GROUP CORINGS 7 0 S. CAENEPEEL, K.JANSSEN,ANDS.H. WANG 0 2 Abstract. We introduce group corings, and study functors between n categories of comodules over group corings, and the relationship to a J graded modules over graded rings. Galois group corings are defined, and aStructureTheorem for theG-comodulesoveraGalois group cor- 1 3 ing is given. We study (graded) Morita contexts associated to a group coring. Ourtheoryisappliedtogroupcoringsassociated toacomodule ] algebra over a Hopf group coalgebra. A R . h t a Introduction m Group coalgebras and Hopf group coalgebras were introduced by Turaev [ [15]. A systematic algebraic study of these new structures has been carried 1 out in recent papers by Virelizier, Zunino, and the third author (see for v example [16, 17, 18, 19, 21, 22]). Many results from classical Hopf algebra 1 3 theory can begeneralized to Hopf group coalgebras; this has been explained 9 in a paper by the first author and De Lombaerde [6], where it was shown 1 thatHopfgroupcoalgebras areinfactHopfalgebras inasuitablesymmetric 0 7 monoidal category. 0 In [20], the third author investigated how Hopf-Galois theory can be devel- / h opedintheframeworkofHopfgroupcoalgebras. AdefinitionofHopf-Galois t extension was presented; the requirement is that a set of canonical maps, a m indexed by the elements of the underlying group, has to be bijective. One : aspect in the present theory that is not satisfactory is the lack of an appro- v priate Structure Theorem: an important result in Hopf-Galois theory states i X that the category of relative Hopf modules over a faithfully flat Hopf-Galois r extension is equivalent to the category of modules over the coinvariants. So a far, no such result is known in the framework of Hopf group coalgebras. Corings were introduced by Sweedler [14], and were revived recently by Brzezin´ski [3]. One of the important observations is that coring theory pro- vides an elegant approach to descent theory and Hopf-Galois theory (see 2000 Mathematics Subject Classification. 16W30, 16W50. Key words and phrases. Corings, group coalgebras, Hopf-Galois extensions, graded rings, graded Morita contexts. ThisresearchwassupportedbytheresearchprojectG.0622.06“Deformationquantiza- tion methods for algebras and categories with applications to quantum mechanics” from FWO-Vlaanderen. The third author was partially supported by the SRF (20060286006) and theFNS (10571026). 1 2 S.CAENEPEEL,K.JANSSEN,ANDS.H.WANG [3, 4, 5]). The aim of this paper is to develop Galois theory for group cor- ings, and to apply it to Hopf group coalgebras. A G-A-coring (or group coring) consists of a set of A-bimodules indexed by a group G, together with a counit map, and a set of diagonal maps indexed by G×G, with appropriate axioms (see Section 1). A first remarkable ob- servation is the fact that we can introduce two different types of comodules over a group coring C. C-comodules consist of a single A-module, with a set of structure maps indexed by G, while G-C-comodules consist of a set of A-modules indexed by G, together with structure maps indexed by G×G. We have a pair of adjoint functors between the two categories of comodules (see Proposition 1.1). This remarkable