flat covers and factorizations

Journal of Algebra 253 (2002) 1–13

www.academicpress.com

Flat covers and factorizations

J. Rosický1

Masaryk University, Janáˇckovo nám. 2a, 662 95 Brno, Czech Republic

Received 20 July 2000

Communicated by Kent R. Fuller

Abstract

The flat cover conjecture, saying that every module has a flat (pre)cover, has beenrecently proved by Bican, El Bashir, and Enochs. We relate flat precovers (and cotorsionpreenvelopes) to weak factorizations and prove that flat monomorphisms form a left partof a weak factorization system. 2002 Elsevier Science (USA). All rights reserved.

Keywords:Flat cover; Weak factorization system

1. Introduction

The flat cover conjecture was formulated by E. Enochs [1] and asks whetherevery module has a flat cover. Enochs also showed that it is equivalent tothe existence of flat precovers. Recall that a flat precover of a moduleM isa homomorphismg :F → M where F is flat and for each homomorphismv :G → M with G flat there isd :G → F such thatg · d = v. J. Xu provedthat flat covers exist over certain commutative rings and presented the status ofthe conjecture in [2]. The flat cover conjecture has been recently proved in [3].This work gives two proofs, one of then (due to Enochs) is based on the resultof P. Eklof and P. Trlifaj [4]. We analyze the Enochs’ proof from the point of

E-mail address:[email protected] Supported by the Ministry of Education of the Czech Republic under the project MSM

143100009.

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0021-8693(02)00043-1

2 J. Rosický / Journal of Algebra 253 (2002) 1–13

view of the theory of accessible categories (cf. [5]) and prove a stronger resultsaying that flat monomorphisms form a left part of a weak factorization systemon the category of modules. Flat precovers (cotorsion preenvelopes) then arisefrom a weak factorization of module homomorphisms having domain (codomain)equal to 0. The concept of a weak factorization has originated in homotopy theory(see [6,7]).

In more detail, a monomorphismf :A→ B is calledflat if coker(f ) :B → F

hasF flat (see [2]). A module homomorphismg :C →D is said to have therightlifting property w.r.t. flat monomorphismsif in each commutative square

A

f

uC

g

B v D

with f a flat monomorphism there is a diagonald :B → C such thatd · f = u

and g · d = v. We show that any module homomorphismh :N → M has afactorizationh = g · f wheref is flat andg has the right lifting property w.r.t.flat monomorphisms. By applying it toh : 0 →M, we get

h : 0 Fg

M

andg is a flat precover ofM. To verify it, one uses the right lifting property asfollows:

0 F

g

G

d

v M.

Since a homomorphismg :C → 0 has the right lifting property w.r.t. flatmonomorphisms iffC is cotorsion, by applying our factorization toh :N → 0we get a cotorsion preenvelope ofN :

h :Nf

C 0 .

Our approach works for an arbitrary classF of modules (satisfying conditionsof Theorem 4.5) in place of flat modules.

2. Weak factorization systems

As we have mentioned in the introduction, weak factorization systems are wellknown in homotopy theory (see [8,9] or [10] for a recent exposition).

J. Rosický / Journal of Algebra 253 (2002) 1–13 3

Definition 2.1. Let K be a category andf :A→ B, g :C →D morphisms suchthat in each commutative square

A

f

uC

g

B v D

there is a diagonald :B →C with d ·f = u andg ·d = v. Then we say thatg hastheright lifting propertyw.r.t.f and thatf has theleft lifting propertyw.r.t. g.

For a classH of morphisms ofK we put

H� = {g | g has the right lifting property w.r.t. eachf ∈H} and�H = {f | f has the left lifting property w.r.t. eachg ∈H}.

Assume thatK has a terminal object 1. Then an objectK of K is injectiveto H iff the morphismK → 1 belongs toH�. In fact, it exactly means that forevery morphismsf :A→ B in H and everyu :A→K there isd :B →K withd ·f = u. We will denote byH� the full subcategory ofK consisting of all objectsinjective toH. Dually, if K has an initial object 0, thenK is projective toH iffthe morphism 0→K belongs to�K.

Definition 2.2 [11]. A weak factorization system(L,R) in a categoryK consistsof two classesL andR of morphisms ofK such that

(1) R = L�, L= �R and(2) any morphismh of K, has a factorizationh= g · f with f ∈L andg ∈ R.

