Relative Extensions of Modules and Homology Groups
We introduce the concepts of relative (co)extensions of modules and explore the relationship between the relative (co)extensions of modules and relative (co)homology groups. Some applications are given.
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| author | Mao, Lixin Zhu, Haiyan Мао, Ліхін Жу, Хайян |
| author_facet | Mao, Lixin Zhu, Haiyan Мао, Ліхін Жу, Хайян |
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| description | We introduce the concepts of relative (co)extensions of modules and explore the relationship between the relative (co)extensions of modules and relative (co)homology groups. Some applications are given. |
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UDC 512.58
Lixin Mao (Nanjing Inst. Technology, China),
Haiyan Zhu (Zhejiang Univ. Technology, Hangzhou, China)
RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS*
ВIДНОСНI РОЗШИРЕННЯ МОДУЛIВ ТА ГОМОЛОГIЧНИХ ГРУП
We introduce the concepts of relative (co)extensions of modules and explore the relationship between the relative
(co)extensions of modules and relative (co)homology groups. Some applications are given.
Введено поняття вiдносних (спiв)розширень модулiв та вивчено взаємозв’язок мiж вiдносними (спiв)розширеннями
модулiв та вiдносними (ко)гомологiчними групами.
1. Introduction. In classical homological algebra, given right R-modules M, N and a left R-module
L, the cohomology group Extn(M,N) is obtained by using a right injective resolution of N or a
left projective resolution of M, and the homology group Torn(M,L) is obtained by using a left
projective (flat) resolution of M or L. In relative homological algebra [5], if G is a preenveloping
class of right R-modules, then we can get the relative cohomology group ExtnG(M,N) computed
by the right G-resolution of N. Similarly, if F is a precovering class of right R-modules, then we
can get the relative cohomology group FExtn(M,N) and the relative homology group FTorn(M,L)
computed by the left F-resolution of M.
The main goal of the present paper is to extend some important properties of classical (co)homology
groups to relative (co)homological groups. We introduce the concepts of an F-extension and a G-
coextension of modules, where F and G denote two classes of right R-modules. It is proven that
the set of all equivalence classes of F-extensions (resp. G-coextensions) of A by C, denoted by
FE(C,A) (resp. EG(C,A)), is an Abelian group. Moreover, we prove that Ext1G(C,A) ∼= EG(C,A)
if G is a monic preenveloping class and FExt1(C,A) ∼= FE(C,A)) if F is an epic precovering
class. As applications, we obtain several properties of relative (co)homology groups. For example,
if F is an epic precovering class of right R-modules, then we prove that: (1) there is a monomor-
phism FExt1(C,A) → Ext1(C,A) for all right R-modules A and C; (2) there is an epimorphism
Tor1(A,B) → FTor1(A,B) for any right R-module A and any left R-module B. In addition, we
give a relative version of Wakamatsu’s lemmas.
We next recall some notions and facts needed in the sequel.
Following [3], we say that a right R-module homomorphism φ : M → G is a G-preenvelope ofM
if G ∈ G and the Abelian group homomorphism φ∗ : Hom(G,G′) → Hom(M,G′) is surjective for
eachG′ ∈ G. A G-preenvelope φ : M → G is said to be a G-envelope ofM if every endomorphism g :
G → G such that gφ = φ is an isomorphism. Dually we have the definitions of an F-precover and
an F-cover. G-envelopes (F-covers) may not exist in general, but if they exist, they are unique up
to isomorphism.
* This paper was supported by NSFC (No. 11171149, 11371187), Jiangsu 333 Project, Jiangsu Six Major Talents Peak
Project, Zhejiang Provincial Natural Science Foundation of China (No. LY12A01026), Nanjing Institute of Technology
Foundation (No. YJK201340).
c© LIXIN MAO, HAIYAN ZHU, 2015
1232 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS 1233
We say that G is a (resp. monic) preenveloping class of right R-modules [8] if every right R-
module has a (resp. monic) G-preenvelope. Dually, F is called a (resp. epic) precovering class of
right R-modules if every right R-module has an (resp. epic) F-precover.
Let G be a preenveloping class. Then any right R-module N has a right G-resolution, i.e., there
is a cocomplex 0 → N → G0 → G1 → . . . with each Gi ∈ G such that . . . → Hom(G1, G) →
→ Hom(G0, G) → Hom(N,G) → 0 is exact for any G ∈ G. Let G. be the deleted cocomplex
corresponding to a right G-resolution ofN, which is unique up to homotopy, then for a rightR-module
M, we obtain the nth cohomology group of the cocomplex Hom(M,G.), denoted by ExtnG(M,N)
(see [5], 8.2). Particularly, if G is the class of injective right R-modules, then ExtnG(M,N) is just the
classical cohomology group Extn(M,N).
