Hopfian Objects, Cohopfian Objects in the Category of

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International Mathematical Forum, Vol. 8, 2013, no. 39, 1903 - 1920
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/imf.2013.37128
Hopfian Objects, Cohopfian Objects in
the Category of Complexes of Left A− Modules
El Hadj Ousseynou Diallo
Département de Mathématiques et Informatique
Faculté des Sciences et Techniques
Université Cheikh Anta Diop de Dakar, Sénégal
[email protected]
Mohamed Ben Faraj Ben Maaouia
UFRSAT, Université Gaston Berger de Saint-Louis, Sénégal
[email protected]
Mamadou Sanghare
Département de Mathématiques et Informatique
Faculté des Sciences et Techniques
Université Cheikh Anta Diop de Dakar, Sénégal
[email protected]
c 2013 El Hadj Ousseynou Diallo, Mohamed Ben Faraj Ben Maaouia and
Copyright Mamadou Sanghare. 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.
Abstract
The main results of this paper are the following theorems :
a) Suppose that C is a chain complex of A− modules with E a non zero
subcomplex of C. If E and C/E are hopfians then C is hopfian
b) If C is a chain complex of A− modules in which all non trivial
subcomplex is cohopfian then C is cohopfian
c) If C is a projective chain complex of A− modules with a subcomplex
E completely invariant and superfluous then E is hopfian if and only if
C/E is hopfian
d) If C is an injective chain complex with a subcomplex completely
1904
El Hadj Ousseynou Diallo et al.
invariant and essential then E is cohopfian if and only if C is cohopfian
e) Any projective and cohopfian or injective and hopfian chain complex
of A− modules is a Fitting chain complex.
Keywords: Chain complex of A-modules, Hopfian complex, Cohopfian
complex, Injective chain complex, Projective chain complex, Fitting chain
complex
1
Introduction
In this paper ,we will study projective,injective ,hopfian ,cohopfian objects of
the category of complexes of A− modules denoted COM P
The objects of COM P are chain complexes and the morphisms are maps of
chains
A chain complex C is a sequence of homomorphisms of A− modules such
as:
dn+1
dn−1
dn
C : . . . → Cn+1 → Cn →
Cn−1 → . . . verifying : dn ◦ dn+1 = 0 ,for all n ∈ Z
A chain map f is defined by:
/
Cn+1
C : ...
f
C : ...
fn+1
/
Cn+1
dn+1
/
fn
dn+1
dn
Cn
/
/
fn−1
dn
Cn
/
...
/
...
Cn−1
/
Cn−1
with dn+1 ◦ fn+1 = fn ◦ dn+1 , ∀n ∈ Z
Given C an object of COMP and f an endomorphism of C .’ C is said hopfian(respectively cohopfian)
if any epimorphism(respectively monomorphism) is an isomorphism
dn+1
d
dn−1
n
A chain complex C : . . . → Cn+1 → Cn →
Cn−1 → . . . is said completely
invariant if and only if each A− module Cn is completely invariant , ∀n ∈ Z
dn+1
dn−1
dn
A chain complex C : . . . → Cn+1 → Cn →
Cn−1 → . . . is said essential if
and only if each A− module Cn is essential , ∀n ∈ Z
A chain complex C is said F itting if for any endomorphism f of C
, there is an integer n such as C = Imf n ⊕ Kerf n
In the section 3 ,we demonstrate the following results:
a)Suppose C is a chain complex with a subcomplex E of C non zero . If E
and C/E are hopfian then C is hopfian
b) If C is a complex with own subcomplex is cohopfian then C is cohopfian
c) If C = ⊕Ci with (Ci ) is a family of complexes then the following assertions
are equivalent :
i) If C is hopf ian (respectively cohopf ian) then Ci is hopf ian (respectively
1905
Hopfian objects, Cohopfian objects in the category of complexes
cohopf ian)
ii)If Ci is completely invariant then C is hopf ian(respectivement cohopf ian)
if and only if Ci is hopf ian (respectively cohopf ian)
In the section 4 ,we demonstrate the following results :
a)If C is a projective chain complex of A− modules with a subcomplex E
completely invariant and superfluous then E is hopfian if and only if C/E is
hopfian
b)If C is an injective chain complex and E a completely invariant and
essential subcomplex of C then E is cohopf ian if and only if C is cohopf ian
c)Theorem: if C is a chain complex of A− modules then
we obtain the following results:
c1 ) If C is projective and cohopf ian then C is hopf ian
c2 ) If C is injective et hopf ian then C is cohopf ian
d)Theorem : Any projective and cohopf ian or injective and hopf ian chain
complex is a F itting chain complex
In this paper A denotes a not inevitably commutative ,unitarian associative
ring and M a left unifere module
2
Definitions and Preliminary results on the
category COM P
Definition et proposition 1 .The category of the complexes of A−modules
denoted COM P is the category objects which are the sequence homomorphisms
of left A− modules (dn ) de A denoted
dn+1
dn−1
d
n
(C, d) : . . . → Cn+1 → Cn →
Cn−1 → . . .
