Group On The Ring In A Finite-type Hopf Group Coalgebra Duality Theory

This dissertation consists of four sections. We shall give a group coring version for the duality theorem for actions and coactions of Hopf-G-coalgebras of finite type.The first and second sections are introduction and preliminaries respectively. In the second section, we show that G = (-)^coC is equivalent to Hom C/A(A,-).In the third section, we introduce the smash product A#H^* and show A#H^* is a associative algebra with the unit 1#1_Hα,α∈G.In the forth section we can describe A#H^* as the dual of the group coring, i.e.,^*(A(?)H)(?)(A(?)H)^*(?)#(H,A)(?)A#H^*.And we give a group coring version for the duality theorem of Hopf-G-coalgebras of homogeneously finite. We state the main results as follows.Theorem 2.3.5 Let (C,x) be a group coring with a family of grouplike elements,T=A^coC={α∈A|ax_α=x_αa,(?)α∈G} is the subring of A.(?) m∈M^?,Hom C/A(A,M)isa right T-module,via (ft)(a) =f(ta),for all f∈Hom C/A(A,M),a∈A,t∈t,ThenHomC/A(A, M) (?)M^coC as right T-module.Theorem 4.1 Let (?) be a G - A- coring, and (?) be homogeneously finite. Then the right dual of the canonical group coring associated to the ring morphism i : Aâ†'R_e is isomorphic to the (opposite) ring HomA(C_e,C_e)^op.Theorem 4.3 Let (?)= (H_α)_α∈G be homogeneously finite over k. A is a rightG- H-comdule algebra. Let A (?) (?) be a G - A-coring. There are ring isomorphisms^*(A(?)(?))(?)(A(?)(?))^*(?)#((?),A)(?)A#(?)^*．Theorem 4.5 Let (?) be a Hopf G-coalgebras of homogeneously finite over k, A be a right G-(?)- comodule algebra,and A is a G-(?)-Galois extension of A^coC. Then...