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 ...
202We now have to prove that this map induces a map ζ :e ∈ E. We have to prove that(∑ ∑ [ζl i,j ⊗ h i,j] ) (∑ ∑Q b · e ⊗ g i = ζijijL⊗H(L⊗H)L +⊗ E H → L. Let[l i,j ⊗ h i,j] b Q⊗ e · g i ).Since e ∈ E, there exist x k , y k ∈ H such that e = [∑ k xk ⊗ y k] . Hence we have toEprove that(∑ ∑ [ζl i,j ⊗ h i,j] ] )bi jQ ·[∑k xk ⊗ y k ⊗ g iE(∑ ∑ [= ζl i,j ⊗ h i,j] ] )Q b ⊗[∑k xk ⊗ y k · g iand by using definition of E µb Qand µ E Qi.e.∑i∑j∑i(∑ ∑ ∑ζi j(∑ ∑ ∑= ζijk li,j S ( h i,j) x k S ( y k) g i = ∑ ijkkwe have to prove that[l i,j S ( h i,j) x k ⊗ y k] )Q b ⊗ g i[l i,j ⊗ h i,j] b Q⊗ x k S ( y k) g i )∑j∑which is true, so that we can conclude that the map ̂ζ :Ek li,j S ( h i,j) x k S ( y k) g iL⊗H(L⊗H)L +⊗ E H → L iswell-defined. Now, we want to prove that ̂ζ is bijective. The inverse of ̂ζ is given byΞ : L →L ⊗ H(L ⊗ H) L ⊗ + E Hl ↦→ [1 H ⊗ 1 H ]b Q⊗ E l.Now we compute(Ξ ◦ ̂ζ) (∑ ∑ [l i,j ⊗ h i,j] ) (∑bi jQ⊗ E g i = Ξi∑j li,j S ( h i,j) )g i∑ ∑= [1 H ⊗ 1 H ]b Q⊗ Ei j li,j S ( h i,j) g i= ∑ ∑[1 [H ⊗ 1 H ]bi j Q⊗ E l i,j ⊗ h i,j] · E gi = ∑ ∑i= ∑ ∑ [l i,j ⊗ h i,j] bi jQ⊗ E g iand ) ()(̂ζ ◦ Ξ (l) = ζ [1 H ⊗ 1 H ]b Q⊗ E l = 1 H S (1 H ) l = l.j [1 H ⊗ 1 H ]b Q[l i,j ⊗ h i,j] E ⊗ E g iLet us show that ̂ζ is an isomorphism of left L-modules. Let a ∈ L and let usconsider̂ζ(a · ∑ ∑ [l i,j ⊗ h i,j] )bi jQ⊗ E g i = ̂ζ(∑ ∑ [( ) a · li,j⊗ h i,j] )bi jQ⊗ E g i= ∑ ∑i j ali,j S ( h i,j) (∑g i = a ·i∑j li,j S ( h i,j) )g i= a · ̂ζ(∑ ∑ [l i,j ⊗ h i,j] )Q b ⊗ E g i .ij
As observed at the beginning of this section, this reproduces what happens in thedual case of the [Scha4] setting where, starting from a Hopf-Galois extension, onecan produce a new Hopf algebra such that the Hopf-Galois object turns into a Hopfbi-Galois object and Hopf algebras are Morita-Takeuchi equivalent. In our setting,coming from a coGalois coextension we get a coherd, which allows us to computethe monads and in particular a new monad together with the new bimodule functor.Following the theory developed in the previous sections, we could then calculate indetails also the equivalence between the module categories with respects to the twomonads.9.3. Galois comodules. Let B Σ A be a B-A-bimodule. Let L = − ⊗ B Σ A , R =Hom A ( B Σ A , −). Let C be an A-coring and let C = (− ⊗ A C, − ⊗ A ∆, r ◦ (− ⊗ A ε)).Assume that (Σ, ρ Σ ) is a B-C-comodule i.e. (Σ, ρ Σ ) is a C-comodule andρ Σ : Σ → Σ ⊗ A Cis a morphism of B-A-bimodules. In particular the mapλ : B → EndC (Mod-A) ((Σ, ρ Σ )) defined by setting λ (b) (x) = bxis well-defined and is a ring morphism. Moreover λ is a monomorphism. In thiscase β = − ⊗ B ρ Σ : − ⊗ B Σ A → − ⊗ B Σ A ⊗ A C is a left C-comodule functor. Theassociated functorial morphism can = ϕ = (Cɛ) ◦ (βR) : LR → C,can : Hom A ( B Σ A , −) ⊗ B Σ AβR→ HomA ( B Σ A , −) ⊗ B Σ A ⊗ A C