Non-commutative separability and group actions

Publicacions Ma emá iques, Vol 36 (1992), 359-367 . A bs ac NON-COMMUTATIVE SEPARABILITY AND GROUP ACTIONS RICARDO ALFARO * Dedica ed o he memo y o Pe e Menal We gi e condi ions o he skew g oup ing S * G o be s ongly sepa able and H-sepa able o e he ing S . In pa icula we show ha he H-sepa abili y is equi alen o S being cen al Galois ex ension . We also look in o he H-sepa abili y o he ing S o e he ixed sub ing R unde a ai h ul ac ion o a g oup G . We show ha such a chain : S * G H-sepa able o e S and S H-sepa able o e R canno occu , and ha he cen alize o R in S is an Azumaya algeb a in he p esen e o a cen al elemen o ace one . In [A] we in oduced he concep o sub ing-Galois ex ensions as a gene aliza ion o cen al Galois ex ensions and gi e a gene aliza ion o he co espondence heo em gi en by DeMeye in [D] and Sze o in [SM] . Simila co espondence heo ems we e gi en by Sugano in [S] using H- sepa abili y . Sepa abili y o non-commu a i e ings was in oduced by Hi a a, and he no ions o H-sepa abili y and "s ong" sepa abili y we e in oduced by Hi a a in [HI] and MacMahon and Mewbo n in [MM] espec i ely . S ong sepa abili y is a weake no ion han H-sepa abili y, bu bo h a e special cases o he gene al no ion o sepa abili y o ing ex ensions . In he case o g oup ac ions we p esen he e condi ions o s ong and H-sepa abili y o skew g oup ings and in pa icula we show ha he skew g oup ing S * G is H-sepa able o e S i and only i S is a cen al Galois ex ension . Fu he mo e, in his case S*G is a Z(S)-Galois *Pa ially suppo ed by a g an om he Facul y De elopmen Fund o he Uni- e si y o Michigan-Flin and a ellowship om he Cen e de Rece ca Ma ema ica, Ba celona, Spain . 360  R . ALFARO ex ension (in he e minology o [A]), allowing us o exp ess S = Z(S)R and S*G = Z(S)I whe e I is he algeb a o G-cen al unc ions . We hen s udy he sepa abili y o CS(R) o e i s ixed sub ing and gi e condi ions o S o be Cs(R)-Galois . All ings he e a e associa i e and Na e a uni y elemen 1 . Z(R) will deno e he cen e o a ing R, and CA(B) will deno e he "cen alize o B in A", Le . he elemen s o he ing A which commu e wi h all he elemen s o he sub ing B o A . 1 . De ini ions and No a ions Le B be a sub ing o a ing A wi h 1 . The ex ension B C A is called sepa able (o A is sepa able o e B) i any o he ollowing equi alen condi ions is sa is ied : 1) The mul iplica ion mapp : A ® B A --> A spli s as an (A - A)- bimodule map . 2) The e exis s an elemen e E A® B A (called a sepa abili y elemen ), such ha ne = ea  o all a E A and p¿(e) = 1 . The ing A is said o be s ongly sepa able o e B i A® B A= K ® L as (A - A)-bimodules, whe e HoMA,A (K, A) = 0 and L (D II - An o some (A - A)-bimodules K, L, H and some posi i e in ege n . In case K = 0 we say ha A is H-sepa able o e B . S ongly sepa able ex ensions a e sepa able bu he con e se is alse, see [MM] . The e is an equi alen de ini ion o his kinds o sepa abili y in e ms o he na u al (A - A) -bimodule map cp : A ® B A - Hom(0~, Aj whe e ~o(a ® b)(x) = axb, C is he cen e o A and A is he cen alize o B in A, CA (B) . The ing A is s