fact can be explained by duality ar- guments. Dualizing the definition of a G-A-coring, we obtain G-A-rings; in the coalgebra case, this was observed in [21]. In contrast to G-corings, G- rings are a well-known concept: in fact there is a categorical correspondence between G-A-rings and G-graded A-rings. The graded ring corresponding to a G-ring is the so-called “packed form” of the G-ring, in the terminology of [21]. We don’t have a similar correspondence between group corings and graded corings, unless the group G in question is finite. This indicates that dualitypropertiesformodulecategories over gradedringshavetobestudied from the point of view of group corings, rather than graded corings. Over a graded ring, one can study ordinary modules and graded modules, and there exists an adjoint pair between the two categories. We also have functors from the categories of C-comodules (resp. G-C-comodules) to mod- ules (resp. graded modules) over the dual graded ring of C. All these functors appear in a commutative diagram of functors (see Proposition 4.5). The functor between the category of G-C-comodules and graded modules is an equivalence if every C (or equivalently, every homogeneous part of the α dual graded ring) is finitely generated and projective as an A-module (see Proposition 4.4). These properties of (co)module categories are studied in Sections 1, 3 and 4. An important class of group corings, called cofree group corings, is investi- gated in Section 2. Basically, these are corings for which all the underlying bimodules are isomorphic. A cofree coring is - up to isomorphism - de- termined by C , its part of degree e; its left dual is the group ring ∗C [G] e e over the left dual of C (Proposition 4.6). The category of G-C-comodules is e equivalent to the category of comodules over C (Theorem 2.2). This is an e analog of the well-known fact that, for a group ring R[G], the category of graded R[G]-modules is equivalent to the category of R-modules. In Section 5, we introduce the notion of grouplike family of a group coring. Grouplike families correspond bijectively to C-comodule structures on A. Fixing a grouplike family, we can introduce the coinvariant subring T of A. Wehavetwopairsofadjointfunctors,oneconnectingmodulesover thecoin- variants to rightC-comodules (Proposition 5.3), andanother oneconnecting modules over the coinvariants to right G-C-comodules (Proposition 5.4). It can be established when the latter adjoint pair, denoted (F ,G ), is a pair 7 7 GROUP CORINGS 3 of inverse equivalences. Given a group coring C with a fixed grouplike fam- ily, we can define a canonical morphism of group corings between the cofree coring built on the Sweedler canonical coring and the coring C. If F is 7 an equivalence, then this canonical morphism is an isomorphism (Proposi- tion 5.7). In this case, we call our group coring a Galois group coring. This is equivalent to C being cofree, and C being a Galois coring. The Structure e Theorem 5.12 is our main result. Basically, if C is Galois, and A is faithfully flat over the coinvariants, then F is an equivalence. 