Pseudosimplisel cebir
Pseudosimplicial algebra
Yazar: SEDAT PAK
Danışman: Y.DOÇ. İBRAHİM İLKER AKÇA
Eskişehir Osmangazi Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı
Derece: Doktora, 2006
Konu: Matematik = Mathematics

Özet:
1Ë Ë ËPSEUDOSIMPLISEL CEBIRSEDAT PAKüOZETInasaridze, H. N.[14], pseudosimplisel grupların homotopi gruplarını kulla-narak, gruplar kategorisindeki değerleri ile birlikte abelyen olmayan uretilmişg ü sfunktorları oluşturmuştur. Bu, asosyatif halkalar kategorisi işindeki cebirsels s cK- teorisinin, GL kovaryant funktorunun sol uretilmiş funktorlar teorisi olarakü selde edilmesine olanak sağlamaktadır. Cebirsel K - teori, modern cebir,gtopoloji, cebirsel geometri, fonksiyonel analiz ve cebirsel sayı teorisi ile ilişkisisolan matematiğin birşok temel alanında ünemli uygulamaları bulunmaktadır.g c oGünümüzde Swan[30], Volodin[15], Karoubi ve Villamayor[24], Gersten[35]uu uve Quillen[7] tarafından oluşturulmuş birşok cebirsel K - teori ile ilgili birs s cşok şalışma vardır.ccsInasaridze, H. N.[14], halka kategorisinden grup kategorisine tanımlı E veGL kovaryant funktorlarını kullanarak elde edilen Kn funktorlarının Swan[30],tarafından tanımlanmış cebirsel K - teoriyi verdiğini ispatlamıştır.s g sBu şalışmada, Inasaridze [14] tarafından tanımlanmış olan pseudosimpliselcs sgrupları ile pseudosimplisel grupların homotopi grupları güzününde bulun-ooudurularak komütatif cebirler işin pseudosimplisel cebir yapısı oluşturuldu. veu c sInasaridze.[14] tarafından pseudosimplisel gruplar işin yapılanlar pseudosim-cplisel cebirler işin incelendi.cTez dürt bülümden oluşmaktadır. Birinci bülüm giriş bülümü olup buo ou s ou s ou uËbülümde tezin yapısı kısaca tanıtılmıştır. Ikinci bülümde tezde kullanılacakou s outemel kavramlar verilmiştir. Projektif sınıf, uretilmiş funktor ve sıralı uclüs ü s üş u2(Cotriple) kavramları incelenmiştir. Sonrasında ise homotopi modülü kavramıs uuve şaprazlanmış modüller hatırlatıldı.c s uü cü u o uUşuncü bülümde Inasaridze [14] tarafından tanımlanmış olan pseudosim-splisel grupların homotopi gruplarını kullanarak pseudosimplisel cebir yapısıtanımlandı ve Inasaridze [14] de pseudosimplisel gruplar işin elde edilen son-cuşlar pseudosimplisel cebirler işin incelendi.c cSon bülümde ise Arvasi [39] nin simplisel komütatif cebirler işin güstermişou u c o solduğu sonuşların bazıları pseudosimplisel cebirler işinde kısmen geşerli olduğug c c c ggüsterildi. Daha sonra şaprazlanmış modüllerin kategorisi ile Moore kom-o c s upleksinin uzunluğu 1 olan pseudosimplisel cebirler kategorisinin denkliği güs-g goterildi. Ayrıca Ellis G.L. [13] den faydalanarak pseudo 2 - şaprazlanmış modülc s uyapısı tanımlandı ve pseudo 2 - şaprazlanmış modül kategorisinin, Moorec s ukompleksinin uzunluğu 2 olan pseudosimplisel cebirler kategorisine denkliğig ggüsterildi. Ayrıca bu bülümde pseudosimplisel cebirlerin n - tipleride ince-o oulendi.Kaynaklar[1] A. Mutlu, Peiï¬er Pairings in the Moore Complex of a Simplicial Group,Ph. D. Thesis, University of Wales, BANGOR, (1997).