Hindawi Publishing CorporationThe Scientific World JournalVolume 2013, Article ID 451091, 4 pageshttp://dx.doi.org/10.1155/2013/451091

Research ArticleOn Flat Objects of Finitely Accessible Categories

Septimiu Crivei

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

Correspondence should be addressed to Septimiu Crivei; crivei@math.ubbcluj.ro

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.

Flat objects of a finitely accessible additive categoryC are described in terms of some objects of the associated functor category ofC, called strongly flat functors. We study closure properties of the class of strongly flat functors, and we use them to deduce theknown 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 partin the development of the theory of module approximations,which has the root in the work of Auslander, Smalø, andEnochs [1, 2]. The conjecture stated that every module hasa flat cover, and it was proved by Bican et al. [3, Theorem 3].Afterwards, the problemwas considered in variousmore gen-eral categories. For instance, Crivei et al. [4] and Rump [5]showed in two different ways that every object of a finitelyaccessible abelian category has a flat cover. Nevertheless, theknowledge about flat objects in such categories is rather lim-ited. The present paper is intended to make a further steptowards a better understanding of flat objects in finitelyaccessible additive categories.

It is well known that every finitely accessible additive cat-egory C has an associated (Grothendieck) functor category(fp(C)op,Ab) consisting of all contravariant additive functorsfrom the full subcategory fp(C) of finitely presented objectsofC to the category Ab of abelian groups. Moreover, Yonedafunctor 𝐻 : C → (fp(C)op,Ab), defined on objects by theassignment 𝑋 → 𝐻

𝑋= HomC(−, 𝑋)|fp(C), induces equiva-

lence between C and the full subcategory of flat objects of(fp(C)op,Ab). We are interested in determining the objectsof the functor category (fp(C)op,Ab) which correspond toflat objects in the original category C via the above equiva-lence. These will be the so-called strongly flat objects of(fp(C)op,Ab). We study some closure properties of the classof strongly flat objects, among which the closure under directlimits and pure epimorphic images. As an application, we use

them to deduce the known result that every object of a finitelyaccessible abelian category has a flat cover. Note that everyfinitely accessible abelian category is alreadyGrothendieck [6,Theorem 3.15].

2. Preliminaries

We recall some further terminology on finitely accessibleadditive categories, mainly following [6, 7]. Throughout thepaper all categories and functors will be additive. An additivecategoryC is called finitely accessible if it has direct limits, theclass fp(C) of finitely presented objects is skeletally small, andevery object is a direct limit of finitely presented objects. LetC be a finitely accessible additive category. A sequence 0 →

𝑋𝑓

→ 𝑌𝑔

→ 𝑍 → 0 in C is a pair of composable morphismswith 𝑔𝑓 = 0. The above sequence in C is called pure exactif it induces an exact sequence of abelian groups 0 →HomC(𝑃,𝑋) → HomC(𝑃, 𝑌) → HomC(𝑃, 𝑍) → 0 forevery finitely presented object 𝑃 ofC.This implies that𝑓 and𝑔 form a kernel-cokernel pair, in which 𝑓 is called a puremonomorphism and 𝑔 a pure epimorphism. The pure exactsequences in C are those which become exact sequences in(fp(C)op,Ab) through Yoneda embedding functor𝐻 : C →(fp(C)op,Ab), defined on objects by 𝑋 → 𝐻

𝑋= HomC

(−,𝑋)|fp(C) and correspondingly on morphisms. The functor𝐻 preserves and reflects purity [6, Corollary 5.11] andcommutes with direct limits. An object 𝑍 of C is called pureprojective if it is projective with respect to every pure exactsequence and flat if every epimorphism 𝑌 → 𝑍 is pure

2 The Scientific World Journal

(e.g., see [6, 8]). If 0 → 𝐴 → 𝐵 → 𝐶 → 0 is a pure exactsequence 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 meana class of objects closed under isomorphisms. Let 𝑀 be anobject 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 homomorphismHom(𝑋, 𝑓) : Hom(𝑋, 𝑋) → Hom(𝑋,𝑀) is an epimor-phism for every𝑋 ∈ X. AnX-precover𝑓 : 𝑋 → 𝑀 of𝑀 isanX-cover if every endomorphism 𝑔 : 𝑋 → 𝑋 with 𝑓𝑔 = 𝑓is an automorphism. The class X is called (pre)covering ifevery object of C has an X-cover. Dually one defines thenotions of relative (pre)envelope and (pre)enveloping class.For instance, every class of modules closed under directproducts and pure submodules is preenveloping [11], whereasevery class of modules closed under direct limits and pureepimorphic images is covering [4, 12].

