COMPARISON OF STANDARD AND (EXTENDED) d-homologies

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1 U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 2, 2018 ISSN COMPARISON OF STANDARD AND (EXTENDED) d-homologies M. Z. Kazemi Baneh* 1 and S. N. Hosseini 2 In this article we compare the standard homology, d-homology and the extended d-homology functors with respect to a kernel transformation d. We also compare the homology and extended homology functors with respect to two kernel transformations. Keywords: abelian category, standard homology, (extended) d-homology, category of R-modules. MSC2010: 18E10, 55N Introduction and Preliminaries The definition of the standard homology functor has been extended from the category of R-modules to abelian categories in [9]. In [5] we have defined the homology with respect to a kernel transformation d, also called the d-homology, and in [7] have defined extended d-homology in more general categories. In this section we have given the definition of d-homology, extended d-homology and some of the results obtained in [5]. In Section 2, we have compared the standard homology as given in [9] and the d-homology, by giving a natural transformation from the standard homology functor to the d-homology functor. Standard homology and extended d-homology have been compared in [4]. In Section 3, we have compared the d- homology and the extended d-homology, by giving a natural transformation from the extended d- homology functor to the d-homology functor. Then we have considered conditions under which this natural transformation is a natural isomorphism. Also we have shown, in an abelian category, the d- homology is the extended d-homology with respect to a particular kernel transformation d. Some other results are also given at the end of this section. In Sections 4 and 5, respectively we have compared the extended homology functors and homology functors with respect to two kernel transformations. Throughout the manuscript we let R-mod be the category of R-modules over a commutative ring with unity. To this end, for a pointed category C, following the notation of [1, 5, 7], we recall: 1 Assistant Professor, Department of Mathematics, Faculty of Sciences, University of Kurdistan, Iran, zaherkazemi@uok.ac.ir 2 Professor, Department of Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Iran, nhoseini@uk.ac.ir 205

2 206 M. Z. Kazemi Baneh*, S. N. Hosseini (i) for f : A B, the maps K f k f A, B c f π 1 C f and P f A are re- π 2 spectively, the kernel, the cokernel and the kernel pair of f; so Equ(f, g) equ(f,g) A denotes the equalizer and B coe(f,g) Coe(f, g) the coequalizer of a pair A f B. (ii) [2, 5]. The image of f is the coequalizer of the kernel pair of f. Furthermore f = m f e f in which e f = coe(π 1, π 2 ). (iii) [5]. Given the diagram below in which the squares are commutative and the rows are coequalizers, i is the unique map making the right square commute. Furthermore, i is a regular epi. A f g r A f B q C g B q For a pointed category C with pullbacks and pushouts, let C be the arrow category and Ĉ be the pair-chain category of C. Pair-chains are composable pairs, (f, g), of morphisms of C, such that gf = 0 and morphisms from (f, g) to (f, g ) are triples (, β, ) making the following squares commutative: A f B g C β A B C f g Let K : C C be the kernel functor and I : C C be the image functor see[5]. Define K =: K pr 2 : Ĉ C and I = I pr 1 : Ĉ C. (iv) The natural transformation j : I K : Ĉ C takes the object (f, g) Ĉ to and the morphism (, β, ) : (f, g) (f, g ) to the following commutative square. I(,β) j f g i C K(β,) In a pointed regular ( homological, semiabelian or abelian ) category is monic. See [2, 5]. (v) The homology functor H s that takes the object (f, g) Ĉ to Hs fg = Coker() is called standard homology functor. See [2, 8, 9] (vi) Let S be the squaring functor. A kernel transformation in a C is a natural transformation d : S K K : C C such that for all (f, g) in Ĉ, the pullback, jfg : R fg Kg 2, of along d g and the coequalizer of the pair g

