notes on several orthogonal classes of flat and fp-injective functors

ISSN 0001-4346, Mathematical Notes, 2014, Vol. 95, No. 1, pp. 78–90. © Pleiades Publishing, Ltd., 2014.Published in Russian in Matematicheskie Zametki, 2014, Vol. 95, No. 1, pp. 93–108.

Notes on Several Orthogonal Classesof Flat and FP-injective Functors*

Lixin Mao**

Institute of Mathematics, Nanjing Institute of Technology, ChinaReceived June 9, 2011; in final form, November 3, 2011

Abstract—In this paper, we develop relative homological algebra in the category of functorsfrom finitely presented modules to Abelian groups. More specifically, we introduce the conceptsof F-injective, F-projective and F-flat functors. Such functors appear when we study coversand envelopes of functors. The relationships among these functors are investigated and someapplications are given.

DOI: 10.1134/S0001434614010088

Keywords: F-injective functor, F-projective functor, F-flat functor, FP -injective functor, flatfunctor, (pre)envelope, (pre)cover.

1. INTRODUCTION

Throughout this paper, R is an associative ring with identity and all modules are unitary. Thecharacter module HomZ(M, Q/Z) of an R-module M is denoted by M+. Given R-modules M and N ,we use the abbreviation (M,N) := HomR(M,N). We denote by mod-R and R-mod the category offinitely presented right (resp. left) R-modules, and by Ab the category of Abelian groups. The categoryof contravariant additive functors (mod-R)op → Ab is denoted by ((mod-R)op,Ab), and the categoryof covariant functors mod-R → Ab, by (mod-R,Ab). If F and G are objects of ((R-mod)op,Ab) or(mod-R,Ab), then the set of morphisms [F,G] consists of the natural transformations from F to G.

It is well known that the two functor categories (mod-R,Ab) and ((mod-R)op,Ab) are both locallyfinitely presented Grothendieck categories. They have received extensive attention since the 1960s.In particular, they play important roles in the investigation of the model theory of modules and therepresentation theory of Artinian algebras (see, e.g., [1]–[12]). By [12, Corollary 7.4, p. 97] therule M → ( · ,M) constitutes a full and faithful left exact functor from the category Mod-R of rightR-modules to ((mod-R)op,Ab). Similarly, there is a full and faithful right exact functor from thecategory R-Mod of left R-modules to (mod-R,Ab), which is given by the rule M → · ⊗ M .

Although not every R-module has a flat envelope in general [13], it is true that every functor in((mod-R)op,Ab) has a flat envelope with the unique mapping property [11]. In this paper, we obtaina dual result, i.e., every functor in (mod-R,Ab) has an FP-injective cover with the unique mappingproperty (Theorem 2.3). So it is natural to study the cokernel of flat envelopes in ((mod-R)op,Ab) andthe kernel of FP-injective covers in (mod-R,Ab). To this aim, we introduce the concepts of F-injective,F-projective, and F-flat functors, which are Ext-orthogonal or Tor-orthogonal classes of FP-injectiveor flat functors. These functors are closely related to the FP-injective (pre)covers in (mod-R,Ab) orthe flat (pre)envelopes in ((mod-R)op,Ab). For example, let F be a functor in (mod-R,Ab), then Fis F-injective if and only if F is a kernel of an FP-injective precover E → L with E injective; F isF-injective and [G,F ] = 0 for any FP-injective functor G in (mod-R,Ab) if and only if F is a kernel of anFP-injective cover with the unique mapping property (Proposition 3.3). Using these results, we obtain adecomposition theorem of F-injective functors (Theorem 3.4). There are dual properties for F-projective

∗The text was submitted by the author in English.**E-mail: [email protected]

78

NOTES ON SEVERAL ORTHOGONAL CLASSES OF FUNCTORS 79

functors. We also clarify the relationships between F-injective, F-projective, F-flat and other functors.Finally, we characterize some important rings in terms of F-injective, F-projective, and F-flat functors.For example, we prove that R is a von Neumann regular ring if and only if every F-injective functor in(mod-R,Ab) is FP-injective if and only if every F-flat functor in ((mod-R)op,Ab) is flat if and onlyif every F-projective functor in ((mod-R)op,Ab) is projective (Theorem 5.1); R is a left hereditary vonNeumann regular ring if and only if every quotient functor of an F-injective functor in (mod-R,Ab) isF-injective if and only if every subfunctor of an F-projective functor in ((R-mod)op,Ab) is F-projective(Theorem 5.2).

