Non-commutative separability and group actions

Publicacions Matemátiques, Vol 36 (1992), 359-367 . A bstract NON-COMMUTATIVE SEPARABILITY AND GROUP ACTIONS RICARDO ALFARO * Dedicated to the memory of Pere Menal We give conditions for the skew group ring S * G to be strongly separable and H-separable over the ring S . In particular we show that the H-separability is equivalent to S being central Galois extension . We also look into the H-separability of the ring S over the fixed subring R under a faithful action of a group G . We show that such a chain : S * G H-separable over S and S H-separable over R cannot occur, and that the centralizer of R in S is an Azumaya algebra in the presente of a central element of trace one . In [A] we introduced the concept of subring-Galois extensions as a generalization of central Galois extensions and give a generalization of the correspondence theorem given by DeMeyer in [D] and Szeto in [SM] . Similar correspondence theorems were given by Sugano in [S] using Hseparability . Separability for non-commutative rings was introduced by Hirata, and the notions of H-separability and "strong" separability were introduced by Hirata in [HI] and MacMahon and Mewborn in [MM] respectively . Strong separability is a weaker notion than H-separability, but both are special cases of the general notion of separability of ring extensions . In the case of group actions we present here conditions for strong and H-separability of skew group rings and in particular we show that the skew group ring S * G is H-separable over S if and only if S is a central Galois extension . Furthermore, in this case S*G is a Z(S)-Galois *Partially supported by a grant from the Faculty Development Fund of the University of Michigan-Flint and a fellowship from the Centre de Recerca Matematica, Barcelona, Spain . 360  R . ALFARO extension (in the terminology of [A]), allowing us to express S = Z(S)R and S*G = Z(S)I where I is the algebra of G-central functions . We then study the separability of CS(R) over its fixed subring and give conditions for S to be Cs(R)-Galois . All rings here are associative and Nave a unity element 1 . Z(R) will denote the center of a ring R, and CA(B) will denote the "centralizer of B in A", Le . the elements of the ring A which commute with all the elements of the subring B of A . 1 . Definitions and Notations Let B be a subring of a ring A with 1 . The extension B C A is called separable (or A is separable over B) if any of the following equivalent conditions is satisfied : 1) The multiplication mapp : A ® B A --> A splits as an (A - A)- bimodule map . 2) There exists an element e E A® B A (called a separability element), such that ne = ea  for all a E A and p¿(e) = 1 . The ring A is said to be strongly separable over B if A® B A= K ® L as (A - A)-bimodules, where HoMA,A (K, A) = 0 and L (D II - An for some (A - A)-bimodules K, L, H and some positive integer n . In case K = 0 we say that A is H-separable over B . Strongly separable extensions are separable but the converse is false, see [MM] . There is an equivalent definition for this kinds of separability in terms of the natural (A - A) -bimodule map cp : A ® B A - Hom(0~, Aj where ~o(a ® b)(x) = axb, C is the center of A and A is the centralizer of B in A, CA (B) . The ring A is strongly separable over B if and only if A, is finitely generated projective C-module and cp is an split epimorphism . Similarly, A is H-separable over B if and only if 0, is finitely generated projective C-module and cp is an isomorphism . For details see [HI] and [MM] . Now let's consider group actions . Let S be a ring with 1, let G be a finite group acting faithfully as automorphisms of S and let R= S G be the fixed ring under G . Writing g(r) = 9 r, the skew group ring S * G is the free left S-module with basis the elements of G and multiplication given by the rule gs = 9sg for all s E S and g E G . Denote by -ff the element E gE S * G . The action of G on S is said to be G-Galois if 9EG S is finitely generated projective right R-module and the