Group corings

0.37 MB·English

by  S. Caenepeel

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.