Next we recall some well-known notions and facts needed in the sequel.Let A be any category and B a class of objects in A. Following [14], [15], we say that a morphism

φ : B → A in A is a B-precover of A if B ∈ B and, for any morphism f : B′ → A with B′ ∈ B thereis a morphism g : B′ → B such that φg = f . A B-precover φ : B → A is called a B-cover of A if everyendomorphism g : B → B such that φg = φ, is an isomorphism. AB-cover φ : B → A is said to have theunique mapping property [16] if for any morphism f : B′ → A with B′ ∈ B, there is a unique morphismg : B′ → B such that φg = f . Dually, we have the definitions of a B-preenvelope and a B-envelope (withthe unique mapping property). The B-covers (B-envelopes) may not exist in general, but if they exist,they are unique up to isomorphism.

A sequence of functors Fμ−→ G

ν−→ H in (C,Ab) with C a skeletally small additive category is calledexact if for every A ∈ C the corresponding sequence of Abelian groups

F (A)μA−−→ G(A) νA−→ H(A)

is exact.A functor X of (C,Ab) with C a skeletally small additive category is called finitely presented if [X,−]

commutes with direct limits, equivalently, if there is an exact sequence

(A, · ) → (B, · ) → X → 0

with A and B in C.A functor F of (C,Ab) with C a skeletally small additive category is said to be FP -injective if

Ext1[X,F ] = 0 for every finitely presented functor X ∈ (C,Ab). By [5, Lemma 1.4], a functor of(mod-R,Ab) is FP-injective if and only if F is isomorphic to · ⊗ M of (mod-R,Ab) for some leftR-module M . Moreover, Gruson and Jensen [7, Sec. 1.2] characterized the injective objects of(mod-R,Ab) as the functors isomorphic to some · ⊗ M , a pure-injective left R-module, where Mis called pure-injective [13, p. 112] if for every pure exact sequence 0 → A → B → C → 0 of leftR-modules, the sequence

0 → Hom(C,M) → Hom(B,M) → Hom(A,M) → 0

is exact.A functor in ((mod-R)op,Ab) is called representable if it is isomorphic to a functor of the form

( · , N), where N is a finitely presented right R-module. Yoneda’s Lemma implies that a functor in((mod-R)op,Ab) is representable if and only if it is a finitely generated projective object of ((mod-R)op,Ab). Thus, a functor in ((mod-R)op,Ab) is projective if and only if it is isomorphic to a direct summandof a direct sum of representable functors, i.e., it is isomorphic to a functor of the form ( · , N) with N apure-projective right R-module, where N is called pure-projective [13, p. 178] if for every pure exactsequence 0 → A → B → C → 0 of right R-modules, the sequence

0 → Hom(N,A) → Hom(N,B) → Hom(N,C) → 0

is exact.Given

F ∈ ((mod-R)op,Ab) and G ∈ (mod-R,Ab),

there is a tensor product F ⊗ G ∈ Ab defined in such a way that the functor

F ⊗ · : (mod-R,Ab) → Ab

MATHEMATICAL NOTES Vol. 95 No. 1 2014

80 LIXIN MAO

becomes the left adjoint of the functor

(F ( · ), · ) : Ab → (mod-R,Ab),

where (F ( · ), · )(C)(D) = (F (D), C) for C ∈ Ab and D ∈ mod-R. Correspondingly, we can also define· ⊗ G : (mod-R)op → Ab as the left adjoint of the functor

(G( · ), · ) :Ab → ((mod-R)op,Ab).

The tensor product of functors has the usual properties, for example,

F ⊗ (X, · ) ∼= F (X) and ( · ,X) ⊗ G ∼= G(X)

for X ∈ mod-R. In the standard way, we may define the left derived functors Torn[F, · ] and Torn[ · , G].A functor F ∈ ((mod-R)op,Ab) is said to be flat if F ⊗ · is an exact functor, equivalently, if

Tor1[F,G] = 0 for any (finitely presented) functor G ∈ (mod-R,Ab). Similarly, we can define a flatfunctor in (mod-R,Ab). By [6, Theorem 1.4], a functor ((mod-R)op,Ab) is flat if and only if F isisomorphic to ( · ,M) for some right R-module M .

For unexplained concepts and notation, we refer the reader to [12], [13], [15], [17], [18].

2. FLATNESS AND FP-INJECTIVITY OF FUNCTORS

We begin with the following lemma. Recall that R is a right coherent ring if every finitely generatedright ideal is finitely presented.

Lemma 2.1. The following assertions for a ring R and n ≥ 0 hold:

(1) (Torn[A,B], C) ∼= Extn[A, (B( · ), C)] ∼= Extn[B, (A( · ), C)] for any A ∈ ((mod-R)op,Ab),B ∈ (mod-R,Ab), and any injective Abelian group C;

(2) Torn[(B( · ), C), A] ∼= (Extn[A,B], C) for any finitely presented functor A ∈ (mod-R,Ab),B ∈ (mod-R,Ab), and any injective Abelian group C;

(3) if R is a right coherent ring, then Torn[A, (B( · ), C)] ∼= (Extn[A,B], C) for any finitelypresented functor A ∈ ((mod-R)op,Ab), B ∈ ((mod-R)op,Ab), and any injective Abeliangroup C.