Dually, let F be a precovering class, then any right R-module M has a left F-resolution, i.e.,
there is a complex . . . → F1 → F0 → M → 0 with each Fi ∈ F such that . . . → Hom(F, F1) →
→ Hom(F, F0) → Hom(F,M) → 0 is exact for any F ∈ F . Let F. be the deleted complex
corresponding to a left F-resolution of M, which is unique up to homotopy. Then for a right
R-module N, we obtain the nth cohomology group of the cocomplex Hom(F., N), denoted by
FExtn(M,N) (see [5], 8.2). In addition, for a left R-module L, we get the nth homology group
of the complex F. ⊗ L, denoted by FTorn(M,L). Particularly, if F is the class of projective right
R-modules, then FExtn(M,N) is just the classical cohomology group Extn(M,N) and FTorn(M,L)
is just the classical homology group Torn(M,L).
Throughout this paper, R is an associative ring with identity and all modules are unitary. All
classes of modules are closed under isomorphisms and direct summands. RM (resp. MR) denotes
a left (resp. right) R-module. The character module HomZ(M,Q/Z) of M is denoted by M+. The
reader is referred to [5, 6, 8, 10, 12, 14] for unexplained concepts and notations.
2. Relative homology groups and relative extensions of modules. Let A and C be two right
R-modules. Then an exact sequence 0→ A→ B → C → 0 is called an extension of A by C [10].
We first introduce the concepts of relative (co)extensions as follows.
Definition 2.1. Given a class F of right R-modules, an exact sequence 0→ A→ B → C → 0
of right R-modules is said to be an F-extension of A by C if
0→ Hom(F,A)→ Hom(F,B)→ Hom(F,C)→ 0
is exact for any F ∈ F .
Dually, given a class G of right R-modules, an exact sequence 0→ A→ B → C → 0 of right R-
modules is called a G-coextension of A by C if 0→ Hom(C,G)→ Hom(B,G)→ Hom(A,G)→ 0
is exact for any G ∈ G.
Remark 2.1. (1) Let F (resp. G) be the class of projective (resp. injective) right R-modules, then
an F-extension (resp. a G-coextension) of A by C is just the usual extension of A by C.
(2) Let F (resp. G) be the class of pure-projective (resp. pure-injective) right R-modules, then an
F-extension (resp. a G-coextension) of A by C is just a pure exact sequence 0 → A → B → C →
→ 0.
(3) Let G be the class of cotorsion right R-modules (A right R-module M is called cotorsion
[4] if Ext1(F,M) = 0 for every flat right R-module F ), then a G-coextension of A by C is just an
exact sequence 0 → A → B → C → 0 with A → B a strongly pure monomorphism in the sense
of [9].
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1234 LIXIN MAO, HAIYAN ZHU
Two F-extensions (G-coextensions) ∆ and ∆′ of A by C are called equivalent if there is σ :
B → B′ such that the following diagram is commutative:
∆ : 0 // A // B //
σ
��
C // 0
∆′ : 0 // A // B′ // C // 0 .
By the Five lemma, the middle homomorphism σ is an isomorphism. So the equivalence of
F-extensions (G-coextensions) is a reflexive, symmetric and transitive relation. We write FE(C,A)
(resp. EG(C,A)) to be the set of all equivalence classes of F-extensions (resp. G-coextensions) of A
by C.
If F (resp. G) is the class of projective (resp. injective) right R-modules, it is well known that
FE(C,A) (resp. EG(C,A)) is an Abelian group using the so called Baer sum (see [10]). We can
extend this result to a more general setting as follows.
Theorem 2.1. The following are true for right R-modules A and C :
(1) EG(C,A) is an Abelian group for any class G of right R-modules.
(2) FE(C,A) is an Abelian group for any class F of right R-modules.
Proof. (1) Let ∆1 : 0 → A
i1→ B1
π1→ C → 0 and ∆2 : 0 → A
i2→ B2
π2→ C → 0 be two
G-coextensions of A by C. Then we get the following pushout diagram:
0
��
0
��
0 // A
i1
//
i2
��
B1
f
��
π1
// C // 0
0 // B2
g
//
π2
��
H12
��
// C // 0 ,
C
��
C
��
0 0
where H12 = (B1 ⊕ B2)/W,W = {(i1(a),−i2(a)) : a ∈ A}, f(b1) = (b1, 0) for b1 ∈ B1, g(b2) =
= (0, b2) for b2 ∈ B2. Let Q = {(x, y) : π1(x) = π2(y), x ∈ B1, y ∈ B2} ⊆ B1⊕B2. Then W ⊆ Q.