3. Strongly Flat Objects in Functor Categories

We are interested in identifying certain objects of a finitelyaccessible additive category C in terms of correspondingobjects of its associated functor category through Yonedafunctor 𝐻 : C → (fp(C)op,Ab). To this end, we introduceand study a specialization of flatness in (fp(C)op,Ab), whichis different from a strongly flat functor in the sense of [13].Recall that every flat object of (fp(C)op,Ab) is of the form𝐻

𝑍

for some object 𝑍 ofC.

Definition 1. Let C be a finitely accessible additive category.A flat object 𝐻

𝑍of (fp(C)op,Ab) is called strongly flat if for

every morphism𝐻𝑔: 𝐻𝑌→ 𝐻𝑍in (fp(C)op,Ab) such that

𝑔 : 𝑌 → 𝑍 is an epimorphism in C, and for every finitelypresented object 𝑃 of (fp(C)op,Ab), the induced abeliangroup homomorphism Hom(𝑃,𝐻

𝑔) : Hom(𝑃,𝐻

𝑌) →

Hom(𝑃,𝐻𝑍) 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 closedunder pure epimorphic images, extensions, direct sums, anddirect limits.

Proof. Let 0 → 𝐴 → 𝐵 → 𝐶 → 0 be a pure exactsequence 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 ≅ 𝐻𝑍for some objects 𝑋, 𝑌, and 𝑍 of C. Then the

initial pure exact sequence has the form

0 → 𝐻𝑋

𝐻𝑓

→ 𝐻𝑌

𝐻𝑔

→ 𝐻𝑍→ 0 (1)

for some morphisms 𝑓, 𝑔 in C. Now let 𝐻𝑤: 𝐻𝑍 → 𝐻

𝑍

be a morphism in (fp(C)op,Ab) such that 𝑤 : 𝑍 → 𝑍 is anepimorphism in C, and let 𝑃 be a finitely presented objectof (fp(C)op,Ab). Consider the pullback of 𝐻

𝑔and 𝐻

𝑤in

(fp(C)op,Ab) in order to obtain the following commutativediagram with exact rows:

0

0

0

0

HX

HX HY HZ

B HZ

𝛽 Hw

Hf Hg

(2)

Since𝐻𝑋and𝐻

𝑍 are flat, so is 𝐵. Hence 𝐵 ≅ 𝐻

𝑌 for some

object𝑌 ofC, and then𝛽 = 𝐻V for somemorphism V : 𝑌→

𝑌 in C. The full and faithful functor 𝐻 reflects pullbacks[14, Chapter II, Theorem 7.1]. Since C is abelian, pullbackspreserve epimorphisms; hence V is an epimorphism in C.Since𝐻

𝑌is strongly flat and is part of a pure exact sequence,

Hom(𝑃, 𝛽) and Hom(𝑃,𝐻𝑔) are epimorphisms. Then the

commutative diagram

Hom(P,Hw)

Hom(P,Hg )

Hom(P,HZ)

Hom(P,HZ)Hom(P,HY)

Hom(P, B)

Hom(P, 𝛽)(3)

shows that Hom(𝑃,𝐻𝑤) 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 𝐴and 𝐶 are flat, and so 𝐵 is also flat. It follows that 𝐴 ≅ 𝐻

𝑋,

𝐵 ≅ 𝐻𝑌, and 𝐶 ≅ 𝐻

𝑍for some objects 𝑋, 𝑌, and 𝑍 of C.

