GALOIS THEORY FOR WEAK HOPF ALGEBRAS

GALOIS THEORY FOR WEAK HOPF ALGEBRAS

S. CAENEPEEL and E. DE GROOT

We develop Hopf-Galois theory for weak Hopf algebras, and recover analogs of classical results for Hopf algebras. Our methods are based on the recently intro- duced Galois theory for corings. We focus on the situation where the weak Hopf algebra is a groupoid algebra or its dual. We obtain weak versions of strongly graded rings, and classical Galois extensions.

AMS 2000 Subject Classification: 16W30.

Key words: Hopf algebra, coring, Galois theory for corings.

INTRODUCTION

Weak bialgebras and Hopf algebras are generalizations of ordinary bial- gebras and Hopf algebras in the following sense: the defining axioms are the same, but the multiplicativity of the counit and comultiplicativity of the unit are replaced by weaker axioms. Perhaps the easiest example of a weak Hopf algebra is a groupoid algebra; other examples are face algebras [15], quantum groupoids [17] and generalized Kac algebras [21]. A purely algebraic approach can be found in [2] and [3].

The aim of this note is to develop Galois theory for weak Hopf algebras.

A possible strategy could be to try to adapt the methods from classical Hopf- Galois theory to the weak situation. This would be unnecessarily complicated, since a more powerful tool became available recently.

Corings and comodules over corings were introduced by Sweedler in [20].

The notion was revived recently by Brzezi´nski [5], who saw the importance of a remark made by Takeuchi, that Hopf modules and many of their generaliza- tions can be viewed as examples of comodules over corings. Many applications came out of this idea, an overview is given in [6].

One of the beautiful applications is a reformulation of descent and Galois theory using the language of corings. This was initiated in [5], and continued by the first author [8] and Wisbauer [22]. An observation in [5] is that weak Hopf modules can be viewed as comodules over a coring, and this implies that the general theory of Galois corings can be applied to weak Hopf algebras.

This is what we will do in Section 2.

REV. ROUMAINE MATH. PURES APPL.,52(2007),2, 151–176

In Section 3, we look at the special case where the weak Hopf algebra is a groupoid algebra. This leads us to the notions of groupoid graded algebras and modules, and the following generalization of a result of Ulbrich: a groupoid graded algebra is Galois if and only if it is strongly graded.

In Section 4, we look at the dual situation, where the weak Hopf algebra is the dual of a finite groupoid algebra. Then we have to deal with finite groupoid actions, as we have to deal with finite group actions in classical Galois theory. The results in this section are similar to the ones obtained in [10], where partial Galois theory (see [14]) is studied using the language of corings. Both are generalizations of classical Galois theory. In partial Galois theory, we look at group actions on an algebra A that are not everywhere defined, and in weak Galois theory, we look at groupoid actions. In both cases, we have a coring that is a direct factor of |G|copies of A, and the left dual of the coring is a Frobenius extension ofA.

1. PRELIMINARY RESULTS 1.1. GALOIS CORINGS

Let A be a ring. An A-coring C is a coalgebra in the category _AMA

of A-bimodules. Thus an A-coringis a triple C = (C,∆_C, ε_C), where C is an A-bimodule, and ∆_C : C → C ⊗AC and ε_C : C → A are A-bimodule maps such that

(1) (∆_C⊗AC)◦∆_C= (C ⊗A∆_C)◦∆_C, and

(2) (C ⊗Aε_C)◦∆_C = (ε_C⊗AC)◦∆_C =C. We use the Sweedler-Heyneman notation

∆_C(c) =c₍₁₎⊗Ac₍₂₎.

for comultiplication. A right C-comodule M = (M, ρ) consists of a right A- moduleM together with a right A-linear mapρ: M →M⊗AC such that (3) (ρ⊗AC)◦ρ= (M ⊗A∆_C)◦ρ,

and

(4) (M ⊗Aε_C)◦ρ=M.

We then say thatC coacts from the right onM, and we write ρ(m) =m_[0]⊗Am_[1].

A rightA-linear mapf :M →N between two rightC-comodulesM and N is called right C-colinear if ρ(f(m)) =f(m_[0])⊗m_[1], for all m ∈M. The category of rightC-comodules andC-colinear maps is denoted byM^C.

x ∈ C is called grouplike if ∆_C(x) = x⊗x and ε_C(x) = 1. Grouplike elements ofCcorrespond bijectively to rightC-coactions onA: ifAis grouplike, then we have the following rightC-coaction ρ on A: ρ(a) =xa.

Let (C, x) be a coring with a fixed grouplike element. For M ∈ M^C, we call

M^coC ={m∈M |ρ(m) =m⊗Ax} the submodule of coinvariants ofM. Observe that

A^coC ={b∈A |bx=xb}

is a subring of A. Let i : B → A be a ring morphism. i factorizes through A^coC if and only if

x∈G(C)^B={x∈G(C) |xb=bx, for all b∈B}.