Put Y12 = Q/W ⊆ H12. Then we get the sequence
Ψ12 : 0→ A
λ12→ Y12
τ12→ C → 0,
where λ12(a) = (i1(a), 0) for a ∈ A and τ12((x, y)) = π1(x) = π2(y) for (x, y) ∈ Q.
We first claim that Ψ12 is exact. In fact, it is clear that λ12 is monic, τ12 is epic and τ12λ12 =
= 0. If τ12((x, y)) = 0, then x = i1(a1) and y = i2(a2) for some a1, a2 ∈ A. Thus (x, y) =
= (i1(a1), i2(a2)) = (i1(a1 + a2), 0) = λ12(a1 + a2). So Ψ12 is exact.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS 1235
We now prove that Ψ12 is a G-coextension of A by C. In fact, let G ∈ G and α ∈ Hom(A,G),
then there exist β1 ∈ Hom(B1, G) and β2 ∈ Hom(B2, G) such that α = β1i1 and α = β2i2 by
hypothesis. Thus by the property of a pushout, there exists ξ ∈ Hom(H12, G) such that the following
diagram is commutative:
A
i1
//
i2
��
B1
f
�� β1
��
B2
g
//
β2 **
H12
ξ
''
G .
Write ε : Y12 → H12 to be the inclusion. Then
α = β1i1 = ξfi1 = (ξε)λ12.
So Hom(Y12, G)→ Hom(A,G) is epic, i.e., Ψ12 is a G-coextension of A by C.
Define [∆1] + [∆2] = [Ψ12]. It is obvious that [∆1] + [∆2] = [∆2] + [∆1].
Let ∆3 : 0 → A
i3→ B3
π3→ C → 0 be a G-coextension of A by C. We next prove that ([∆1] +
+ [∆2]) + [∆3] = [∆1] + ([∆2] + [∆3]).
Let ([∆1] + [∆2]) + [∆3] = [Ξ], where Ξ : 0 → A
ω→ U/V
ρ→ C → 0 is a G-coextension
of A by C with U = {((x, y), z) : τ12((x, y)) = π3(z), (x, y) ∈ Y12, z ∈ B3} ⊆ Y12 ⊕ B3, V =
= {(λ12(a),−i3(a)) : a ∈ A}. Let [∆2] + [∆3] = [ψ23] and [∆1] + ([∆2] + [∆3]) = [Λ], where ψ23 :
0 → A
λ23→ Y23
τ23→ C → 0 is a G-coextension of A by C and Λ : 0 → A
µ→ M/N
ν→ C → 0 is
a G-coextension of A by C with M = {(x, (y, z)) : π1(x) = τ23((y, z)), x ∈ B1, (y, z) ∈ Y23} ⊆
⊆ B1 ⊕ Y23, N = {(i1(a),−λ23(a)) : a ∈ A}.
Define σ : U/V → M/N by σ(((x, y), z)) = (x, (y, z)) for (x, y) ∈ Y12, z ∈ B3. We claim
that σ is well defined. In fact, if ((x, y), z) = 0, then ((x, y), z) = (λ12(a),−i3(a)) for some
a ∈ A. So (x, y) = (i1(a), 0), z = −i3(a). Thus (x, y) − (i1(a), 0) = (i1(b),−i2(b)) for some
b ∈ A. Hence x = i1(a + b), y = −i2(b). So (x, (y, z)) = (i1(a + b), (−i2(b),−i3(a)) = (i1(a +
+ b), (−i2(a+ b), 0) = (i1(a+ b), −λ23(a+ b)) ∈ N. Thus (x, (y, z)) = 0. Moreover, it is easy to
verify that the following diagram is commutative:
Ξ : 0 // A
ω
// U/V
ρ
//
σ
��
C // 0
Λ : 0 // A
µ
// M/N
ν
// C // 0 .
So ([∆1] + [∆2]) + [∆3] = [∆1] + ([∆2] + [∆3]).
On the other hand, the split exact sequence 0 : 0 → A
ι→ A ⊕ C κ→ C → 0 is clearly a G-
coextension of A by C. We claim that [∆1] + [0] = [∆1]. In fact, let [∆1] + [0] = [ψ1], where ψ1 :
0 → A
λ1→ Q1/W1
τ1→ C → 0 is a G-coextension of A by C with W1 = {(i1(a),−(a, 0)) :
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1236 LIXIN MAO, HAIYAN ZHU
a ∈ A}, Q1 = {(x, (a, π1(x))) : x ∈ B1, a ∈ A}.Define σ1 : Q1/W1 → B1 by σ1((x, (a, π1(x)))) =
= x + i1(a). It is easy to verify that σ1 is well defined, σ1λ1 = i1 and π1σ1 = τ1. Thus [0] is the
zero element in EG(C,A).
