On Stable Orthogonal Decompositions of Idempotents

Journal of Algebra 217, 711᎐735 Ž1999. Article ID jabr.1998.7820, available online at http:rrwww.idealibrary.com on

On Stable Orthogonal Decompositions of Idempotents Yun Fan Department of Mathematics, Wuhan Uni¨ ersity, Wuhan, 430072, People’s Republic of China

and Burkhard Kulshammer ¨ Mathematisches Institut, Uni¨ ersitat ¨ Jena, Jena, D-07740, Germany Communicated by Michel Broue´ Received October 1, 1998

1. INTRODUCTION The well-known Green’s indecomposability theorem says that the induced module of an absolutely indecomposable module from a subnormal subgroup of p-primary index is absolutely indecomposable. Later a related result appeared in w7x which shows that for a p-solvable group a projective module over large enough ground-fields is an induced module from a Hall p⬘-subgroup. Working on G-algebras, w9x gave a systematic approach for these subjects, where the hypothesis so-called ‘‘deployee’’ ´ is assumed; for module case, this condition is equivalent to the absolute indecomposability. Reference w8x extended results of w7x and some others to the G-algebra version. On the other hand, the indecomposability theorem was considered for perfect ground-fields in w5x; later, w4x considered the arbitrary groundfields and showed what happens for induced modules of indecomposable modules from subnormal groups of p-primary index. In this paper we are concerned with Puig’s approach, but over arbitrary ground-fields. Usually we follow the terminology in w12x. Here we introduce necessary notations so that we can state our main ideas and results. 1.1. Let p be a prime integer. Let G be a finite group. Let O be a complete discrete valuation ring with residue field k of characteristic p. Note that O is arbitrary, and the case O s k is allowed. 711 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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An O-algebra A always means an O-free unitary algebra of finite O-rank; but subalgebras are not necessarily unitary; A* denotes the multiplicative group consisting of all the invertible elements of A. An A-module means an O-free left A-module of finite O-rank. We say that A is a G-ring if it is endowed with a group homomorphism from G to the automorphism group of the ring A; in this case we denote y1 the image of a g A under the action of x g G by a x . Further, if A is an O-algebra and the G-action on A is O-linear, then we say that A is a G-algebra over O . For a G-algebra A and a subgroup H of G, A H denotes the subalgebra consisting of all the H-fixed elements; and the map y1 TrHG : A H ª AG , TrHG Ž a. s Ý x g G r H a x , is called the trace map; for G GŽ H . brevity, we denote A H s TrH A which is clearly an ideal of AG . We call orthogonal set of idempotents on an algebra A any set of non-zero idempotents which are pairwise orthogonal. Considering all the orthogonal sets I of idempotents, we have a natural partial relation on them: I⬘ is said to be a refinement of I if O I⬘ > O I, where O I denotes the O-submodule of A generated by I, O I is in fact a subalgebra of A obviously. By irkŽ A., called the idempotent rank of A in w9x, we denote the cardinality of a maximal orthogonal set of idempotents on A; this is well defined Ževen if A is not unitary., see 2.4 below. In particular, A is called a local algebra if irkŽ A. s 1. As usual, CG Ž I . and NG Ž I . stand for the centralizer and stabilizer, resp., of I in G for a G-set S and I : S. 1.2. Let A be a G-algebra and H a subgroup of G. A H is said to be deployee ´ if every primitive idempotent in A H is still primitive in any coefficient extensions O ⬘ mO A H . Assume that G is a p-group. In w9x the following were proved, cf. w12, Sects. 23 and 24x. 1.2.1. If A H is deployee, ´ then AGH is deployee. ´ 1.2.2. If the hypothesis in 1.2.1 holds, then for any idempotent y1 i g AGH there is an idempotent j g A H such that i s TrHG Ž j . s Ý x g G r H j x is an orthogonal decomposition of idempotents. 1.2.3. If the hypothesis in 1.2.1 holds for any subgroup H of G, then all maximal G-stable orthogonal sets of idempotents on A are conjugate by Ž A*. G , and any such maximal one I has the properties: irkŽŽ O I . G . s irkŽ AG . and every i g I satisfies i f ACRG Ž i. for any proper subgroup R of CG Ž i .. Remark. The maximal G-stable orthogonal sets of idempotents on A must exist since A is of finite O-rank. We are concerned with whether they are conjugate and how to characterize them. A G-stable orthogonal set I of idempotents on A which has the properties in 1.2.3 is called a local decomposition on 1 A , in notation of the so-called local theory; see w12, Sect.

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24x. Thus, it is reasonable to work on the pointed groups on G-algebras A, which are pairs H␤ of a subgroup H of G and a conjugacy class ␤ of primitive idempotents on A H. In this notation, the property ‘‘i f ACRG Ž i. for any proper subgroup R of CG Ž i .’’ is just to say that the pointed group Ž CG Ž i ..␥ with i g ␥ is local; this is also related to the following concept of defect groups; see 2.3 below and w12, 14.3 and Sect. 18x. 1.3. DEFINITION. groups on A.

