Can Be Divided Into The Nature Of The Functor Ring. Group

Let (?)=(C_α)α∈G be a G - A-Coring,where A is a ring with unit,and Gis a group.Theunit element of G will be denoted by e. In, there is the conception of a group coring. And the author can give the tensor product of the corings and show that bicomodules are in fact comodules. In, for an A-coring C, the author study when two functors, the forgetfulfunctor: M^Câ†'M^A and induction functor -(?)C : M_Aâ†'M^C are separable.This paper wants to show that when the forgetful functor: M^?â†'M^S And its right adjoint functor are separable.In section 3, we define a bicomdule (?)M(?), where (?) is a G - A -coring and (?) is a G - B- coring. We also show that the bicomodules are in fact comodules.In.section 4, we prove that there is a pair of adjoint functor (F₄, G₄) between the categories M(?) and M_A. Then we prove the two main results of this paper:Theorem 4.5 Let (?) be a cofree group coring,and G is finite.Then the functor (?)_α∈G^vα(-(?)_AC_e) is separable if and only if there exists an invarint (?) = (q_α)_α∈G∈(?)^A={q_α∈C_α|q_α·a=a·q_α,α∈G such thatε(q_e) = 1.Theorem 4.8 Let G - A-coring be a cofree group coring,and G is finite.Then the forgetful functor F : M^?â†'M_A is separable if and only if there exists anτ∈Hom_A((?)_α∈Gv_α(C_e(?)AC_e),A), such thatÏ„o(∑_α∈Gi_αov_αoâ–³_e,e)=ε,and...