7 Moritatheoryplays animportantroleinthistheory. Toagroupcoringwith a fixed grouplike family, we can associate several Morita contexts. Two of them are classical Morita contexts, and have been studied in a special sit- uation (see Section 10) in [20]. But the natural Morita contexts are in fact graded Morita contexts. In Section 6, we give some generalities on graded Morita contexts; in Sections 7 and 8, we discuss the Morita contexts and their relationship. In some situations, the Galois property of a group coring can be charac- terized by the graded Morita contexts associated to it. We study this in Section 9. In the final Section 10, we briefly discuss the situation where C is a group coring A⊗H associated to a right H-comodule algebra A, where H is a Hopf group coalgebra, as introduced in [15]. We show that this group coring is Galois if and only ifA is an H-Galois extension in the sense of [20]. This entails a Structure Theorem for relative group Hopf modules; we also describe the dual of the group coring A⊗H. Throughout this paper, we will adopt the following notational conventions. For an object M in a category, M will also denote the identity morphism on M. Let G be a group and M a (right) A-module. We will often need collections of A-modules isomorphic to M and indexed by G. We will consider these modulesas isomorphic, butdistinct. Let M×{α} bethe modulewith index α. We then have isomorphisms µ : M → M ×{α}, µ (m) = (m,α). α α We can then write M ×{α} = µ (M). µ can be considered as a dummy α variable, we will also use the symbols γ,ν,.... We will identify M and M ×{e} using µ . e 1. Group corings and comodules Let G be a group, and A a ring with unit. The unit element of G will be denoted by e. A G-group A-coring (or shortly a G-A-coring) C is a family (C ) of A-bimodules together with a family of bimodule maps α α∈G ∆ : C → C ⊗ C ; ε: C → A, α,β αβ α A β e such that (1) (∆ ⊗ C )◦∆ = (C ⊗ ∆ )◦∆ α,β A γ αβ,γ α A β,γ α,βγ 4 S.CAENEPEEL,K.JANSSEN,ANDS.H.WANG and (2) (C ⊗ ε)◦∆ = C = (ε⊗ C )◦∆ , α A α,e α A α e,α for all α,β,γ ∈ G. We use the following Sweedler-type notation for the comultiplication maps ∆ : α,β ∆ (c) = c ⊗ c , α,β (1,α) A (2,β) for all c ∈C . Then (2) takes the form αβ (3) c ε(c ) = c= ε(c )c . (1,α) (2,e) (1,e) (2,α) (1) justifies the following notation: ((∆ ⊗ C )◦∆ )(c) = ((C ⊗ ∆ )◦∆ )(c) =c ⊗ c ⊗ c , α,β A γ αβ,γ α A β,γ α,βγ (1,α) A (2,β) A (3,γ) for all c ∈ C . If C is a G-A-coring, then C = C is an A-coring, with αβγ e comultiplication ∆ and counit ε. e,e A morphism between two G-A-corings C and D consists of a family of A- bimodule maps (f ) , f : C → D such that α α∈G α α α (f ⊗ f )◦∆ = ∆ ◦f and ε◦f = ε. α A β α,β α,β αβ e If A is a commutative ring, and ac = ca, for all α ∈ G, a ∈ A and c ∈ C , α then C is called a G-coalgebra, cf. [15]. Overagroupcoring, wecan definetwo differenttypes ofcomodules. Aright C-comodule is a right A-module M together with a family of right A-linear maps (ρ ) , ρ : M → M ⊗ C , such that α α∈G α A α (4) (M ⊗ ∆ )◦ρ =(ρ ⊗ C )◦ρ A α,β αβ α A β β and (5) (M ⊗ ε)◦ρ = M. A e We use the following Sweedler-type notation: ρ (m) =m ⊗ m . α [0] A [1,α] (4) justifies the notation ((M ⊗ ∆ )◦ρ )(m) = ((ρ ⊗ C )◦ρ )(m) = m ⊗ m ⊗ m , A α,β αβ α A β β [0] A [1,α] A [2,β] and (5) is equivalent to m ε(m ) = m, for all