遗传预扭模类与遗传扭模类
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摘要
在研究*-模的过程中，R．Colpi等人研究了*-模的生成类Gen(R~P)何时成为一个好的范畴，即何时取得子模封闭或者扩张封闭．本文去掉*-模条件的限制，研究了Gen(R~P)成为一个取子模封闭和扩张封闭的范畴(即成为遗传扭模类)，模R~P需满足的条件与具有的相关性质。本文共分三章：
第一章，给出了文章的背景及文中要用到的一些基本概念。
第二章讨论了Gen(R~P)成为遗传预扭模类和遗传扭模类的充分必要条件以及此时它的一些性质。主要结果如下：
定理2．1．4设R~P，S=End(R~P)，那么以下条件等价：
(1)Gen(R~P)是遗传预扭模类．
(2)P_S是平坦的S-Mittag-Leffler模，Gen(R~P)=Pres(R~P)，且TH保持Gen(R~P)中的满射．
(3)P_3是平坦模，且Stat(P)=Gen(R~P)．
定理2．2．1设R~P，S=End(R~P)，Gen(R~P)是遗传预扭模类，则Gen(R~P)是遗传扭模类当且仅当Gen(R~P)(?)Ker P(?)_SExt_R~1(P，-)．
定理2．2．2设R~P，S=End(R~P)，则Gen(R~P)是遗传扭模类当且仅当以下两个条件同时成立：
(1)P_S是平坦的S-Mittag-Leiffler模；
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P，-)．当以上成立时，Gen(R~P)对应的无扭模类是Ker Hom_R(P，-)．
定理2．2．10设R环，R~P是R上的有限长度模，则Gen(R~P)是遗传扭模类当且仅当：存在拟投射生成子R~T使得Gen(R~P)=Gen(R~T)(?)Ker Ext_R~1(T，-)．
定理2．3．5设R是左Artin环，R~P是任意模，S=End(R~P)，则Gen(R~P)是遗传扭模类当且仅当以下两个条件同时成立：
(1)P_S是有限生成投射模；
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P，-)．
定理2．3．10设R是Artin环，R~P是R上的有限生成模，Gen(R~P)是遗传扭模类．那么对所有M∈Gen(R~P)，都有P_0-res．dim(M)=pd_SH_(P_0)M．(其中P_0为定理2．2．10中得到的拟投射生成子，S=End(R~(P_0))．
推论2．3．11设R是Artin环，R~P是R上的有限生成模，Gen(R~P)是遗传扭模类．如果对所有M∈Gen(R~P)，都有P_0-res．dim(M)≤n，那么gdS≤n(P_0为定理2．2．10中得到的拟投射生成子，S=End(R~(P_0))．
第三章研究了当Gen(R~P)成为遗传(预)扭模类时，环S=End(R~P)上的模类Cogen(S~P*)具有的性质和环B=End(P_S)=BiEnd(R~P)上的模类Gen(B~P)的性质．主要结论如下：
定理3．1．2设R~P，S=End(R~P)，若Gen(R~P)是遗传预扭模类，那么：
(1)(Ker P(?)_S-，Cogen(S~P*))是由Copres(S~P*)余生成的扭论；
(2)(Ker P(?)_S-，Cogen(S~P*))是遗传扭论．
定理3．2．1若Gen(R~P)是R-Mod中的遗传预扭模类，那么Gen(B~P)是B-Mod中的遗传预扭模类(B=BiEnd(R~P))．
R.Colpi gave some conditions that Gen(R~P) is closed under submodules and extension when R~P is a *-module. We gave the coditions that Gen(R~P) is closed under submodules and extension to be a hereditary torsion class, when R~P is an arbitrary module. And we studied some properties of it. This paper is divided into three chapters.
In the first chapter, wo introduced the background of this paper and some definitions.
In the second chapter, we discussed the necessary and sufficient conditions for that Gen(R~P) is a hereditary (pre)torsion class. And then gave some properties of it.The main results as following:
Theorem 2.1.4 Let P be a left R-module and lot S=End(R~P), then the following conditions are equivalent:
(1)Gen(R~P) is a hereditary pretorsion class.
(2)P_S is flat and S-Mittag-Leffler, Gen(R~P)=Pres(R~P), and TH preserves cpimorphismin Gen(R~P).
