Separable Functors for the Category of Doi-hopf Modules. Applications

We prove a Maschke type Theorem for the category of Doi-Hopf modules. In fact, we give necessary and sufficient conditions for the functor forgetting the C-coaction to be separable. This leads to a generalized notion of integrals. Our results can be applied to obtain Maschke type Theorems for Yetter-Drinfel’d modules, Long dimodules and modules graded by G-sets. Existing Maschke type Theorems due to Doi and the authors are recovered as special cases. 0 Introduction One of the key results in classical representation theory is Maschke’s Theorem, stating that a finite group algebra kG is semisimple if and only if the characteristic of k does not divide the order of G. Many similar results (we call them Maschke type Theorems) exist in the literature, and the problem is always to find a necessary and sufficient condition for a certain object O to be reducible or semisimple. These objects occur in several disciplines in mathematics, they can be for example groups, affine algebraic groups, Lie groups, locally compact groups, or Hopf algebras, and, in a sense, this last example contains the previous ones as special cases. The general idea is to consider representations of O. Roughly stated, O is reducible if and only if all representations of O are completely reducible, and this comes to the fact that any monomorphism between two represetations splits. The idea behind the proof of a Maschke type Theorem is then the following: one applies a functor to the category of representations of O, forgetting some of the structure, and in such a way that we obtain objects in a more handable category, for instance vector spaces over a field or modules over a commutative ring. Then the strategy is to look for a deformation of a splitting map of a Tempus visitor at UIA Research supported by the bilateral project “Hopf algebras and co-Galois theory” of the Flemish and Romanian governments. Research supported by the “FWO Flanders research network WO.011.96N” 1 monomorphism in the “easy” category in such a way that it becomes a splitting in the category of representations. In a Hopf algebraic setting, the tool that is applied to find such a deformation is often called an integral. In fact there is more: the deformation turns out to be functorial in all cases that are known, and this can be restated in a categorical setting: the Maschke type Theorem comes down to the fact that the forgetful functor is separable in the sense of [27]. A separable functor F : C → D has more properties: information about semisimplicity, injectivity, projectivity of objects in D yields information about the corresponding properties in C. At first sight, it seems to be a very difficult problem to decide whether a functor is separable. However, if a functor F has a right adjoint G, then there is an easy criterium for the separability of F : the unit ρ : 1 → GF has to be split (see [28]), and this criterium will play a crucial role in this paper. The classical “Hopf algebraic Maschke Theorem” of Larson and Sweedler ([21]) tells us that a finite dimensional Hopf algebra is semisimple if and only if there exists a integral that is not annihilated by the augmentation map. Several generalizations to categories of (generalized) Hopf modules have been presented in the literature, see Cohen and Fishman ([7] and [8]), Doi ([11] and [12]), Ştefan and Van Oystaeyen ([34]) and the authors ([4]). The results in all these papers can be reformulated in terms of separable functors, in fact they give sufficient conditions for a forgetful functor to be separable. An unsatisfactory aspect is that these conditions are sufficient, but not necessary. In this paper, we return to the setting of [4]: we consider a Doi-Hopf datum (H,A,C), and