verifying dn ◦ dn+1 = 0 , for all n ∈ Z and the morphism are chain maps
A chain map f : (C, d) → (C , d ) is a sequence of homomorphisms (fn ) of left
A− modules such as:
/
Cn+1
(C, d) : . . .
f
(C , d ) : . . .
fn+1
/
Cn+1
dn+1
/
Cn
fn
dn+1
/
Cn
dn
/
...
/
...
Cn−1
fn−1
dn
verifying: dn+1 ◦ fn+1 = fn ◦ dn+1 , for all n ∈ Z
Remark 1 A chain complex
dn+1
dn−1
dn
Cn−1 → . . .
(C, d) : . . . → Cn+1 → Cn →
/
/
Cn−1
1906
El Hadj Ousseynou Diallo et al.
is said to be zero either all terms Cn zero or the map chain zero(all dn zero) ,
∀n ∈ Z
Proposition 1 Let be a pair of chain maps f , g : of (C, d) into (C , d)
.The morphism denoted : f + g = (fn + gn )n∈ and named sum of f and g is
a chain
with f = (fn )n∈ and g = (gn )n∈
Proof 1 Let the following diagramm:
Cn
fn +gn
Cn
dn
dn
/
/
Cn−1
fn−1 +gn−1
Cn−1
We have:
dn ◦ (fn + gn ) = dn ◦ fn + dn ◦ gn
but dn ◦ fn = fn−1 ◦ dn et dn ◦ gn = gn−1 ◦ dn
Hence dn ◦ (fn + gn ) = (fn−1 + gn−1 ) ◦ dn
then f + g is a chain map of (C, d) into (C , d)
Proposition 2 Let be two chains complexes (C, d) and (C , d) of A− modules
Their direct sum denoted by (C ⊕ C , d ⊕ d ) is:
(C ⊕ C )n = Cn ⊕ Cn et (d ⊕ d )n = dn ⊕ dn , where
dn ⊕ dn : Cn ⊕ Cn → Cn−1 ⊕ Cn−1
(xn , xn ) → (dn (xn ), dn (xn ))
Proof 2 It’s clear that dn ⊕ dn is a morphism.We show (dn ⊕ dn ) ◦ (dn+1 ⊕
dn+1 ) = 0
It’s known that (dn+1 ⊕ dn+1 )(xn+1 , xn+1 ) = (dn+1 (xn+1 ), dn+1 (xn+1 ))
Hence (dn ⊕ dn ) ◦ ((dn+1 ⊕ dn+1 )(xn+1 , xn+1 ))=(dn ◦ dn+1 (xn+1 ), dn ◦ dn+1 (xn+1 ))
But dn ◦ dn+1 (xn+1 ) = 0 and dn ◦ dn+1 (xn+1 ) = 0
Then (dn ⊕ dn ) ◦ (dn+1 ⊕ dn+1 ) = 0, for all n ∈ Z
dn+1
d
dn−1
n
Cn−1 → . . . a chain
Proposition 3 Let be (C, d) : . . . → Cn+1 → Cn →
complex of A− modules. and (En ) a family of A− modules such that n ,En is
un sub-module of Cn . Hence :
if dn (En ) ⊆ En−1 then sequence of induced morphism (dn : En → En−1 ) is a
chain complex of A− modules denoted (E, d) and called subcomplex of (C, d)
. The sequence of (in : En → Cn ) canonical monomorphism is a chain map
i : (E, d) → (C, d)
1907
Hopfian objects, Cohopfian objects in the category of complexes
Proof 3
- Suppose that δn : En → En−1 the induced morphism of dn , We show that
δn is well defined:
Suppose that x ∈ En , hence dn (x) ∈ En−1 Then δn is well defined
δn is a morphism because composite of two morphisms
Let us verify that δn ◦ δn+1 = 0
Let be x ∈ En+1 , hence δn+1 (x) = dn+1 (x) ∈ En and δn ((dn+1 )(x)) = dn ◦
dn+1 (x) = 0 then δn ◦ δn+1 = 0 , for all n ∈ Z
So (E, δ) the sequence (dn : En → En−1 ) is a chain complex
-We show that i : (E, δ) → (C, d) is a chain map
E : ...
i
C : ...
/
En+1
in+1
/
Cn+1
δn+1
/E
δn
n
in
dn+1
/
Cn
/
in−1
dn
/
...
/
...