[2] Barr, M., and J. Beck: Homology and standard constructions.Springer Lecture Notes in Mathematics No. 80 (1969)[3] D. Conduche, Modules Croises Generalises de Longueur, 2. Jour. PureAppl. Algebra, 34 (1984) 155-178.[4] D. M. Kan, A Combinatorial Deï¬nition of Homotopy Groups, Annalsof Maths., 61, 288-312, (1958).[5] D. Quillen, On the homology of commutative rings, Porc. Sympos.Pure Math 17, (1970).[6] D. G. Quillen, Spectral sequences of a double semi-simplicial group,Topology 5 (1966) , 155 -157. MR 33 # 3302.[7] D. G. Quillen, Cohomology of groups and algebraic K - theory, In-vited hour adress presented at the 77th Annual Meeting of the Amer.Math. Soc., Atlantic City, N. J., January 1971.[8] D. Quillen, Homotopical Algebra, Lecture Notes in Math. 43 (Springer,New York, 1967).34[9] E.R. Aznar, Cohomologia no abeliana in categorieas de interÂ´s, Ph.D.eThesis, Universidad de Santiago de Compostela, Alxebra 33 (1981)[10] F. Keune, Homotopical Algebra and Algebraic K-theory, Thesis, Uni-versiteit van Amsterdam, (1972).[11] G.J. Ellis , Crossed Modules and Their Higher Dimensional Ana-logues, Ph.D. Thesis, U.C.N.W., (1984)[12] G.J. Ellis and R. Steiner, Higher Dimensional Crossed Modules andthe Homotopy groups of (n+1)-ads. , J. Pure and Applied Algebra, 46,117-136, (1987).[13] G.J. Ellis , Homotopical Aspects of Lie Algebras, J.Austral. Math.Soc. (SeriesA), 54 (1993) 393 - 419[14] H. N. Inasaridze, Homotopy of pseudosimplicial groups and non-abelian derived functors, Sakharth. SSR Mecn. Akad. Moambe 76 (1974),533-536. (Russian)[15] I. A. Volodin, Algebraic K - theory as an extraordinary homologytheory on the category of associative rings with a unit, Izv. Akad. NaukSSSR Ser. Mat 35 (1971), 884 - 873 = Math. USSR Izv. 5 (1971), 859 -887. MR 45 # 5201.[16] J. L. Castiglioni and M. Ladra, Peiï¬er Elements in SimplicialGroups and Algebras, Preprint submitted to Elsevier Science, 24 Febru-ary 2006.[17] J. H. C. Whitehead, Combinatorial Homotopy I and II, Bull. Amer.Math. Soc., 55, 231-245 and 453-496, (1949).5[18] J.L. LODAY., Spaces with ï¬nitely many non - trivial homotopy groups,J. Pure Appl. Algebra, 24 (1982) 179 - 202.ü[19] K. Reidemeister, Uber Identitaten von Relationen, Abh. Math. Sem.Univ. hamburg 16, pp. (114-118), (1949).Â´[20] M. Andre,.e, Homologie des algÂ´bres commutatives, Die Grundlehreneder mathematischen Wisssenschaften in Einzeldarstellungen, Band 206,Springer-Verlag, (1974).