We then have a pair of adjoint functors (F, G), respectively between the cate- gories MB and M^C and the categories BM and ^CM. For N ∈ MB and M ∈ M^C,

F(N) =N⊗BA and G(M) =M^coC. The unit and counit of the adjunction are

ν_N :N →(N ⊗BA)^coC, ν_N(n) =n⊗B1, and

ζM :M^coC⊗BA→M, ζM(m⊗Ba) =ma.

We have a similar pair of adjoint functors (F, G) between the categories of leftB-modules and left C-comodules.

Leti: B →A be a morphism of rings. The associated canonical coring isD=A⊗BA, with comultiplication and counit given by the formulas

∆_D: D → D ⊗AD ∼=A⊗BA⊗BA, ∆_D(a⊗Ba) =a⊗B1⊗Ba and

ε_D: D=A⊗BA→A, ε_D(a⊗Ba) =aa.

If i : B → A is pure as a morphism of left and right B-modules, then the categoriesMB and M^D are equivalent.

Let (C, x) be a coring with a fixed grouplike element, andi: B →A^coC a ring morphism. We then have a morphism of corings

can :D=A⊗BA→ C, can(a⊗Ba) =axa.

If F is fully faithful, then B ∼= A^coC; if G is fully faithful, then can is an isomorphism. (C, x) is called a Galois coring if can :A⊗A^coCA→ Cis bijective.

From [8], we recall the following results.

Theorem 1.1. Let (C, x) be an A-coring with fixed grouplike element, andT =A^coC. Then the following statements are equivalent.

(1) (C, x) is Galois andA is faithfully flat as a left T-module;

(2) (F, G) is an equivalence andA is flat as a left T-module.

Let (C, x) be a coring with a fixed grouplike element, and takeT =A^coC. Then^∗C=_AHom(C, A) is a ring, with multiplication given by

(5) (f#g)(c) =g(c₍₁₎f(c₍₂₎)).

We have a morphism of ringsj : A→^∗C, given by j(a)(c) =ε_C(c)a.

This makes^∗C into an A-bimodule, via the formula (af b)(c) =f(ca)b.

Consider the left dual of the canonical map:

∗can :^∗C →^∗D ∼=TEnd(A)^op, ^∗can(f)(a) =f(xa).

We then have the following result.

Proposition 1.2. If (C, x) is Galois, then ^∗can is an isomorphism.The converse property holds ifC andAare finitely generatedprojective, respectively as a left A-module, and a left T-module.

Let Q={q ∈ ^∗C |c₍₁₎q(c₍₂₎) = q(c)x, for all c∈ C}. A straightforward computation shows that Q is a (^∗C, T)-bimodule. Also A is a left (T,^∗C)- bimodule; the right^∗C-action isinduced by the rightC-coaction: a·f =f(xa).

Now, consider the maps

τ : A⊗^∗_CQ→T, τ(a⊗^∗_Cq) =q(xa), (6)

µ: Q⊗T A→^∗C, µ(q⊗T a) =q#i(a).

(7)

With this notation, we have the following property (see [12]).

Proposition 1.3. (T,^∗C, A, Q, τ, µ) is a Morita context. The map τ is surjective if and only if there existsq ∈Q such that q(x) = 1.

We also have (see [8])

Theorem 1.4. Let (C, x) be a coring with fixed grouplike element, and assume thatCis a left A-progenerator. We take a subringB ofT =A^coC, and consider the map

can : D=A⊗BA→ C, can(a⊗T a) =axa. Then the following statements are equivalent:

(1) • can is an isomorphism;

• A is faithfully flat as a left B-module.

(2) • ^∗can is an isomorphism;

• A is a left B-progenerator.

(3) • B =T;

• the Morita context (B,^∗C, A, Q, τ, µ) is strict.

(4) • B =T;

• (F, G) is an equivalence of categories.

1.2. WEAK HOPF ALGEBRAS

Let k be a commutative ring. Recall that a weak k-bialgebra is a k- module with a k-algebra structure (µ, η) and a k-coalgebra structure (∆, ε) such that

∆(hk) = ∆(h)∆(k), for allh, k ∈H, and

∆²(1) = 1₍₁₎⊗1₍₂₎1₍₁)⊗1₍₂)= 1₍₁₎⊗1₍₁)1₍₂₎⊗1₍₂), (8)

ε(hkl) = ε(hk₍₁₎)ε(k₍₂₎l) =ε(hk₍₂₎)ε(k₍₁₎l), (9)

for allh, k, l ∈ H. We use the Sweedler-Heyneman notation for the comulti- plication, namely

∆(h) =h₍₁₎⊗h₍₂₎ =h₍₁)⊗h₍₂).