ongly sepa able o e B i and only i A, is ini ely gene a ed p ojec i e C-module and cp is an spli epimo phism . Simila ly, A is H-sepa able o e B i and only i 0, is ini ely gene a ed p ojec i e C-module and cp is an isomo phism . Fo de ails see [HI] and [MM] . Now le 's conside g oup ac ions . Le S be a ing wi h 1, le G be a ini e g oup ac ing ai h ully as au omo phisms o S and le R= S G be he ixed ing unde G . W i ing g( ) = 9 , he skew g oup ing S * G is he ee le S-module wi h basis he elemen s o G and mul iplica ion gi en by he ule gs = 9sg o all s E S and g E G . Deno e by - he elemen E gE S * G . The ac ion o G on S is said o be G-Galois i 9EG S is ini ely gene a ed p ojec i e igh R-module and he na u al map 0 : S * G -+ EndRS gi en by O( g)(x) = ( 9 x) is a ing isomo phism ; o equi alen ly, he e exis elemen s ai, bi (called a G-Galois basis) such NON-COMMUTATIVE SEPARABILITY  36 1 ha Y" al g bi = 1i g = 1 and he sum is 0 i g =~ 1 (Le ., Si S = S * G) . i The " ace map", : S --> R is gi en by (x) _  gx which is an geG (R - R)-bimodule homomo phism . Le T be a G-s able sub ing o S ( ha is g E T o all E T, g E G), we say ha S is a T-Galois ex ension o R i he ac ion o G on T is G-Galois . Fo de ails and p ope ies, see [A] . I X is a subse o S, le I(X) = {g E G/ gx = x bx E X} be he "ine ia g oup" o X, (I(X) is always a subg oup o G) . 2 . Sepa abili y and skew g oup ings In [MS, heo ems 2 .2 and 2 .3] i is shown ha i S is a simple ing, G a ini e ou e g oup o au omo phisms o S and F = I(Z(S)), hen S* G is H-sepa able o e S * F and S * G is H-sepa able o e S i and only i F is i ial . Bu in his case S * G is simple and hence he ac ion o G on S is G-Galois . We'll gi e a gene al esul ela ing G-Galois ac ions wi h s ong and H-sepa abili y . Le D = CS*C(S) and C = Z (S * G) . The ac ion o G on S induces a ai h ul ac ion o G on S * G ia conjuga ion, ga = gag -1 o a E S * G ; and G also ac s on D . Le M be he iné ia g oup o D, hus G/M ac s ai h ully on D by h a = 9 a o any g E h . Lemma 2 .1 . D G = DGIM = C . P oo . The i s equali y is ob ious since M is he ine ia g oup o D . Now le a E D G , hen ag = ga  dg E G and by de ini ion o D, sa = as  b's E S ; hence a E C . Con e sely, i a E C, ag = ga  dg E G and hence a E D G , (is clea ha C C D) . Theo em 2 .2 . Le M be he ine ia g oup o D = CS*G(S) and le C be he cen e o S * G . Assumme he e is a cen al elemen w in S wi h m(w) = 1 . I D is G/M-Galois o e C, hen S * G is s ongly sepa able o e S . P oo . Le cp : S * G ® s S * G -~ Hom(Dc, S * Gc) be he na u al (S * G -S * G)-bimodule map, and le {ai, bi} be a G/M-Galois basis o D o e C ; hen de ine he maps i by i(x) = ,/,,(bix), hus i E Hom(DC,CC) and {ai, i} o m a dual p ojec i e basis o D o e C . Fi s  we show ha  { i}  is  a basis  o Hom(DC, S * Gc)  as (S * G -S * G)-bimodule .  