NON-COMMUTATIVE SEPARABILITY AND GROUP ACTIONS

NON-COMMUTATIVE SEPARABILITY AND GROUP ACTIONS

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 .

. 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 map p : A ®B A --> A splits as an (A -A)bimodule map .
2) There exists an element e E A®BA (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®BA = 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 = SG be the fixed ring under G. Writing g(r) = 9r, 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 g E 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 (9x) is a ring isomorphism ; or equivalently, there exist elements ai, bi (called a G-Galois basis) such that Y" al gbi = 1 if 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) .

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.f = E f(aá)fi = E fif(ai) .Now we prove that cp is an epimorphism.
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, . . ., mq} and let lk = 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 Og = {r E S/ r = sr ds E S} .If 4'g =,L 0 g is said to be w-inner, and if Og = 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 .Now let rg E Wg, so x = rgg E D, and hence trGI M (x) = 1: hrg gh -1 = 1: hrghgh-1 E C C_ S. Thus h rg = 0 heGIM hEGIM if hgh-1 0 1, this is if g z/~1 and so rg = 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 Proof.. (=~) Assume the same notation as in the proof of theorem 2.2 ; (~-=) 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 e E D and a pair h 0 g E G such that 9 xe = hxe for all x E D. If we let e' = 9 e, we have e' 7~0 and xe' = 9-lhxe' = e' s -lhx.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 ai7rG bi = 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.
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.

. 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) .
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: Remark .Note that in proposition 2 .3we 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 EG (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 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(9x -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 = EG 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 EG .Furthermore, by [S2], E _ Og is a direct sum and 9 Og = Cxg , 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 (EG) and EG = 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 .
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 ha = 9a for any g E h.Lemma 2.1 .DG = DGIM = C. Proof.The first equality is obvious since M is the inertia group of D. Now let a E DG , 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 DG , (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(Dc , S * Gc), then f (a) = f aifi(a) _ f(ai)fi(a) _ fi(a)f(ai) ; thus i i i himj)-1 E S * G®,S S * G. Then 'P(1k) = m 'w h¡m'bk ;P(himj ® (himj)-1) and by (*) mj whi bkW(hi(9hi 1 ) i h¡-, w I ~o (hibk ® h% 1~= fk .Hence we may 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 a E 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: himj)-'r = lkr, ghimjwbk ® (himj) -1 = >~(ghi)mjwbk ® ((ghi)mj)-1g, 1,7 áj but {ghi} is another transversal of M in G, hence by (*) glk = lkg and therefore 0 is an (S * G -S * G)-bimodule map.
cp = ids*co Ss*G and (p is an isomorphism.