and 𝑔 in 𝒞: (8) Since 𝒞 is abelian, 𝑔^′ is an epimorphism in 𝒞. Since ⊕_𝑖∈𝐹𝐻_𝑍𝑖 is strongly flat, it follows that Hom(𝑃,𝐻_𝑔′)∶Hom(𝑃,𝐻_𝑌′)→Hom(𝑃,⊕_𝑖∈𝐹𝐻_𝑍𝑖) is an epimorphism. Then the induced commutative diagram (9) implies that Hom(𝑃,𝐻_𝑔) is an epimorphism. Hence ⊕_𝑖∈𝐼𝐻_𝑍𝑖 is strongly flat.
Finally, let (𝐻_𝑍𝑖,𝑓_𝑖𝑗)_𝐼 be a direct system of strongly flat objects of (fp(𝒞)^op,Ab). Then there is a pure epimorphism _𝑖∈𝐼𝐻_𝑍𝑖⟶lim_→𝐻_𝑍𝑖 (10) in (fp(𝒞)^op,Ab) (e.g., see [9, Example 33.9]). We have already proved that the class of strongly flat objects of (fp(𝒞)^op,Ab) is closed under direct sums and pure epimorphic images. Hence the direct limit lim_→𝐻_𝑍𝑖 is strongly flat.

4. Flat Objects in Finitely Accessible Categories

Now let us relate flat objects of a finitely accessible additive category 𝒞 and strongly flat objects of its associated functor category (fp(𝒞)^op,Ab).

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

Proof. (1) By [7, Lemma 3.1].
(2) Assume first that 𝑍 is a flat object of 𝒞. Let 𝐻_𝑔∶𝐻_𝑌→𝐻_Z be a morphism in (fp(𝒞)^op,Ab) such that 𝑔∶𝑌→𝑍 is an epimorphism in 𝒞, and let 𝛾∶𝑃→𝐻_𝑍 be a morphism in (fp(𝒞)^op,Ab) with 𝑃 finitely presented. Since 𝑍 is flat in 𝒞, 𝑔 is a pure epimorphism, and so there is a pure exact sequence 0⟶𝑋⟶𝑌⟶^𝑔𝑍⟶0 (11) in 𝒞. Then the induced sequence 0⟶𝐻_𝑋⟶𝐻⟶_𝑌^𝐻𝑔𝐻_𝑍⟶0 (12) is pure exact in (fp(𝒞)^op,Ab). Now 𝛾 lifts to a morphism 𝑃→𝐻_𝑌, showing that 𝐻_𝑍 is strongly flat in (fp(𝒞)^op,Ab).
Conversely, assume that 𝐻_𝑍 is a strongly flat object of (fp(𝒞)^op,Ab). Consider in 𝒞 an epimorphism 𝑔∶𝑌→𝑍, a finitely presented object 𝐿, and a morphism 𝑤∶𝐿→𝑍. Then 𝐻_𝐿 is finitely generated projective and so finitely presented in (fp(𝒞)^op,Ab) (e.g., see [15, Theorem 1.1]). Since 𝐻_𝑍 is strongly flat in (fp(𝒞)^op,Ab), there is a morphism 𝜑∶𝐻_𝐿→𝐻_𝑌 such that 𝐻_𝑔𝜑=𝐻_𝑤. Now we have 𝜑=𝐻_ℎ for some morphism ℎ∶𝐿→𝑌 in 𝒞. Then 𝑔ℎ=𝑤, showing that 𝑔∶𝑌→𝑍 is a pure epimorphism in 𝒞, and so 𝑍 is flat in 𝒞.
(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 𝒞 be a finitely accessible abelian category. Then the class of flat objects of 𝒞 is covering.

Proof. The class of strongly flat objects of the functor category (fp(𝒞)^op,Ab) is closed under direct limits and pure epimorphic images by Theorem 2. Then it is a covering class in (fp(𝒞)^op,Ab) 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 𝒞 is a covering class.