the category of so-called Doi-Hopf modules M(H)A consisting of modules with an algebra action and a coalgebra coaction. As explained in previous publications ([2], [13],. . . ) M(H)A unifies modules, comodules, Sweedler’s Hopf modules, Takeuchi’s relative Hopf modules, graded modules, modules graded by G-sets, Long dimodules and Yetter-Drinfel’d modules. We consider the functor F forgetting the C-coaction, and we give necessary and sufficient conditions for this functor to be separable. To this end, we study natural transformations ν : GF → 1, and the clue result is the following: the natural transformation ν is completely determined if we know the map νC⊗A : C⊗C⊗A → C⊗A. We recall that C ⊗ A plays a special role in the category of Doi-Hopf modules (although it is not a generator). Conversely, a map C ⊗ C ⊗ A → C ⊗ A in the category of Doi-Hopf modules can be used to construct a natural transformation, provided it suffices two additional properties. The next step is then to show that the natural transformation splits the unit ρ if and only if the map νC⊗A satisfies a certain normalizing condition. We obtain a necessary and sufficient condition for the functor F to be separable, and this condition can be restated in several ways. Actually we prove that the k-algebra V consisting of all natural transformations ν : GF → 1 is isomorphic to five different k-algebras, consisting of k-linear maps satisfying certain properties. One of these algebras, named V4, consists of right C -linear maps γ : C → Hom(C,A) that are centralized by a left and right A-actions. We have called these maps γ A-integrals, because they are closely related to Doi’s total integrals (see Section 3.1). The normalized A-integrals (also called total A-integrals) are then right units in V4, and our main Theorem takes the following form: the forgetful functor is separable if and only if there exists a total A-integral γ : C → Hom(C,A). Our technique can also be applied to the right adjoint G of the functor F (see Section 2.2), and this leads to the notion of dual A-integral. In Section 3, we give several applications and examples: we explain how the results of [4], [11] and [12] are special cases (Sections 3.1 and 3.2), and apply our results to some Hopf module categories that are special cases of the Doi-Hopf module category: Yetter-Drinfel’d modules, Long dimodules and modules graded by G-sets (see Sections 3.5-3.6). In the Yetter-Drinfel’d case, this leads to a generalization of the Drinfel’d double in the case of 2 an infinite dimensional Hopf algebra, using Koppinen’s generalized smash product [17] and the results of [2]. Finally, if C is finitely generated and projective, then we find necessary and sufficient conditions for the extension A → A#C to be separable (Section 3.4). In the situation where C = H, we recover some existing results of Cohen and Fischman [7], Doi and Takeuchi [14], and Van Oystaeyen, Xu and Zhang [37]. 1 Preliminary results Throughout this paper, k will be a commutative ring with unit. Unless specified otherwise, all modules, algebras, coalgebras, bialgebras, tensor products and homomorphisms are over k. H will be a bialgebra over k, and we will extensively use Sweedler’s sigma-notation. For example, if (C,∆C) is a coalgebra, then for all c ∈ C we write ∆C(c) = ∑ c(1) ⊗ c(2) ∈ C ⊗ C. If (M,ρM ) is a left C-comodule, then we write ρM (m) = ∑ m<−1> ⊗m<0>, for m ∈ M . M will be the category of left C-comodules and C-colinear maps. For a k-algebra A, MA (resp. AM) will be the category of right (resp. left) A-modules and A-linear maps. The dual C = Hom(C, k) of a k-coalgebra C is a k-algebra. The multiplication on C is given by the convolution 〈f ∗ g, c〉 = ∑ 〈f, c(1)〉〈g, c(2)〉, for all f, g ∈ C and c ∈ C. C is a C-bimodule. The left and right action