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Descent Cohomology and CoringsTomasz Brzeziński a

a Department of Mathematics , University of Swansea , Swansea, UKPublished online: 20 Jun 2008.

To cite this article: Tomasz Brzeziński (2008) Descent Cohomology and Corings, Communications inAlgebra, 36:5, 1894-1900, DOI: 10.1080/00927870801941523

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Communications in Algebra®, 36: 1894–1900, 2008Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870801941523

DESCENT COHOMOLOGY AND CORINGS

Tomasz BrzezinskiDepartment of Mathematics, University of Swansea, Swansea, UK

A coring approach to non-Abelian descent cohomology of Nuss and Wambst (2007) isdescribed and a definition of a Galois cohomology for partial group actions is proposed.

Key Words: Coring; Descent cohomology; Galois comodule.

2000 Mathematics Subject Classification: 16W30.

1. INTRODUCTION

1.1. Motivation and Aims

In a recent article Nuss and Wambst (2007) have introduced a non-Abeliandescent cohomology for Hopf modules and related it to classes of twisted formsof modules corresponding to a faithfully flat Hopf-Galois extension. Hopf modulescan be understood as a special class of entwined modules and hence comodulesof a coring (Brzezinski, 2002). The aim of this note is to show how the descentcohomology introduced in Nuss and Wambst (2007) fits into recent developments inthe descent theory for corings (El Kaoutit and Gómez-Torrecillas, 2003; Caenepeel,2004; Caenepeel et al., 2007). In particular we construct the zeroth and first descentcohomology sets for a coring with values in a comodule and relate it to isomorphismclasses of module-twisted forms. We then use this general framework to proposea definition of a non-Abelian Galois cohomology for idempotent partial Galoisactions on noncommutative rings introduced in Caenepeel and De Groot (2005).

1.2. Notation and Conventions

We work over a commutative associative ring k with unit. All algebras areassociative, unital, and over k. The identity map on a k-module M is denotedby M . Given an algebra A, the coproduct in an A-coring � is denoted by �� andthe counit by ��. A (fixed) coaction in a right �-comodule M is denoted by �M .Hom��−�−�, End��−� and Aut��−� denote the homomorphisms, endomorphisms,and automorphisms of right �-comodules, respectively. End��M� is a ring with theproduct given by the composition, M is a left End��M�-module with the product

Received December 8, 2006. Communicated by R. Wisbauer.Address correspondence to Tomasz Brzezinski, Department of Mathematics, University of

Swansea, Singleton Park, Swansea SA2 8PP, UK; E-mail: T.Brzezinski@swansea.ac.uk

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DESCENT COHOMOLOGY AND CORINGS 1895

given by evaluation, and Hom��M�N� is a right End��M�-module with the productgiven by composition.

For an algebra A, ��A� denotes the group of units in A, and for an A-coring�, ���� is the set of grouplike elements of �, i.e., elements g ∈ G such that���g� = g ⊗A g and ���g� = 1. ���� is a right ��A�-set with the action given by theconjugation g · u �= u−1gu, for all u ∈ ��A� and g ∈ ����.

More details on corings and comodules can be found in Brzezinski andWisbauer (2003).

2. DESCENT AND PARTIAL GALOIS COHOMOLOGIES

2.1. Construction of Descent Cohomology Sets

Given an A-coring � and a right �-comodule M with fixed coaction �M � M →M ⊗A �, define the zeroth descent cohomology group of � with values in M as thegroup of �-comodule automorphisms, i.e.,

D0���M� �= Aut��M��

Any isomorphism of right �-comodules, f � M → M , induces an isomorphism ofcohomology groups

f ∗ � D0���M� → D0��� M�� �→ f−1 � � f�

The set Z1���M� of descent 1-cocycles on � with values in M is defined as a set ofall �-coactions F � M → M ⊗A �. Since M comes equipped with the right coaction�M , Z1���M� is a pointed set with a distinguished point �M .

