Some Special Modules, Subcategories And Coalgebras

Tilting theory was introduced in the early eighties in the context of finitely generated modules over artin algebras by Brenner and Butler [5] and by Happel and Ringel [6] ;and since then it has played a central role in the development of represention theory of artin algebras. As the generalation of tilting thery , some author consider the wakamatsu tilting modules ,*-module and *ⁿ-module. In [7], Robin Haetshorne re-seach the coherent functor and some property . In [8],[9],and [10] , Robert Wisbauer intoduced subcategryσ[M] of left R modules category ,and given the equivlent condition ofσ[M] closed under extension. In [11], J.den Berg and P.Wisbauer discuss the property of modules whose hereditary pretorsion classes are closed under products. In [12],Lixin Mao and Nanqing Ding introduce Relative copure injective and copure flat modules,and consider their equivelent conditions. In [23] , Lifang introduce the Green equivalent of coalgera ,as the generalation of groups.As the farther reseach of theories as above, we discuss five main problems in this thesis. In the second charter, we study the properties of wakamatsu tilting mod-ules,costar modules ,co-*ⁿ-modules and n-coherent functorIn the third section, we discuss the property of subcategoryÏ€[M}. In the forth chapter, we reseach the property of some special modules and and Tor-torsion pair ; In the last chapter, we defined the Green equivelent ofÏ€-coalgebra,and reseach the and Green equivlent on tensor products ofÏ€-coalgebras.For tilting module T, Trlifj defined the tlilting cotorsion pair C = (A, B) and find a close relation among the class A, B and the kenel of C. we generete the result to wakamatsu tilting module in the section 1 of Chapter 1 . we defind the wakamatsu tilting modules and find some close relation similily. The main result as followsLet∧be an Artin algebra, T∈Mod∧be a wakamatsu tilting module we denote by Gen*T the class of module N∈Mod∧such that there exsit an exact sequencewith T_i∈Add T and Ext¹(T, ker f_i) = 0 for any i∈N. we denote the T^⊥∩Gen*T by X_T The main result as followsTheorem 1.1.3Let∧be an Artin algebra, T∈Mod∧be a wakamatsu tilting module with pd T = n ,Then(a) (^⊥X_T, X_T) is a comepele cotorsion pair with the kernel Add T. (b)^⊥X_T coincide with the class of all modules M such that there is an exact sequencewhere T_i∈G AddT, kerg_i∈X_T.In the second section of Chapter 2, we consider the costar modules and co-*^N-module over ring homomorphism.Theorem 2.2.3 Letξ: A→R be a ring homomorphism , If _AU is an costar mdoule with Hom_A(_AR_A,_AU))∈Cogen_AU , then Hom_A(_AR_R,_AU)) is a costar R-module.Theorem 2.2.7 Letξ: A→R be a ring homomorphism. If _AU is an co-*ⁿ- module with Hom_A(_AR_A,_AU))∈Cogen_AU, then Rom_A(_AR_R,_AU)) is an co-*ⁿ-module. In the third section of Chapter 2 we defind n-coherent functor,which generated the coherent functor in [7],and study some property.Definition 2.3.2 . F∈F is finitely generated if there exists a exact sequencem h_m→F→0 for some M∈R-mod. F is n-coherent if there exist exact sequence h_M-n→h_Mn-1→…→h_M1→h_M0→F→0 for some M₁∈R-mod ,i = 0…n.Theorem 2.2.3. (a) Let n = 3k + 1, k∈Z, F and G are n-coherent functors, i.e. there exist exact sequencefor some M_i, A∈R-mod ,i = 0…n.If F→|f G is a morphism of n-coherent functors and h_Mi(?) h_Ai for i = 3k+ 2 and0≤i≤n, k∈N . then coker f is also n-coherent.(b) If 0→F→G→H→0 is an exact sequence of functors with F, H n-coherent, then G is n-coherent.In the first section of Charter 3, we consider the dulity of [9], and find the equivlent condition ofÏ€[M] closed under extension.Proposition 3.1.7 For U∈π[M] the following assertions are equivalent .(a)Ï€[U] is closed under extensions inÏ€[M].