H-Galois extensions with normal basis for weak Hopf algebras

International Electronic Journal of Algebra Volume 21 (2017) 23-38 H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS J. N. Alonso ´ Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez Received: 15 January 2016; Revised: 9 July 2016 Communicated by A. C¸i˘gdem ¨ Ozcan Abstract. Let Hbe a weak Hopf algebra and let Abe an H-comodule algebra with subalgebra of coinvariants AH. In this paper we introduce the notion of H-Galois extension with normal basis and we prove that AH,→A is an H-Galois extension with normal basis if and only if AH,→Ais an H-cleft extension which admits a convolution invertible total integral. As a consequence, if His cocommutative and Acommutative, we obtain a bijective correspondence between the second cohomology group H2 ϕAH(H, AH) and the set of isomorphism classes of H-Galois extensions with normal basis whose left action over AHis ϕAH. Mathematics Subject Classification (2010): 18D10, 16T05 Keywords:H-Galois extensions, normal basis, weak Hopf algebra 1. Introduction It is a well-known fact in classical Galois theory that if B⊂Ais a finite Galois extension of fields with Galois group H, then A/B has a normal basis, i.e., there exists a∈Asuch that the set {x.a ;x∈H}is a basis for Aover B. Generalizing finite Galois extension of fields, Kreimer and Takeuchi introduce in [13] the notion of H-Galois extension with normal basis, associated to a Hopf algebra Hin a category of modules over a commutative ring, and in [10] Doi and Takeuchi show that there exists an equivalence between the notion of H-Galois extension with normal basis and the one of H-cleft extension for H. This result can be generalized to symmetric closed categories [11] and in [7] we find a more general formulation in the context of entwining structures that was extended to the weak setting in [2] by using the notion of weak C-cleft extensions defined in [1]. On the other hand, being Aan algebra, Ca coalgebra and ΓA H:C⊗A→A⊗Ca morphism in a strict monoidal category with equalizers and coequalizers, such that (A, C, ΓA H) is a weak entwining structure, we have introduced in [2] the notion of weak C-Galois extension with This work was supported by Ministerio de Econom´ıa y Competitividad and by Feder founds. Grant MTM2013-43687-P: Homolog´ıa, homotop´ıa e invariantes categ´oricos en grupos y ´algebras no asociativas. 24 ALONSO ´ ALVAREZ, FERN ´ ANDEZ VILABOA AND GONZ´ ALEZ RODR´ IGUEZ normal basis and we proved that, if A⊗ − preserves coequalizers, there exists an equivalence between weak C-Galois extensions and weak C-cleft extensions. Taking into account that every right comodule algebra over a weak Hopf algebra Hinduces a weak entwining structure, the results obtained in [1] and [2] can be applied for the study of Galois theory for weak Hopf algebras. In [5] we introduce the notion of H-cleft extension for a weak Hopf algebra H and we prove that this kind of extensions are examples of weak H-cleft extensions like the ones introduced in [1] and satisfying the classical notion of cleftness when particularizing to the Hopf setting. Assuming cocommutativity for H, we give in [5] a bijective correspondence between the equivalence classes of H-cleft extensions AH,→Band the equivalence classes of crossed systems for Hover AHwhere AHdenotes the subalgebra of coinvariants of the H-comodule algebra (A, ρA) in the weak context. This result permits to generalize the ones proved