• No results found

On Azumaya Galois extensions and skew group rings

N/A
N/A
Protected

Academic year: 2020

Share "On Azumaya Galois extensions and skew group rings"

Copied!
5
0
0

Loading.... (view fulltext now)

Full text

(1)

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

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,SGthe 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 ofSGif it is aG-Galois extension ofSGwhich is an AzumayaCG-algebra. It was shown thatSis an Azumaya Galois extension if and only ifS∗Gis an AzumayaCG-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 CG-module of rankn. Moreover, for the skew group ringSG, whereSis a separable CG-algebra, an expression of the commutator subring ofCinSGis obtained by using Sand its commutator subring inS∗G. Furthermore, letHbe a normal subgroup ofG, Kthe commutator subgroup ofHinG, andHthe inner automorphism group ofSG induced by the elements ofH(Kand(G/H)are similarly defined). Then, it is shown that(S∗G)K

is a(G/K)-Galois extension with a Galois system{m1gj, g1

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 in-tegerninvertible inS, C the center ofS,SG the subring of the elements fixed un-der each element inG, andS∗G the skew group ring ofGover S. LetB be a sub-ring of a sub-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 andaai⊗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

(2)

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)= 11, i=1,...,v and

j=1,...,u. The ringSis called aG-Galois extension ofSGif there exist{c

i,diinS,i= 1,...,kfor some integerk}such thatcidi=1 andaigj(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 separa-ble extension. A skew group ringS∗Gis a ring with a free basis{g

i}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)=gxg1forginHand allxinSG}), 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 com-mutator subring ofCinS∗GwhenSis a separableCG-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(SG)G

, and ZG is a finitely generated and projectiveCG-module ofrankn.

Proof. (i)(ii). SinceS is an Azumaya Galois extension,S∗Gis an AzumayaCG -algebra (that is, Z= CG) and SG 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(VSG(S))=S[6, Prop. 1.2].

(ii)(i). SinceVS∗G(VSG(S))=S,Zis contained inS; and soZis contained inC. But

thenZ=CG. This implies that SG is an AzumayaCG-algebra by (ii). Thus,S is an Azumaya Galois extension [3, Thm. 3.1].

(i)(iii). Since the restriction ofGtoSisG,SGis aG-Galois extension of(SG)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 AzumayaCG-algebra [3, Thm. 3.1]. Moreover, sinceZ=CG,ZGis a freeZ-module of rankn.

(iii)(i). SinceS∗Gis a G-Galois extension of(SG)G

with an inner Galois group

G, it is an H-separable extension of (SG)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(SG)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)=VSG(VSG(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 ele-ments{gi}ofGas generators ofZG, it is not difficult to show that{gi}are free over

(3)

n=n(rank ofZoverCG). This implies thatZ=CG. Therefore,SGis an Azumaya CG-algebra; and soSis an Azumaya Galois extension [3, Thm. 3.1].

Corollary3.2. LetS be a separableCG-algebra. IfVSG(S)is aG-Galois exten-sion, whereGis the inner automorphism group ofV

S∗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}forVSG(S). Then, it is straightforward to check that

{cj;gjdi⊗g−j1, i=1,...,kandj=1,...,mfor some integerskandm}is anH -sep-arable 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, soSGis a separable exten-sion ofS. By hypothesis,Sis a separableCG-algebra, soSGis a separableCG-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 separableCG-algebraS, we next give an expression ofVS∗G(C)in terms ofS andVSG(S)(for more aboutVSG(S), see [1]).

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

(ii) SZ,VS∗G(S), andVSG(C)are Azumaya CZ-algebras contained inS∗G, such that

VS∗G(C)SZ⊗VSG(S), where⊗is over CZ.

Proof. (i) Since S is a separable CG-algebra, C is also a separableCG-algebra. Hence,C⊗Zis a separableZ-algebra, whereis overCG; 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,SGis a separable CG- algebra by the transitivity of separable extensions; and soSGis an AzumayaZ -algebra. But thenVS∗G(CZ)is a separable subalgebra ofS∗Gsuch thatVSG(VSG(CZ))

=CZ [4, Thm. 4.3] (forCZ is a separable subalgebra ofS∗Gby (i)). This implies that the center ofVS∗G(CZ)isCZ. Thus,VSG(CZ)is an AzumayaCZ-algebra. By

hypothe-sis again,Sis a separableCG-algebra, so it is an AzumayaC-algebra. Hence,SCZis an AzumayaCZ-algebra, whereis overC. Thus,SZ is also an AzumayaCZ-algebra. Noting thatSZ⊂VS∗G(CZ), we conclude thatVSG(CZ)SZ⊗VSG(SZ), whereis

overCZ [7, Thm. 4.3]. Moreover, sinceVS∗G(CZ)=VSG(C)andVSG(SZ)=VSG(S),

we conclude thatVS∗G(C)SZ⊗VSG(S), whereis overCZ.

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

Corollary3.4. IfS is an Azumaya Galois extension, thenVS∗G(C)S⊗VSG(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

exten-sion ofVS∗G(CG). In fact, sinceS is aG-Galois extension andS⊂VSG(C),VSG(C)

is aG-Galois extension with the same Galois system asS by noting thatV

(4)

G-invariant (forGis the restriction ofGtoS). Moreover, it is clear that(V

S∗G(C))G=

VS∗G(CG).