Let A be a G-algebra; let G␣ and H␤ be pointed

1.3.1. G␣ is said to be projecti¨ e relati¨ e to H␤ if for i g ␣ there are j g ␤ and a, b g A H such that i s TrHG Ž ajb.; while G␣ is said to be projective relative to H if ␣ ; AGH . The minimal H such that G␣ is projective relative to H is called a defect group of G␣ . 1.3.2. G␣ is said to be free relati¨ e to H␤ if for i g ␣ there is a j g ␤ such that i s TrHG Ž j . is an orthogonal decomposition of idempotents; while G␣ is said to be free relative to H if such an idempotent j g A H exists. In fact, the two relative projectivities of a pointed group are equivalent in some sense; and it is the same for the two relative freeness; see 2.2 and 2.2⬘ below. It is clear that the relative freeness implies the relative projectivity. The converse is certainly false. To some extent, our aim is just to look for the conditions which make the converse true. If H is normal in G, then a necessary and sufficient condition and some more information are stated in Proposition 1.5 below in the following notation; and, we have another more general result; see Proposition 3.7 below. 1.4. DEFINITION. For any pointed group H␤ on a G-algebra A, we have an NG Ž H␤ .-algebra AŽ H␤ . which is simple, where NG Ž H␤ . s NG Ž H␤ .rH; let V Ž H␤ . be the unique simple AŽ H␤ .-module, and DŽ H␤ . s End AŽ H␤ .Ž V Ž H␤ .. which is a division algebra; then a unique twisted group algebra DŽ H␤ .# NˆG Ž H␤ . is determined and V Ž H␤ . becomes a DŽ H␤ .# NˆG Ž H␤ .-module, which is called the multiplicity module of H␤ . For details see the subsection 3.6 below; and here the twisted group algebras are in a generalized sense, see the subsection 2.7 below. By the way we remark that any division algebra D is in fact over k if O / k; moreover, let < Ds : k < stand for the degree over k of a separable maximal subfield of D; see 2.10 below. PROPOSITION 1.5. Let A be a G-algebra and H be a normal subgroup of G; let G␣ be a pointed group on A projecti¨ e relati¨ e to H. Then G␣ is free relati¨ e to H if and only if DŽ H␤ .# NˆG Ž H␤ . is a local algebra for some Ž and then any . pointed group H␤ ; G␣ . Moreo¨ er, if this is the case, then

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1.5.1. for i g ␣ and any two orthogonal decompositions i s TrHG Ž j . s j⬘. with idempotents j, j⬘ g A H , there are a g Ž A*. G and y g G such a y y1 that j s j⬘; y1 1.5.2. for any H␤ ; G␣ and any j g ␤ , the idempotents j x , x g GrH, form a G-stable orthogonal set of idempotents and TrHG Ž j . g ␣ . TrHG Ž

For the case that H is subnormal in G, however, we have to request the subnormal series of pointed groups. Recall that a series H s H0 F H1 F ⭈⭈⭈ F Hn s G of subgroups is said to be subnormal from H to G if every Ht is normal in Htq1 for t s 0, 1, . . . , n y 1. Similarly, on a G-algebra A, a series H␤ s Ž H0 .␤ 0 ; Ž H1 .␤ ; ⭈⭈⭈ ; Ž Hn .␤ s G␣ of pointed groups is said to be subnormal 1 n projecti¨ e from H␤ to G␣ if the corresponding series of subgroups is subnormal and every Ž Htq1 .␤ tq 1 is projective relative to Ž Ht .␤ t. LEMMA 1.6. Let G␣ and H␤ be pointed groups on a G-algebra A; and assume that H is subnormal in G. If G␣ is projecti¨ e relati¨ e to H␤ , then for any subnormal series of subgroups from H to G there is a corresponding subnormal projecti¨ e series of pointed groups from H␤ to G␣ . Con¨ ersely, if there is a subnormal projecti¨ e series of pointed groups from H␤ to G␣ , then G␣ is projecti¨ e relati¨ e to H␤ . THEOREM 1.7. Let A be a G-algebra, let H be a subnormal subgroup of G, let G␣ and H␤ be pointed groups on A. Then the following three statements are equi¨ alent: 1.7.1. G␣ is free relati¨ e to H␤ ; 1.7.2. for any subnormal series of subgroups from H to G there is a corresponding subnormal projecti¨ e series H␤ s Ž H0 .␤ 0 ; Ž H1 .␤ 1 ; ⭈⭈⭈ ; Ž Hn .␤ s G␣ of pointed groups such that DŽŽ Ht .␤ .# NˆH ŽŽ Ht .␤ . is a local n t tq 1 t algebra for t s 0, 1, . . . , n y 1. 1.7.3. there is a subnormal projecti¨ e series H␤ s Ž H0 .␤ 0 ; Ž H1 .␤ 1 ; ⭈⭈⭈ ; Ž Hn .␤ n s G␣ of pointed groups such that DŽŽ Ht .␤ t .# NˆH tq 1ŽŽ Ht .␤ t . is a local algebra for t s 0, 1, . . . , n y 1. Further, if this is the case and i s TrHG Ž j . s TrHG Ž j⬘. for i g ␣ and j, j⬘ g ␤ both are orthogonal decompositions of idempotents, then there is an a g Ž A*. G such that ajay1 s j⬘. COROLLARY 1.8. Notations as in 1.7 abo¨ e. If H is a defect group of G␣ and both i s TrHG Ž j . s TrHG Ž j⬘. with i g ␣ and j, j⬘ g A H are orthogonal decompositions of idempotents, then there are a g Ž A*. G and y g NG Ž H . y1 y1 y1 y1 such that j a y s j⬘, or equi¨ alently,  j x < x g GrH 4 a s Ž j⬘. x < x g GrH 4 .