3 Comparison of Standard and (Extended) d-homologies 207 j 1 = pr 1 j fg and j 2 = pr 2 j fg exist, where pr 1 and pr 2 are the projection maps. Let R be a commutative ring with unity. Any kernel transformation in R-mod is of the form d = rpr 1 + spr 2, for some r, s R. (vii) The d-homology functor H d : Ĉ C takes (f, g) Ĉ to Hd fg = Coe(j 1, j 2 ) and the morphism (, β, ) to H d (, β, ). We have the following commutative diagram. K(β,) q 1 q 1 H d fg H d f g H d (,β,) (viii) Let m : A C and j : B C be two maps in C. Define A + C B also denoted by A+B by the pushout of the pair (, ) where B P jm A is the pullback of (j, m). Let d be a natural transformation from S K to K, (f, g) Ĉ and be the diagonal map. we have the maps m dg g :I dg g (such that m dg g e dg g = d g g ) and :. The sum I dg g + is therefore obtained by the following diagrams: and P jm P jm pb I dg g m d g g po i I dg g h I dg g + Commutativity of the left diagram implies that there is a unique map β : I dg g + making the following diagram commute. P jm I dg g po i h I dg g + m dg g β With H fg d = C β, the cokernel of β, we have: (ix) For each morphism (σ, δ, ζ) : (f, g) (f, g ) in Ĉ, there is a unique map H d (σ, δ, ζ) : H fg d H f d g, such that the following diagram commutes: c β H d fg K(δ,ζ) c β Hd f g H d (σ,δ,ζ)

4 208 M. Z. Kazemi Baneh*, S. N. Hosseini The mapping H d : Ĉ C that takes the object (f, g) Ĉ to H fg d and the morphism (, β, ) to H d (, β, ) is a functor. The functor H d :Ĉ C is called the extended d-homology or the extended homology functor with respect to the kernel transformation d. 2. Standard homology versus d-homology In this section, unless stated otherwise, we assume C is a pointed category with pullbacks, cokernels and coequalizer of kernel pairs, and we investigate the relation between the standard homology and the d-homology. For two morphisms f : C A and g : C B there is a unique morphism f, g : C A B such that pr 1 f, g = f and pr 2 f, g = g. Theorem 2.1. [6]. Let d be a kernel transformation in C. There is a pointwise regular epi natural transformation p :H s H d :Ĉ C. Proof. Since j d h 1, 0 = d g j, 0 for h : 0 and R fg is a pullback of the pair (, d g ), there are unique maps ψ and p ψ fg such that j ψ = j, 0 and the following diagram commutes. ψ j 0 c j H s fg j 1 R fg H q fg d j 2 Furthermore by (iii) of 1, p fg is regular epic. Simillarly there are unique maps ϕ and p ϕ fg such that j 2 ϕ = j and j 1 ϕ = 0. Since P ψ fg c j = q = p ϕ fg c j, p ψ fg = pϕ fg and we denote it by p fg. Lemma 2.1. If for (f, g) Ĉ, ψ : R fg or ϕ : R fg is epic, then Hfg s = Hfg d. p ψ fg Proof. H d fg = Coe(j 1, j 2 ) = Coe(j 1 ψ, j 2 ψ) = Coker(j) = H s fg. Theorem 2.2. [5]. In an abelian category C if d = pr 1 pr 2, then H d = H s. 3. d-homology versus Extended d-homology In this section we assume C is a pointed category with pullback and pushout and for a given natural transformation d :S K K we find relations between the d-homology H d and the extended d-homology H d. We know there are natural transformations p : H s H d and u : H s H d which are pointwise regular epic. Furthermore for (f, g) Ĉ there are ψ : R fg and i : + I dg g such that β i = j and j 1 ψ = j. Then we have: Lemma 3.1. If i is epic, then there is an epi n fg : H d fg Hd fg. Proof. By Lemma 3.2 u fg in [4] is an isomorphism. Define n fg = p fg u 1 fg.

