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

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

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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 ′ ) .

Moreover the unit is u = σ : B → A. Then A = ( )− ⊗ B A, − ⊗ B m, (− ⊗ R u) ◦ r−−1is a monad on the category of right B-modules, Mod-B, as in Example 3.3. Notethat we have an iso of categories K : Mod-A → A (Mod-B) given byMod-A −→ A (Mod-B)( )X, µAX ↦→( )X, µ A Xwhere µ A X : X ⊗ B A → X is well-defined starting from µ A X : X ⊗ A → X. In fact wehave(µ A X (xb ⊗ a) = µ A X µAX (x ⊗ σ (b)) ⊗ a )XisA-mod= µ A X (x ⊗ m A (σ (b) ⊗ a)) = µ A X (x ⊗ ba) .Now, since A = ( )− ⊗ B A, − ⊗ B m, (− ⊗ R u) ◦ r−−1 is a monad, we can considerAF = − ⊗ B A : Mod-B → A (Mod-B) ≃ Mod-A and A U = − ⊗ A A B : A (Mod-B) ≃Mod-A → Mod-B so that C = A F A U = −⊗ A A⊗ B A is a comonad on A (Mod-B) ≃Mod-A associated to the A-coring C = A ⊗ B A. The category of comodules for thecomonad C = A F A U = − ⊗ A A ⊗ B A is then the category of right comodules for theA-coring C = A ⊗ B AAF A U ( A (Mod-B)) = C ( A (Mod-B)) = ( A (Mod-B)) C ≃ (Mod-A) C= C (Mod-A) = A F A U (Mod-A)and it is the category of right descent data from A to B, usually denoted byD (A ↓ B) .Corollary 9.21 ([Scha4, Theorem 4.5.2] Faithfully flat descent). Let A be a k-algebra and let B ⊆ A be a k-algebra extension. Let C = A ⊗ B A be the canonicalA-coring. The following statements are equivalent:(a) B A is flat and the functor − ⊗ B A : Mod-B → D (A ↓ B) is an equivalenceof categories;(b) B A is faithfully flat.Proof. Apply Corollary 9.17 to the case ” B Σ A ” = B A A , noting that by Example9.20 C (Mod-A) = D (A ↓ B) where C = − ⊗ A C = − ⊗ A A ⊗ B A. □Remark 9.2ong>2.ong> The inverse equivalence of the induction functor − ⊗ B A : Mod-B → D (A ↓ B) = C (Mod-A) where C = − ⊗ A C = − ⊗ A A ⊗ B A, maps a descentdatum (M, δ) into M coδ = {m ∈ M | δ (m) = m ⊗ B 1 A } ≃ M coC . Moreover, sincewe have an equivalence, in particular the counit is an isomorphism, so that the mapis an isomorphism with inverse given byM coδ ⊗ B A ɛ M−→ Mm ⊗ B a ↦→ maM → M coδ ⊗ B Am ↦→ δ (m) .In fact we have [(δ ⊗ B A) ◦ δ] (m) = m 00 ⊗ B m 01 ⊗ B m 1 = m 0 ⊗ B 1 A ⊗ B m 1 so thatδ (m) ∈ M coδ ⊗ B A.209

Moreover the unit is u = σ : B → A. Then A = ( )− ⊗ B A, − ⊗ B m, (− ⊗ R u) ◦ r−−1is a m<strong>on</strong>ad <strong>on</strong> the category of right B-modules, Mod-B, as in Example 3.3. Notethat we have an iso of categories K : Mod-A → A (Mod-B) given byMod-A −→ A (Mod-B)( )X, µAX ↦→( )X, µ A Xwhere µ A X : X ⊗ B A → X is well-defined starting from µ A X : X ⊗ A → X. In fact wehave(µ A X (xb ⊗ a) = µ A X µAX (x ⊗ σ (b)) ⊗ a )XisA-mod= µ A X (x ⊗ m A (σ (b) ⊗ a)) = µ A X (x ⊗ ba) .Now, since A = ( )− ⊗ B A, − ⊗ B m, (− ⊗ R u) ◦ r−−1 is a m<strong>on</strong>ad, we can c<strong>on</strong>siderAF = − ⊗ B A : Mod-B → A (Mod-B) ≃ Mod-A and A U = − ⊗ A A B : A (Mod-B) ≃Mod-A → Mod-B so that C = A F A U = −⊗ A A⊗ B A is a com<strong>on</strong>ad <strong>on</strong> A (Mod-B) ≃Mod-A associated to the A-coring C = A ⊗ B A. The category of comodules for thecom<strong>on</strong>ad C = A F A U = − ⊗ A A ⊗ B A is then the category of right comodules for theA-coring C = A ⊗ B AAF A U ( A (Mod-B)) = C ( A (Mod-B)) = ( A (Mod-B)) C ≃ (Mod-A) C= C (Mod-A) = A F A U (Mod-A)and it is the category of right descent data from A to B, usually denoted byD (A ↓ B) .Corollary 9.21 ([Scha4, Theorem 4.5.2] Faithfully flat descent). Let A be a k-algebra and let B ⊆ A be a k-algebra extensi<strong>on</strong>. Let C = A ⊗ B A be the can<strong>on</strong>icalA-coring. The following statements are equivalent:(a) B A is flat and the functor − ⊗ B A : Mod-B → D (A ↓ B) is an equivalenceof categories;(b) B A is faithfully flat.Proof. Apply Corollary 9.17 to the case ” B Σ A ” = B A A , noting that by Example9.20 C (Mod-A) = D (A ↓ B) where C = − ⊗ A C = − ⊗ A A ⊗ B A. □Remark 9.2<str<strong>on</strong>g>2.</str<strong>on</strong>g> The inverse equivalence of the inducti<strong>on</strong> functor − ⊗ B A : Mod-B → D (A ↓ B) = C (Mod-A) where C = − ⊗ A C = − ⊗ A A ⊗ B A, maps a descentdatum (M, δ) into M coδ = {m ∈ M | δ (m) = m ⊗ B 1 A } ≃ M coC . Moreover, sincewe have an equivalence, in particular the counit is an isomorphism, so that the mapis an isomorphism with inverse given byM coδ ⊗ B A ɛ M−→ Mm ⊗ B a ↦→ maM → M coδ ⊗ B Am ↦→ δ (m) .In fact we have [(δ ⊗ B A) ◦ δ] (m) = m 00 ⊗ B m 01 ⊗ B m 1 = m 0 ⊗ B 1 A ⊗ B m 1 so thatδ (m) ∈ M coδ ⊗ B A.209

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