natural map 0 : S * G -+ EndRS given by O(rg)(x) = r ( 9 x) is a ring isomorphism ; or equivalently, there exist elements ai, bi (called a G-Galois basis) such NON-COMMUTATIVE SEPARABILITY  36 1 that Y" al g bi = 1if g = 1 and the sum is 0 if g =~ 1 (Le ., SirS = S * G) . i The "trace map", tr : S --> R is given by tr(x) _  gx which is an geG (R - R)-bimodule homomorphism . Let T be a G-stable subring of S (that is gt E T for all t E T, g E G), we say that S is a T-Galois extension of R if the action of G on T is G-Galois . For details and properties, see [A] . If X is a subset of S, let I(X) = {g E G/ gx = x bx E X} be the "inertia group" of X, (I(X) is always a subgroup of G) . 2 . Separability and skew group rings In [MS, theorems 2 .2 and 2 .3] it is shown that if S is a simple ring, G a finite outer group of automorphisms of S and F = I(Z(S)), then S* G is H-separable over S * F and S * G is H-separable over S if and only if F is trivial . But in this case S * G is simple and hence the action of G on S is G-Galois . We'll give a general result relating G-Galois actions with strong and H-separability . Let D = CS*C(S) and C = Z (S * G) . The action of G on S induces a faithful action of G on S * G via conjugation, ga = gag -1 for a E S * G ; and G also acts on D . Let M be the inértia group of D, thus G/M acts faithfully on D by h a = 9 a for any g E h . Lemma 2 .1 . D G = DGIM = C . Proof . The first equality is obvious since M is the inertia group of D . Now let a E D G , then ag = ga  dg E G and by definition of D, sa = as  b's E S ; hence a E C . Conversely, if a E C, ag = ga  dg E G and hence a E D G , (is clear that C C D) . Theorem 2 .2 . Let M be the inertia group of D = CS*G(S) and let C be the center of S * G . Assumme there is a central element w in S with trm(w) = 1 . If D is G/M-Galois over C, then S * G is strongly separable over S . Proof . Let cp : S * G ® s S * G -~ Hom(Dc, S * Gc) be the natural (S * G -S * G)-bimodule map, and let {ai, bi} be a G/M-Galois basis for D over C ; then define the maps fi by fi(x) = tr,/,,(bix), thus fi E Hom(DC,CC) and {ai, fi} form a dual projective basis for D over C . First  we show that  {fi}  is  a basis  for Hom(DC, S * Gc)  as (S * G -S * G)-bimodule .  For, let a E D,  f E Hom(D c , S * Gc), 36 2  R . ALFARO then f (a) = f  aifi(a)  _  f(ai)fi(a) _  fi(a)f(ai) ; thus i  i  i f = E f (aá)fi = E fif (ai) . Now we prove that cp is an epimorphism . Note that W(g ® g') (a) = gctig -1 , thus cp(g (9 g -1 ) acts as g E G/M on D and cp(g (9 g -1 ) = ~ o(h ® h -1 )whenever g = h in G/M(*) . Choose {h1, ... , h p} a transversal of M in G, then fj(x) = tr .IM (bjx) _ 1 : h¡(bjx) = L . hibjhix h ¡ h¡ _  hi bj<P(hi (9 hi 1 )(x) = E ;P(hibj ® hi 1 )(x) . Therefore fj E Im( ;P) and hence cP is epic . Notice that the expression of fj above is independent of the choice of the transversal of M in G by (*) . It is only left to show that cp splits as (S * G -S * G)-bimodule homomorphism . Let M be given by the set {m1, . . . , m q} and let 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~ = fk . Hence wemay define the map 0 : Hom(DC, S * Gc) - S * G®, S * G by linearity with 0(fk) = lk . To show that 0 is an (S * G -S * G)-bimodule map, we need to show alk = lka for all aE S * G . Let r E S, since bk E D and w is central in S we have : and if g E G, we have : glk = rlk = 57r(him j )wbk ® (himj)-1 ?,j Y~ (himj)(h i m, ) -l rwbk ® (himj)-1 ?j 217 him j wbk ® (him')-lr(h¡mj)-1 himjwbk ® (himj)-'r = lkr, ghimjwbk ® (himj) -1 = >~( ghi)mjwbk ® ((ghi)mj)-1g, 1,7  áj NON-COMMUTATIVE SEPARABILITY  36 3 but {ghi} is another transversal of M in G, hence by (*) glk = lkg and therefore 0 is an (S * G -S * G)-bimodule map . We then have ~0 Cj :f (ak)fk l  = ~o ( ~ k  k  f (ak)fk f (ak)W(lk)  f (ak)fk = f . and so 0 splits cp . k Now we want to show an equivalent condition for the skew group ring S * G to be H-separable over S . We start by giving some notation and some neccesary conditions assuming all the notation as in theorem 2 .2 . For every g E G define O g = {r E S/ r gs = sr  ds E S} . If 4'g =,L 0  g is said to be w-inner, and if O g = 0 for every g z/~ 1 G is said to be w-outer . It is not difficult to see that D = 1 : Ogg . gEG For the proof of the main theorem we will need a result that appears in [A], and we reproduce here for completeness . Proposition 2 .3 . ([A, prop . 