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

192=(Qε C H C) ( ∑ )−□ C k i ⊗ h iso that the counitality conditions are also satisfied, i.e. χ is really a coherd. SinceH k is faithfully flat, we have that ( k, ε H) (= Coequ Mod-k H ⊗ ε H , ε H ⊗ H ) and sinceH C is faithfully coflat, by [Schn1, Proposition <strong>1.</strong>1], we also have that(C, π) =(C, ε C H C) ()= Coequ Comod-C H□ C ε C H C , ε C H C □ C H= Coequ Comod-C (H□ C π, π□ C H)so that X is a regular formal codual structure and thus χ is a regular coherd. FollowingTheorem 6.29, we calculate the monad(A, x) = Coequ Fun(w l , w r)where w l = (χP ) ◦ (QP δ C ) and w r = QP ε C : QP C → QP . In our caseand− ⊗ ∑ i− ⊗ ∑ i∑∑w l : − ⊗ H ⊗ H□ C H → − ⊗ H□ C H∑j ki,j ⊗ ∑ h i,j ⊗ g i ↦→ − ⊗ ∑ iw r : − ⊗ H ⊗ H□ C H → − ⊗ H□ C H∑j ki,j ⊗ ∑ h i,j ⊗ g i ↦→ − ⊗ ∑ ij ki,j (1) ⊗ ki,j (2) S ( h i , j ) g ij εH ( k i,j) h i,j ⊗ g i .Assume now that k is a field, so that everything is flat over k. Hence, for everyX ∈ Mod-kand thusAX ==X ⊗ H□ C HIm (X ⊗ w l − X ⊗ w r ) = X ⊗ H□ C HIm (X ⊗ (w l − w r ))X ⊗ H□ C HX ⊗ Im (w l − w r ) = X ⊗ H□ CHIm (w l − w r )A = − ⊗ H□ CH〈 ∑where I w =i∑j ki,j (1) ⊗ ki,j (2) S (hi , j) g i − ∑ 〉i∑j εH (k i,j ) h i,j ⊗ g i . In the sequel,given elements ∑ i hi ⊗ g i ∈ H□ C H, we will use the notation][∑i hi ⊗ g i = ∑ i hi ⊗ g i + I w .AWe will prove that this new monad A on the category Mod-k is isomorphic to themonad coming from the algebra L. Consider the following mapI wϕ : H□ CH−→ LI] w[∑i hi ⊗ g i ↦→ ∑ S ( h i) g iiA