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With these facts, we can show the following result. COROLLARY 1.9. Let A be a G-algebra and H be a subnormal subgroup of G of p-primary index. 1.9.1. If p ¦ < DŽ H␤ .s : k < for any pointed group H␤ on A, then p ¦ < DŽ G␣ .s : k < for any pointed group G␣ on A which is projecti¨ e relati¨ e to H; 1.9.2. If the condition in 1.9.1 holds, then any pointed group G␣ which is projecti¨ e relati¨ e to H is free relati¨ e to H. 1.9.3. If G is a p-group and the condition in 1.9.1 holds for any subgroup H of G, then all maximal G-stable orthogonal sets of idempotents on A are conjugate by Ž A*. G ; and a G-stable orthogonal set I of idempotents on A is maximal if and only if I has the properties: irkŽŽ O I . G . s irkŽ AG . and e¨ ery i g I satisfies that i f ACRG Ž i. for any proper subgroup R of CG Ž i .. The above result contains Puig’s result 1.2, because the condition ‘‘ A H is deployee’’ ´ implies that DŽ H␤ . is a purely inseparable extension over k, i.e., < DŽ H␤ .s : k < s 1. The necessary preliminaries are sketched in Section 2. In Section 3 we relate a G-algebra of a special type to a twisted group algebra of G over the endomorphism algebra of a projective module of the G-algebra, such a machine plays an important role for our purpose, see 3.7; by the way, a G-algebra version of the stable Clifford theory is described, see 3.5. All the results announced in this Introduction are proven in Section 4. 2. PRELIMINARIES The notations introduced in 1.1 are preserved throughout this paper. We begin with algebras. For convenience, we call the quotient algebra ArJ Ž A. of an algebra A by its radical the head of A. Algebra homomorphisms in this paper are not necessarily unitary though we say that an algebra always means unitary algebra. An algebra homomorphism f : A ª A⬘ is said to be an embedding if f is injective and its image f Ž A. s f Ž1 A . ⭈ A⬘ ⭈ f Ž1 A ., where f Ž1 A . is in fact an idempotent of A⬘; and in that case A is called an embedded subalgebra of A⬘. Modifying the notation for modules and for pointed groups, e.g., notations in w1x or w12x, we introduce the following definitions. 2.1. DEFINITION.

Let A be a G-algebra, and H F G be a subgroup.

2.1.1. A is said to be projective relative to H, if AG s AGH , or equivalently, there is an a g A H such that 1 A s TrHG Ž a.. 2.1.2. A is said to be free relative to H, if there is an idempotent y1 f g A H such that 1 A s TrHG Ž f . s Ý x g G r H f x is an orthogonal decomposition of idempotents.

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Let A be a G-algebra. Recall from w10x that a pointed group H␤ on A is a pair of a subgroup H of G and a conjugacy class ␤ of primitive idempotents in A H ; and we say that a pointed group H␤ contains a pointed group K␥ , denoted H␤ > K␥ , if H G K and there are i g ␤ and j g ␥ such that ij s j s ji. For a pointed group H␤ on A, we denote A␤ s iAi for an i g ␤ ; A␤ is an H-algebra, and it is well defined up to natural isomorphism. On the other hand, in the head A H rJ Ž A H . there is a unique simple direct factor, denoted by AŽ H␤ ., such that the image of ␤ is contained all in AŽ H␤ .. It is clear that, K␥ ; H␤ if and only if K F H and ␥ l A␤ / ⭋, if and only if the image of ␤ in AŽ K␥ . is non-zero. For all of these, see w10, 12x. The following is an easy fact, cf. w12, Sect. 14x, which connects the relative projectivity of pointed groups and the relative projectivity of algebras, 2.2. LEMMA. For a pointed group G␣ on a G-algebra A and H F G, the following four statements are equi¨ alent: 2.2.1. G␣ is projecti¨ e relati¨ e to H, i.e., ␣ ; AGH ; 2.2.2. there is a pointed group H␤ on A such that G␣ is projecti¨ e relati¨ e to H␤ ; 2.2.3. there is a pointed group H␤ on A such that G␣ is projecti¨ e relati¨ e to H␤ and G␣ > H␤ ; 2.2.4.

A␣ is projecti¨ e relati¨ e to H.

We remark that there are such cases that G␣ is projective relative to H␤ but G␣ r H␤ ; w2, Corollary 7.3x and w12, Sect. 10x discussed this question carefully, though both the references assumed that the ground-fields are algebraically closed. Additionally, the results 1.6 and 4.3 of this paper are also related to this question. On the other hand, it is similar and easier for the relative freeness. 2.2⬘. LEMMA. For a pointed group G␣ on a G-algebra A and H F G, the following three statements are equi¨ alent: 2.2⬘.1. G␣ is free relati¨ e to H; 2.2⬘.2. G␣ is free relati¨ e to a pointed group H␤ on A; 2.2⬘.3.

A␣ is free relati¨ e to H.

2.3. Recall that a pointed group H␤ on a G-algebra A is said to be local if ␤ ­ A H R for any proper subgroup R of H; in this case H must be a p-subgroup. On the other hand, H␤ is said to be a defect pointed group of a pointed group G␣ if G␣ is projective relative to H␤ but not relative to any

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pointed group R␦ which is properly contained in H␤ . A well-known fact is that Žsee w10, 12, Sect. 18x.: 2.3.1. LEMMA. H␤ is a defect pointed group of G␣ if and only if H␤ is a maximal local pointed group which is contained in G␣ . In addition, all defect pointed groups of G␣ are conjugate by G. Recall that we have defined a partial relation on the orthogonal sets of idempotents on an algebra A in 1.1. Moreover, an orthogonal set I of idempotents on A is said to be complete if 1 A s Ý i g I i. Note that a maximal orthogonal set of idempotents on A must be complete. But, if we consider only a subalgebra B of A, then a maximal orthogonal set of idempotents on B may not be complete, since B may not be unitary. Moreover, if A is a G-algebra, then a maximal G-stable orthogonal set of idempotents on A is complete, but not necessarily a maximal orthogonal set of idempotents on A. The following fact is in fact a part of w9, Lemma 1.4x. 2.4. LEMMA.

Let A be an algebra and B be an ideal of A.

