若干Gorenstein同调维数性质的研究
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摘要
同调代数是代数学的一个重要分支,它的兴起对群、李代数与结合代数的研究起了非常重要的作用。其中环的同调维数是近代环论一个重要的研究领域。自上世纪六十年代以来,同调维数一直是环论研究的重要课题,特别是非交换环的同调维数的研究极大地丰富和发展了同调代数理论,它的理论和方法对代数学和其他相关学科研究起着重要作用。
1969年Auslander和Bridger在[2]中探讨了当R是左右Noetherian环时R-模M的G-维数。给出了当R-模M是有限生成时M的G-维数的性质。证明了不等式G-dim_RM≤pd_RM,且当pd_RM<∞时G-dim_RM=pd_RM。1995年,E.E.Enochs和O.M.G.Jenda在[13]中定义了一般环R上的Gorenstein投射模。2000年L.W.Christensen在[6]中证明了当R是左右Noetherian环且M是有限的Gorenstein投射模时,[6]中G-维数与[2]中G-维数相同,并且给出了Noetherian环上的有限模M是Gorenstein投射的当且仅当G-dim_RM=0.2004年H.Holm在[17]中研究了Gorenstein投射维数、Gorenstein内射维数及Gorenstein平坦维数。
本文分为两部分:第一部分致力于Gorenstein内射模、Gorenstein平坦模、Gorenstein余平坦模以及Gorenstein投射模的研究,并且讨论了Gorenstein内射模、Gorenstein平坦模与Gorenstein投射模之间的关系;第二部分主要研究了左右内射维数与左右Gorenstein内射维数之间的关系。
第一章绪论部分是预备知识,首先介绍了Gorenstein投射模、Gorenstein
Homological Algebra is an important part of Algebra. The development of Homological Algebra brought out a great important in studying the group, Lie algebra and associative ring. One important research field of the recent ring theory is the homological dimension of the ring. From 1960s the study about the homological dimension of the ring over noncommutative notherian ring enriched the classical result about the homological dimension of the ring. Its theories and methods influenced the Algebra and other subjects.When R is a two-sided Noetherian ring, Auslander and Bridger[2]introduce in 1969 the G-dimension, G — dim_RM, for every finite, that is, finitely generated R-module M. They proved the inequality G — dim_RM ≤ pd_RM, with equality G — dim_R = pd_RM whenpd_RM < ∞. In 1995 Over a general ring R, Enochs and Jenda defined in [13] Gorenstein projective modules. In 200C L.W:Christensen proved in [6]that if R is two-sided Noetherian, and G is a finite Gorenstein projective module, then the new definition agree with that of Auslander and Bridger.And he also proved that a finite module over a notherian Ring is Gorenstein projective if and only if G — dim_R = 0. In 2004 in [17], Henrik.Holm studied the closely related about Gorenstein projective, Gorenstein injective and Gorenstein flat dimensions.This paper has two part. In the first part it devotes to research the
definition and character of Gorenstein injective module, Goernstein flat module and Gorenstein projective module. And we also deduce the relation about Gorenstein injective module, Gorenstein flat module and Gorenstein projective module. In the second part through research the relation of left and right injective dimension we get the relation of left and right Gorenstein injective dimension.In section 1, preliminaries. It introduces the definition and the elementary properties of Gorenstein projective modules and Gorenstein flat modules. Then introduces the Gorenstein divided function. And the relation between Gorenstein EXT function and EXT function, the Gorenstein TOR function and TOR function, further more, it introduces the selforthogonal module of finite injective dimension.In section 2, it discusses the characters of Gorenstein injective module and it proves that for every module M with finite Gorenstein injective dimension admits a nice Gorenstein injective preenvelopes. Further more it discusses the relation between Gorenstein injective module and injective module.In section 3, it defines the definition of coflat, and similarly it also define the definition of Gorenstein coflat. It introduces the definition of Gorenstein coflat, Gorenstein coflat resolution, Gorenstein coflat dimension. Then it introduces the characters of Gorenstein coflat modules. At last it mainly discusses the relation of Gorenstein flat module Gorenstein injective module, Gorenstein projective module.In section 4, it discusses when A is a k-Gorenstein ring and a^a is a faithfully balanced selforthogonal bimodule and if the Gorenstein injective di-
mension of u\ is finite then the Gorenstein injective dimension of a^ is also finite. Further more we get l.Gid\U) = r.