En−1
/
Cn−1
If x ∈ En+1 , hence δn+1 (x) = dn+1 (x) therefore in ◦ δn+1 (x) = in (dn+1 (x)) =
dn+1 (x)
On the other hand if x ∈ En+1 we obtain in+1 (x) = x hence dn+1 ◦ in+1 (x) =
dn+1 (x)
therefore x ∈ En+1 we get in ◦ δn+1 (x) = dn+1 ◦ in+1 (x)
So i : (E, δ) → (C, d) is a chain map that is i : (E, d) → (C, d)
dn+1
d
dn−1
n
Proposition 4 Let be (C, d) : . . . → Cn+1 → Cn →
Cn−1 → . . . a chain
complex and
dn+1
dn−1
dn+1
dn
(E, d) : . . . → En+1 → En →
En−1 → . . . , (F, d) : . . . → Fn+1 →
d
dn−1
n
Fn →
Fn−1 → . . . two subcomplexes of C
Then this family of modules is a chain complex:
dn+1
dn−1
dn
En−1 Fn−1 → . . .
E F :. . . → En+1 Fn+1 → En Fn →
It’s named intersection of E and F
Proof 4 We have E and F subcomplexes of C then: dn (En ) En−1 and
dn (Fn ) Fn−1
therefore dn (En Fn ) dn (En ) En−1
and dn (En Fn ) dn (Fn ) Fn−1
hence dn (En Fn ) En−1 Fn−1
So E F is a subcomplex named intersection of E and F
Proposition 5 f is a chain of (C, d) into (C , d ) . Suppose kerf = (kerfn )n∈
the family of left A− modules
Δn+1
Δ
kerf : . . . → kerfn+1 → kerfn →n kerfn−1 → . . .
with Δn (x) = dn (x),for all x ∈ kerfn
1908
El Hadj Ousseynou Diallo et al.
then (kerf, Δ) is a subcomplex of (C, d)
Proof 5
/
Cn+1
C : ...
f
C : ...
fn+1
/
Cn+1
dn+1
/
Cn
fn
dn+1
/
Cn
dn
/
fn−1
dn
/
...
/
...
Cn−1
/
Cn−1
On a: Δn+1 : kerfn+1 → kerfn
Let us verify that Δn+1 is well defined.
Let be x ∈ kerfn+1 donc fn+1 (x) = 0
Hence dn+1 ◦ fn+1 (x) = 0
The diagramm commutes:
dn+1 ◦ fn+1 (x) = fn ◦ dn+1 (x) = 0 then dn+1 (x) ∈ kerfn
but Δn+1 (x) = dn+1 (x) therefore Δn+1 (x) ∈ Kerfn .
On another hand if x = y then dn+1 (x) = dn+1 (y) and Δn+1 (x) = Δn+1 (y) ,
so Δn+1 is well defined and it it clear it is a morphism
we calculate Δn ◦ Δn+1
for x ∈ kerfn+1 whe have Δn ◦Δn+1 (x) = Δn (dn+1 (x)) = dn ◦dn+1 (x) therefore
Δn ◦ Δn+1 = 0
Proposition 6 Considering a chain map f of (C, d) into (C , d ) .
Then (Imf, α) = (Imfn , αn ) is a chain complex defined by : for all n ∈ Z we
have αn : Imfn → Imfn−1 with αn (y) = dn (y)
αn+1
α
((Imf, α) : . . . → Imfn+1 → Imfn →n Imfn−1 → . . . )
(C , d)
Proof 6
Let be αn+1 : Imfn+1 → Imfn
Imfn+1
y
→
→
Imfn
dn+1 (y)
..Considering y ∈ Imfn+1 there is x ∈ Cn+1 such that fn+1 (x) = y then
◦ fn+1 (x) = dn+1 (y)
but dn+1 ◦ fn+1 = fn ◦ dn+1 hence dn+1 (y) = fn (dn+1 (x))
therefore αn+1 (y) = dn+1 (y) ∈ Imfn
so αn+1 is well defined
...If y=y’ ,dn+1 (y) = dn+1 (y ) then αn+1 (y) = αn+1 (y )
....We calculate αn ◦ αn+1
