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

218Let us compute, for every d ∈ D,[ ]ψ ◦ (D ⊗ uA ) ◦ r −1D (d) = [ψ ◦ (D ⊗ uA )] (d ⊗ 1 k ) = ψ (d ⊗ 1 A )so that we have(A,ρ H A)H-com algDisHmodcoal= (1 A ) 0⊗ d · (1 A ) 1= 1 A ⊗ d · 1 H = 1 A ⊗ d= (u A ⊗ D) (1 k ⊗ d) = [ ](u A ⊗ D) ◦ l −1D (d)ψ ◦ (D ⊗ u A ) ◦ r −1D= (u A ⊗ D) ◦ l −1D .Let us prove (223) . Let us compute, for every a ∈ A, d ∈ D[(ψ ⊗ D) ◦ (D ⊗ ψ) ◦(∆ D ⊗ A )] (d ⊗ a)= [(ψ ⊗ D) ◦ (D ⊗ ψ)] ( d (1) ⊗ d (2) ⊗ a )= (ψ ⊗ D) ( d (1) ⊗ a 0 ⊗ d (2) · a 1)= a00 ⊗ d (1) · a 01 ⊗ d (2) · a 1A isHcomod= a 0 ⊗ d (1) · a 1(1) ⊗ d (2) · a 1(2)DisHmodcoalg= a 0 ⊗ (d · a 1 ) (1)⊗ (d · a 1 ) (2)so that we getandso that we get= ( A ⊗ ∆ D) (a 0 ⊗ d · a 1 ) = [( A ⊗ ∆ D) ◦ ψ ] (d ⊗ a)(ψ ⊗ D) ◦ (D ⊗ ψ) ◦ ( ∆ D ⊗ A ) = ( A ⊗ ∆ D) ◦ ψ[rA ◦ ( A ⊗ ε D) ◦ ψ ] (d ⊗ a) = [ r A ◦ ( A ⊗ ε D)] (a 0 ⊗ d · a 1 )(= r A a0 ⊗ ε D (d · a 1 ) ) DisHmodcoal (= r A a0 ⊗ ε D (d) ε H (a 1 ) )= aε D (d) = ε D (d) a = l A(ε D (d) ⊗ a ) = [ l A ◦ ( ε D ⊗ A )] (d ⊗ a)r A ◦ ( A ⊗ ε D) ◦ ψ = l A ◦ ( ε D ⊗ A ) .Then by Proposition 9.30, ( D = A ⊗ D, ∆ D , ε D) is the A-coring associated to theentwining (A, D, ψ) and M D A = (Mod-A)D ≃ M D A (ψ). Note that M ∈ MD A (ψ) , issuch that ( ( )M, µ M) A is a right A-module, M, ρDM is a right D-comodule satisfying(µAM ⊗ D ) ◦ (M ⊗ ψ) ◦ ( ρ D M ⊗ A ) = ρ D M ◦ µ A Mi.e. for every m ∈ M and for every a ∈ A( )ρDM ◦ µ A M (m ⊗ a) = µAM (m 0 ⊗ a 0 ) ⊗ µ H D (m 1 ⊗ a 1 )which is exactly the condition (230) for M ∈ M D A (H) . Since morphisms in bothcategories M D A (ψ) and MD A (H) are right A-linear and right D-colinear morphismswe deduce thatM D A (ψ) ≃ M D A (H) .Since by Lemma 9.33 A ∈ M D A (H) ≃ MD A (ψ) , we can apply Theorem 9.31 to thecase ”C” = D and ”M C A (ψ) ” = MD A (ψ) ≃ MD A (H) .□