galois theory for hopf categories applied …homepages.vub.ac.be/~scaenepe/galoisbis4.pdfgalois...

GALOIS THEORY FOR HOPF CATEGORIES APPLIED TO

PARTIAL ACTIONS BY GROUPOIDS

S. CAENEPEEL AND T. FIEREMANS

Abstract. We define rings and corings over diagonal linear categories,whose underlying class of objects is a set X. To a coring with a fixedgrouplike matrix, we associate a pair of adjoint functors, and investigatewhen this forms a pair of inverse equivalences. A necessary condition isthat a certain family of canonical maps consists of isomorphisms; in thiscase the coring is called a Galois coring. To find sufficient conditions wehave to impose some finiteness conditions; a first possible condition isthat the set of objects is finite. In this case, we can put the coring inquestion in packed form, and reduce the question to a similar questionabout corings over an algebra, that is the case when X is a singleton. ForX infinite, we have to assume that the pieces of the diagonal algebraare finitely generated projective over the algebra of coinvariants. Weapply our results to groupoids acting partially on algebras (or diagonallinear categories), as introduced by Bagio and Paques [2]. We associatea coring to such a partial action, and study when it is a Galois coring.We relate this to the Galois theory introduced by Bagio and Paques.The two notions are equivalent when the diagonal category involved iscommutative. In this situation, we present several equivalent conditionsfor the Galois property.

Introduction

In [3], enriched categorical arguments were applied in order to introduceHopf categories as multi-objected versions of Hopf algebras. Many resultson Hopf algebras can be generalized to Hopf categories, for example thefundamental theorem on Hopf algebras, see [3, Sec. 10]. This was the start-ing point for the development of a Galois theory for Hopf categories in [10].This theory works well, but some of its aspects remain unsatisfactory, as wasexplained in the final remark 8.3 in [10]. In order to explain this, we go backto Hopf-Galois theory as developed by Chase and Sweedler in [12]. Basicexamples of Hopf algebras are group algebras and their duals (in the casewhere the underlying group is finite). Chase and Sweedler consider coactionsrather than actions, an approach that appears to be more general. ThenHopf-Galois extensions over a group algebra correspond to strongly gradedrings. In order to recover classical Galois theory of commutative rings, as

2010 Mathematics Subject Classification. 16T05.Key words and phrases. Enriched category, Hopf category, Descent theory, Hopf-Galois

extension.

1

2 S. CAENEPEEL AND T. FIEREMANS

originally introduced by Auslander and Goldman in [1], one observes thatactions by a finite group, equivalently by a finite group algebra, correspondto coactions by the dual of that group algebra, and then it turns out thatHopf-Galois extensions over the dual of a finite group algebra correspondto classical Galois extensions. In the multi-object situation, the situationis more involved since the notion of Hopf category is not self-dual, see [3,Sec. 5]. Examples of Hopf categories arise from groupoids (viewed as cate-gories in which every morphism is invertible). The theory in [10] works wellwith respect to groupoid gradings, but groupoid actions are not covered. Apossible way out could be to develop the Galois theory over the oppositecategory of vector spaces; this is possible, but gives actions by groupoidson categories with a comultiplication structure instead of a multiplicationstructures.Bagio and Paques ([2], see also [21]) introduced Galois theory for finitegroupoids acting (partially) on algebras. The aim of the present paper is topresent an alternative version of Galois theory that can be matched with thetheory of Bagio and Paques. Essentially our paper is divided into two parts.In Sections 2-5, we develop the general theory; in Section 6, we apply thisto partial actions by locally finite groupoids, and recover the Galois theoryof [2].An essential difference with the approach in our previous article [10] is thatwe are dealing with different types of objects: the Galois categories con-sidered in [10] are linear categories with an additional structure, while theones from the present paper are diagonal linear categories with an additionalstructure, creating a difference in dimension. The underlying arguments areparallel, in the sense that the strategy is to determine when a certain adjointpair of functors provides a pair of inverse equivalences, that can be refor-mulated as a descent type result. Actually, the storyline of both theoriesfollows the theory of Galois corings. Following an observation by Takeuchi,Brzezinski [6] observed that the language of corings provided a unified andsimplified approach to several notions related to Hopf algebra. In particular,various generalizations of Galois theory could be unified using the theory ofcorings. A survey was presented by the first author in [8], which served asa kind of guideline for [10] and the present paper. A major difference isthat there exists - as far as we know - no appropriate notion of coring thatexplains the results in [10]. In the present paper, the corings make theircomeback. The line-up of the first part of the paper is as follows. The cate-gories that are considered are always small, in the sense that the underlyingclass of objects X is a set. Corings over a diagonal k-linear category andgrouplike matrices are introduced in Section 2. To a coring with a fixedgrouplike element, we can associate a pair of adjoint functors. Then we findsome necessary conditions for this pair to be a pair of inverse equivalences,see Proposition 3.2. One condition concerns the coinvariants, the other onestates that a family of maps, called the canonical maps, need to be isomor-phisms. If these conditions are satisfied, then we call our coring a Galois

GALOIS THEORY AND PARTIAL ACTIONS 3

coring. The canonical maps determine a morphism between the coring inquestion and the so-called Sweedler canonical coring. Thus the problem canbe reduced to the situation where we work with the Sweedler coring. Hereit turns out that some finiteness assumption have to be made, and here wehave two possibilities. In Section 3, we consider the situation where everycomponent of the diagonal k-linear category is finitely generated and pro-jective over the coinvariants, and where a finite number of them generatethe category of modules over the coinvariants. Then we can provide explicitformulas for the inverses of the unit and couint of the adjunction. In Sec-tion 4, we consider the case where X is finite. Now we can reduce to thesituation studied in [6, 8], by putting the diagonal category and the coringin packed form.In Section 6, we recall the definition of (unital) partial action by a locallyfinite groupoid on an algebra, and associate a diagonal k-linear category anda coring with a grouplike matrix to it. Then we investigate when this coringis Galois. In the situation where the groupoid is finite, we can compare thisto the notion of partial Galois extension from [2]. It turns out that the twonotions coincide in the case where the components of the diagonal algebraare commutative, but that our notion is stronger in the general situation.This phenomenon is not new, as it finds its origins in the Galois theory ofcommutative ring extensions initiated by Auslander and Goldman in [1]. In[11], six equivalent definition of Galois extensions of commutative rings arepresented (see also [14] for a concise treatment). These conditions are nolonger equivalent (some of them no longer make sense) if one considers themore general situations in either of the following directions: the group canbe replaced by a Hopf algebra, or the rings can be taken non-commutative.Bagio and Pacques take a different condition for their generalization, whichcomes down to the surjectivity of the canonical map, while we insist on thebijectivity, which is the right approach in our opinion, as it leads to Galoisdescent theory.As we have remarked, the six equivalent definitions of Galois extension from[11], have no counterpart in the more general versions of Galois theory thatappeared afterwards. However, if one considers partial actions of locallyfinite groupoids on commutative algebras, then we can generalize the mainresult [11]. This is the content of our main Theorem 6.5.Our approach is more general than the one in [2], in the sense that we con-sider partial actions by locally finite groupoids, while the groupoids in [2] arefinite. This means that the situation considered in [2] falls under Section 4,where X is considered to be finite. This means that the Galois theory of [2]reduces to a special case of the Galois theory for corings studied in [6, 8], ofcourse modulo the remark made above about the surjectivity of the canon-ical map. This was already observed in [9], where partial Galois theory forgroups was explained using Galois corings.


1. Preliminary results

1.1. V-categories. Let V be a monoidal category. From [4, Sec. 6.2], werecall the notion of V-category. In particular, in the case where V = Mk,the category of vector spaces over a field k (or, more generally, the categoryof modules over a commutative ring k), a V-category is a k-linear category.In [3], the notion of (semi-)Hopf V-category is introduced. In this paper, wewill work over V =Mk and over V =Mop

k . A Hopf Mk-category is calleda k-linear Hopf category, while a Hopf Mop

k -category is called a k-linearHopf opcategory. Let us specify the definitions from [3] to this particularsituation.Let A = (Ax)x∈X be a k-linear category, and let X be the class of objects inA. For x, y ∈ X, we write Axy for the k-module of morphisms from y to x.For all x, y, z ∈ X, we then have the composition maps mxyz : Axy⊗Ayz →Axz, mxyz(a ⊗ b) = ab, for all a ∈ Axy and b ∈ Ayz. The unit element ofAxx is denoted by 1x.Let A and B be k-linear categories with the same underlying class of objectsX. A k-linear functor f : A → B that is the identity on X is called a k-linear X-functor: for all x, y ∈ X, fxy : Axy → Bxy is a k-linear mappreserving multiplication and unit.For a class X, we introduce the category Mk(X). An object is a family ofobjectsM = (Mxy)x,y∈X inMk indexed byX×X. A morphism ϕ : M → Nconsists of a family of k-linear maps ϕxy : Mxy → Nxy indexed by X ×X.We will also need the category Dk(X). Objects are families of k-modulesN = (Nx)x∈X indexed by X, and a morphism N → N ′ consists of a familyof k-linear maps Nx → N ′x. Dk(X) can be viewed as the full subcategory ofMk(X), consisting of objects M satisfying Mxy = 0 if x 6= y.Let A be a k-linear category. A diagonal right A-module is an object M inDk(X) together with a family of k-linear maps

ψxy : Mx ⊗Axy →My, ψxy(m⊗ a) = ma

such that the following associativity and unit conditions hold: (ma)b =m(ab); m1y = m, for all m ∈Mx, a ∈ Axy and b ∈ Ayz.Let M and N be right A-modules. A morphism ϕ : M → N in Dk(X)is called right A-linear if ϕy(ma) = ϕx(m)a, for all m ∈ Mx and a ∈ Axy.The category of diagonal right A-modules and right A-linear morphisms isdenoted by Dk(X)A.Dk(X) is a symmetric monoidal category, and an algebra B in Dk(X) con-sists of a family of k-algebras (Bx)x∈X indexed by X. We can consider Bas a k-linear category: Bxy = {0} if x 6= y and Bxx = Bx. B is then calleda diagonal k-linear category. We can consider the category Dk(X)B.Notation: Take Y,Z ⊂ X finite. For M ∈ Dk(X) and N ∈Mk(X), let

MY = ⊕y∈YMy ; NY Z = ⊕y∈Y ⊕z∈Z Nyz.