Remark 2.3. (1) Let(L,R) be a weak factorization system in a categoryK havinga terminal object 1. Then any morphismK → 1 has a factorization

Kf K 1

with f ∈ L and K ∈ L�. It means thatK has enoughL-injective objects.Dually, if K has an initial object 0 then 0→K has a factorization

0 K∗ gK

with g ∈ R andK∗ R-projective. It means thatK has enoughR-projectiveobjects.

(2) In general, a morphismg :C →D lies inL� iff as an object of the commacategoryK ↓ D, g is L-injective for the classL of all morphismsf of K ↓ Dsuch thatP(f ) ∈ L whereP :K ↓ D → K is the projection (see [8, 12.4.2]).Then, factorizations

h :Af

Cg

B


yield thatK ↓ B has enoughL-injectives for eachB in K.Dually,A ↓ K has enoughR-projectives for eachA in K.

For any classC of morphisms ofK we getL = �C andR = L� satisfyingDefinition 2.2(1).

Remark 2.4. For any classC of morphisms ofK, �C contains all isomorphismsand is

(a) stable under pushouts,(b) closed under transfinite compositions, and(c) closed under retracts in the comma categoryA ↓ K (of objects underA).

It means that

(a) If

Bg

D

A

f

g C

f

is a pushout andf ∈ �C thenf ∈ �C.(b1) If f1 :A→B andf2 :B → C are in�C thenf2 · f1 ∈ �C.(b2) If (fij :Ai → Aj)i�j�λ is a smooth chain (i.e.,λ is a limit ordinal,

(fij :Ai → Aj)i<j is a colimit for any limit ordinalj < λ) andfij ∈ �Cfor i � j < λ thenf0λ ∈ �C.

(c) If f :A→B is in �C andg :A→ C is a retract off in A ↓ K theng ∈ �C.(The retract assumption means an existence ofs :C → B and r :B → C

with r · s = idc, s · g = f andr · f = g.)

A morphism is called aC-cofibration (see [11]) if it is retract (in the senseof (1c)) of an isomorphism or of a transfinite composition of pushouts of elementsof C. The class ofC-cofibrations is denoted by cof(C). The following basic resultgoes back to Quillen [7]. In the present form it can be found in [11] (see also [10]).

Theorem 2.5. LetK be a locally presentable category andC a set of morphismsof K. ThenR = C� and L = �R form a weak factorization system inK.Moreover,L = cof(C).

A weak factorization system(L,R) is calledcofibrantly generatedif thereexists a setC of morphisms such thatL= cof(C) andR = C�.


3. Accessible categories of modules

Recall that a category isλ-accessible, whereλ is a regular cardinal, providedthat

(i) K hasλ-directed colimits, and(ii) K has a setA of λ-presentable objects such that every object inK is a

λ-directed colimit of objects fromA.

K is calledaccessibleif it is λ-accessible for some regular cardinalλ. ω-accessible categories are calledfinitely accessible. A monomorphismf :A→ B

is said to bepureprovided that in each commutative square

X

u

hY

v

Af

B

with X andY finitely presentable,u factorizes throughh; i.e.,u= t · h for somet :Y →A (see [5]).

For a ringR, the categoryR-Mod of R-modules is finitely accessible and puremonomorphisms are the usual ones. Every full subcategoryK of R-Mod closedunder directed colimits and pure submodules is accessible (see [5, 2.36]). A well-known result is that a monomorphismf :A→B in R-Mod is pure if each finitelypresentable module is projective w.r.t. coker(f ). Epimorphisms coker(f ), wheref is pure, are calledpure epimorphisms(cf. [12]).

Lemma 3.1. In R-Mod, pure monomorphisms are stable under pullbacks alongpure epimorphisms.

Proof. Consider pure monomorphismsf :A→ B, g :D→ C and a pullback

E

h

gB

coker(f )

D g C.

(1)

We have to prove thatg is pure. But (2) is a pushout andh is an epimorphism(cf. [13, 1.76]). Therefore

coker(g)= coker(g) · coker(f )

(see [14, 12.3.4]). Henceg is pure (because coker(g) is a pure epimorphism).✷The following result presents the core of the Enochs’ proof of the flat cover

conjecture.


Proposition 3.2. Let K be a full subcategory ofR-Mod closed under directedcolimits and pure submodules. There is a regular cardinalλ such that everyK ∈ K is a union of a smooth chain of pure submodulesKi ∈ K, i < α, suchthat |K0|< λ and|Ki+1/Ki |< λ for all i < α.