Proof. (1) First of all, for any A ∈ ((mod-R)op,Ab), B ∈ (mod-R,Ab), and any injective Abeliangroup C, by definition, we have the isomorphisms

(A ⊗ B,C) ∼= [A, (B( · ), C)] ∼= [B, (A( · ), C)].

Now let A. : · · · → P1 → P0 → 0 be a deleted projective resolution of A. Applying · ⊗ B, we obtainthe complex

A. ⊗ B : · · · → P1 ⊗ B → P0 ⊗ B → 0.

By [18, Exercise 6.4, p. 170], since ( · , C) is an exact functor, we have

(Torn[A,B], C) = (Hn(A. ⊗ B), C) ∼= Hn((A. ⊗ B), C)∼= Hn([A.(B( · ), C)]) = Extn[A, (B( · ), C)].

Similarly, we can show that (Torn[A,B], C) ∼= Extn[B, (A( · ), C)], by choosing the projective resolu-tion of B.

(2) Let A ∈ (mod-R,Ab) be finitely presented, i.e., there is an exact sequence

(X, · ) → (Y, · ) → A → 0

in (mod-R,Ab) with X and Y finitely presented right R-modules. For any B ∈ (mod-R,Ab) oneobtains the exact sequence

0 → [A,B] → [(Y, · ), B] → [(X, · ), B].



By Yoneda’s Lemma, we obtain the exact sequence

0 → [A,B] → B(Y ) → B(X).

So for any injective Abelian group C, we have the exact sequence

(B(X), C) → (B(Y ), C) → ([A,B], C) → 0.

Thus, we obtain the following commutative diagram with exact rows:

(B( · ), C) ⊗ (X, · ) ��

∼=��

(B( · ), C) ⊗ (Y, · )∼=

��

�� (B( · ), C) ⊗ A

��

�� 0

(B(X), C) �� (B(Y ), C) �� ([A,B], C) �� 0.

By the Five Lemma, (B( · ), C) ⊗ A ∼= ([A,B], C).By [5, Sec. 1.2], there exists a deleted projective resolution of A:

A. : · · · → P1 → P0 → 0

with each Pi finitely generated. Applying (B( · ), C) ⊗ · , we obtain the complex

(B( · ), C) ⊗ A. : · · · → (B( · ), C) ⊗ P1 → (B( · ), C) ⊗ P0 → 0.

Since ( · , C) is an exact functor, we have

Torn[(B( · ), C), A] = Hn((B( · ), C) ⊗ A.) ∼= Hn(([A., B], C))∼= (Hn([A., B]), C) = (Extn[A,B], C).

(3) Since A is finitely presented, there is an exact sequence

( · ,X) → ( · , Y ) → A → 0

in ((mod-R)op,Ab) with X and Y finitely presented right R-modules.For any B ∈ ((mod-R)op,Ab) and injective Abelian group C, we have the following commutative

diagram with exact rows:

( · ,X) ⊗ (B( · ), C) ��

∼=��

( · , Y ) ⊗ (B( · ), C)

∼=��

�� A ⊗ (B( · ), C)

��

�� 0

(B(X), C) �� (B(Y ), C) �� ([A,B], C) �� 0.

So by the Five Lemma, A ⊗ (B( · ), C) ∼= ([A,B], C).Since R is a right coherent ring, the subcategory of finitely presented functors in ((mod-R)op,Ab) is

an Abelian category (see [1]). Thus, A admits a deleted projective resolution: A. : · · · → P1 → P0 → 0with each Pi finitely generated. Applying · ⊗ (B( · ), C), we obtain the complex

A. ⊗ (B( · ), C) : · · · → P1 ⊗ (B( · ), C) → P0 ⊗ (B( · ), C) → 0.

So the exactness of the functor ( · , C) implies

Torn[A, (B( · ), C)] = Hn(A. ⊗ (B( · ), C)) ∼= Hn(([A., B], C))∼= (Hn([A., B]), C) = (Extn[A,B], C).

This completes the proof.

Take an additive functor F from an additive category C to Ab. We denote by F+ the additive functorfrom Cop to Ab defined by F+(M) = (F (M))+ for any M ∈ C.