1.2. Bimodules. Let A and B be rings, and consider M ∈ AMB. Then∗M = AHom(M,A) ∈ BMA, with

(1) (b · f · a)(m) = f(mb)a,

for b ∈ B, a ∈ A, f ∈ ∗M and m ∈ M . This construction is functorial: ifϕ : M → M ′ in AMB, then we have ∗ϕ : ∗M ′ → ∗M in BMA, given by∗ϕ(f) = f ◦ ϕ.Assume that M ∈ AM is finitely generated projective, with finite dual basisf (·) ⊗A m(·) ∈ ∗M ⊗A M . This is a formal notation, where summation isimplicitly understood. The upper dots will be replaced by indices whereneeded in the sequel. This means that

(2) m = f (·)(m)m(·) and f = f (·) · f(m(·)),

for all m ∈M and f ∈ ∗M . We also have that

(3) b · f (·) ⊗A m(·) = f (·) ⊗A m(·)b,

for all b ∈ B. In a similar way, for N ∈ BMA, we have that N∗ =HomA(N,A) ∈ AMB, with action (a · h · b)(n) = ah(bn). In particular,(∗M)∗ ∈ AMB, and we have an (A,B)-bimodule map

(4) i : M → (∗M)∗, i(m)(f) = f(m).

If M is finitely generated projective as a left A-module, then i is an isomor-phism of bimodules, with inverse given by the formula i−1(h) = h(f (·))m(·).

2. Corings and Rings

2.1. Corings and comodules. Let k be a commutative ring, let X be aclass, and let A be a diagonal k-linear category with underlying class X. AnA-coring C is an object of Mk(X) with the following additional structure.

• Every Cxy is an (Ax, Ay)-bimodule;• we have bimodule maps ∆xyz : Cxz → Cxy⊗AyCyz and εx : Cxx → Ax,

called the comultiplication and counit maps satisfying the coassocia-tivity and counit conditions

∆xuz(c(1,x,z))⊗Az c(2,z,y) = c(1,x,u) ⊗Au ∆u,z,y(c(2,u,y));(5)

c = εx(c(1,x,x))c(2,x,y) = c(1,x,y)εy(c(2,y,y)),(6)

for c ∈ Cxy where we use the Sweedler-type notation ∆xyz(c) = c(1,x,y) ⊗Ay

c(2,y,z), for x, y, z ∈ X and c ∈ Cxz.A right C-comodule M is an object M ∈ Dk(X)A, together with a collectionof maps ρxy : My → Mx ⊗Ax Cxy satisfying the following properties: ρxy isright Ay-linear, and

ρxz(m[0,z])⊗Az m[1,z,y] = m[0,x] ⊗Ax ∆x,z,y(m[1,x,y])(7)

m = m[0,y]εy(m[1,y,y]),(8)

for all m ∈ My, where we use the Sweedler notation ρxy(m) = m[0,x] ⊗Ax

m[1,x,y]. A morphism between right C-comodules M and N is a morphism


f : M → N in Dk(X)A that preserves the right C-coaction, in the sensethat (fx ⊗Ax Cxy) ◦ ρxy = ρxy ◦ fy, for all x, y ∈ X. The category of rightC-comodules is denoted by Dk(X)C .

2.2. grouplike matrices. Let C be an A-coring. A grouplike matrix g is afamily of elements gxy ∈ Cxy satisfying

(9) ∆xyz(gxz) = gxy ⊗Ay gyz and εx(gxx) = 1x,

for all x, y, z ∈ X.

Proposition 2.1. For an A-coring C, there is a bijective correspondencebetween grouplike matrices and right C-coactions on A.

Proof. Suppose that A ∈ Dk(X)C , with coaction ρxy : Ay → Ax ⊗Ax Cxy ∼=Cxy. Then gxy = ρxy(1y) satisfies (9), and g is a grouplike matrix.Conversely, if g is a grouplike matrix, then ρxy : Ay → Ax ⊗Ax Cxy ∼= Cxy isdefined as follows: ρxy(a) = 1x ⊗Ax gxyay

∼= gxyay. It is clear that the twoconstructions are inverses. �

2.3. Coinvariants. Assume that X is a (possibly infinite) set. Let (C, g)

be an A-coring, with a fixed grouplike matrix g, and let ρAxy be the corre-

sponding C-coaction on A. The coinvariants of M ∈ Dk(X)C are defined bythe formula

(10) M coC = {(mz)z ∈∏z∈X

Mz | ρxy(my) = mx ⊗Ax gxy, for all x, y ∈ X}.

In particular, it is easy to see that

(11) AcoC = {(bz)z ∈∏z∈X

Az | gxyby = bxgxy, for all x, y ∈ X}.

The proof of Proposition 2.2 is straightforward, and is left to the reader.

Proposition 2.2. Let X be a set, let A be a diagonal k-category, and let(C, g) be an A-coring, with a fixed grouplike matrix g. Then AcoC is a k-

algebra. If M is a right C-comodule, then M coC is a right AcoC-module. Thiscreates the coinvariants functor G : Dk(X)C →MAcoC .

2.4. A pair of adjoint functors. Let B be a k-algebra, and let i : B →AcoC be an algebra morphism. We have algebra morphisms

Bi // AcoC ⊂ //

∏z∈X Az

py // Ay

and it follows that every Ay is a B-bimodule via restriction of scalars. Thecomposition

Dk(X)CG //MAcoC //MB

is still denoted by G.

Proposition 2.3. Let X be a set, and let (C, g) be an A-coring, with a fixedgrouplike matrix g. Then G : Dk(X)C →MB has a left adjoint F .


Proof. For N ∈MB, let F (N) be given by F (N)x = N⊗BAx, with coactionρxy : N ⊗B Ay → N ⊗B Ax ⊗Ax Cxy ∼= N ⊗B Cxy, defined by the formulaρxy(n⊗B a) = n⊗B gxyay, for all x, y ∈ X. The unit and the counit of theadjunction are given by the formulas

(12)ηN : N → GF (N) = F (N)coC , ηN (n) = (n⊗B 1x)xεxM : FG(M)x = M coC ⊗B Ax →Mx, ε

xM ((mz)z ⊗B a) = mxa

�

2.5. Rings and modules. An A-ring A is an object of Mk(X) with thefollowing additional structure. Every Axy is an (Ax, Ay)-bimodule, and wehave bimodule maps mxyz : Axy ⊗Ay Ayz → Axz and ηx : Ax → Axxsatisfying the obvious associativity and unit conditions. For a ∈ Axy andb ∈ Ayz, we write ab = mxyz(a ⊗Ay b). A right A-module consists of M ∈Dk(X)A together with maps ψxy : Mx⊗AyAxy →My, satisfying the obviousassociativity and unit conditions. The category of right A-modules andmorphisms in Dk(X)A preserving the right A-action is denoted by Dk(X)A.Let C be an A-coring. ∗C = A is the A-ring given by the following data:

Ayx = ∗Cxy = AxHom(Cxy, Ax) ∈ AyMAx .

For g ∈ Azy, f ∈ Ayx, we have g#f ∈ Azx given by the formula

(13) (g#f)(c) = f(c(1,x,y)g(c(2,y,z))),

for c ∈ Cxz. It is straightforward to show that this establishes maps

mzyx : Azy ⊗Ay Ayx → Azx,satisfying the necessary associativity conditions. The unit maps are givenby putting ηx(1x) = εx ∈ Axx, so we obtain that A is an A-ring.Now we assume that every Cxy is finitely generated projective as a left Ax-module, with finite dual basis

f (yx) ⊗Ax c(xy) ∈ Ayx ⊗Ax Cxy.

The comultiplication on C can be recovered from the multiplication on A.Indeed, for c ∈ Cxz, we have

∆xyz(c) = c(1,x,y) ⊗Ay c(2,y,z)(2)= f (yx)(c(1,x,y))c

(xy) ⊗Ay f(zy)(c(2,y,z))c

(yz)

(3)= f (yx)(c(1,x,y))c

(xy)f (zy)(c(2,y,z))⊗Ay c(yz)

= (f (zy)(c(2,y,z)) · f (yx))(c(1,x,y))c(xy) ⊗Ay c

(yz)

= f (yx)(c(1,x,y)f

(zy)(c(2,y,z)))c(xy) ⊗Ay c

(yz)

= (f (zy)#f (yx))(c)c(xy) ⊗Ay c(yz)

Proposition 2.4. Let C be an A-coring, and assume that every Cxy is finitelygenerated and projective as a left Ax-module. Let ∗C = A be the correspond-ing A-ring. Then the categories Dk(X)C and Dk(X)A are isomorphic.


Proof. Take M ∈ Dk(X)C . M is a right A-module, under the followingaction

m · f = m[0,x]f(m[1,x,y]),

for m ∈My and f ∈ Ayx.Conversely, M ∈ Dk(X)A is a right C-cocomodule, with coaction given by

ρxy(m) = m · f (yx) ⊗Ax c(xy),

for all m ∈My. Further details are left to the reader. �

Let A be an A-ring. A character family consists of a family χ = χxy ∈A∗yx (that is, χxy : Ayx → Ax is right Ax-linear), such that the followingconditions hold,

(14) χxz(g#f) = χxy(χyz(g)f),

for g ∈ Azy and f ∈ Azx, and

(15) χxx(ηx(1x)) = 1x.

Proposition 2.5. Let C be an A-coring, and assume that every Cxy is finitelygenerated projective as a left Ax-module. There is a bijective correspondencebetween grouplike matrices in C and grouplike character families in A = ∗C.

Proof. For a grouplike matrix g, let χ be given by the formula χxy = i(gxy),with i as in (4). We leave it to the reader to show that χ is a grouplikecharacter family. i is invertible because of the assumption that Cxy is finitelygenerated projective, and this leads to the converse construction. �

Let Let (A, χ) be an A-ring with a character family. The invariants ofM ∈ Dk(X)A are defined as

MA = {(mz)z ∈∏z∈X

Mz |my ·f = mxχyx(f), for all x, y ∈ X and f ∈ Ayx}.

