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

182(208)(209)(210)(211)(h1 ) 0 ⊗ h2 ⊗ ( h 1) 1= (h 2 ) 1 ⊗ (h 2 ) 2 ⊗ S (h 1 )h 1 h 2 = ε H (h) 1 A(hl) 1 ⊗ (hl) 2 = l 1 h 1 ⊗ h 2 l 2a 0 (a 1 ) 1 ⊗ (a 1 ) 2 = 1 A ⊗ a.9.5. The Schauenburg situation is the particular case when T = A co(H) = k. Hencewe haveA = Mod-k,B = Mod-k where T = End eC (Σ) = A co(H) = k.A = − ⊗ k A : A = Mod-k −→ A = Mod-kC = − ⊗ k H : A = Mod-k −→ A = Mod-kB = − ⊗ k A = A : B = Mod-k −→ B = Mod-k where A = Hom A (A A , A A )Ψ = − ⊗ k ψ : AC = − ⊗ k H ⊗ k A −→ CA = − ⊗ k A ⊗ k H(ψ : H ⊗ k A → A ⊗ k H, h ⊗ k a ↦→ can C can−1C (1 A ⊗ k h) a )L = − ⊗ k A : Mod-k −→ Mod-A ∼ = A AW = Hom A (A, −) : Mod-A −→ Mod-k˜C = − ⊗ A A ⊗ k H ∼ = ⊗ k H : Mod-A −→ Mod-Al = eC ρ L : L = − ⊗ k A −→ ˜CL = − ⊗ k A ⊗ A A ⊗ k H ∼ = − ⊗ k A ⊗ k HLet us assume thatThen,k = commutative ringH = Hopf algebraA/k = faithfully flat H-Galois extension with ρ H A : A → A ⊗ HA co(H) = k1 Acan : A ⊗ A → A ⊗ H defined by setting can (a ⊗ b) = ab 0 ⊗ b 1γ : H → A ⊗ A defined by setting γ (h) := can −1 (1 A ⊗ h) = h 1 ⊗ h 2τ : A → A ⊗ A ⊗ Aa ↦→ a 0 ⊗ γ (a 1 ) = a 0 ⊗ a 1 1 ⊗ a 2 1is a pretorsor. We want to construct the comonads C and D as in Theorem 6.5where A = B = Mod-k and P = Q = − ⊗ A. First, let us consider ω l = − ⊗ k ̂ωland ω r = − ⊗ k ̂ω r : − ⊗ A ⊗ A → − ⊗ A ⊗ A ⊗ A wherêω l = ( σ A ⊗ A ⊗ A ) ◦ (A ⊗ τ) : A ⊗ A → A ⊗ A ⊗ Âω l (a ⊗ b) = ab 0 ⊗ b 1 1 ⊗ b 2 1̂ω r = (u A ⊗ A ⊗ A) : A ⊗ A → A ⊗ A ⊗ Âω r (a ⊗ b) = 1 A ⊗ a ⊗ b.