Considering y ∈ Imfn+1 donc αn+1 (y) = dn+1 (y)
dn+1
Hopfian objects, Cohopfian objects in the category of complexes
1909
αn ◦ αn+1 (y) = αn (dn+1 (y)) like αn (x) = dn (x) we have αn ◦ αn+1 (y) = dn ◦
dn+1 (y)
So αn ◦ αn+1 = 0
that proove (Imf, α) is a subcomplex of (C , d )
Proposition 7 Considering (E, δ) a subcomplex of (C, d) .Suppose that K =
(Kn )n∈ o? Kn = Cn |En .Then (K, α)) is a chain complex named quotient
complex of C by E and denoted C/E with α = (αn ) and αn : Kn → Kn−1
Proof 7
Considering pn : Cn → Kn the canonical epimorphism
Suppose that αn = pn−1 ◦ dn ◦ p−1
n
p−1
n
d
pn−1
n
Kn → C n →
Cn−1 → Kn−1
ȳ → y → dn (y) → pn−1 (dn (y))
αn (ȳ) = pn−1 ◦ dn ◦ p−1
n (ȳ)
..Demonstrate that αn have a sense
−1
−1
Let be ȳ ∈ Kn then p−1
n (ȳ) ∈ Cn and dn (pn (ỹ)) ∈ Cn−1 , pn−1 (dn (pn (ȳ))) ∈
Kn−1
...Demonstrate that αn is well defined
Considering ȳ = ȳ then ȳ − ȳ = 0̄ therefore y − y = 0̄ so y −y ∈ Kerpn = En
so pn−1 ◦ dn (y − y ) = 0
This proove that pn−1 ◦ dn (y) − pn−1 ◦ dn (y ) = 0
−1 hence pn−1 ◦ dn (y) = pn−1 ◦ dn (y ) then pn−1 ◦ dn ◦ p−1
n (ȳ) = pn−1 ◦ αn ◦ pn (ȳ )
so αn (ȳ) = αn (ȳ ) then αn well defined
..Demonstrate that αn ◦ αn+1 = 0
−1
αn = pn−1 ◦ dn ◦ p−1
n et αn+1 = pn ◦ dn+1 ◦ pn+1
−1
αn ◦ αn+1 = (pn−1 ◦ dn ◦ p−1
n ) ◦ (pn ◦ dn+1 ◦ pn+1 )
−1
αn ◦ αn+1 = pn−1 ◦ dn ◦ (idCn ) ◦ dn+1 ◦ pn+1
So αn ◦ αn+1 = pn−1 ◦ (dn ◦ dn+1 ) ◦ p−1
n+1 = 0 then αn ◦ αn+1 = 0
therefore (K, α) a quotient complex of C by E
Corollary 1 Considering f : (C, d) → (C , d) a chain map.There is C /Imf
a quotient complex .It is named the cokernel of f and denoted cokerf =
(cokerfn )n∈
Proof 8 Imf is a subcomlpex of C Soit αn : Imfn → Imfn−1
y → αn (y) = dn (y)
/Imfn−1
u : Cn /Imfn → Cn−1
Considering n
z̄
→ un (z̄)
1910
El Hadj Ousseynou Diallo et al.
Considering Kn = Cn /Imfn et pn : Cn → Kn the canonical surjection
we have un (z̄) = pn−1 ◦ d n ◦ pn −1 (z̄)
un is well defined d’après the proposition 2.0.6 et (K , u) is a chain complex
Corollary 2 Let be f a chain map of (C, d) into (C , d ).The chain complex
C/Kerf is named coimage of f et denoted coimf
Proof 9
Suppose : Δn : kerfn → kerfn−1 with Δn (x) = dn (x), x ∈ kerfn for all
n∈Z
pn−1
p−1
dn
n
Given Kn = Cn /kerfn et Kn → Cn →
Cn−1 → Kn−1 et βn = pn−1 ◦ dn ◦ p−1
n
βn : Kn → Kn−1 have a sense and is well defined ,satisfy βn ◦ βn+1 = 0 .So
(K, β) is a chain complex
Theorem 1 Caracterisation of a monomorphism of COM P Considering f a
chain map of (C, d) into (C , d).
