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

187Now(ϕ ⊗ A) |(H⊗A)co(H) = (ϕ ⊗ A) ◦ i (H⊗A)co(H)where i (H⊗A)co(H) : (H ⊗ A) co(H) → H ⊗ A is the canonical inclusion and hence weget(ϕ −1 ⊗ A ⊗ A ) []◦ [(ϕ ⊗ A) ⊗ A] ◦ i (H⊗A)co(H) ⊗ A ◦ δ C ◦ ϕ= ( ϕ −1 ⊗ A ⊗ A ) ◦ (ϕ ⊗ A ⊗ A) ◦ (H ⊗ γ) ◦ ∆ Hi.e. []i (H⊗A)co(H) ⊗ A ◦ δ C ◦ ϕ = (H ⊗ γ) ◦ ∆ H .Now we have[() ] (hi (H⊗A)co(H) ⊗ A ◦ δ 1 C ⊗ h 2) [() ]= i (H⊗A)co(H) ⊗ A ◦ δ C ◦ ϕ (h)= ( (H ⊗ γ) ◦ ∆ H) (h) = h 1 ⊗ h 1 2 ⊗ h 2 2i.e. (i (H⊗A)co(H) ⊗ A) (δC (h 1 ⊗ h 2)) = h 1 ⊗ h 1 2 ⊗ h 2 2Let us compute δ D : D → QQ = ⊗A⊗(H ⊗ A) co(H) following Proposition 7.2 whichneeds to satisfy(κ ′ 0Q) ◦ δ D = (Dj) ◦ ∆ D .In our case this meanswhereand(jP Q) ◦ (κ ′ 0Q) ◦ δ D = (jP Q) ◦ (Dj) ◦ ∆ D = (jj) ◦ ∆ D = (τ ⊗ A)κ ′ 0 (h ⊗ a) = h 1 ⊗ h 2 ⊗ a(jj) ◦ ∆ D (a ⊗ b) = [(τ ⊗ A) ◦ j] (a ⊗ b) = (τ ⊗ A) (a ⊗ b) = a 0 ⊗ a 1 1 ⊗ a 2 1 ⊗ b.so thatδ D(∑a i ⊗ b i )= ∑ (ai ) 0 ⊗ ( a i) 1 ⊗ bi .In this more specific situation we could compute both the comonads C and D andthe functor Q so that we obtained a coherd. We now would like to compute themonads corresponding to the coherd following Theorem 6.29. But the computationsare not straightforward and it is not clear what these new monads are.9.<strong>2.</strong> H-Galois coextension. This is the most clear example of a coherd that wecould give. It gives also a description of the dual case of the Morita-Takeuchiequivalence studied by Schauenburg in [Scha4]. In fact we could understand theequivalence between the module categories over the two monads constructed fromthe coherd.Let H = ( H, m H , u H , ∆ H , ε H , S ) be a Hopf algebra and let L ⊆ H be a rightcoideal subalgebra i.e. ∆ H (L) ⊆ L ⊗ H. We can consider ε H : H → k as acharacter so that)J = J ε H =〈(hy) (1)ε((hy) H (2)− h (1) ε ( H h (2) y ) 〉| h ∈ H, y ∈ L