2.4.1. Any maximal orthogonal set of idempotents on B can be extended to a maximal orthogonal set of idempotents on A. 2.4.2. An A*-conjugate of a maximal orthogonal set of idempotents on B is still maximal on B; two maximal orthogonal sets of idempotents on B are conjugate by A*. Proof. The first fact and the first half of the second one are clear. For two maximal orthogonal sets J and J⬘ of idempotents on B, extending them to maximal ones I and I⬘ on A, resp., we have an a g A* such that I a s I⬘; hence J a j J⬘ is an orthogonal set of idempotents on B; by the maximality of both J a and J⬘, we have J a s J⬘. On the lifting of idempotents, the following result was essentially first proven in w11x; later it was reproven in the following version by the first author and revised by Lluis Puig; w8x gives a directly elementary proof Žand simplified by Puig., including the conjugation part. Here we sketch a proof in notation of projective covers. 2.5. LEMMA. Let A be a G-algebra and N be a G-stable ideal of A which is contained in the radical of A. Assume that I is a G-stable complete orthogonal set of idempotents on the quotient algebra A s ArN. Then there is a G-stable orthogonal set I of idempotents on A which lifts I if and only if for any i g I there is an idempotent i g AC G Ž i. which lifts i. Moreo¨ er, in that case, any two such liftings I and I⬘ of I are conjugate by 1 A q N G .

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The condition is satisfied if A H s A H for all H F G; in particular, if one of the following holds: 2.5.1.

AG s AG 1;

2.5.2. A has a G-stable O-basis ⍀ and a G-subset ⍀⬘ : ⍀ such that ⍀⬘ is an O-basis of A. Proof. If I is a G-stable lifting of I and i g I is the lifting of i g I, then, by the G-stability of both I and I, it is clear that the orbit in I containing i is the lifting of the orbit in I containing i, hence i and i have the same stabilizer in G. Conversely, for every G-orbit It of I and a representative i t g It , we y1 have an idempotent iXt g AC G Ž i t . which lifts i t ; and obviously, Ž iXt . x for y1 X X xy 1 x x g G is an idempotent lifting Ž i t . ; setting It s Ž i t . < x g G4 , we get a lifting Žbut not orthogonal in general. of It . Additionally, running over the orbits of I, we get a G-stable lifting I⬘ of I. For every i⬘ g I⬘ which lifts i g I, obviously, Ai⬘ is a projective cover of the A-module Ai; hence [i⬘g I⬘ Ai⬘ is a projective cover of A s Ý i g I Ai. On the other hand, A is of course a projective cover of A. But A s J Ž A. q Ý i⬘g I⬘ Ai⬘, hence A s Ý i⬘g I⬘ Ai⬘ by the definition of the radical. By the uniqueness of the projective covers we get rank O Ž A. s rank O Ž[i⬘g I⬘ Ai⬘.. So the sum A s Ý i⬘g I⬘ Ai⬘ has to be a direct sum, i.e., A s [i⬘g I⬘ Ai⬘. This direct decomposition is stable under the G-action since I⬘ is G-stable. Now take I to be the set of the idempotents corresponding to this direct sum, i.e., 1 A s Ý i g I i and i g Ai⬘. Noting that the set of idempotents determined by a direct decomposition of an algebra is unique, we conclude that I is a G-stable orthogonal set of idempotents on A which lifts I. Suppose both I and I⬘ are such G-stable liftings of I. For i g I by i g I and i⬘ g I⬘ we denote the liftings of i in I and in I⬘, resp. Set u s Ý i g I ii⬘. Then obviously 1 A s u g A, hence u g 1 A q N; in particular, u is an invertible element of A. Further, by the G-stabilities on the three sets, it is clear that G fixes u, hence u g 1 A q N G. Now it is a routine to check that, for any i 0 g I, we have i 0 u s uiX0 , i.e., uiX0 uy1 s i 0 . At last, if we have a surjective homomorphism AC G Ž i. ª Ž A. C G Ž i., then i can be lifted to an idempotent i g AC G Ž i.. In particular, if 2.5.1 holds, the following lemma shows A H ª A H is surjective for all H F G. On the other hand, these natural maps are clearly surjective when 2.5.2 holds. 2.6. LEMMA. Let A be a G-algebra and N be a G-stable ideal contained in Ž ArN . G s Ž ArN .1G . In that the radical of A. Then AG s AG 1 if and only if H H case, the natural map A ª Ž ArN . is surjecti¨ e for any subgroup H of G. GŽ . Proof. Denote A s ArN. If AG s AG 1 , then 1 A s Tr1 a for a g A, so GŽ . G G 1 A s Tr1 a , hence Ž A. s Ž A.1 .

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Conversely, if Ž A. G s Ž A.1G , then 1 A s Tr1G Ž a. for a g Ž A., so 1 A g Tr1G Ž a. q N; hence Tr1G Ž a. is an invertible element of A, and AG s AG 1. If the conditions hold, then, by the Mackey decomposition of the trace maps, for any H F G we have A H s A1H , hence Ž A. H s Ž A.1H ; thus Ž A. H s Ž A.1H s A1H s A H . 2.7. We introduce the twisted group algebras, in a generalized sense, of a group over an algebra. Let L be an O-algebra and L* denote the multiplicative group of L consisting of all the invertible elements. Let

ˆª G ª 1 1 ª L* ª G ˆ and let Gˆ act be a group extension with O * contained in the center of G, ˆ on L such that the G-action is compatible with the L*-conjugation. We construct an algebra, called a twisted group algebra of G over L and ˆ whose O-module is the induced module L#Gˆ s denoted by L#G, ˆ L mO L* O G, and whose multiplication is defined by y1

Ž a m ˆx . Ž a⬘ m ˆx⬘ . s Ž aŽ a⬘ . ˆx

xx⬘ . . m Ž ˆˆ

ˆ for all a, a⬘ g L, ˆ x, ˆ x⬘ g G.