(3)P_S is fiat and Gen(R~P)=Stat(P)
Theorem 2.2.1 Let. P be a left R-module and let S=End(R~P), if Gen(R~P) is a hereditary pretorsion class, then Gen(R~P) is a hereditary torsion class if only if Gen(R~P)(?)Ker P(?)Ext_R~1(P,-).
Theorem 2.2.2 Let P be a left R-module and let S=End(R~P), then Gen(R~P) is a hereditary torsion class if and only if the following coditions are satisfied:
(1)P_S is flat and S-Mittag-Leffler;
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P, -).The acording torsion-free class is Ker Hom_R(P, -) when those conditions above arc satisfied.
Theorem 2.2.10 Let R be a ring and let P be a module of finite length over R, the Gen(R~P) is a hereditary torsion class if and only if there is a quasi-progenerators R~T satisfying Gcn(R~P)=Gen(R~T)(?)Ker ExtR_~1(T,- )
Theorem 2.3.5 Let R be a Artin ring and let P be a left R-module, S=End(R~P), then Gen(R~P) is a hereditary torsion class if and only if the following coditions are satisfied:
(1)Ps is a finitely generated projective module;
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P,-).
Theorem 2.3.10 Let R be a Artin ring and let P bo a finitely generated moduleover R. If Gen(R~P) is a hereditary torsion class, then P_0-res.dim(M)=pd_SH_(p_0)M for all M∈Gen(R~P) (P_0is the quasi-progenerator in Theorem 2.2.10).
Corollary 2.3.11 let P be a finitely generated module over a Artin ring R, and Gen(R~P) is a hereditary torsion class. If P_0-res.dim(M)≤n for all M∈Gen(R~P) (Pois the quasi-progenerator in Theorem 2.2.10). then gdS≤n (S=End(R~P_0)).
In the third chapter, we inverstigated some other hereditary (pre)torsion classes over the endomorphism and the biendomorphism ring of R~P. The main results as following:
Corollary 3.1.2 Let P be a left R-module and let S=End(R~P), if Gen(R~P) is a hereditary pretorsion class, then:
(1)(Ker P(?)_S-,Cogen(S~P*)) is a torsion class cogenerated by Copres(S~P*);
(2)(Ker P(?)_S-,Cogen(S~P*)) is a hereditary torsion class.Corollary 3.1.2 If Gen(R~P) is a hereditary pretorsion class in R-Mod, then Gen(R~P) is a hereditary pretorsion class in B-Mod (B=BiEnd(R~P)).
第一章，给出了文章的背景及文中要用到的一些基本概念。
第二章讨论了Gen(R~P)成为遗传预扭模类和遗传扭模类的充分必要条件以及此时它的一些性质。主要结果如下：
定理2．1．4设R~P，S=End(R~P)，那么以下条件等价：
(1)Gen(R~P)是遗传预扭模类．
(2)P_S是平坦的S-Mittag-Leffler模，Gen(R~P)=Pres(R~P)，且TH保持Gen(R~P)中的满射．
(3)P_3是平坦模，且Stat(P)=Gen(R~P)．
定理2．2．1设R~P，S=End(R~P)，Gen(R~P)是遗传预扭模类，则Gen(R~P)是遗传扭模类当且仅当Gen(R~P)(?)Ker P(?)_SExt_R~1(P，-)．
定理2．2．2设R~P，S=End(R~P)，则Gen(R~P)是遗传扭模类当且仅当以下两个条件同时成立：
(1)P_S是平坦的S-Mittag-Leiffler模；
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P，-)．当以上成立时，Gen(R~P)对应的无扭模类是Ker Hom_R(P，-)．
定理2．2．10设R环，R~P是R上的有限长度模，则Gen(R~P)是遗传扭模类当且仅当：存在拟投射生成子R~T使得Gen(R~P)=Gen(R~T)(?)Ker Ext_R~1(T，-)．
定理2．3．5设R是左Artin环，R~P是任意模，S=End(R~P)，则Gen(R~P)是遗传扭模类当且仅当以下两个条件同时成立：
(1)P_S是有限生成投射模；
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P，-)．
定理2．3．10设R是Artin环，R~P是R上的有限生成模，Gen(R~P)是遗传扭模类．那么对所有M∈Gen(R~P)，都有P_0-res．dim(M)=pd_SH_(P_0)M．(其中P_0为定理2．2．10中得到的拟投射生成子，S=End(R~(P_0))．
推论2．3．11设R是Artin环，R~P是R上的有限生成模，Gen(R~P)是遗传扭模类．如果对所有M∈Gen(R~P)，都有P_0-res．dim(M)≤n，那么gdS≤n(P_0为定理2．2．10中得到的拟投射生成子，S=End(R~(P_0))．
第三章研究了当Gen(R~P)成为遗传(预)扭模类时，环S=End(R~P)上的模类Cogen(S~P*)具有的性质和环B=End(P_S)=BiEnd(R~P)上的模类Gen(B~P)的性质．主要结论如下：