IfHis a weak bialgebra, then we have idempotent maps Π^L,Π^R,Π^L,Π^R: H→H given by

Π^L(h) =ε(1₍₁₎h)1₍₂₎, Π^R(h) = 1₍₁₎ε(h1₍₂₎), Π^L(h) = 1₍₁₎ε(1₍₂₎h), Π^R(h) =ε(h1₍₁₎)1₍₂₎. We haveH^L= Im (Π^L) = Im (Π^R) and H^R= Im (Π^R) = Im (Π^L).

Let H be a weak bialgebra. Recall from [1] and [9] that a right H- comodule algebra A is ak-algebra with a right H-comodule structure ρ such that ρ(a)ρ(b) =ρ(ab) for all a, b∈A, and such that the following equivalent statements hold (see [9, Proposition 4.10]):

ρ²(1) =

1_[0]⊗1_[1]1₍₁₎⊗1₍₂₎, (10)

ρ²(1) =

1_[0]⊗1₍₁₎1_[1]⊗1₍₂₎, (11)

a_[0]⊗Π^R(a_[1]) =a1_[0]⊗1_[1], (12)

a_[0]⊗Π^L(a_[1]) = 1_[0]a⊗1_[1], (13)

1_[0]⊗Π^R(1_[1]) =ρ^r(1), (14)

1_[0]⊗Π^L(1_[1]) =ρ^r(1), (15)

ρ^r(1)∈A⊗H^L. (16)

Lemma1.5. Let H be a weak bialgebra, and A a right H-comodule alge- bra. Then

ε(h₍₁₎1_[1])1_[0]⊗h₍₂₎ = 1_[0]⊗h1_[1], (17)

ε(h1_[1])1_[0]a=ε(ha_[1])a_[0]

(18)

for allh∈H and a∈A.

Proof. This is a special case of [9, Theorem 4.14]. Details are as follows:

ε(h₍₁₎1_[1])1_[0]⊗h₍₂₎ = ε(h₍₁₎1₍₁₎1_[1])1_[0]⊗h₍₂₎1₍₂₎

= ε(h₍₁₎1_[1])1_[0]⊗h₍₂₎1_[2] = 1_[0]⊗h1_[1]; ε(h1_[1])1_[0]a = ε(h1_[2])ε(1_[1]a_[1])1_[0]a_[0]

= ε(h1_[1]a_[1])1_[0]a_[0] =ε(ha_[1])a_[0].

A weak Hopf algebra is a weak bialgebra together with a mapS:H →H, called the antipode, satisfying for allh∈H the conditions

(19) h₍₁₎S(h₍₂₎) = Π^L(h); S(h₍₁₎)h₍₂₎ = Π^R(h);

(20) S(h₍₁₎)h₍₂₎S(h₍₃₎) =S(h).

2. WEAK HOPF-GALOIS EXTENSIONS

LetH be a weak bialgebra, andAa rightH-comodule algebra. We then have a projection

g: A⊗H →A⊗H, g(a⊗h) =a1_[0]⊗h1_[1]. Observe that, for anya∈A,

ρ(a) =a_[0]⊗a_[1] =a_[0]1_[0]⊗a_[1]1_[1] =g(ρ(a))∈Im (g).

Lemma2.1. LetH be a weak bialgebra. Then C= Im (g) is anA-coring, with structure maps

b(a1_[0]⊗h1_[1])b = bab_[0]⊗hb_[1],

∆_C(1_[0]⊗h1_[1]) = (1_[0]⊗h₍₁₎1_[1])⊗A(1⊗h₍₂₎), ε_C(1_[0]⊗h1_[1]) = 1_[0]ε(h1_[1]).

ρ(1) = 1_[0]⊗1_[1] is a grouplike element in C.

Proof. This is a special case of a result in [4, Proposition 2.3]. The fact that ∆_C(1_[0]⊗h1_[1])∈ C ⊗AC follows from

(21) ∆_C(1_[0]⊗h1_[1]) = (1_[0]⊗h₍₁₎1_[1])⊗A((1_[0]⊗h₍₂₎1_[1]).

Indeed,

(1_[0]⊗h₍₁₎1_[1])⊗A((1_[0]⊗h₍₂₎1_[1])

= (1_[0]1_[0]⊗h₍₁₎1_[1]1_[1])⊗A(1⊗h₍₂₎1_[2])

= (1_[0]1_[0]⊗h₍₁₎1_[1]1_[1]1₍₁₎)⊗A(1⊗h₍₂₎1₍₂₎)

= (1_[0]⊗h₍₁₎1_[1]1₍₁₎)⊗A(1⊗h₍₂₎1₍₂₎)

= (1_[0]⊗h₍₁₎1₍₁₎1_[1])⊗A(1⊗h₍₂₎1₍₂₎) =ρ(a).

Taking h = 1 in (21) and (21), we see that ε_C(ρ(1)) = 1 and ∆_C(ρ(1)) = ρ(1)⊗Aρ(1).