Finally consider the exact sequence ∆
′
1 : 0 → A
−i1→ B1
π1→ C → 0, which is obviously a G-
coextension of A by C. We claim that [∆1] + [∆
′
1] = [0]. In fact, let [∆1] + [∆
′
1] = [ψ′], where
ψ′ : 0 → A
λ′→ Q′/W ′
τ ′→ C → 0 is a G-coextension of A by C with W ′ = {(i1(a), i1(a)) :
a ∈ A}, Q′ = {(x, y) : π1(x) = π1(y), x, y ∈ B1} = {(y + i1(a), y) : a ∈ A, y ∈ B1}. Define σ′ :
Q′/W ′ → A ⊕ C by σ′((y + i1(a), y)) = (a, π1(y)). It is easy to verify that σ′ is well defined,
σ′λ′ = ι and κσ′ = τ ′. So ∆
′
1 is the negative element of ∆1 in EG(C,A).
It follows that EG(C,A) is an Abelian group.
(2) can be proved dually.
Theorem 2.1 is proved.
It is well known that, in standard homological algebra, the cohomological group Ext1(C,A)
is isomorphic to the group of all equivalence classes of extensions of A by C. This result can be
generalized as follows.
Theorem 2.2. The following are true:
(1) If G is a monic preenveloping class of right R-modules, then there is an Abelian group
isomorphism Ext1G(C,A) ∼= EG(C,A) for all right R-modules A and C.
(2) If F is an epic precovering class of right R-modules, then there is an Abelian group isomor-
phism FExt1(C,A) ∼= FE(C,A) for all right R-modules A and C.
Proof. (1) Let 0→ A
d0→ G0 d1→ G1 d2→ G2 → . . . be a right G-resolution of A. Then we get the
cocomplex
0→ Hom(C,G0)
d1∗→ Hom(C,G1)
d2∗→ Hom(C,G2)→ . . . .
So Ext1G(C,A) = ker(d2∗)/im(d1∗).
Let Γ : 0→ A
i→ B
π→ C → 0 be a G-coextension of A by C, then there exist ε0 : B → G0 and
ε1 : C → G1 such that the following diagram with exact rows is commutative:
0 // A
i
// B
ε0
��
π
// C //
ε1
��
0
0 // A
d0
// G0
d1
// G1 // G2.
Note that d2ε1π = d2d1ε0 = 0. Thus d2ε1 = 0, and so ε1 ∈ ker(d2∗).
Define Θ : EG(C,A) → Ext1G(C,A) by Θ([Γ]) = ε1. We claim that Θ is well defined. In
fact, if there exist ε′0 : B → G0 and ε′1 : C → G1 such that the above diagram also commutes,
then (ε′0 − ε0)i = 0, so there exists χ : C → G0 such that ε′0 − ε0 = χπ. Thus (ε′1 − ε1)π =
= d1(ε′0 − ε0) = d1χπ. So ε′1 − ε1 = d1χ ∈ im(d1∗). Hence ε1 = ε′1.
We now prove that Θ is a group homomorphism.
Let Υ : 0 → A
ι→ H
ρ→ C → 0 be a G-coextension of A by C. Then there is the following
commutative diagram with exact rows:
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS 1237
0 // A
ι
// H
γ0
��
ρ
// C //
γ1
��
0
0 // A
d0
// G0
d1
// G1 // G2 .
Let [Γ] + [Υ] = [Ψ], where Ψ : 0 → A
λ→ Q/W
τ→ C → 0 is a G-coextension of A by C with
Q = {(x, y) : π(x) = ρ(y), x ∈ B, y ∈ H} and W = {(i(a),−ι(a)) : a ∈ A} by Theorem 2.1.
Define η : Q/W → G0 by η((x, y)) = ε0(x) + γ0(y) for (x, y) ∈ Q. Then η is well defined and
ηλ(a) = η((i(a), 0)) = ε0i(a) = d0(a), (ε1+γ1)τ((x, y)) = (ε1+γ1)(π(x)) = d1ε0(x)+d1γ0(y) =
= d1η((x, y)). So we have the following commutative diagram with exact rows:
0 // A
λ
// Q/W
η
��
τ
// C //
ε1+γ1
��
0
0 // A
d0
// G0
d1
// G1 // G2 .
Thus Θ([Γ] + [Υ]) = Θ([Γ]) + Θ([Υ]). We next prove that Θ is a group isomorphism.