ˆ is well defined. If the It is easy to check that the multiplication on L#G algebra L is generated by L* Že.g., if L is a division algebra., then the ˆ is determined up to graded-isomorphism by the equivalence algebra L#G ˆ of L* by G. Further, we denote a m ˆx s axˆ class of the group extension G ˆ s [x g G Lxˆ where ˆx is a lifting of for short. Then it is clear that L#G x g G. The following statement is easily verified. ˆ be as abo¨ e. If E is an algebra and there are a unitary 2.7.1. Let L#G algebra homomorphism ␪ : L ª E and a group homomorphism ␩ ˆ : Gˆy1ª E* which are compatible Ž i.e., ␪ < L* s ␩ < L* and ␩ ˆ Ž ˆx .␪ Ž a.␩ˆ Ž ˆx .y1 s ␪ Ž a ˆx . for ˆx g Gˆ and a g L., then we ha¨ e a unitary algebra homomorphism Ž denoted also by ␩ ˆ. ␩ ˆ : L#Gˆ ª E,

axˆ ¬ ␪ Ž a . ␩ ˆ Ž ˆx . .

ˆ be a twisted group algebra of G over 2.8. DEFINITION. Let L#G ˆ an algebra L as above, and H be a subgroup of G. Let V be an L#Gmodule. 2.8.1. V is said to be projective relative to H, if any surjective ˆ ˆ L#G-homomorphism W ª V is split provided it is split as an L# H-homomorphism. 2.8.2. V is said to be free relative to H, if V ( Ind GH ŽU . s ˆ. mL# Hˆ U for an L# H-module ˆ Ž L#G U.

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Most fundamentals on the relative projectivity and the relative freeness for usual group algebras can be extended to the twisted group algebras. For example, we state the Higman’s criterion in 3.4.1 in the next section, but in a version convenient for our quotations in this paper. For the moment, we just mention the following criterion for relative freeness, which is easy to check.

ˆ 2.9. LEMMA. An L#G-module V is free relati¨ e to H if and only if V has ˆ an L# H-submodule V0 such that as L-module V has a direct decomposition V s [x g G r H ˆ xV0 , with liftings ˆ x of x. Remark. For arbitrary algebra L, it is convenient to consider the ˆ L-free L#G-modules, so that, ‘‘free relative to 1’’ really implies ‘‘free,’’ and ‘‘projective relative to 1’’ really implies ‘‘projective.’’ Note that this is always the case if L is a division algebra. ˆ is a twisted group algebra, then as we see in 2.7, Gˆ acts on L, If L#G hence G acts on the center ZŽ L. of L. 2.10. Remark. Let D be a division algebra. Then its center ZŽ D . is a field extension of k; and all the maximal subfields of D are extensions over ZŽ D . with a fixed degree s called the Schur index of the division algebra; moreover, there must be a maximal subfield of D which is separable over ZŽ D .. In particular, the degree over k of a maximal separable subfield of D is independent of the choice of the maximal separable subfield; we call this degree the separable degree of D over k, and denote it by < Ds : k <. 2.11. LEMMA. Let D be a di¨ ision algebra and P be a finite p-group, and let D# Pˆ be a twisted group algebra. Then D# Pˆ has a unique indecomposable projecti¨ e module; let m and D⬘ denote the multiplicity and the head of the endomorphism algebra, resp., of the indecomposable projecti¨ e module, then m is p-primary and m ⭈ < DsX : k < s < Ds : k <. In particular, D# Pˆ is a local algebra if p ¦ < Ds : k <. Proof. This fact is implicitly included in w4, 6x; we sketch it. First assume P is of order p; then by w4x one of the following holds: Ža. P acts on ZŽ D . faithfully and D# Pˆ is a division algebra, and ZŽ D# Pˆ. s ZŽ D . P ; Žb. P acts on ZŽ D . faithfully and D# Pˆ ( M p Ž D⬘., where D⬘ is a division algebra with ZŽ D⬘. ( ZŽ D . P and M p Ž D⬘. denotes the matrix algebra; Žc. P acts on ZŽ D . trivially and D# Pˆ ( DP is the usual group algebra; Žd. P acts on ZŽ D . trivially and D# Pˆ is a division algebra, and ZŽ D# Pˆ. is a purely inseparable extension of ZŽ D . of degree p;

STABLE DECOMPOSITIONS

721

Že. P acts on ZŽ D . trivially and D# Pˆ ( M p Ž D⬘. where D⬘ is a division algebra with ZŽ D⬘. being a purely inseparable extension of ZŽ D . of degree p. Checking each case, we see that m s 1 or p, and m ⭈ < DsX : k < s < Ds : k <. The result can be obtained in general by induction on a subnormal series of P with each factor of order p, just as in w6x.

3. G-ALGEBRAS AND MODULES OVER TWISTED GROUP ALGEBRAS We show a relationship between the two objects mentioned in the title of this section. First, we describe how to get a twisted group algebra and a module over it from a G-algebra of special type. Then we describe an inverse construction. 3.1. Let A be a G-algebra satisfying the following condition: 3.1.1. Condition. The left regular module A A s V [ ⭈⭈⭈ [ V is a direct sum of copies of a projective A-module V. y1 Then V x ( V for all x g G as A-modules. Let E s End O Ž V . and L s End AŽ V .. Then V is both an A-module and an L-module. Regarding V as an L-module, it is clear by the construction that L maps injectively into E. On the other hand, A ( MmŽ L. obviously; hence, the A-module structure of V maps A injectively into E. In the following, for convenience, we identify both A and L with their images in E, i.e., A : E and L : E. Moreover, by the isomorphism A ( MmŽ L., we have the following fact: 3.1.2. A and L are the centralizers of each other in E; i.e., L s End AŽ V . and A s End LŽ V .. y1

Further, consider the G-action on A. For any x g G, since V ( V x as A-modules, there is an ˜ x g E* which gives the isomorphism; i.e., for V we have y1

˜x Ž a¨ . s a x ⭈ ˜x Ž ¨ . for all a g A and ¨ g V ; y1

in other words, in E we have ˜ xa s a x ˜ x, or equivalently, ax

y1

s˜ xax˜y1

for all a g A.