(b) every N∈π[M] which allows an exact sequence U^∧→N→U^∧(for some set A) belongs toÏ€[U].In [11], J.den Berg and P.Wisbauer discuss the property of modules whose hereditary pretorsion classes are closed under products. In the second section of the Charter 3, we get the dual result and give the property of modules whose cohereditary pretor-sionfree classes are closed under coproducts. the main Theorem is as followsTheorem 3.2.6Let M be a left R-module, if F is a M-codominated cohereditary pretorsionfree class inÏ€[M], then F is weak subgencrated by the class of all F-torsionfree submodule of M. In the first section of the Chapter 4, we difind the copure projective moduls and give the equivlent condition, which is dual to the conclusion in [12].Definition 4.1.1. Let R be a ring, n a fixed nonnegative integer and P_n the class of all left R-modules of projective dimension at most n. A left R-module M is called n-copure projective if Ext¹ (M, N) = 0 for any N∈P_n.Theorem 4.1.4.Let R be a ring. The following are equivalent for a left .R-module M:(1) M is n-copure projective.(2) For every exact sequence 0→K→P→M→0 with P∈P_n, K→P is an P_n -preenvelope of K.(3) M is a cokernel of an P_n-preenvelope f : A→B with B projective.(4) M is projective with respect to every exact sequence 0→A→B→C→0 with A∈P-n.Let(?)₀ stand for all injective left R-modules. we have the followng result.Theorem 4.1.12. Let R be ring with sup{pdE|E∈(?)₀}≤n and n≥1. If M is an (n - 1)-copure projective left R-module, then there is an exact sequence 0→M→P→K→0 such that P is projective and K is n-copure projective.In the second section of the Chapter 4, we dsicuss the n-cotorsion modules and strongly cotorsion modules and corresponding cover and envelope. Let F_n be a class of all modules whose flat dimension are at most n, F^＜∞ be a class of all modules of finite falt dimension. A left R-module M is called n-cotorsion if Ext¹ (N,M) = 0 for any N∈F_n. A left R-module P is called strongly cotorsion if Ext¹ (N, P) = 0 for any N∈F^＜∞ . The main result:Thorem 4.2.4Let R be an IF ring , then the following are equivalent for a left R-module M:(1) M is n-cotorsion module.(2) For any exact sequence 0→M→F→L→0 .where F∈F_n Then F→L is a F_n-precover of L .(3) M is the kenel of F_n-precover f : A→B, where A is flat.(4)M is projective with respect to every exact sequence 0→A→B→C→0 ,where C∈F_n.Theorem 4.2.15For any ring R satisfy (*), every left R-module has a strongly cotorsion envelope if and only if every left R-moudle has a F^＜∞-cover.In [16],X.jinzhong study the property of cotorsion modules and show that any R module has the cotorsion envelope. Duality, in the third section of the Charter 4, we consder what ring satisfy that the ^⊥F and⊥^P is a precover class, which F and P stand for the class of all flat moduls and all projective modules . the main resultTheorem 4.3.4 Let R be a commutative coherent IF-ring ,and gldim≤2. then every modules has a ^⊥F-precover.Theorem 4.3.5 If R satisfies the following conditions 1)(^⊥P)^⊥= P2) every left R module has a mononoriphic projective envelope. Then enery left R modules has a ^⊥P-precover.In the forth section of the Charter 4, We introduce several weak global dimensions associated with a Tor-torsion pair of classes of left R-modules and right R-modules and consider the relation between them.Theorem 4.4.6 Let (F, C) be a Tor-torsion pair of R modules, and if m,l,n,p are definded in the paper , then max{m, l}≤n + p.In the forth section of the Charter 5, we defind the Green equivalent onÏ€-coalgebra and Green equivlent on tensor products ofÏ€-coalgebras , and the relaion betweenÏ€-coalgebras and their coidea.Let be aÏ€-coalgebra is a dulily of C . We define C⁸fC act C :→and←as follow for anyfor every are aright coidea, left coidea and idea,We denote [c>β,< c]β< c>βDefinition 5.5 Let C= ({C_α}α∈π,â–³,ε) be aÏ€-coalgebra,We define theÏ€reen relate on C as follows,for every C,d∈C_αfor allβ∈π Theorem 5.16 Let c₁,c₂∈C_α,d₁,d₂∈D_α. Then andTheorem 5.24 Letφ: C→D beÏ€-coalgebra homomorphism, if for anyα∈C_βsuch that . Then...