[21] M. Andre, Homologie des Algebres Commutatives, Lecture Notes inMath., Springer, 206, (1970).[22] M. ArTIN and B. MAZUR, Etale homotopy, Lecture Notes in Maths,Springer-Verlag, 100, (1968).[23] M.Gerstenhaber. On the Deformation of Rings and Algebras, Ann.Math. 84 (1966).[24] Max Karoubi and Orlando Villamayor, Foncteurs K n en alge-bre et en topologie, C. R. Acad. Sci. Paris Ser. A-B 269 (1969) , A416- A419. MR 40 # 4944.[25] C. Morgenegg, Sur les invariants d'un anneau local Â´ corps rÂ´siduela ede caractÂ´ristique 2, Ph.D. thesis, Lausanne EPFL (1979).e[26] Myles Tierney and Wolfgang Vogel, Simplicial resolutions andderived functors, Math.Z. 111 (1969), 1-14. MR 40 # 2733.[27] P. Carrasco and A.M. Cegarra, Group-theoretic Algebraic Modelsfor Homotopy Types, Jour. Pure Applied Algebra, 75, 195-235, (1991).[28] P. J: Ehlers and T. Porter, Varietes of Simplicial Gruboids, I:Crossed Complexes, Jour. Pure Appl. Algebra, (to appear).6[29] R. G. Swan, Some relations between higher K - functors, J. Algebra21 (1972), 113 - 136. MR 47 # 1916.[30] R. G. Swan, Nonabelian homological algebra and K - theory, Proc.Sympos. Pure Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1968,pp. 88 - 123. MR 41 # 1839.ü[31] R. Peiffer, Uber Identitaten zwischen Relationen, Math. Ann. 121,pp. (67-99), (1949).[32] R. Brown, and J. L. Loday, Van Kampen Theorems for diagram ofspaces, Topology, 26, 311-335, (1987).[33] Samuel Eilenberg and J. C. Moore, Foundations of relative ho-mological algebra, Mem. Amer. Math. Soc., No. 55 (1965). MR 31 #2294.[34] S.Lichtenbaum and M.Schlessinger The Cotangent Complex of aMorphism, Trans. American Society, 128, 41-70, (1967).[35] S. M. Gersten, On the functors K2 . I. J. Algebra 17 (1971), 212 - 237.MR 45 # 1992.[36] S. M. Gersten, On Mayer - Vietoris functors and algebraic K - theory,J. Algebra 18 (1971), 51 -88. MR 43 # 6290.[37] T.Porter. Homology of Commutative Algebras and an Invariant ofSimis and Vasconceles, J. Algebra 99, 458-465, (1986).[38] T.Porter. Some categorical results in the theory of crossed modulesincommutative algebras, J. Algebra 109, 415-429, (1987).[39] Z. Arvasi, Applications in commutative algebra of Moore complex ofsimplicial algebra, Ph.D. Thesis, University of Wales, Bangor, (1994).7[40] Z. Arvasi,.and T. Porter, Simplicial and Crossed Resolutions ofCommutative Algebras, Journal of Algebra, 181, 426-448, (1996).[41] Z. Arvasi and T. Porter, Freness Conditions for 2-Crossed Modules ofCommutative Algebras, Applied Categorical Structures, Vol. 6, 455-471,(1998).[42] Z. Arvasi,M. KOCAK and M. ALP, A Combınatorıal deï¬nition of n- types of sımplicial commutative algebras, Tr. Journal of Mathematics,22, 243-271, (1998).