Cɛ→ − ⊗ A Cf ⊗ B x ↦→ f ⊗ B x 0 ⊗ A x 1 ↦→ f (x 0 ) ⊗ A x 1can M : Hom A ( B Σ A , M) ⊗ B Σ A → M ⊗ A Cf ⊗ B x ↦→ f (x 0 ) ⊗ A x 1203We havecan = (Cɛ) ◦ (βR) = (ɛ ⊗ A C) ◦ Hom A ( B Σ A , −) ⊗ B ρ Σcan M = ϕ M (f ⊗ B t) = (ɛ ⊗ A C) (f ⊗ B t 0 ⊗ A t 1 ) = f (t 0 ) ⊗ A t 1 .K ϕ : Mod-B → C (Mod-A) = Comod-CM ↦→ (M ⊗ B Σ, M ⊗ B ρ Σ ) .Since Mod-B has all equalizers, K ϕ has a right adjointD ϕ (X, x) = Equ ((− ⊗ A C) ◦ ρ Σ , Hom A ( B Σ A , x))= {f ∈ Hom A ( B Σ A , X) | x ◦ f = (f ⊗ A C) ◦ ρ Σ }= HomC (Mod-A) ((Σ, ρ Σ ) , (X, x))Hence D ϕ = HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) = Comod-C → Mod-B has a leftadjoint K ϕ = (− ⊗ B Σ, − ⊗ B ρ Σ ) .Theorem 9.6 ([GT, Theorem 3.1] ). HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) →Mod-B is full and faithful if and only if
- Page 152 and 153: 152Thus we obtainσ B ◦ ( ) (P µ
- Page 154 and 155: 154Thus hQ is an isomorphism with i
- Page 156 and 157: 156) ( )l=(pb Q AQ B ◦ ̂QA µ QB
- Page 158 and 159: 158In fact we haveTherefore we dedu
- Page 160 and 161: 160χ= h 1 ◦ (P xQ B ) ◦ (P QP
- Page 162 and 163: 162so that we obtain:(190)We comput
- Page 164 and 165: 164(194)=) )(p QB ̂QA ◦(Qpb Q◦
- Page 166 and 167: 166= Ξ ◦ (A A U A λ) ◦ (xx A
- Page 168 and 169: 168)(155)= k 2 ◦(Qpb Q◦ (Ql A U
- Page 170 and 171: 170) ) (χ= ρ ◦(p QB ̂QA ◦(Qp
- Page 172 and 173: 172Theorem 8.13. Let A and B be cat
- Page 174 and 175: 174l = eC ρ L : L = − ⊗ B A
- Page 176 and 177: and[µBQ ◦ ( Qσ B)] (− ⊗ T x
- Page 178 and 179: 178so that− ⊗ R 1 A ⊗ R c = (
- Page 180 and 181: 180− ⊗ T x ⊗ R 1 A ⊗ A f
- Page 182 and 183: 182(208)(209)(210)(211)(h1 ) 0 ⊗
- Page 184 and 185: 184= abd 0 ⊗ d 1 1 ⊗ d 2 1b⊗d
- Page 186 and 187: 186so that h 1 ⊗ h 2 ⊗ a ∈ A
- Page 188 and 189: 188= 〈( h (1) y (1))εH ( h (2) y
- Page 190 and 191: 190H C is faithfully coflat. Assume
- Page 192 and 193: 192=(Qε C H C) ( ∑ )−□ C k i
- Page 194 and 195: 194Following Theorem 6.29, we now c
- Page 196 and 197: 196)û E(ε C H C (h) = û E (π (h
- Page 198 and 199: 198Letandα l = (ϕ ⊗ H) ( (x ⊗
- Page 200 and 201: 200This map is well-defined, in fac
- Page 204 and 205: 2041) − ⊗ B Σ A preserves the
- Page 206 and 207: 206functorial isomorphism. In parti
- Page 208 and 209: 208coaction ρ C Σ : Σ → Σ ⊗
- Page 210 and 211: 210Now, we consider a particular ca
- Page 212 and 213: 212Definition 9.27. Let k be a comm
- Page 214 and 215: 214∆coass= a ⊗ c (1) ⊗ A 1 A
- Page 216 and 217: 216Definition 9.32.</strong
- Page 218 and 219: 218Let us compute, for every d ∈
- Page 220 and 221: 220• 2-cells: monad functor trans
- Page 222 and 223: 222We now want to prove that ρ Q·
- Page 224 and 225: 224Proof. Let us consider the follo
- Page 226 and 227: 226and since p Q•B Q ′ ,Q ′
- Page 228 and 229: 228(241)= (1 Q • B l Q ′) ζ Q,
- Page 230 and 231: 230On the other hand, we can first
