Skew group rings which are Galois

©Hindawi Publishing Corp.

SKEW GROUP RINGS WHICH ARE GALOIS

GEORGE SZETO and LIANYONG XUE

(Received 20 March 1998)

Abstract.LetS∗Gbe a skew group ring of a finite groupGover a ringS. It is shown that ifS∗Gis anG_{-Galois extension of(S∗G)}G

, whereG_{is the inner automorphism group} ofS∗Ginduced by the elements inG, thenSis aG-Galois extension ofSG_{. A necessary} and sufficient condition is also given for the commutator subring of(S∗G)G

inS∗G to be a Galois extension, where(S∗G)G

is the subring of the elements fixed under each element inG_.

Keywords and phrases. Skew group rings, Azumaya algebras, Galois extensions,H -separa-ble extensions.

2000 Mathematics Subject Classification. Primary 16S35; Secondary 16W20.

1. Introduction. LetSbe a ring with 1,Cthe center ofS,Ga finite automorphism group of S of ordern invertible in S, SG _{the subring of the elements fixed under} each element inG, S∗G a skew group ring of groupG over S, and G _{the inner} automorphism group ofS∗Ginduced by the elements inG, that is,g_(x)=gxg−1 for eachginGandxinS∗G, so the restriction ofG_{toS isG. In [3, 2], aG-Galois} extensionS ofSG_{which is an AzumayaC}G_{-algebra is characterized in terms of the} AzumayaCG_{-algebra S∗G and theH-separable extensionS∗G ofS, respectively,} and the properties of the commutator subring of S in S∗G are given in [1]. It is clear thatSis aG-Galois extension ofSG_{implies thatS∗Gis aG-Galois extension of}

(S∗G)G_{with the same Galois system asS. In the present paper, we prove the converse} theorem: ifS∗Gis aG_{-Galois extension of(S∗G)}G

, thenS is aG-Galois extension ofSG_{. Moreover, for a G-Galois extensionS∗Gof (S∗G)}G

which is a projective separableCG_{-algebra,Scan be shown to be aG-Galois extension ofS}G_{which is also a} projective separableCG_{-algebra. Then a sufficient condition on(S∗G)}G

is given forS to be aG-Galois extension ofSG_{which is an AzumayaC}G_{-algebra, and an equivalent} condition onSG _{is obtained for the commutator subring of(S∗G)}G

inS∗Gto be a G-Galois extension.

2. Preliminaries. Throughout, we keep the notation as given in the introduction. LetBbe a subring of a ringAwith 1. Following [3, 2],Ais called a separable extension of Bif there exist{ai,biinA, i=1,2,...,mfor some integerm}such thataibi=1, and sai⊗bi=ai⊗bis for all s inA, where⊗ is overB. An Azumaya algebra is a separable extension of its center. A ringA is called anH-separable extension ofB ifA⊗BAis isomorphic to a direct summand of a finite direct sum of Aas an

Then it is called a G-Galois extension ofSG _{if there exist elements{c}

i,diinS, i= 1,2,...,kfor some integerk}such thatcigj(di)=δ1,j, whereG= {g1,g2,...,gn} with identityg1, for eachgj∈G. Such a set{ci,di}is called aG-Galois system forS.

3. Galois skew group rings. In this section, we show that a G_{-Galois extension} skew group ringS∗Gimplies aG-Galois extensionS. More results are obtained forSG when(S∗G)G

is a projective separableCG_{-algebra, and anH-separableS}G_-extension, respectively.

Theorem3.1. IfS∗G is aG_{-Galois extension of (S∗G)}G

, thenS is aG-Galois extension ofSG_.

Proof. Let{ui,vi|i=1,2,...,m}be aG-Galois system ofS∗Gover(S∗G)G, that is,ui andviare elements ofS∗Gsatisfyingmi=1uig(vi)=mi=1uigvig−1=

δ1,g. Letwi=h∈Ghvi, i=1,2,...,m. Thengwi=h∈Gghvi=wi. Since{h|h∈

G}is a basis of S∗Gover S, we haveui=h∈Gsh(ui)hand wi=h∈Gsh(wi)h, i= 1,2,...,m, for somes(ui)

h ,s(wh i)inS. Letxi=h∈Gsh(ui)andyi=s(w1 i), i=1,2,...,m. We prove that{xi,yi|i=1,2,...,m}is aG-Galois system for S overSG. First, we prove that

(1) gs(wi) h

=s(wi)

gh for alli=1,2,...,mand allg,h∈G, (2) mi=1uiwi=1.

For (1), sincewi=gwi, we have

k∈G

s(wi) k k=

h∈G

s(wi) h h=g

h∈G

s(wi) h h=

h∈G

gs(wi) h h=

h∈G

gs(wi) h

gh. (3.1)

Since{k|k∈G}is a basis ofS∗GoverS,gs(wi) h

=s(wi) gh .

