H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS

Let H be a weak Hopf algebra and let A be 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 ↪→ A is an H-cleft extension which admits a convolution invertible total integral. As a consequence, if H is cocommutative and A commutative, we obtain a bijective correspondence between the second cohomology group H2 AH (H,AH) and the set of isomorphism classes of H-Galois extensions with normal basis whose left action over AH is φAH . Mathematics Subject Classification (2010): 18D10, 16T05


Introduction
It is a well-known fact in classical Galois theory that if B ⊂ A is a finite Galois extension of fields with Galois group H, then A/B has a normal basis, i.e., there exists a ∈ A such that the set {x.a ; x ∈ H} is a basis for A over 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 H in 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 A an algebra, C a coalgebra and Γ A H : C ⊗ A → A ⊗ C a morphism in a strict monoidal category with equalizers and coequalizers, such that (A, C, Γ A H ) is a weak entwining structure, we have introduced in [2] the notion of weak C-Galois extension with 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 H induces 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 A H → B and the equivalence classes of crossed systems for H over A H where A H denotes the subalgebra of coinvariants of the H-comodule algebra (A, ρ A ) 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 is called the associative constraint, and the natural isomorphisms l A : K ⊗ A → A and r A : 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 C is 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 C admits equalizers then every idempotent morphism in C splits, i.e., for every morphism q : Y → Y such that q = q • q there exists an object Z (image of q) and morphisms i : Z → Y and p : Y → Z such that q = i • p and p • i = id Z .
For each object M in C, we denote the identity morphism by id M : M → M and for simplicity of notation, given objects M , N , P in C and a morphism f : M → N , we write P ⊗ f for id P ⊗ f and f ⊗ P for f ⊗ id P .
Let D = (D, ε D , δ D ) be a coalgebra, with counit ε D : D → K and coproduct δ D : D → D ⊗ D, and let A = (A, η A , µ A ) be an algebra with unit η A : K → A and product µ A : A ⊗ A → A. If f, g : D → A in C are morphisms in C, f * g denotes the usual convolution product in the category, that is, For an algebra A, the category of right (resp. left) A-modules will be denoted by M A (resp. A M). Similarly, if D is a coalgebra we denote by M D (resp. D M) the category of right (resp. left) D-comodules.
If moreover, (a4) there exists a morphism λ H : H → H in C (called antipode of H) satisfying: we say that the weak bialgebra H is 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 µ H and the coassociativity of the coproduct δ H .
In a similar way to the Hopf algebra case, the antipode λ H of a weak Hopf In what follows we denote by H L the image of the target morphism and by p L and i L the morphisms such that i L • p L = Π L H and p L • i L = id H L . Finally, we have that (see [6]), Definition 2.2. Let H be a weak bialgebra and let A be an algebra with coaction if one of the following equivalent conditions holds (see [8], Proposition 4.10): is a right-right weak entwining structure (see [8], Theorem 4.14), i.e., it satisfies where Let A and H be fixed. We denote by M H A (Γ H A ) the category of right-right weak entwined modules, i.e., the objects M in C together with two morphisms is a right H-comodule and the following equality . Let (A, ρ A ) be a right H-comodule algebra. We define the subalgebra of coinvariants of A by the equalizer: • ρ A and also, by (1) and (2) Note that the weak Hopf algebra H is a right H-comodule algebra with comodule structure giving by ρ H = δ H and subalgebra of coinvariants H H = H L . In this case is an idempotent and, as a consequence, there exist an object A H and morphisms is a weak entwined module and a left A-module with action On the other hand, the equality comes directly from (5).
As a consequence, there exists a unique morphism, called the canonical morphism, Note that, if C is a closed category, the functor A⊗− preserves coequalizers. Also, if A is a finite object, i.e., there exists an object A * and an adjunction A ⊗ − A * ⊗ −, we have that A ⊗ − preserves coequalizers.
Let H be a weak Hopf algebra and let (A, ρ A ) be a right H-comodule algebra. In Definition 1.8 of [1] we introduce the set Reg W R (H, A) as the one whose elements are the morphisms h : H → A such that there exists h −1 : H → A, called the left weak inverse of h, satisfying h −1 * h = e A where e A is the morphism defined in (7) for the right-right weak entwining structure Γ H A associated to (A, ρ A ).
Also, by (2.9) of [5], we can assume without loss of generality that e A * h −1 = h −1 and as a consequence (10) can be expressed as   and ω A = (p A ⊗ H) • ρ A satisfy the equality ω A • ω A = id A . As a consequence, the morphism Ω A H ⊗H = ω A • ω A is idempotent and we have a commutative diagram In the second section of [5], Trivially, the inverse is unique and we get that f * f −1 * f = f (see Definition 2.4 of [5]