- Page 232 and 233: 232so that we define the map φ F (
- Page 234 and 235: 234Since we have(B • B (Q · A) ,
- Page 236 and 237: 2362-cells. This means that a comon
- Page 238 and 239: 238defined by settingu Q·A = ( u (
- Page 240 and 241: 240the unique A-bimodule morphism s
- Page 242 and 243: 242Let F be a finite subset of Hom
- Page 244 and 245: 244Lemma A.4. Let A be an abelian c
- Page 246 and 247: 246We haveT (ζ) ◦ ξ ◦ T H (p)
- Page 248 and 249: 248where k : Ker (Coker (f ◦ p))
- Page 250 and 251: 250be the codiagonal map of the ρ
As observed at the beginning of this secti<strong>on</strong>, this reproduces what happens in thedual case of the [Scha4] setting where, starting from a Hopf-Galois extensi<strong>on</strong>, <strong>on</strong>ecan produce a new Hopf algebra such that the Hopf-Galois object turns into a Hopfbi-Galois object and Hopf algebras are Morita-Takeuchi equivalent. In our setting,coming from a coGalois coextensi<strong>on</strong> we get a coherd, which allows us to computethe m<strong>on</strong>ads and in particular a new m<strong>on</strong>ad together with the new bimodule functor.Following the theory developed in the previous secti<strong>on</strong>s, we could then calculate indetails also the equivalence between the module categories with respects to the twom<strong>on</strong>ads.9.3. Galois comodules. Let B Σ A be a B-A-bimodule. Let L = − ⊗ B Σ A , R =Hom A ( B Σ A , −). Let C be an A-coring and let C = (− ⊗ A C, − ⊗ A ∆, r ◦ (− ⊗ A ε)).Assume that (Σ, ρ Σ ) is a B-C-comodule i.e. (Σ, ρ Σ ) is a C-comodule andρ Σ : Σ → Σ ⊗ A Cis a morphism of B-A-bimodules. In particular the mapλ : B → EndC (Mod-A) ((Σ, ρ Σ )) defined by setting λ (b) (x) = bxis well-defined and is a ring morphism. Moreover λ is a m<strong>on</strong>omorphism. In thiscase β = − ⊗ B ρ Σ : − ⊗ B Σ A → − ⊗ B Σ A ⊗ A C is a left C-comodule functor. Theassociated functorial morphism can = ϕ = (Cɛ) ◦ (βR) : LR → C,can : Hom A ( B Σ A , −) ⊗ B Σ AβR→ HomA ( B Σ A , −) ⊗ B Σ A ⊗ A C Cɛ→ − ⊗ A Cf ⊗ B x ↦→ f ⊗ B x 0 ⊗ A x 1 ↦→ f (x 0 ) ⊗ A x 1can M : Hom A ( B Σ A , M) ⊗ B Σ A → M ⊗ A Cf ⊗ B x ↦→ f (x 0 ) ⊗ A x 1203We havecan = (Cɛ) ◦ (βR) = (ɛ ⊗ A C) ◦ Hom A ( B Σ A , −) ⊗ B ρ Σcan M = ϕ M (f ⊗ B t) = (ɛ ⊗ A C) (f ⊗ B t 0 ⊗ A t 1 ) = f (t 0 ) ⊗ A t 1 .K ϕ : Mod-B → C (Mod-A) = Comod-CM ↦→ (M ⊗ B Σ, M ⊗ B ρ Σ ) .Since Mod-B has all equalizers, K ϕ has a right adjointD ϕ (X, x) = Equ ((− ⊗ A C) ◦ ρ Σ , Hom A ( B Σ A , x))= {f ∈ Hom A ( B Σ A , X) | x ◦ f = (f ⊗ A C) ◦ ρ Σ }= HomC (Mod-A) ((Σ, ρ Σ ) , (X, x))Hence D ϕ = HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) = Comod-C → Mod-B has a leftadjoint K ϕ = (− ⊗ B Σ, − ⊗ B ρ Σ ) .Theorem 9.6 ([GT, Theorem 3.1] ). HomC (Mod-A) ((Σ, ρ Σ ) , −) : C (Mod-A) →Mod-B is full and faithful if and <strong>on</strong>ly if
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