5 Comparison of Standard and (Extended) d-homologies 209 Theorem 3.1. If for all (f, g) Ĉ, i is epic, then there is a pointwise regular epi natural transformation n : H d H d. Proof. By hypothesis u is a natural isomorphism. Define n = p u 1. p is a regular epi, so is n. To the end of this section we assume C is an abelian category. For (f, g) Ĉ, let j be the pullback of j along d g and q be the coequalizer (pr 1 j, pr 2 j ). Then q = coker(pr 1 j pr 2 j ) = coker(δj ) in which δ = pr 1 pr 2. Factoring δj as δj = me, since e is epic, q = coker(me) = coker(m). Lemma 3.2. Let (f, g) Ĉ. qd g g = 0 if and only if d g g factors through m, i.e. there is : I δj such that d g g = m. Proof. Since m is monic, m = ker(coker(m)) = ker(q). The result then follows. Note: In the proof of theorem 2.1 there exists a unique map ψ, such that j ψ = j, 0. Furthermore j = δ j, 0 = δj ψ so that qj = qδj ψ = 0. Lemma 3.3. Let (f, g) Ĉ. With the map β :I d g g + and δj = me, qβ = 0 if and only if d g g factors through m. Proof. Suppose qβ = 0. Since βh = m dg g and m dg g e dg g = d g g, qd g g = qm dg g e dg g = qβhe dg g = 0. By the above lemma d g g factors through m. Conversely suppose d g g factors through m, so by the above lemma qd g g = 0. So qm dg g e dg g = 0. Since e dg g is epic, qm dg g = 0. Since m dg g = βh, qβh = 0. On the other hand since qj = 0 and j = βi, qβi = 0. The pushoutness of I dg g + implies qβ = 0. Proposition 3.1. Let (f, g) Ĉ. If d g g factors through m, then there is an epi n fg : H d fg Hd fg. Furthermore Hd fg = C k nfg. Proof. By the above theorem qβ = 0, so there is a unique map n fg : H fg d Hd fg, such that n fg c β = q. Since q is epi, so is n fg. The last assertion follows from the fact that C is an abelian category. Theorem 3.2. If for all (f, g) Ĉ, d g g factors through m, then there is a pointwise regular epi natural transformation n : H d H d. Proof. By the above proposition, there is a unique n fg : H fg d Hd fg such that n fg c β = q. To show n = {n fg } is a natural transformation, given (σ, δ, ζ) :(f, g) (f, g ) in Ĉ, by (vii) of 1, H d (σ, δ, ζ)q = q K(δ, ζ), and by (ix) of 1, Hd (σ, δ, ζ)c β = c β K(δ, ζ). So: H d (σ, δ, ζ)n fg c β = H d (σ, δ, ζ)q = q K(δ, ζ) = n f g c β k(δ, ζ) = n f g Hd (σ, δ, ζ)c β Since c β is epic, the following diagram commutes. H d (σ,δ,ζ) H d fg n fg H d fg H f d g H d n f g f g H d (σ,δ,ζ)

6 210 M. Z. Kazemi Baneh*, S. N. Hosseini and so n : H d H d :Ĉ C is a natural transformation. Corollary 3.1. For d = rpr 1 +spr 2, with r, s Z, there is a natural transformation n : H d H d. Proof. Let (f, g) Ĉ. Since d g s, r = 0, there is a unique morphism η : R fg such that the following triangles commute. s, r 0 η R fg j K 2 g d g pb d g j We have q(r+s) = qδ s, r = qδj η = 0η = 0. But d g g = r+s = m r+s e r+s and so qm r+s e r+s = 0. Since e r+s is epic, qm r+s = 0. But m r+s = m dg g = βh, hence qβh = 0. We know qβi = qj = 0 too. It follows that qβ = 0. The result then follows from Lemma 3.3 and Theorem 3.2 Let d = rpr 1 + spr 2 and (f, g) Ĉ. With d the pullback of d along j, we have: c β d g j = c β jd g = c β βid = 0id = 0 and so c β rpr 1 j + c β spr 2 j = 0. On the other hand, c β (r + s) = c β m r+s e r+s = c β βhe r+s = 0he r+s = 0 and so c β rpr 1 j + c β spr 1 j = 0. It follows that c β sδj = 0. Since the square s c β c β commutes, sc β δj = 0. H d fg s Hd fg Corollary 3.2. Let d = rpr 1 + spr 2. If for all A, s :A A is monic, then n : H d H d is an isomorphism. Proof. Let (f, g) Ĉ. Since s is monic, c βδj = 0. Since q is a cokernel, there exists a unique map n : H d H d such that n q = c β. It then follows that n n = 1 and nn = 1. Hence n fg : H fg d = Hfg d is an isomorphism. Remark 3.1. For C = R-mod Corollaries 3.1 and 3.2 can be generalized to the case r, s R. Also if for all R- module A, s :A A by s(a) = sa is monic, then H d = H d. Example 3.1. Let C be the category of R-modules and (f, g) Ĉ. For d = rpr 1+pr 2 or d = pr 1 + rpr 2 with r R we have: Hfg d = H fg d = (1+r)+