Proof. Following [5, 2.33], there is a regular cardinalλ > |R| such that everysubmoduleA⊆ B with A λ-presentable is contained in a pure submoduleA⊆ B

with A alsoλ-presentable. Since|R| < λ, anR-moduleX is λ-presentable iff|X|< λ.

Let K ∈ K and construct a smooth chain of pure submodulesKi ⊆ K asfollows. LetK0 be anyλ-presentable pure submodule ofK. For a limit ordinali,we takeKi = ⋃

j<i Kj . Havingi with Ki �=K, take aλ-presentable submodule0 �=M ⊆K/Ki and a pullback

K K/Ki

Ki+1 M.

The pullback property yields the embeddingKi ⊆Ki+1 and, by Lemma 3.1,Ki+1is pure inK. ClearlyK = ⋃

i<α Ki for someα andKi ∈K for eachi < α. ✷

4. Flat monomorphisms

Definition 4.1. LetF be a full subcategory ofR-Mod. A monomorphismf :A→B will be called anF -monomorphismif coker(f ) :B → F hasF in F . The classof all F -monomorphisms will be denoted byF -Mono.

Some of the following proofs are written in a way showing that they are validin categories more general thanR-Mod.

Lemma 4.2. LetF a full subcategory ofR-Mod containing0 and closed underdirected colimits and extensions. Thencof(F -Mono)=F -Mono.

Proof. (a)F -Mono is stable under pushouts. Consider a pushout

Bg

D

A

f

g C

f

wheref is anF -monomorphism. Since cokerf · g = coker(f ) (see [14, 12.3.4]),f is anF -monomorphism.


(b)F -Mono is closed under transfinite compositions.(b1) Let f :A → B and g :B → C be F -monomorphisms. Consider the

following diagram:

Af

B

coker(f )

gC

coker(gf )coker(g)

B uC v C

where u and v are the induced homomorphisms. Following [14, 12.3.4], themiddle trapezoid is a pushout and thusu is a monomorphism. Hencev =coker(u). SinceF is closed under extensions andB andC are inF , C is in F .Thereforeg · f is anF -monomorphism.

(b2) Let (fij :Ai → Aj)i�j�λ be a smooth chain in whichfij , i � j < λ areF -monomorphisms. Consider

Ai

coker(f0i)

fijAj

coker(f0j )

Fi tijFj

wheretij is given by coker(f0j ) · fij · f0i = coker(f0j ) · f0j = 0. SinceFi is inF for i < λ, Fλ = colimFi andF is closed under directed colimits, we get thatf0λ is anF -monomorphism.

(c) F -Mono is closed under retracts. LetF :A→ B be anF -monomorphismandg :A→ C a retract off in A ↓ R-Mod. Consider

Af

g

Bcoker(f )

r

Br

C

s

coker(g)Cs

wherer ands makeg a retract off and r , s are the induced homomorphisms.They makeC a retract ofB and sinceF is closed under retracts inR-Mod (see[5, 2.4]), we have thatC is in F ; i.e.,g is anF -monomorphisms.

(d) F -Mono contains all isomorphisms. Iff :A→ B is an isomorphism thencoker(f ) :B → 0. ✷Remark 4.3. We have

(F − Mono)� = {C | Ext(F,C)= 0 for all F ∈ F

}.


Indeed, Ext(F,C) = 0 for all F ∈ F is clearly equivalent to the fact that everyF -monomorphismC → B splits. This property has everyC ∈ (F -Mono)�.Conversely, consider anR-moduleC having this property and form a pushout

Af

u

B

u

Cf

D

wheref is anF -monomorphism andu a homomorphism. Sincef is anF -mono-morphism by Lemma 4.2,f splits and thusu factorizes throughf . HenceC ∈ (F -Mono)�.

Lemma 4.4. Let F be a full subcategory ofR-Mod containing all freeR-mod-ules. Theng :C → D belongs to(F -Mono)� iff it is an epimorphism andker(g) :E→C hasE in (F -Mono)�.

Proof. I. Let g :C →D be in(F -Mono)� and consider

Eker(g)

Cg

D .

ThenE ∈ (F -Mono)�.To verify it, consider anF -monomorphismf :A→ B andu :A→ E. In the

diagram

A

f

uE

ker(g)

C

g

B

v

t

0 D

we get a diagonalt with g · t = 0 andt ·f = ker(g) ·u (becauseg ∈ (F -Mono)�)andv with ker(g) · v = t (becauseg · t = 0). Hence

ker(g) · v · f = t · f = ker(g) · uand thusv · f = u.

