Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

• R = associative unital algebra• A = R-ring• C = R-coring• ψ : C ⊗ R A → A ⊗ R C a right entwining• ˜C = A ⊗ R C the induced A-coring• (Σ A , ρ e CΣ ) = right ˜C-comodule = right entwined module.• T = End eC (Σ) .Note that if T ⊆ A is a right C-Galois extension i.e.comodule and the canonical Galois mapcan C : A ⊗ T A → A ⊗ R Ct ⊗ T t ′ ↦→ tρ C A (t ′ ) = tt ′ 0 ⊗ R t ′ 1is an isomorphism, then we can consider the right entwiningψ : C ⊗ R A → A ⊗ R C(c ⊗ R t ↦→ can C can−1C (1 A ⊗ R c) t )173( )AR , ρ C A is a right C-and hence the A-coring A ⊗ R C, which turns out to be a right Galois coring i.e. Ais a right comodule over the A-coring A ⊗ R C via ρ e CA defined byA ∼ = A ⊗ A A A⊗ Aρ C A−→ A ⊗ A A ⊗ R C ∼ = A ⊗ R C,t ↦→ 1 A ⊗ A t 0 ⊗ R t 1 .The coinvariants of A with respect to this coaction is still T and the canonical Galoismap iscan A⊗R C : A ⊗ T A → A ⊗ A A ⊗ R Ct ⊗ T t ′ ↦→ tρ A⊗ RCA(t ′ ) = t ⊗ A t ′ 0 ⊗ R t ′ 1 = 1 ⊗ A tt ′ 0 ⊗ R t ′ 1 = 1 ⊗ A can C (t ⊗ T t ′ )so that can A⊗R C is still an isomorphism. Therefore we can consider this case as aparticular case of the previous one, where(Σ A , ρ e CΣ ) =(A A , ρ e )CA .Let A be a right Galois extension of B over the Hopf algebra H. In thiscase we haveA = Mod-R,B = Mod-B where B = A co(H)A = − ⊗ R A : A = Mod-R −→ A = Mod-RB = − ⊗ B A : B = Mod-B −→ B = Mod-BQ = − ⊗ B A : B = Mod-B → A = Mod-RP = − ⊗ R A B : A = Mod-R → B = Mod-BQP = − ⊗ R A ⊗ B A m A→ − ⊗ R AP Q = − ⊗ B A ⊗ R A m A→ − ⊗ B AC = − ⊗ R H : A = Mod-R −→ A = Mod-R