ON AZUMAYA GALOIS EXTENSIONS AND SKEW GROUP RINGS

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Vol. 22, No. 1 (1999) 91–95 S 0161-17129922091-1

ON AZUMAYA GALOIS EXTENSIONS AND SKEW GROUP RINGS

GEORGE SZETO

(Received 6 November 1997 and in revised form 22 December 1997)

Abstract.Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

Keywords and phrases. Azumaya algebras, Galois extensions, H-separable extensions, skew group rings.

1991 Mathematics Subject Classiﬁcation. 16S30, 16W20.

1. Introduction. LetSbe a ring with 1,Ga finite automorphism group ofSof order nfor some integerninvertible inS,S^Gthe subring of the elements fixed under each element inG,Cthe center ofS, andS^∗Gthe skew group ring ofGoverS. In [3] and [2], Sis called an Azumaya Galois extension ofS^Gif it is aG-Galois extension ofS^Gwhich is an AzumayaC^G-algebra. It was shown thatSis an Azumaya Galois extension if and only ifS^∗Gis an AzumayaC^G-algebra. The purpose of the present paper is to give two more characterizations of an Azumaya Galois extension in terms of the Azumaya skew group ringS^∗G. We show thatS is an AzumayaG-Galois extension if and only ifS^∗Gis an Azumaya algebra over its center Z, aG-Galois extension with an inner Galois groupGinduced by the elements ofG, andZGis a finitely generated projective C^G-module of rankn. Moreover, for the skew group ringS^∗G, whereSis a separable C^G-algebra, an expression of the commutator subring ofCinS^∗Gis obtained by using Sand its commutator subring inS^∗G. Furthermore, letHbe a normal subgroup ofG, Kthe commutator subgroup ofHinG, andHthe inner automorphism group ofS^∗G induced by the elements ofH(Kand(G/H)are similarly defined). Then, it is shown that(S^∗G)^Kis a(G/K)-Galois extension with a Galois system{m⁻¹gj, g⁻¹_j /gjinH}

if and only if TrG(gi)=0 for eachginot inK, wheremis the order ofHfor some integermand TrG(gi)is the trace ofGatgi.

2. Preliminaries. Throughout, letSbe a ring with 1,G= {g1,...,gn}for some integerninvertible inS, C the center ofS,S^G the subring of the elements ﬁxed under each element inG, andS^∗G the skew group ring ofGover S. LetB be a subring of a ring A. We callA a separable extension of B if there exist {ai,bi} in A, i=1,...,mfor some integer m, such that

aibi=1 and

aai⊗bi =

ai⊗bia for all a in A, where ⊗ is over B and {ai,bi}is called a separable system for A.

An Azumaya algebra is a separable extension over its center. A ringA is called an H-separable extension ofB ifA⊗A is a direct summand of a ﬁnite direct sum of Aas anA-bimodule, where⊗ overB. Denote the commutator subring ofBinAby

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VA(B). An H-separable extensionA over B is equivalent to the existence of anH- separable system{diinVA(B);

(xij⊗yij)inVA⊗A(A)},j=1,...,uandi=1,...,v for some integers u and v such that

di

(xij⊗yij)

= 1⊗1, i=1,...,v and j=1,...,u. The ringSis called aG-Galois extension ofS^Gif there exist{ci,diinS,i= 1,...,kfor some integerk}such that

cidi=1 and

aigj(bi)=0 for eachgj=1, where{ci,di}is called a G-Galois system forS. It is well known that an Azumaya algebra is anH-separable extension and that anH-separable extension is a separable extension. A skew group ringS^∗Gis a ring with a free basis{gi}overSsuch that gis=(gi(s))gifor eachgiinGandsinS. We denote the center ofS^∗GbyZ, the inner automorphism group ofS^∗Ginduced by the elements of the subgroupHofGbyH (= {g/g(x)=gxg⁻¹forginHand allxinS^∗G}), and the commutator subgroup ofHinGbyVG(H).