Considerd ∈ D and p :R → D with p(1) = d . Since 0→ R belongs toF -Mono, there is a diagonalt in the square

0 C

g

R p D.


Henced = g(t (1)) and thusg is an epimorphism.II. At first, let g :C → D be an epimorphism such thatD is in F and, in

ker(g) :E→ C, E is in (F -Mono)�. We will prove thatg ∈ (F -Mono)�.Consider anF -monomorphismf :A→B and a commutative square

A

f

uC

g

B v D.

Sinceg = coker(ker(g)), ker(g) is anF -monomorphism and thus ker(g) splits(becauseE ∈ (F -Mono)�). Henceg splits;g · p = idC . Consider the morphismpvf − u :A→ C. We have

g(pvf − u)= gpvf − gu= vf − gu= 0.

Thus there iss :A→ E with ker(g)s = pvf − u. SinceE ∈ (F -Mono)�, thereis t :B → E with t · f = s. Thenpv − ker(g)t :B → C is a required diagonal.Indeed,

g(pv − ker(g)t

) = v − 0 = v

and(pv − ker(g)t

)f = pvf − ker(g)s = pvf − (pvf − u)= u.

Now, letg :C →D be an arbitrary epimorphism such that ker(g) :E→ C hasE ∈F -Mono�. Consider a commutative square

A

f

uC

g

B v D

wheref is anF -monomorphism.We will complete it to the commutative diagram

A

f

uC

g

A

p

f

u C

q

g

B∗

p

t

v∗D∗

q

B v D


wherep :B∗ → B and q :D∗ → D are epimorphism withB∗ andD∗ free,q · v∗ = v · p, the trapezoids on the left and on the right side are pullbacks andu

is the induced homomorphism. Since

Eker(g) ker(g)

C qC

commutes,g is an epimorphism andD∗ ∈ F , we know thatg ∈ (F -Mono)�.Since the trapezoid on the left side is a pushout as well, we get thatf ∈F -Mono.It yields a diagonalt :B∗ → C in the square in the middle. Since

q · t · f = u · pthere is a uniques :B →C such that

s · p = q · t and s · f = u

(using the pushout property atB). Since

g · s · p= g · q · t = q · g · t = q · v∗ = v · p,we have

g · s = v.

Hences is the needed diagonal.✷Theorem 4.5. Let F be a full subcategory ofR-Mod containing all freemodules and closed under directed colimits, pure submodules, pure quotientsand extensions. Then(F -Mono, F -Mono�) is a cofibrantly generated weakfactorization system.

Proof. Following Proposition 3.2, there is a regular cardinalλ such that everyF ∈ F is a union of a smooth chain of pure submodulesFi ∈ F , i < α, such that|F0|< λ and|Fi+1/Fi |< λ for all i < α. By the standard result (cf. [15, XII.1.14]or [4]), the following statements are equivalent for eachR-moduleC:

(a) Ext(F,C)= 0 for all F ∈F ;(b) Ext(F,C)= 0 for all F ∈F with |F |< λ.

Let M = {ker(pF ) | F ∈ F , |F |< λ} ∪ {0→ R} where

0 Aker(pF )

F ∗ pFF 0 (∗)

presentsF as a quotient of a freeR-moduleF ∗. SinceM ⊆ F -Mono, we have(F -Mono)� ⊆M�. Assume thatC ∈M� and consider the long exact sequence


0 → Hom(F,C)→ Hom(F ∗,C)→ Hom(A,C)→ Ext(F,C)

→ Ext(F ∗,C)→ ·· ·induced by (∗). Since Ext(F ∗,C) = 0 and Hom(F ∗,C) → Hom(A,C) issurjective, we get that Ext(F,C) = 0. HenceC satisfies (b) and, followingRemark 4.3, we get thatC ∈ (F -Mono)�. Hence

(F -Mono)� =M�.

Following Lemma 4.4,

(F -Mono)� =M�.

In fact,F -Mono� ⊆ M� and, following the proof of Lemma 4.4, anyg ∈ M�is an epimorphism with ker(g) :E→C havingE in F -Mono�.

We have

F -Mono⊆ �(M�) = cof(M)⊆F -Mono

by Lemma 4.2. SinceM is a set, (F -Mono,F -Mono�) is a cofibrantly generatedweak factorization system.✷

We will denote by Flat the class of all flat monomorphisms and by Cot theclass of all epimorphismsg :C →D having, in ker(g) :E → C, E ∈ Flat� (i.e.,C cotorsion).