3 .3]) Assume S*G is H-separable over S . Then G is w-outer and D = Z(S) . Proof . - Since S * G - E ® (So g) as S-S-bimodules, C, (D) = S gEG by [S, proposition 1 .3] . Hence Z(D) = C,, G(D) n D C_ S and therefore C C_ Z(D) C_ Z(S) . Now let r g E Wg, so x = r g g E D, and hence trGI M (x) =  1 :  hr g gh -1 =  1 :  hrg hgh -1 E C C_ S . Thus h rg = 0 heGIM hEGIM if hgh-1 0 1, this is if g z/~ 1 and so r g = 0 if g z,~= 1 . Therefore Y'g = 0 if g :~É 1, and so G is w-outer . By the comment above D= ~ 1 1, so D= Z(S) . Theorem 2 .4 . Let D, M, C, S, G and w as in theorem 2 .2 . D is GGalois over C and M is trivial if and only if S * G is H-separable over S . Proof . . (=~) Assume the same notation as in the proof of theorem 2 .2 ; hibk ® h% 1 , and hence so now we have lk = h ti 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 isomorphism . (~-=) Assume m E M and n E D, then cp(m ® m -1 )(a) = mam -1 = a = cp(1 ® 1) (a), but cp is an isomorphism, hence M = 1 . Now we will show D is G-Galois over C . By proposition 2 .3 D is commutative, and by [S, proposition 1 .3] D is a separable C-algebra . Assume that there exists a non zero idempotent eE D and a pair h0gE G such that 9 xe = h xe for all xE D . If we let e' = 9 e, we have e' 7~ 0 and xe' = 9-lh xe' = e' s -lh x . But G is w-outer, hence g -1 h = 1, thus g = h, a contradiction . Therefore D is G-Galois over C by [DI, proposition III . 1 .2] . If S is a simple ring and G is outer, then Z(S) is a field, and hence G/M is G/M-Galois over Z(S) where M = I(Z(S)) . Therefore applying the previous theorems we obtain an improvement of [MS, Theorem 2 .3 and Theorem 2 .2,ii)] Corollary 2 .5 . Let S be a simple ring and G be outer . i) If 3w E Z(S) such that trm(w) = 1, then S * G is strongly separable over S . ii) S * G is H-separable over S if and only if M = 1 . We can see now a relationship between H-separability and T-Galois extensions in the following corollaries : Corollary 2 .6 . S * G is H-separable over S if and only if S is a central Galois extension of R . Proof . (~) 9 ai, bi E Z(S) such that  E a i 7r G b i = 1, but Z(S) C_ Cs * G(S) =D and D is G-invariant, hence D is G-Galois over DG=C and by theorem 2 .4 S * G is H-separable over S . (=) Obvious from the theorem 2 .4 and proposition 2 .3 . The case of commutative rings is now determined : Corollary 2 .7 . Let S be a commutative ring . S * G is H-separable over S if and only if S is G-Galois over R . Consider again the action of G on S * G by conjugation . It follows that the centralizar of G in S * G is precisely equal to the fixed ring (S * G) G = I, which in the language of C*-algebas is callad the algebra of G-central functions, (see [OP]) . Hence we obtain : Proposition 2 .8 . Let S * G be H-separable over S . Then S * G is a Z(S)-Galois extension of I and therefore S * G = Z(S)I . NON-COMMUTATIVE SEPARABILITY  36 5 3 . H-separability and fixed ring Now we study some neccesary conditions for the ring S to be Hseparable over the fixed ring R . The centralizer of R in S will be denoted by E and all the notation from Section 2 will be assumed . Let X be a G-invariant subset of S . It can be easily seen that CS(X ) is a G-invariant subring of S and thus G acts on it . Flzrthermore we have that (CS(X))G = CR(X) . Hence, if we take X =R we get the following relation : EG= Z(R) C Z(E) . On the other hand it is obvious that Z(S) C Z(E) . Proposition 3 .1 . Let S be H-separable over R . Then : 1) G is w-inner . 2) R = CS(E) 3) E G = Z(R) = Z(E) Proof . 1) Recall that 4'9 = {r E S/ r gs = sr  ds E S} . Consider the (S - S)-bimodule Sg . Then Eg= Cs,~g(R) and Ogg = Csg(S), therefore we get Eg= E OZ(S) Ogg and hence 4'g =~ 0 . 