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

208coaction ρ C Σ : Σ → Σ ⊗ A Σ ∗ ⊗ B Σ defined by setting ρ C Σ (s) = ∑ n x i ⊗ A x ∗ i ⊗ B s where(x i , x ∗ i ) i=1,...,nis a dual basis for Σ A . Then we can apply Corollary 9.15 (a) ⇔ (b) tothe case ”C” = Σ ∗ ⊗ B Σ so that can : Σ ∗ ⊗ B Σ → C is the identity map. □9.18. Let B → A a k-algebra extension. Let ” B Σ A ” = B A A in 9.13. Then thecomatrix coring becomes C = A ⊗ B A which is an A-coring with coproduct ∆ C : C =A ⊗ B A → C ⊗ A C = A ⊗ B A ⊗ A A ⊗ B A defined by setting∆ C (a ⊗ B a ′ ) = a ⊗ B 1 A ⊗ A 1 A ⊗ B a ′and counit ε C : C = A ⊗ B A → A defined by settingε C (a ⊗ B a ′ ) = aa ′for every a, a ′ ∈ A. Such A-coring C = A ⊗ B A is called canonical coring or Sweedlercoring associated to the algebra extension B → A.Definition 9.19. Let B be a k-algebra and let B → σ A be an algebra extension.A right descent datum from A to B is a right A-module M together with a rightA-module morphism δ : M → M ⊗ B A such that(219) (δ ⊗ B A) ◦ δ = (M ⊗ B σ ⊗ B A) ◦ ( M ⊗ B l −1Aand(220) µ M ◦ δ = Mi=1)◦ δwhere l −1A: B ⊗ B A → A is the canonical isomorphism and µ M : M ⊗ B A → M isinduced by the A-module structure of M, µ A M : M ⊗ A → A. Given (M, δ) , (M ′ , δ ′ )two right descent data from A to B, a morphism of right descent data from A to Bis a right A-module map f : M → M ′ such thatδ ′ ◦ f = (f ⊗ B A) ◦ δ.We will denote by D (A ↓ B) the category of right descent data. Similarly one candefine left descent data from A to B and their category (A ↓ B) D.Let A = (A, m A , u A ) be a monad on a category A. Then we can consider theadjunction ( A F , A U) , where A F : A → A A and A U : AA → A, with unit u Awhich is the unit of the monad and counit λ A determined by A U ( λ A(X, A µ X))=A µ X for every ( X, A µ X)∈ A A. Then A F A U is a comonad on the category A A byProposition 4.4. Hence we can consider the category of comodules for the comonadC = A F A U, A F A U ( A A) = C ( A A) which is the category of descent data with respect tothe monad A and it is denoted by Des A (A) .Example 9.20. Let B σ → A be a k-algebra extension. Then A is a B-ring. In factm A : A ⊗ A → A induces m : A ⊗ B A → A as follows. We have to prove thatm A (ab ⊗ a ′ ) = m A (a ⊗ ba ′ ) . We computem A (ab ⊗ a ′ ) = m A (aσ (b) ⊗ a ′ ) = (aσ (b)) a ′= a (σ (b) a ′ ) = m A (a ⊗ σ (b) a ′ ) = m A (a ⊗ ba ′ ) .