For (2), since{ui,vi|i=1,2,...,m}is aG-Galois system forS∗Gover(S∗G)G,

m

i=1uih(vi)mi=1uihvih−1=δ1,h. Therefore, 1=

h∈G

δ1,hh=

h∈G

 m

i=1

uihvih−1

 _h=

h∈G m

i=1 uihvi=

m i=1 ui h∈G

hvi= m

i=1

uiwi. (3.2)

Next, we prove that{xi,yi|i=1,2,...,m}is aG-Galois system forSoverSG. By using (1) and (2), we get

1= m

i=1 uiwi=

m i=1   h∈G

s(ui) h h     k∈G

s(wi) k k   = m i=1 h∈G k∈G

s(ui)

h hsk(wi)k= m i=1 h∈G k∈G

s(ui) h h

s(wi) k hk = m i=1 g∈G hk=g

s(ui) h h

s(wi) k hk= m i=1 g∈G hk=g

s(ui)

h s(whki)hk by (1)

= m i=1 g∈G h∈G

s(ui)

h shh(wi−)1_ghh−1g

sincehk=g, k=h−1_g

= m i=1 g∈G h∈G

s(ui)

h sg(wi)g=

g∈G

 m

i=1

h∈G

s(ui) h s(wg i)

 _g.

Hence,mi=1h∈Gsh(ui)sg(wi)=δ1,g. Butxi=h∈Gsh(ui),yi=s1(wi), andg

s(wi)

1

=s(wi) g by (1). So,

m

i=1

xigyi= m

i=1

h∈G

s(ui) h g

s(wi)

1

= m

i=1

h∈G

s(ui)

h sg(wi)=δ1,g. (3.4)

We show more properties of theG-Galois extensionSofSG_{whenS∗Gis aG-Galois} extension of(S∗G)G

which possesses a property.

Theorem3.2. IfS∗Gis aG_{-Galois extension of (S∗G)}G

which is a projective separableCG_{-algebra, thenS is aG-Galois extension ofS}G _{which is also a projective}

separableCG_-algebra.

Proof. SinceS∗Gis aG_{-Galois extension of(S∗G)}G

,Sis aG-Galois extension ofSG_{by Theorem 3.1. Again, sinceS∗Gis aG-Galois extension of(S∗G)}G

, it is a separable extension [5]. Also,(S∗G)G

is a separableCG_{-algebra, soS∗Gis a separable}

CG_{-algebra by the transitivity of separable extensions. Next, we claim that S is also} a separableCG_{-algebra. In fact, sincenis a unit inS, the trace map:(1/n)(tr}

G( )):

S→SG_{→0 is a splitting homomorphism of the imbedding homomorphism ofS}G_into

S as a two sidedSG_{-module. Hence,S}G_{is a direct summand ofS. SinceS is a direct} summand ofS∗Gas anSG_-bimodule,SG _{is so ofS∗Gas anS}G_{-module. Moreover,}

S is a finitely generated and projective SG_{-module (for S is a G-Galois extension of}

SG_{), soS∗Gis a finitely generated and projective S}G_{-module by the transitivity of} the finitely generated and projective modules. This implies that SG _{is a projective} separableCG_{-algebra by [5, proof of Lem. 2, p. 120].}

Theorem3.3. If

(i) S∗Gis aG_{-Galois extension of(S∗G)}G (ii) (S∗G)G

is anH-separable extension ofSG _{which is a separableC}G_-algebra,

thenS is aG-Galois extension ofSG_{which is an AzumayaC}G_-algebra. Proof. SinceS∗Gis aG_{-Galois extension of(S∗G)}G

with an inner Galois group G_{,S∗Gis anH-separable extension of(S∗G)}G

[7, Prop. 4]. By hypothesis,(S∗G)G is anH-separable extension ofSG_{, soS∗Gis anH-separable extension ofS}G_{by the} transitivity ofH-separable extensions. Noting thatnis a unit ofS, we haveSG_{is an}

SG_{-direct summand ofS. ButS is a direct summand ofS∗G as anS}G_{-module, so}

SG _{is a direct summand ofS∗G as anS}G_{-module. Thus, V}

S∗G(VS∗G(SG))=SG [6, Prop. 1.2]. This implies that the center ofS∗Gis contained inSG_{, and so the center} ofS∗GisCG_{. Therefore,S∗Gis an AzumayaC}G_{-algebra. Thus,S}G _{is an Azumaya}

CG_{-algebra. Consequently,Sis aG-Galois extension ofS}G _{which is an AzumayaC}G -algebra [2, Thm. 3.1].

4. Galois commutator subrings. In [7], the class ofG-Galois andH-separable exten-sion was studied. LetAbe aG-Galois andH-separable extension ofAG_{and letV}

A(AG) be the commutator subring ofAG_{inA. Then,V}

In the following, we denote the center ofG byP and the centerS∗G byZ. By a direct computation, we have the following.

Lemma4.1. (1)LetI= {gi∈G|gi(d)=dfor eachd∈ZG}. ThenI=P. (2)Letxbe an element in(S∗G)G

. Thenx=ni=1sigisuch thatgj(si)=skwhenever

gjgig−j1=gj(gi)=gk∈G.

Lemma4.2. Assume thatS∗Gis aG_{-Galois extension of(S∗G)}G

andan Azumaya Z-algebra. ThenVS∗G((S∗G)G)is a central(G/P)-Galois algebra if andonly ifSPG=

(S∗G)G G.

