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

174l = eC ρ L : L = − ⊗ B A −→ ˜CL = − ⊗ B Σ ⊗ A A ⊗ R H ∼ = − ⊗ B A ⊗ R CA second particular case of this situation we have the one where R = kis a commutative ring, C = H is a Hopf algebra over k and A is a rightGalois extension of T = A co(H) .Now let us setA = Mod-R,B = Mod-T where T = End eC (Σ)A = − ⊗ R A : A = Mod-R −→ A = Mod-RC = − ⊗ R C : A = Mod-R −→ A = Mod-RB = − ⊗ T B : B = Mod-T −→ B = Mod-T where B = Hom A (Σ A , Σ A )Ψ = − ⊗ R ψ : AC = − ⊗ R C ⊗ R A −→ CA = − ⊗ R A ⊗ R CL = − ⊗ T Σ : Mod-T −→ Mod-A ∼ = A AW = Hom A (Σ, −) : Mod-A −→ Mod-T˜C = − ⊗ A A ⊗ R C : Mod-A −→ Mod-Al = eC ρ L : L = − ⊗ T Σ −→ ˜CL = − ⊗ T Σ ⊗ A A ⊗ R C ∼ = − ⊗ T Σ ⊗ R CWhen Σ A is f.g.p., we setA (Σ ∗ ) B= Hom A ( B Σ, A A)and we consider the following formal dual structure M = (A, B, P, Q, σ A , σ B ) on thecategories A and B .• A = (− ⊗ R A, − ⊗ R m A , − ⊗ R u A ) is a monad on A = Mod-R• B = (− ⊗ T B, − ⊗ T m B , − ⊗ T u B ) is a monad on B = Mod-T where T =End eC (Σ)• P = − ⊗ R Σ ∗ T : Mod-R → Mod-T• Q = A U ◦ L = − ⊗ T Σ R : Mod − T → Mod − R − ⊗ T Σ : Mod-T → Mod-R• σ A : QP → A is defined by• σ B : P Q → B is defined byσ A : QP = − ⊗ R Σ ∗ ⊗ T Σ → − ⊗ R A− ⊗ R f ⊗ T x ↦→ − ⊗ R f (x)σ B : P Q = − ⊗ T Σ R ⊗ R Σ ∗ T → B = − ⊗ T B ∼ = − ⊗ T Σ R ⊗ A Σ ∗ T− ⊗ T y ⊗ R γ ↦→ − ⊗ T y · γ () ≃ − ⊗ T yγ (x i ) ⊗ A x ∗ i= − ⊗ T y ⊗ A γ (x i ) x ∗ i = − ⊗ T y ⊗ A γ• ( P : A → B, B µ P : BP → P, µ A P : P A → P ) is a bimodule functor• ( Q : B → A, A µ Q : AQ → Q, µ B Q : QB → Q) is a bimodule functor• σ A : QP → A is A-bilinear• σ B : P Q → B is B-bilinearσ A ◦ (A µ Q P ) = m A ◦ ( Aσ A) and σ A ◦ ( )Qµ A P = mA ◦ ( σ A A )σ B ◦ (B µ P Q ) = m B ◦ ( Bσ B) and σ B ◦ ( )P µ B Q = mB ◦ ( σ B B )