Let C = Q/Z in Lemma 2.1. We have the following

Corollary 2.2. The following assertions for a ring R hold:


82 LIXIN MAO

(1) A functor F ∈ ((mod-R)op,Ab) (resp., (mod-R,Ab)) is flat if and only if F+ is FP -injectivein (mod-R,Ab) (resp., ((mod-R)op,Ab));

(2) A functor G ∈ (mod-R,Ab) is FP -injective if and only if G+ is flat in ((mod-R)op,Ab);

(3) If R is a right coherent ring, then a functor G ∈ ((mod-R)op,Ab) is FP -injective if and onlyif G+ is flat in (mod-R, Ab).

It is known that every functor in ((mod-R)op,Ab) has a flat envelope with the unique mappingproperty (see [11, Proposition 2.4]). The next theorem may be viewed as a dual of the above result.

Theorem 2.3. Every functor in (mod-R,Ab) has an FP -injective cover with the unique mappingproperty.

Proof. Let F ∈ (mod-R,Ab). We define ϕ : · ⊗ F (RR) → F by

ϕM (x ⊗ t) = F (ξM (x))(t) ∈ F (M)

for any M ∈ mod-R, x ∈ M and t ∈ F (RR), where ξM : M → (R,M) is the standard natural isomor-phism. Let A ∈ R-Mod and δA : A → R ⊗ A be the standard natural isomorphism. Then the map

[ · ⊗ A, · ⊗ F (RR)]ϕ∗−→ [ · ⊗ A,F ]

has an inverse given by τ → · ⊗ τRδA for any τ ∈ [ · ⊗A,F ]. Thus, ϕ is an FP-injective cover with theunique mapping property.

3. F-INJECTIVE, F-PROJECTIVE, AND F-FLAT FUNCTORS

In order to describe the structure of FP-injective covers in (mod-R,Ab) or flat envelopes in((mod-R)op,Ab), we introduce several orthogonal functors of FP-injective or flat functors as follows.

Definition 3.1. Given a ring R, a functor F ∈ (mod-R,Ab) is said to be F-injective if Ext1[H,F ] = 0for any FP-injective functor H ∈ (mod-R,Ab).

A functor Q ∈ ((mod-R)op,Ab) is said to be F-projective if Ext1[Q,T ] = 0 for any flat functorT ∈ ((mod-R)op,Ab).

A functor G ∈ ((mod-R)op,Ab) is said to be F-flat if Tor1[G,H] = 0 for any FP-injective functorH ∈ (mod-R,Ab).

Remark 3.2. Clearly, the class of F-injective functors of (mod-R,Ab) is closed under direct products,direct summands and extensions. The class of F-projective functors of ((mod-R)op,Ab) is closed underdirect sums, direct summands and extensions. The class of F-flat functors of ((mod-R)op,Ab) is closedunder direct limits, direct summands and extensions.

Now we deal with properties of F-injective, F-projective, and F-flat functors. As we will see, thesefunctors are closely related to the kernel of an FP-injective (pre)cover in (mod-R,Ab) or the cokernelof a flat (pre)envelope in ((mod-R)op,Ab).

Statement 3.3. The following assertions hold for a functor F ∈ (mod-R,Ab):

(1) F is F-injective if and only if F is a kernel of an FP -injective precover E → L in(mod-R,Ab) with E injective;

(2) F is F-injective satisfying [G,F ] = 0 for any FP -injective functor G ∈ (mod-R,Ab) ifand only if F is a kernel of an FP -injective cover with the unique mapping property in(mod-R,Ab).



Proof. (1) If F is F-injective, then there exists an exact sequence 0 → F → E → L → 0 with Einjective. For any FP-injective functor G, the above sequence induces the exact sequence

[G,E] → [G,L] → Ext1[G,F ] = 0.

So E → L is an FP-injective precover in (mod-R,Ab).Conversely, if F is a kernel of an FP-injective precover f : E → L with E injective, then we have the

exact sequence 0 → F → E → E/F → 0.For any FP-injective functor G in (mod-R,Ab) we obtain the exact sequence

[G,E] → [G,E/F ] → Ext1[G,F ] → 0.

Since [G,E] → [G,L] → 0 is exact, it is easy to check that the sequence [G,E] → [G,E/F ] → 0 is alsoexact. Thus, Ext1[G,F ] = 0 and F is F-injective.

(2) If F is F-injective and [G,F ] = 0 for any FP-injective functor G ∈ (mod-R,Ab), then thereexists an exact sequence 0 → F → E → L → 0 with E injective.

For any FP-injective functor G, we obtain the induced exact sequence

0 = [G,F ] → [G,E] → [G,L] → Ext1[G,F ] = 0.

So [G,E] ∼= [G,L] which implies that E → L is an FP-injective cover with the unique mappingproperty in (mod-R,Ab).

Conversely, suppose that F is a kernel of an FP-injective cover f : E → L with the unique mappingproperty in (mod-R,Ab). For any FP-injective functor G, the exact sequence 0 → F → E → L yieldsthe exact sequence

0 → [G,F ] → [G,E] → [G,L].