We call (M, ρ) a relative right (A, H)-Hopf module if M is a right A- module, (M, ρ) is a rightH-comodule, and

(22) ρ(ma) =m_[0]a_[0]⊗m_[1]a_[1],

for allm ∈M and a∈A. M^H_A is the category of relative Hopf modules and rightA-linear H-colinear maps.

The following result is a special case of [4, Proposition 2.3]. Since it is essential in what follows, we give a self-contained proof.

Proposition2.2. The category M^H_A of relative Hopf modules is isomor- phic to the categoryM^C of right C-comodules over the coring C= Img.

Proof. Take (M,ρ)˜ ∈ M^C. Letρ be the composition ρ: M−→^ρ˜ M ⊗AC−→M ⊗A(A⊗H)−→^∼= M⊗H.

If

ρ(m) =˜

i

m_i⊗Aa_i1_[0]⊗h_i1_[1], then

ρ(m) =m_[0]⊗m_[1] =

i

m_ia_i1_[0]⊗h_i1_[1]. So, ˜ρ is determined byρ since

ρ(m) =˜

i

m_i⊗Aa_i1_[0]⊗h_i1_[1] =

i

m_i⊗Aa_i1_[0]1_[0]⊗h_i1_[1]1_[0]

(23)

=

i

m_ia_i1_[0]⊗A1_[0]⊗h_i1_[1]1_[0]=m_[0]⊗A1_[0]⊗m_[1]1_[1]. From the fact that ˜ρ is right A-linear, it follows that

ρ(ma) = ˜˜ ρ(m)a=m_[0]⊗A(1_[0]⊗m_[1]1_[1])a=m_[0]⊗Aa_[0]⊗m_[1]a_[1],

so,

ρ(ma) =m_[0]a_[0]⊗m_[1]a_[1].

From the fact that (M⊗Aε_C)(˜ρ(m)) =m and (23), it follows that m=m_[0]1_[0]⊗ε(m_[1]1_[1]) =m_[0]⊗ε(m_[1]).

For allm∈M, inM⊗A(A⊗H)⊗A(A⊗H) we have the equation (24) (M ⊗A∆_C)(˜ρ(m)) = (˜ρ⊗ C)(˜ρ(m)).

The left hand side of (24) is

m_[0][0]⊗A(1_[0]⊗m_[0][1]1_[1])⊗A(1_[0]⊗m_[1]1_[1]).

The image inM ⊗A(A⊗H⊗H) is

m_[0][0]⊗A(1_[0]1_[0]⊗m_[0][1]1_[1]1_[1]1₍₁₎⊗m_[1]1₍₂₎) =

= m_[0][0]⊗A(1_[0]⊗m_[0][1]1_[1]1₍₁₎⊗m_[1]1₍₂₎) while inM ⊗H⊗Hthis is

m_[0][0]1_[0]⊗m_[0][1]1_[1]1₍₁₎⊗m_[1]1₍₂₎ =m_[0][0]1_[0]⊗m_[0][1]1_[1]⊗m_[1]1_[2] =

=m_[0][0]1_[0][0]⊗m_[0][1]1_[0][1]⊗m_[1]1_[1] =m_[0][0]⊗m_[0][1]⊗m_[1]. The right hand side of (24) is

m_[0]⊗A(1_[0]⊗m_[1](1)1_[1])⊗A(1_[0]⊗m_[1](2)1_[1]).

The image inM ⊗A(A⊗H⊗H) is

m_[0]⊗A(1_[0]1_[0]⊗m_[1](1)1_[1]1_[1]1₍₁₎⊗m_[1](2)1₍₂₎) =

= m_[0]⊗A(1_[0]⊗m_[1](1)1_[1]1₍₁₎⊗m_[1](2)1₍₂₎) while inM ⊗H⊗H this is

m_[0]1_[0]⊗m_[1](1)1_[1]1₍₁₎⊗m_[1](2)1₍₂₎ =m_[0]1_[0]⊗m_[1](1)1_[1]⊗m_[1](2)1_[2] =

=m_[0]1_[0]⊗m_[1](1)1_[1](1)⊗m_[1](2)1_[1](2)=m_[0]⊗m_[1](1)⊗m_[1](2). It follows that

m_[0][0]⊗m_[0][1]⊗m_[1]=m_[0]⊗m_[1](1)⊗m_[1](2),

and ρ is coassociative. Conversely, given a relative Hopf module (M, ρ), we define ˜ρ: M →M⊗AC using (23). Straightforward computations show that (M,ρ)˜ ∈ M^C.

Let (M, ρ) be a relative Hopf module, and (M,ρ) the corresponding˜ C- comodule. Thenm∈M^coC if and only if

ρ(m) =˜ m⊗A1_[0]⊗1_[1]

if and only if

ρ(m) =m1_[0]⊗1_[1].