Let NE* Ž A. denote the normalizer of A in the multiplicative group E*. In addition, by 3.1.2, we have the centralizer CE* Ž A. s L*. Thus we reach the

FAN AND KULSHAMMER ¨

722

first exact row of the following diagram:

6

L* 6

ˆ G

G

. 6

6

6

6

L*

1

␰

␰ˆ

1

NE* Ž A .

6

NE* Ž A .

6

L*

6

6

1

Ž 3.1.3.

1

Additionally, the above argument also shows that the G-action on A induces a group homomorphism ␰ : G ª NE* Ž A.rL* indicated by the right vertical arrow of the above diagram Ž3.1.3.. So, by pull-back, we get the second exact row of the diagram and the diagram Ž3.1.3. is commutative. ˆ normalizes A and, in fact, induces the Note that the image in E of G G-action on A; also it normalizes L since L is the centralizer of A in E, ˆ hence induces a G-action on L. Therefore, by 2.7, we have a twisted group ˆ algebra L#G of G over L determined by the diagram Ž3.1.3.. But note that, originally, the algebra L is independent of the G-action on A since L s End AŽ V .. Further, from 2.7.1, we get an algebra homomorphism also denoted by ␰ˆ:

ˆ ª E s End O Ž V . ; ␰ˆ: L#G

Ž 3.1.4.

and from 3.1.2 we see that End LŽ V . s A, hence we have

ˆ 3.1.5. V is an L#G-module and End L#GˆŽ V . s AG . 3.2. Remark. The above constructing procedure in 3.1 is reversible. Assume we have:

ˆ 3.2.1. Condition. An O-algebra L and a twisted group algebra L#G ˆ are given, and V is an L-free L#G-module. Let A s End LŽ V .. By the L-freeness of V it is easy to see that the left regular A-module A A s V [ ⭈⭈⭈ [ V, a direct sum of rank LŽ V . copies of V, and End AŽ V . s L. Further, let E s End O Ž V .. Then, just like that in 3.1, both A and L are injectively mapped into E and, identifying them with their images in E, we see that A and L are centralizers of each other ˆ in E. Moreover, the L#G-module structure of V is a unitary algebra ˆ ª E, and ␰ˆ< L just coincides with the injection homomorphism ␰ˆ: L#G ˆ Žin Gˆ which sends L into E. For x g G, let ˆ x be a lifting of x in L#G ˆ precisely, see 2.7., then ␰ Ž ˆ x . stabilizes Žby conjugation in E . L in E, hence stabilizes A in E as A is the centralizer of L in E; and if x s 1,

STABLE DECOMPOSITIONS

723

then ˆ x g L*, hence ˆ x centralizes A. Therefore we have a group homomorˆ phism from G s GrL* to NE* Ž A.rCE* Ž A.; the latter one is a subgroup of the automorphism group of A. Now the following conclusion is clear. 3.2.2. A is a G-algebra, and AG s End L#GˆŽ V .; more precisely, G acts on A in the following way: y1

a x Ž ¨ . s ␰ˆŽ ˆ x . ( a( ␰ˆŽ ˆ x.

y1

Ž¨.

for all a g A and ¨ g V .

It is a routine, though it is not a short routine, to check the following assertion. 3.3. LEMMA. Let A be a G-algebra satisfying the condition 3.1.1 and L, ˆ and V be constructed as in 3.1. If the G-algebra A⬘ is constructed by the L#G, ˆ abo¨ e 3.2.2 from the L#G-module V, then A⬘ ( A as G-algebras. ˆ Con¨ ersely, let L, L#G, and V be as in 3.2.1 and A be the G-algebra as in X ˆ and V⬘ are constructed by 3.1 from the G-algebra A and 3.2.2. If L⬘, L#G, X ˆ ( L#G, ˆ and with this isomorphism, the A-module V, then L⬘ ( L and L#G V⬘ ( V. Return to the construction 3.1.

ˆ 3.4. LEMMA. Let A be a G-algebra satisfying 3.1.1 and V be the L#Gmodule constructed in 3.1.5. Then for any subgroup H of G the following hold: 3.4.1. to H; 3.4.2.

A is projecti¨ e relati¨ e to H if and only if V is projecti¨ e relati¨ e A is free relati¨ e to H if and only if V is free relati¨ e to H.

Proof. First we prove 3.4.2. Assume V is free relative to H. By Lemma ˆ 2.9, there is an L# H-submodule V0 of V such that we have an L-module decomposition V s [x g G r H ˆ xV0 ; let f g A s End LŽ V . be the projection from V to V0 determined by this decomposition; then it is easy to check y1 that f x is the projection from V to the direct summand ˆ xV0 , hence y1 x  f < x g GrH 4 is a complete orthogonal set of idempotents in A; in other words, the G-algebra A is free relative to H. Conversely, assume A is free relative to H. That is, there is an y1 idempotent f g A H such that  f x < x g GrH 4 is a complete orthogonal set of idempotents in A. Since A s End LŽ V ., the sum V s y1 [x g G r H f x V is an L-module direct decomposition; further, it is clear y1 ˆ that f x V s ˆ x ⭈ Ž fV . and fV is an L# H-submodule of V since f g A H . Thus, by Lemma 2.9 again, V is free relative to H. Now we turn to prove 3.4.1. From the relative projectivity of A we have ˆ 1 A s TrHG Ž a. for an a g A H . For an L#G-surjection ␳ : W ª V, if ␭: ˆ ˆ V ª W is an L# H-lifting for ␳ , then ␰ s TrHG Ž ␭ a. is an L#G-lifting for ␳ . These are as the same as for the usual group algebras.