Note that for the Sweedler coring A⊗B A associated to a ring extension B →A, Z1�A⊗B A�M� is the set of non-commutative descent data on M (cf., Brzezinskiand Wisbauer, 2003, Section 25.4). This motivates the name descent cocycles.

Lemma 2.1. Let M� M be right �-comodules. Any right A-linear isomorphismf � M → M induces a bijection f ∗ � Z1���M� → Z1��� M� defined by

f ∗�F� �= �f−1 ⊗A �� � F � f�

The operation �−�∗ maps the identity map into the identity map and reverses the orderof the composition, i.e., for all right A-module isomorphisms f � M → N , g � N → M ,�g � f�∗ = f ∗ � g∗. Furthermore, if f � M → M is an isomorphism of �-comodules, thenf ∗ is an isomorphism of pointed sets.

Proof. If F � M → M ⊗A � is a right coaction, and f � M → M is a right A-linearisomorphism, then

�f ∗�F�⊗A �� � f ∗�F� = �f−1 ⊗A �⊗A �� � �F ⊗A �� � �f ⊗A �� � �f−1 ⊗A �� � F � f= �f−1 ⊗A �⊗A �� � �M ⊗A ��� � F � f= �M ⊗A ��� � �f−1 ⊗A �� � F � f = �M ⊗A ��� � f ∗�F��

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where the second equality follows by the coassociativity of F . This proves that f ∗�F�is a coassociative coaction. The counitality of F immediately implies that also f ∗�F�is a counital map. Thus f ∗ is a well-defined map, and it is obviously a bijection asstated. The proofs of remaining statements are straightforward. �

Lemma 2.1 immediately implies that there is a right action of the A-linearautomorphism group AutA�M� on Z1���M� given by

Z1���M�×AutA�M� → Z1���M�� �F� f� �→ f ∗�F��

Definition 2.2. The first descent cohomology set of � with values in the �-comoduleM is defined as the quotient of Z1���M� by the action of AutA�M�, and is denotedby D1���M�.

Since Z1���M� is a pointed set, so is D1���M� with the class of �M as adistinguished point.

2.2. M-Torsors

Given a right �-comodule M , an M-torsor is a triple �X� �X� �, where X isa right �-comodule with the coaction �X and � M → X is an isomorphism ofright A-modules. Two M-torsors �X� �X� � and �Y� �Y � �� are said to be equivalent if�X� �X� and �Y� �Y � are isomorphic as comodules. Equivalence classes of M-torsorsare denoted by Tors�M�. Tors�M� is a pointed set with the class of the M-torsor�M� �M�M� as a distinguished point. Since Tors�M� is a set of isomorphism classesof comodules which are isomorphic to M as modules, one obtains the followingdescription of Tors�M�.

Proposition 2.3. For all �-comodules M , there is an isomorphism of pointed sets

D1���M� � Tors�M��

Proof. Explicitly, the isomorphism is constructed as follows. Given a coactionF � M → M ⊗A �, consider an M-torsor T�F� �= �M� F�M�. Conversely, given anM-torsor �X� �X� �, define the coaction on M ,

D�X� �X� � �= ∗��X� = �−1 ⊗A �� � �X �

(cf., Lemma 2.1). �

2.3. Module-Twisted Forms

For k-algebras A and B, fix a �B�A�-bimodule � and a right B-module N .A right B-module P is called a �-twisted form of N in case there exists a rightA-module isomorphism � P ⊗B � → N ⊗B �. �-twisted forms �P� �, �Q��� ofN are said to be equivalent if P and Q are isomorphic as right B-modules. Theequivalence classes of �-twisted forms of N are denoted by Twist���N�. Twist���N�is a pointed set with the class of �N�N ⊗B �� as a distinguished point.

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2.4. Descent Cohomology and the Galois-Comodule Twisted Forms.