Fo , le a E D,  E Hom(D c , S * Gc), 36 2  R . ALFARO hen (a) =  ai i(a)  _  (ai) i(a) _  i(a) (ai) ; hus i  i  i = E (aá) i = E i (ai) . Now we p o e ha cp is an epimo phism . No e ha W(g ® g') (a) = gc ig -1 , hus cp(g (9 g -1 ) ac s as g E G/M on D and cp(g (9 g -1 ) = ~ o(h ® h -1 )whene e g = h in G/M(*) . Choose {h1, ... , h p} a ans e sal o M in G, hen j(x) = .IM (bjx) _ 1 : h¡(bjx) = L . hibjhix h ¡ h¡ _  hi bj<P(hi (9 hi 1 )(x) = E ;P(hibj ® hi 1 )(x) . The e o e j E Im( ;P) and hence cP is epic . No ice ha he exp ession o j abo e is independen o he choice o he ans e sal o M in G by (*) . I is only le o show ha cp spli s as (S * G -S * G)-bimodule homomo phism . Le M be gi en by he se {m1, . . . , m q} and le l k = 2,3 himjwbk ® (himj)-1 E S * G®, S S * G . Then 'P(1k) = m 'w h¡m 'bk ;P(himj ® (himj)-1)  and by (*) mjw hi bkW(hi(9hi 1)i h¡-, w I ~o (hibk ® h% 1~ = k . Hence wemay de ine he map 0 : Hom(DC, S * Gc) - S * G®, S * G by linea i y wi h 0( k) = lk . To show ha 0 is an (S * G -S * G)-bimodule map, we need o show alk = lka o all aE S * G . Le E S, since bk E D and w is cen al in S we ha e : and i g E G, we ha e : glk = lk = 57 (him j )wbk ® (himj)-1 ?,j Y~ (himj)(h i m, ) -l wbk ® (himj)-1 ?j 217 him j wbk ® (him')-l (h¡mj)-1 himjwbk ® (himj)-' = lk , ghimjwbk ® (himj) -1 = >~( ghi)mjwbk ® ((ghi)mj)-1g, 1,7  áj NON-COMMUTATIVE SEPARABILITY  36 3 bu {ghi} is ano he ans e sal o M in G, hence by (*) glk = lkg and he e o e 0 is an (S * G -S * G)-bimodule map . We hen ha e ~0 Cj : (ak) k l  = ~o ( ~ k  k  (ak) k (ak)W(lk)  (ak) k = . and so 0 spli s cp . k Now we wan o show an equi alen condi ion o he skew g oup ing S * G o be H-sepa able o e S . We s a by gi ing some no a ion and some neccesa y condi ions assuming all he no a ion as in heo em 2 .2 . Fo e e y g E G de ine O g = { E S/ gs = s  ds E S} . I 4'g =,L 0  g is said o be w-inne , and i O g = 0 o e e y g z/~ 1 G is said o be w-ou e . I is no di icul o see ha D = 1 : Ogg . gEG Fo he p oo o he main heo em we will need a esul ha appea s in [A], and we ep oduce he e o comple eness . P oposi ion 2 .3 . ([A, p op . 3 .3]) Assume S*G is H-sepa able o e S . Then G is w-ou e and D = Z(S) . P oo . - Since S * G - E ® (So g) as S-S-bimodules, C, (D) = S gEG by [S, p oposi ion 1 .3] . Hence Z(D) = C,, G(D) n D C_ S and he e o e C C_ Z(D) C_ Z(S) . Now le g E Wg, so x = g g E D, and hence GI M (x) =  1 :  h g gh -1 =  1 :  h g hgh -1 E C C_ S . Thus h g = 0 heGIM hEGIM i hgh-1 0 1, his is i g z/~ 1 and so g = 0 i g z,~= 1 . The e o e Y'g = 0 i g :~É 1, and so G is w-ou e . By he commen abo e D= ~ 1 1, so D= Z(S) . Theo em 2 .4 . Le D, M, C, S, G and w as in heo em 2 .2 . D is G- Galois o e C and M is i ial i and only i S * G is H-sepa able o e S . P oo . . (=~) Assume he same no a ion as in he p oo o heo em 2 .2 ; hibk ® h% 1 , and hence so now we ha e lk = h i wb k ® (h i ) - I = cp(1 (9 1) = 1 : cp(1 (S 1) (ak)lk = E a k  k hibk 1 h i ®h2 1 =1®1 . hibk ® h2 I 36 4  R . ALFARO Thus 0 - cp = ids*co S s*G and (p is an isomo phism . (~-=) Assume m E M and n E D, hen cp(m ® m -1 )(a) = mam -1 = a = cp(1 ® 1) (a), bu cp is an isomo phism, hence M = 1 . Now we will show D is G-Galois o e C . By p oposi ion 2 .3 D is commu a i e, and by [S, p oposi ion 1 .3] D is a sepa able C-algeb a . Assume ha he e exis s a non ze o idempo en eE D and a pai h0gE G such ha 9 xe = h xe o all xE D . I we le e' = 9 e, we ha e e' 7~ 0 and xe' = 9-lh xe' = e' s -lh x . Bu G is w-ou e , hence g -1 h = 1, hus g = h, a con adic ion . The e o e D is G-Galois o e C by [DI, p oposi ion III . 