Then the initial short exact sequence has the form

0 → 𝐻𝑋

𝐻𝑓

→ 𝐻𝑌

𝐻𝑔

→ 𝐻𝑍→ 0 (4)

for somemorphisms𝑓, 𝑔 inC, and it is pure by the flatness of𝐻𝑍. Now let𝐻V : 𝐻𝑌 → 𝐻𝑌 be amorphism in (fp(C)

op,Ab)

such that V : 𝑌 → 𝑌 is an epimorphism in C, and let 𝑃be a finitely presented object of (fp(C)op,Ab). Consider thepullback of𝐻

𝑓and𝐻V in (fp(C)

op,Ab) in order to obtain the

following commutative diagram with exact rows:

0

0

0

0HX HY HZ

HZA HY

Hv

Hf Hg

𝛼 (5)

Since 𝐻𝑍is flat, the upper row of the diagram is pure. Since

𝐻𝑌 is flat, it follows that 𝐴 is also flat. Hence 𝐴 ≅ 𝐻

𝑋 for

some object 𝑋 of C, and then 𝛼 = 𝐻𝑢for some morphism

𝑢 : 𝑋 → 𝑋 in C. Using that𝐻 is full and faithful and C isabelian, one deduces as in the first part of the proof that 𝑢 isan epimorphism in C. Since 𝐻

𝑋is strongly flat, Hom(𝑃, 𝛼)

is an epimorphism. Then the induced commutative diagramwith exact rows

The Scientific World Journal 3

0

0

0

0

Hom(P, A) Hom(P,HY )

Hom(P,HZ)

Hom(P,HZ)

Hom(P,HX) Hom(P,HY)

Hom(P, 𝛼) Hom(P,H�) (6)

implies that Hom(𝑃,𝐻V) is an epimorphism. Hence 𝐵 ≅ 𝐻𝑌is strongly flat.

The closure of the class of strongly flat objects of(fp(C)op,Ab) under extensions implies its closure underfinite direct sums. Now let ⊕

𝑖∈𝐼𝐻𝑍𝑖≅ 𝐻⊕𝑖∈𝐼𝑍𝑖

be a direct sumof strongly flat objects of (fp(C)op,Ab). Let 𝐻

𝑔: 𝐻𝑌

→

𝐻⊕𝑖∈𝐼𝑍𝑖

be a morphism in (fp(C)op,Ab) such that 𝑔 : 𝑌 →⊕𝑖∈𝐼𝑍𝑖is an epimorphism in C, and let 𝑃 be a finitely

presented object of (fp(C)op,Ab).Then there is a finite subset𝐹 of 𝐼 such that

Hom (𝑃, 𝜑) : Hom(𝑃,⨁𝑖∈𝐹

𝐻𝑍𝑖) → Hom(𝑃,⨁

𝑖∈𝐼

𝐻𝑍𝑖)

(7)

is an epimorphism, where 𝜑 : ⊕𝑖∈𝐹𝐻𝑍𝑖

→ ⊕𝑖∈𝐼𝐻𝑍𝑖

is theinclusion morphism. Then 𝜑 ≅ 𝐻

𝑢: 𝐻⊕𝑖∈𝐹𝑍𝑖

→ 𝐻⊕𝑖∈𝐼𝑍𝑖

,where 𝑢 : ⊕

𝑖∈𝐹𝑍𝑖→ ⊕

𝑖∈𝐼𝑍𝑖is the inclusion morphism.

Consider the pullback of 𝑢 and 𝑔 inC:

Y Y

g

u

g

⨁i∈F

Zi ⨁i∈I

Zi

(8)

SinceC is abelian, 𝑔 is an epimorphism inC. Since ⊕𝑖∈𝐹𝐻𝑍𝑖

is strongly flat, it follows that Hom(𝑃,𝐻𝑔) : Hom(𝑃,𝐻

𝑌) →

Hom(𝑃, ⊕𝑖∈𝐹𝐻𝑍𝑖) is an epimorphism. Then the induced

commutative diagram

Hom(P,Hg ) Hom(P,Hg )

Hom(P,HY ) Hom(P,HY)