Recall that a right �-comodule � is called a Galois comodule or ��� �� is calleda Galois coring in case � is finitely generated and projective as an A-module and theevaluation map

Hom������⊗B � → �� f ⊗ s �→ f�s��

is an isomorphism of right �-comodules. Here and in the remainder of thissubsection B is the endomorphism ring B �= End����.

Theorem 2.4. Let � be a Galois right �-comodule that is faithfully flat as a leftB-module. Then, for all right B-modules N , there is an isomorphism of pointed sets

D1��� N ⊗B �� � Twist���N��

where N ⊗B � is a comodule with the induced coaction N ⊗B ��.

Proof. First recall that by the Galois comodule structure theorem (El Kaoutitand Gómez-Torrecillas, 2003, Theorem 3.2) (cf. Brzezinski and Wisbauer, 2003,18.27) the functors Hom����−� and −⊗B � are inverse equivalences between thecategories of right �-comodules and right B-modules. Take any right B-module N .Note that Tors�N ⊗A �� are �-comodule isomorphism classes of comodules whichare isomorphic to N ⊗B � as right A-modules, while Twist���N� are isomorphismclasses of B-modules P such that P ⊗B � is isomorphic to N ⊗B � as a rightA-module (and hence as a �-comodule, with induced coactions). Since −⊗B � is anequivalence, there is an isomorphism of pointed sets Tors�N ⊗A �� � Twist���N�,and the assertion follows by Proposition 2.3. �

2.5. Beyond Faithfully Flat and Finite Galois Comodules

Start with two algebras A and B, an A-coring � and a �B�A�-bimodule � witha left B-linear right �-coaction �� � � → �⊗A �. This defines the functor −⊗B � �MB → M�, where MB is the category of right B-modules and M� is the category ofright �-comodules. If this functor is an equivalence, then the same reasoning as inthe proof of Theorem 2.4 yields an isomorphism of pointed sets

D1��� N ⊗B �� � Twist���N��

where N ⊗B � is a comodule with the induced coaction N ⊗B ��.

Let B = End����. For Galois comodules which are finitely generated andprojective as right A-modules but are not faithfully flat as B-modules sufficientconditions for −⊗B � to be an equivalence are found in Böhm and Vercruysse(2007, Theorem 4.6). In particular D1��� N ⊗B �� � Twist���N�� for a cleftbicomodule � for a right coring extension � of � provided � has a grouplikeelement (cf. Böhm and Vercruysse, 2007, Definition 5.1, Corollary 5.5). For Galoiscomodules in the sense of Wisbauer (2006) (i.e., comodules � such that themap of functors HomA���−⊗B� → −⊗A �, ⊗ s �→ � ⊗A ������s�� is a natural

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isomorphism, no finiteness assumption on �A) the functor −⊗B � � MB → M� is anequivalence if and only if the functor −⊗B � � MB → MA is comonadic (cf. Böhmand Vercruysse, 2007, Section 1).

Even more generally, �-twisted forms can be defined if the algebra B isnonunital (e.g., B can be an ideal in End����). If B is firm in the sense that theproduct map B ⊗B B → B is an isomorphism, and � is a firm B-module, then inthe case of infinite comatrix corings the conditions for −⊗B � to be an equivalenceare given in El Kaoutit (2004, Theorem 5.9) and Gómez-Torrecillas and Vercruysse(2007, Theorem 4.15) (cf. Caenepeel et al., 2006, Theorem 1.3).