Then f is a monomorphism ofCOM P if and only if Kerf = 0
Proof 10 Suppose that f is a monomorphism of (C, d) into (C , d) hence f ◦
u = f ◦ v implies u = v then for all n ∈ Z fn ◦ un = fn ◦ vn implies un = vn
so fn is a monomorphism kerfn = 0, for all n ∈ Z d’o? kerf = 0
Reciprocally suppose kerf = 0 hence kerf is a zero complex so each term is
zero so fn is a monomorphism of A− modules then for all n ∈ Z it gives if
fn ◦ un = fn ◦ vn hence un = vn so (f ◦ u)n = (f ◦ v)n implies (u)n = (v)n
finally f is a monomorphism of chain complexes
Theorem 2 Given f : (C, d) → (C , d) a morphism of chain complexes .Then
f is an epimorphism of chain complexes if and only if Imf = C Proof 11 Suppose that f is a epimorphism of COMP then u ◦ f = v ◦ f it
gives u = v hence for all n ∈ Z (u ◦ f )n = (v ◦ f )n implies un = vn fn is an
epimorphism ofA− modules then Imfn = Cn , for all n ∈ Z so Imf = C Reciprocally suppose Imf = C hence ,for all n ∈ Z ,Imfn = Cn then fn
is an epimorphism of A− modules so for all n ∈ Z,un ◦ fn = vn ◦ fn alors
(u◦f )n = (v◦f )n implies un = vn then f is an epimorphism of chain complexes
3
hopf ian objects , cohophian objects, injectiveobjects, projective objects in category COM P
Definition 1 Given C an object category COM P .Then C is named simple
if it is different from zero and it’s unique subcomplex is itself
Hopfian objects, Cohopfian objects in the category of complexes
1911
Definition 2 Considering C an object of COM P and f a chain map of C into
C . Then C is said to be hopf ian (respectively cohopf ian ) if any epimorphism
f (respectively any monomorphism) is an isomorphism
Proposition 8 Any simple complex is hopf ian and cohopf ian
Proof 12 Given C a simple object of COM P it means that ∀n ∈ Z, Cn is
simple Suppose fn is surjective We obtain imfn = Cn et Cn = 0 suppose
kerfn = 0 then kerfn is a submodule of Cn different from 0 and different from
Cn what contradicts the hypothesis Cn simple then C is hopfian
In the other hand suppose C is an simple object and suppose Cn simple with fn
injective then kerfn = 0 suppose imfn = Cn we would have imfn = 0 because
fn = 0 et imfn = Cn what contradicts the hypothesis Cn simple
Proposition 9 Given C a simple complexe and f a morphism of C in itself
.Then the set of morphisms C provided with this operations + et ◦ is a field
Proof 13 Like Cis simple means that ∀n ∈ Z , Cn is simple It is known
that (End(Cn ), +, ◦) is an associative and unitarian ring Considering fn ∈
End(Cn )|0 and fn = 0 alors kerfn = Cn and kerfn are submodules of Cn then
kerfn = 0 therefore fn is injective
In the other hand f (Cn ) ≤ Cn et fn = 0then fn (Cn ) = Cn therefore fn issurjective
Proposition 10 Considering C an object of COM P with E subcomplex different from zero of C . If C|E hopf ian alors C is an hopf ian object of COM P
Proof 14 Assume that C is a non hopfian complex hence there is k ∈ Z such
fk : Ck → Ck as a surjection different from an isomorphism As Ek = kerfk
we have Ek = 0 because fk is not injective Then fk induces an isomorphism
f¯k : Ck |Ek → Ek If πk : Ck → Ck |Ek demonstrate the canonical surjection the
compositeπk ◦ f¯k : Ck |Ek → Ck |Ek is a surjection which is not injective then
the kernel of the composite is not reduced to 0 . what contradicts the hypothesis
Ck |Ek hopf ian
Proposition 11 Given C an object of COMP which own proper subcomplex
E is cohopf ian then C is cohopf ian
1912
El Hadj Ousseynou Diallo et al.
Proof 15 By using the method of proof by contradiction we get :assume that C
is not cohopf ian then there are k ∈ Z and an injective morphism uk : Ck → Ck
not surjective Soit Ek = imuk As Ek Ck alors uk induces an isomorphisme
u¯k : Ck → Ek with u¯k |Ek : Ek → Ek is a non surjective injection .What
contradicts Ek cohopf ian
Definition 3 A A sub module N of M is said to be completely invariant if
for any endomorphism f of M we have f (N ) N
Definition 4 Given E, N, M A− modules . E is said to be injective if for
any monomorphism α : N → M and for any morphism φ : N → E, there is a
morphism ψ : M → E verifying φ = ψ ◦ α
Definition 5 Considering E, R, S chains complexes of A− modules .Then E
is an injective chain complex if for any monomorphism of chains complexesα :
S → R and for any morphism of chains φ; S → E there is a map chain
ψ : R → E such as ψ ◦ α = φ
/R
~
~~
φ
~~ψ
~
~
S
α
E
Remark:If E injective chain complex then E is an injective object of category of complexes of A− modules
wn+1
w
Theorem 3 Given (E, w) : . . . → En+1 → En →n En−1 → . . . a chain
complex of A− modules.Then E isinjective if and only if for all n ∈ Z, En is
injective
Proof 16 Suppose E is injective and considering f : M → N a monomorphism of A− modules
and φn : M → En a morphism of A− modules
Given S et R two chains complexes of A− modules and α a chain map of S
into R such as:
S : ...
α
R : ...
/
Sn+1
αn+1
un+1
/
Sn
αn
/ Rn+1 vn+1 /
Rn
un
/
αn−1
vn
/
/
...
/
...