Write µ : im(d1)→ G1 to be the inclusion. Then there exists ν : G0 → im(d1) such that µν = d1.
Let β ∈ ker(d2∗). Then d2β = 0. So im(β) ⊆ ker(d2) = im(d1). Thus there exists β̂ : C → im(d1)
such that β = µβ̂. We obtain the following pullback diagram:
0 // A
f
// D
g
//
ω
��
C //
β̂
��
0
0 // A
d0
// G0
ν
// im(d1) // 0 .
For any M ∈ G and any homomorphism h : A → M, there is j : G0 → M such that h = jd0.
So (jw)f = jd0 = h. Thus the sequence Hom(D,M) → Hom(A,M) → 0 is exact. Hence the
exact sequence ∆ : 0 → A
f→ D
g→ C → 0 is a G-coextension of A by C. Since β = µβ̂, we have
Θ([∆]) = β. So Θ is an epimorphism.
On the other hand, let Θ([Γ]) = ε1 = 0, then ε1 = d1κ for some κ ∈ Hom(C,G0). Since
d2ε1 = d2d1κ = 0, im(ε1) ⊆ ker(d2) = im(d1). Thus there exists ε̂1 : C → im(d1) such that
ε1 = µε̂1. So µνκ = d1κ = ε1 = µε̂1. Thus νκ = ε̂1.
Consider the following diagram with exact rows:
0 // A
i
// B
π
//
ε0
��
C
κ
||
//
ε̂1
��
0
0 // A
d0
// G0
ν
// im(d1) // 0.
Then there exists δ : B → A such that δi = 1 by [6, p. 44] (Lemma 8.4). Therefore Γ is a split exact
sequence, and so [Γ] = 0. Hence Θ is a monomorphism.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1238 LIXIN MAO, HAIYAN ZHU
(2) can be proved dually.
Theorem 2.2 is proved.
As an immediate consequence of Theorem 2.2, we have the following corollary.
Corollary 2.1. The following are true:
(1) If G is a monic preenveloping class of right R-modules, then there is a monomorphism
Ext1G(C,A)→ Ext1(C,A) for all right R-modules A and C.
(2) If F is an epic precovering class of right R-modules, then there is a monomorphism
FExt1(C,A)→ Ext1(C,A) for all right R-modules A and C.
Obviously, a preenveloping class G of right R-modules is monic if and only if G contains all
injective right R-modules and a precovering class F of right R-modules is epic if and only if F
contains all projective right R-modules. Furthermore, we have the following result.
Corollary 2.2. The following are true:
(1) Let G be a monic preenveloping class of right R-modules, then Ext1G(C,A) ∼= Ext1(C,A) for
all right R-modules A and C if and only if G is the class of injective right R-modules.
(2) Let F be an epic precovering class of right R-modules, then FExt1(C,A) ∼= Ext1(C,A) for
all right R-modules A and C if and only if F is the class of projective right R-modules.
Proof. (1) ⇒. For any M ∈ G, there is an exact sequence 0 → M → E → C → 0 with E
injective. By Theorem 2.2(1), the exact sequence is a G-coextension of M by C, and so is split.
Thus M is injective.
⇐ is trivial.
(2) can be proved dually.
Corollary 2.2 is proved.
Now we characterize when ExtnG(−,−) and FExtn(−,−) (n = 1, 2) vanish.
Proposition 2.1. The following are true:
(1) Let G be a monic preenveloping class of right R-modules, then any G-coextension of A by C
0→ A→ B → C → 0 is split if and only if Ext1G(C,A) = 0.
(2) Let F be an epic precovering class of right R-modules, then any F-extension of A by C
0→ A→ B → C → 0 is split if and only if FExt1(C,A) = 0.
Proof. (1)⇐. Since G is a monic preenveloping class of rightR-modules, we have Ext0G(C,−) ∼=
∼= Hom(C,−) (see [5, p. 170]) and G is closed under finite direct sums by [1] (Lemma 1). Thus by
[5] (Theorem 8.2.5(1)), the G-coextension of A by C 0 → A → B → C → 0 induces the exact
sequence
0→ Hom(C,A)→ Hom(C,B)→ Hom(C,C)→ Ext1G(C,A) = 0.
So 0→ A→ B → C → 0 is split.
⇒. Since any G-coextension of A by C 0 → A → B → C → 0 is equivalent to the exact
sequence 0 → A → A ⊕ C → C → 0, EG(C,A) = 0. So Ext1G(C,A) ∼= EG(C,A) = 0 by
Theorem 2.2(1).