3. Skew group rings. In this section, keeping the notations of Section 2, we give two characterizations of an Azumaya Galois extension and an expression of the commutator subring ofCinS^∗GwhenSis a separableC^G-algebra.

Theorem3.1. The following statements are equivalent:

(i) S is an Azumaya Galois extension,

(ii) S^∗Gis an AzumayaZ-algebra andS satisﬁes the double centralizer property in S^∗G, and

(iii) S^∗Gis an AzumayaZ-algebra and aG-Galois extension of(S^∗G)^G, and ZG is a ﬁnitely generated and projectiveC^G-module ofrankn.

Proof. (i)⇒(ii). SinceS is an Azumaya Galois extension,S^∗Gis an AzumayaC^G- algebra (that is, Z= C^G) and S^∗G is an H-separable extension of S [3, Thm. 3.1].

Noting that S is a direct summand of S^∗G as a left S-module, we conclude that VS^∗G(VS^∗G(S))=S[6, Prop. 1.2].

(ii)⇒(i). SinceVS^∗G(VS^∗G(S))=S,Zis contained inS; and soZis contained inC. But thenZ=C^G. This implies that S^∗G is an AzumayaC^G-algebra by (ii). Thus,S is an Azumaya Galois extension [3, Thm. 3.1].

(i)⇒(iii). Since the restriction ofGtoSisG,S^∗Gis aG-Galois extension of(S^∗G)^G with the same Galois system as S (for S is G-Galois). Also, by hypothesis, S is an Azumaya Galois extension, soS^∗Gis an AzumayaC^G-algebra [3, Thm. 3.1]. Moreover, sinceZ=C^G,ZGis a freeZ-module of rankn.

(iii)⇒(i). SinceS^∗Gis a G-Galois extension of(S^∗G)^G with an inner Galois group G, it is an H-separable extension of (S^∗G)^G [7, Cor. 3]. But n is a unit in S, so VS^∗G((S^∗G)^G)is a separable Z-algebra and a finitely generated and projectiveZ- module of rankn[7, Prop. 4]. Moreover,S^∗Gis aG-Galois extension of(S^∗G)^G, so it is finitely generated and projective(S^∗G)^G-module. Sincenis a unit inS,ZGis a sep- arableZ-algebra. But thenVS^∗G((S^∗G)^G)=VS^∗G(VS^∗G(ZG))=ZGby the commutator theorem for Azumaya algebras [4, Thm. 4.3]. Therefore, ZG is a finitely generated and projectiveZ-module of rankn[1, Prop. 4]. From the fact that there aren elements{gi}ofGas generators ofZG, it is not difficult to show that{gi}are free over Z. Hence,Z is a finitely generated and projective C^G-module. Thus, the rank ofZG over C^G is a product of the rank of ZG overZ and the rank ofZ overC^G; that is,

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n=n(rank ofZoverC^G). This implies thatZ=C^G. Therefore,S^∗Gis an Azumaya C^G-algebra; and soSis an Azumaya Galois extension [3, Thm. 3.1].

Corollary3.2. LetS be a separableC^G-algebra. IfVS^∗G(S)is aG-Galois exten- sion, whereGis the inner automorphism group ofVS^∗G(S)induced by and isomorphic withG, thenSis an Azumaya Galois algebra.

Proof. SinceVS^∗G(S) is aG-Galois extension, there exists a G-Galois system {ci,diinVS^∗G(S)|i=1,...,k}forVS^∗G(S). Then, it is straightforward to check that {cj;

gjdi⊗g⁻¹_j , i=1,...,kandj=1,...,mfor some integerskandm}is anH-separable system forS^∗G over S [1, Thm. 1]. Hence, S satisﬁes the double centralizer property inS^∗G[7, Prop. 1.2]. Moreover,nis a unit inS, soS^∗Gis a separable extension ofS. By hypothesis,Sis a separableC^G-algebra, soS^∗Gis a separableC^G-algebra by the transitivity of separable extensions. But thenS^∗G is an AzumayaZ-algebra.

Therefore,Sis an Azumaya Galois extension by Theorem 3.1.

