THE GALOIS ALGEBRAS AND THE AZUMAYA GALOIS EXTENSIONS

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THE GALOIS ALGEBRAS AND THE AZUMAYA GALOIS EXTENSIONS

GEORGE SZETO and LIANYONG XUE Received 26 October 2001

LetB be a Galois algebra over a commutative ringR with Galois groupG,Cthe center ofB,K= {g∈G|g(c)=cfor allc∈C},J_g= {b∈B|bx=g(x)bfor allx∈B}for eachg∈K, andB_K=(⊕

g∈KJ_g). ThenB_Kis a central weakly Galois algebra with Galois group induced byK. Moreover, an Azumaya Galois extensionBwith Galois groupKis characterized by usingB_K.

2000 Mathematics Subject Classification: 16S35, 16W20.

1. Introduction. LetBbe a Galois algebra over a commutative ringR with Galois group Gand C the center ofB. The class of Galois algebras has been investigated by DeMeyer [2], Kanzaki [6], Harada [4,5], and the authors [7]. In [2], it was shown that if R contains no idempotents but 0 and 1, thenB is a central Galois algebra with Galois groupK andC is a commutative Galois algebra with Galois groupG/K whereK= {g∈G|g(c)=cfor allc∈C}[2, Theorem 1]. This fact was extended to the Galois algebraBoverRcontaining more than two idempotents [6, Proposition 3], and generalized to any Galois algebraB [7, Theorem 3.8] by using the Boolean algebraBagenerated by{0,eg|g∈Gfor a central idempotenteg}whereBJg=Beg

andJg= {b∈B|bx=g(x)bfor allx∈B}for eachg∈G [6]. The purpose of this paper is to show that there exists a subalgebraB_K ofB such that B_K is a central weakly Galois algebra with Galois groupK|BK induced byK where a weakly Galois algebra was defined in [8] and thatBKB^Kis an Azumaya weakly Galois extension with Galois groupK|BKB^K where an Azumaya Galois extension was studied in [1]. Thus some characterizations of an Azumaya Galois extensionBofB^Kwith Galois groupK are obtained, and the results as given in [2,6] are generalized.

2. Definitions and notations. Throughout, letB be a Galois algebra over a com- mutative ring R with Galois groupG, C the center of B, and K= {g∈G |g(c)= cfor allc∈C}. We keep the definitions of a Galois extension, a Galois algebra, a cen- tral Galois algebra, a separable extension, and an Azumaya algebra as defined in [7].

An Azumaya Galois extensionAwith Galois group Gis a Galois extensionAofA^G which is aC^G-Azumaya algebra whereCthe center ofA[1]. A weakly Galois exten- sionAwith Galois groupGis a finitely generated projective left moduleAoverA^G such thatA_lGHom_AG(A,A)whereA_l= {a_l,a left multiplication map bya∈A}[8].

We call thatAis a weakly Galois algebra with Galois groupGifAis a weakly Galois extension with Galois groupGsuch thatA^G is contained in the center ofAand that

Ais a central weakly Galois algebra with Galois groupG ifAis a weakly Galois ex- tension with Galois group G such thatA^G is the center of A. An Azumaya weakly Galois extensionAwith Galois groupGis a weakly Galois extensionAofA^Gwhich is aC^G-Azumaya algebra whereCthe center ofA.

3. A weakly Galois algebra. In this section, letBbe a Galois algebra overRwith Galois groupG,Cthe center ofB,B^G= {b∈B|g(b)=bfor allg∈G}, andK= {g∈ G|g(c)=cfor allc∈C}. Then, B= ⊕

g∈GJg =(⊕

g∈KJg)⊕(⊕

g∈KJg) where J_g= {b∈B|bx=g(x)bfor allx∈B}[6, Theorem 1] . We denote⊕

g∈KJ_g byB_K and the center ofB_KbyZ. Clearly,Kis a normal subgroup ofG. We show thatB_Kis an Azumaya algebra overZand a central weakly Galois algebra with Galois groupK|BK.