Abstract:
1PSEUDOSIMPLICIAL ALGEBRASEDAT PAKSUMMARYUsing the homotopy groups of pseudosimplicial groups, Inasaridze [14]construct nonabelian derived functors with values in the category of groups.This enables him to obtain algebraic K - theory in the category of associativerings as the theory of left derived functors of the covariant functor GL. Alge-braic K - theory is a modern and perspective branch of algebra having manyimportant applications in fundamental areas of mathematics connected withalgebra, topology, algebraic geometry, functional analysis and algebraic num-ber theory. There are several algebraic K - theories in existence, constructedSwan [30], Karoubi and Villamayor [?], Volodin [15], Gersten [36] and Quillen[7], that enlist homotopical methods into the study of algebraic objects.Inasaridze, H. N. [14], proved that the functors Kn give the algebraic K- theory of Swan using two covariant functors GL and E from the categoryof rings to the category of groups. In this thesis we deï¬ne pseudosimplicialalgebra considering pseudosimplicial groups. Then we give some results ofpseudosimplicial algebras.This thesis consists of four chapter. In the ï¬rst chapter we introducestructure of thesis. In the second chapter we give some basic information.We recall projective class, derived functors, cotriples, homotopy modules andcrossed modules. In the next chapter we deï¬ne pseudosimplicial algebra con-sidering pseudosimplicial groups and homotopy groups of that.2In the last chapter we work on the higer order Peiï¬er elements of pseu-dosimplicial algebras which are given by Z. Arvasi for simlicial algebras ver-sion. [39] and we show that this results are partially correct for pseudosimpli-cial algebras. Then we show that the category of crossed module is equivalentto the category of pseudosimplicial algebras with Moore compleks of length1. Also we deï¬ne pseudo 2 - crossed module and we prove that the categoryof pseudo 2 - crossed module is equivalent to the category of pseudosimplicialalgebras with Moore compleks of length 2. Then we examine n - types ofpseudosimplicial algebras.3Kaynaklar[1] A. Mutlu, Peiï¬er Pairings in the Moore Complex of a Simplicial Group,Ph. D. Thesis, University of Wales, BANGOR, (1997).[2] Barr, M., and J. Beck: Homology and standard constructions.Springer Lecture Notes in Mathematics No. 80 (1969)[3] D. Conduche, Modules Croises Generalises de Longueur, 2. Jour. PureAppl. Algebra, 34 (1984) 155-178.[4] D. M. Kan, A Combinatorial Deï¬nition of Homotopy Groups, Annalsof Maths., 61, 288-312, (1958).[5] D. Quillen, On the homology of commutative rings, Porc. Sympos.Pure Math 17, (1970).[6] D. G. Quillen, Spectral sequences of a double semi-simplicial group,Topology 5 (1966) , 155 -157. MR 33 # 3302.[7] D. G. Quillen, Cohomology of groups and algebraic K - theory, In-vited hour adress presented at the 77th Annual Meeting of the Amer.Math. Soc., Atlantic City, N. J., January 1971.[8] D. Quillen, Homotopical Algebra, Lecture Notes in Math. 43 (Springer,New York, 1967).45[9] E.R. Aznar, Cohomologia no abeliana in categorieas de interÂ´s, Ph.D.eThesis, Universidad de Santiago de Compostela, Alxebra 33 (1981)[10] F. Keune, Homotopical Algebra and Algebraic K-theory, Thesis, Uni-versiteit van Amsterdam, (1972).[11] G.J. Ellis , Crossed Modules and Their Higher Dimensional Ana-logues, Ph.D. Thesis, U.C.N.W., (1984)[12] G.J. Ellis and R. Steiner, Higher Dimensional Crossed Modules andthe Homotopy groups of (n+1)-ads. , J. Pure and Applied Algebra, 46,117-136, (1987).