by Doi [9] about the characterization of equivalence classes of crossed systems as the second Sweedler cohomology group in the cocommutative Hopf algebra setting. To obtain this generalization we need the cohomology theory of algebras over cocommutative weak Hopf algebras we developed in [4] and used in [5] in order to give the weak Hopf version of Doi’s result, i.e., a bijection between the isomorphism classes of H-cleft extensions AH,→B, the equivalence classes of crossed systems for Hover AHand the second cohomology group H2 ϕZ(AH)(H, Z(AH)), where Z(AH) is the center of AHand ϕZ(AH)the corresponding associated action. As we have pointed above, H-cleft extensions are a kind of weak H-cleft extensions, and these are equivalent to weak H-Galois extensions with normal basis. This leads naturally to the following question: Is there a special class of weak H-Galois extensions with normal basis equivalent to H-cleft extensions? In order to give an affirmative response to this question we introduce the notion of H-Galois extension with normal basis like a special kind of weak H-Galois extension with normal basis and we prove that if A⊗ − preserves coequalizers, the following assertions are equivalent: (i) AH,→Ais an H-cleft extension that admits a convolution invertible total integral. (ii) AH,→Ais an H-Galois extension with normal basis. As a consequence, taking into account that, if His cocommutative, every Hcleft extension AH,→Aadmits a convolution invertible total integral, we obtain that AH,→Ais an H-cleft extension if and only if AH,→Ais an H-Galois extension with normal basis. Therefore, if Ais commutative, we obtain a bijective correspondence between the second cohomology group H2 ϕAH(H, AH) and the set H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS 25 of equivalence classes of H-Galois extensions with normal basis with associated left action over the subalgebra of coinvariants ϕAH. 2. Galois extensions with normal basis and Cleft extensions in a weak setting Throughout this paper C= (C,⊗, K, c) is a symmetric monoidal category with equalizers and coequalizers, where ⊗is the tensor product, Kthe base object and cM,N :M⊗N→N⊗Mthe natural isomorphism of symmetry. For any objects A,Band Cin Cthe natural isomorphism aA,B,C : (A⊗B)⊗C→A⊗(B⊗C) is called the associative constraint, and the natural isomorphisms lA:K⊗A→A and rA:A⊗K→A, are known as the left and right unit constraints, respectively. Moreover, by Theorem XI.5.3 of [12] we know that every monoidal category is monoidally equivalent to a strict one (i.e., a category such that the constraint isomorphisms are identities), and then there is no loss of generality in assuming that Cis strict. We assume that the reader is familiar with the notions of (co)algebra and (co)module and morphisms between them in this monoidal setting (see [1], [2]). Note that if Cadmits equalizers then every idempotent morphism in Csplits, i.e., for every morphism q:Y→Ysuch that q=q◦qthere exists an object Z(image of q) and morphisms i:Z→Yand p:Y→Zsuch that q=i◦pand p◦i=idZ. For each object Min C, we denote the identity morphism by idM:M→Mand for simplicity of notation, given objects M,N,Pin Cand a morphism f:M→N, we write P⊗ffor idP⊗fand f⊗Pfor f⊗idP. Let D= (D, εD, δD) be a coalgebra, with counit εD:D→Kand coproduct δD:D→D⊗D, and let A= (A, ηA, µA) be an algebra with unit ηA:K→Aand product µA:A⊗A→A. If f, g :D→Ain Care morphisms in C,f∗gdenotes the usual convolution product in the