Theorem3.1. The algebraBKis an Azumaya algebra overZ.

Proof. By the definition ofB_K,B_K= ⊕

g∈KJ_g, soC(=J1)⊂B_K. SinceBis a Galois algebra with Galois groupGandK= {g∈G|g(c)=cfor allc∈C}, the order ofKis a unit inCby [6, Proposition 5]. Moreover,Kis anC-automorphism group ofB, soBK

is aC-separable algebra by [5, Proposition 5]. ThusB_Kis an Azumaya algebra overZ.

In order to show thatBKis a central weakly Galois algebra with Galois groupK|BK, we need two lemmas.

Lemma3.2. LetL= {g∈K|g(a)=afor alla∈B_K}. Then,Lis a normal subgroup ofKsuch thatK(=K/L)is an automorphism group ofB_Kinduced byK(i.e.,K|BKK).

Proof. Clearly,Lis a normal subgroup ofK, so for anyh∈K, h

BK

= ⊕

g∈K

h Jg

= ⊕

g∈K

J_hgh−1= ⊕

g∈hKh⁻¹

Jg= ⊕

g∈K

Jg=BK. (3.1)

ThusK|BKK.

Lemma3.3. The fixed ring ofBKunderK,(BK)^K=Z.

Proof. Letxbe any element in(B_K)^Kandbany element inB_K. Thenb=

g∈Kb_g whereb_g∈J_g for eachg∈K. Hencebx=

g∈Kb_gx=

g∈Kg(x)b_g=

g∈Kxb_g= x

g∈Kbg =xb. Thereforex ∈Z. Thus (BK)^K ⊂Z. Conversely, for any z∈Z and g∈K, we have thatzx=xz=g(z)xfor anyx∈Jg, so(g(z)−z)x=0 for anyx∈Jg. Hence(g(z)−z)J_g= {0}. Noting thatBJ_g=J_gB=B, we have that(g(z)−z)B= {0}, sog(z)=zfor anyz∈Zandg∈K. ThusZ⊂(BK)^K. Therefore(BK)^K=Z.

Theorem3.4. The algebraBKis a central weakly Galois algebra with Galois group K|BKK.

Proof. ByLemma 3.3, it suffices to show that (1)BK is a finitely generated pro- jective module overZ, and (2)(BK)lKHomZ(BK,BK). Part (1) is a consequence of Theorem 3.1. For part (2), sinceB_K is an Azumaya algebra overZ byTheorem 3.1 again, BK⊗ZB^o_K HomZ(BK,BK) [3, Theorem 3.4, page 52] by extending the map (a⊗b)(x)=axb linearly for a⊗b ∈BK⊗ZB^o_K and eachx ∈BK where B_K^o is the

opposite algebra ofBK. By denoting the left multiplication map witha∈BKbyaland the right multiplication map withb∈BKbybr,(a⊗b)(x)=(albr)(x)=axb. Since B_K= ⊕

g∈KJ_g,B_K⊗ZB_K^o=

g∈K(B_K)_l(J_g)_r. Observing that(J_g)_r =(J_g)_lg⁻¹ where g=g|BK∈K|BKK, we have thatB_K⊗ZB_K^o=

g∈K(B_K)_l(J_g)_r=

g∈K(B_K)_l(J_g)_lg⁻¹=

g∈K(B_KJ_g)_lg⁻¹. Moreover, sinceBJ_g=B for eachg∈KandB= ⊕

h∈GJ_h=B_K⊕ (⊕

h∈KJh),BK⊕(⊕

h∈KJh)=B=BJg=BKJg⊕(⊕

h∈KJhJg)such thatBKJg⊂BK

and⊕

h∈KJhJg⊂ ⊕

h∈KJh. HenceBKJg=BKfor eachg∈K. ThereforeBK⊗ZB_K^o=

g∈K(B_KJ_g)_lg⁻¹=

g∈K(B_K)_lg⁻¹=(B_K)_lK. Thus(B_K)_lKHom_Z(B_K,B_K). This com- pletes the proof of part (2). ThusBK is a central weakly Galois algebra with Galois groupK|BKK.