Sn−1
Rn−1
Hopfian objects, Cohopfian objects in the category of complexes
1913
where for all n ∈ Z
Sn = M and un = IdM
with Rn = N and vn = IdN
αn is a monomorphism of A− modules ,for all n ∈ Z
Given φ a chain map of S into E such as :
/
Sn+1
S : ...
φn+1
φ
/
/
Sn
wn+1
un
/
/E
wn
/
...
/
...
/
...
/
...
Sn−1
φn−1
φn
En+1
E : ...
un+1
/
En−1
n
where φk : M → Ek any morphisms of A−modules
E are injective ther is ψ a chain map such as:
/
Sn+1
R : ...
ψn+1
ψ
/
/
Sn
wn+1
vn
/
Sn−1
ψn−1
ψn
En+1
E : ...
vn+1
/ En
wn
/
En−1
such as ψ◦α = φ donc ψn ◦f = φn .Hence there is ψn : Rn → En tel ψn ◦f = φn
En est injectif
Reciprocally suppose for all n ∈ Z, En is an A− injective module and demonstrate that E is an injective chain complex
Considering γ a monomorphism of chains complexes
/
Sn+1
S : ...
γn+1
γ
/
/
Sn
wn+1
un
/
/
...
En−1
/
...
Sn−1
/
...
/
...
Sn−1
γn−1
γn
En+1
E : ...
un+1
/E
wn
/
un
n
/
Given β a chain complex such as:
/
R : ...
β
E : ...
Sn+1
βn+1
/
En+1
un+1
Sn
βn
wn+1
/ En
/
βn−1
wn
/
En−1
Then for all n ∈ Z, En is injective and γn a monomorphism of A−modules so
1914
El Hadj Ousseynou Diallo et al.
there is λn : Rn → En verifying βn = λn ◦ γn Considering λ such as:
/
R : ...
λ
Rn+1
λn+1
/
E : ...
vn+1
/
λn
En+1
wn+1
vn
Rn
/
λn−1
wn
/ En
/
...
/
...
Rn−1
/
En−1
Demonstrate λ is a chain map
/
S ...
Sn+1
β
R...
$
/
Rn+1
λ
E ...
/
/
Sn
γn+1
γ
un+1
vn+1
λn+1
Rn
En+1
wn+1
/
vn
/
Rn−1
λn
En
wn
/
...
γn−1
γn
/
/ Sn−1 un−1 /
un
vn−1
/
...
/
...
λn−1
En−1
wn−1
It is enough wn+1 ◦ λn+1 = λn ◦ vn+1
We have wn+1 ◦ βn+1 = βn un+1 because β is a chain
But βn = λn ◦ γn
then wn+1 ◦ (λn+1 ◦ γn+1 ) = (λn ◦ γn ) ◦ un+1
implies (wn+1 ◦ λn+1 ) ◦ γn+1 = λn ◦ (γn ◦ un+1 )
So (wn+1 ) ◦ λn+1 ) ◦ γn+1 = λn ◦ (vn+1 ◦ γn+1 )
Finally (wn+1 ◦ λn+1 ) ◦ γn+1 = (λn ◦ vn+1 ) ◦ γn+1
Or γn+1 is a monomorphism of A− modules so wn+1 ◦ λn+1 = λn ◦ vn+1 , for
all n ∈ Z
d’o? λ is a chain
Let us verify λ ◦ γ = β
We know that n ∈ Z, βn = λn ◦ γn avec β = (βn ) ,γ = (γn ) et λ = (λn ) then
λ ◦ γ = β.This prooveE is injective
Definition 6 Given N a A− submodule of a module M is said to be completely
invariant if for any endomorphism f of M we have f (N) N
Definition 7 Given C : . . . Cn+1 → Cn → Cn−1 → . . .
a chain complex of A− modules and E : . . . En+1 → En → En−1 → . . .
a subcomplex of C
Then E is said to be completely invariant subcomplex of C if for all n ∈ Z, En
is completely invariant
Definition 8 A A− submodule E of M is essential in M if for any submodule
Hopfian objects, Cohopfian objects in the category of complexes
1915
K different from zero of M , we have E K = 0; M is named essential
extension of E
Definition 9 Considering C : . . . Cn+1 → Cn → Cn−1 → . . . a chain complex
of A− modules and E : . . . En+1 → En → En−1 → . . . a subcomplex of C .
Then E is said to be essential chain complex in C if for all n ∈ Z, En is
essential in Cn
3.0.1
projective
Definition 10 Given M a A− module . An A− module P is said to be
projective if for any epimorphism g : M → N and for all morphism f : P → N
, there is a morphism h : P → M verifying f = g ◦ h
Definition 11 Considering E, S, R three chains complexes of A− modules.