1969年Auslander和Bridger在[2]中探讨了当R是左右Noetherian环时R-模M的G-维数。给出了当R-模M是有限生成时M的G-维数的性质。证明了不等式G-dim_RM≤pd_RM,且当pd_RM<∞时G-dim_RM=pd_RM。1995年,E.E.Enochs和O.M.G.Jenda在[13]中定义了一般环R上的Gorenstein投射模。2000年L.W.Christensen在[6]中证明了当R是左右Noetherian环且M是有限的Gorenstein投射模时,[6]中G-维数与[2]中G-维数相同,并且给出了Noetherian环上的有限模M是Gorenstein投射的当且仅当G-dim_RM=0.2004年H.Holm在[17]中研究了Gorenstein投射维数、Gorenstein内射维数及Gorenstein平坦维数。
本文分为两部分:第一部分致力于Gorenstein内射模、Gorenstein平坦模、Gorenstein余平坦模以及Gorenstein投射模的研究,并且讨论了Gorenstein内射模、Gorenstein平坦模与Gorenstein投射模之间的关系;第二部分主要研究了左右内射维数与左右Gorenstein内射维数之间的关系。
第一章绪论部分是预备知识,首先介绍了Gorenstein投射模、Gorenstein
Homological Algebra is an important part of Algebra. The development of Homological Algebra brought out a great important in studying the group, Lie algebra and associative ring. One important research field of the recent ring theory is the homological dimension of the ring. From 1960s the study about the homological dimension of the ring over noncommutative notherian ring enriched the classical result about the homological dimension of the ring. Its theories and methods influenced the Algebra and other subjects.When R is a two-sided Noetherian ring, Auslander and Bridger[2]introduce in 1969 the G-dimension, G — dim_RM, for every finite, that is, finitely generated R-module M. They proved the inequality G — dim_RM ≤ pd_RM, with equality G — dim_R = pd_RM whenpd_RM < ∞. In 1995 Over a general ring R, Enochs and Jenda defined in [13] Gorenstein projective modules. In 200C L.W:Christensen proved in [6]that if R is two-sided Noetherian, and G is a finite Gorenstein projective module, then the new definition agree with that of Auslander and Bridger.And he also proved that a finite module over a notherian Ring is Gorenstein projective if and only if G — dim_R = 0. In 2004 in [17], Henrik.Holm studied the closely related about Gorenstein projective, Gorenstein injective and Gorenstein flat dimensions.This paper has two part. In the first part it devotes to research the
definition and character of Gorenstein injective module, Goernstein flat module and Gorenstein projective module. And we also deduce the relation about Gorenstein injective module, Gorenstein flat module and Gorenstein projective module. In the second part through research the relation of left and right injective dimension we get the relation of left and right Gorenstein injective dimension.In section 1, preliminaries. It introduces the definition and the elementary properties of Gorenstein projective modules and Gorenstein flat modules. Then introduces the Gorenstein divided function. And the relation between Gorenstein EXT function and EXT function, the Gorenstein TOR function and TOR function, further more, it introduces the selforthogonal module of finite injective dimension.In section 2, it discusses the characters of Gorenstein injective module and it proves that for every module M with finite Gorenstein injective dimension admits a nice Gorenstein injective preenvelopes. Further more it discusses the relation between Gorenstein injective module and injective module.In section 3, it defines the definition of coflat, and similarly it also define the definition of Gorenstein coflat. It introduces the definition of Gorenstein coflat, Gorenstein coflat resolution, Gorenstein coflat dimension. Then it introduces the characters of Gorenstein coflat modules. At last it mainly discusses the relation of Gorenstein flat module Gorenstein injective module, Gorenstein projective module.In section 4, it discusses when A is a k-Gorenstein ring and a^a is a faithfully balanced selforthogonal bimodule and if the Gorenstein injective di-
mension of u\ is finite then the Gorenstein injective dimension of a^ is also finite. Further more we get l.Gid\U) = r.
引文
[1] F.W.Anderson, K.R.Fuller. Rings and Categoties of Modules, Springer-Verlag, Berlin and New York (1974).
[2] M.Auslander, M.Bridger, Stable modules theory, Memoirs Amer. Math. Soc. No.94. 1969.
[3] L.L.Avramov, H.-B. Foxby, Ring homorphisms and finite Gorenstein dimension, Proc. London Math. Soc. (3)75(2)(1997) 241-270.
[4] L.L.Avramov, A.Martsinkovsky, Absolute, relative and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3)85 (2) (2002)393-440.
[5] H.Bass. Injective dimensions in Noertian rings, Trans. Amer. Math. Soc. 102(1962)18-29.
[6] L.W.Christensen. Gorenstein Dimensions, Lecture Notes in mathematics, Vol.1747, Springer, Berlin, 2000.
[7] R.F.Damiano, Coflat rings and modules, Pacific J. Math., 81 (2)(1979)349-369
[8] E.E.Enochs. Injective and flat covers, envelopesand resolvents Israel J. Math. 39(3)(1981), 189-209.
[9] E.E.Enochs, O.M.G.Jenda, Balanced functors applied to modules, J. Algebra 92(2)(1985), 303-310.
[10] E.E.Enochs, O.M.G.Jenda. Resolvents and dimensions of modules, Arch. Math.56(1991)528-532.
[11] E.E.Enochs, O.M.G.Jenda. On Gorenstein injective modules, Comm. Alg. 2(10)(1993), 3489-3501.
[12] E.E.Enochs, O.M.G.Jenda. B.Torrecillas, Gorenstein flat modules, J, Nanjing University, 10(1)(1993), 1-9.
[13] E.E.Enochs, O.M.G.Jenda, Gorenstein injective and projective modules, Math. Zeit. 220(1995),611-633.