Proposition 2.6. Let C be an A-coring, and assume that Cxy is finitelygenerated projective as a left Ax-module, for all x, y ∈ X, and let ∗C = A bethe corresponding A-ring. Let g be a grouplike matrix, with corresponding

character family χ. For M ∈ Dk(X)C ∼= Dk(X)A, we have that

MA = M coC .

Proof. Take (mz)z ∈M coC . For all x, y ∈ X, we have that

ρxy(my) = mx ⊗Ax gxy.

Consequently, we have, for all f ∈ Ayx that

my · f = my[0,x]f(my[1,x,y]) = mxf(gxy) = mxχxy(f),

and it follows that (mz)z ∈ MA. The proof of the converse is left to thereader. �


2.6. Examples. In the examples below, we always work over a set X.

Example 2.7. Sweedler canonical coring Let A be a diagonal k-linearcategory, and let B be a k-algebra, such that we have an algebra morphismix : B → Ax, for every x ∈ X. The Sweedler canonical coring D is given bythe data: Dxy = Ax ⊗B Ay, and

∆xyz : Dxz = Ax ⊗B Az → Dxy ⊗Ay Dyz ∼= Ax ⊗B Ay ⊗B Az,∆xyz(a⊗B a′) = a⊗B 1y ⊗B a′;

εx : Dxx = Ax ⊗B Ax → Ax, εx(a⊗B a′) = aa′.

It is obvious that 1x ⊗B 1y is a grouplike matrix of D.An object M ∈Mk(X)D will also be called a descent datum. It consists ofM = (Mx)x ∈ Dk(X)A together with a collection of right Ay-linear k-linearmaps ρxy : My → Mx ⊗Ax Dxy ∼= Mx ⊗B Ay, ρxy(m) = m<0,x> ⊗B m<1,y>

in Dk(X)A satisfying the following coassociativity and counit conditions

ρxy(m<0,y>)⊗B m<1,z> = m<0,x> ⊗B 1y ⊗B m<1,z>;(16)

m<0,z>m<1,z> = m,(17)

for all m ∈ Mz. The fact that ρxy is right Ay-linear is expressed by theformula

(18) ρxy(ma) = m<0,x> ⊗B m<1,y>a,

for all m ∈My and a ∈ Ay. The coinvariants of M ∈Mk(X)D are given bythe formula

(19) M coD = {(mz)z ∈∏z∈X

Mz | ρxy(my) = mx ⊗B 1y, for all x, y ∈ X}.

In particular,

AcoD = {(bz)z ∈∏z∈X

Az | 1x ⊗B by = bx ⊗B 1y, for all x, y ∈ X}.

The induction functor K : MB →Mk(X)D is given by the following data:K(N)x = N ⊗B Ax, with coaction ρxy : N ⊗B Ay → N ⊗B Ax ⊗B Ay,ρxy(n⊗B a) = n⊗B 1x⊗B a. The unit ν and the counit ζ of the adjunction(K,R = (−)coD) between MB and Mk(X)D are given by the formulas

νN : N → K(N)coD, νN (n)y = n⊗B 1y;ζxM : M coD ⊗B Ax →Mx, ζ

zM ((mz)z ⊗B a) = mxa.

The Sweedler canonical coring plays a central role in the development ofthe theory: given a grouplike matrix g in an arbitrary coring, we have aunique morphisms of corings from the Sweedler coring to the arbitary coring,mapping 1x ⊗B 1y to gxy.

Proposition 2.8. Let (C, g) be an A-coring together with a fixed grouplikematrix g. The collection of maps

canxy : Dxy = Ax ⊗B Ay → Cxy, canxy(a⊗B b) = agxyb

define a morphism of A-corings can : D → C.


Example 2.9. Diagonal k-linear K-comodule categories. Let K bea k-linear (semi-)Hopf opcategory. We have comultiplication maps δxyz :Kxz → Kxy ⊗Kyz and counit maps εx : Kxx → k, and every Kxy is a k-algebra. A diagonal k-linear category A is called a k-linear right K-comodulecategory if we have k-algebra maps

ρxy : Ay → Ax ⊗Kxy, ρxy(a) = a[0,x] ⊗ a[1,x,y]

such that the following coassociativity and counit properties hold:

(Ax ⊗ δxzy) ◦ ρxy = (ρxz ⊗Kzy) ◦ ρzy ; (Ay ◦ εy) ◦ ρyy = Ay.

We define an A-coring C as follows. Cxy = Ax⊗Kxy is an (Ax, Ay)-bimodule,with left Ax-action and right Ay-action given by

a′(a⊗ k)a′′ = a′aa′′[0,x] ⊗ ka′′[1,x,y].

The comultiplication and counit are given by the formulas

∆xyz = Ax ⊗ δxyz : Cxz = Ax ⊗Kxz → Cxy ⊗Ay Cyz ∼= Ax ⊗Kxy ⊗Kyz,∆xyz(a⊗ k) = a⊗ k(1,x,y) ⊗ k(2,y,z);

εx = Ax ⊗ εx : Cxx = Ax ⊗Kxx → Ax, εx(a⊗ k) = aεx(k).

(gxy =)1x ⊗ 1xy)x,y∈X is a grouplike matrix of C.

Example 2.10. Enwining Structures Let A be a diagonal algebra, andlet C be a k-linear opcategory, that is, a category enriched in the oppositeof the category of modules over k. We have comultiplication maps δxyz :Cxz → Cxy ⊗ Cyz and counit maps εx : Cxx → k. An entwining between Aand C consists of a collection ψ of k-linear maps

ψxy : Cxy ⊗Ay → Ax ⊗ Cxy, ψxy(c⊗ a) = aψ ⊗ cψ = aΨ ⊗ cΨ,

such that the following formulas hold:

ψxy(c⊗ ab) = aψbΨ ⊗ cψΨ, for all c ∈ Cxy and a, b ∈ Ay;aψδxzy(c) = aψΨ ⊗ cΨ

(1,x,z) ⊗ cψ(2,z,y), for all c ∈ Cxy and a ∈ Ay;

ψxy(c⊗ 1y) = 1x ⊗ c, for all c ∈ Cxy;aψεx(cψ) = εx(a)c, for all c ∈ Cxx and a ∈ Ax.

(A,C, ψ) is called an entwining structure. To an entwining structure (A,C, ψ),we can associate an A-coring C. Cxy = Ax ⊗ Cxy, with (Ax, Ay)-bimodulestructure given by the formula

a′(a⊗ c)a′′ = a′aa′′ψ ⊗ cψ,

with a′, a ∈ Ax, a′′ ∈ Ay and c ∈ Cxy. The comultiplication and counit aregiven by the formulas

∆xyz = Ax ⊗ δxyz : Cxz = Ax ⊗ Cxz → Cxy ⊗Ay Cyz ∼= Ax ⊗ Cxy ⊗ Cyz,εx = Ax ⊗ εx : Cxx = Ax ⊗Kxx → Ax.


3. Galois corings

Our aim is to study when the adjoint pair (F,G) from Proposition 2.3 isa pair of inverse equivalences. In this Section, we present some necessaryconditions. In Sections 4 and 5, we will present sufficient conditions assum-ing an appropriate finiteness condition. Let (C, g) be a coring with a fixed

grouplike matrix. We fix x ∈ X, and consider M ∈ Dk(X)C given by thefollowing data: My = Cxy ∈MAy , with coaction

ρyz = ∆xyz : Mz →My ⊗Ay Cyz.

Lemma 3.1. With notation as above, we have an isomorphism M coC ∼= Ax.

Proof. For a ∈ Ax, (agxy)y ∈M coC . Indeed,

ρyz(agxz) = ∆xyz(agxz)(9)=agxy ⊗Ay gyz,

for all y, z ∈ X. Now consider the maps

f : Ax →M coC , f(a) = (agxy)y,

g : M coC → Ax, g((cy)y) = εx(cx).

For a ∈ Ax, we easily compute that

(g ◦ f)(a) = g((agxy)y) = εx(agxx)(9)=a.

If (cy)y ∈M coC , then

ρyz(cz) = cz(1,x,y) ⊗Ay cz(2,y,z)(10)= cy ⊗Ay gyz,

for all y, z ∈ X. Taking y = x, and applying εx to the first tensor factor, weobtain that cz = εx(cx)gxz, and

(cy)y = f(εx(cx)) = (f ◦ g)((cy)y).

This shows that g = f−1, proving the result. �

Proposition 3.2. Let X be a set and let (C, g) be an A-coring with a fixedgrouplike matrix. Consider the adjunction (F,G) from Proposition 2.3.

(1) If F is fully faithful, then i : B → AcoC is an isomorphism.(2) If G is fully faithful, then the morphism of corings can : D → C

from Proposition 2.8 is an isomorphism of corings.

Proof. 1) It suffices to observe that ηB = i : B → AcoC .2) Fix x ∈ X and consider M ∈ Dk(X)C as in Lemma 3.1. Take y ∈ X,a ∈ Ax and b ∈ Ay.(

εyM ◦ (f ⊗B Ay))(a⊗B b) = εyM

((agxz)z ⊗B b)

(12)= agxyb = canxy(a⊗ b)

We know from Lemma 3.1 that f is an isomorphism, and εyM is an isomor-

phism since G is fully faithful. Therefore canxy = εyM ◦ (f ⊗B Ay) is an

isomorphism, and can is an isomorphism. �

Proposition 3.2 motivates the following definition.


Definition 3.3. Let X be a set and let (C, g) be an A-coring with a fixed

grouplike matrix. Let B = AcoC , and let D be the associated Sweedlercanonical coring. C is called a Galois coring if can : D → C is an isomorphismof corings.

