ON HOPF DEMEYER-KANZAKI GALOIS EXTENSIONS

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ON HOPF DEMEYER-KANZAKI GALOIS EXTENSIONS

GEORGE SZETO and LIANYONG XUE Received 15 October 2002

LetHbe a finite-dimensional Hopf algebra over a fieldk,Ba leftH-module algebra, andH^∗the dual Hopf algebra ofH. For anH^∗-Azumaya Galois extensionBwith centerC, it is shown that Bis anH^∗-DeMeyer-Kanzaki Galois extension if and only ifCis a maximal commutative separable subalgebra of the smash product B#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

2000 Mathematics Subject Classification: 16W30, 16H05.

1. Introduction. LetHbe a finite-dimensional Hopf algebra over a fieldk, Ba leftH-module algebra, andH^∗the dual Hopf algebra ofH. In [7], the class of Azumaya Galois extensions of a ring as studied in [1,2] was generalized to H^∗-Azumaya Galois extensions. AnH^∗-Azumaya Galois extensionBwas char- acterized in terms of the smash productB#Hsee [7, Theorem 3.4]. Observing that the commutatorVB(B^H)ofB^H inBis also anH^∗-Azumaya Galois exten- sion (see [7, Lemma 4.1]), in the present paper, we will give a characterization of anH^∗-Azumaya Galois extensionBin terms ofVB(B^H). Moreover, we will in- vestigate the class ofH^∗-Azumaya Galois extensionsBsuch thatVB(B^H)=C, whereC is the center ofB. We note that when H=kG, whereG is a finite automorphism group ofB, such aBis precisely a DeMeyer-Kanzaki Galois ex- tension with Galois groupG[3,6,8,9]. Several equivalent conditions are then given for anH^∗-Azumaya Galois extension being anH^∗-DeMeyer-Kanzaki Ga- lois extension, and the characterization of a commutative Galois algebra as given by Ikehata [5, Theorem 2] is generalized to anH^∗-DeMeyer-Kanzaki Ga- lois extension.

2. Basic definitions and notation. Throughout,H denotes a finite-dimen- sional Hopf algebra over a fieldkwith comultiplication∆and counitε,H^∗the dual Hopf algebra ofH,B a left H-module algebra,C the center ofB, B^H= {b∈B|hb=ε(h)bfor allh∈H}, andB#H the smash product ofBwithH, whereB#H=B⊗kHsuch that, for allb#handb#hinB#H,(b#h)(b#h)= b(h1b)#h2h, where∆(h)=

h1⊗h2.

For a subringAofB with the same identity 1, we denote the commutator subring of A in B by VB(A). We call B a separable extension of A if there

exist{ai,biinB, i=1,2,...,mfor some integerm}such that

aibi=1 and bai⊗bi=

ai⊗bib for allbinBwhere⊗is overA. An Azumaya algebra is a separable extension of its center. A ring B is called a Hirata separable extension ofAifB⊗AB is isomorphic to a direct summand of a finite direct sum ofBas aB-bimodule. A ringBis called anH^∗-Galois extension ofB^HifB is a rightH^∗-comodule algebra with structure mapρ:B→B⊗kH^∗such that β:B⊗B^HB→B⊗kH^∗is a bijection whereβ(a⊗b)=(a⊗1)ρ(b). AnH^∗-Galois extensionB is called anH^∗-Azumaya Galois extension ifB is separable over B^Gwhich is an Azumaya algebra overC^G, and anH^∗-DeMeyer-Kanzaki Galois extension ifBis anH^∗-Azumaya Galois extension andVB(B^H)=C.

Let P be a finitely generated and projective module over a commutative ringR. Then for a prime idealp ofR,Pp (=P⊗RRp)is a free module over Rp (= the local ring ofRatp), and the rank ofPpoverRpis the number of copies ofRpin Pp, that is, rank_Rp(Pp)=mfor some integerm. It is known that the rankR(P )is a continuous function (rankR(P )(p)=rankR_p(Pp)=m) from Spec(R)to the set of nonnegative integers with the discrete topology (see [4, Corollary 4.11, page 31]). We will use the rank_R(P )-function for a finitely generated and projective modulePover a commutative ringR.