[14] E.E.Enochs, O.M.G.Jenda. Gorenstein blance of Horn and Tensor, Tsukuba J. Math. 19(1995),1-13.
[15] E. E. Enochs, O. M. G. Jenda. Gorenstein Injective and Flat Dimension, Math. Japonica 44(2) (1996), 261-268.
[16] E. E. Enochs, O. M. G. Jenda, Jinzhong XU. Foxby Duality and Gorenstein Injectire and Projective Modules, Trans. Amer. Math. Soc. , 348 (8) (1996) 3223-3234.
[17] H. Holm. Gorenstein homological dimensions, J. Pure and Applied Algebra 189 (2004) 167-193.
[18] H. Holm. Gorenstein Derived Functors, Proc. AMS. 132 (7) (2004) 1913-1923.
[19] Zhaoyong Huang and Gaohua Tang. Self-orthogonal modules over coherent rings, J. Pure and Appl. Algebra 161 (2001) 167-176.
[20] Zhaoyong huang. Selforthogonal modules with finite injective dimension Ⅱ, J. Algebra 264 (2003) 262-268.
[21] T. W. Hungerford. 代数学(冯克勤聂灵召校),湖南教育出版社(1984).
[22] Y. Iwanaga, On rings with finite self-injective dimension Ⅱ, Tsukuba J. Math. 4 (1) (1980) 107-113.
[23] 程福长,易忠.环的同调维数,广西师范大学出版社.
[24] 陈家鼐.环与模,北京师范大学出版社(1989).
[25] 陈志杰.模范畴同调代数与层,上海科学技术出版社.
[26] 李桃生.范畴与同调代数基础.华中师范大学出版社.
[2] M.Auslander, M.Bridger, Stable modules theory, Memoirs Amer. Math. Soc. No.94. 1969.
[3] L.L.Avramov, H.-B. Foxby, Ring homorphisms and finite Gorenstein dimension, Proc. London Math. Soc. (3)75(2)(1997) 241-270.
[4] L.L.Avramov, A.Martsinkovsky, Absolute, relative and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3)85 (2) (2002)393-440.
[5] H.Bass. Injective dimensions in Noertian rings, Trans. Amer. Math. Soc. 102(1962)18-29.
[6] L.W.Christensen. Gorenstein Dimensions, Lecture Notes in mathematics, Vol.1747, Springer, Berlin, 2000.
[7] R.F.Damiano, Coflat rings and modules, Pacific J. Math., 81 (2)(1979)349-369
[8] E.E.Enochs. Injective and flat covers, envelopesand resolvents Israel J. Math. 39(3)(1981), 189-209.
[9] E.E.Enochs, O.M.G.Jenda, Balanced functors applied to modules, J. Algebra 92(2)(1985), 303-310.
[10] E.E.Enochs, O.M.G.Jenda. Resolvents and dimensions of modules, Arch. Math.56(1991)528-532.
[11] E.E.Enochs, O.M.G.Jenda. On Gorenstein injective modules, Comm. Alg. 2(10)(1993), 3489-3501.
[12] E.E.Enochs, O.M.G.Jenda. B.Torrecillas, Gorenstein flat modules, J, Nanjing University, 10(1)(1993), 1-9.
[13] E.E.Enochs, O.M.G.Jenda, Gorenstein injective and projective modules, Math. Zeit. 220(1995),611-633.
[14] E.E.Enochs, O.M.G.Jenda. Gorenstein blance of Horn and Tensor, Tsukuba J. Math. 19(1995),1-13.
[15] E. E. Enochs, O. M. G. Jenda. Gorenstein Injective and Flat Dimension, Math. Japonica 44(2) (1996), 261-268.
[16] E. E. Enochs, O. M. G. Jenda, Jinzhong XU. Foxby Duality and Gorenstein Injectire and Projective Modules, Trans. Amer. Math. Soc. , 348 (8) (1996) 3223-3234.
[17] H. Holm. Gorenstein homological dimensions, J. Pure and Applied Algebra 189 (2004) 167-193.
[18] H. Holm. Gorenstein Derived Functors, Proc. AMS. 132 (7) (2004) 1913-1923.
[19] Zhaoyong Huang and Gaohua Tang. Self-orthogonal modules over coherent rings, J. Pure and Appl. Algebra 161 (2001) 167-176.
[20] Zhaoyong huang. Selforthogonal modules with finite injective dimension Ⅱ, J. Algebra 264 (2003) 262-268.
[21] T. W. Hungerford. 代数学(冯克勤聂灵召校),湖南教育出版社(1984).
[22] Y. Iwanaga, On rings with finite self-injective dimension Ⅱ, Tsukuba J. Math. 4 (1) (1980) 107-113.
[23] 程福长,易忠.环的同调维数,广西师范大学出版社.
[24] 陈家鼐.环与模,北京师范大学出版社(1989).
[25] 陈志杰.模范畴同调代数与层,上海科学技术出版社.
[26] 李桃生.范畴与同调代数基础.华中师范大学出版社.