1 .2] . I S is a simple ing and G is ou e , hen Z(S) is a ield, and hence G/M is G/M-Galois o e Z(S) whe e M = I(Z(S)) . The e o e applying he p e ious heo ems we ob ain an imp o emen o [MS, Theo em 2 .3 and Theo em 2 .2,ii)] Co olla y 2 .5 . Le S be a simple ing and G be ou e . i) I 3w E Z(S) such ha m(w) = 1, hen S * G is s ongly sepa able o e S . ii) S * G is H-sepa able o e S i and only i M = 1 . We can see now a ela ionship be ween H-sepa abili y and T-Galois ex ensions in he ollowing co olla ies : Co olla y 2 .6 . S * G is H-sepa able o e S i and only i S is a cen al Galois ex ension o R . P oo . (~) 9 ai, bi E Z(S) such ha  E a i 7 G b i = 1, bu Z(S) C_ Cs * G(S) =D and D is G-in a ian , hence D is G-Galois o e DG=C and by heo em 2 .4 S * G is H-sepa able o e S . (=) Ob ious om he heo em 2 .4 and p oposi ion 2 .3 . The case o commu a i e ings is now de e mined : Co olla y 2 .7 . Le S be a commu a i e ing . S * G is H-sepa able o e S i and only i S is G-Galois o e R . Conside again he ac ion o G on S * G by conjuga ion . I ollows ha he cen aliza o G in S * G is p ecisely equal o he ixed ing (S * G) G = I, which in he language o C*-algebas is callad he algeb a o G-cen al unc ions, (see [OP]) . Hence we ob ain : P oposi ion 2 .8 . Le S * G be H-sepa able o e S . Then S * G is a Z(S)-Galois ex ension o I and he e o e S * G = Z(S)I . NON-COMMUTATIVE SEPARABILITY  36 5 3 . H-sepa abili y and ixed ing Now we s udy some neccesa y condi ions o he ing S o be H- sepa able o e he ixed ing R . The cen alize o R in S will be deno ed by E and all he no a ion om Sec ion 2 will be assumed . Le X be a G-in a ian subse o S . I can be easily seen ha CS(X ) is a G-in a ian sub ing o S and hus G ac s on i . Flz he mo e we ha e ha (CS(X))G = CR(X) . Hence, i we ake X =R we ge he ollowing ela ion : EG= Z(R) C Z(E) . On he o he hand i is ob ious ha Z(S) C Z(E) . P oposi ion 3 .1 . Le S be H-sepa able o e R . Then : 1) G is w-inne . 2) R = CS(E) 3) E G = Z(R) = Z(E) P oo . 1) Recall ha 4'9 = { E S/ gs = s  ds E S} . Conside he (S - S)-bimodule Sg . Then Eg= Cs,~g(R) and Ogg = Csg(S), he e o e we ge Eg= E OZ(S) Ogg and hence 4'g =~ 0 . 2) I is clea ha R C_ CS(E) . Now, le E CS(E) and le g E G . We can see g as an elemen o HOMR-R (S, S) which is isomo phic o E%(S) E by [H2, p oposi ion 4 .7] . Thus he e exis s elemen s di, el E E such ha gx =J :i dixei o all x E S, and he e o e g di ei = j : i diei = ; so G R . 3) By he commen s abo e, i is only neccesa y o show he second equali y . Bu , by pa 2) we ha e : Z(R) =R n Cs(R) = R n E _ Cs(E) n E= Z(E) . a Rema k . No e ha in p oposi ion 2 .3 we showed ha i he skew g oup ing S * G is H-sepa able o e he base ing S, hen he ac ion o G mus be w-ou e . He e we ob ain he opposi e condi ion, i he ing S is H-sepa able o e he ixed ing R, he ac ion o G mus be w- inne . The e o e we canno ha e a "chain" o H-sepa abble ex ensions in ai h ul g oup ac ions . P oposi ion 3 .2 . Le S be H-sepa able