3. H^∗-Azumaya Galois extensions. In this section, keeping all notations as given inSection 2, we will characterize anH^∗-Azumaya Galois extensionBin terms of the commutatorVB(B^H)ofB^HinB.

Theorem3.1. IfB=B^H·VB(B^H), then(VB(B^H))^H=C^H.

Proof. SinceC⊂VB(B^H),C^H⊂(VB(B^H))^H. Conversely, sinceVB(B^H)⊂B, (VB(B^H))^H⊂B^H. Hence(VB(B^H))^H⊂B^H∩VB(B^H)⊂the center ofVB(B^H). But B=B^H·VB(B^H), so the center ofVB(B^H)isC. Thus,(VB(B^H))^H⊂C^H.

Theorem3.2. A ringBis anH^∗-Azumaya Galois extension ofB^Hif and only ifB=B^H·VB(B^H)such thatVB(B^H)is anH^∗-Azumaya Galois extension ofC^H andB^His an AzumayaC^H-algebra.

Proof. (⇒)SinceBis anH^∗-Azumaya Galois extension ofB^H, thenVB(B^H) is an H^∗-Azumaya Galois extension of (VB(B^H))^H (see [7, Lemma 4.1]) and B^H is an AzumayaC^H-algebra (see [7, Theorem 3.4]). Moreover, by the proof of [7, Lemma 4.1],B#His an Azumaya C^H-algebra such thatB#H B^H⊗C^H

(VB(B^H)#H) B^H(VB(B^H)#H), where B^H and VB(B^H)#H are Azumaya C^H- algebras. But H is a finite-dimensional Hopf algebra over a fieldk, so B B^H⊗C^HVB(B^H) from the isomorphism B#H B^H⊗C^H(VB(B^H)#H), and so B=B^H·VB(B^H). Hence(VB(B^H))^H=C^H byTheorem 3.1. ThusVB(B^H)is an H^∗-Azumaya GaloisC^H-algebra.

(⇐)SinceVB(B^H)is anH^∗-Azumaya Galois algebra overC^H,VB(B^H)#His an AzumayaC^H-algebra [7, Theorem 3.4]. By hypothesis,B^His an AzumayaC^H- algebra, soB^H⊗C^H(VB(B^H)#H) B^HVB(B^H)#H=B#Hwhich is an Azumaya

C^H-algebra. ThusB#His a Hirata separable extension ofB(see [5, Theorem 1]).

Moreover,VB(B^H)is a separableC^H-algebra (see [7, Theorem 3.4]) andB^H is an AzumayaC^H-algebra by hypothesis, soB^H·VB(B^H) (=B)is also a separable C^H-algebra. ThusBis anH^∗-Azumaya Galois extension ofB^H[7, Theorem 3.4].

Next we generalize the characterization of a commutative Galois algebra as given by Ikehata (see [5, Theorem 2]) to a commutativeH^∗-Galois algebra.

Lemma3.3. IfC is a commutativeH^∗-Galois algebra overC^H, thenC is a maximal commutative subalgebra ofC#H.

Proof. Since C is a commutative H^∗-Galois algebra over C^H, C#H Hom_CH(C,C)[6, Theorem 1.7]. Hence it suffices to show thatVHom_CH(C,C)(CL)

=CL whereCL= {cL,the left multiplication map induced byc∈C}. In fact, CL⊂VHom_CH(C,C)(CL)is clear. Conversely, let f∈VHom_CH(C,C)(CL). Then, for eachc∈C, (cf )(x)=(f c)(x)for allx∈C. Hencecf (x)=f (cx), and so cf (1)=f (c)for allc∈C. Thusf (c)=df(c)for allc∈C, wheredf =f (1)∈ C, that is,f=(df)L∈CL.