7 Comparison of Standard and (Extended) d-homologies Extended homologies with respect to Two Kernel Transformations In this section we let C be a pointed with pullbacks and pushouts and we investigate the relation between extended homologies with respect to two kernel transformations. Theorem 4.1. Let d and d be two kernel transformations. If for (f, g) Ĉ, m d g g factors through m d g g, then there is a unique morphism p fg : H fg d d H fg such that p fg c βd = c βd. In addition p fg is regular epic. Proof. By hypothesis, there is l :I dg g I d g g such that m l = m, where m = and m = m d g g. So we have the following pullbacks: m dg g P jm µ I dg g l /// P jm j m I d g g /// Since in the diagram: P jm µ m P jm i 1 I h dg g l + I dg g I d g g h + I d g g (i) the front and back squares are pushouts and the left and the top squares are commutative, there exists unique map ν such that the right and the bottom squares are commutative. Since β d i = j = β d i = β d νi and β d h = m = m l = β d h l = β d νh, by pushoutness of the back square, β d = β d ν. Since c βd β d = c βd β d ν = 0, there is a unique map p fg such that the following diagram commutes. + I dg g β d ν c βd Hd fg i ν + I d g g β d c βd p fg Hd fg

8 212 M. Z. Kazemi Baneh*, S. N. Hosseini By (iii) of 1 p fg is regular epic. We know m :I cod : C C is a natural transformation. Theorem 4.2. Let d and d be two kernel transformations. If for all g Ĉ, m d g g factors through m d g g, then there is a pointwise regular epi natural transformation p : H d H d. Proof. Let (σ, δ, ζ) :(f, g) (f, g ) be in Ĉ. By (ix) of 1, we have H d (σ, δ, ζ)c βd = c β d K(δ, ζ) and H d (σ, δ, ζ)c βd = c β K(δ, ζ). By Theorem 5.1, p fgc d βd = c βd and p f g c β d = c β. So p d f g Hd (σ, δ, ζ)c βd = H d (σ, δ, ζ)p fg c βd. Since c βd is epic, the following square commutes. H d fg p fg H d fg H d (σ,δ,ζ) H d f g p f g Hd f g H d (σ,δ,ζ) Corollary 4.1. Let d and d be two kernel transformations. If for all g Ĉ, m d g g factors through m d g g by l :I dg g I d g g and l is epic, then p : H d H d is a natural isomorphism. Proof. Let (f, g) Ĉ. Since l is epic and the front square in the diagram (i) of Theorem 5.1 is a pushout, the morphism ν obtained in that diagram is epic. It follows that c βd = coker(β d ) = coker(β d ν) = coker(β d ) = c βd and so p fg : Hd fg = Hd fg is an isomorphism. Example 4.1. Let C be an abelian category, d = rpr 1 + spr 2 and d = r pr 1 + s pr 2 such that r + s = r + s. Then H d = H d. Proof. Since d = r + s = r + s = d, the result follows. 5. Homologies with respect to Two Kernel Transformations In this section we let C be a pointed regular category with coequalizers and we investigate the relation between homologies with respect to two kernel transformations. Recall [2] for each morphism f : A B in the category C, there is a functor f 1 : Sub(B) Sub(A) in which f 1 (m) : f 1 (M) A for a subobject m : M B, is the pulback of m along f. Since K : Ĉ C is a functor, we can rewrite the natural transformation d : S K K : C C to d : S K K : Ĉ C. We know d 1 ( ) = jfg : d 1 ( ) = R fg Kg 2 for (f, g) Ĉ. Then the following diagram is