4. A Galois system. It is well known that{n−1gi,g1

i |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)G

with a Galois system{m−1g

j, g−j1|gjinH}, wheremis the order ofH.

Proof. (i) We want to show that giKgi−1⊂K for eachgi in G. For anyx in K and y in H, gixgi−1y=gixgi−1ygigi−1=gixzgi−1, where z=g−i1ygi. SinceH is normal inG,zis inH. Hence,xz=zx. But thengixgi−1y=gixzgi−1=gizxgi−1=

gigi−1ygixgi−1=ygixgi−1. This implies thatgixgi−1is inK. Thus,Kis normal inG. (ii) Assume that TrH(gi)=0 for eachginot inK. Thengjgigj1=0, whereH=

{gj|j=1,...,mfor some integerm}; that is,gjgigj−1gi−1gi=gj(gi)(g−j1)

gi

=0,j=1,...,m. Hence,(m−1)g

j(gi)(g−j1)

=0 for eachginot inK. Clearly, for eachgi inK,(m−1)gj(gi)(g−j1)

=1. Thus,{m−1g

j, gj−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)(g1

j )

=0 for eachginot inK, sogjgigj−1gi−1=0. Hence,gjgig−j1=0; that is, TrH(gi)=0 for eachginot inK.

We derive the following corollaries.

Corollary4.2. S∗Ghas a(G/K)-Galois system{n1gi,g1

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 imme-diately from the theorem.

Corollary4.3. S∗G has aG-Galois system{n1g

i,g−i1|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 ofSGandH= {giinG|gi(a)= afor eachainA}. Then

(i) His normal inGand (ii) Denoting (S∗G)H

byB andVG(H)byK,TrK(gi)=0for each gi not in H if

and only if {m−1g

(5)

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 overCG 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{21,21j;1,j}. Hence,SGis a G-Galois extension with the same Galois system.

(2) SinceS∗Gis a separable extension ofS andSis an AzumayaR-algebra,SGis 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=RandCG=C=R.

(6)Zis a freeR-module of rank 2 andZGis a freeCG-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.

References

[1] R. Alfaro and G. Szeto,The centralizer onH-separable skew group rings, Rings, extensions, and cohomology (New York), Lecture Notes in Pure and Appl. Math., vol. 159, Marcel Dekker, Inc., 1994, Basal Hong Kong, pp. 1–7. MR 95g:16027. Zbl 812.16038. [2] ,Skew grouprings which are Azumaya, Comm. Algebra23(1995), no. 6, 2255–2261.

MR 96b:16027. Zbl 828.16030.

[3] ,On Galois extensions of an Azumaya algebra, Comm. Algebra25 (1997), no. 6, 1873–1882. CMP 97 11. Zbl 890.16017.

[4] F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, vol. 181, Springer-Verlag, New York, 1971. MR 43#6199. Zbl 215.36602.

[5] F. R. DeMeyer,Some notes on the general Galois theory of rings, Osaka J. Math.2(1965), 117–127. MR 32#128. Zbl 143.05602.

[6] K. Sugano,Note on semisimple extensions and separable extensions, Osaka J. Math.4(1967), 265–270. MR 37#1412. Zbl 199.07901.

[7] ,On a special type of Galois extensions, Hokkaido Math. J.9(1980), no. 2, 123–128. MR 82c:16036. Zbl 467.16005.

[8] G. Szeto,Separable Subalgebras of a Class of Azumaya Algebras, to appear in Internat. J. Math. Math. Sci.

[9] G. Szeto and Y. F. Wong,On Azumaya projective group rings, Azumaya algebras, ac-tions, and modules, Contemp. Math., vol. 124, Amer. Math. Soc., 1992, pp. 251–256. MR 92m:16024. Zbl 749.16013.

References

Related documents

While DPs may show typical susceptibility to the composite face effect, other effects attributed to holistic face processing may be aberrant; for example, many DPs may show

CSEIT172629 CSEIT172629 | Received 01 Nov 2017 | Accepted 10 Nov 2017 | November December 2017 [(2)6 59 66] International Journal of Scientific Research in Computer Science, Engineering

Furthermore, Benson (2011) conducted a study on the relationship between individual ownership structure and non performing loans of companies listed at the

There are results on the existence and asymptotic esti- mates of solutions for third order ordinary differential equations with singularly perturbed boundary value problems, which

Wiwatanapataphee, “Positive solutions of singular boundary value problems for systems of nonlinear fourth order di ff erential equations,” Nonlinear Analysis: Theory, Methods

In this article, a Quad-Rectangular Shaped Microstrip Antenna (QRSMA) has been designed, fabricated and simulated using HFSS software and tested by Agilent vector analyser model no..

In this paper we presented several classes of cubic Boolean functions which shows good behaviour with respect to second order nonlinearities and obtained efficient lower bounds

Strong association between time watching television and blood glucose control in children and adolescents with type 1 diabetes: response to Margeirsdottir et al... Parvanta SA,