We conclude thatM^coC =M^coH, which is by definition M^coH ={m∈M |ρ(m) =m1_[0]⊗1_[1]}.

The grouplike element ρ(1) induces a right C-coaction ˜ρ on A given by ρ(a) = 1˜ ⊗A(1_[0]⊗1_[1])a= 1⊗A(a_[0]⊗a_[1]).

The correspondingH-coaction on A is the originalρ, that is, ρ(a) =a_[0]⊗a_[1].

Observe that

T =A^coH ={a∈A |a1_[0]⊗1_[1]}.

Leti:B →T be a ring morphism. We have seen in Section 1.1 that we have a pair of adjoint functors (F, G) defined by

F :MB → M^HA, F(N) =N ⊗BA, G:M^HA → MB, G(N) =N^coH. F(N) =N⊗BAis a relative Hopf module via

ρ(n⊗Ba) =n⊗Ba_[0]⊗a_[1]. The canonical map can : A⊗BA→ C takes the form (25) can(a⊗Bb) =a(1_[0]⊗1_[1])b=ab_[0]⊗b_[1].

We call A a weak H-Galois extension of A^coH if (C, ρ(1)) is a Galois coring, that is, can : A⊗A^coH A → C is an isomorphism. From Theorem 1.1, we immediately have the following result.

Proposition2.3. Let H be a weak bialgebra, and Aa rightH-comodule algebra. Then the following assertions are equivalent:

(1) A is A a weak H-Galois extension of T and faithfully flat as a left B-module;

(2) (F, G) is an equivalence andA is flat as a left T-module.

Our next aim is to compute^∗C.

Proposition 2.4. Let H be a weak bialgebra, A a right H-comodule algebra, and C= Im (g) the A-coring that we introduced above. Then

∗C =AHom(C, A)∼= Hom(H, A) =

={f ∈Hom(H, A) |f(h) = 1_[0]f(h1_[1]) for all h∈H}, with multiplication rule

(26) (f#g)(h) =f(h₍₂₎)_[0]g(h₍₁₎f(h₍₂₎)_[1]).

Proof. Define α: ^∗C →Hom(H, A) by

α(ϕ)(h) =ϕ(1_[0]⊗h1_[1]).

We have α(ϕ) =f ∈Hom(H, A) since 0 = α(ϕ)((1⊗h)(ρ(1)−ρ(1)²))

= α(ϕ)(1_[0]⊗h1_[1])−α(ϕ)(1_[0]1_[0]⊗h1_[1]1_[1])

= f(h)−1_[0]f(h1_[1]).

Now, defineβ : Hom(H, A)→^∗C by

β(f)(a1_[0]⊗h1_[1]) =af(h).

We have to show thatβ is well-defined. Assume that

ia_i1_[0]⊗h_i1_[1] = 0 in A⊗H. Then

i

a_i⊗h_i =

i

a_i⊗h_i−

i

a_i1_[0]⊗h_i1_[1], hence

i

a_if(h_i) =

i

a_if(h_i)−

i

a_i1_[0]f(h_i1_[1]) = 0.

A straightforward verification shows that α is the inverse of β. Using α andβ, we can transport the multiplication on ^∗C to Hom(H, A). This gives

(f#g)(h) = (ϕ#ψ)(1_[0] ⊗h1_[1]) =ψ((1_[0]⊗h₍₁₎1_[1])ϕ(1_[0]⊗h₍₂₎1_[1]) =

=ψ((1_[0]⊗h₍₁₎1_[1])f(h₍₂₎) =ψ(f(h₍₂₎)_[0]⊗h₍₁₎f(h₍₂₎)_[1]) =

=f(h₍₂₎)_[0]g(h₍₁₎f(h₍₂₎)_[1]), as needed.

The dual of the canonical map^∗can : ^∗C →AHom(A⊗BA, A) is given by

∗can(ϕ)(b⊗Aa) =ϕ(can(b⊗Aa)) =ϕ(ba_[0]⊗a_[1], and transports to^∗can : Hom(H, A)→BEnd(A)^op given by (27) ^∗can(f)(a) =^∗can(ϕ)(1⊗Ba) =a_[0]f(a_[1]).

Remarks 2.5. 1) We have a projection p : Hom(H, A) → Hom(H, A), p(f) = ˜f given by ˜f(h) = 1_[0]f(1_[1]).

2) The unit element of Hom(H, A) is not ε, but ˜ε=α(ε_C). This can be verified directly as follows:

(˜ε#g)(h) = 1_[0]ε(h₍₂₎1_[2])g(h₍₁₎1_[1]) = 1_[0]ε(h₍₂₎1₍₂₎)g(h₍₁₎1₍₁₎1_[1]) =

= 1_[0]ε(h₍₂₎g(h₍₁₎1_[1]) = 1_[0]g(h1_[1]) =g(h);

(f#˜ε)(h) = f(h₍₂₎)_[0]ε(h˜ ₍₁₎f(h₍₂₎)_[1]) =f(h₍₂₎)_[0]1_[0]ε(h₍₁₎f(h₍₂₎)_[1]1_[0]) =

= f(h₍₂₎)_[0]ε(h₍₁₎f(h₍₂₎)_[1])⁽¹⁸⁾= 1_[0]ε(h₍₁₎1_[1])f(h₍₂₎) =

(17)= 1_[0]f(h1_[1]) =f(h).