So [G,F ] = 0 since [G,E] → [G,L] is monic. In addition, F is F-injective by [15, Proposition 1.2.2].

Theorem 3.4. A functor F ∈ (mod-R,Ab) is F-injective if and only if F = X ⊕ H , where X isan injective functor, H is an F-injective functor with [G,H] = 0 for any FP -injective functorG ∈ (mod-R,Ab).

Proof. “⇐” is clear.

“⇒” Let F be an F-injective functor. As in the proof of Proposition 3.3 (1), there exists an exactsequence 0 → F → E → L → 0, where E is injective and E → L is an FP-injective precover. LetQ → L be an FP-injective cover with the unique mapping property in (mod-R,Ab). Then Q → L isepic. So we have the following commutative diagram with exact rows:

0 �� K ��

φ

��

Q

γ

��

�� L �� 0

0 �� F

σ

��

�� E

β��

�� L �� 0

0 �� K �� Q �� L �� 0.

Note that K is K F-injective and [G,K] = 0 for any FP-injective functor G by Proposition 3.3 (2).Since βγ = 1, E = ker(β) ⊕ im(γ). Thus, ker(β) is injective. By the Snake Lemma, ker(σ) ∼= ker(β)is injective. It is clear that σφ = 1 by the above diagram. So F = ker(σ) ⊕ H with H ∼= K, as desired.

Statement 3.5. If R is a right coherent ring and Q is a finitely presented functor in the category((mod-R)op,Ab) such that Ext1[Q,T ] = 0 for any representable functor T ∈ ((mod-R)op,Ab),then Q is F-projective.


84 LIXIN MAO

Proof. There is an exact sequence

( · , A)( · ,f)−−−→ ( · , B) → Q → 0

in ((mod-R)op,Ab) with A and B finitely presented right R-modules. Since R is right coherent,K = ker(f) is finitely presented. Thus, we obtain the projective resolution of Q

0 → ( · ,K) → ( · , A)( · ,f)−−−→ ( · , B) → Q → 0.

For any flat functor G ∈ ((mod-R)op,Ab), G = lim−→Ti with Ti representable by [6, Theorem 1.3].Consider the complex

0 → [( · , B), lim−→Ti] → [( · , A), lim−→Ti] → [( · ,K), lim−→Ti] → 0,

which is isomorphic to the complex

0 → lim−→[( · , B), Ti] → lim−→[( · , A), Ti] → lim−→[( · ,K), Ti] → 0.

Since lim−→ commutes with homology, we have

Ext1[Q,G] = Ext1[Q, lim−→Ti] ∼= lim−→Ext1[Q,Ti] = 0.

So Q is F-projective.

The following result can be proved in a way dual to the proof of Proposition 3.3 and Theorem 3.4.

Statement 3.6. The following assertions hold for a functor Q in ((mod-R)op,Ab):

(1) Q is F-projective if and only if Q is a cokernel of a flat preenvelope A → B in the category((mod-R)op,Ab) with B projective;

(2) Q is F-projective and [Q,G] = 0 for any flat functor G ∈ ((mod-R)op,Ab) if and only if Q isa cokernel of a flat envelope with the unique mapping property in ((mod-R)op,Ab);

(3) Q is F-projective if and only if Q = U ⊕ V , where U is a projective functor, and V is anF-projective functor with [V,G] = 0 for any flat functor G in ((mod-R)op,Ab).

Statement 3.7. The following are equivalent for a functor A ∈ ((mod-R)op,Ab):

(1) A is F-flat;

(2) Tor1[A,H] = 0 for any injective functor H ∈ (mod-R,Ab);

(3) A+ ∈ (mod-R,Ab) is F-injective.

Proof. (1) ⇒ (2) is trivial.(2) ⇒ (1) Let · ⊗ M be an FP-injective functor in (mod-R,Ab). By [13, Proposition 5.3.9], there

exists a pure exact sequence 0 → M → N → L → 0 of left R-modules with N pure-injective, whichinduces the exact sequence in (mod-R,Ab)

0 → · ⊗ M → · ⊗ N → · ⊗ L → 0.Thus, we obtain the split exact sequence

0 → ( · ⊗ L)+ → ( · ⊗ N)+ → ( · ⊗ M)+ → 0.

By (2) and Lemma 2.1 (1), we have

Ext1[A, ( · ⊗ N)+] ∼= (Tor1[A, · ⊗ N ])+ = 0.

Thus, Ext1[A, ( · ⊗ M)+] = 0, and so

(Tor1[A, · ⊗ M ])+ ∼= Ext1[A, ( · ⊗ M)+] = 0.