(2) ⇐. Since F is an epic precovering class of right R-modules, we have FExt0(−, A) ∼=
∼= Hom(−, A) (see [5, p. 170]) and F is closed under direct sums by [7] (Proposition 1). So by [5]
(Theorem 8.2.3(2)), the F-extension of A by C 0→ A→ B → C → 0 induces the exact sequence
0→ Hom(C,A)→ Hom(B,A)→ Hom(A,A)→ FExt1(C,A) = 0.
Thus 0→ A→ B → C → 0 is split.
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RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS 1239
⇒. Since any F-extension of A by C 0→ A→ B → C → 0 is equivalent to the exact sequence
0 → A → A ⊕ C → C → 0, we have FE(C,A) = 0. Thus FExt1(C,A) ∼= FE(C,A) = 0 by
Theorem 2.2(2).
Proposition 2.1 is proved.
Corollary 2.3. The following are true:
(1) Let G be a monic preenveloping class of right R-modules, then a right R-module A belongs
to G if and only if Ext1G(C,A) = 0 for any right R-module C.
(2) Let F be an epic precovering class of right R-modules, then a right R-module C belongs to
F if and only if FExt1(C,A) = 0 for any right R-module A.
Proof. It is easy by Proposition 2.1.
Proposition 2.2. The following are true:
(1) Let G be a monic preenveloping class of right R-modules, then Ext2G(N,M) = 0 for all right
R-modules M and N if and only if C ∈ G for any G-coextension 0 → A → B → C → 0 with
B ∈ G.
(2) Let F be an epic precovering class of right R-modules, then FExt2(N,M) = 0 for all right
R-modules M and N if and only if A ∈ F for any F-extension 0→ A→ B → C → 0 with B ∈ F .
Proof. (1) ⇒. By [5] (Theorem 8.2.5(1)), for any right R-module N, any G-coextension
0→ A→ B → C → 0 with B ∈ G induces the exact sequence
0 = Ext1G(N,B)→ Ext1G(N,C)→ Ext2G(N,A) = 0.
So Ext1G(N,C) = 0. Thus C ∈ G by Corollary 2.3(1).
⇐. For any right R-module M, by hypothesis, there exists a G-coextension 0 → M → B →
→ C → 0 with B ∈ G. So C ∈ G. Thus by [5] (Theorem 8.2.5(1)), for any right R-module N, we
get the induced exact sequence
0 = Ext1G(N,C)→ Ext2G(N,M)→ Ext2G(N,B) = 0.
So Ext2G(N,M) = 0.
(2) can be proved dually.
Proposition 2.2 is proved.
The following result may be viewed as a relative version of Wakamatsu’s lemmas.
Theorem 2.3. The following are true:
(1) Suppose that G is a monic preenveloping class of right R-modules and C is a class of right R-
modules closed under G-coextensions. If α : N →M is a C-envelope ofN, then Ext1G(coker(α), C) =
= 0 for any C ∈ C.
(2) Suppose that F is an epic precovering class of right R-modules and D is a class of right R-
modules closed under F-extensions. If α : N → M is a D-cover of M, then FExt1(D, ker(α)) = 0
for any D ∈ D.
Proof. (1) By Proposition 2.1(1), it is enough to show that any G-coextension 0→ C → B
ρ→
ρ→ coker(α)→ 0 with C ∈ C is split.
Let λ : im(α)→M be the inclusion and π : M → coker(α) the canonical map. Then there exists
γ : N → im(α) such that λγ = α.
Consider the following pullback diagram:
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1240 LIXIN MAO, HAIYAN ZHU
0
��
0
��
im(α)
i
��
im(α)
λ
��
0 // C // X
θ
��
β
// M //
π
��
0
0 // C // B
��
ρ
// coker(α)
��
// 0 .
0 0
Since 0→ C → B
ρ→ coker(α)→ 0 is a G-coextension, it is easy to see that 0→ C → X →M → 0
is also a G-coextension. Thus X ∈ C since C is closed under G-coextensions. Because α : N → M
is a C-envelope, there exists g : M → X such that iγ = gα. Thus α = λγ = βiγ = βgα. Hence βg
is an isomorphism.
Define ϕ : coker(α) → B by ϕ(x) = θg(βg)−1(x) for x ∈ M. Since θg(βg)−1α = θgα =
= θiγ = 0, ϕ is well defined. Note that
ρϕ(x) = ρθg(βg)−1(x) = πβg(βg)−1(x) = π(x) = x
for x ∈ M. Thus ρϕ = 1. Hence 0 → C → B
ρ→ coker(α) → 0 is split, and so Ext1G(coker(α),
C) = 0.
(2) can be proved dually.
Theorem 2.3 is proved.
