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

210Now, we consider a particular case of the setting investigated above.Lemma 9.23. Let C be an A-coring. Then A can be endowed with a right C-comodulestructure ρ C A if and only if C has a grouplike element, namely [( l A C ◦ ρC A)(1A ) ] .Proof. Assume first that A has a right C-comodule structure given by ρ C A . We wantto prove that g = [( lC A ◦ A) ρC (1A ) ] is a grouplike element for C. First, fromg = [( )lC A ◦ ρ C A (1A ) ]we deduce that(221) ρ C A (1 A ) = ( l A C) −1(g) = 1A ⊗ A gLet us compute∆ (( C lC A ◦ ρA) C (1A ) ) = ( ∆ C ◦ lC A ◦ ρA) C (1A ) = [( lC A ⊗ A C ) ◦ ( A ⊗ A ∆ C) ◦ ρA] C (1A )[(lAC ⊗ A C ) ◦ ( ρ C A ⊗ A C ) ◦ ρA] C (1A ) = [( lC A ⊗ A C ) ◦ ( ρ C A ⊗ A C )] ( ρ C A (1 A ) )ArightC-com=Moreover(221)= [( l A C ⊗ A C ) ◦ ( ρ C A ⊗ A C )] (1 A ⊗ A g) = [( l A C ◦ ρ C A)⊗A C ] (1 A ⊗ A g)= ( l A C ◦ ρ C A)(1A ) ⊗ A g = g ⊗ A g.ε (( C lC A ◦ ρA) C (1A ) ) = ( ε C ◦ lC A ◦ ρA) C (1A )= [ l A ◦ ( A ⊗ A ε C) ]◦ ρ C A (1A ) ArightCcom= 1 A .Conversely, let us assume that g ∈ C is a grouplike element and let us define ρ C A :A → A ⊗ A C by settingρ C A (a) = 1 A ⊗ A g · a.We have to check that it defines a C-comodule structure on A. We compute, forevery a ∈ A,[(A ⊗A ∆ C) ◦ ρ C A](a) =(A ⊗A ∆ C) (1 A ⊗ A g · a) ∆C Alin= 1 A ⊗ A g ⊗ A g · aso that= ( ρ C A ⊗ A C ) (1 A ⊗ A g · a) = [( ρ C A ⊗ A C ) ◦ ρ C A](a)(A ⊗A ∆ C) ◦ ρ C A = ( ρ C A ⊗ A C ) ◦ ρ C A.We also have, for every a ∈ A,[rAA ◦ ( A ⊗ A ε C) ] [◦ ρ C A (a) = rAA ◦ ( A ⊗ A ε C)] (1 A ⊗ A g · a)so that= r A A(1A ⊗ A ε C (g · a) ) ε C Alin= 1 A ε C (g) a = ar A A ◦ ( A ⊗ A ε C) ◦ ρ C A = Id A .□