are given by the formulas c⇀c = ∑ 〈c, c(2)〉c(1) and c↼c ∗ = ∑ 〈c, c(1)〉c(2) (1) for c ∈ C and c ∈ C. This also holds for C-comodules. For example if (M,ρM ) is a left C-comodule, then it becomes a right C-module by m · c = ∑ 〈c,m<−1>〉m<0>, for all m ∈ M and c ∈ C. If C is projective as a k-module, then a k-linear map f : M → N between two left C-comodules is C-colinear if and only if it is right C-linear. An algebra A that is also a left H-comodule is called a left H-comodule algebra if the comodule structure map ρA is an algebra map. This means that ρA(ab) = ∑ a<−1>b<−1> ⊗ a<0>b<0> and ρA(1A) = 1H ⊗ 1A for all a, b ∈ A. Similarly, a coalgebra that is also a right H-module is called a right H-module coalgebra if ∆C(c · h) = ∑ c(1) · h(1) ⊗ c(2) · h(2) and εC(c · h) = εC(c)εH (h), for all c ∈ C, h ∈ H. We recall that a functor F : C → D is called fully faithful if the maps HomC(M,N) → HomD(FM,FN) are isomorphisms for all objects M , N ∈ C. 3 1.1 Doi-Hopf modules Let H be a bialgebra, A a left H-comodule algebra and C a right H-module coalgebra. We will always assume that C is flat as a k-module. Following [2], we will call the threetuple (H,A,C) a Doi-Hopf datum. A right-left Doi-Hopf module is a k-module M that has the structure of right A-module and left C-comodule such that the following compatibility relation holds ρM (ma) = ∑ m<−1> · a<−1> ⊗m<0>a<0>, (2) for all a ∈ A, m ∈ M . M(H)A will be the category of right-left Doi-Hopf modules and A-linear, C-colinear maps. In [6], induction functors between categories of Doi-Hopf modules are studied. It follows from [6, Theorem 1.3] that the forgetful functor F : C = M(H)A → MA has a right adjoint G : MA → M(H)A given by G(M) = C ⊗M with structure maps (c⊗m) · a = ∑ c · a<−1> ⊗ma<0> (3) ρC⊗M (c⊗m) = ∑ c(1) ⊗ c(2) ⊗m (4) for any c ∈ C, a ∈ A and m ∈ M . It is easy to see that the unit ρ : 1C → GF of the adjoint pair (F,G) is given by the C-coaction ρM : M → M ⊗C on any Doi-Hopf module M . The counit δ : FG → 1C is given by δN : C ⊗N → N, δN (c⊗ n) = ε(c)n for any right A-module N . A is a right A-module, so G(A) = C ⊗ A and GFG(A) = C ⊗ C ⊗ A are Doi-Hopf modules. For later use, we give the action and coaction explicitely: (c⊗ b)a = ∑ ca<−1> ⊗ ba<0> (5) ρ(c⊗ b) = ∑ c(1) ⊗ c(2) ⊗ b (6) (c⊗ d⊗ b)a = ∑ ca<−2> ⊗ da<−1> ⊗ ba<0> (7) ρ(c⊗ d⊗ b) = ∑ c(1) ⊗ c(2) ⊗ d⊗ b (8) ρ is the unit of the adjoint pair (F,G), and therefore the coaction ρC⊗A : C ⊗ A → C ⊗ C ⊗ A is A-linear and C-colinear. Now assume that H is a Hopf algebra. Then we can also consider the category AM(H) C of left A-modules that are also right C-comodules, with the additional compatibility relation (see [6]) ρ(am) = ∑ a<0>m<0> ⊗m<1>S(a<−1>) (9) Now the forgetful functor G : AM(H) C → M has a left adjoint F ′ given by F (M) = M ⊗ A and (see [6]) a(m⊗ b) = m⊗ ab (10) ρ(m⊗ b) = ∑ m<0> ⊗ b<0> ⊗m<1>S(b<−1>) (11) 4 It follows in particular that C⊗A = F (C) ∈ AM(H) C , and this makes C⊗A into a left A-module and a right C-comodule. The algebra C is a left H-module algebra; the H-action is given by the formula 〈h · c, c〉 = 〈c, c · h〉 for all h ∈ H, c ∈ C and c ∈ C. The smash product A#C is equal to A ⊗ C as a k-module, with multiplication defined by (a#c)(b#d) = ∑ a<0>b#c ∗ ∗ (a<−1> · d ), (12) for all a, b ∈ A, c, d ∈ D. Recall that we have a natural functor P : M(H)A → MA#C∗ sending a Doi-Hopf module M to itself, with right A#C-action given by m · (a#c) = ∑ 〈c,m<−1>〉m<0>a (13) for any m ∈ M , a ∈ A and c ∈ C. P is fully faithful if C is projective as a k-module, and P is an equivalence of categories if C is finitely generated and projective as a k-module (cf. [13]). In [17], Koppinen introduced the following version of the smash product: #(C,A) is equal to Hom(C,A) as a k-module, with multiplication given by the formula (f • g)(c) = ∑