Assume that Ay ∈ BM and Cxy ∈ AxM are finitely generated projective, forall x, y ∈ X. Then Dxy = Ax ⊗B Ay ∈ AxM is finitely generated projective,and canxy : Dxy → Cxy is bijective if and only if nacyx = ∗canxy : Ayx =∗Cxy → ∗Dxy is bijective, see Section 1.2.Observe that AxHom(Ax⊗B Ay, Ax) ∼= BHom(Ay, Ax) = Eyx, with multipli-cation maps given by opposite composition

mxyz : Exy ⊗Ay Eyz → Exz, mxyz(f ⊗Ay g) = g ◦ f.

We compute easily that

nacyx : Ayx → Eyx, nacyx(f)(a) = f(gxya),

for f ∈ Ayx and a ∈ Ay.

4. Descent Theory I: X is an arbitrary set

It follows from Proposition 3.2 that C is isomorphic to the Sweedler canon-ical coring if the adjunction (F,G) from Proposition 2.3 is a pair of inverseequivalences. Therefore, it suffices to investigate the Sweedler canonicalcoring, that is, we want to study when the adjoint pair (K,R) from Exam-ple 2.7 is a pair of inverse equivalences. We begin with some remarks aboutclassical faithfully flat descent theory.Consider a ring extension i : B → A, and let D = A⊗B A be the Sweedlercanonical coring; the reader may consult the literature, see e.g. [6, 7, 8] formore detail, or else may keep in mind that it is just the Sweedler coringfrom Example 2.7 in the case where X is a singleton. The faithfully flatdescent theorem can then be stated as follows, see [8, Proposition 2.5]: if Ais faithfully flat as a left B-module, then the adjoint pair (K,R) is a pairof inverse equivalences. Note that this is the formulation of faithfully flatdescent in terms of corings, its history goes back to Grothendieck [17] forschemes, Knus and Ojanguren [18, 19] for commutative ring extensions andCipolla [13] for arbitrary morphisms of rings.It could be conjectured that (K,R) from Example 2.7 is a pair of inverseequivalences if the Ax are faithfully flat as a left B-module. We were notable to prove this result in full generality. Actually, it turns out that somefiniteness assumptions are needed, on X or on A.The case where X is finite is discussed in Section 5. In the current Section,X is an arbitrary set.Coming back to the classical case (where X is a singleton): the proof of thefaithfully flat theorem consists in proving that the unit and counit of theadjunction are bijective. This relies on flatness arguments, and is thereforenot constructive. The arguments do not pass to our situation, due to the


fact that infinite products do not commute with tensor products.We escape this problem in the following manner: if we assume that A is aleft B-progenerator, then we can give some explicit formulas for the inverseof the counit (in terms of the finite dual basis of A as a left B-module, Abeing finitely generated projective) and of the unit (in terms of the elementof trace 1, A being a generator of the category of left B-modules. Theseformulas can be extended to our situation. As far as we know, these formu-las do not appear in the literature, and we will present them below. Oncethe formulas are on paper, it is easy to prove them, by verifying that theydescribe the inverses of the unit and counit morphisms of the adjunction.This proof can be obtained by specializing the argument presented in thesequel to the case where X is a singleton. The formulas were obtained in thefollowing way: if A is a B-progenerator, then the category of D-comodulesis isomorphic to the category of modules over BEnd(A)op. This gives anadjunction between MB and M

BEnd(A)op , which is the adjunction comingfrom the Morita context associated to the left B-module A. The coinvari-ants functor and the right adjoint functor coming from the Morita contextare isomorphic, because of uniqueness of adjoints. This isomorphism canbe computed explicitly. The Morita context is strict, and the (co)unit mor-phisms and their inverses are described in terms of the finite dual basis andthe trace 1 element. Combining everything, one obtains the following.For M ∈MD, the counit map of the adjunction is

ζM : M coD ⊗B A→M, ζM (m⊗B a) = ma.

Assume that A is finitely generated projective as a left B-module, with finitedual basis q(·) ⊗B a(·) ∈ ∗A⊗B A, which means that

(20) a = q(·)(a)a(·)

for all a ∈ A. Then ζM is bijective, with inverse given by the formula

(21) ζ−1M (m) =

∑k

m<0>q(·)(m<1>)⊗B a(·).

for all m ∈M .For N ∈MB, the unit of the adjunction is

νN : N → (N ⊗B A)coD, νN (n) = n⊗B 1.

Assume that A is a generator of BM. This means that there exists rl ∈ ∗Aand cl ∈ A such that

(22)∑l

rl(cl) = 1B.

Then νN is bijective, with inverse given by the formula

(23) ν−1N (∑i

ni ⊗B bi) =∑i,l

nirl(bicl),


for every∑

i ni ⊗B bi ∈ (N ⊗B A)coD. Formulas (21-22) are the startingpoints for our next result, in which we return to the categorical setting.

Proposition 4.1. Let X be a set, let A be a diagonal k-linear category, andlet B be a k-algebra, such that we have an algebra morphism ix : B → Ax,for every x ∈ X. Let D be the Sweedler canonical coring, as in Example 2.7.If Au ∈ BM is finitely generated projective for a given u ∈ X, then

ζuM : M coD ⊗B Au →Mu, ζuM

((mz)z ⊗B a

)= mua,

is bijective, for all M ∈ Dk(X)D. The inverse (ζuM )−1 is described as follows:

if q(u) ⊗B a(u) ∈ ∗Au ⊗B Au is the finite dual basis of Au ∈ BM, then

(24) (ζuM )−1(m) =(m<0,z>q

(u)(m<1,u>))z⊗ a(u),

for all m ∈ Mu. Consequently the coinvariants functor R = (−)coD is fullyfaithful if Au ∈ BM is finitely generated projective for all u ∈ X.

Proof. For u ∈ X, fix a tensor representation q(u)⊗B a(u) =∑

k qk⊗ak, andfix an index k. We will first show that(

m<0,z>qk(m<1,u>))z∈M coD.

For all x, y ∈ X, we have that

ρxy(m<0,y>qk(m<1,u>))(18)= ρxy(m<0,y>)qk(m<1,u>)

(16)= m<0,x> ⊗B 1yqk(m<1,u>) = m<0,x>qk(m<1,u>)⊗B 1y,

so that (19) holds, proving the assertion. This shows that the right handside of (24) is an element of M coD ⊗B Au, as needed.For all m ∈Mu, we have that(

ζuM ◦ (ζuM )−1)(m) = m<0,u>q

(u)(m<1,u>)a(u)(20)= m<0,u>m<1,u>

(17)= m.

Now take (mz)z ∈M coD and a ∈ Au. Then((ζuM )−1 ◦ ζuM

)((mz)z ⊗B a

)= (ζuM )−1(mua)

(18,24)=

∑k

(mu<0,z>q

(u)(mu<1,u>a))z⊗B a(u)

(19)=

∑k

(mzq

(u)(1ua))z⊗B a(u) =

∑k

(mz)zq(u)(a)⊗B a(u)

=∑k

(mz)z ⊗B q(u)(a)a(u)(20)= (mz)z ⊗B a.

We conclude that (ζuM )−1 is the inverse of ζuM . �

Proposition 4.2. Let X be a set, let A be a diagonal k-linear category, andlet B be a k-algebra, such that we have an algebra morphism ix : B → Ax,for every x ∈ X. Let D be the Sweedler canonical coring, as in Example 2.7.Assume that (Ax)x∈X is a family of generators of BM. Then for every x


in some finite Z ⊂ X there exist rx,lx ∈ ∗Ax and cx,lx ∈ Ax such that∑x∈Z

∑lxrx,lx(cx,lx) = 1B. For all N ∈MB, the counit map

νN : N → K(N)coD, νN (n) = (n⊗B 1z)z

is bijective, with inverse given by the formula

(25) ν−1N

((∑jz

njz ⊗B bjz)z)

=∑x∈Z

∑lx,jx

njxrx,lx(bjxcx,lx),

for all (∑

jznjz ⊗B bjz)z ∈ K(N)coD. Consequently the functor K : MB →

Mk(X)D from Example 2.7 is fully faithful if (Ax)x∈X is a family of gener-ators of BM.

Proof. For all n ∈ N , we have that

(ν−1N ◦ νN )(n) = ν−1

N

((n⊗B 1z)z

)=∑x∈Z

∑lx

nrx,lx(1xcx,lx) = n.

Take p =(∑

jz(njz ⊗B bjz)

)z∈ K(N)coD. It follows from (19) that∑

jx

njx ⊗B 1z ⊗B bjx =∑jz

njz ⊗B bjz ⊗B 1x.

For x ∈ X, and a fixed index lx, we have that∑jx

njx ⊗B 1z ⊗B bjxcx,lx =∑jz

njz ⊗B bjz ⊗B cx,lx .

Since rx,lx is left B-linear, we can apply it to the third tensor factor. Thenwe make summation over x and lx, multiply the second and third tensorfactor, and obtain subsequently∑

x∈Y

∑jx,lx

njx ⊗B 1zrx,lx(bjxcx,lx) =∑x∈Y

∑jz ,lx

njz ⊗B bjzrx,lx(cx,lx);

∑x∈Y

∑jx,lx

njxrx,lx(bjxcx,lx)⊗B 1z =∑jz

njz ⊗B bjz .

This means that the z-components of p and (νN ◦ ν−1N )(p) coincide, and,

consequently p = (νN ◦ ν−1N )(p). �

Combining Propositions 4.1 and 4.2, we obtain the following result.

Theorem 4.3. Let X be a set, let A be a diagonal k-linear category, and letB be a k-algebra, such that we have an algebra morphism ix : B → Ax, forevery x ∈ X. Let D be the Sweedler canonical coring, as in Example 2.7.Assume that Au ∈ BM is finitely generated projective for all u ∈ X, andthat (Ax)x∈X is a family of generators of BM. Then the adjoint pair (K,R)between MB and Dk(X)D is a pair of inverse equivalences.


Proposition 4.4. Let X be an arbitrary set and let A = (Ax)x be a diag-onal algebra, let C be an A-coring and let B = AcoC. Then the adjunction(F,G) from Proposition 2.3 is an equivalence of categories if the followingconditions are satisfied

(1) C is a Galois coring in the sense of Definition 3.3;(2) Ax ∈ BM is finitely generated projective, for all x ∈ x;(3) (Ax)x∈X is a family of generators of BM.

