On Equivalence And Duality For Some Categories

Morita theory gives some characterizations of equivalences between module categories. The theory of equivalences and dualities between module subcategories, originating in the theory of Morita theorems, had been studied extensively. In the early eighties, the notion of tilting module was introduced and tilting theory may be viewed as a generalization of Morita equivalence theory. In [20], Puller gave the notion of quasi-progenerator and gave another generalization of Morita theory. Menni and Orsatti introduced a generalization of these modules that have come to be called *-modules. In [29], Golpi studied the relation between the classical tilting modules and *-modules, namely, P_R is a classical tilting module if and only if P_R is a faithful finendo *-module if and only if P_R is a *-module and E(R)∈Gen(P_R), where E(R) is the injective envelope of R.Miyashita [62] gave the definition of finitely generated tilting modules of projective dimension≤n over any ring and then Hugel and Coelho [24] considered infinitely generated tilting modules of projective dimension≤n over any ring. Recently, Wei and other authors [52] generalized *-modules to *ⁿ-modules and gave a connection between *ⁿ-modules and finitely generated tilting modules of projective dimension≤n. Wei also introduced the notion of *^s-modules, where s denotes static, by replacing the subcategory Gen(P) in the theory of *-module with the subcategory Stat(P) and some results on *-module are successively extended to *^s-modules.On the other side, quasi-duality modules are dual to quasi-progenerators and cotilting modules are dual to tilting modules. Quasi-duality modules and cotilting modules have been a central topic of recent investigation in module theory. Colby and Fuller generalized these modules to costar modules which may be viewed as dual to *-modules in a sense, cf. [39]. Category-theoretic and homological methods are intrinsically related to the role of tilting modules. Tilting theory has been extended to abstract categories, like the case of derived categories, cf. [8]. Most of results about tilting theory need finiteness assumptions. Colpi [31] has considered 1-tilting objects in Grothendieck categories avoiding any finiteness condition.In [27], Takeuchi developed a theorem that characterizes equivalences of comodule categories over fields, dualizing Morita results on equivalences of module categories. W. Mingyi introduced the classical tilting comodules over coalgebras over fields and prove the tilting theorem for classical tilting comodules over right semiperfect right conoetherian coalgebras over fields(see [59]). Recently, many authors study coalgebras over rings. Al-Takhman [21] generalized the Morita-Takeuchi theory to coalgebras over rings.In this thesis, we get the following results.In chapter 1, we give introduction and preliminaries.In chapter 2, We give the definition of co-*-modules and investigate the relation between 1-cotilting modules and co-*-modules. Colpi [29] characterized 1-tilting modules by *-modules. We shall characterize 1-cotilting modules by co-*-modules in chapter 2. Moreover, we shall discuss the little finitistic dimension on left A-module category and right R-module category, where A is the endmorphism ring of a finitistic cotilting bimodule. We get the following main results:Theorem 2.2.2 Let P_R∈Mod-R. The following conditions are equivalent:(1) P_R is a 1-cotilting module;(2) P_R is a co-*-module and Proj_R (?) Cogen(P_R);(3) P_R is a co-*-module and R∈Cogen(P_R);(4) P_R is a faithful co-*-module;(5) P_R is a faithful cofinendo co-*-module and Cogen(P_R)-injective.Theorem 2.4.7 Let _AP_R be a finitistic cotilting bimodule, where A = End_R(P).(1) If fin .dimR= d <∞, then fin.dimA≤d + 1.(2) If fin.dimA = d <∞, then fin.dimR≤d + 1.(3) If fin.dimR<∞or fin.dimA <∞, then |fin.dim.R - fin.dimA|≤1.In chapter 3, We give the definition of co-*ⁿ-modules and investigate the relation between n-cotilting modules and co-*ⁿ-modules. We characterize n-cotilting modules by co-*ⁿ-modules. We get the following main result:Theorem 3.2.4 Let P_R∈Mod-R and Cogen_n(P_R) is closed under submodules. Denote by Proj_R the class of all projective R-modules. The following conditions are equivalent.