3) ^∗C is an A-bimodule, hence Hom(H, A) is also an A-bimodule. The structure is given by the formula

(28) (af b)(h) =a_[0]f(ha_[1])b.

Let us next compute the Morita context (T,Hom(H, A), A, Q, τ, µ) from Proposition 1.3. We have a ring morphismj=α◦i: A→^∗C →Hom(H, A), given by

j(a)(h) = 1_[0]ε(h1_[1])a.

Take ϕ ∈ Q = {ϕ ∈ ^∗C | c₍₁₎ϕ(c₍₂₎) = ϕ(c)ρ(1), for all c ∈ C}, and let α(ϕ) =q. Then

c₍₁₎ϕ(c₍₂₎) = (1_[0]⊗h₍₁₎1_[1])f(h₍₂₎) =f(h₍₂₎)_[0]⊗h₍₁₎f(h₍₂₎)_[1]

and

ϕ(c)ρ(1) =f(h)1_[0]⊗1_[1],

henceQis the subset of Hom(H, A) consisting of mapsf : H→A satisfying (29) f(h₍₂₎)_[0]⊗h₍₁₎f(h₍₂₎)_[1] =f(h)1_[0]⊗1_[1],

for allh∈H;Qis an (Hom(H, A), T)-bimodule: f ·q·a=f#q#j(a), for all f ∈Hom(H, A), q∈Q and a∈A.

Ais a (T,Hom(H, A))-bimodule; the left T-action is given by left multi- plication, and the right Hom(H, A))-action is given by a·q=a_[0]q(a_[1]). The connecting maps

τ : A⊗_Hom(H,A)Q→T andµ: Q⊗T A→Hom(H, A) are given by the formulas

τ(a⊗q) =a_[0]q(a_[1]),

µ(q⊗a) =q#j(a), that is, µ(q⊗a)(h) =q(h)a.

It follows from Proposition 1.2 thatτ is surjective if and only if there exists q∈Q such that 1_[0]q(1_[1]) = 1. Theorem 1.4 takes the following form.

Theorem 2.6. Let H be a finitely generated projective weak bialgebra, and A a right H-comodule algebra. Take a subring B of T = A^coH. Then, with notation as above, the following assertions are equivalent:

(1) • can is an isomorphism;

• A is faithfully flat as a left B-module.

(2) • ^∗can is an isomorphism;

• A is a left B-progenerator.

(3) • B =T;

• the Morita context (B,Hom(H, A), A, Q, τ, µ) is strict.

(4) • B =T;

• (F, G) is an equivalence of categories.

Proposition 2.7. Let H be a weak Hopf algebra. Then H is a weak H-Galois extension of H^coH =H^L.

Proof. Let us first show that H^coH = H^L. If h ∈ H^coH then ∆(h) = h1₍₁₎⊗1₍₂₎, hence h=ε(h1₍₁₎)1₍₂₎ = Π^R(h)∈H^L.

Conversely, ifh∈H^L thenh= Π^R(h) =ε(h1₍₁₎)1₍₂₎, so

∆(h) =ε(h1₍₁₎)1₍₂₎⊗1₍₃₎ =ε(h1₍₁₎)1₍₂₎1₍₁)⊗1₍₂)=h1₍₁₎⊗1₍₂₎, henceh∈H^coH.

Let us next show that can is invertible, with inverse can⁻¹(1₍₁₎⊗h1₍₂₎) = 1₍₁₎S(h₍₁₎1₍₂₎⊗H^Lh₍₂₎1₍₃₎.

can⁻¹(can(h⊗k)) = can⁻¹(hk₍₁₎⊗k₍₂₎) =hk₍₁₎S(k₍₂₎)⊗H^Lk₍₃₎

= hΠ^L(k₍₁₎)⊗H^Lk₍₂₎ =h⊗H^LΠ^L(k₍₁₎)k₍₂₎

= h⊗H^Lε(1₍₁₎k₍₁₎)1₍₂₎k₍₂₎ =h⊗H^L(k);

can(can⁻¹(1₍₁₎⊗h1₍₂₎)) = can(1₍₁₎S(h₍₁₎1₍₂₎⊗_HLh₍₂₎1₍₃₎)