Recall that an algebraAis called an Azumaya weakly Galois extension ofA^Kwith Galois groupKifAis a weakly Galois extension ofA^Kwhich is aC^K-Azumaya algebra whereC is the center ofA. Next, we show thatBKB^K is an Azumaya weakly Galois extension with Galois groupK|BKB^K K. We begin with the following two lemmas aboutBK.

Lemma3.5. The fixed ring ofBunderK,B^K=VB(BK).

Proof. For anyb∈B^Kandx∈Jgfor anyg∈K, we have thatxb=g(b)x=bx, so b∈V_B(J_g)for anyg∈K. Thusb∈V_B(B_K). Conversely, for anyb∈V_B(B_K)andg∈K, we have thatbx=xb=g(b)x for anyx∈Jg, so(g(b)−b)x=0 for anyx∈Jg. Hence(g(b)−b)Jg= {0}. ButBJg=JgB=Bfor anyg∈K, so(g(b)−b)B= {0}. Thus g(b)=bfor anyg∈K; and sob∈B^K. ThereforeB^K=V_B(B_K).

Lemma3.6. The algebraB^Kis an Azumaya algebra overZwhereZis the center of BK.

Proof. SinceBis a Galois algebra overRwith Galois groupG,B is an Azumaya algebra over its centerC. By the proof ofTheorem 3.1,BKis aC-separable subalgebra ofB, soVB(BK)is aC-separable subalgebra ofBandVB(VB(BK))=BKby the commu- tator theorem for Azumaya algebras [3, Theorem 4.3, page 57]. This implies thatB_K andVB(BK)have the same centerZ. ThusVB(BK)is an Azumaya algebra overZ. But, byLemma 3.5,B^K=VB(BK), soB^Kis an Azumaya algebra overZ.

Theorem3.7. LetA=B_KB^K. ThenAis an Azumaya weakly Galois extension with Galois groupK|AK.

Proof. SinceBK is a central weakly Galois algebra with Galois groupK|BK K by Theorem 3.4, B_K is a finitely generated projective module overZ and (B_K)_lK Hom_Z(B_K,B_K). ByLemma 3.6,B^Kis an Azumaya algebra overZ, soA(B_K⊗ZB^K)is a finitely generated projective module overB^K(=A^K). Moreover, sinceB^K=VB(BK)by Lemma 3.5and(B_K)_lKHom_Z(B_K,B_K),

A_lK=B_KB^K

lK=B_K

lKB^K

rB_KK⊗ZB^KHom_ZB_K,B_K

⊗ZB^K Hom_BK

BK⊗ZB^K,BK⊗ZB^K

Hom_BK

BKB^K,BKB^K

=Hom_AK(A,A).

(3.2)

ThusAis a weakly Galois extension ofA^Kwith Galois groupK|AK. Next, we claim thatAhas centerZ andA^K is an Azumaya algebra over Z^K. In fact,BK andB^K are Azumaya algebras overZbyTheorem 3.1andLemma 3.6, respectively, soA(=B_KB^K) has centerZandA^K=(BKB^K)^K=B^K. Noting thatB^K is an Azumaya algebra overZ, we conclude thatA^K is an Azumaya algebra overZ^K. ThusAis an Azumaya weakly Galois extension with Galois groupK|AK.

4. An Azumaya Galois extension. In this section, we give several characterizations of an Azumaya Galois extensionB by usingBK. This generalizes the results in [2,6].

TheZ-module{b∈BK|bx=g(x)bfor allx∈BK} is denoted byJ_g^(BK) forg∈K whereK(=K/L)is defined inLemma 3.2.

Lemma4.1. The algebraB_Kis a central Galois algebra with Galois groupK|BKK if and only ifJ_g^(BK)= ⊕

l∈LJ_glfor eachg∈K.