For the skew group ringS^∗GofGover a separableC^G-algebraS, we next give an expression ofVS^∗G(C)in terms ofS andVS^∗G(S)(for more aboutVS^∗G(S), see [1]).

Theorem3.3. IfSis a separableC^G-algebra, then (i) CZ is a commutative separable subalgebra ofS^∗Gand

(ii) SZ,VS^∗G(S), andVS^∗G(C)are Azumaya CZ-algebras contained inS^∗G, such that VS^∗G(C)SZ⊗VS^∗G(S), where⊗is over CZ.

Proof. (i) Since S is a separable C^G-algebra, C is also a separableC^G-algebra.

Hence,C⊗Zis a separableZ-algebra, where⊗is overC^G; and so the homomorphic imageCZ ofC⊗Zis also a separableZ-algebra. Clearly,CZ is commutative.

(ii) Sincenis a unit inS,S^∗Gis a separableS-extension. Hence,S^∗Gis a separable C^G- algebra by the transitivity of separable extensions; and soS^∗Gis an AzumayaZ- algebra. But thenVS^∗G(CZ)is a separable subalgebra ofS^∗Gsuch thatVS^∗G(VS^∗G(CZ))

=CZ [4, Thm. 4.3] (forCZ is a separable subalgebra ofS^∗Gby (i)). This implies that the center ofVS^∗G(CZ)isCZ. Thus,VS^∗G(CZ)is an AzumayaCZ-algebra. By hypothesis again,Sis a separableC^G-algebra, so it is an AzumayaC-algebra. Hence,S⊗CZis an AzumayaCZ-algebra, where⊗is overC. Thus,SZ is also an AzumayaCZ-algebra.

Noting thatSZ⊂VS^∗G(CZ), we conclude thatVS^∗G(CZ)SZ⊗VS^∗G(SZ), where⊗is overCZ [7, Thm. 4.3]. Moreover, sinceVS^∗G(CZ)=VS^∗G(C)andVS^∗G(SZ)=VS^∗G(S), we conclude thatVS^∗G(C)SZ⊗VS^∗G(S), where⊗is overCZ.

By [3, Thm. 3.1], ifS is an Azumaya Galois extension, thenS^∗Gis an AzumayaC^G- algebra (that is,Z=C^G) andS is a separableC^G-algebra. Thus, we have the following result.

Corollary3.4. IfS is an Azumaya Galois extension, thenVS^∗G(C)S⊗VS^∗G(S) as AzumayaC-algebras, where⊗is overCsuch thatVS^∗G(C)is aG-Galois extension ofVS^∗G(CG).

Proof. By the above remark, it suﬃces to show thatVS^∗G(C)is aG-Galois extension ofVS^∗G(CG). In fact, sinceS is aG-Galois extension andS⊂VS^∗G(C),VS^∗G(C) is aG-Galois extension with the same Galois system asS by noting thatVS^∗G(C)is

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G-invariant (forGis the restriction ofGtoS). Moreover, it is clear that(VS^∗G(C))^G= VS^∗G(CG).

4. A Galois system. It is well known that{n⁻¹gi,g_i⁻¹|giinG}is a separable system for a separable group ringRGover a ringR with 1, whereG= {gi|i=1,...,n}for some integerninvertible inR, for a separable skew group ringS^∗GoverSand for a separable projective group ringRGfoverRas deﬁned in [9]. In this section, we give an equivalent condition for(S^∗G)^K to have a(G/K)-Galois system similar to the above separable system for a normal subgroupKofG.

Theorem4.1. LetHbe a normal subgroupofGandVG(H)=K. Then (i) Kis a normal subgroupofGand

(ii) TrH(gi)=0for eachgi not inK if and only if(S^∗G)^K is a(G/K)-Galois ex- tension of(S^∗G)^Gwith a Galois system{m⁻¹gj, g⁻¹_j |gjinH}, wheremis the order ofH.