Proof. Since n is a unit in Z and S∗G is an Azumaya Z-algebra, VS∗G((S∗

G)G

)=VS∗G(VS∗G(ZG))=ZG by the commutator theorem for Azumaya algebras [4, Thm. 4.3] (forZGis a separableZ-subalgebra). Moreover, sinceS∗Gis aG_-Galois extension of(S∗G)G

with an inner Galois groupG_{, it is anH-separable extension} of(S∗G)G

[7, Prop. 4]. Hence,VS∗G((S∗G)G)(=ZG)is a central(G/P)-Galois al-gebra if and only if(S∗G)P

=(S∗G)G

ZGby [7, Lem. 4.1(1) and Thm. 6.3]. Clearly, Z⊂(S∗G)G

, and so(S∗G)G

ZG=(S∗G)G

G. Noting thatP is the center ofG, we have(S∗G)P

=SP_{G. Thus, the lemma holds.}

Theorem4.4. Assume thatS∗Gis aG_{-Galois extension of(S∗G)}G

andan Azu-mayaZ-algebra. ThenZGis a central(G_{/P)-Galois algebra if andonly if, for every}

s∈SP_{, there exists ann×nmatrix[s}

k,h]k,h∈Gfor somesk,hinS such that (1) h∈Gsgh−1_,h=δ1_,gs(therefore,s=_h∈Gs_h−1_,h), and

(2) g(sk,h)=sgkg−1_,hfor everyg∈G.

Proof. (⇒) Assume thatZGis a central(G_{/P)-Galois algebra. Then by Lemma} 4.2,SP_G=(S∗G)G

G. Therefore, for everys∈SP_,s=s1∈SP_G=(S∗G)G

G. Hence, there existsk∈Gsk,hk∈(S∗G)Gfor eachh∈Gsuch that

s=s1= h∈G

 

k∈G

sk,hk

 _h=

g∈G

 

kh=g

sk,h

 _g=

g∈G

 

h∈G

sgh−1_,h



_{g. (4.1)}

Since{g|g∈G}is a basis ofS∗GoverS, we haveh∈Gsgh−1_,h=δ1,gsand, therefore,

h∈Gsh−1_,h=s. Furthermore, for eachh∈G,_k∈Gsk,hk∈(S∗G)G, i.e.,k∈Gsk,hk=

gk∈Gsk,hkg−1=k∈Gg(sk,h)gkg−1for everyg∈G. Therefore,g(sk,h)=sgkg−1_,hfor

everyg∈Gsince{k|k∈G}is a basis ofS∗GoverS.

(⇐) Assume that, for everys∈SP_{, there exists ann×nmatrix[sk,h]k,h∈Gsuch that}

h∈Gsgh−1_,h=δ1,gsandg(sk,h)=sgkg−1_,hfor everyg∈G. Then

g

 

k∈G

sk,hk



_g−1₌

k∈G

g(sk,h)gkg−1=

k∈G

sgkg−1_,hgkg−1=

k∈G

sk,hk, (4.2)

that isk∈Gsk,hk∈(S∗G)Gfor everyh∈G. Therefore,

s= g∈G

δ1,gsg=

g∈G

 

h∈G

sgh−1_,h

 _g=

g∈G

 

kh=g

sk,h

 _g

=

h∈G

 

k∈G

sk,hk



_h∈(S∗G)G G.

Hence, for everys ∈SP _{and every g∈G, sg∈(S∗G)}G

GG=(S∗G)G

G, that is SP_G⊆(S∗G)G

G.

On the other hand, for anyk∈Gskk∈(S∗G)G, we have

k∈G

skk=g

k∈G

skkg−1=

k∈G

g(sk)gkg−1 for everyg∈G. (4.4)

Therefore,g(sk)=sgkg−1for everyg∈Gsince{k|k∈G}is a basis ofS∗GoverS.

In particular, for everyp∈P, p(sk)=spkp−1 =sk, i.e., sk∈SP for everyk∈G and, therefore,k∈Gskk∈SPGifk∈Gskk∈(S∗G)G. Hence,(S∗G)G⊆SPG. Therefore,

(S∗G)G

G⊆ SP_{GG= S}P_{G. Hence, S}P_G=(S∗G)G

G. So,(S∗G)P

=SP_{G =(S∗}

G)G

G=(S∗G)G

ZG. Consequently, by Lemma 4.2,ZG is a central(G_/P)-Galois algebra.

Acknowledgement. This work was supported by a Caterpillar Fellowship at Bradley University. We would like to thank Caterpillar Inc. for the support.

References

[1] R. Alfaro and G. Szeto,The centralizer onH-separable skew group rings, Rings, Exten-sions, and Cohomology (Evanston, IL, 1993), Dekker, New York, 1994, pp. 1–7. MR 95g:16027. Zbl 812.16038.

[2] ,Skew group rings 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. MR 98h:13007. Zbl 890.16017.

[4] F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, vol. 181, Springer-Verlag, Berlin, 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 andseparable 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.