Proof. LetBK be a central Galois algebra with Galois groupK|BKK. ThenBK=

⊕

g∈KJ_g^(BK) [6, Theorem 1]. Next it is easy to check that ⊕

l∈LJ_gl ⊂ J_g^(BK). But BK = ⊕

g∈KJg, so⊕

g∈KJg= ⊕

g∈KJ_g^(BK) where⊕

l∈LJgl ⊂J_g^(BK). Thus J_g^(BK)=

⊕

l∈LJ_gl for each g ∈ K. Conversely, since J^(B_g^K) = ⊕

l∈LJ_gl for each g ∈ K, B_K = ⊕

g∈KJ_g = ⊕

g∈KJ_g^(BK). Moreover, by Lemma 3.3, (B_K)^K = Z, so K is a Z-automorphism group ofBK. HenceJ_g^(BK)J_g^(B₋₁^K)=Zfor eachg∈K. ThusBKis a cen- tral Galois algebra with Galois groupK|BKKbecauseB_Kis an AzumayaZ-algebra byTheorem 3.1(see [4, Theorem 1]).

Next, we characterize an Azumaya Galois extensionBwith Galois groupK.

Theorem4.2. The following statements are equivalent:

(1) Bis an Azumaya Galois extension with Galois groupK;

(2) Z=C;

(3) B=B_KB^K;

(4) B_Kis a central Galois algebra overCwith Galois groupK|BKK.

Proof. (1)⇒(2). SinceBis an Azumaya Galois extension with Galois groupK,B^K is aC^K-Azumaya algebra. But, byLemma 3.6,B^K is an Azumaya algebra overZ, so Z=C^K. HenceC⊂Z=C^K⊂C. ThusZ=C.

(2)⇒(3). Suppose thatZ=C. Then, byTheorem 3.1,BKis an Azumaya algebra over C. Hence by the commutator theorem for Azumaya algebras,B=B_KV_B(B_K)[3, Theo- rem 4.3, page 57]. But, byLemma 3.6,B^K=V_B(B_K), soB=B_KB^K.

(3)⇒(4). By hypothesis,B=BKB^K, soL= {1}whereLis given inLemma 3.2. By the proofs ofTheorem 3.1andLemma 3.6,B_KandB^KareC-separable subalgebras of the AzumayaC-algebraBsuch thatB=B_KB^K, soB_KandB^Kare Azumaya algebras overC [3, Theorem 4.4, page 58]. ThusCis the center ofBK. Next, we claim thatJg=Jg^(BK)for eachg∈K. In fact, it is clear thatJg⊂Jg^(BK). Conversely, for eacha∈Jg^(BK)andx∈B such thatx=yzfor somey∈B_Kandz∈B^K, noting thatB^K=V_B(B_K), we have that ax=ayz=g(y)az=g(y)za=g(yz)a=g(x)a. ThusJ^(Bg^K)⊂J_g. This proves that Jg=J^(Bg^K)(=J_g^(BK)sinceL= {1})for eachg∈K. Hence,BKis a central Galois algebra overCwith Galois groupK|BKKbyLemma 4.1.

(4)⇒(1). SinceBis a Galois algebra with Galois groupG,Bis a Galois extension with Galois groupK. By hypothesis,BKis a central Galois algebra overCwith Galois group K|BKK, so the center ofB_KisC, that is,Z=C. HenceB^Kis an Azumaya algebra over C(=C^K)byLemma 3.6. ThusBis an Azumaya Galois extension with Galois groupK.

Theorem 4.2generalizes the following result of Kanzaki [6, Proposition 3].

Corollary4.3. IfJ_g= {0}for eachg∈K, thenBis a central Galois algebra with Galois groupKandCis a Galois algebra with Galois groupG/K.

Proof. This is the case inTheorem 4.2thatB=BKB^K=BKwhereB^K=C.

We conclude the present paper with two examples, one to illustrate the result in Theorem 4.2, and another to show thatZ≠C.