= 1₍₁₎S(h₍₁₎1₍₂₎)h₍₂₎1₍₃₎⊗h₍₃₎1₍₄₎

= 1₍₁₎1₍₁)ε(h₍₁₎1₍₂₎1₍₂))⊗h₍₂₎1₍₃₎

= 1₍₁₎1₍₁)ε(h₍₁₎1₍₃₎)ε(1₍₂₎1₍₂))⊗h₍₂₎1₍₄₎

= 1₍₁₎1₍₁)ε(1₍₂₎1₍₂))⊗ε(h₍₁₎1₍₃₎)h₍₂₎1₍₄₎

= 1₍₁₎1₍₁)ε(1₍₂₎1₍₂))⊗εh1₍₃₎) = 1₍₁₎⊗h1₍₂₎. 3. GROUPOID GRADINGS

Recall that a groupoid G is a category in which every morphism is an isomorphism. In this section, we consider finite groupoids, i.e., groupoids with a finite number of objects. The set of objects ofGwill be denoted byG₀, and the set of morphisms by G₁. The identity morphism on x ∈ G₀ will also be

denoted byx. For σ: x→y in G₁, we write s(σ) =x andt(σ) =y,

respectively for the source and the target of σ. For everyx ∈G, G_x ={σ ∈ G|s(σ) =t(σ) =x}is a group.

Let Gbe a groupoid, and k a commutative ring. The groupoid algebra is the direct product

kG=

σ∈G₁

ku_σ,

with multiplication defined by the formula uσuτ =

u_στ if t(τ) =s(σ), 0 if t(τ)=s(σ).

The unit element is 1 =

x∈G₀ux;kG is a weak Hopf algebra, with comulti- plication, counit and antipode given by the formulas

∆(u_σ) =u_σ⊗u_σ, ε(u_σ) = 1 andS(u_σ) =u_σ−1. Using the formula

∆(1) =

x∈G₀

u_x⊗u_x,

we can compute that Π^L: kG→kGis given by the formula Π^L(u_σ) =

x∈G₀

ε(u_xu_σ) =u_t(σ), hence

(kG)^L=

x∈G₀

ku_x.

Let k be a commutative ring. A G-graded k-algebra is a k-algebra A together with a direct sum decomposition

A=

σ∈G₁

A_σ,

such that

(30) A_σA_τ

⊂A_στ if t(τ) =s(σ),

= 0 if t(τ)=s(σ).

and

(31) 1A∈

x∈G₀

A_x.

Proposition 3.1. Let Gbe a finite groupoid and ka commutative ring.

We have an isomorphism between the categories ofkG-comodule algebras and G-graded k-algebras.

Proof. Let (A, ρ) be akG-comodule algebra, and defineρ: A→A⊗kG by

A_σ ={a∈A |ρ(a) =a⊗u_σ}.

From the fact thatA⊗kGis a free left A-module with basis {u_σ |σ ∈G₁}, it follows thatA_σ∩A_τ ={0} ifσ=τ.

Fora∈A, we can write

ρ(a) =

σ∈G₁

aσ⊗uσ. It follows from the coassociativity ofρ that

σ∈G₁

ρ(a_σ)⊗u_σ =

σ∈G₁

a_σ⊗u_σ⊗u_σ, henceρ(a_σ)a_σ⊗u_σ, soa_σ ∈A_σ and

a=

σ∈G₁

aσε(uσ) =

σ∈G₁

aσ ∈σ∈G₁ Aσ. Ifa∈A_σ andb∈A_τ, thenρ(ab) =ab⊗u_σu_τ, and (30) follows.

(16) tells us that ρ(1_A) ∈ A⊗(kG)^L =

x∈G₀A⊗u_x, hence we can write

ρ(1A) =

x∈G₀

1x⊗ux

and

1_A=

x∈G₀

1_x∈

x∈G₀

A_x.

Conversely, let A be a G-graded algebra. For a ∈ A, let a_σ be the projection ofA on A_σ. Then define ρ(a) =

σ∈G₁a_σ⊗u_σ; (16) then follows immediately from (31).

Proposition 3.2. Let A be a G-graded algebra, and take x∈G₀. Then

σ∈G_xAσ is a Gx-graded algebra, with unit 1x. In particular, Ax is a k- algebra with unit 1_x.

Proof. Ifa∈Aσ with t(σ) =x, then a⊗σ=ρ(a) =ρ(1_Aa) =

y∈G₀

1_ya⊗u_yu_σ = 1_xa⊗σ,

hencea= 1_xa. In a similar way, we can prove that a=a1_x if s(a) =x. The result then follows easily.

Let A be a G-graded algebra. A G-graded right A-module is a right A-module together with a direct sum decomposition

M =

σ∈G₁

M_σ such that

M_σA_τ

⊂M_στ if t(τ) =s(σ),

= 0 if t(τ)=s(σ).

A rightA-linear mapf : M →N between twoG-graded rightA-modules is called graded if f(Mσ) ⊂ Nσ, for all σ ∈ G₁. The category of G-graded right A-modules and graded A-linear maps is denoted by M^GA. The proof of the next result is then similar to the proof of Proposition 3.1.