Then E is said to be projective complexif for any epimorphism α : R → S
and for all chain map φ : E → S , there is a chain map ψ : E → R suvh as
α ◦ ψ = φ ,illustrated by the following diagramme:
E
R
3.0.2

ψ
α
/
φ
S
remark
IfE is a projective chain of A− modules then E an projective object of the
category of complexes of A− modules
vn+1
v
n
Theorem 4 Considering E : . . . → En+1 → En →
En−1 → . . . a chain of
A− modules .E is a projective chain complex if and only if for all n ∈ Z ,En
is a A− projective module .
Proof 17 Suppose E is projective
Considering f : N → M a epimorphism of A− modules and φn : En → M a
morphism of A− modules
Given S and R two chains complexes and α a chain map of R into S such as:
/ Rn+1 vn+1 / Rn vn / Rn−1
/ ...
R : ...
α
S : ...
αn+1
/
Sn+1
αn
un+1
/
Sn
αn−1
un
/
Sn−1
/ ...
1916
El Hadj Ousseynou Diallo et al.
o? Rn = M et vn = IdM
Sn = N et un = IdN
Given αn an epimorphism of A− modules
Considering φ a chain map of E into S verifying:
/
En+1
E : ...
φn+1
φ
/
/E
wn
/
φn−1
un+1
/
Sn
/
...
/
...
En−1
n
φn
Sn+1
S : ...
wn+1
un
/
Sn−1
With φk : Ek → M are any morphisms of A− modules.
Like E is projective then there is a chain map ψ such as :
/
En+1
E : ...
ψn+1
ψ
R : ...
/
wn+1
/E
Rn+1
/
vn+1
/
Rn
vn
/
...
/
...
En−1
ψn−1
ψn
wn
n
/
Rn−1
o? α ◦ ψ = φ so for all n ∈ Z, αn ◦ ψn = φn .Then there is ψn : En → Rn such
as f ◦ ψn = φn .So En is projective
Reciprocally let us suppose for all n ∈ Z , En is a A− projective module and
let us demonstrate E is a projective chain complex
Considering γ an epimorphisme such as:
R : ...
/ Rn+1 vn+1 /
γn+1
γ
/
S : ...
Rn
γn
Sn+1
vn
/
γn−1
un+1
/
wn+1
/E
Sn
/
...
/
...
Rn−1
un
/
Sn−1
Given β a chain map verifying:
/
E : ...
β
En+1
βn+1
/
S : ...
Sn+1
wn
/
βn−1
βn
un+1
/
Sn
/
...
/
...
En−1
n
un
/
Sn−1
Then like for all n ∈ Z ,En is projective and γn is an epimorphism of A−
modules so there is λn : En → Rn verifying γn ◦ λn
Given λ such as:
/
E : ...
λ
R : ...
En+1
λn+1
/
Rn+1
wn+1
/E
wn
/
n
λn
vn+1
/ vn
λn−1
vn
/
/
...
/
...
En−1
Rn−1
1917
Hopfian objects, Cohopfian objects in the category of complexes
Let us demonstrate that λ is a chain map
R...
/
Rn+1
/
S ...
β
$
E ...
/
/
Rn
γn+1
γ
λ
vn+1
Sn+1
un+1
βn+1
/
En+1
wn+1
/
vn
/
...
/ Sn−1 un−1 /
...
γn−1
γn
Sn
un
βn
En
wn
/
vn−1
/
Rn−1
βn−1
En−1
wn−1
/
...
It is enough that λn+1 ◦ wn+1 = vn+1 ◦ λn
but βn+1 ◦ wn+1 = un+1 ◦ βn because β is a chain map
Like γn ◦ λn = βn then :
γn+1 ◦ λn+1 ◦ wn+1 = un+1 ◦ (γn ◦ λn )
So (γn+1 ◦ λn+1 ) ◦ wn+1 = un+1 ◦ (γn ◦ λn )
Hence γn+1 ◦ (λn+1 ◦ wn+1 ) = γn+1 ◦ (vn+1 ◦ λn )
But γn+1 and a epimorphism so λn+1 ◦ wn+1 = vn+1 ◦ λn what justifies λ is a
chain map
Let us verify that γ ◦ λ = β
We know that for all n ∈ Z , βn = γn ◦ λn d’o? β = γ ◦ λ. which prooves that
E is an injective chain complex
Proposition 12 Considering (Ci ) a family of chains complexes ⊕Ci is projectiveif and only if Ci is projective:
Proof 18 In this proof we use the caracterisation theorem of projectives chains
complexes
Let us suppose that ⊕Ci is projective then for all n ∈ Z , ⊕Cin is a A− module
projective therefore Cin is an A− projective module
So Ci is a projective chain complex for all i
Reciprocally let us suppose that Ci is a projecive chain complex then for all
n ∈ Z ,Cin is an A− projective module therefore ⊕Cin is projective finally ⊕Ci
est projective
Definition 12 A A− sub- module E ofM est said to be superfluous in M if
for any submodule K of M such as K + E = M then K = M
Definition 13 Given C : . . . → Cn+1 → Cn → Cn−1 → . . . a chain complex
of A− modules and E : . . . → En+1 → En → En−1 → . . . a subcomplex of C
1918
El Hadj Ousseynou Diallo et al.
