The Cofficient Extends Of Induction Functor Of Quantum Algebras

We Let A = Z [v]_? ,where v is an indeterminate and (?) is an ideal in Z [v] generated by v-1 and a fixed odd prime p ,A' = Q(v) is the fraction field of A, let U' is a quantum algebra over A' associated to symmetry Car tan matrix (a_ij)_n×n. U is the A-subalgebra of U' generated by E_i^N,F_i^N,K_i,K_i^-1(i= 1,2,…,N≥0), so is a A-Hopf algebra.In this paper,the author gives several features of induction functor D_Γ(-),H_Γ(-) and H_Γ⁰(U_Γ/U_Γ^b,-) afterextending coefficient from A which is the basic ring of functors above to any A -algebraΓ. Let A-algebraΓis a field, then we have a U_Γ-module isomorphism D(λ)(?)_AΓ(?) D_Γ(λ). If category U-module M is a finite free A-module, then we haveD(λ(?)M)(?)Γ(?)D_Γ((λ(?)M)_Γ).Furthor more ,we have proved that H_Γ(-) is an exactfuntor and the quantum coordinate algebra H_Γ(Γ) (?)Γ[U_Γ] is a freeΓ-moduel. last, we have a U_Γ-module isomorphism H⁰(U/U^b,λ)(?)_AΓ(?)H_Γ⁰(U_Γ/U_Γ^b,λ),Whereλ∈X⁺,and provethat H_Γ⁰(U_Γ/U_Γ⁰,-) is an exact funtor if A -algebraΓis a flat A -moduel.