FAN AND KULSHAMMER ¨

724

Conversely, applying the relative projectivity of V, we see that V is an ˆ L#G-direct summand of Ind GH Ž V .. Let A˜ s End LŽInd GH Ž V ... By Remark 3.2, A˜ is a G-algebra and, by 3.4.2, it is free relative to H. In particular, ˆ A˜G s A˜GH . On the other hand, let V ª Ind GH Ž V . be an L#G-split injection, ˜ Then from A˜G s A˜GH with it we can get a G-algebra embedding A ª A. and this G-algebra embedding we can reach the desired conclusion AG s AGH . As an application, we show a G-algebra version of the stable Clifford’s theory. 3.5. PROPOSITION. Let H be a normal subgroup of G and A be a G-algebra with an orthogonal decomposition 1 A s TrHG Ž f ., where f g A H is y1 an idempotent. If f x for any x g$ G is conjugate to f by Ž$ A*. H , then there is H o G a twisted group algebra Ž fA f .#ŽGrH. ( A , where ŽGrH. o denotes the opposite group. Proof. Consider the G-algebra A H where G s GrH. By the hypothey1 sis we have an A H -module decomposition A H s [x g G A H f x which H H satisfies the condition 3.1.1. In addition, End A H Ž A f . s Ž fA f . o. Thus o ˆ by 3.1 we get the twisted group algebra Ž fA H f .#G, and A H f is a module H o ˆ over Ž fA f .#G, which has the following decomposition: AH f s

[f xgG

xy1 H

A fs

[ ˆx ⭈ fA

H

H f s Ind G 1 Ž fA f . .

xgG

o ˆ That is, as an Ž fA H f .#G-module, A H f is isomorphic to the regular o ˆ Ž fA H f .#G-module. So G

H o ˆŽ A f . ( AG s Ž A H . ( End Ž f A H f .#G

ž Ž fA

H

ˆ f . o#G

o

/

o

ˆ. . ( Ž fA H f . # Ž G

3.6. Let A be a G-algebra and H␤ be a pointed group on A. Recall that AŽ H␤ . denotes the simple factor of A H associated with ␤ ; i.e., the natural composition map A H ª A H rJ Ž A H . ª AŽ H␤ . sends ␤ to a nonzero set. It is clear that AŽ H␤ . is an NG Ž H␤ .-algebra where NG Ž H␤ . s NG Ž H␤ .rH, hence it is a simple NG Ž H␤ .-algebra; in particular, AŽ H␤ . has a unique simple module V Ž H␤ ., which is also a projective AŽ H␤ .-module; i.e., the condition 3.1.1 is satisfied. Thus the construction 3.1 can be applied to determine a division algebra DŽ H␤ . and a twisted group algebra DŽ H␤ .# NˆG Ž H␤ ., and V Ž H␤ . becomes a DŽ H␤ .# NˆG Ž H␤ .-module. 3.6.1. DEFINITION. The above NG Ž H␤ .-algebra AŽ H␤ . is called the multiplicity algebra of the pointed group H␤ on A, the DŽ H␤ .# NˆG Ž H␤ .module V Ž H␤ . is called the multiplicity module of the pointed group H␤ on A.

725

STABLE DECOMPOSITIONS

Note that, if the ground-field k is algebraically closed, these notations are the same as the usual ones, e.g., described carefully in w12, Sect. 19x; but only the k*-groups are involved and k# NˆG Ž H␤ . are indeed the usual twisted group algebras; in particular, they are just the usual group algebras if < G : H < is p-primary. Returning to our general case, from 3.1.5 we have the following formula at once: A Ž H␤ .

N G Ž H␤ .

s End DŽ H␤ .# NˆG Ž H␤ . Ž V Ž H␤ . . .

Ž 3.6.2.

Now we come to the main result of this section. 3.7. PROPOSITION. of G.

Let A be a G-algebra and H be a normal subgroup

3.7.1. A is a G-algebra projecti¨ e relati¨ e to H if and only if the multiplicity module V Ž H␤ . is a projecti¨ e DŽ H␤ .# NˆG Ž H␤ .-module for any pointed group H␤ on A. 3.7.2. A is a G-algebra free relati¨ e to H if and only if the multiplicity module V Ž H␤ . is a free DŽ H␤ .# NˆG Ž H␤ .-module for any pointed group H␤ on A. Proof. We make several reductions, and then prove the two assertions. First, since H is normal in G, A H is a GrH-algebra which is in fact the real object of the proposition. So in the following we can assume H s 1; then pointed groups H␤ are just pointed groups 1␤ . Second, if the results hold for ArJ Ž A. where J Ž A. denotes the radical, then it is easy to see, by Lemmas 2.5 and 2.6, they also hold for A. So in the following we can assume that A is a semisimple G-algebra; in this case, A is a direct product of simple algebras and every simple factor corresponds to exactly one pointed group 1␤ on A; that is, As

Ł A Ž1␤ . 1␤

with 1␤ running over the pointed groups on A.

Then it is clear that the G-action on A permutes the simple factors. Third, for every G-orbit ⍀ of the simple factors we collect the members in ⍀ to get a G-stable factor A ⍀ of A then A s Ł ⍀ A ⍀ is a direct product of the G-algebras A ⍀ . Obviously, the proposition holds for A if and only if it holds for every A ⍀ . So in the following we can assume that G permutes the simple factors of A transitively. Pick up a simple factor AŽ1␤ ., the stabilizer of AŽ1␤ . in G is just NG Ž1␤ .; and the cosets GrNG Ž1␤ . correspond bijectively onto the simple factors of A. For brevity, denote N s NG Ž1␤ . and S s AŽ1␤ .. Then it is

FAN AND KULSHAMMER ¨

726 easy to check that As

Sx

Ł

y1

Ž algebra direct product. ,

Ž 3.7.3.

xgGrN

and the trace map induces a unitary algebra isomorphism as: (

TrNG : S N ª AG .