Proof. (i) We want to show that giKg_i⁻¹⊂K for eachgi in G. For anyx in K and y in H, gixg⁻¹_i y=gixg_i⁻¹ygig_i⁻¹=gixzg_i⁻¹, where z=g⁻¹_i ygi. SinceH is normal inG,zis inH. Hence,xz=zx. But thengixg⁻¹_i y=gixzg⁻¹_i =gizxg_i⁻¹= gig_i⁻¹ygixg_i⁻¹=ygixg⁻¹_i . This implies thatgixg_i⁻¹is inK. Thus,Kis normal inG.

(ii) Assume that TrH(gi)=0 for eachginot inK. Then

gjgig_j⁻¹=0, whereH= {gj|j=1,...,mfor some integerm}; that is,

gjgig_j⁻¹g_i⁻¹gi= gj

(gi)(g⁻¹_j ) gi

=0,j=1,...,m. Hence,(m⁻¹) gj

(gi)(g⁻¹_j )

=0 for eachginot inK. Clearly, for eachgi inK,(m⁻¹)

gj

(gi)(g⁻¹_j )

=1. Thus,{m⁻¹gj, g_j⁻¹|gjinH}is a (G/K)- Galois system for(S^∗G)^K(forH⊂(S^∗G)^K), wheremis the order ofH.

Conversely,(m⁻¹) gj

(gi)(g_j⁻¹)

=0 for eachginot inK, so

gjgig_j⁻¹g_i⁻¹=0.

Hence,

gjgig⁻¹_j =0; that is, TrH(gi)=0 for eachginot inK.

We derive the following corollaries.

Corollary4.2. S^∗Ghas a(G/K)-Galois system{n⁻¹gi,g⁻¹_i |giinG}, whereKis the center ofG, if and only ifTrG(gi)=0for eachginot inK.

Proof. LetHbeG. ThenK=the center ofG; and so the corollary follows immediately from the theorem.

Corollary4.3. S^∗G has aG-Galois system{n⁻¹gi,g⁻¹_i |giinG}if and only if TrG(gi)=0for eachgi=1.

Proof. This is the case of the theorem that the center ofGis trivial.

We derive an equivalent condition for a Galois subring ofS^∗G arising from aG- invariant subring.

Corollary4.4. LetAbe aG-invariant subring ofS^∗GandH= {giinG|gi(a)= afor eachainA}. Then

(i) His normal inGand

(ii) Denoting (S^∗G)^HbyB andVG(H)byK,TrK(gi)=0for each gi not in H if and only if {m⁻¹gj,g⁻¹_j |gjinK}is a(G/H)-Galois system for B, where m is the

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order ofK.

Proof. Part (i) is straightforward and part (ii) follows immediately from Theo- rem 4.1.

We conclude the present paper with an example of an Azumaya skew group ring S^∗G which is aG-Galois extension such that the rank of ZG overC^G is notn(see Theorem 3.1(iii)). Hence,S is not an Azumaya Galois extension by Theorem 3.1.

LetR be the real ﬁeld,S=R[i,j,k]the quaternion algebra overR, andG= {1,g| g(x)=ix(i)⁻¹for eachxinS}. Then

(1)S is aG-Galois extension with a Galois system{2⁻¹,2⁻¹j;1,−j}. Hence,S^∗Gis a G-Galois extension with the same Galois system.

(2) SinceS^∗Gis a separable extension ofS andSis an AzumayaR-algebra,S^∗Gis a separableR-algebra. Hence,S^∗Gis an AzumayaZ-algebra.

(3) The centerZofS^∗Gis(R+Ri)by direct computation.

(4)ZGis free overZby direct veriﬁcation.

(5)C=RandC^G=C=R.

(6)Zis a freeR-module of rank 2 andZGis a freeC^G-module of rank 4 (=2=the order ofG), so one of the three conditions in Theorem 3.1(iii) does not hold.

Acknowledgement. This work was supported by the Caterpillar Fellowship at Bradley University during the author’s sabbatical leave in Fall, 1997. The present paper was revised according to the suggestions of the referee. The author would like to thank the referee for his valuable suggestions.

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Szeto: Department of Mathematics, Bradley University, Peoria, Illinois61625, USA

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