m ∈M. [0] [1,e] A morphism of right C-comodules is a right A-linear map f : M → N satisfying the condition (6) (f ⊗ C )◦ρ = ρ ◦f, A α α α forallα∈ G. MC willbeournotationforthecategory ofrightC-comodules. A right G-C-comodule M is a family of right A-modules (M ) , together α α∈G with a family of right A-linear maps ρ : M → M ⊗ C α,β αβ α A β GROUP CORINGS 5 such that (7) (M ⊗ ∆ )◦ρ = (ρ ⊗ C )◦ρ α A β,γ α,βγ α,β A γ αβ,γ and (8) (M ⊗ ε)◦ρ = M α A α,e α for all α,β,γ ∈ G. We now use the following Sweedler-type notation: ρ (m) =m ⊗ m , α,β [0,α] A [1,β] for m ∈ M . (7) justifies the notation αβ ((M ⊗ ∆ )◦ρ )(m) = ((ρ ⊗ C )◦ρ )(m) α A β,γ α,βγ α,β A γ αβ,γ = m ⊗ m ⊗ m , [0,α] A [1,β] A [2,γ] for m ∈ M . (8) implies that m ε(m ) = m, for all m ∈ M . A αβγ [0,α] [1,e] α morphism between two right G-C-comodules M and N is a family of right A-linear maps f : M → N such that α α α (f ⊗ C )◦ρ = ρ ◦f . α A β α,β α,β αβ The category of right G-C-comodules will be denoted by MG,C. Proposition 1.1. We have a pair of adjoint functors (F ,G ) between the 1 1 categories MG,C and MC. Moreover, if G is a finite group, then (F ,G ) is 1 1 a Frobenius pair of functors, i.e. F is also a right adjoint of G . 1 1 Proof. Take M = (M ) ∈ MG,C, and define α α∈G F (M)= M = M. 1 M α α∈G The coaction maps ρ : M → M⊗ C are defined as follows: for m ∈ M , α A α β let (9) ρα(m)= m[0,βα−1]⊗Am[1,α]. Otherwise stated, ρα = Lβ∈Gρβα−1,α. Let us show that (4,5) hold. For all m ∈ M , we compute that γ ((ρα⊗ACβ)◦ρβ)(m) =(ρα ⊗ACβ)(m[0,γβ−1]⊗Am[1,β]) = m[0,γβ−1α−1]⊗Am[1,α]⊗Am[2,β] = ((M ⊗ ∆ )◦ρ )(m); A α,β αβ ((M ⊗ ε)◦ρ )(m) = m ε(m ) = m. A e [0,γ] [1,e] For a morphism f : M → N in MG,C, we simply define F (f)= f . 1 M α α∈G Let us now define G . For M ∈ MC, let G (M) = µ (M), where we use 1 1 α α the notation introduced at the end of the introduction. The coaction maps ρ : µ (M) → µ (M)⊗ C are defined by α,β αβ α A β (10) ρ (µ (m)) =µ (m )⊗ m , α,β αβ α [0] A [1,β] 6 S.CAENEPEEL,K.JANSSEN,ANDS.H.WANG for all m ∈ M. The formulas (7,8) hold since ((M ⊗ ∆ )◦ρ )(µ (m)) α A β,γ α,βγ αβγ = (M ⊗ ∆ )(µ (m )⊗ m ) α A β,γ α [0] A [1,βγ] = µ (m )⊗ m ⊗ m α [0] A [1,β] A [2,γ] = ρ (µ (m ))⊗ m α,β αβ [0] A [1,γ] = ((ρ ⊗ C )◦ρ )(µ (m)); α,β A γ αβ,γ αβγ ((M ⊗ ε)◦ρ )(µ (m)) α A α,e α = (M ⊗ ε)(µ (m )⊗ m ) α A α [0] A [1,e] = µ (m )ε(m )= µ (m ε(m )) = µ (m). α [0] [1,e] α [0] [1,e] α On the morphisms, G is defined as follows: for f : M → N in MC, we put 1 G (f)= (ν ◦f ◦µ−1) . 1 α α α∈G Take M ∈ MG,C and N ∈MC, and consider the map ψ : HomC(F (M),N) → HomG,C(M,G (N)) 1 1 defined as follows. For f : M → N, let Lα∈G α ψ(f) = ν ◦f ◦i : M → G (N) = ν (N), α α α α 1 α α where i : M → M is the canonical injection. Now consider the α α Lα∈G α map φ: HomG,C(M,G (N)) → HomC(F (M),N), 1 1 defined as follows: for g = (g ) : M → G (N), let α α∈G 1 φ(g)(m) = (ν−1◦g ◦p )(m), X α α α α∈G where now p : M → M is the canonical projection. Straightfor- α Lα∈G α α ward computations show that ψ and φ are well-defined. They are inverses, since φ(ψ(f))(m) = (ν−1◦ν ◦f ◦i ◦p )(m) = f( (i ◦p )(m)) = f(m), X α α α α X α α α∈G α∈G for all m ∈ M , and Lα∈G α ψ(φ(g)) (m) = (ν ◦φ(g)◦i )(m) α α α = (ν ◦ν−1◦g ◦p ◦i )(m) = (ν ◦ν−1◦g )(m) = g (m), X