Hom(P, 𝜑)Hom(P,⨁

i∈F

) Hom(P,⨁i∈I

HZ𝑖 )HZ𝑖

(9)

implies that Hom(𝑃,𝐻𝑔) is an epimorphism. Hence ⊕

𝑖∈𝐼𝐻𝑍𝑖

is strongly flat.Finally, let (𝐻

𝑍𝑖, 𝑓𝑖𝑗)𝐼be a direct system of strongly flat

objects of (fp(C)op,Ab). Then there is a pure epimorphism

⨁𝑖∈𝐼

𝐻𝑍𝑖→ lim→𝐻𝑍𝑖 (10)

in (fp(C)op,Ab) (e.g., see [9, Example 33.9]).We have alreadyproved that the class of strongly flat objects of (fp(C)op,Ab) isclosed under direct sums and pure epimorphic images.Hencethe direct limit lim

→𝐻𝑍𝑖is strongly flat.

4. Flat Objects in FinitelyAccessible Categories

Now let us relate flat objects of a finitely accessible additivecategory C and strongly flat objects of its associated functorcategory (fp(C)op,Ab).

Theorem 3. Let C be a finitely accessible additive category.Then the equivalence induced by the Yoneda functor𝐻 : C →(fp(C)op,Ab) betweenC and the full subcategory of flat objectsof (fp(C)op,Ab) restricts to equivalences between the followingfull 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 projectiveobjects of (fp(C)op,Ab).

Proof. (1) By [7, Lemma 3.1].(2)Assumefirst that𝑍 is a flat object ofC. Let𝐻

𝑔: 𝐻𝑌→

𝐻Z be a morphism in (fp(C)op,Ab) such that 𝑔 : 𝑌 → 𝑍 is

an epimorphism inC, and let 𝛾 : 𝑃 → 𝐻𝑍be a morphism in

(fp(C)op,Ab) with 𝑃 finitely presented. Since 𝑍 is flat inC, 𝑔is a pure epimorphism, and so there is a pure exact sequence

0 → 𝑋 → 𝑌𝑔

→ 𝑍 → 0 (11)

inC. Then the induced sequence

0 → 𝐻𝑋→ 𝐻

𝑌

𝐻𝑔

→ 𝐻𝑍→ 0 (12)

is pure exact in (fp(C)op,Ab). Now 𝛾 lifts to amorphism𝑃 →𝐻𝑌, showing that𝐻

𝑍is strongly flat in (fp(C)op,Ab).

Conversely, assume that 𝐻𝑍is a strongly flat object of

(fp(C)op,Ab). Consider inC an epimorphism 𝑔 : 𝑌 → 𝑍, afinitely presented object𝐿, and amorphism𝑤 : 𝐿 → 𝑍.Then𝐻𝐿is finitely generated projective and so finitely presented in

(fp(C)op,Ab) (e.g., see [15,Theorem 1.1]). Since𝐻𝑍is strongly

flat in (fp(C)op,Ab), there is a morphism 𝜑 : 𝐻𝐿→ 𝐻𝑌such

that 𝐻𝑔𝜑 = 𝐻

𝑤. Now we have 𝜑 = 𝐻

ℎfor some morphism

4 The Scientific World Journal

ℎ : 𝐿 → 𝑌 inC. Then 𝑔ℎ = 𝑤, showing that 𝑔 : 𝑌 → 𝑍 is apure epimorphism inC, and so 𝑍 is flat inC.

(3) This follows by (1) and (2).

Using the above theorems we may deduce the followingknown result on the existence of flat covers in finitely accessi-ble 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 ofC is covering.

Proof. Theclass of strongly flat objects of the functor category(fp(C)op,Ab) is closed under direct limits and pure epi-morphic images by Theorem 2. Then it is a covering class in(fp(C)op,Ab) by [4,Theorem2.4] (also see [12,Theorem2.5]).By Theorem 3 and [4, Lemma 2.5] it follows that the class offlat objects ofC is a covering class.

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[11] J. Rada and M. Saoŕın, “Rings characterized by (pre)envelopesand (pre)covers of their modules,” Communications in Algebra,vol. 26, no. 3, pp. 899–912, 1998.

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[13] L. Mao, “On strongly flat and Ω-Mittag-Leffler objects in thecategory ((𝑅 −mod)op,Ab),” Mediterranean Journal of Mathe-matics, vol. 10, no. 2, pp. 655–676, 2013.

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