Hence Tor1[A, · ⊗ M ] = 0, i.e., A is F-flat.(1) ⇔ (3) is obvious if, in Lemma 2.1 (1), we take B to be FP-injective, C = Q/Z, and n = 1.



Theorem 3.8. The following assertions hold for a ring R:

(1) The cokernel of a flat preenvelope in ((mod-R)op,Ab) is F-flat;

(2) A finitely presented F-flat functor in ((mod-R)op,Ab) is the cokernel of a flat preenvelopeQ → P with P projective.

Proof. (1) Let f : K → F be a flat preenvelope in ((mod-R)op,Ab) with L = coker(f). There is anexact sequence

0 → im(f) i−→ F → L → 0.

It is clear that i : im(f) → F is a flat preenvelope.

For any FP-injective functor G ∈ (mod-R,Ab), G+ is flat in ((mod-R)op,Ab) by Corollary 2.2 (2).Thus, we obtain the exact sequence

[F,G+] → [im(f), G+] → 0,

which yields the exactness of

(F ⊗ G)+ → (im(f) ⊗ G)+ → 0.

So the sequence 0 → im(f) ⊗ G → F ⊗ G is exact. Thus, the exactness of

0 → Tor1[L,G] → im(f) ⊗ G → F ⊗ G

implies that Tor1[L,G] = 0, i.e., L is F-flat.

(2) Let F be a finitely presented F-flat functor in ((mod-R)op,Ab), i.e., there is an exact sequence( · , A) → ( · , B) → F → 0 in ((mod-R)op,Ab) with A and B finitely presented right R-modules. So weobtain two exact sequences ( · , A) → K → 0 and 0 → K → ( · , B) → F → 0.

For any flat functor G in ((mod-R)op,Ab), we have the following commutative diagram:

( · , A) ⊗ G+

��

�� K ⊗ G+

α

��[( · , A), G]+

γ �� [K,G]+.

By Yoneda’s Lemma, we have

( · , A) ⊗ G+ ∼= G+(A) = G(A)+ ∼= [( · , A), G]+.

Note that γ is epic, and so α is epic.Since G+ is FP-injective by Corollary 2.2 (1), Tor1[F,G+] = 0. Thus, β : K ⊗ G+ → ( · , B) ⊗ G+

is monic. Consider the following commutative diagram:

K ⊗ G+

α

��

β �� ( · , B) ⊗ G+

∼=��

[K,G]+ θ �� [( · , B), G]+.

So α is an isomorphism, and hence θ is monic. Therefore [( · , B), G] → [K,G] is epic. Thus, K →( · , B) is a flat preenvelope.

Corollary 3.9. Any F-projective functor F in ((mod-R)op,Ab) is F-flat. The converse holds if F isfinitely presented.

Proof. It is an immediate consequence of Proposition 3.6 (1) and Theorem 3.8.


86 LIXIN MAO

4. INJECTIVE, PROJECTIVE, AND FLAT FUNCTORS

In this section, we turn our attention to the connection between injective (resp., projective, flat)functors and F-injective (resp., F-projective, F-flat) functors.

The implications: injective ⇒ F-injective, projective ⇒ F-projective, flat ⇒ F-flat are immediate,we characterize the situations where these implications are reversible as follows.

Statement 4.1. The following are equivalent for a functor F in (mod-R,Ab):

(1) F is injective;

(2) F is F-injective and FP -injective;

(3) F is F-injective and is a quotient of an FP -injective functor in (mod-R,Ab);

(4) Exti[H,F ] = 0 for any FP -injective functor H ∈ (mod-R,Ab) and i ≥ 1.

Proof. (1) ⇒ (2) ⇒ (3) and (1) ⇒ (4) are trivial.

(3) ⇒ (1) There is an exact sequence 0 → F → · ⊗ N with · ⊗N injective. By (3) there is an exactsequence · ⊗ M → F → 0. Thus, the composition · ⊗ M → F → · ⊗ N is given by · ⊗ f for somef : M → N . Put L = coker(f). Then we have the following exact commutative diagram:

· ⊗ M

��

��

�f �� · ⊗ N �� · ⊗ L �� 0

F

��

��

��

��

0

��0

Since F is F-injective, the exact sequence 0 → F → · ⊗ N → · ⊗ L → 0 is split, and so F is injective.

(4) ⇒ (1) Let A be any functor in (mod-R,Ab). There is an exact sequence

0 → A → · ⊗ M0 → · ⊗ M1.

Suppose that · ⊗ M0 → · ⊗ M1 is given by · ⊗ f for some f : M0 → M1. Then we obtain the exactsequence in (mod-R,Ab)

0 → A → · ⊗ M0· ⊗f−−−→ · ⊗ M1 → · ⊗ coker(f) → 0.