Example 4.5. Infinite comatrices Let X be an arbitrary set. Let B = k,and let A be the diagonal algebra with Ax = k1x ∼= k, for every x ∈ X.ix : k → k1x is given by the formula ix(r) = r1x. Then Ax⊗Ay = k1x⊗1y.Setting 1x ⊗ 1y = fxy, we obtain that the associated Sweedler comatrixcoring D is described as follows: Dxy = kfxy, with comultiplication andcounit maps given by the formulas

∆xyz(fxz) = fxy ⊗ fyz and εx(fxx) = 1.

The definition of a descent datum (or D-comodule) boils down to the fol-lowing. It consists of a collection of k-modules M = (Mx)x indexed by theelements of X, together with a family of k-linear maps ρxy : My →Mx suchthat ρxy ◦ ρyz = ρxz and ρxx = Mx. Remark that this implies that all theρxy are isomorphisms.The coinvariants of a descent datum M are given by the following formula:

M coD = {(mx)x ∈∏x

Mx | ρxy(my) = mx, for all x, y ∈ X}.

The descent datum K(N) associated to a k-module N is defined as follows:K(N)x = N1x, and ρxy : N1y → N1x is the natural isomorphism definedby setting ρxy(n1y) = n1x, for all n ∈ N . The conditions of Theorem 4.3are satisfied, so that (K, (−)coD) is a pair of inverse equivalences betweenMk and Mk(X)D. This can also be proved directly.

Now let X = {1, · · · , n} be finite. The matrix coalgebra Mn(k) is thecoalgebra with free k-basis {fij | i, j = 1 · · · , n} and comultiplication andcounit given by the formulas

∆(fij) =n∑k=1

fik ⊗ fkj and ε(fij) = δij ,

extended linearly. It is well-known that Mn(k) is the dual coalgebra of thematrix algebra Mn(k). It is easy to show that the category of descent datain the above sense is isomorphic to the category of comodules over Mn(k).The descent theory tells us that Mk is equivalent to Mk(X)D, which is

isomorphic toMMn(k) and Mn(k)M. We recover the classical result that thecategory of modules over a matrix algebra is isomorphic to the category ofmodules over k.


5. Descent Theory II: X is a finite set

Assume that X is finite, and let A = (Ax)x be a diagonal k-linear category,as introduced in Section 1. Then A =

⊕xAx is a k-algebra, called the

diagonal algebra (Ax)x in packed form. Let M be a right A-module. Then

M ∼= M ⊗A A ∼=⊕x

M ⊗A Ax ∼=⊕x

M1x.

Write Mx = M1x. Mx is a right Ax-module, hence U(M) = (Mx)x ∈Dk(X)A, called the unpacking of M . Conversely, if (Mx)x ∈ Dk(X)A, thenP (M) = M =

⊕xMx ∈ MA. If f : M → N is a right A-linear map, then

for all m ∈ Mx, we have that f(m) = f(m1x) = f(m)1x ∈ Nx, hence frestricts to fx : Mx → Nx, and U(f) = (fx)x is a morphism U(M)→ U(N)in Dk(X)A. If f : M → N in Dk(X)A, then we have P (f) : P (M)→ P (N)in MA, defined as follows: for m =

∑xmx ∈ M , P (f)(m) =

∑x fx(mx).

This clearly defines two inverse functors, so we have the following result.

Proposition 5.1. Let X be finite and let A = (Ax)x be a diagonal algebra.We have an isomorphism of categories U : MA

∼= Dk(X)A.

In a similar way, we have an isomorphism of categories

U : AMA∼= AMk(X)A.

If C is an A-bimodule, then the corresponding U(C) = C ∈ AMk(X)A is(Cx,y = 1xC1y)x,y.For x, y ∈ X, Ax and Ay are A-bimodules. If x 6= y, then Ax ⊗A Ay = 0.Indeed, for a ∈ Ax and b ∈ Ay, we have that

a⊗A b = a1x ⊗A 1yb = a⊗A 1x1yb = 0.

We claim that the canonical surjection p : Ax ⊗Ax Ax → Ax ⊗A Ax isbijective. The map

f : Ax ⊗Ax → Ax ⊗Ax Ax, f(a⊗ b) = a⊗Ax b

satisfies the following property, for all c =∑

x cx ∈ A:

ac⊗Ax b = acx ⊗Ax b = a⊗Ax cxb = a⊗Ax cb,

so that we have a well-defined map q : Ax⊗AAx → Ax⊗Ax Ax, which is theinverse of p. We conclude that Ax ⊗A Ax ∼= Ax ⊗Ax Ax

∼= Ax.Now take M ∈MA and N ∈ AM. Then

M ⊗A N ∼=⊕x,y

M ⊗A Ax ⊗A Ay ⊗AM

∼=⊕x

M ⊗A Ax ⊗Ax Ax ⊗A Nx =⊕x

Mx ⊗Ax Nx.(26)

Now let C be an A-coring (see for example [6, 7] for more detail on corings).Then C is an A-bimodule, and U(C) = C = (Cxy)x,y with Cxy an (Ax, Ay)-bimodule. The comultiplication ∆ : C → C ⊗A C is an A-bimodule map.


Applying (26)

C ⊗A C =⊕x,u,z

Cxu ⊗Au Cuz.

Take c ∈ Cxz. Then c = 1xc1z, and

∆(c) = ∆(1xc1z) = 1x∆(c)1z ∈⊕u

Cxu ⊗Au Cuz,

so ∆ restricts to a map Cxz →⊕

u Cxu ⊗Au Cuz. Composing this map withthe projection

⊕u Cxu ⊗Au Cuz → Cxy ⊗Ay Cyz, we obtain a map ∆xyz :

Cxz → Cxy ⊗Ay Cyz. It follows from the coassociativity of ∆ that the ∆xyz

satisfy the coassociativity conditions (5).ε : C → A is an A-bimodule map. For c ∈ Cxy, we have that

ε(c) = ε(1xc1y) = 1xε(c)1y = ε(c)1x1y.

If x 6= y, then ε(c) = 0. If x = y, then ε(c) = ε(c)1x ∈ Ax. Define εx :Cxx → Ax as the restriction of ε to Cxx. It is clear that ε(c) =

∑x εx(1xc1x).

It is easy to prove that the maps εx satisfy the counit conditions (6). Thisproves that C = (Cxy)xy is an A = (Ax)x-coring in the sense of Section 2.1.Conversely, given an A-coring C, it is easy to show that P (C) = C =

⊕x,y Cxy

is an A-coring. This establishes an isomorphism between the categories ofA-corings and A-corings.Let C be an A-coring, and let M ∈MC be a right C-comodule, in the senseof [6, 7, 8]. The coaction ρ : M → M ⊗A C =

⊕x,yMx ⊗Ax Cxy is right

A-linear. For m ∈Mz, we have that

ρ(m) = ρ(m1z) = ρ(m)1z ∈⊕x

Mx ⊗Ax Cxz.

So ρ restricts to a map Mz →⊕

xMx ⊗Ax Cxz. Composing this map withthe projection

⊕xMx ⊗Ax Cxz →My ⊗Ay Cyz, we obtain a map

ρyz : Mz →My ⊗Ay Cyz.

It is now easy to prove that the maps ρyz satisfy the axioms (7-8), provingthat M is a right C-comodule in the sense of Section 2.1. Conversely, if M isa right C-comodule, then P (M) is a right P (C)-comodule. This shows thatthe packing and unpacking functors induce an isomorphism of categories

U : Dk(X)C →MC .

Grouplike elements of C correspond to right C-comodule structures on A; bythe above category isomorphism, these correspond to right (Cxy)xy-comodulestructures on (Ax)x, which correspond to grouplike matrices g in C, in thesense of Section 2.2. Let us make this correspondence explicit. For a grou-plike element g ∈ G(C), let gxy = 1xg1y ∈ Cxy. Then

∆(gxy) = 1x∆(g)1y = 1xg ⊗A g1y =∑u,v

1xg1u ⊗A 1vg1y


=∑u

1xg1u ⊗A 1ug1y =∑u

1xg1u ⊗Au 1ug1y ∈⊕u

Cxu ⊗Au Cuz.

Take the projection of ∆(gxy) on Cxy ⊗Ay Cyz. This gives ∆xyz(gxz) =gxy ⊗Ay gyz. Finally ε(gxy) = 1xε(g)1y = 1x1y = δxy1x. In particular,εx(gxx) = 1x, and this proves that (gxy)x,y is a grouplike matrix. Conversely,if (gxy)x,y is a grouplike matrix, then g =

∑x,y gxy is a grouplike element of

C.Now let D = A ⊗B A be the Sweedler canonical coring associated to theringmorphism i =

⊕x ix : B → A =

⊕xAx. Then U(D) = (Dxy)x,y is

precisely the Sweedler canonical coring from Example 2.7. For a grouplikeelement g ∈ G(C), we have the canonical morphism of A-corings

can : D → C, can(a⊗B a′) = aga′,

see [6, 7, 8]. It is easy to establish that U(can) : U(D)→ U(C) is preciselythe canonical morphism can from Proposition 2.8.We have pairs of adjoint functors (F,G = (−)coC) betweenMB and Dk(X)C

(Proposition 2.3) and (F1 = − ⊗B A,G1 = (−)coC) between MB and MC[8, Sec. 1]. It is easy to see that P ◦ F = F1, and G1 = G ◦ U . It followsthat (F,G) is a pair of inverse equivalences if and only (F1, G1) is a pair ofinverse equivalences.

Proposition 5.2. Let X be a finite set and let A = (Ax)x be a diagonalalgebra, let C be an A-coring and let B = AcoC. Then the adjunction (F,G)from Proposition 2.3 is an equivalence of categories if the following condi-tions are satisfied

(1) C is a Galois coring in the sense of Definition 3.3;(2) A ∈ BM is faithfully flat.