Corollary 4.6. LetR be a ring. Then(Flat,Cot) is a weak factorization system inR-Mod.

Proof. The full subcategory ofR-Mod consisting of flatR-modules is closedunder directed colimits, extensions and contains all freeR-modules. Sinceflat R-modules are precisely directed colimits of finitely presentable projectiveR-modules, any pure submodule and any pure quotients of a flatR-module areflat. Hence the result follows from Theorem 4.5 and Lemma 4.4.✷Remark 4.7. We have shown that Theorem 2.5 can replace [1, Theorem 10] inthe proof of the flat cover conjecture.

5. Flat precovers in varieties

Under avariety we mean an equationally defined class of finitary universalalgebras (cf. [5]).R-modules form a variety for any ringR. Even the commacategoryA ↓ R-Mod is a variety for anyR-module A. It suffices to takeelements ofA as new constant symbols and equalities true inA as new equations.A ↓R-Mod is no longer an additive category.


In any varietyV we can defineflat algebrasas directed colimits of (regular)projective algebras. Then, inR-Mod, we get the usual flat modules.

Lemma 5.1. LetA be anR-module. Thenf :A→X is flat inA ↓ R-Mod iff itis a flat monomorphism inR-Mod.

Proof. Since the projectionP :A ↓ R-Mod → R-Mod preserves colimits, itpreserves regular epimorphisms. Conversely, consider

A

x y

Xf

Y

wheref :X→ Y is an epimorphism inR-Mod. Hencef is a coequalizer of itskernel pair

Zu

vX

in R-Mod. It is easy to see thatf is a coequalizer of the pair

Ai1 x

A+Z〈x,u〉〈x,v〉 X

in A ↓ R-Mod. Hencef is a regular epimorphism inA ↓ R-Mod iff P(f ) is anepimorphism inR-Mod.

Since (monomorphisms with a projective kernel, epimorphisms) is a weak fac-torization system inR-Mod (see [16]),p :A→ X is projective inA ↓R-Modiff it is a monomorphism and coker(p) :X → P has P projective (cf. Re-mark 2.3(2)). Consequently,p :A→ Y is flat inA ↓R-Mod iff it is a monomor-phism and coker(p) :Y →Q hasQ flat. ✷Corollary 5.2. Let R be a ring andA an R-module. Then any algebra inA ↓R-Mod has a flat precover.

Proof immediately follows from Definition 4.1 and Lemma 5.1.The author does not know any example of a variety in which flat precovers do

not exist.

References

[1] E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981) 189–209.[2] J. Xu, Flat Covers of Modules, Lecture Notes in Math., Vol. 1634, Springer-Verlag, Berlin, 1996.


[3] L. Bican, R. El Bashir, E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33(2001), to appear.

[4] P.C. Eklof, A.H. Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001) 41–51.[5] J. Adámek, J. Rosický, Locally Presentable and Accessible Categories, Cambridge Univ. Press,

1994.[6] A.K. Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9

(1976/7) 207–220.[7] D. Quillen, Homotopical Algebra, Lecture Notes in Math., Vol. 43, Springer-Verlag, 1967.[8] P. Hirschhorn, Localizations of model categories, preprint, 1998, http://www.math.mit.edu/psh.[9] M. Hovey, Model Categories, Amer. Math. Society, Providence, 1998.

[10] J. Adámek, H. Herrlich, J. Rosický, W. Tholen, On a generalized small-object argument for theinjective subcategory problem, submitted.

[11] T. Beke, Sheafiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000)447–475.

[12] P. Rothmaler, Purity in model theory, in: M. Droste, R. Göbel (Eds.), Advances in Algebra andModel Theory, Gordon and Breach, 1997, pp. 445–469.

[13] F. Borceux, in: in: Handbook of Categorical Algebra, Vol. 2, Cambridge Univ. Press, 1994.[14] H. Schubert, Kategorien, Akademie-Verlag, Berlin, 1970.[15] P.C. Eklof, A.H. Mekler, Almost Free Modules: Set-theoretic Methods, North-Holland, 1990.[16] J. Adámek, H. Herrlich, J. Rosický, W. Tholen, Weak factorization systems and topological

functors, Appl. Cat. Struct., to appear.

flat covers and factorizations

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