α β β β α α α α α β∈G for all α ∈ G and m ∈ M . It is easy to show that ψ and φ define nat- α ural transformations. Let us describe the unit η and the counit ε of the 1 1 adjunction. For M ∈ MG,C, we have η = µ ◦i : M → µ ( M ); 1,M,β β β β β M α α∈G GROUP CORINGS 7 for N ∈ MC, we have ε = µ−1◦p : µ (N)→ N. 1,N X α α M α α∈G α∈G Toprovethefinalstatement, let usassumethatGisfinite. TakeM ∈ MG,C and N ∈MC, and consider the map Φ : HomG,C(G (N),M)→ HomC(N,F (M)), 1 1 defined by Φ(g)(n) = (i ◦g ◦µ )(n) ∈ M , X α α α M α α∈G α∈G for all morphisms g = (g ) : G (N) → M in MG,C. Now consider the α α∈G 1 map Ψ : HomC(N,F (M)) → HomG,C(G (N),M), 1 1 defined by Ψ(f) = p ◦f ◦µ−1 :µ (N) → M , α α α α α for all right C-colinear maps f :N → M . One can check that Φ and Lα∈G α Ψ are well-defined. Let us check that Φ and Ψ are mutually inverse: Φ(Ψ(f))(n) = (i ◦Ψ(f) ◦µ )(n) X α α α α∈G = (i ◦p ◦f ◦µ−1◦µ )(n) = (i ◦p )(f(n)) =f(n), X α α α α X α α α∈G α∈G for all n ∈ N, and, for all α∈ G, Ψ(Φ(g)) = p ◦Φ(g)◦µ−1 = (p ◦i ◦g ◦µ ◦µ−1) α α α X α β β β α β∈G = g ◦µ ◦µ−1 = g . α α α α Let us finally describe the unit ν and the counit ζ of this adjunction. For 1 1 N ∈ MC, we have ν = i ◦µ : N → µ (N); 1,N X α α M α α∈G α∈G for M ∈ MG,C, we have ζ = p ◦µ−1 : µ ( M )→ M . 1,M,β β β β M α β α∈G (cid:3) 8 S.CAENEPEEL,K.JANSSEN,ANDS.H.WANG 2. Cofree group corings Definition 2.1. A G-A-coring C is called cofree if there exist A-bimodule isomorphisms γ : C = C → C such that α e α (11) ∆ (γ (c)) = γ (c )⊗ γ (c ), α,β αβ α (1) A β (2) for all c ∈C. From (2) and (11), it follows that (ε◦γ−1)(γ (c) )γ (c) = γ (c), α αβ (1,α) αβ (2,β) β for all c ∈ C. This can be restated as follows: for all c ∈ γ (C) = C , we αβ αβ have (12) (ε◦γ−1)(c )c = (γ ◦γ−1)(c). α (1,α) (2,β) β αβ In a similar way, we obtain the formula (13) c (ε◦γ−1)(c ) = (γ ◦γ−1)(c). (1,α) β (2,β) α αβ A cofree group coring C is defined up to isomorphism by C . We will write e C = C hGi. e Theorem 2.2. If C is a cofree group coring, then the categories MCe and MG,C are equivalent. Proof. We define a functor F : MCe → MG,C as follows: F (N) = ν (N) 2 2 α α is an isomorphic copy of N; the coaction maps are ρ : ν (N) → ν (N)⊗ γ (C ), ρ (ν (n)) = ν (n )⊗ γ (n ). α,β αβ α A β e α,β αβ α [0] A β [1] We also have a functor G : MG,C → MCe, G (M) = M , with coaction 2 2 e ρ = ρ. It is then clear that G (F (N)) = N, for all N ∈ MCe. For e,e 2 2 M ∈ MG,C, we have that F (G (M)) = (ν (M )) . 2 2 α e α∈G It is clear that the map ϕ : M → ν (M ), ϕ (m) = ν (m )ε(γ−1(m )) α α α e α α [0,e] α [1,α] is right A-linear. ϕ= (ϕ ) is a morphism in MG,C, since α α∈G ((ϕ ⊗ C )◦ρ )(m) = ϕ (m )⊗ m α A β α,β α [0,α] A [1,β] = ν (m )(ε◦γ−1)(m )⊗ m α [0,e] α [1,α] A [2,β] (1=2) ν (m )⊗ (γ ◦γ−1)(m ) α [0,e] A β αβ [1,αβ] (1=3) ν (m )⊗ γ m (ε◦γ−1)(m ) α [0,e] A β(cid:16) [1,e] αβ [2,αβ] (cid:17) = ν (m )⊗ γ (m )(ε◦γ−1)(m ) α [0,e] A β [1,e] αβ [2,αβ] = ρ (ν (m ))(ε◦γ−1)(m ) α,β αβ [0,e] αβ [1,αβ] = (ρ ◦ϕ )(m), α,β αβ GROUP CORINGS 9 for all m ∈ M . Next we define αβ ψα : να(Me)→ Mα, ψα(να(m)) = m[0,α](ε◦γα−−11)(m[1,α−1]). For all m ∈ M , we compute α (ψ ◦ϕ )(m) = ψ (ν (m )ε(f−1(m ))) α α α α [0,e] α [1,α] = m[0,α](ε◦γα−−11)(m[1,α−1])(ε◦γα−1)(m[2,α]) = m[0,α](ε◦γα−−11)(cid:16)m[1,α−1](ε◦γα−1)(m[2,α])(cid:17) (1=3) m[0,α](ε◦γα−−11)(cid:16)(γα−1 ◦γe−1)(m[1,e])(cid:17) = m ε(m ) = m. [0,α] [1,e] For all m ∈ M , we have e (ϕα ◦ψα)(να(m)) = ϕα(cid:16)m[0,α](ε◦γα−−11)(m[1,α−1])(cid:17) = να(m[0,e])(ε◦γα−1)(m[1,α])(ε◦γα−−11)(m[2,α−1]) = να(m[0,e])(ε◦γα−1)(cid:16)m[1,α](ε◦γα−−11)(m[2,α−1])(cid:17) (1=3) ν (m )(ε◦γ−1◦γ ◦γ−1)(m ) α [0,e] α α e [1,e] = ν (m ε(m )) = ν (m). α [0,e] [1,e] α This shows that ψ is inverse to ϕ , and our result follows. (cid:3) α α 3. Graded corings and comodules Let C be an A-coring, C is called a G-graded A-coring if there exists a direct sum decomposition C = C as A-bimodules such that ∆(C ) ⊂ Lα∈G α α Lβ∈GCαβ−1 ⊗ACβ and ε(Cα) = 0 if α 6= e. If A is a commutative ring, and ac = ca, for all a ∈ A and c ∈ C, then C is called a G-graded coalgebra, cf. [12]. To a G-graded A-coring, we can associate a G-A-coring C = (C ) . The α α∈G counit is the restriction of ε to C , and ∆ is the composition e α,β ∆ p Cαβ−→MCαβγ−1 ⊗ACγ−→Cα⊗ACβ, γ∈G where p is the obvious projection. Let (M,ρ) be a right C-comodule. For each α ∈ G, we consider the map (M ⊗ p )◦ρ : M → M ⊗ C , A α A α where p : C → C is the projection. Then M is a right C-comodule, and α α we obtain a functor MC → MC. (M,ρ) is called a G-graded right C-comodule if we have a decomposition 10 S.CAENEPEEL,K.JANSSEN,ANDS.H.WANG M = M as right A-modules, such that Lα∈G α ρ(Mα)⊂ MMαβ−1 ⊗ACβ. β∈G Now consider the maps ρ p ρα,β : Mαβ−→MMαβγ−1 ⊗ACγ−→Mα⊗ACβ. γ∈G Then M = (M ) is a right G-C-comodule, and we have a functor from α α∈G the category of graded C-comodules to MG,C. If G is finite, then there is a one-to-one correspondence between graded corings and group corings: if C is a group coring, then C is a graded Lα∈G α coring. In this situation the two functors between (G-graded) C-comodules and (G-)C-comodules are isomorphisms of categories. 4. Graded rings and modules Let A be a ring and R = R a G-graded ring. Suppose that we have Lα∈G α a ring morphism i : A → R . Then we call R a G-graded A-ring. Every e R is then an A-bimodule, via restriction of scalars, and the decomposition α of R is a decomposition of A-bimodules. The category of G-graded right R-modules will be denoted by MG. R Let C be a G-A-coring. For every α ∈ G, Rα = ∗Cα−1 = AHom(Cα−1,A) is an A-bimodule, with (a·f ·b)(c) = f(ca)b, for all f ∈Rα, a,b ∈ A and c ∈ Cα−1. Take f ∈R , g ∈ R and define f #g ∈ R as the composition α α β β α β αβ C(αβ)−1∆β−−→1,α−1Cβ−1 ⊗ACα−1Cβ−−1→⊗AfαCβ−1−g→β A, that is, (fα#gβ)(c) = gβ(c(1,β−1)fα(c(2,α−1))), for all c ∈ C(αβ)−1. This defines maps mα,β : Rα⊗ARβ → Rαβ, which make R = R into a G-graded A-ring. Let us show that the multiplication Lα∈G α is associative: take hγ ∈ Rγ and c∈ C(αβγ)−1. We then compute that ((fα#gβ)#hγ)(c) = hγ(c(1,γ−1)(fα#gβ)(c(2,(αβ)−1))) = hγ(c(1,γ−1)gβ(c(2,β−1)fα(c(3,α−1)))) = (fα#(gβ#hγ))(c). ε∈ R is a unit for the multiplication; i: A → R , i(a)(c) = ε(c)a is a ring e e homomorphism, since (i(a)#i(b))(c) = i(b)(c i(a)(c )) = i(b)(c ε(c )a) (1,e) (2,e) (1,e) (2,e) = i(b)(ca) = ε(ca)b = ε(c)ab = i(ab)(c). We conclude that R = R is a G-graded A-ring, called the (left) dual Lα∈G α (graded) ring of the group coring C. We will also write ∗C = R.