2) It is clear that R C_ CS(E) . Now, let r E CS(E) and let g E G . We can see g as an element of HOMR-R (S, S) which is isomorphic to E%(S) E by [H2, proposition 4 .7] . Thus there exists elements di, el E E such that gx =J :i dixei for all x E S, and therefore gr direi = r j : i diei =r ; sor G R . 3) By the comments above, it is only neccesary to show the second equality . But, by part 2) we have : Z(R) =R n Cs(R) = R n E _ Cs(E) n E= Z(E) . a Remark . Note that in proposition 2 .3 we showed that if the skew group ring S * G is H-separable over the base ring S, then the action of G must be w-outer . Here we obtain the opposite condition, if the ring S is H-separable over the fixed ring R, the action of G must be winner . Therefore we cannot have a "chain" of H-separabble extensions in faithful group actions . Proposition 3 .2 . Let S be H-separable over the fixed ring R and assume there exists a central element in S of trace one . Then E is separable over Z(S) and H-separable over E G (so E is an Azumaya algebra) . Proof . . The existente of a central element of trace 1 makes the trace map tr : S -4 R split as a (R - R)-bimodule map . Hence R is a direct summand of S as (R - R)-bimodules and by [S, proposition 1 .3] E is 36 6  R . ALFARO separable over Z(S) . Furthermore, since Z(S) C_ Z(E), the theorem of Azumaya for separable extension over commutative rings implies that E is separable over its centar Z(E) and Z(E) is separable over Z(S) . Therefore, E is H-separable over Z(E), which by proposition 3 .1 is equal to the fixed subring EG . The action of G on S induces an action on E, but we need to consider the inertia subgroup K = I(E) . In this way G/K acts faithfully on E . We now describe conditions for E to be a Galois extension of EG . Proposition 3 .3 . g E K if and only if Og C Z(E) . Proo£ Since Og C_ E the neccesary condition is obvious . Now let a E Og C Z(E) ; then a( 9 x - x) = 0 for all x E E and therefore gx = x forallxEE . Theorem 3 .4 . Let S be H-separable over R and assume there is a central element of trace 1 . S is an E-Galois extension of R if and only if C = E G and K is trivial . Proof . (=) By definition of E-Galois extension, K is trivial and the action of G on E is G-Galois, moreover by proposition 3 .2 E is Hseparable over E G . Furthermore, by [S2], E _  Og is a direct sum and 9 Og = Cx g , thus proposition 3 .3 implies that Z(E) = C, so proposition 3 .1 gives us the result . (~) Since K is trivial and the fixed elements in E coincide exactly with the central elements we have that the sum  ~9 is direct ; moreover 9 in this case E =CE (E G ) and E G = Z(E) giving us CE(EG ) equal to the direct sum of the correspondent 09 . Thus by [S2, theorem 1 .2] the action of G on E is G-Galois . References [A] ALFARO R ., T-Galois Extensions on Rings and a submodule correspondence, Comm . i n Algebra, to appear . [D] DEMEYER F ., Some notes in the general Galois Theory of rings, Osaka J . Math . 2 (1965), 117-127 . [DI] DEMEYER F . AND INGRAHAM E ., "Separable Algebras over Commutative Rings," Zecture Notes in Mathematics 181, Springer-Verlag, 1971 . NON-COMMUTATIVE SEPARABILITY  36 7 [HI] HIRATA K ., Some types of separable extension of rings, Nagoya Math . J . 33 (1968), 107-115 . [H2] HIRATA K ., Separable extensions and centralizers of rings, Nagoya Math . J . 35 (1969), 31-45 . [MS] MCMAHON E . AND SHAPIRO J ., On strong and H-separability in ordinary and Skew group rings, Houston J . Math . 15, no . 3 (1989) . [MM] MCMAHON E . AND MEWBORN A . C ., Separable extensions of non-commutative rings, Hokkaido Math . J . XIII, 1 (1984) . [OP] OSTERBURG J . AND PELIGRAD C ., A Strong Connes Spectrum for finite group actions of simple rings, J . Algebra, to appear . [S] SUGANo K ., On centralizers in separable extensions, Osaka J . Math . 7 (1970), 29-40 . [S2] SUGANo K ., On a special type of Galois extensions, Hokkaido Math . Journal9 (1980), 123-128 . [SM] SZETO G . AND MA L ., On center-Galois extensions over a ring, Glasnik Matematicki 24 (44) (1989), 11-16 . University of Michigan-Flint MI 48502 U .S .A . Rebut el 25 de Novembre de 1991