category, that is, f∗g=µA◦(f⊗g)◦δD. For an algebra A, the category of right (resp. left) A-modules will be denoted by MA(resp. AM). Similarly, if Dis a coalgebra we denote by MD(resp. DM) the category of right (resp. left) D-comodules. Definition 2.1. A weak bialgebra Hin Cis an algebra (H, ηH, µH)and a coalgebra (H, εH, δH)satisfying: (a1) δH◦µH= (µH⊗µH)◦(H⊗cH,H ⊗H)◦(δH⊗δH), (a2) εH◦µH◦(µH⊗H)=(εH⊗εH)◦(µH⊗µH)◦(H⊗δH⊗H) = (εH⊗εH)◦(µH⊗µH)◦(H⊗(cH,H ◦δH)⊗H), (a3) (δH⊗H)◦δH◦ηH= (H⊗µH⊗H)◦(δH⊗δH)◦(ηH⊗ηH) = (H⊗(µH◦cH,H )⊗H)◦(δH⊗δH)◦(ηH⊗ηH). 26 ALONSO ´ ALVAREZ, FERN ´ ANDEZ VILABOA AND GONZ´ ALEZ RODR´ IGUEZ If moreover, (a4) there exists a morphism λH:H→Hin C(called antipode of H) satisfying: (a4-1) idH∗λH= ((εH◦µH)⊗H)◦(H⊗cH,H )◦((δH◦ηH)⊗H), (a4-2) λH∗idH= (H⊗(εH◦µH)) ◦(cH,H ⊗H)◦(H⊗(δH◦ηH)), (a4-3) λH∗idH∗λH=λH. we say that the weak bialgebra His a weak Hopf algebra in the category C. Note that in a strict monoidal category the associativity of the convolution product follows by the associativity of the product µHand the coassociativity of the coproduct δH. In a similar way to the Hopf algebra case, the antipode λHof a weak Hopf algebra His unique, antimultiplicative (λH◦µH=µH◦(λH⊗λH)◦cH,H ), anticomultiplicative (δH◦λH=cH,H ◦(λH⊗λH)◦δH) and leaves the unit ηHand the counit εHinvariable (λH◦ηH=ηH, εH◦λH=εH). Moreover, we can define the idempotent morphisms ΠL H(target), ΠR H(source), ΠL Hand ΠR Hby ΠL H= ((εH◦µH)⊗H)◦(H⊗cH,H )◦((δH◦ηH)⊗H); ΠR H= (H⊗(εH◦µH)) ◦(cH,H ⊗H)◦(H⊗(δH◦ηH)); ΠL H= (H⊗(εH◦µH)) ◦((δH◦ηH)⊗H); ΠR H= ((εH◦µH)⊗H)◦(H⊗(δH◦ηH)); which satisfy the equalities ΠL H=idH∗λH, ΠR H=λH∗idHand then ΠL H∗ΠL H= ΠL H, ΠR H∗ΠR H= ΠR H. In what follows we denote by HLthe image of the target morphism and by pLand iLthe morphisms such that iL◦pL= ΠL Hand pL◦iL=idHL. Finally, we have that (see [6]), ΠL H◦ΠL H= ΠL H; ΠL H◦ΠR H= ΠR H; ΠR H◦ΠL H= ΠL H; ΠR H◦ΠR H= ΠR H; (1) ΠL H◦ΠL H= ΠL H; ΠL H◦ΠR H= ΠR H; ΠR H◦ΠL H= ΠL H; ΠR H◦ΠR H= ΠR H.(2) Definition 2.2. Let Hbe a weak bialgebra and let Abe an algebra with coaction ρA:A→A⊗Hsuch that (A, ρA)is a right H-comodule satisfying the equality µA⊗H◦(ρA⊗ρA) = ρA◦µA. The object (A, ρA)is called a right H-comodule algebra if one of the following equivalent conditions holds (see [8], Proposition 4.10): (b1) (ρA⊗H)◦ρA◦ηA= (A⊗(µH◦cH,H )⊗H)◦((ρA◦ηA)⊗(δH◦ηH)), (b2) (ρA⊗H)◦ρA◦ηA= (A⊗µH⊗H)◦((ρA◦ηA)⊗(δH◦ηH)), (b3) (A⊗ΠR H)◦ρA= (µA⊗H)◦(A⊗(ρA◦ηA)), (b4) (A⊗ΠL H)◦ρA= (µA⊗H)◦(A⊗cH,A)◦((ρA◦ηA)⊗A), (b5) (A⊗ΠR H)◦ρA◦ηA=ρA◦ηA, (b6) (A⊗ΠL H)◦ρA◦ηA=ρA◦ηA. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS 27 For a right H-comodule algebra Awe define ΓH A:H⊗A→A⊗Has ΓH A= (A⊗µH)◦(cH,A ⊗H)◦(H⊗ρA).Then the triple (A, H, ΓH A) is a right-right weak entwining structure (see [8], Theorem 4.14), i.e., it satisfies ΓH A◦(H⊗µA)=(µA⊗H)◦(A⊗ΓH A)◦(ΓH A⊗A),(3) (A⊗δH)◦ΓH A= (ΓH A⊗H)◦(H⊗ΓH A)◦(δH⊗A),(4) ΓH A◦(H⊗ηA) = (eA⊗H)◦δH,(5) (A⊗εH)◦ΓH A=µA◦(eA⊗A),(6) where eA= (A⊗εH)◦ΓH A◦(H⊗ηA).(7) Let Aand Hbe fixed. We denote by MH A(ΓH A) the category of right-right weak entwined modules, i.e., the objects Min Ctogether with two morphisms φM:M⊗A→Aand ρM:M→M⊗Hsuch that (M, φM) is a right A-module, (M, ρM) is a right H-comodule and the following equality ρM◦φM= (φM⊗H)◦(M⊗ΓH A)◦(ρM⊗A) (8) holds. Obviously, if (A, ρA) is a right H-comodule algebra, (A, µA, ρA) is an object of MH A(ΓH A). Let (A, ρA) be a right H-comodule algebra. We define the subalgebra of coinvariants of Aby the equalizer: -- - AHAA⊗H iAρA ζA where