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

190H C is faithfully coflat. Assume also C H coflat. Then we can consider thefollowing formal codual structure X = (C, D, Q, P, δ C , δ D ) whereA = Mod-kB = Comod-CC = ( − ⊗ H, − ⊗ ∆ H , − ⊗ ε H) : A = Mod-k −→ A = Mod-k(D = −□ C H, −□ C ∆ H , −□ C ε C H C) : B = Comod-C −→ B = Comod-CQ = −□ C H : B = Comod-C → A = Mod-kP = − ⊗ H C : A = Mod-k → B = Comod-Cδ C : = − ⊗ ∆ H : C = − ⊗ H → QP = − ⊗ H□ C Hδ D : = −□ C ∆ H : D = −□ C H → P Q = −□ C H ⊗ H.Now, for every ∑ ∑i j ki,j ⊗ h i,j ⊗ g i ∈ (H ⊗ H) □ C H, we have that∑ ∑( )(214)i j ki,j ⊗ h i,j(1) ⊗ π h i,j(2)⊗ g i = ∑ ∑i j ki,j ⊗ h i,j ⊗ π ( )g(1)i ⊗ gi(2) .We want to prove that ∑ ∑i j ki,j ⊗ S (h i,j ) g i ∈ H ⊗ L. We compute, using the leftH-linearity of π and (214)∑ ∑ (i j ki,j ⊗ π S ( h i,j) )(1) gi (1) ⊗ S ( h i,j) (2) gi (2)= ∑ ∑ ( )i j ki,j ⊗ S h i,j(2)π ( ) )g(1)i ⊗ S(h i,j(1)g(2)i= ∑ ∑ ( ) ( ) ( )i j ki,j ⊗ S h i,j(2)π h i,j(3)⊗ S h i,j(1)g i= ∑ ∑ ( ( ) ) ( )i j ki,j ⊗ π S h i,j(2)h i,j(3)⊗ S h i,j(1)g i= ∑ ∑i j ki,j ⊗ π (1 H ) ⊗ S ( h i,j) g iso that we get∑ ∑ij ki,j ⊗πwhich means∑ ∑i(S ( h i,j) )(1) gi (1) ⊗S ( h i,j) (2) gi (2) = ∑ ∑i j ki,j ⊗π (1 H )⊗S ( h i,j) g ij ki,j ⊗ S ( h i,j) g i ∈ Ker ( H ⊗ [C ρ H − π (1 H ) ⊗ (−) ]) = H ⊗ co(C) H = H ⊗ Li.e.∑ ∑(215)i j ki,j ⊗ S ( h i,j) g i ∈ H ⊗ L.Similarly, for every ∑ ∑i j li,j ⊗ h i,j ⊗ g i ∈ (L ⊗ H) □ C H, we have that∑ ∑i j li,j ⊗ S ( h i,j) g i ∈ Ker ( L ⊗ [C ρ H − π (1 H ) ⊗ (−) ]) = L ⊗ co(C) H = L ⊗ Li.e.(216)∑i∑j li,j ⊗ S ( h i,j) g i ∈ L ⊗ L