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

Let C be an A-coring and assume that g ∈ C is a grouplike element. Then we canconsider the map ρ A : A → A ⊗ A C defined by settingρ A (a) = 1 A ⊗ A (g · a) for every a ∈ A.We denote by C = − ⊗ A C. Then, by Lemma 9.23, (A, ρ A ) is a right C-comoduleandEndC (Mod-A) ((A, ρ A )) ≃ {b ∈ A | 1 A ⊗ A (g · b) = b ⊗ A g}= {b ∈ A | 1 A ⊗ A (g · b) = 1 A ⊗ A bg} = {b ∈ A | g · b = bg} = A coC .In this case the mapcan : Σ ∗ ⊗ B Σ = A ⊗ B A → Cis defined by settingcan (a ⊗ B a ′ ) = aga ′and C is called a Galois coring iff can is an isomorphism and B = A coC .Proposition 9.24. Let C be an A-coring and assume that g ∈ C is a grouplikeelement. Let B ⊆ A coC . Then the following statements are equivalent:(a) A C is flat and the functor − ⊗ B A : Mod-B → C (Mod-A) = (Mod-A) C is anequivalence of categories;(b) the canonical map can : A ⊗ B A → C is an isomorphism and B A is faithfullyflat;(c) A C is flat, A is a finitely generated projective generator of C (Mod-A) andλ : B → T = EndC (Mod-A) ((A, ρ A )) = A coC is an isomorphism.Theorem 9.25. [BRZ2002, Theorem 5.6]Let C be an A-coring and assume thatg ∈ C is a grouplike element.1) If C is a Galois coring and A coCA is faithfully flat, then the functor −⊗ A coC A :Mod-A coC → C (Mod-A) = (Mod-A) C is an equivalence of categories and A Cis flat.2) If the functor − ⊗ A coC A : Mod-A coC → C (Mod-A) is an equivalence ofcategories, then C is a Galois coring.3) If A C is flat and the functor − ⊗ A coC A : Mod-A coC → C (Mod-A) is anequivalence of categories, then A coCA is faithfully flat.Proof. 1) follows from Proposition 9.24 (b) ⇒ (a).2) follows from Theorem 9.7.3) follows from Proposition 9.24 (a) ⇒ (b). □Corollary 9.26. Let C be an A-coring and assume that g ∈ C is a grouplikeelement. Assume that A C is flat. Let B ⊆ A coC . Then the following statements areequivalent:(a) the functor − ⊗ B A : Mod-B → C (Mod-A) = (Mod-A) C is an equivalenceof categories;(b) the canonical map can : A ⊗ B A → C is an isomorphism and B A is faithfullyflat;(c) A is a finitely generated projective generator of C (Mod-A) and λ : B → T =EndC (Mod-A) ((A, ρ A )) = A coC is an isomorphism.211