Proof. (1) implies that P (C) = C is a Galois coring in the sense of [8, Def.3.2]. It follows from [8, Prop. 3.8] that (F1, G1) is a pair of inverse equiva-lences. We have seen above that this implies that (F,G) is a pair of inverseequivalences. �

6. Partial actions by groupoids

Historically, two equivalent approaches to groupoids are possible. An olderapproach goes back to Brandt [5], and defines a groupoid as a set witha partially defined operation satisfying certain axioms. The definition isdesigned in such a way that a groupoid with an operation that is definedeverywhere is a group. The second approach is categorical: a groupoid is acategory in which all morphisms are isomorphisms. From a small groupoidin the categorical sense, one can construct a groupoid in the original senseas follows: consider the set of all (iso)morphisms of the groupoid, and letthe partial operation be given by composition. This establishes a one to onecorrespondence between small categorical groupoids and Brandt groupoids,we refer to [20] for more detail.


In [2], the classical definition of a groupoid is used. We will treat groupoidsas (small) categories in which every morphism is invertible, or, if one prefers,as Hopf categories over the category of sets, see [3]. The objects of a groupoidG will typically be denoted as x, y, z, · · · , and the morphisms as g, k, l, · · · .The set of objects will be denoted by X. The identity on x will also bedenoted by x. We will denote Gxy = HomG(y, x), so that the compositionyields maps Gxy ×Gyz → Gxz.In [2], the notion of partial action of a groupoid G on a (not necessarilyunital) associative algebra A is introduced. Under certain conditions, theyintroduce the subring of invariants R [2, Sec. 4], and then define when A isa partial Galois of R [2, Sec. 5]. In the sequel, we will associate a coringC in the sense of Section 2 to a partial action by a groupoid. Then we willshow that if C is Galois in the sense Definition 3.3, then A is a partial Galoisextension in the sense of [2], and we will give sufficient conditions in orderthat the two notions are equivalent.From [2, Sec. 1], we recall the definition of a partial action by a groupoidG.

Definition 6.1. A partial action of G on a (not necessarily unital) algebraA consists of a pair

α = (D = {Dg | g ∈ G}, α = {αg | g ∈ G}),where Dg is a two-sided ideal of A, and αg : Dg−1 → Dg is a ring isomor-phism, for every g ∈ G. The following axioms have to be satisfied, for allx, y, z ∈ X, g ∈ Gxy and h ∈ Gyz. Dg is an ideal in Dx, and

(i) αx : Dx → Dx is the identity, for x ∈ X;(ii) α−1

h (Dg−1 ∩Dh) ⊂ D(gh)−1 ;

(iii) (αg ◦ αh)(a) = αgh(a), for all a ∈ α−1h (Dg−1 ∩Dh),

or, equivalently, condition (i) and

(iia) αh restricts to an isomorphism

D(gh)−1 ∩Dh−1 → Dg−1 ∩Dh.

(iiia) (αg ◦ αh)(a) = αgh(a), for all a ∈ D(gh)−1 ∩Dh−1 .

The triple (A,D,α) is then called a partial G-action.

Proof. For the sake completeness, we show that the two sets of axioms areequivalent. First assume that ((i-iii)) hold. It follows from (ii) that

α−1h (Dg−1 ∩Dh) ⊂ D(gh)−1 ∩Dh−1 .

Since (gh)h−1 is defined, it follows from (ii) that

α−1h−1(D(gh)−1 ∩Dh−1) ⊂ Dg−1 ∩Dh,

andα−1h (Dg−1 ∩Dh) = D(gh)−1 ∩Dh−1 .

(iia) follows and so does (iiia). Conversely, (ii) follows immediately from(iia), and (iii) follows from (iia) and (iiia). �


Taking g = h−1 in (iiia), we find that (αh−1 ◦ αh)(a) = a, for all a ∈Dy ∩Dh−1 = Dh−1 , and we can conclude that

(27) α−1h = αh−1 ,

for all h ∈ G.

Assume that a two-sided ideal D of A has a unit element e. Then for alla ∈ A, we have that ae, ea ∈ D, and ae = eae = ea, so that e is a centralidempotent in A. If a ∈ D, then a = ae ∈ Ae, and we conclude that D = Aeis a principal ideal generated by a central idempotent.If e and f are central idempotents, then Ae∩Af = Aef . ef = e if and onlyif Ae is an ideal of Af .

Proposition 6.2. Let G be a groupoid, and let A be a ring. For eachg ∈ G, let Dg = Aeg be a two-sided ideal with unit element eg, and letαg : Dg−1 → Dg be a ring automorphism. Then (A,D,α) is a partialG-action if and only if egex = eg, and the axioms (i), (iib) and (iiib), or,equivalently, (i), (iib) and (iiic) are satisfied, for all x, y, z ∈ X, g ∈ Gxyand h ∈ Gyz.

(iib) αh restricts to an isomorphism

e(gh)−1eh−1Dz = e(gh)−1eh−1A→ eg−1ehDy = eg−1ehA.

(iiib) αg(αh(aeh−1)eg−1

)= αgh(aeh−1g−1)eg, for all a ∈ A.

(iiic) αg(αh(a)eg−1

)= αgh(aeh−1g−1), for all a ∈ Dh−1.

In this situation (A,D,α) is called a unital partial G-action.

Proof. It follows from the observations above that Dg is an ideal in Dx ifand only if egex = eg, and that (iia) is equivalent to (iib). (iia) implies that

(28) αh(ekeh−1) = ehkeh.

Assume that egex = eg, and that (i) and (iia) are satisfied. For all a ∈ A,we have that

αg(αh(aeh−1eh−1g−1

)= αg

(αh(aeh−1)αh(eh−1eh−1g−1)

)(28)= αg

(αh(aeh−1)eheg−1

)= αg

(αh(aeh−1)eg−1

)αgh(aeh−1eh−1g−1) = αgh(aeh−1g−1)αgh(eh−1eh−1g−1eh−1)

(28)= αgh(aeh−1g−1)egheg = αgh(aeh−1g−1)eg.

The left hand sides of both equalities are equal if and only if the right handsides are equal, which is precisely the equivalence of (iiia) and (iiib).Assume that (i), (iib) and (iiib) hold. In (iib), we make the substitution{

k = g−1

l = ghor

{g = k−1

h = kl

It follows that αkl restricts to an isomorphism

αkl : Ael−1el−1k−1 → Aekekl,


or, mutatis mutandis, αgh restricts to an isomorphism

αgh : Aeh−1eh−1g−1 → Aegegh.

It follows that αgh(aeh−1g−1) ∈ Aeg, and αgh(aeh−1g−1) = αgh(aeh−1g−1)eg,and (iiic) follows.Conversely, assume that (i), (iib) and (iiic) hold. For all a ∈ A, we computethat

αgh(aeh−1g−1)eg = αg(αg−1(αgh(aeh−1g−1)eg)(iiic)= αg(αh(aeh−1g−1eh−1)) = αg(αh(aeh−1)αh(eh−1g−1eh−1))

(iib)= αg(αh(aeh−1)eg−1eh) = αg(αh(aeh−1)eg−1),

and (iiib) follows. �

We extend the maps αg : Dg−1 → Dg to αg : Dy → Dx as follows: for alla ∈ Dy, put

αg(a) = αg(aeg−1).

Then we can rephrase (iiib):

(29) (αg ◦ αh)(a) = αg(αh(aeh−1)eg−1)(iiib)= αgh(aeh−1g−1)eg = αgh(a)eg.

It is easy to see that αg is multiplicative:

(30) αg(ab) = αg(aeg−1beg−1) = αg(aeg−1)αg(beg−1) = αg(a)αg(b).

The ideals Dg play the central role in the definition of a partial action. Itis not really important how they are place within the ring A. Let GPar bethe category with partial G-actions as objects. A morphism (A,D,α) →(B,E, β) consists of a family ϕ = {ϕx | x ∈ X}, where ϕx : Dx → Ex isa ring morphism such that ϕx(Dx) is an ideal in Ex, ϕx restricts to a mapDg → Eg, for for all g ∈ Gxy, and the following diagrams commute:

Dg−1

ϕy //

αx

��

Eg−1

βx��

Dgϕx // Eg

Let (A,D,α) be a partial G-action, and consider A′ = ⊕x∈XDx. It is clearthat (A′, D, α) is also a partial G-action, and (A,α) ∼= (A′, α) in GPar.As in [2, Sec. 4-5], we restrict to partial G-actions of the form (A =⊕x∈XDx, D, α). We consider the associated (categorical) groupoid G andthe diagonal k-linear category A = (Ax)x∈X . We say that (D,α) is a partialaction of the groupoid G on A, allowing us to switch from the ring theoret-ical language from [2] to the categorical language that we used in the firstpart of this paper.

From now on, let us also assume that Gxy is finite, for all x, y ∈ X. Forthe moment we allow X to be infinite, but, at the time that we make com-parison with the results in [2] in Section 6.1, we will make the additional


assumption that X is finite, as in [2]. We consider unital partial actions ofthe form (G,A,D, α).

A partial G-action (A,D,α) is called global if Dg = Dx, for all x ∈ X andg ∈ Gxy.Consider a unital global G-action. The condition Dg = Dx then reduces toeg = ex. For g ∈ Gxy, h ∈ Gyz, we have αg : Dy → Dx and αgh = αg ◦ αh.This can be restated in the language of the preceding Sections: A = (Dx)x∈Xis a diagonal k-linear category, and the groupoid G acts on A from the left.By linearity, the Hopf category H = kG acts on A.

Consider a diagonal category A = (Dx)x∈X . Our next aim is to associatean A-coring to a unital partial G-action of the form (A = ⊕x∈XDx, D, α).For x, y ∈ X, let

Cxy =⊕g∈Gxy

Dg =⊕g∈Gxy

Dxeg.

Let vg be the element of Cxy with entry eg at position g, and 0 elsewhere.Then we can write

Cxy =⊕g∈Gxy

Dxegvg =⊕g∈Gxy

Dgvg.