9 Comparison of Standard and (Extended) d-homologies 213 a pullback in the category F unct(ĉ, C) of functors from Ĉ to C. d 1 (I) d I d 1 (j) j S K K d Theorem 5.1. Let d and d be two kernel transformations and (f, g) Ĉ. If j j i.e. d 1 ( ) = j factors through d 1 ( ) = j, then there is a unique map p fg : Hfg d Hd fg such that p fg q = q. In addition p fg is regular epic. Proof. By assumption there is a map ϕ such that j ϕ = j. Then j 1 ϕ = j 1 and j 2 ϕ = j 2. Therefore there is a unique map p fg : Hfg d Hd fg such that the following diagram commutes. R fg j 1 ϕ R fg j 2 j 1 j 2 q H d fg p fg H d q fg Theorem 5.2. Let d and d be two kernel transformations. If d 1 (j) d 1 (j), then there is a pointwise regular epi natural transformation p : H d H d. Proof. Let (σ, δ, ζ) :(f, g) (f, g ) be in Ĉ. By (vii) of 1, we have Hd (σ, δ, ζ)q fg = q f g K(δ, ζ) and Hd (σ, δ, ζ)q fg = q f g K(δ, ζ). By the above theorem, p fg q fg = q fg and p f g q f g = q f g. So H d (σ, δ, ζ)p fg q fg = p f g Hd (σ, δ, ζ)q fg. Since q fg is epic, the following square commutes. H d fg p fg H d fg H d (σ,δ,ζ) H d f g p f g H d f g H d (σ,δ,ζ) Corollary 5.1. Let d and d be two kernel transformations. If d 1 d 1, then there is a pointwise regular epi natural transformation p : H d H d. Corollary 5.2. Let C be an abelian category and let k be an integer. If d = kd, then there is a natural transformation p : H d H d. Proof. Since for (f, g) in Ĉ, k = k, there is a unique ϕ such that the following diagram commutes.

10 214 M. Z. Kazemi Baneh*, S. N. Hosseini R fg ϕ d g k j fg R fg d g K 2 g d g k K 2 g d g So jfg j fg and hence there is a unique map p fg : Hfg d Hd fg epic. which is regular Corollary 5.3. Let C = R-mod and k R. If d = kd, then there is a natural transformation p : H d H d. 6. Conclusions The d-homology and the extended d-homology functors are introduced and studied in [5, 6, 4] and [7]. In order to further study and investigate the properties of these homology functors, here we have done a comparison of the standard homology, the d-homology and the extended d-homology functors. R E F E R E N C E S [1] J. Adamek, H. Herrlich, G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, [2] F. Borceux, D. Bourn, MalCev, Protomodular, Homological and Semi-Abelian Categories, Kluwer Academic Publishers, [3] P. Freyd, Abelian Categories, Harper and Row, [4] S.N. Hosseini, M.Z. Kazemi-Baneh, Extended d-homology, Italian Journal of Pure and Applied Mathematics, N0. 38 (2017), pp [5] S.N. Hosseini, M.Z. Kazemi Baneh, Homology with respect to a Kernel Transformation, Turk J of Math, 35 (2011) [6] M.Z. Kazemi-Baneh, Homotopic Chain Maps Have Equal s-homology and d-homology, International Journal of Mathematics and Mathematical Sciences, Volume 2016, [7] M.Z. Kazemi Baneh, S.N. Hosseini, Exact Sequences of Extended d-homology, Algebraic Structures and Their Applications, Vol. 3 No. 1 ( 2017 ), pp [8] S. MacLane, Categories for the Working Mathematician, 2nd edition, Springer-Verlag, [9] M.S. Osborne, Basic Homological Algebra, Springer-Verlag, 2000.