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

201= Coequ Fun(mL X, L L µ X) 3.14= ( X, L µ X).Thus to this aim it is enough to construct an isomorphism of left L-modules ̂ζ :L⊗H⊗(L⊗H)L + E H → L. This will imply that ζ = −⊗ ̂ζ : Q EE ̂Q = −⊗L⊗H⊗(L⊗H)L + E H →− ⊗ L gives rise to a functorial isomorphism L Q EE ̂QL ∼ = IdL A. We want to showthat ̂ζ is the following morphismLet us consider∑iL ⊗ Ĥζ :(L ⊗ H) L ⊗ + E H → L∑ [l i,j ⊗ h i,j] Q b ⊗ E g i ↦→ ∑ ∑i∑ij∑ζ : L ⊗ H□ C H → L∑j li,j ⊗ h i,j ⊗ g i ↦→ ∑ ij li,j S ( h i,j) g i .j li,j S ( h i,j) g iand let us prove that it is well-defined. By (217) we get that ∑ iNow we use that C H is coflat. Let us prove thatζ [ (L ⊗ H) L + □ C H ] = 0.∑j li,j S (h i,j ) g i ∈ L.Let ∑ i zi ⊗ h i ∈ (L ⊗ H) L + □ C H where, for each i, z i ∈ (L ⊗ H) L + . This meansthat there exist elements w i,j ∈ L ⊗ H and elements t i,j ∈ L + such thatz i = ∑ j wi,j · t i,jSince w i,j ∈ L ⊗ H there exist l i,j,k ∈ L and g i,j,k ∈ H such thatw i,j = ∑ k li,j,k ⊗ g i,j,k .Hence we have∑i zi ⊗ h i = ∑ ∑ ∑ (l i,j,k ⊗ g i,j,k) · t i,j ⊗ h ii j k= ∑ ∑ ∑ [l i,j,k t i,ji j k(1) ⊗ gi,j,k t i,j(2) − ( l i,j,k ⊗ g i,j,k) (ε )] H ti,j⊗ h i= ∑ ∑ ∑i j k li,j,k t i,j(1) ⊗ gi,j,k t i,j(2) ⊗ hi − ( l i,j,k ⊗ g i,j,k) (ε ) H ti,j⊗ h iso that)ζ(∑i zi ⊗ h i = ∑ ∑ ∑ ( )i j k li,j,k t i,j(1) S g i,j,k t i,j(2)h i − ( l i,j,k S ( g i,j,k)) (ε ) H ti,jh i= ∑ ∑ ∑ ( )i j k li,j,k t i,j(1) S t i,j(2)S ( g i,j,k) h i − ( l i,j,k S ( g i,j,k)) (ε ) H ti,jh i= ∑ ∑ ∑ (i j k li,j,k ε ) H ti,jS ( g i,j,k) h i − ( l i,j,k S ( g i,j,k)) (ε ) H ti,jh i = 0.L⊗Hhence we have a well defined map ζ : □(L⊗H)L + C H → L defined by setting(∑ ∑ [ζl i,j ⊗ h i,j] )Q b ⊗ g i = ∑ ∑i j li,j S ( h i,j) g i .ij