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

Theorem 9.10 ([GT]). Let C be an A-coring and let Σ be a right C-comodule.Assume that A C is flat. Set T = EndC (Mod-A) ((Σ, ρ Σ )). Then the following areequivalent:(a) The functor HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) → Mod-T is full andfaithful where C = −⊗ A C.(b) ɛ : HomC (Mod-A) ((Σ, ρ Σ ) , − ⊗ T Σ) → C (Mod-A) is an isomorphism.(c) (Σ, ρ Σ ) is a generator of C (Mod-A).(d) can : HomC (Mod-A) ((Σ, ρ Σ ) , −) ⊗ T Σ → − ⊗ A C is an isomorphism and T Σ isflat.Proof. By Proposition A.12, A C is flat if and only if (Mod-A) C is a Grothendieckcategory and the forgetful functor U : (Mod-A) C → Mod-A is left exact. Also,by the foregoing, D ϕ = HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) → Mod-T has a leftadjoint K ϕ = (− ⊗ T Σ, − ⊗ T ρ Σ ) .(a) ⇔ (b) It follows by Proposition <strong>2.</strong>3<strong>2.</strong>(a) ⇔ (c) It follows by Proposition A.3.(c) ⇒ (d) Since (Σ, ρ Σ ) is a generator of C (Mod-A) and since (− ⊗ T Σ, − ⊗ T ρ Σ ) :Mod-T → C (Mod-A) is a left adjoint of HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) →Mod-T, by Gabriel-Popescu Theorem A.9, (− ⊗ T Σ, − ⊗ T ρ Σ ) is a left exact functor.Since the forgetful functor U : (Mod-A) C → Mod-A is also left exact, wededuce that − ⊗ T Σ : Mod-T → Mod-A is left exact i.e. T Σ is flat. SinceHomC (Mod-A) ((Σ, ρ Σ ) , −) is full and faithful, by Theorem 9.6, can is an isomorphism.(d) ⇒ (a) It follows by Theorem 9.6.□Theorem 9.11 ([GT]). Let C be an A-coring, let B be a ring and assume that A Cis flat. Let (Σ, ρ Σ ) be a B-C-comodule. Then the following are equivalent:(a) The functor − ⊗ B Σ A : Mod-B → C (Mod-A) is an equivalence of categorieswhere C = − ⊗ A C.(b) can : Hom A ( B Σ A , −) → − ⊗ A C is an isomorphism and B Σ is faithfully flat.(c) (Σ, ρ Σ ) is a generator of C (Mod-A) and the functor − ⊗ B Σ : Mod-B →C (Mod-A) is full and faithful.(d) (Σ, ρ Σ ) is a generator of C (Mod-A), the functor − ⊗ B Σ : Mod-B →C (Mod-A) is faithful and λ : B → T = EndC (Mod-A) ((Σ, ρ Σ )) is an isomorphism.Proof. (a) ⇒ (b) By Theorem 9.7, can is an isomorphism. Since A C is flat, byProposition A.12, the forgetful functor U : C (Mod-A) → Mod-A is exact. Since Uis also faithful, we get that the functor− ⊗ B Σ A : Mod-B → Mod-Ais faithful and exact.(b) ⇒ (a) It follows by Theorem 9.7.(a) ⇒ (c) Since B is a generator of Mod-B, B Σ ≃ B ⊗ B Σ is a generator ofC (Mod-A).(c) ⇒ (d) Since − ⊗ B Σ : Mod-B → C (Mod-A) is full and faithful and it isthe left adjoint of the adjunction ( − ⊗ B Σ, HomC (Mod-A) ((Σ, ρ Σ ) , −) ) , the unit is a205