Hence by (4), we have

Ext1[A,F ] ∼= Ext3[ · ⊗ coker(f), F ] = 0.

Thus, F is injective.

Statement 4.2. The following are equivalent for a functor G in ((mod-R)op,Ab):

(1) G is flat;

(2) G is F-flat and embeds in a flat functor in ((mod-R)op,Ab);

(3) Tori[G,H] = 0 for any FP -injective functor H ∈ (mod-R,Ab) and i ≥ 1.



Proof. (1) ⇒ (2) and (1) ⇒ (3) are trivial.

(2) ⇒ (1) By (2), Proposition 3.7 and Corollary 2.2 (1), G+ is F-injective and is a quotient of anFP-injective functor in (mod-R,Ab). So G+ is injective by Proposition 4.1. Thus, G is flat byCorollary 2.2 (1).

(3) ⇒ (1) By Lemma 2.1 (1), for any FP-injective functor H ∈ (mod-R,Ab) and i ≥ 1 we have

Exti[H,G+] ∼= (Tori[G,H])+ = 0.

Thus, G+ is injective by Proposition 4.1. So G is flat by Corollary 2.2 (1).

Statement 4.3. The following are equivalent for a functor G in ((mod-R)op,Ab):

(1) G is projective;

(2) G is F-projective and flat;

(3) G is F-projective and embeds in a flat functor in ((mod-R)op,Ab);

(4) Exti[G,T ] = 0 for any flat functor T ∈ ((mod-R)op,Ab) and i ≥ 1.


(2) ⇒ (1) There is the exact sequence 0 → F → P → G → 0 with P projective. Note that F is flatsince G is flat. Hence Ext1[G,F ] = 0. Thus, the above sequence is split, and so G is projective.

(2) ⇔ (3) follows from Proposition 4.2 and Corollary 3.9.

(4) ⇒ (1) Let A be any functor in ((mod-R)op,Ab). By [9, p. 356], there is an exact sequence in((mod-R)op,Ab)

0 → F2 → F1 → F0 → A → 0

where each Fi is flat. Hence. Ext1[G,A] ∼= Ext3[G,F2] = 0 by (4). Thus, G is projective.

5. SPECIAL CLASSES OF RINGS

As a consequence of the above results, we give new characterizations of some special rings in termsof F-injective, F-projective, and F-flat functors.

By [4, Proposition 4.1], R will be a von Neumann regular ring if and only if every functor in((mod-R)op,Ab) is flat. Thus, by Corollary 2.2 (1) and (2) R is a von Neumann regular ring if andonly if every functor in (mod-R,Ab) is FP-injective.

Theorem 5.1. The following are equivalent for a ring R:

(1) R is a von Neumann regular ring;

(2) every F-injective functor in (mod-R,Ab) is FP -injective;

(3) every F-projective functor in ((mod-R)op,Ab) is projective;

(4) every F-flat functor in ((mod-R)op,Ab) is flat;

(5) every functor in ((mod-R)op,Ab) is F-flat;

(6) for any exact sequence 0 → A → B → C → 0 in ((mod-R)op,Ab) with B and C F-flat, A isalso F-flat;


88 LIXIN MAO

(7) for any exact sequence 0 → A → B → C → 0 in ((mod-R)op,Ab) where B and C are F-projective, A is also F-projective;

(8) for any exact sequence 0 → A → B → C → 0 in (mod-R,Ab) with A and B F-injective, C isalso F-injective.

Proof. (1) ⇒ (2) Since every functor in (mod-R,Ab) is FP-injective by (1), every F-injective functorin (mod-R,Ab) is injective by Proposition 4.1.

(2) ⇒ (4) Let F be an F-flat functor in ((mod-R)op,Ab). Then F+ is F-injective by Proposition 3.7.So F+ is FP-injective by (2). Thus, F is flat by Corollary 2.2 (1).

(4) ⇒ (3) Since every F-projective functor F in ((mod-R)op,Ab) is F-flat by Corollary 3.9, F is flatby (4). Thus, F is projective by Proposition 4.3.

(3) ⇒ (1) Let F be any functor in ((mod-R)op,Ab). There is an exact sequence 0 → A → G →F → 0 with G flat. Let f : A → H be a flat envelope with the unique mapping property. Then f is amonomorphism. By Proposition 3.6 (2), coker(f) is F-projective, and so it is projective by (3). Thus, Ais flat. By [9, Proposition 6], F embeds in a flat functor. So F has a monic flat envelope g : F → L withthe unique mapping property. Since coker(g) is F-projective, and so is projective, F is flat. Thus, R is avon Neumann regular ring.

(1) ⇒ (5) ⇒ (6) are trivial.