Theorem 3.4. Let C be a commutative separable C^H-algebra containing C^Has a direct summand as aC^H-module. Then,Cis a commutativeH^∗-Galois algebra overC^H if and only ifC⊗C^H(C#H) Mn(C), the matrix algebra over Cof ordernwherenis the dimension ofHoverk.

Proof. (⇒)SinceCis anH^∗-Galois algebra overC^H,C#H Hom_CH(C,C) such thatCis finitely generated and projective overC^H[6, Theorem 1.7]. Hence C#His an AzumayaC^H-algebra andCis a maximal commutative subalgebra of the Azumaya C^H-algebra C#H byLemma 3.3. By hypothesis,C is also a separableC^H-algebra, soCis a splitting ring for the AzumayaC^H-algebraC#H such thatC⊗C^H(C#H) Hom_C(C#H,C#H)(see the proof of [4, Theorem 5.5, page 64]). Noting thatC#H=C⊗kHwhich is a freeC-module of ranknwhere n=dim_k(H), we have thatC⊗C^H(C#H) Mn(C).

(⇐)SinceC⊗C^H(C#H) Mn(C), C⊗C^H(C#H)is an AzumayaC-algebra.

By hypothesis,C^H is a direct summand ofC as aC^H-module, soC#H is an AzumayaC^H-algebra [4, Corollary 1.10, page 45]. HenceC#His a Hirata sep- arable extension ofC. ButC is a separableC^H-algebra by hypothesis, soC is anH^∗-Galois algebra overC^H[7, Theorem 3.4].

We remark that the necessity does not need the hypothesis thatC^H is a direct summand ofC.

4. H^∗-DeMeyer-Kanzaki Galois extensions. We recall that B is an H^∗- DeMeyer-Kanzaki Galois extension ofB^HifBis anH^∗-Azumaya Galois exten- sion ofB^H andVB(B^H)=C. In this section, we characterize anH^∗-DeMeyer- Kanzaki Galois extension in terms of the smash productVB(B^H)#Hand prove thatCis a splitting ring for the AzumayaC^H-algebrasVB(B^H)#HandB#H.

Theorem4.1. LetB be an H^∗-Azumaya Galois extension of B^H. Then the following statements are equivalent:

(1) Bis anH^∗-DeMeyer-Kanzaki Galois extension ofB^H; (2) rank_CH(VB(B^H))=rank_CH(C);

(3) Cis a maximal commutative separable subalgebra ofVB(B^H)#H. Proof. (1)⇒(2). It is clear.

(2)⇒(1). Since B is an H^∗-Azumaya Galois extension of B^H, VB(B^H) is an H^∗-Azumaya Galois algebra over C^H by Theorem 3.2such that VB(B^H)is a separable and finitely generated projective module overC^H(see [7, Theorem 3.4]). Hence the rank function rank_CH(VB(B^H))is defined and VB(B^H) is an Azumaya algebra over its center [4, Theorem 3.8, page 55]. ButB=B^H·VB(B^H) byTheorem 3.2, so the center ofVB(B^H)isC. ThusVB(B^H)is an AzumayaC- algebra; and soCis a direct summandVB(B^H)as aC-module. This implies that C is a direct summandVB(B^H)as aC^H-module. Therefore the rank function rank_CH(C)is also defined. Now by hypothesis, rank_CH(VB(B^H))=rank_CH(C), soVB(B^H)=C, that is,Bis anH^∗-DeMeyer-Kanzaki Galois extension ofB^H.

(1)⇒(3). SinceB is anH^∗-DeMeyer-Kanzaki Galois extension ofB^H,B is an H^∗-Azumaya Galois extension such thatVB(B^H)=C. HenceB=B^H·VB(B^H) B^H⊗C^HCsuch thatCis anH^∗-Galois algebra overC^HbyTheorem 3.2, and soC is a separableC^H-algebra containingC^Has a direct summand as aC^H-module [7, Theorem 3.4]. HenceCis a maximal commutative separable subalgebra of C#HwhereC=VB(B^H)byLemma 3.3.