Remark 2.2. (1) Let F (resp. G) in Theorem 2.3 be the class of projective (resp. injective) right
R-modules, then Theorem 2.3 is just the usual Wakamatsu’s lemmas (see [5], Corollary 7.2.3 and
Proposition 7.2.4 or [14], Section 2.1).
(2) Following [13], an exact sequence 0 → A → B → C → 0 of left R-modules is called
RD-exact if the sequence Hom(R/Ra,B)→ Hom(R/Ra,C)→ 0 is exact for every a ∈ R. A left
R-module G is called RD-injective if for every RD-exact sequence 0 → A → B → C → 0 of left
R-modules, the sequence 0 → Hom(C,G) → Hom(B,G) → Hom(A,G) → 0 is exact. According
to [2], a right R-module F is called RD-flat if for every RD-exact sequence 0→ A→ B → C → 0
of left R-modules, the sequence 0→ F ⊗A→ F ⊗B → F ⊗ C → 0 is exact.
Let F be the class of pure-projective right R-modules. It is well known that F is an epic
precovering class of right R-modules (see [5], Example 8.3.2). Let 0 → X → Y → Z → 0 be a
pure exact sequence of right R-modules with X and Z RD-flat, then we get the split exact sequence
0 → Z+ → Y + → X+ → 0. Since X+ and Z+ are RD-injective by [2] (Proposition 1.1), we
have Y + is RD-injective. Hence Y is RD-flat. So the class of RD-flat right R-modules is closed
under F-extensions by Remark 2.1(2). Note that any right R-module M has an RD-flat cover α :
N →M by [11] (Theorem 2.6(2)). Thus FExt1(D, ker(α)) = 0 for any RD-flat right R-module D
by Theorem 2.3(2).
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RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS 1241
We next give some isomorphism formulas about relative (co)homological groups.
Theorem 2.4. The following are true:
(1) Let G be a preenveloping class of right R-modules, AS a projective right S-module, SBR an
(S,R)-bimodule, CR a right R-module and n ≥ 0. Then
ExtnG(A⊗S B,C) ∼= HomS(A,ExtnG(B,C)).
(2) Let F be a precovering class of right R-modules, AR a right R-module, RBS an (R,S)-
bimodule, ES an injective right S-module and n ≥ 0. Then
FExtn(A,HomS(B,E)) ∼= HomS(FTorn(A,B), E).
Proof. (1) Let G. : 0→ G0 → G1 → . . . be a deleted right G-resolution of C. Then we obtain
the cocomplex HomR(A⊗S B,G.):
0→ HomR(A⊗S B,G0)→ HomR(A⊗S B,G1)→ . . . ,
which is isomorphic to the cocomplex HomS(A,HomR(B,G.)) :
0→ HomS(A,HomR(B,G0))→ HomS(A,HomR(B,G1))→ . . . .
Note that HomS(A,−) is an exact functor. So by [12, p. 170] (Exercise 6.4), we have ExtnG(A ⊗
⊗ SB,C) = Hn(HomR(A⊗S B,G.)) ∼= Hn(HomS(A,HomR(B,G.))) ∼= HomS(A,Hn(HomR(B,
G.))) = HomS(A,ExtnG(B,C)).
(2) Let F. : . . . → F1 → F0 → 0 be a deleted left F-resolution of A. Then we obtain the
cocomplex HomS(F. ⊗R B,E):
0→ HomS(F0 ⊗R B,E)→ HomS(F1 ⊗R B,E)→ . . . ,
which is isomorphic to the cocomplex HomR(F.,HomS(B,E)) :
0→ HomR(F0,HomS(B,E))→ HomR(F1,HomS(B,E))→ . . . .
Note that HomS(−, E) is an exact functor. So by [12, p. 170] (Exercise 6.4), we have HomS(FTorn(A,
B), E) = HomS(Hn(F. ⊗R B), E) ∼= Hn(HomS(F. ⊗R B), E) ∼= Hn(HomR(F.,HomS(B,E))) =
= FExtn(A,HomS(B,E)).
Theorem 2.4 is proved.
Corollary 2.4. Let F be a precovering class of right R-modules, AR a right R-module, RB a
left R-module and n ≥ 0. Then FExtn(A,B+) ∼= FTorn(A,B)+.
Proof. Let S = Z and E = Q/Z in Theorem 2.4(2). Then we get the isomorphism
FExtn(A,B+) ∼= FTorn(A,B)+.
Finally we discuss the relationship between Torn(A,B) and FTorn(A,B).
