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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 [1] Anderson, f. W. And fuller, K. R., Rings and categories of modules, New York, Springer - Verlag (1973) [2] Ben Maaouia Mohamed Ben Fraj, Dea - Faculté des Sciences et Technique(juillet 1998) 1920 El Hadj Ousseynou Diallo et al. [3] Ben Maaouia Mohamed Ben Fraj, Doctorat de 3ème cycle - Faculté des Sciences et Technique UCAD - Dakar (Juillet 2003) [4] Goutam Mukherjee, Hopﬁan and Cohopﬁan G-CW-complexes ,proceedings of the American Mathematical Society Volume 125, Number 4, April 1997, Pages 12291236 S 0002-9939(97)03778-7 [5] Haghany. A Journal of Algebra 243,765-779(2001) Modules whose Injective Endomorphismes are Essential [6] Hmaimou. A and al/Journal of algebra Generalized Fitting modules and rings 308(2007)199- 214 [7] Hiremath, Hopﬁan rings and co-Hopﬁan modules, Indian J. Pure and Appl. Math., 17, 1986, 895-900. MR 87i :163 [8] Rotman Joseph J., Notes on homological algebra, University of Illinois, Uraba (1968) [9] Rotman Joseph J., An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman (Aug 17, 1988) [10] Rotman Joseph J., An introduction to homological Algebra Academic Press, New York [11] Sanghare Mamadou ,Thèse d’état ,Faculté des Sciences et Techniques, UCAD, Dakar, ( 17 décembre 1993.) [12] Varadarajan. K , Hopﬁan and co-Hopﬁan objects, Publicacions Matematiques, 36, 1992, 293- 317. MR 93i :16002 Received: July 7, 2013