(6) ⇒ (4) Let F be any F-flat functor in ((mod-R)op,Ab). Then there is an exact sequence0 → K → P → F → 0 with P projective. Thus, K is F-flat by (6). For any FP-injective functor Hin (mod-R,Ab), one obtains the exact sequence

0 = Tor2[P,H] → Tor2[F,H] → Tor1[K,H] = 0.

So Tor2[F,H] = 0. By induction, Tori[F,H] = 0 for any i ≥ 3. Thus, F is flat by Proposition 4.2.

(2) ⇒ (8) and (3) ⇒ (7) are clear.

(7) ⇒ (3) Let F be any F-projective functor in ((R-mod)op,Ab). Then there is an exact sequence0 → K → P → F → 0 with P projective. So K is F-projective by (7). For any flat functor T in((R-mod)op,Ab) we obtain the exact sequence

0 = Ext1[K,T ] → Ext2[F, T ] → Ext2[P, T ] = 0.

Hence Ext2[F, T ] = 0. By induction, Exti[F, T ] = 0 for any i ≥ 3. Thus, F is projective by Proposi-tion 4.3.

(8) ⇒ (6) For any exact sequence 0 → A → B → C → 0 in ((mod-R)op,Ab) with B and C F-flat,we have the exact sequence in (mod-R,Ab)

0 → C+ → B+ → A+ → 0.

Since C+ and B+ are F-injective by Proposition 3.7, A+ is F-injective by (8). So A is F-flat.

Theorem 5.2. The following are equivalent for a ring R:

(1) R is a left hereditary von Neumann regular ring;

(2) every quotient functor of an F-injective functor in (mod-R,Ab) is F-injective;

(3) every subfunctor of an F-projective functor in ((R-mod)op,Ab) is F-projective.



Proof. (1) ⇒ (2) Let F → G be an epimorphism in (mod-R,Ab) with F F-injective. Then F ∼= · ⊗Mand G ∼= · ⊗ N for some left R-modules M and N by (1). Note that F is injective by Theorem 5.1.Thus, M is a pure-injective left R-module, and so is injective. Since R is left hereditary and N is aquotient of M , N is injective. Thus, G is (F-)injective.

(2) ⇒ (1) R is a von Neumann regular ring by (2) and Theorem 5.1.

Let M → N be an epimorphism of left R-modules with M injective. Then · ⊗ M → · ⊗ N is anepimorphism in (mod-R,Ab). Hence · ⊗ N is F-injective by (2) and so is injective by Theorem 5.1.Thus, N is pure-injective, and hence is injective. Therefore R is left hereditary.

(1) ⇒ (3) Let A → B be a monomorphism in ((R-mod)op,Ab) wit B F-projective. Since R is vonNeumann regular, we have A ∼= ( · ,M) and B ∼= ( · , N) for some left R-modules M and N . Note that Bis projective by Theorem 5.1. So N is pure-projective. But N is also flat, and hence is projective.Because R is left hereditary and M embeds in N , M is also projective. Thus, A is (F-)projective.

(3) ⇒ (1) R is a von Neumann regular ring by (3) and Theorem 5.1.

Let M → N be a monomorphism of left R-modules with N projective. Then ( · ,M) → ( · , N) is amonomorphism in ((R-mod)op,Ab). Since ( · , N) is projective, ( · ,M) is F-projective by (3), and sois projective by Theorem 5.1. Therefore M is pure-projective, and hence is projective. Thus, R is lefthereditary.

Corollary 5.3. The following are equivalent for a ring R:

(1) R is an Artinian semisimple ring;

(2) every functor in (mod-R,Ab) is F-injective;

(3) every functor in ((mod-R)op,Ab) is F-projective.


(2) ⇒ (1) By Theorem 5.1, R is a von Neumann regular ring, and so every functor in (mod-R,Ab) isinjective. Thus, R is an Artinian semisimple ring.

(3) ⇒ (1) By Theorem 5.1, R is a von Neumann regular ring, and hence every functor in ((mod-R)op,Ab) is projective. So R is Artinian semisimple.

Remark 5.4. Although every F-projective functor F in ((mod-R)op,Ab) is F-flat, the converse is nottrue in general. In fact, let R be a von Neumann regular ring which is not Artinian semisimple. Thenevery functor in ((mod-R)op,Ab) is F-flat, but not every functor in ((mod-R)op,Ab) is F-projective byCorollary 5.3.

ACKNOWLEDGMENTS

The author wishes to thank the referee for the very helpful comments and suggestions.

This work was supported by NSFC (grants nos. 11071111 and 11171149), NSF of Jiangsu Provinceof China (grant no. BK2011068), the Jiangsu 333 Project, and the Jiangsu Six Major Talents PeakProject.


90 LIXIN MAO

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notes on several orthogonal classes of flat and fp-injective functors

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