定理3．1．2设R~P，S=End(R~P)，若Gen(R~P)是遗传预扭模类，那么：
(1)(Ker P(?)_S-，Cogen(S~P*))是由Copres(S~P*)余生成的扭论；
(2)(Ker P(?)_S-，Cogen(S~P*))是遗传扭论．
定理3．2．1若Gen(R~P)是R-Mod中的遗传预扭模类，那么Gen(B~P)是B-Mod中的遗传预扭模类(B=BiEnd(R~P))．
R.Colpi gave some conditions that Gen(R~P) is closed under submodules and extension when R~P is a *-module. We gave the coditions that Gen(R~P) is closed under submodules and extension to be a hereditary torsion class, when R~P is an arbitrary module. And we studied some properties of it. This paper is divided into three chapters.
In the first chapter, wo introduced the background of this paper and some definitions.
In the second chapter, we discussed the necessary and sufficient conditions for that Gen(R~P) is a hereditary (pre)torsion class. And then gave some properties of it.The main results as following:
Theorem 2.1.4 Let P be a left R-module and lot S=End(R~P), then the following conditions are equivalent:
(1)Gen(R~P) is a hereditary pretorsion class.
(2)P_S is flat and S-Mittag-Leffler, Gen(R~P)=Pres(R~P), and TH preserves cpimorphismin Gen(R~P).
(3)P_S is fiat and Gen(R~P)=Stat(P)
Theorem 2.2.1 Let. P be a left R-module and let S=End(R~P), if Gen(R~P) is a hereditary pretorsion class, then Gen(R~P) is a hereditary torsion class if only if Gen(R~P)(?)Ker P(?)Ext_R~1(P,-).
Theorem 2.2.2 Let P be a left R-module and let S=End(R~P), then Gen(R~P) is a hereditary torsion class if and only if the following coditions are satisfied:
(1)P_S is flat and S-Mittag-Leffler;
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P, -).The acording torsion-free class is Ker Hom_R(P, -) when those conditions above arc satisfied.
Theorem 2.2.10 Let R be a ring and let P be a module of finite length over R, the Gen(R~P) is a hereditary torsion class if and only if there is a quasi-progenerators R~T satisfying Gcn(R~P)=Gen(R~T)(?)Ker ExtR_~1(T,- )
Theorem 2.3.5 Let R be a Artin ring and let P be a left R-module, S=End(R~P), then Gen(R~P) is a hereditary torsion class if and only if the following coditions are satisfied:
(1)Ps is a finitely generated projective module;
(2)Gen(R~P)=Pres(R~P)(?)Ker P(?)_SExt_R~1(P,-).
Theorem 2.3.10 Let R be a Artin ring and let P bo a finitely generated moduleover R. If Gen(R~P) is a hereditary torsion class, then P_0-res.dim(M)=pd_SH_(p_0)M for all M∈Gen(R~P) (P_0is the quasi-progenerator in Theorem 2.2.10).
Corollary 2.3.11 let P be a finitely generated module over a Artin ring R, and Gen(R~P) is a hereditary torsion class. If P_0-res.dim(M)≤n for all M∈Gen(R~P) (Pois the quasi-progenerator in Theorem 2.2.10). then gdS≤n (S=End(R~P_0)).
In the third chapter, we inverstigated some other hereditary (pre)torsion classes over the endomorphism and the biendomorphism ring of R~P. The main results as following:
Corollary 3.1.2 Let P be a left R-module and let S=End(R~P), if Gen(R~P) is a hereditary pretorsion class, then:
(1)(Ker P(?)_S-,Cogen(S~P*)) is a torsion class cogenerated by Copres(S~P*);
(2)(Ker P(?)_S-,Cogen(S~P*)) is a hereditary torsion class.Corollary 3.1.2 If Gen(R~P) is a hereditary pretorsion class in R-Mod, then Gen(R~P) is a hereditary pretorsion class in B-Mod (B=BiEnd(R~P)).