[13] G.J. Ellis , Homotopical Aspects of Lie Algebras, J.Austral. Math.Soc. (SeriesA), 54 (1993) 393 - 419[14] H. N. Inasaridze, Homotopy of pseudosimplicial groups and non-abelian derived functors, Sakharth. SSR Mecn. Akad. Moambe 76 (1974),533-536. (Russian)[15] I. A. Volodin, Algebraic K - theory as an extraordinary homologytheory on the category of associative rings with a unit, Izv. Akad. NaukSSSR Ser. Mat 35 (1971), 884 - 873 = Math. USSR Izv. 5 (1971), 859 -887. MR 45 # 5201.[16] J. L. Castiglioni and M. Ladra, Peiï¬er Elements in SimplicialGroups and Algebras, Preprint submitted to Elsevier Science, 24 Febru-ary 2006.[17] J. H. C. Whitehead, Combinatorial Homotopy I and II, Bull. Amer.Math. Soc., 55, 231-245 and 453-496, (1949).6[18] J.L. LODAY., Spaces with ï¬nitely many non - trivial homotopy groups,J. Pure Appl. Algebra, 24 (1982) 179 - 202.ü[19] K. Reidemeister, Uber Identitaten von Relationen, Abh. Math. Sem.Univ. hamburg 16, pp. (114-118), (1949).Â´[20] M. Andre,.e, Homologie des algÂ´bres commutatives, Die Grundlehreneder mathematischen Wisssenschaften in Einzeldarstellungen, Band 206,Springer-Verlag, (1974).[21] M. Andre, Homologie des Algebres Commutatives, Lecture Notes inMath., Springer, 206, (1970).[22] M. ArTIN and B. MAZUR, Etale homotopy, Lecture Notes in Maths,Springer-Verlag, 100, (1968).[23] M.Gerstenhaber. On the Deformation of Rings and Algebras, Ann.Math. 84 (1966).[24] Max Karoubi and Orlando Villamayor, Foncteurs K n en alge-bre et en topologie, C. R. Acad. Sci. Paris Ser. A-B 269 (1969) , A416- A419. MR 40 # 4944.[25] C. Morgenegg, Sur les invariants d?un anneau local Â´ corps rÂ´siduela ede caractÂ´ristique 2, Ph.D. thesis, Lausanne EPFL (1979).e[26] Myles Tierney and Wolfgang Vogel, Simplicial resolutions andderived functors, Math.Z. 111 (1969), 1-14. MR 40 # 2733.[27] P. Carrasco and A.M. Cegarra, Group-theoretic Algebraic Modelsfor Homotopy Types, Jour. Pure Applied Algebra, 75, 195-235, (1991).[28] P. J: Ehlers and T. Porter, Varietes of Simplicial Gruboids, I:Crossed Complexes, Jour. Pure Appl. Algebra, (to appear).7[29] R. G. Swan, Some relations between higher K - functors, J. Algebra21 (1972), 113 - 136. MR 47 # 1916.[30] R. G. Swan, Nonabelian homological algebra and K - theory, Proc.Sympos. Pure Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1968,pp. 88 - 123. MR 41 # 1839.ü[31] R. Peiffer, Uber Identitaten zwischen Relationen, Math. Ann. 121,pp. (67-99), (1949).[32] R. Brown, and J. L. Loday, Van Kampen Theorems for diagram ofspaces, Topology, 26, 311-335, (1987).[33] Samuel Eilenberg and J. C. Moore, Foundations of relative ho-mological algebra, Mem. Amer. Math. Soc., No. 55 (1965). MR 31 #2294.[34] S.Lichtenbaum and M.Schlessinger The Cotangent Complex of aMorphism, Trans. American Society, 128, 41-70, (1967).[35] S. M. Gersten, On the functors K2 . I. J. Algebra 17 (1971), 212 - 237.MR 45 # 1992.[36] S. M. Gersten, On Mayer - Vietoris functors and algebraic K - theory,J. Algebra 18 (1971), 51 -88. MR 43 # 6290.[37] T.Porter. Homology of Commutative Algebras and an Invariant ofSimis and Vasconceles, J. Algebra 99, 458-465, (1986).[38] T.Porter. Some categorical results in the theory of crossed modulesincommutative algebras, J. Algebra 109, 415-429, (1987).[39] Z. Arvasi, Applications in commutative algebra of Moore complex ofsimplicial algebra, Ph.D. Thesis, University of Wales, Bangor, (1994).8[40] Z. Arvasi,.and T. Porter, Simplicial and Crossed Resolutions ofCommutative Algebras, Journal of Algebra, 181, 426-448, (1996).[41] Z. Arvasi and T. Porter, Freness Conditions for 2-Crossed Modules ofCommutative Algebras, Applied Categorical Structures, Vol. 6, 455-471,(1998).[42] Z. Arvasi,M. KOCAK and M. ALP, A Combınatorıal deï¬nition of n- types of sımplicial commutative algebras, Tr. Journal of Mathematics,22, 243-271, (1998).