(1) P_R is an n-cotilting module;(2) P_R is a co-*ⁿ-module and Proj_R (?) Cogen_n(P_R) (?)^⊥P_R.In chapter 4, We give the definition of r-costar modules under artin algebra situation. Colby and Fuller [39] gave the definition of costar modules, and costar modules can induce dualities of module categories. r-Costars are obtained by replacing the subcategory cogen(P_Λ) in the theory of costar modules with the subcategory Ref(P_Λ). We also give characterizations of r-costar modules and discuss r-costar modules with some special properties. We get the following main result:Theorem 4.4.2 Let P∈mod-Λ, the following are equivalent.(1) P is an r-costar module such that Ref(P_Λ) is a resolving subcategory.(2) Ref(P_Λ) = KerExt_Λ^{i≥1}(-,P).(3) Ref(P_Λ) (?) KerExt_Λ^{i≥1}(-,P)(?)cogen(P_Λ).(4) proj_Λ(?) Ref(P_Λ) (?) KerExt_Λ^{i≥1}(-,P) and proj_A (?) Ref(_AP) (?) KerExt_A^{i≥1}(-,P).In chapter 5, Given any Grothendieck category y, we give the definitions of n-tilting objects and *ⁿ-objects. We also get some results of n-tilting objects in this general setting. This may be viewed as a generalization of 1-tilting object in [31] and n-tilting modules in [52]. We get the following main result:Theorem 5.2.5 Let V be a *ⁿ-object and A = End_y(V). Then the following conditions are equivalent.(1) V is a *ⁿ-object.(2) V is selfsmall and for any exact sequence 0→L→M→N→0 with M,N∈Gen_n(V), we have L∈Gen_n(V) if and only if the induced sequence 0→H_V(L)→H_V(M)→H_V(N)→0 is exact.(3) V is selfsmall and for any exact sequence 0→N→V^X→M→0 with M∈Gen_n(V) and X is a set, we have N∈Gen_n(V) if and only if the induced sequence 0→H_V(N)→H_V(V^X)→H_V(M)→0 is exact.(4) V induces an equivalence T_V : KerT_V^{i≥1}(?) Gen_n(V) : H_V.In chapter 6, We give the definitions of n-self-cotilting comodules and n-cotilting comodules in comodule categories, where n-self-cotilting comodules and n-cotilting comodules may be viewed as a generalization of self-cotilting comodules given by Wis-bauer [11] and tilting comodules given by Wang [58] respectively. We also obtain that n-self-cotilting comodules and n-cotilting comodules can induce equivalences between categories of comodules. The results on equivalences of module categories in [52] are dualized to comodule categories. In section 6.5, we give example of n-self-cotilting comodules. We get the following main result:Theorem 6.2.5 The following conditions are equivalent.(1) T is a quasi-finite self-co-small n-self-cotilting comodule.(2) T is a quasi-finite self-co-small and for any exact sequence 0→L→M→N→0 with L,M∈Cogen_n(T), we have N∈Cogen_n(T) if and only if the induced sequence 0→F(L)→F(M)→F(N)→0 is exact.(3) T is a quasi-finite self-co-small and for any exact sequence 0→N→T^X→M→0 with N∈Cogen_n(T) and X is a set, we have M∈Cogen_n(T) if and only if the induced sequence 0→F(N)→F(T^X)→F(M)→0 is exact.(4) T is quasi-finite and T induces an equivalence G : KerG(i≥1) (?) Cogen_n(T) : F.In chapter 7, We give the definition of (n, t)-quasi-injective comodules in M^C over coalgebra C and obtain that (n, t)-quasi-injective comodules can induce equivalences of categories of comodules. The results on equivalences of module categories in [56] are dualized to comodule categories. We get the following main result:Theorem 7.1.6 If T_C is a quasi-finite self-co-small (n, t)-quasi-injective comodule with n≥2 and 1≤t≤n - 1, then for each 0≤i≤t - 1, there is an equivalence F : T_i(?) A_i : G.In chpater 8, We give the definitions of oo-quasi-injective comodules and oo-cotilting comodules in M^C. We also investigate equivalences induced by these comodules. The results on equivalences of module categories in [57] are dualized to comodule categories. We get the following main result:Theorem 8.1.4 Let T_C be a quasi-finite self-co-small comodule and D = h_C(T_C, T_C). Then the following conditions are equivalent.(1) T is an∞-quasi-injective comodule.(2) For any exact sequence 0→L→M→N→0 with L,M∈Cogen_{∞}(T), we have N∈Cogen_{∞}(T) if and only if the induced sequence 0→F(L)→F(M)→ F(N) → 0 is exact.(3) For any exact sequence 0→N→T^X→M→0 with N ∈ Cogen_{∞}(T) and X is a set, we have M ∈ Cogen_{∞}(T) if and only if the induced sequence 0 → F(N)→ F(T^X) → F(M) → 0 is exact.(4) T induces an equivalence G : KerG^{i≥1} Cogen_{∞}(T) : F.