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

Proof. (a) ⇒ (c) Apply Proposition A.19 to the functor T = − ⊗ B Σ. Since B is afinitely generated and projective generator of Mod-B, Σ C ≃ B ⊗ B Σ C is a finitelygenerated and projective generator of C (Mod-A). By the equivalence (a) ⇔ (d) ofTheorem 9.11 we get that λ : B → T = EndC (Mod-A) ((Σ, ρ Σ )) is an isomorphism.(c) ⇒ (b) Let us consider U : C (Mod-A) → Mod-A which is the left adjointof the free functor − ⊗ A C : Mod-A → C (Mod-A). We have to prove that Σ A isfinitely generated and projective. Now, by Proposition A.18, we prove that Σ A isfinite, i.e. that Hom Mod-A (Σ A , −) preserves coproducts. Let us consider a family(A i ) i∈I∈ Mod-A. We have the followingΣfinite≃∐i∈IHomC (Mod-A)Hom Mod-A (U (Σ) , A i ) (U,−⊗ AC)adj≃(Σ, ∐ i∈I(A i ⊗ A C))∐i∈I−⊗ A Cright adj≃HomC (Mod-A) (Σ, A i ⊗ A C)HomC (Mod-A)((U,−⊗ A C)adj≃ Hom Mod-A U (Σ) , ∐ )A ii∈I(Σ,( ∐i∈IA i)207⊗ A C)Since Σ A = U (Σ) we deduce that Hom Mod-A (Σ A , −) preserves coproducts. Since byassumption A C is flat, by Theorem 9.10 (c) ⇒ (d) we get thatcan : HomC (Mod-A) ((Σ, ρ Σ ) , −)⊗ B Σ → −⊗ A C is an isomorphism and B Σ is flat. Bydiagram 218 we obtain that can is also an isomorphism. Since Σ is a finitely generatedprojective generator of C (Mod-A), by Corollary A.21 HomC (Mod-A) ((Σ, ρ Σ ) , −)is an equivalence of categories, hence so is − ⊗ B Σ : Mod-B → C (Mod-A) so thatBΣ is faithfully flat.(b) ⇒ (a) Since can is an isomorphism, we have that A C is flat if and only ifAΣ ∗ ⊗ B Σ is flat. By assumption we know that B Σ is flat. Since Σ A is finitelygenerated and projective, also A Σ ∗ is finitely generated and projective so A Σ ∗ is flat.Therefore the functor − ⊗ A Σ ∗ ⊗ B Σ is left exact and, since can is an isomorphism,−⊗ A C is also left exact. By diagram 218, since can is an isomorphism, can is also anisomorphism. Now, A C is flat and B Σ is faithfully flat, then we can apply Theorem9.11 (b) ⇒ (a) to deduce that − ⊗ B Σ : Mod-B → C (Mod-A) is an equivalence ofcategories.□Remark 9.16. By Corollary 9.15 we deduce that if A C is flat and − ⊗ B Σ A : Mod-B → C (Mod-A) is an equivalence of categories, then Σ is a Galois C-comodule.Corollary 9.17 ([GT, Theorem 3.10] Generalized Descent for Modules). Let B Σ Abe a B-A-bimodule such that Σ A is finitely generated and projective. Let Σ ∗ =Hom A (Σ, A) . Then the following are equivalent:(a) A (Σ ∗ ⊗ B Σ) is flat and the functor − ⊗ B Σ : Mod-B → C (Mod-A) is anequivalence of categories where C = − ⊗ A Σ ∗ ⊗ B Σ(b) B Σ is faithfully flat.Proof. By (9.13) we have that Σ ∗ ⊗ B Σ is an A-coring and thus C = − ⊗ A Σ ∗ ⊗ B Σis a comonad on Mod-A. Note that Σ is a B-C-bicomodule via a canonical right