(3)⇒(2). SinceBis anH^∗-Azumaya Galois extension ofB^H,B=B^H·VB(B^H) B^H⊗C^HVB(B^H)such thatVB(B^H)is anH^∗-Azumaya Galois algebra overC^Hby Theorem 3.2. HenceVB(B^H)#H is an AzumayaC^H-algebra andVB(B^H)is an AzumayaC-algebra [7, Theorem 3.4]. By hypothesis,Cis a maximal commu- tative separable subalgebra ofVB(B^H)#H, so

C⊗C^H

VB B^H

#H

Hom_C VB

B^H

#H,VB B^H

#H

(4.1)

(see [4, Theorem 5.5, page 64]). On the other hand,VB(B^H)#H Hom_CH(VB(B^H), VB(B^H))(see [7, Theorem 3.4]). Thus

C⊗C^H

VB B^H

#H

C⊗C^HHom_CH VB

B^H ,VB

B^H Hom_C

C⊗C^HVB

B^H

,C⊗C^HVB

B^H

; (4.2)

and so Hom_C(VB(B^H)#H,VB(B^H)#H) Hom_C(C⊗C^HVB(B^H),C⊗C^HVB(B^H)). This implies thatVB(B^H)#H P⊗C(C⊗C^HVB(B^H))for some finitely gener- ated projectiveC-modulePof rank 1, that is,VB(B^H)#H P⊗C^HVB(B^H). Tak- ing rank_CH( ) both sides, we have thatn·rank_CH(VB(B^H))=(rank_CH(P ))· (rank_CH(VB(B^H))) where n = dim_k(H). But rank_CH(VB(B^H)) is also n, so rank_CH(C)=rank_CH(P )=n=rank_CH(VB(B^H)).

Theorem 4.1 implies that the Azumaya C^H-algebras VB(B^H)#H and B#H have a nice splitting ringCwhich is anH^∗-Galois algebra overC^Hand separa- ble overC^Hsuch thatC⊗C^H(VB(B^H)#H)andC⊗C^H(B#H)are matrix algebras.

Corollary4.2. IfBis anH^∗-DeMeyer-Kanzaki Galois extension ofB^H, then C⊗C^H(VB(B^H)#H) Mn(C), the matrix algebra overC of ordernwheren= dim_k(H).

Proof. By hypothesis,Bis anH^∗-DeMeyer-Kanzaki Galois extension ofB^H, soC (=VB(B^H))is anH^∗-Galois algebra overC^HbyTheorem 3.2. HenceC is a separableC^H-algebra andC#His an AzumayaC^H-algebra [7, Theorem 3.4].

ThusC^His a direct summand ofCas aC^H-module. Therefore,C⊗C^H(C#H) Mn(C)byTheorem 3.4.

Corollary4.3. IfBis anH^∗-DeMeyer-Kanzaki Galois extension ofB^H, then C⊗C^H(B#H) Mn(B), the matrix algebra overBof ordernwheren=dim_k(H).

Proof. ByCorollary 4.2,C⊗C^H(C#H) Mn(C), so

B^H⊗C^HC⊗C^H(C#H) B^H⊗C^HMn(C). (4.3) SinceB=B^H·VB(B^H) B^H⊗C^HVB(B^H)=B^H⊗C^HC, we have that

C⊗C^H(B#H) C⊗C^H

B^H⊗C^HC

#H C⊗C^HB^H⊗C^H(C#H) B^H⊗C^HC⊗C^H(C#H) B^H⊗C^HMn(C) Mn

B^H⊗C^HC Mn(B).

(4.4)

Acknowledgments. This paper was written under the support of a Cater- pillar Fellowship at Bradley University. The authors would like to thank Cater- pillar Inc. for the support.

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George Szeto: Department of Mathematics, Bradley University, Peoria, IL 61625, USA E-mail address:[email protected]

Lianyong Xue: Department of Mathematics, Bradley University, Peoria, IL 61625, USA E-mail address:[email protected]