引文
[1] R. Colpi, Some remarks on equivalences between categories of modules. Comm Algebra. 1990, 18(6): 1935-1951.
[2] Colpi R, Menini C. On the structure of *-modules. .1 Algebra, 1993, 158(2): 400-419.
[3] Wisbauer R. Static modules and equivalences. In: Oystaeyen.V, Saorin.M. Interactions Between Ring Theory and Representations of Algebras, LNPAM 210. New York: Marcel Decker, 2000. 423-449.
[4] Sato M. On equivalences between module subcategories. J Algebra, 1979, 59: 412-420.
[5] Liu M, Wang M Y. Equivalence and duality between subcategories. Journalof Southwest Jiaotong University, 2004, 12(2): 193-196.
[6] Bongartz K, Tilted algebras. In: Auslander M, Lluis E. Representations of Algebras, LNM 903. Berlin: Springer-Verlag, 1980, 26-38.
[7] Goodearl K R. Distributing tensor product over direct product. Pacific J Math, 1972, 43(1): 107-110.
[8] Lenzing H. Endlich prasentierbare moduln. Arch der Math, 1969, 20: 262-266.
[9] Beaehy J A, Blair W D. Finitely annihilated modules and orders in artinianrings. Comm Algebra, 1978, 6(1):1-34.
[10] Rotman J. An Introduction to Homological Algebra. New York, San Francisco, London: Academic Press, 1979. 85-85
[11] I. Assom, D. Simson, A. Skowronski. Elements of the Representation Theoryof Associative Algebras. USA: Cambridge University Press, 2006. 184-201
[12] R. Colpi, J. Trlifaj, Tilting modules and tilting torsion theories. J Algebra,1995, 178: 614-634.
[13] Wei Jiaqun, Global dimension of the endomorphism ring and *~n-modules. J Algebra, 2005, 291: 238-249.
[14] R. Colby, K. Fuller, Hereditary torsion theory counter equivalences. J Algebra, 1996, 183: 217-230.
[15] P. Bland, Topics in torsion theory. Math.Research, 103, Verlag, Berlin, 1998. 217-230.
[16]魏加群，倾斜理论及其推广．南京大学博士毕业论文，2001．
[2] Colpi R, Menini C. On the structure of *-modules. .1 Algebra, 1993, 158(2): 400-419.
[3] Wisbauer R. Static modules and equivalences. In: Oystaeyen.V, Saorin.M. Interactions Between Ring Theory and Representations of Algebras, LNPAM 210. New York: Marcel Decker, 2000. 423-449.
[4] Sato M. On equivalences between module subcategories. J Algebra, 1979, 59: 412-420.
[5] Liu M, Wang M Y. Equivalence and duality between subcategories. Journalof Southwest Jiaotong University, 2004, 12(2): 193-196.
[6] Bongartz K, Tilted algebras. In: Auslander M, Lluis E. Representations of Algebras, LNM 903. Berlin: Springer-Verlag, 1980, 26-38.
[7] Goodearl K R. Distributing tensor product over direct product. Pacific J Math, 1972, 43(1): 107-110.
[8] Lenzing H. Endlich prasentierbare moduln. Arch der Math, 1969, 20: 262-266.
[9] Beaehy J A, Blair W D. Finitely annihilated modules and orders in artinianrings. Comm Algebra, 1978, 6(1):1-34.
[10] Rotman J. An Introduction to Homological Algebra. New York, San Francisco, London: Academic Press, 1979. 85-85
[11] I. Assom, D. Simson, A. Skowronski. Elements of the Representation Theoryof Associative Algebras. USA: Cambridge University Press, 2006. 184-201
[12] R. Colpi, J. Trlifaj, Tilting modules and tilting torsion theories. J Algebra,1995, 178: 614-634.
[13] Wei Jiaqun, Global dimension of the endomorphism ring and *~n-modules. J Algebra, 2005, 291: 238-249.
[14] R. Colby, K. Fuller, Hereditary torsion theory counter equivalences. J Algebra, 1996, 183: 217-230.
[15] P. Bland, Topics in torsion theory. Math.Research, 103, Verlag, Berlin, 1998. 217-230.
[16]魏加群，倾斜理论及其推广．南京大学博士毕业论文，2001．