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

which is well-defined by (213), i.e. ∑ S (h i ) g i ∈ L. Note that, since L = co(C) H, forevery b ∈ L, we haveThe inverse of this map is given by1 H ⊗ π (1 H ) ⊗ b = 1 H ⊗ ∑ π (b 1 ) ⊗ b 2 .ϕ −1 : L −→ H□ CHI wb ↦→ [1 H ⊗ b] A.In fact we have (ϕ −1 ◦ ϕ) ( [ ∑ i hi ⊗ g i ] A)= ϕ −1 ( ∑ i S (hi ) g i ) = [1 H ⊗ ∑ i S (hi ) g i ] A=[ ∑ i hi ⊗ g i ] Aby definition of I w and (ϕ ◦ ϕ −1 ) (b) = ϕ ([1 H ⊗ b] A) = S (1 H ) b = band thus ϕ is bijective so thatA = − ⊗ H□ CH≃ − ⊗ L : Mod-k → Mod-k.I wThe functorial morphisms m A and u A of the monad A are uniquely determined byx ◦ (χP ) = m A ◦ (xx) and x ◦ δ C = u A ◦ ε Cwhere x : H□ C H → H□ CHI wdenotes the canonical projection. In our case we have( [ ]] )m H□ C HIw∑i hi ⊗ g i ⊗A[∑j kj ⊗ l j =A[∑i,j hi ⊗ g i S ( k j) ]l j A=[∑i,j 1 H ⊗ S ( h i) g i S ( k j) ] (∑l j = ϕ −1 S ( h i) g i S ( k j) )l jAi,j( (∑= ϕ −1 m L S ( h i) g i ⊗ ∑ S ( k j) ))l ji j= ( ϕ −1 ◦ m L ◦ (ϕ ⊗ ϕ) ) ( [∑]i hi ⊗ g]Ai ⊗[∑j kj ⊗ l jfrom which we deduce thatMoreoverso thatu H□ C HIwu H□ C HIwϕ ◦ m H□ C HIw= m L ◦ (ϕ ⊗ ϕ) .(ε H (h) ) = (x ◦ δ C ) (h) = [ h (1) ⊗ h (2)]((1 k ) = u H□ C H ε H (1 H ) ) = (x ◦ δ C ) (1 H ) = [ ]1 H(1) ⊗ 1 H(2) = [1 A H ⊗ 1 H ] AIw= [1 H ⊗ 1 L ] A= ϕ −1 (1 L ) = ( )ϕ −1 ◦ u L (1k )from which we deduce thatϕ ◦ u H□ C H = u L .IwThe two relations obtained say that ϕ : H□ CH−→ L is an algebra isomorphism sothatA = − ⊗ H□ CHI wI w≃ − ⊗ L as monads.AA)193