2.6. Comparison with the Results of Nuss and Wambst (2007)

To a given Hopf algebra H and a right H-comodule algebra A, one associatesan A-coring � �= A⊗k H with the obvious left A-action, diagonal right A-action andthe coproduct and counit A⊗k �H and A⊗k �H (cf. Brzezinski and Wisbauer, 2003,33.2). The category of �A�H�-Hopf modules is isomorphic to the category of right�-comodules (cf. Brzezinski, 2002, Proposition 2.2). Since A is a right H-comodulealgebra, it is an �A�H�-Hopf module, hence a �-comodule. The endomorphism ringEnd��A� can be identified with the subalgebra of coinvariants B �= �b ∈ A � �A�b� =b ⊗ 1�. A is a Galois �-comodule if and only if B ⊆ A is a Hopf-Galois H-extension.With these identifications in mind, the definition of the descent cohomology in Nussand Wambst (2007) is a special case of Definition 2.2, Nuss and Wambst (2007,Lemma 2.2) can be derived from Lemma 2.1, Nuss and Wambst (2007, Theorem 2.6)follows by Theorem 2.4, while Nuss and Wambst (2007, Proposition 2.8) is a specialcase of Proposition 2.3.

2.7. Galois Cohomology for Partial Group Actions

As the theory of corings covers all known examples of Hopf-type modules,such as Yetter–Drinfeld modules, Doi–Koppinen Hopf modules, entwined, andweak entwined modules, the results of the present note can be easily appliedto all these special cases (cf. Brzezinski and Wisbauer, 2003 for more details).One of the special cases of the descent cohomology for corings is that of partialGalois actions for noncommutative rings studied in Caenepeel and De Groot(2005) (cf. Dokuchaev et al., 2007). In this section, we propose to use the descentcohomology to define the Galois cohomology for partial group actions, thusextending the classical Galois cohomology (Serre, 1997, Section I.5).

Take a finite group G and an algebra A. To any element � ∈ G associatea central idempotent e� ∈ A and an isomorphism of ideals � � Ae�−1 → Ae�.Following Caenepeel and De Groot (2005), we say that the collection �e�� ���∈G isan idempotent partial action of G on A if Ae1 = A, 1 = A and, for all a ∈ A, �� � ∈ G,

����ae�−1�e�−1� = ���ae����−1�e��

(see Dokuchaev and Exel, 2005, Definition 1.1 for the most general definition ofa partial group action). Given an idempotent partial action �e�� ���∈G of G on A,define the invariant subalgebra of A,

AG �= �a ∈ A � ∀� ∈ G� ��ae�−1� = ae���

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The extension AG ⊆ A is said to be G-Galois if and only if the map

A⊗AG A → ⊕�∈G

Ae�� a⊗ a′ �→ ∑�∈G

a��a′e�−1�v��

is bijective. Here v� denotes the element of⊕

�∈G Ae� which is equal to e� at position� and to zero elsewhere. For example, if there exists a convolution invertible rightcolinear map k�G� → A, where k�G� is the Hopf algebra of functions on G, thenAG ⊆ A is a G-Galois extension known as a cleft extension (Böhm and Vercruysse,2007, Section 6.5).

By Caenepeel and De Groot (2005, Proposition 2.2) (or as a matter ofdefinition), �e�� ���∈G is an idempotent partial action of G on A if and only if� �= ⊕

�∈G Ae� is an A-coring with the following A-actions, coproduct, and counit:

a�a′v��a′′ = aa′��a

′′e�−1�v�� ���av�� =∑�∈G

av� ⊗A v�−1�� ���av�� = a���1�

for all a� a′� a′′ ∈ A and � ∈ G. Furthermore, the extension AG ⊆ A is G-Galois ifand only if

⊕�∈G Ae� is a Galois coring (with respect to the grouplike element∑

�∈G v�). Following Caenepeel and De Groot (2005, Definition 2.4), a rightA-module M together with right A-module maps ��� � M → M���∈G is called apartial Galois descent datum, provided �1 = M and each of the �� restricted to Me�−1

is an isomorphism. In view of Caenepeel and De Groot (2005, Proposition 2.5),partial Galois descent data on a right A-module M are in bijective correspondencewith descent cycles Z1�

⊕�∈G Ae��M�. In particular, for any right AG-module N ,

there is a partial Galois descent datum

M = N ⊗AG A� �� � N ⊗AG A → N ⊗AG Ae�� n⊗ a �→ n⊗ ��ae�−1�e��

This is simply the right �-coaction induced from the coaction on A given by thegrouplike element

∑�∈G v�. A partial Galois cohomology of the G-Galois extension

AG ⊆ A with values in the automorphism group AutA�M� of the right A-moduleM = N ⊗AG A is defined as

Hi�G�AutA�M�� �= Di

(⊕�∈G

Ae��M

)� i = 0� 1�

By Theorem 2.4, if A is a faithfully flat AG-module, then Hi�G�AutA�M�� describesequivalence classes of A-twisted forms of N .

As an example take N = AG. In this case M = A and the automorphismgroup AutA�A� can be identified with the group of units ��A� and Aut��A� canbe identified with ��AG�. Furthermore, there is a bijective correspondence betweenright coactions of

⊕�∈G Ae� on A and the set of grouplike elements in

⊕�∈G Ae�

(cf. Brzezinski, 2002, Lemma 5.1). In this way, we obtain

H0�G���A�� = ��AG�� H1�G���A�� = �(⊕

�∈GAe�

)/��A��

where ��A� acts on A from the right by conjugation as in Section 1.2.

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ACKNOWLEDGMENT

I would like to thank Gabriella Böhm for discussions and for invitation andwarm hospitality in Budapest, where the first version of this note was completed.

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Böhm, G., Vercruysse, J. (2007). Morita theory for coring extensions and cleft bicomodules.Adv. Math. 209:611–648.

Brzezinski, T. (2002). The structure of corings. Induction functors, Maschke-type theorem,and Frobenius and Galois-type properties. Algebr. Represent. Theory 5:389–410.

Brzezinski, T., Wisbauer, R. (2003). Corings and Comodules. Cambridge: CambridgeUniversity Press.

Caenepeel, S. (2004). Galois corings from the descent theory point of view. In: Galois Theory,Hopf Algebras, and Semiabelian Categories. Fields Inst. Commun., 43, Providence, RI:Amer. Math. Soc., pp. 163–186.

Caenepeel, S., De Groot, E. (2005). Galois corings applied to partial Galois theory.In: Kalla, S. L., Chawla, M. M., eds. Proceedings of the International Conference onMathematics and Applications, ICMA 2004. Kuwait: Kuwait University.

Caenepeel, S., De Groot, E., Vercruysse, J. (2006). Constructing infinite comatrix coringsfrom colimits. Appl. Categ. Str. 14:539–565.

Caenepeel, S., De Groot, E., Vercruysse, J. (2007). Galois theory for comatrix corings:Descent theory, Morita theory, Frobenius and separability properties. Trans. Amer.Math. Soc. 359:185–226.

Dokuchaev, M., Exel, R. (2005). Associativity of crossed products by partial actions,enveloping actions and partial representations. Trans. Amer. Math. Soc. 357:1931–1952.

Dokuchaev, M., Ferrero, M., Pacques, A. (2007). Partial actions and Galois theory. J. PureAppl. Alg. 208:77–87.

El Kaoutit, L., Gómez-Torrecillas, J. (2003). Comatrix corings: Galois corings, descenttheory, and a structure theorem for cosemisimple corings. Math. Z. 244:887–906.

El Kaoutit, L., Gómez-Torrecillas, J. (2004). Infinite comatrix corings. Int. Math. Res. Not.39:2017–2037.

Gómez-Torrecillas, J., Vercruysse, J. (2007). Comatrix corings and Galois comodules overfirm rings. Algebr. Represent. Theory 10:271–306.

Nuss, P., Wambst, M. (2007). Non-Abelian Hopf cohomology. J. Algebra 312:733–754.Serre, J.-P. (1997). Galois Cohomology. Berlin: Springer.Wisbauer, R. (2006). On Galois comodules. Comm. Algebra 34:2683–2711.

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