Abelian Property of the Category of U-Complexes

Transcription

1 Iteratioal Joural of Mathematical Aalysis Vol. 10, 2016, o. 17, HIKARI Lt, Abelia Property of the Category of U-Complexes Gustia Elfiyati Faculty of Scieces a Techology, UIN Jakarta Ita Muchtai - Alamsyah Dellavitha Nasutio Utih Amartiwi Faculty of Scieces a Techology, UIN Jakarta, Ioesia Copyright c 2015 Gustia Elfiyati, Ita Muchtai-Alamsyah, Dellavitha Nasutio a Utih Amartiwi. This article is istribute uer the Creative Commos Attributio Licese, which permits urestricte use, istributio, a reprouctio i ay meium, provie the origial work is properly cite. Abstract The otio of a chai U complex a chai (U, U ) map were itrouce by Davvaz a Shabbai as a geeralizatio of a chai complex a a chai map respectively. I this paper we cotiue their research by proposig a category of U complexes as a geeralizatio of the category of complexes. We show that the category of U complexes is a abelia category.

2 850 G. Elfiyati, I. Muchtai-Alamsyah, D. Nasutio a U. Amartiwi Mathematics Subject Classificatio: 55U15, 18E10 Keywors: chai U complex, chai (U, U ) map, category of U complexes, abelia category 1 Itrouctio A sequece of R moules a R homomorphism (1) is calle exact sequece if Im( +1 ) = 1 (0). It is a atural questio what if 0 is replace by U 1, a submoul of 1. Davvaz a Paria [1] moifie the efiitio of exact sequece of moules which is calle U exact sequeces a geeralize some results from existig oes to the moifie case. Their research was motivate by the exact sequece of hypergroups which geerally has o zero elemet, itrouce by Frei a Elerberry i [2]. Davvaz a Shabbai cotiue workig o this topic a propose the cocept of U complex as a geeralizatio of complex [3]. They efie the cocepts of chai U complex, U homology, chai (U, U ) map, chai (U, U ) homotopy a U fuctor a use the cocepts to fi a geeralizatio of several results i homological algebra. This paper aims to apply the previous results to examie the cocepts category of U complexes. We show that the category of U complexes is a abelia category. 2 Chai of U-Complexes I this sectio we review some results itrouce by Davvaz a Shabbai. Defiitio 2.1 Give a family = (, U, ) = (, U, ) Z where, U are R moules a each of cosists U a : 1. A chai (, U, ) : is calle U complex if for all Z we have: 1. +1( +1 ) U 1 a 2. Im ( ) U 1 The efiitio implisitly say that a chai complex is a chai 0 complex.

3 Abelia property of the category of U complexes 851 Defiitio 2.2 Let (, U, ) be a U complex a (Y, U Y, Y ) be a U Y complex. The sequece f = (f : Y ) Z is calle chai (U, U Y ) map if followig iagram is commutative a f (U ) U Y for each Z f+1 f f 1 Y Y Y Y Y 1 Propositio 2.3 Let (, U, ) be a U complex such that +1( +1 ) = U 1 a (Y, U, Y ) is a chai U Y complex. If f = (f : Y ) Z is a chai map the it is also a chai (U, U Y ) map. Defiitio 2.4 Let Let (, U, ) a (Y, U, Y ) be a chai U complex a U Y complex respectively. A chai (U, U Y ) map f = (f ) Z is a isomorphism if f is R moules isomorphism for all Z a f 1 = (f 1 ) Z is a chai (U Y, U ) map. If there exists a isomorphism from (, U, ) to (Y, U, Y ) we say that (, U, ) isomorphic to (Y, U, Y ). The isomorphism of chai U complexes is a equivalece relatio. Propositio 2.5 If chai U complex a U Y complex are isomorphic the U U Y for all Z. 3 The Category of U-Complexes I this sectio we itrouce the cocept of a category of U complexes a stuy its property. Let A be a abelia category R Mo. Defiitio 3.1 The category of U complexes C(A, U) is a category whose objects are chai U complexes i A, the morphisms are chai (U, U ) map a the compositio operatio is the usual compositio fuctio. Theorem 3.2 The category of U-complexes C(A, U) is a abelia category Proof A1 Let (, U, ) a (Y, U, Y ) be a chai U complex a U Y complex respectively. Assume that f = (f ) Z a g = (g ) Z are two chai (U, U Y ) maps. By efiig f +g = (f +g ) Z it is easy to prove that Hom C(A,U) (, Y ) is a abelia group a the compositio of morphisms Hom C(A,U) (Y, Z) Hom C(A,U) (, Y ) Hom C(A,U) (, Z) is biliier over iteger.