For later use, we make the following observation. Take ag, bg ∈ Dx, for allg ∈ Gxy. Then ∑

g∈Gxy

agvg =∑g∈Gxy

bgvg in Cxy

if and only if

(31) ageg = bgeg, for all g ∈ Gxy.Cxy is a (Dx, Dy)-bimodule, with action

a(a′vg)b = aa′αg(b)vg,

for all a, a′ ∈ Dx and b ∈ Dy. The associativity of the right Dy-actionfollows from (30). We define structure maps

∆xyz : Cxz → Cxy ⊗Dy Cyz, ∆xyz(avh) =∑g∈Gxy

avg ⊗Dy vg−1h;

εx : Cxx → Dx, εx(∑g∈Gxx

agvg) = ax.

Proposition 6.3. With notation as above, C = (Cxy)x,y∈X is an A-coring,with grouplike matrix given by γxy =

∑h∈Gxy

vh.

Proof. We first show that ∆xyz is right Az-linear and that εx is right Az-linear (left linearity is obvious). For a ∈ Az, we have that

∆xyz(vh)a =∑g∈Gxy

vg ⊗Dy αg−1h(a)vg−1h


=∑g∈Gxy

αg(αg−1h(a))vg ⊗Dy vg−1h(29)=

∑g∈Gxy

αh(a)egvg ⊗Dy vg−1h

=∑g∈Gxy

αh(a)vg ⊗Dy vg−1h = ∆xyz(αh(a)vh) = ∆xyz(vha).

εx(∑g∈Gxx

agvga) = εx(∑g∈Gxx

agαg(a)vg)

= axαx(a) = axa = εx(∑g∈Gxx

agvg)a.

We next prove the coassociativity property.((∆xyu ⊗Dy Cyz) ◦∆xyz

)(vh) = (∆xyu ⊗Dy Cyz)

( ∑g∈Gxy

vg ⊗Dy vg−1h

)=

∑g∈Gxy

∑l∈Gxu

vl ⊗Du vl−1g ⊗Dy vg−1h

is equal to((Cxu ⊗Du ∆uyz) ◦∆xyz

)(vh) = (Cxu ⊗Du ∆uyz)

( ∑l∈Gxu

vl ⊗Du vl−1h

)=

∑l∈Gxu

∑k∈Guy

vl ⊗Du vk ⊗Dy vk−1l−1h.

This can be seen using the substitution g = lk, k = l−1g.Let us next prove the left counit property:

∆xxz(vh) =∑g∈Gxx

vg ⊗Dx vg−1h =∑g∈Gxx

egvg ⊗Dx vg−1h;

((εx ⊗Dx Cxy) ◦∆xxz

)(vh) = ex ⊗Dx vxh = vh.

Finally γ is a grouplike matrix:

εx(γxx) = εx(∑g∈Gxx

egvg) = ex;

∆xyz(γxz) =∑h∈Gxz

∑g∈Gxy

vg ⊗Dy vg−1h

=∑g∈Gxy

∑l∈Gyz

vg ⊗Dy vl = γxy ⊗Dy γyz,

where we carried out the substitution l = g−1h ∈ Gyz. �

Following (11), we compute the coinvariants AcoC . (bx)x∈X ∈∏x∈X Dx is

coinvariant if and only if

γxyby =∑h∈Gxy

αh(by)vh = bxγxy =∑h∈Gxy

bxvh,


for all x, y ∈ X, or, equivalently, in view of (31),

(32) αh(by) = bxeh,

for all x, y ∈ X and h ∈ Gxy.

Let B be a k-algebra, and let i : B → AcoC be an algebra homomorphism.Also consider the composition ix : B → AcoC → Ax, for every x ∈ X. Thecanonical maps canxy : Dxy = Dx ⊗B Dy → Cxy are given by the formula

canxy(a⊗B b) =∑g∈Gxy

aαg(b)vg.

We now describeAyx = ∗Cxy ∈ DyMDx ,

with bimodule structure given by formula (1). For g ∈ Gxy, let ug : Cxy →Dg ⊂ Dx be the left Dx-linear map defined by the formula

ug(vh) = δgheg,

for h ∈ Gxy. Then ug ∈ Ayx. For c ∈ C, we have that

c =∑h∈Gxy

uh(c)vh,

hence ∑h∈Gxy

uh ⊗Dx vh ∈ Ayx ⊗Dx Cxy

is a finite dual basis of Cxy. In particular, it follows that

f =∑h∈Gxy

uhf(vh),

for all f ∈ Ayx. Observe that ugeg = ug. Indeed, for all h ∈ Gxy, we havethat

(ugeg)(vh) = ug(vh)eg = δgheg = ug(vh).

We conclude thatAyx =

⊕g∈Gxy

ugDg.

We compute the lef Dy-action on Ayx. For b ∈ Dy and g, h ∈ Gxy, we havethat

(bug)(vh)(1)= ug(vhb) = ug(αh(beh−1)vh)

= αg(beg−1) = ug(vh)αg(beg−1) = (ugαg(beg−1))(vh),

and we conclude thatbug = ugαg(beg−1).

Now we determine the multiplication on A. For g ∈ Gxy, h ∈ Gyz, andl ∈ Gxz, we have that

(uh#ug)(vl)(13)= ug

( ∑m∈Gyz

vlm−1uh(vm))


(32)= ug(vlh−1eh) = ug(αlh−1(ehehl−1)vlh−1)

(28)= ug(elehl−1)vlh−1) = ug(elvlh−1)

= elδgh,leg = eghδgh,leg = (ugheg)(vl),

and we conclude thatuh#ug = ugheg.

Now take b ∈ Dh ⊂ Dy and a ∈ Dg ⊂ Dx. Then

(33) uhb#uga = ughαg(beg−1)a.

Indeed,

uhb#uga = uh#buga = uh#ugαg(beg−1)a = ughegαg(beg−1)a = ughαg(beg−1)a.

Let us now compute the duals of the canonical maps

nacyx = ∗canxy : Ayx → Eyx = BHom(Dy, Dx).

For all g ∈ Gxy, a ∈ Dx and b ∈ Dy, we have

nacyx(uga)(b) = (uga)(∑h∈Gxy

vhb)

= ug( ∑h∈Gxy

αh(b)vh)a

= αg(b)a.

Proposition 6.4. For x ∈ X, vx ∈ Im(canxx) if and only if there existai, bi ∈ Dx such that

(34)∑i

aiαg(bi) = δg,xex.

for all g ∈ Gx,x. In this situation, the following statements hold.

(1) canxy is surjective, for all y ∈ X.

(2) Dx is finitely generated projective as a right AcoC-module.

Proof. The first assertion is obvious.(1) Take h, g ∈ Gxy. It follows from (29) and (34) that∑

i

aiαg(αh−1(bi)) =∑i

aiαgh−1(bi)eg = δg,hexeg = δg,heg

and

canxy(∑i

ai ⊗B αh−1(bi)) =∑g∈Gxy

∑i

aiαg(αh−1(bi))vg

=∑g∈Gxy

δg,hegvg =∑g∈Gxy

δg,hvg = vh.

(2) For a ∈ Dx and y ∈ X, let

by =∑g∈Gyx

αg(a) ∈ Dy.


Then b = (by)y∈Y ∈ AcoC . Indeed, for all h ∈ Gzy, we have that

αh(by) =∑g∈Gyx

αhg(a) =∑l∈Gzx

αl(a) = bz.

Then T x : Dx → AcoC , T x(a) = b is left and right AcoC-linear: for c ∈ AcoC ,we have that

T x(c · a)y = T x(cxa)y =∑g∈Gyx

αg(cx)αg(a) = cyby = (cb)y.

The right AcoC-linearity is proved in a similar way. Now∑

i ai ⊗B T x(bi−)

is the finite dual basis of Dx as a right AcoC-module. Indeed, for all a ∈ Dx,we have that∑

i

aiTx(bia) =

∑i

ai∑g∈Gxx

αg(bi)αg(a)

=∑g∈Gxx

(∑i

aiαg(bi))αg(a)

=∑g∈Gxx

δg,xexαg(a) = exαx(a) = a.

�

Theorem 6.5 is the main result of this paper. It is the appropriate gener-alization of [11, Theorem 1.3] and [14, Prop. III.1.2] to partial actions bygroupoids on commutative rings.

Theorem 6.5. Let A be a diagonal k-linear category such that every Axis commutative and consider a partial action (G,A,D, α). Let C be theassociated A-coring, as in Proposition 6.3. Let B be a k-algebra, and leti : B → AcoC be an algebra homomorphism. Also consider the compositionix : B → AcoC → Ax. We assume that there exists a finite subset Z ⊂ Xsuch that iZ = ⊕z∈Ziz : B → DZ is injective. Then B is commutative, andDZ is faithful as a B-module. Then the following assertions are equivalent.

(1) • i : B → AcoC is an isomorphism;• for all x, y ∈ X, for all idempotent e ∈ Dx, and for all g ∈ Dxy:

if αg(a)e = ae for all a ∈ Dx, then g = x or e = 0.• Dx is separable as a B-algebra, for all x ∈ X.

(2) • i : B → AcoC is an isomorphism;• for all x ∈ X, there exist ai, bi ∈ Dx such that (34) holds.

(3) C is a Galois coring in the sense of Definition 3.3, that is• i : B → AcoC is an isomorphism;• can : D → C is an isomorphism of A-corings.

(4) • Dx is finitely generated and projective as a left B-module, forall x ∈ X;• nac : A → E is an isomorphism of A-rings.

(5) • i : B → AcoC is an isomorphism;


• for all x ∈ X, for all M ∈ max(Dx), for all g 6= x ∈ Gxx, thereexists a ∈ Dx such that αg(a)− a ∈ Dx \M ;

(6) the pair of adjoint functors (F,G) between MB and Dk(X)C fromProposition 2.3 is a pair of inverse equivalences.

Proof. (1) =⇒ (2). Let e =∑

i ai ⊗B bi ∈ Dx ⊗B Dx be a separability

idempotent of Dx as a B-algebra. For g 6= x ∈ Gxx, f =∑

i aiαg(bi) ∈ Dx

is an idempotent, and

af =∑i

aaiαg(bi) =∑i

aiαg(bia)

=∑i

aiαg(bi)αg(a) = αg(a)f,

for all a ∈ Dx, and it follows that f = 0, since g 6= x. (34) follows since∑i aibi = x, as e is a separability idempotent.