Proposition3.3. LetGbe a groupoid, andAa G-graded algebra. Then the categories M^G_A and M^kg_A are isomorphic.

Ifm∈Mσ, withs(σ) =x, then ρ(m) =m⊗σ =ρ(m1) =

y∈G₀

m1y ⊗σx=m1x⊗σ, hencem1x =m.

LetM ∈ M^G_A ∼=M^kg_A. Thenm∈M^cokG if and only if ρ(m) =

σ∈G₁

mσ⊗σ =

x∈G₀

m1x⊗x, if and only ifm∈

x∈G₀M_x. We conclude that

(32) M^cokG=

x∈G₀

M_x. In particular,

T =A^cokG =

x∈G₀

A_x. If N ∈ MT, then N =

x∈GN_x, with N_x =N1_x ∈ MA_x. We have a pair of adjoint functors

F =− ⊗T A: MT → M^G_A, G= (−)^cokG: M^G_A→ MT, with unit and counit given by the formulas

ν_N : N →(N ⊗T A)^cokG, ν_N(n) =

x∈G₀

n_x⊗T 1_x, ζM : M^cokG⊗T A→M, ζM(m⊗T a) =ma.

A G-graded k-algebra is called strongly graded if AσAτ =Aστ if t(τ) =s(σ).

Proposition 3.4. Let A be a G-graded k-algebra. With notation as above, ν_N : N →(N ⊗T A)^cokG is bijective for every N ∈ MT.

Proof. First observe that

(N ⊗T A)σ =N⊗T Aσ =N_t(σ)⊗T Aσ, hence

(N⊗T A)^cokG=

x∈G₀

(N⊗T A)_x =

x∈G₀

N_x⊗T A_x. The map

f : (N ⊗T A)^cokG→N, f(n_x⊗a_x) =n_xa_x is the inverse ofν_N: it is obvious that f◦ν_N =N. We also have

(ν_N ◦f)(n_x⊗T a_x) =n_xa_x⊗T 1x=n_x⊗T a_x.

Let A be an algebra graded by a group G. It is a classical result from graded ring theory (see e.g. [16]) that A is strongly graded if and only if the categories ofG-gradedA-modules andA₁-modules are equivalent. This is also equivalent toA being a kG-Galois extension ofA_e. We now present the groupoid version of this result.

Theorem 3.5. Let Gbe a groupoid, and A a G-graded k-algebra. Then the following assertions are equivalent.

(1) A is strongly graded;

(2) (F, G) is a category equivalence;

(3) (A⊗kG, ρ(1) =

x∈G₀ex⊗x) is a Galois coring;

(4) can : A⊗T A→A⊗kG, can(a⊗b) =

σ∈G₁ab_σ⊗σ is surjective.

Proof. 1)⇒2). On account of Proposition 3.4, we only have to show thatζM is bijective, for every graded A-moduleM.

Takem∈M_σ, withs(σ) =x,t(σ) =y. Then σσ⁻¹ =x, and there exist a_i ∈A_σ−1,a_i ∈A_σ such that

ia_ia_i= 1x. We havema_i ∈M_σA_σ−1 ⊂M_y ⊂ M^cokG, and

ζ_M

i

ma_i⊗a_i

=

i

ma_ia_i =m, and it follows thatζM is surjective.

Take

m_j =

x∈G₀

m_j,x ∈

x∈G₀

M_x and c_j ∈A.

Assume

ζ_M

j

m_j⊗T c_j

=

j

m_jc_j = 0, and takeσ∈G₁ withs(σ) =xand t(σ) =y. Then

0 =

j

mjcj

σ =

j

mjcj,σ, hence

j

mj⊗T cj,σ=

i,j

mj⊗T cj,σa_iai =

i,j

mjcj,σa_i⊗T ai= 0,

and

j

m_j⊗T c_j =

σ∈G₁

j

m_j⊗T c_j,σ = 0, and it follows thatζ_M is injective.

2)⇒3) follows from the observation made before Theorem 1.1 (see [8, Proposition 3.1]).

3)⇒4) is trivial.

4)⇒1). Take σ, τ ∈G₁ such that s(σ) = t(τ), and c ∈A_στ. Since can is surjective, there exist homogeneous a_i, b_i ∈Asuch that

can

i

a_i⊗b_i

=

i,j

a_ib_j⊗deg(b_j) =c⊗τ.

On the left hand side, we can delete all the terms for which deg(b_j) =τ. So we find

i,j

a_ib_j⊗τ =c⊗τ

and

i,j

a_ib_j =c.

On the left hand side, we can now delete all terms of degree different fromστ, since the degree of the right hand side isστ. This means that we can delete all terms for which deg(a_i)=σ. So we find that

i,ja_ib_j =c, with deg(a_i) =σ, and deg(b_i) =τ, and A_σA_τ =A_στ.