4 852 G. Elfiyati, I. Muchtai-Alamsyah, D. Nasutio a U. Amartiwi A2 The zero object i C(A, U) is the chai of 0 complex which all moules are zero. A3 A coprouct of two objects = (, U, ) a Y = (Y, U Y, Y ) is a object Y = ( Y, U Y, ) Y = ( ) Y, U Y, Y Z where ( U U Y = U Y ) ( a Y = 0 0 Y together with chai ( U, U ) Y map ι a chai ( U Y, U ) Y map ι Y satisfyig the uiversal property: for every objects Z i C (A, U), chai ( U, U Z) map f a chai ( U Y, U Z) map f Y there is a uique chai ( U Y, U Z) map f makig followig iagram commutative. ) Z (f ) f (f Y ) (2) (ι ) Y (ι Y ) Y A4 Let f = (f : Y ) Z be a chai ( U, U ) Y map, the each f is a morphism i A. We show the existece of a cokerel a leave the ual. Sice A is a abelia category, each f has a cokerel C = Y /Im (f ) i A together with a morphism c : Y C such that c f = 0 satisfyig the uiversal property of cokerel, i.e. there is a uique morphism C : C C 1 such that c 1 C = Y c. Let f = (f : Y ) Z be a chai ( U, U ) Y map, the each f is a morphism i A. Hece the followig iagram is commutative f+1 f f 1 Y Y +1 Y +1 Y Y Y 1 c+1 c c 1 C C C C C C 1 By choosig U C = U Y / Im (f ), it is easy to check that C = ( ) C, C, U C is a chai U C complex a satisfyig the uiversal property of cokerel for f. A5 Let U Im(f) ( ) = f U coim(f) a U = U /ker(f ) the coim(f) a Im(f) are objects i C (A, U). Cosier the atural morphism coim(f) coim(f), sice A is abelia the for every the atural morphism (3)

5 Abelia property of the category of U complexes 853 g : coim(f ) Im(f ) is a isomophism, hece the ivers g 1 : coim(f ) Im(f ) is also isomorphism a g 1 = (g 1 ) Z is a chai map. As U Im(f) ( ) ( ) = f U we have g 1 U Im(f) = U, hece g 1 is a chai ( U Y, U ) map. By the Defiitio 2.4, g is a isomorphism of U complexes. Ackowlegemets. This research is supporte by ITB a UIN Jakarta Research Grat. Refereces [1] B. Davvaz a Y.A. Paria-Gramaleky, A Note o Exact Sequeces, Bull. Malaysia Math. Soc., 22 (1999), o. 2, [2] D. Frei a Y. Sureau, Hypergroupes e type U et homologie e complexes, Algebra Uiversalis, 35 (1996), [3] B. Davvaz a H. Shabai-Solt, A Geeralizatio of Homological Algebra, J. Korea Math. Soc., 39 (2002), o. 6, [4] S.I. Gelfa a Y.I. Mai, Methos of Homological Algebra, 2 Eitio, Spriger-Verlag, Heielberg, [5] C.A. Weibel, A Itrouctio to Homological Algebra, Cambrige Uiversity Press, Uite Kigom, [6] T. Holm, P. Jörgese a R. Rouquier, Triagulate Categories, Loo Math. Soc. Lecture Note Series, o. 375, Cambrige Uiversity Press, Uite Kigom, [7] S.M. Lae, Categories for the Workig Mathematicia, 2 Eitio, Spriger, New York, [8] J.-L Verier, Des catégories érivées es catégories abélees, Astésque o. 239, Receive: November 1, 2015; Publishe: Jue 20, 2016