Example4.4. LetA=R[i,j,k], the real quaternion algebra over the field of real numbersR,B=(A⊗RA)⊕A⊕A⊕A⊕A, andGthe group generated by the elements in {g1,k_i,k_j,k_k,h_i,h_j,h_k}whereg1is the identity ofGand for all(a⊗b,a1,a2,a3,a4)∈ B,

k_ia⊗b,a1,a2,a3,a4

=iai⁻¹⊗b,ia1i⁻¹,ia2i⁻¹,ia3i⁻¹,ia4i⁻¹, kj

a⊗b,a1,a2,a3,a4

=

jaj⁻¹⊗b,ja1j⁻¹,ja2j⁻¹,ja3j⁻¹,ja4j⁻¹ , kk

a⊗b,a1,a2,a3,a4

=

kak⁻¹⊗b,ka1k⁻¹,ka2k⁻¹,ka3k⁻¹,ka4k⁻¹ , h_i

a⊗b,a1,a2,a3,a4

=

a⊗ibi⁻¹,a2,a1,a4,a3

,

h_ja⊗b,a1,a2,a3,a4

=a⊗jbj⁻¹,a3,a4,a1,a2, h_ka⊗b,a1,a2,a3,a4

=a⊗kbk⁻¹,a4,a3,a2,a1.

(4.1)

Then,

(1) we can check that B is a Galois algebra overB^G with Galois group G where B^G= {(r1⊗r2,r ,r ,r ,r )|r1,r2, r∈R} ⊂C, andC=(R⊗R)⊕R⊕R⊕R⊕R, the center ofB;

(2) K= {g∈G|g(c)=cfor allc∈C} = {g1,ki,kj,kk};

(3) J1=C, J_ki =(Ri⊗1)⊕Ri⊕Ri⊕Ri⊕Ri, J_kj =(Rj⊗1)⊕Rj⊕Rj⊕Ri⊕Rj, J_kk=(Rk⊗1)⊕Rk⊕Rk⊕Ri⊕Rk, soB_K=(A⊗RR)⊕A⊕A⊕A⊕A. HenceB_K has centerC, that isZ=C, andBKis a central Galois algebra overCwith Galois groupK|BKK;

(4) B^K=(R⊗A)⊕R⊕R⊕R⊕R and B=B_KB^K, that is,B is an Azumaya Galois extension with Galois groupK.

Example4.5. LetA=R[i,j,k], the real quaternion algebra over the field of real numbersR,B=A⊕A⊕A,G= {1,gi,g_j,g_k}, and for all(a₁,a₂,a₃)∈B,

gi

a1,a2,a3

=

ia1i⁻¹,ia2i⁻¹,ia3i⁻¹ , gj

a1,a2,a3

=

ja1j⁻¹,ja3j⁻¹,ja2j⁻¹ , g_ka1,a2,a3

=ka1k⁻¹,ka3k⁻¹,ka2k⁻¹.

(4.2)

Then,

(1) Bis a Galois algebra overB^G whereB^G= {(r1,r ,r )|r1, r∈R} ⊂C, andC= R⊕R⊕R, the center ofB. TheG-Galois system is{a_i;b_i|i=1,2,...,8}where

a1=(1,0,0), a2=(i,0,0), a3=(j,0,0), a4=(k,0,0), a5=(0,1,0), a6=(0,j,0), a7=(0,0,1), a8=(0,0,k);

b1=1

4a1, b2= −1

4a2, b3= −1

4a3, b4= −1 4a4, b5=1

2a5, b6= −1

2a6, b7=1

2a7, b8= −1 2a8,

(4.3)

(2) K= {g∈G|g(c)=cfor allc∈C} = {1,gi}whereJ_gi=Ri⊕Ri⊕Ri, soB_K= R[i]⊕R[i]⊕R[i]which is a commutative ring not equal toC, that is,Z≠C.

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

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George Szeto: Department of Mathematics, Bradley University, Peoria, IL61625, USA E-mail address:[email protected]

Lianyong Xue: Department of Mathematics, Bradley University, Peoria, IL61625, USA E-mail address:[email protected]