Galois and Cleft extensions and cohomology
In this section we give the main results of the paper and a cohomological interpretation of Cleft extensions.
Proof. First of all, note that by the definition of the morphism Π L H and the properties of the (co)unit η H (ε H ), it is easy to see that Π L H • η H = η H . Now, by composing (d2) of Definition 2.7 with η H and using (d1) we have that Therefore, holds and as a consequence we have: where the first equality follows by the properties of the unit η A , the second one by (15), the third one by the properties of µ A H , the fourth one because Ω A H ⊗H is a morphism of left A H -modules.
Then, using that i A is a monomorphism we conclude the proof.  Proof. (i) ⇒ (ii) By Corollary 2.1 of [5] we know that if A H → A is an H-cleft extension with convolution invertible integral h then it is a weak H-cleft extension with cleaving morphism h. Therefore, by Theorem 2.11 of [2], we obtain that for Ω A H ⊗H we have the following identity: Indeed: where the first equality follows by the structure of right H-comodule algebra of A, the second one by the definition of A H , the third one by the associativity of µ A , the fourth one by the structure of right H-comodule algebra of A and the properties of the unity η A . In the fifth equality we used that h is a right H-comodule morphism and finally, the last one follows by (e1) of Lemma 3.11 of [3]. Then, where the first equality follows by the definition of b A , the second one by (16), the third one by the properties of ρ A , the fourth one by the equality (d2) of Lemma 3.9 On the other hand, using (16), the properties of η A H and the condition of total integral for h we have Therefore, we get (d1), because i A H ,H is a monomorphism. Moreover, and (d2) holds. Note that in the first equality we used (16), in the fourth one we applied that h is total and in the fifth one we use that Therefore, A H → A is an H-Galois extension with normal basis.
By (13) and (d2) of Definition 2.7, On the other hand, using that b A and Ω A H ⊗H are morphisms of right Hcomodules and (d1) of Definition 2.7 we have: where the third equality uses the definitions of Ω A H ⊗H and ρ A H ×H .
As a consequence, and h is a total integral.
To finish the proof it only remains to show that h −1 * h * h −1 = h −1 . First we proceed by showing the identity Indeed, composing with the coequalizer q A,A we have that and then (17) holds. In the last equalities, the first one follows by the definition of ρ A⊗ A H A , the second one by (12) and in the third one we used that ω A is a morphism of right H-comodules. The fourth equality follows by the properties of the counit and the fifth one by the identity obtained in the proof that h * h −1 = h•Π L H . The sixth one relies on the definition of µ A H , in the seventh one we applied that Ω A H ⊗H is a morphism of left A H -modules and in the eighth one we used that Ω A H ⊗H = ω A • ω A and ω A • ω A = id A . Finally, the last one follows by (12). Therefore, where the first equality follows using that h * h −1 = h • Π L H , the second one because ∆ A⊗H is a morphism of right H-comodules, the third one by the definition of ρ A H and in the fourth one we applied that γ −1 A is a morphism of right H-comodules. Finally, the fifth equality follows by (17) and the last one by the definition of As a consequence, by Propositions 2.2 and 2.3 of [5], we have the following corollary.  To finish this paper, we will give a cohomological interpretation of Cleft extensions. For clarity, we briefly describe the construction of the cohomology groups in the weak setting. The interested reader can find the details in [4]. Assume that H is a cocommutative weak Hopf algebra and let (A, ϕ A ) be a commutative weak left H-module algebra. Let H 0 be the unit object of C and for n ≥ 1 denote by H n the n-fold tensor power H ⊗· · ·⊗H. If n ≥ 2, m n H denotes the morphism m n H : H n → H defined by m 2 H = µ H and by m 3 H = m 2 H • (H ⊗ µ H ), · · · , m n H = m n−1 H • (H n−2 ⊗ µ H ) for k > 2. Analogously, with δ H n we denote the coproduct defined for the coalgebra H n . Finally, ϕ n A will be the morphism ϕ n A : H n ⊗ A → A defined as ϕ 1 A = ϕ A and ϕ n A = ϕ A • (H ⊗ ϕ n−1 A ). For brevity, we denote the morphisms ϕ A • (m n H ⊗ η A ) and ϕ A • (H ⊗ η A ) by u n and u 1 , respectively.
For n ≥ 1, let Reg ϕ A (H n , A) be the set of morphisms σ : H n → A such that there exists a morphism σ −1 : H n → A (the convolution inverse of σ) satisfying the following equalities: and Reg ϕ A (H L , A) will be the set of morphisms g : H L → A such that there exists a morphism g −1 : H L → A satisfying The sets Reg ϕ A (H L , A), Reg ϕ A (H n , A) are abelian groups with neutral elements u 0 and u n respectively. Moreover, we can define a cosimplicial complex of abelian groups with coface operators defined by be the coboundary morphisms of the cochain complex Then, (Reg ϕ A (H • , A), D • ϕ A ) gives the Sweedler cohomology of H in (A, ϕ A ). Therefore, the kth group will be defined by   if H is a finite Hopf algebra, the last bijection is the isomorphism between the second cohomology group H 2 (H, K), introduced by Sweedler [14], and the group of isomorphism classes of Galois H-objects with normal basis.