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

and[µBQ ◦ ( Qσ B)] (− ⊗ T x ⊗ R f ⊗ T y) = µ B Q (− ⊗ T x · f () ⊗ T y)176Now we have[ Aµ Q ◦ ( σ A Q )] (− ⊗ T x ⊗ R f ⊗ T y) = A µ Q (− ⊗ T x ⊗ R f (y)) = − ⊗ T xf (y)so that= − ⊗ T x · f () (y) = − ⊗ T xf (y)A µ Q ◦ ( σ A Q ) = µ B Q ◦ ( Qσ B) .Finally we compute[ Bµ P ◦ ( σ B P )] (− ⊗ R f ⊗ T x ⊗ R g) = B µ P (− ⊗ R f ⊗ T x · g ()) = − ⊗ R f (x · g ())and[µAP ◦ ( P σ A)] (− ⊗ R f ⊗ T x ⊗ R g) = µ A P (− ⊗ R f (x) ⊗ R g) = − ⊗ R f (x) g ()so that, for every y ∈ Σ we haveand[− ⊗ R f (x · g ())] (y) = − ⊗ R f (xg (y)) = − ⊗ R f (x) g (y)so that we get[− ⊗ R f (x) g ()] (y) = − ⊗ R f (x) g (y)B µ P ◦ ( σ B P ) = µ A P ◦ ( P σ A) .Note that, in the case R A is faithfully flat,by Corollary 9.2,(A, u A ) = (Mod-R, − ⊗ R u A ) = Ker (− ⊗ R γ)= Equ Fun (− ⊗ R u A ⊗ R A, − ⊗ R u A ⊗ R A) .Analogously if T B is faithfully flat we have(B, u B ) = (Mod-T, − ⊗ T u B ) = Equ Fun (− ⊗ T u B ⊗ T B, − ⊗ T u B ⊗ T B).Thus, in the following we will assume that both R A and T B are faithfullyflat so that we have a regular formal dual structure.The counit ɛ of the adjunction (L, W ) is given byɛ M : Hom A (Σ, M) ⊗ T Σ −→ Mf ⊗ T x ↦→ f (x)for each M ∈ Mod-A. Therefore we get that) ( ) eCcan =(˜Cɛ ◦ ρ L W : LW = Hom A (Σ, −) ⊗ T Σ −→ ˜C = − ⊗ A A ⊗ R C Ris defined bycan M : Hom A (Σ, M) ⊗ T Σ −→ M ⊗ A A ⊗ R Cγ ⊗ T x ↦→ γ (x 0 ) ⊗ R x 1)for each M ∈ Mod-A. Hence we deduce that(L, eC ρ L is a left ˜C-Galois functorif and only if (Σ A , ρ e CΣ ) is a right Galois comodule.