(2) =⇒ (1). Taking g = x in (34), we find that∑

i aibi = x. For all a ∈ Dx,we have that ∑

j

aj ⊗B bja =∑i,j

aj ⊗B aiT x(bibja)

=∑i,j

ajTx(bjabi)⊗B ai

=∑i

abi ⊗B ai.

Taking a = ex, we find that∑

j aj ⊗B bj =∑

j bj ⊗B aj , and then we findthat

(35)∑j

aj ⊗B bja =∑j

aaj ⊗B bj .

for all a ∈ Dx. This proves that Dx is separable as a B-algebra.If αg(a)e = ae for all a ∈ Dx, then

e =∑i

aibie =∑i

aiαg(bi)e(34)= δg,xexe = δg,xe.

If g 6= x, then it follows that e = 0, proving the second assertion of (1).

(2) =⇒ (3). We define γ : Cxy → Dx ⊗B Dy using the formula

γ(avh) =∑i

aai ⊗B αh−1(bi),

for all a ∈ Dx and h ∈ Gxy. We first show that γ is right Dy-linear. For alla ∈ Dy, we have that

γ(vh)a =∑i

ai ⊗B αh−1(bi)a


=∑i

ai ⊗B αh−1(bi)aeh−1(29)=∑i

ai ⊗B αh−1(bi)αh−1(αh(a))

(30)=

∑i

ai ⊗B αh−1

(biαh(a)

)(35)=∑i

αh(a)ai ⊗B αh−1(bi)

= γ(αh(a)vh) = γ(vha).

Now γ is the inverse of canxy. We have seen in the proof of Proposition 6.4(1)that

(canxy ◦ γ)(avh) = avh.

Finally

(γ ◦ canx,y)(ex ⊗B ey) = γ(∑g∈Gxy

vg) =∑g∈Gxy

∑i

ai ⊗B αg−1(bi)

=∑i

ai ⊗B T x(bi)ey =∑i

aiTx(bi)⊗B ey = ex ⊗B ey.

It follows from the fact that canxy and γ are (Dx, Dy)-bimodule maps thatγ ◦ canx,y is the identity on Dx ⊗B Dy.

(3) =⇒ (2) is obvious.

(3) =⇒ (4). (2) holds, and it follows from Proposition 6.4(2) that Dx isfinitely generated and projective as a right, and, a fortiori, as a left B-module. If canxy is bijective, then nacyx = ∗canxy is bijective.

(4) =⇒ (3). Take a finite Y ⊂ X containing Z. Consider pY : AcoC → DY ,

given by the formula pY ((ax)x) =∑

x∈Y ax. The diagram

Bi //

iY &&

AcoC

pY

��DY

commutes. From the fact that iY is injective, we easily deduce that i isinjective.Then DY is finitely generated projective and faithful as a (left) B-module,so it is a left B-progenerator. This implies that EY Y = BEnd(DY ) is acentral B-algebra. It also follows from (4) that

nacY Y = ⊕x,y∈Y nacxy : AY Y → EY Y

is an isomorphism. For (ax)x ∈ AcoC , we have that∑y∈Y

uyay ∈ Z(AY Y ).


Indeed, for x, y ∈ Y , h ∈ Gxy and b ∈ Dh, we have that(∑z∈Y

uzaz)#uhb = uyay#uhb

(33)= uhαh(ay)b

(32)= uhaxehb = uhbax

(33)= uhb#uxax = uhb#

(∑z∈Y

uzaz).

It follows that nacY Y(∑

z∈Y uzaz)

is central in EY Y , so it is given by mul-tiplication by iY (b), for some b ∈ B. For d =

∑z∈Y dz ∈ DY , we have

that

iY (b)d =∑z∈Y

iz(b)dz =∑z∈Y

i(b)zdz = nacY Y(∑z∈Y

uzi(b)z)(d),

hence

nacY Y(∑z∈Y

uzaz)

= nacY Y(∑z∈Y

uzi(b)z),

and, in view of the bijectivity of nacY Y , we obtain that∑z∈Y

uzaz =∑z∈Y

uzi(b)z.

We conclude that az = i(b)z, for all z ∈ Y . For a given x ∈ X, let Y be afinite subset of X containing x and Z. Then ax = i(b)x, hence a = i(b) ∈AcoC , and i is surjective. This concludes the proof of the first statement of(3).For g ∈ Gxy, Dg = Dxeg is finitely generated projective as a left Dx-module.This implies that Cxy = ⊕g∈GxyDgvg is finitely generated projective as a leftDx-module. Dy is finitely generated projective as a left B-module, and thisimplies that Dxy = Dx ⊗B Dy is finitely generated projective as a left Dx-module. We have seen in Section 1.2, (4), that the vertical morphisms inthe commutative diagram

Dxy

��

canxy // Cxy

��E∗yx

nac∗yx // A∗yx

are isomorphisms. nacyx and nac∗yx are isomorphisms, and it follows thatcanxy is an isomorphism, for all x, y ∈ X.

(2) =⇒ (5). Suppose that αg(a) − a ∈ M , for all a ∈ Dx. Then ex =∑i ai(bi − αg(bi)) ∈M , which is a contradiction.

(5) =⇒ (2). Take g 6= x ∈ Gxx.The ideal of Dx generated by {αg(a)−a | a ∈Dx} is not contained in any maximal ideal of Dx, hence it is the whole of


Dx. This means that there exist a1, · · · , an, b1, · · · , bn ∈ Dx such that

n∑i=1

ai(αg(bi)− bi) = ex.

Now let an+1 = −∑n

i=1 aiαg(bi) and bn+1 = ex. Then we easily computethat

n+1∑i=1

aiαg(bi) = 0 andn+1∑i=1

aibi = ex.

Now assume that V, V ′ ⊂ Gx contain x, and that there exists ai, bi, a′j , b′j ∈

Dx such that∑i

aiαg(bi) = δg,xex and∑j

a′jαg′(b′j) = δg′,xex,

for all g ∈ V and g′ ∈ V ′. For all h ∈ V ∪ V ′, we have that∑i,j

aia′iαh(bib

′i) =

(∑i

aiαh(bi))(∑

j

a′jαh(b′j))

= δh,xex,

and (2) follows after we observe that Gxx = ∪g∈Gxx\{x}{x, g}.

(6) =⇒ (3) follows immediately from Proposition 3.2.

(3) =⇒ (6). (3) expresses that C is a Galois coring. From Proposition 6.4(2),it follows that Dx is finitely generated and projective as a left, and a fortiorias a right, B-module, since B is commutative, for all x ∈ X. By assump-tion, DZ is faitfhul as a left B-module, for some Z ⊂ X finite, hence DZ isfinitely generated projective and faithful as a left B-module, and is there-fore a left B-progenerator since B is commutative. The three conditions ofProposition 4.4 are fulfilled, and it follows that (F,G) is a pair of inverseequivalences. �

6.1. Galois corings versus partial Galois extensions. We assume thatX is finite. Let (G,A = ⊕x∈XDx, D, α) be a partial action. As above, wealso consider the associated categorical groupoid G etc. Following [2, Sec.4], an element a ∈ A is called invariant under α if

(36) αg(aeg−1) = aeg,

for all g ∈ G. According to [2], the subring of A consisting of elementsinvariant under α is denoted as Aα. For every x ∈ X, let ax = aex, so that(ax)x∈X ∈ A is the element in A corresponding to a ∈ A. For g ∈ Gxy, (36)can be rewritten as

αg(ayeg−1) = axeg,

expressing that (ax)x∈X ∈ AcoC , see (32). We conclude that AcoC ∼= Aα.


Definition 6.6. [2, Sec. 4] Let (G,A = ⊕x∈XDx, D, α) be a partial action.A is called an α-partial Galois extension of Aα = B if there exist ri, si ∈ Asuch that

(37)∑i

riαg(sieg−1) =∑x∈X

δx,gex

for all g ∈ G.1

Proposition 6.7. Let (G,A = ⊕x∈XDx, D, α) be a partial action. A is anα-partial Galois extension of Aα = B if and only if canxx : Dx⊗BDx → Cxxis surjective, for all x ∈ X.

Proof. First assume that A is an α-partial Galois extension of B. For g ∈Gyy, take the projection of both sides of (37) onto Dy. This gives∑

i

rieyαg(sieg−1) =∑i

rieyαg(siey) = δy,gey.

This means that (34) holds, with ai = riey and bi = siey, and canyy issurjective.Conversely, assume that canxx is surjective, for all x ∈ X. It follows fromProposition 6.4(1) that canxy is surjective, for all x, y ∈ X. For all x, y ∈ X,there exists aixy ∈ Dx, bixy ∈ Dy such that

(38)∑ixy

aixyαg(bixyeg−1) = δgxex,

for all g ∈ Gxy. Now take g ∈ G. There exists u, v ∈ X such that g ∈ Guv.We will show that

A =∑x,y∈X

∑ixy

aixyαg(bixyeg−1) =∑x∈X

δx,gex.

Since bixy ∈ Dy and eg−1 ∈ Dv, bixyeg−1 = 0 if y 6= v, hence

A =∑x∈X

∑ixv

aixvαg(bixveg−1).

Since aixv ∈ Dx and αg(bixveg−1) ∈ Du, aixvαg(bixveg−1) = 0 if x 6= u, and

A =∑iuv

aiuvαg(biuveg−1)(38)= δgueu =

∑x∈X

δgxex,

as needed. �

1In [2] the summation over X in the right hand side of (37) is missing.


References

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Department of Mathematics and Data Science WIDS, Vrije Universiteit Brus-sel VUB, Pleinlaan 2, B-1050 Brussels, BelgiumE-mail address: [email protected]

URL: http://homepages.vub.ac.be/~scaenepe/

Department of Mathematics and Data Science WIDS, Vrije Universiteit Brus-sel VUB, Pleinlaan 2, B-1050 Brussels, BelgiumE-mail address: [email protected]

URL: http://homepages.vub.ac.be/~tfierema/

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