ζA= (µA⊗H)◦(A⊗(ρA◦ηA)). Note that, by (b3), ζA= (A⊗ΠR H)◦ρA and also, by (1) and (2), (AH, iA) is the equalizer of ρAand (A⊗ΠL H)◦ρA. It is not difficult to see that (AH, ηAH, µAH) is an algebra, being ηAHand µAH the factorizations through the equalizer iAof the morphisms ηAand µA◦(iA⊗iA), respectively. As a consequence, ϕA=µA◦(iA⊗A) (respectively φA=µA◦(A⊗iA)) defines a left (right) AH-module structure for A. Note that the weak Hopf algebra His a right H-comodule algebra with comodule structure giving by ρH=δHand subalgebra of coinvariants HH=HL. In this case iH=iL. The morphism ∆A⊗H= (µA⊗H)◦(A⊗ΓH A)◦(A⊗H⊗ηA) : A⊗H→A⊗His an idempotent and, as a consequence, there exist an object AHand morphisms iA⊗H:AH→A⊗H,pA⊗H:A⊗H→AHsuch that ∆A⊗H=iA⊗H◦ pA⊗Hand idAH=pA⊗H◦iA⊗H.Moreover AHis a right A-module, where the action is defined by φAH=pA⊗H◦(µA⊗H)◦(A⊗ΓH A)◦(iA⊗H⊗A), and a 28 ALONSO ´ ALVAREZ, FERN ´ ANDEZ VILABOA AND GONZ´ ALEZ RODR´ IGUEZ right H-comodule, with coaction ρAH= (pA⊗H⊗H)◦(A⊗δH)◦iA⊗H, and (AH, φAH, ρAH) is a weak entwined module and a left A-module with action ϕAH=pA⊗H◦(µA⊗H)◦(A⊗iA⊗H). On the other hand, the equality ∆A⊗H= (µA⊗H)◦(A⊗((eA⊗H)◦δH)),(9) comes directly from (5). The morphism (lifted canonical morphism) rA=pA⊗H◦(µA⊗H)◦(A⊗ρA) : A⊗A→AHfactorizes through the coequalizer morphism qA,A :A⊗A→A⊗AHA of the morphisms θ1 A,A =A⊗ϕAand θ2 A,A =φA⊗A. As a consequence, there exists a unique morphism, called the canonical morphism, γA:A⊗AHA→AHsuch that γA◦qA,A =rA.Further, rAand γAare morphisms of right H-comodules being ρA⊗A=A⊗ρAand ρA⊗AHAthe factorization of (qA,A ⊗H)◦(A⊗ρA) through the coequalizer qA,A.If the functor A⊗ − preserves coequalizers, γAis a morphism of left A-modules where ϕA⊗BAis the factorization of qA,A ◦(µA⊗A) through the coequalizer A⊗qA,A. Finally, γAis a morphism of right A-modules where φA⊗BA is the factorization of qA,A ◦(A⊗µA) through the coequalizer qA,A ⊗A. Definition 2.3. If the functor A⊗ − preserves coequalizers, we say that AH,→A is a weak H-Galois extension if the canonical morphism γAis an isomorphism. Note that, if Cis a closed category, the functor A⊗− preserves coequalizers. Also, if Ais a finite object, i.e., there exists an object A∗and an adjunction A⊗ − a A∗⊗ −, we have that A⊗ − preserves coequalizers. Let Hbe a weak Hopf algebra and let (A, ρA) be a right H-comodule algebra. In Definition 1.8 of [1] we introduce the set RegW R(H, A) as the one whose elements are the morphisms h:H→Asuch that there exists h−1:H→A, called the left weak inverse of h, satisfying h−1∗h=eAwhere eAis the morphism defined in (7) for the right-right weak entwining structure ΓH Aassociated to (A, ρA). Definition 2.4. We say that AH,→Ais a weak H-cleft extension if there exists a morphism h:H→Ain RegW R(H, A)(called the cleaving morphism) of right H-comodules such that ΓH A◦(H⊗h−1)◦δH=ζA◦(eA∗h−1).(10) Also, by (2.9) of [5], we can assume without loss of generality that eA∗h−1=h−1 and as a consequence (10) can be expressed as ΓH A◦(H⊗h−1)◦δH=ζA◦h−1.