such as:
E is said to be superfluous in C if for all n ∈ Z , En is superfluous in Cn
Theorem 5 Considering a projective complexe and E a completely, invariant
et superfluous subcomplex of C then E is hopf ian if only if C/E is hopfian
Proof 19 and En is completely invariant and superfluous in Cn
Then any epimorphism of E in itelf is an isomorphism so En is hopf ian
So Cn /En is hopf ian for all n ∈ Z , in conclusion C/E est hopfian
Reciprocallylet us suppose C/E is hopf ien donc Cn /En est hopf ien avec
Cn projectif et En est un A− sous- module complètement invariant et superflu
dans Cn donc pour tout n ∈ Z , En is hopf ian then E is hopf ian
Theorem 6 Considering C an un injective chain complexe and E a completely
invariant and essential subcomplex of C in Cthen E is cohopf ian if and only
if C is cohopf ian
Proof 20 Let us suppose that Eis cohopfian then En is cohopf ian and En is a
A−completely invariant and essential submodule in Cn then Cn est cohopf ian
for all n ∈ Z , so C is cohopf ian
Reciprocally let us suppose that C is cohopf ian that Cn is cohopf ian and
En is an A− completely invariant and essential submodule in Cn
. Then En is cohopf ian for all n ∈ Z therefore E is cohopf ian
dn+1
d
dn−1
n
Cn−1 →
Definition 14 Given a chain complex C : . . . → Cn+1 → Cn →
. . . said to be noetherian (respectively artinian ) if for all n ∈ Z ,Cn is
noetherian (respectivement artinian )
Theorem 7 Considering C a chain complex of A-modules then :
- if C is projective and cohopf ian then C is hopf ian
- if Cis injective and hopf ian then C is cohopf ian
Proof 21 Let us suppose that C is a projective and cohopf ian complexe then
Cn is an A− projective and cohopf ian module
so Cn is hopf ian for all n ∈ Z so C is hopf ian
Let us suppose that C an injective and hopf ian chain complexe then Cn is a
A− injective and hopfian module
then Cn is cohopfian for all n ∈ Z so C is cohopfian
1919
Hopfian objects, Cohopfian objects in the category of complexes
4
FITTING chains complexes
Definition 15 An A− module M is said to be of F IT T ING if for any endomorphism f of M ,there is an integer n ≥ 1 such as M = Kerf n ⊕ Imf n
Definition 16 A chain complex is said to be of F IT T ING if for all chain
map f of C in itself , there is an integer n ≥ 1 tel C = Kerf n ⊕ Imf n
Theorem 8 Considering C : . . . → Cn+1 → Cn → Cn−1 → . . . a chain
complex of A− modules. C is F IT T ING chain if and only if for all n ∈ Z ,
Cn is an F IT T ING A− module
Proof 22 C is a Fitting complex then there is an integer n non nul such as
C=Kerf n ⊕ Imf n
hence for all k ∈ Z we have Ck =Kerfkn ⊕ Imfkn then k ∈ Z ,Ck is a F itting
module
Reciprocally if Ck is a F itting module then there is an iteger n different from
zero such as
Ck =Kerfkn ⊕ Imfkn
so for all k ∈ Z ,Ck is a F itting module then C is a F itting chain complex
Theorem 9 Any projective and cohopf ian or injective and hopf ian is a
F itting chain complex
dn+1
d
dn−1
n
Cn−1 → . . .
Proof 23 Given a chain complex C : . . . → Cn+1 → Cn →
using previous theorem 6 t if C is projective andcohopf ian then C is hopf ian
.Then C is hopf ian et cohopf ian
therefore for all n ∈ Z , Cn is an A− hopf ian and cohopf ian module so Cn
est de F itting
So as previous theorem6 if C is an injective and hopf ian complex then C is
cohopf ian .So C is hopf ian et cohopf ian
therefore for all n ∈ Z , Cn is an A− module hopf ian and cohopf ian then Cn
is a F itting module then C is a F itting chain complexe
References
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1920
El Hadj Ousseynou Diallo et al.
[3] Ben Maaouia Mohamed Ben Fraj, Doctorat de 3ème cycle - Faculté des
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[10] Rotman Joseph J., An introduction to homological Algebra Academic
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Received: July 7, 2013
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