Suppose that F is an epic precovering class of right R-modules. Let
. . .→ F2 → F1 → F0 → A→ 0
be a left F-resolution of a right R-module A and let
. . .→ P2 → P1 → P0 → A→ 0
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 9
1242 LIXIN MAO, HAIYAN ZHU
be a left projective resolution of A. Then there exist fi : Pi → Fi such that the following diagram is
commutative:
. . . // P2
f2
��
// P1
//
f1
��
P0
f0
��
// A // 0
. . . // F2
// F1
// F0
// A // 0.
Applying −⊗B to the above diagram, we have the following commutative diagram of complexes:
. . . // P2 ⊗B //
f2⊗1
��
P1 ⊗B
f1⊗1
��
// P0 ⊗B
f0⊗1
��
// 0
. . . // F2 ⊗B // F1 ⊗B // F0 ⊗B // 0.
Then it is easy to check that there exist group homomorphisms:
ηn : Torn(A,B)→ FTorn(A,B), n ≥ 0.
Theorem 2.5. If F is an epic precovering class of right R-modules, then η1 : Tor1(A,B) →
→ FTor1(A,B) is an epimorphism for any right R-module A and any left R-module B.
Proof. Consider the following commutative diagram:
FExt1(A,B+)
γ
//
α
��
Ext1(A,B+)
β
��
FTor1(A,B)+
η+1
// Tor1(A,B)+.
Note that α and β are isomorphisms by Corollary 2.4 and γ is a monomorphism by Corollary 2.1(2).
So η+1 : FTor1(A,B)+ → Tor1(A,B)+ is a monomorphism. Thus η1 : Tor1(A,B) → FTor1(A,B)
is an epimorphism.
Theorem 2.5 is proved.
Remark 2.3. LetF be an epic precovering class of rightR-modules. Although η1 : Tor1(A,B)→
→ FTor1(A,B) is an epimorphism by Theorem 2.5, this is not an isomorphism in general. For exam-
ple, if F is the class of pure-projective Z-modules, then FTor1(Z2,Z2) = 0, but Tor1(Z2,Z2) ∼= Z2.
Note that Tor0(A,B) ∼= FTor0(A,B) ∼= A⊗B for any right R-module A and any left R-module
B. It is natural to ask when Torn(A,B)→ FTorn(A,B) is an isomorphism. We give the following
answer which is easy to verify.
Proposition 2.3. Let F be an epic precovering class of right R-modules. Then the following are
equivalent:
(1) Torn(A,B) ∼= FTorn(A,B) for any right R-module A, left R-module B and n ≥ 1.
(2) Tor1(A,B) ∼= FTor1(A,B) for any right R-module A and left R-module B.
(3) Every M ∈ F is flat.
1. Chen J. L., Ding N. Q. A note on existence of envelopes and covers // Bull. Austral. Math. Soc. – 1996. – 54. –
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RELATIVE EXTENSIONS OF MODULES AND HOMOLOGY GROUPS 1243
2. Couchot F. RD-flatness and RD-injectivity // Communs Algebra. – 2006. – 34. – P. 3675 – 3689.
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Received 20.01.13
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|
| id | umjimathkievua-article-2061 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:17:57Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/a8/2bc0e9456c1732beca7a57e17c8edfa8.pdf |
| spelling | umjimathkievua-article-20612019-12-05T09:49:43Z Relative Extensions of Modules and Homology Groups Відносні розширення модулів та гомолопчних груп Mao, Lixin Zhu, Haiyan Мао, Ліхін Жу, Хайян We introduce the concepts of relative (co)extensions of modules and explore the relationship between the relative (co)extensions of modules and relative (co)homology groups. Some applications are given. Введено поняття відносних (спів)розширень модулів та вивчено взаємозв'язок між відносними (спів)розширеннями модулів та відносними (ко)гомологічними групами. Institute of Mathematics, NAS of Ukraine 2015-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2061 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 9 (2015); 1232–1243 Український математичний журнал; Том 67 № 9 (2015); 1232–1243 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2061/1138 https://umj.imath.kiev.ua/index.php/umj/article/view/2061/1139 Copyright (c) 2015 Mao Lixin; Zhu Haiyan |
| spellingShingle | Mao, Lixin Zhu, Haiyan Мао, Ліхін Жу, Хайян Relative Extensions of Modules and Homology Groups |
| title | Relative Extensions of Modules and Homology Groups |
| title_alt | Відносні розширення модулів та гомолопчних груп |
| title_full | Relative Extensions of Modules and Homology Groups |
| title_fullStr | Relative Extensions of Modules and Homology Groups |
| title_full_unstemmed | Relative Extensions of Modules and Homology Groups |
| title_short | Relative Extensions of Modules and Homology Groups |
| title_sort | relative extensions of modules and homology groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2061 |
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