o e he ixed ing R and assume he e exis s a cen al elemen in S o ace one . Then E is sepa able o e Z(S) and H-sepa able o e E G (so E is an Azumaya algeb a) . P oo . . The exis en e o a cen al elemen o ace 1 makes he ace map : S -4 R spli as a (R - R)-bimodule map . Hence R is a di ec summand o S as (R - R)-bimodules and by [S, p oposi ion 1 .3] E is 36 6  R . ALFARO sepa able o e Z(S) . Fu he mo e, since Z(S) C_ Z(E), he heo em o Azumaya o sepa able ex ension o e commu a i e ings implies ha E is sepa able o e i s cen a Z(E) and Z(E) is sepa able o e Z(S) . The e o e, E is H-sepa able o e Z(E), which by p oposi ion 3 .1 is equal o he ixed sub ing EG . The ac ion o G on S induces an ac ion on E, bu we need o conside he ine ia subg oup K = I(E) . In his way G/K ac s ai h ully on E . We now desc ibe condi ions o E o be a Galois ex ension o EG . P oposi ion 3 .3 . g E K i and only i Og C Z(E) . P oo£ Since Og C_ E he neccesa y condi ion is ob ious . Now le a E Og C Z(E) ; hen a( 9 x - x) = 0 o all x E E and he e o e gx = x o allxEE . Theo em 3 .4 . Le S be H-sepa able o e R and assume he e is a cen al elemen o ace 1 . S is an E-Galois ex ension o R i and only i C = E G and K is i ial . P oo . (=) By de ini ion o E-Galois ex ension, K is i ial and he ac ion o G on E is G-Galois, mo eo e by p oposi ion 3 .2 E is H- sepa able o e E G . Fu he mo e, by [S2], E _  Og is a di ec sum and 9 Og = Cx g , hus p oposi ion 3 .3 implies ha Z(E) = C, so p oposi ion 3 .1 gi es us he esul . (~) Since K is i ial and he ixed elemen s in E coincide exac ly wi h he cen al elemen s we ha e ha he sum  ~9 is di ec ; mo eo e 9 in his case E =CE (E G ) and E G = Z(E) gi ing us CE(EG ) equal o he di ec sum o he co esponden 09 . Thus by [S2, heo em 1 .2] he ac ion o G on E is G-Galois . Re e ences [A] ALFARO R ., T-Galois Ex ensions on Rings and a submodule co - espondence, Comm . i n Algeb a, o appea . [D] DEMEYER F ., Some no es in he gene al Galois Theo y o ings, Osaka J . Ma h . 2 (1965), 117-127 . [DI] DEMEYER F . AND INGRAHAM E ., "Sepa able Algeb as o e Commu a i e Rings," Zec u e No es in Ma hema ics 181, Sp inge -Ve lag, 1971 . NON-COMMUTATIVE SEPARABILITY  36 7 [HI] HIRATA K ., Some ypes o sepa able ex ension o ings, Nagoya Ma h . J . 33 (1968), 107-115 . [H2] HIRATA K ., Sepa able ex ensions and cen alize s o ings, Nagoya Ma h . J . 35 (1969), 31-45 . [MS] MCMAHON E . AND SHAPIRO J ., On s ong and H-sepa abili y in o dina y and Skew g oup ings, Hous on J . Ma h . 15, no . 3 (1989) . [MM] MCMAHON E . AND MEWBORN A . C ., Sepa able ex ensions o non-commu a i e ings, Hokkaido Ma h . J . XIII, 1 (1984) . [OP] OSTERBURG J . AND PELIGRAD C ., A S ong Connes Spec um o ini e g oup ac ions o simple ings, J . Algeb a, o appea . [S] SUGANo K ., On cen alize s in sepa able ex ensions, Osaka J . Ma h . 7 (1970), 29-40 . [S2] SUGANo K ., On a special ype o Galois ex ensions, Hokkaido Ma h . Jou nal9 (1980), 123-128 . [SM] SZETO G . AND MA L ., On cen e -Galois ex ensions o e a ing, Glasnik Ma ema icki 24 (44) (1989), 11-16 . Uni e si y o Michigan-Flin MI 48502 U .S .A . Rebu el 25 de No emb e de 1991