(11) Then, if the extension AH,→Ais weak H-cleft, by Proposition 1.12 of [1], we get that qA=µA◦(A⊗h−1)◦ρA:A→Afactorizes through iA. Therefore, H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS 29 there exists a unique morphism pA:A→AHsuch that qA=iA◦pA. Then, h∗h−1=qA◦hand, as a consequence, h∗h−1admits a factorization through iA. Now we recall the definition of H-Galois extension with normal basis we formulate in [2]. Definition 2.5. A weak H-Galois extension AH,→Ahas a normal basis if there exists an idempotent morphism of left AH-modules and right H-comodules ΩAH⊗H: AH⊗H→AH⊗H(ϕAH⊗H=µAH⊗H,ρAH⊗H=AH⊗δH) and an isomorphism of left AH-modules and right H-comodules bA:A→AH×H, where AH×His the image of ΩAH⊗Hand ϕAH×H=pAH,H ◦ϕAH⊗H◦(AH⊗iAH,H ), ρAH×H= (pAH,H ⊗ H)◦ρAH⊗H◦iAH,H ,being iAH,H :AH×H→AH⊗Hand pAH,H :AH⊗H→AH× Hthe morphisms such that iAH,H ◦pAH,H = ΩAH⊗Hand pAH,H ◦iAH,H =idAH×H. For a weak H-Galois extension with normal basis, if we define ωA=b−1 A◦pAH,H : AH⊗H→Aand ω0 A=iAH,H ◦bA:A→AH⊗H, the morphism ω0 A◦ωA= ΩAH⊗H, ωA◦ω0 A=idAand m0 A=µA◦(A⊗((iA⊗εH)◦ω0 A)) : A⊗A→Afactorizes through the coequalizer qA,A. Then there exists a unique morphism of left Amodules mA:A⊗AHA→Asuch that mA◦qA,A =µA◦(A⊗((iA⊗εH)◦ω0 A)) (12) (see Lemma 1.9 of [2]). Note that, in these conditions, we have that pAH,H ,iAH,H , ωAand ω0 Aare also morphisms of left AH-modules and right H-comodules. As we have showed in [1], there is a close connection between weak H-cleft extensions and weak H-Galois extensions with normal basis. More precisely, the main result in [1] establishes that, if A⊗ − preserves coequalizers, AH,→Ais a weak H-cleft extension if and only if AH,→Ais a weak H-Galois extension with normal basis. For clarity we briefly review the proof: Let AH,→Abe a weak H-Galois extension with normal basis. We define the cleaving morphism hA=ωA◦(ηAH⊗H) : H→Aand its left weak inverse is h−1 A=mA◦γ−1 A◦pA⊗H◦(ηA⊗H) : H→A. Note that in this part of the proof we obtain that mA◦γ−1 A◦pA⊗C◦ρA= ((iA C⊗εC)◦ω0 Aand hA∗h−1 A= (iA⊗εH)◦ΩAH⊗H◦(ηAH⊗H).(13) Conversely, if AH,→Ais a weak H-cleft extension with cleaving morphism h, the morphisms of left AH-modules and right H-comodules defined by ωA=µA◦(iA⊗h) and ω0 A= (pA⊗H)◦ρAsatisfy the equality ωA◦ω0 A=idA. As a consequence, the morphism ΩAH⊗H=ω0 A◦ωAis idempotent and we have a commutative diagram 30 ALONSO ´ ALVAREZ, FERN ´ ANDEZ VILABOA AND GONZ´ ALEZ RODR´ IGUEZ - ZZZZ~  3 ZZZ~ AH⊗H AH⊗H A AH×H ωAω0 A pAH,H iAH,H ΩAH⊗H where pAH,H ◦iAH,H =idAH×H. Therefore, the morphism bA=pAH,H ◦ω0 Ais an isomorphism of right H-comodules and left AH-modules with inverse b−1 A= ωA◦iAH,H . Moreover, the inverse of the canonical morphism γAis γ−1 A=qA,A ◦ (µA⊗A)◦(A⊗h−1 A⊗hA)◦(A⊗δH)◦iA⊗H. In the second section of [5], we introduce the notion of H-cleft extension for a weak Hopf algebra Hand we prove that this kind of extensions are examples of weak H-cleft extensions. To define H-cleft extensions we need convolution invertible integrals. As in the Hopf setting, for a weak Hopf algebra Hand a right H-comodule algebra (A, ρA), an integral is a morphism of right H-comodules f:H→A. If moreover f◦ηH=ηAwe will say that the integral is total. An integral f:H→Ais convolution invertible if there exists a morphism f−1:H→A(called the convolution inverse of f) such that (c1) f−1∗f=eA. (c2) f∗f−1= (A⊗(εH◦µH)) ◦((ρA◦ηA)⊗H). (c3) f−1∗f∗f−1=f−1. Trivially, the inverse is unique and we get that f∗f−1∗f=f(see Definition 2.4 of [5]). Note that, when fis a total integral, we can rewrite (c1) as f−1∗f=f◦ΠR H and (c2) as f∗f−1=f◦ΠL H. Definition 2.6. We say that AH,→Ais an H-cleft extension if there exists a convolution invertible integral h:H→Asuch that the morphism h∗h−1factorizes through the equalizer iA. Obviously, HL,→His a weak H-cleft extension with h=idHand h−1=λH. By Proposition 2.2 of [5] we know that if His a cocommutative weak Hopf algebra and there exists a convolution invertible integral f:H→Athen AH,→Ais an H-cleft extension. Also, by Corollary 2.1 of [5], we have that an H-cleft extension is also a weak H-cleft extension. Finally, Proposition 2.3 of [5] asserts that, if the antipode of His an isomorphism and AH,→Ais an H-cleft extension with convolution invertible integral f, then h=µA◦(f⊗(f−1◦ηH))) is a total integral. Moreover, if His cocommutative his convolution invertible. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS 31 In the following definition we introduce the notion of H-Galois extension with normal basis. Definition 2.7. Let Hbe a weak Hopf algebra and let (A, ρA)be a right H-comodule algebra such that A⊗ − preserves coequalizers. We say that a weak H-Galois extension with normal basis AH,→Ais an H-Galois extension with normal basis if the following identities hold: (d1) bA◦ηA=pAH,H ◦(ηAH⊗ηH). (d2) ((bA◦iA)⊗εH)◦ΩAH⊗H◦(ηAH⊗H) = pAH,H ◦(ηAH⊗ΠL H). 3. Galois and Cleft extensions and cohomology In this section we give the main results of the paper and a cohomological interpretation of Cleft extensions. Lemma 3.1. Let Hbe a weak Hopf algebra and let (A, ρA)be a right H-comodule algebra such that A⊗− preserves coequalizers. If AH,→Ais an H-Galois extension with normal basis the following equality holds: (AH⊗εH)◦ΩAH⊗H◦(AH⊗ηH) = idAH.(14) Proof. First of all, note that by the definition of the morphism ΠL Hand the properties of the (co)unit ηH(εH), it is easy to see that ΠL H◦ηH=ηH. Now, by composing (d2) of Definition 2.7 with ηHand using (d1) we have that ((bA◦iA)⊗εH)◦ΩAH⊗H◦(ηAH⊗ηH) = pAH,H ◦(ηAH⊗ηH) = bA◦ηA. Therefore, (iA⊗εH)◦ΩAH⊗H◦(ηAH⊗ηH) = ηA(15) holds and as a consequence we have: iA =µA◦(iA⊗ηA) = ((µA◦(iA⊗iA)⊗εH)◦(AH⊗(ΩAH⊗H◦(ηAH⊗ηH))) = (iA⊗εH)◦(µAH⊗H)◦(AH⊗(ΩAH⊗H◦(ηAH⊗ηH))) = (iA⊗εH)◦ΩAH⊗H◦(AH⊗ηH). where the first equality follows by the properties of the unit ηA, the second one by (15), the third one by the properties of µAH, the fourth one because ΩAH⊗His a morphism of left AH-modules. Then, using that iAis a monomorphism we conclude the proof.  Theorem 3.2. Let Hbe a weak Hopf algebra and let (A, ρA)be a right H-comodule algebra such that A⊗ − preserves coequalizers. The following are equivalent. 38 ALONSO ´ ALVAREZ, FERN ´ ANDEZ VILABOA AND GONZ´ ALEZ RODR´ IGUEZ [8] S. Caenepeel and E. de Groot, Modules over weak entwining structures, Contemp. Math., 267 (2000), 31-54. [9] Y. Doi, Equivalent crossed products for a Hopf algebra, Comm. Algebra, 17(12) (1989), 3053-3085. [10] Y. Doi and M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Algebra, 14(5) (1986), 801-817. [11] J. M. Fern´andez Vilaboa and E. Villanueva Novoa, A characterization of the cleft comodule triples, Comm. Algebra, 16(3) (1988), 613-622. [12] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, SpringerVerlag, New York, 1995. [13] H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J., 30(5) (1981), 675-692. [14] M. E. Sweedler, Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc., 133 (1968), 205-239. J. N. Alonso ´ Alvarez (Corresponding Author) Departamento de Matem´aticas Universidad de Vigo Campus Universitario Lagoas-Marcosende E-36280 Vigo, Spain e-mail: [email protected] J. M. Fern´andez Vilaboa Departamento de ´ Alxebra Universidad de Santiago de Compostela Campus Sur E-15771 Santiago de Compostela, Spain e-mail: joseman[email protected] R. Gonz´alez Rodr´ıguez Departamento de Matem´atica Aplicada II Universidad de Vigo Campus Universitario Lagoas-Marcosende E-36310 Vigo, Spain e-mail: [email protected]