Ž 3.7.4.

y1

Further, since for any simple factor S x of A it is easy to check that y1 Tr1G Ž S x . s Tr1G Ž S ., we have Tr1G Ž A. s Tr1G Ž S .. In other words, the restriction map of Ž3.7.4. gives an isomorphism (

TrNG < S1N : S1N ª AG 1 .

Ž 3.7.5.

Now the assertion 3.7.1 follows from Ž3.7.4., Ž3.7.5., and Ž3.4.1. at once. For 3.7.2, first assume S is an N-algebra free relative to 1 Žcf. 3.4.2.; i.e., we have an orthogonal decomposition 1 S s Tr1N Ž f . for an idempotent f g S. Then by the unitary isomorphism Ž3.7.4., we get 1 A s TrNG Ž1 S . s TrNG ŽTr1N Ž f .. s Tr1G Ž f ., which is also an orthogonal decomposition of idempotents by the direct product Ž3.7.3.. That is, A is free relative to 1. Conversely, assume 1 A s Tr1G Ž f . with an idempotent f g A is an orthogonal decomposition of idempotents. Let R be a set of representatives for GrN. Then 1S s 1S ⭈

Ý

fx

y1

s

xgG

s

ž

tgR

y1

xgG

Ý Ý 1S ⭈ f t ygN

1S ⭈ f x

Ý

s

Ý Ý 1S ⭈ f t y

y1

ygN tgR

yy1

/

.

y1

Since  f x < x g G4 is an orthogonal set of idempotents and 1 S is a central idempotent of A, we see that the idempotent es

Ý 1S ⭈ f t tgR

belongs to S; and, for the same reason, given any two y 1 / y 2 g N we have < t g R 4 l  tyy1 < t g R 4 s ⭋.  tyy1 1 2 y1

Hence  e y < y g N 4 is an orthogonal set of idempotents on S and 1 S s y1 Ý y g N e y s Tr1N Ž e .. That is, S is an N-algebra free relative to 1.

STABLE DECOMPOSITIONS

727

3.8. LEMMA. Let A be a simple G-algebra and V be the unique simple A-module; let i g AG be an idempotent and A⬘ s iAi which is a simple ˆ and V be G-algebra again, and V ⬘ be the unique A⬘-module. Let D, D#G, X ˆ and V ⬘ be constructed from constructed from A and V by 3.1; and D⬘, D#G, ( A⬘ and V ⬘ by 3.1. Then there are an isomorphism D ª D⬘ and a compatible ( X ˆª ˆ and a corresponding D#G-module ˆ isomorphism D#G D#G surjecti¨ e homomorphism V ª V⬘ which is split. Proof. Let j be a primitive idempotent of A such that ij s j s ji; then, since j g A⬘ is still primitive, we can assume that V s Aj and V⬘ s AX j s iAj. Let E s End O Ž V ., then A and D are unitary subalgebras of E and are centralizers each other in E. Note that, by construction of 3.1, we have 3.8.1. Any a g A as an element of E is just the left multiplication by a on Aj; and, with D ( Ž jAj . o , any d g jAj as an element of E is just the right multiplication by d on Aj, where Ž jAj . o denotes the opposite algebra of jAj. Since V⬘ s iAj, E⬘ s End O Ž V⬘. is an embedded subalgebra of E; i.e., E⬘ s iEi; and by 3.8.1, A⬘, as a subalgebra of E⬘, is also an embedded subalgebra of A ; E. In particular, D⬘ ( Ž jA⬘ j . o s Ž jiAij . o s Ž jAj . o ( D Žnote that this isomorphism is not enough for the requirements of the lemma.. Additionally, as the same as 3.8.1, we have 3.8.2. Any a⬘ g A⬘ as an element of E⬘ is just the left multiplication by a⬘ on iAj; and, with D⬘ ( Ž jAj . o , any d⬘ g jAj as an element of E⬘ is just the right multiplication by d⬘ on iAj. Noting that i g A : E and d g D : E commute each other, we get an algebra homomorphism:

␦ : D ª D⬘,

d ¬ di;

and from 3.8.1 and 3.8.2 it is easy to check the following diagram is commutative: Ž jAj . o s Ž jAj . o 6

6

D

(

␦

6

(

D⬘.

FAN AND KULSHAMMER ¨

728

ˆ Thus ␦ is an isomorphism. Recall from Ž3.1.3. that V becomes a D#Gmodule through the following commutative diagram

ˆ G

G

1

␰

6

6

6

D*

1

6

6

D* 6

␰ˆ

1

NE* Ž A .

6

NE⬘ Ž A .

6

D*

6

6

1

and the algebra homomorphism determined by the group homomorphism ˆ ª E; see Ž3.1.4.; and ␰ˆŽ D#Gˆ. s D ⭈ ␰ˆŽ Gˆ.. ␰ˆ is also denoted by ␰ˆ: D#G ˆ., Since i g AG hence i commutes with the elements in both D and ␰ˆŽ G multiplying by i, from the above commutative diagram we get the following commutative one

6

6

ˆ G

G

6

ˆ G

␰⬘

6

6

D⬘* 6

6

1

1

6

␰ˆ⬘

NŽ E⬘.* Ž A⬘.rD* 6

6

NŽ E⬘.* Ž A⬘. 6

6

D⬘*

6

1

G

1

1

␦N D* (

6

D*

6

1

where the first two rows are just the commutative diagram which deterX ˆ cf. Ž3.1.3.. Thus we reach the mines the twisted group algebra D#G, following isomorphism which is compatible with the isomorphism ␦ : D ª D⬘: (

X ˆ ª D#G, ˆ ␦ˆ: D#G

dxˆ ¬ ␦ Ž d . ˆ x.

X ˆ ª E and ␰ˆ⬘: D#G ˆª Note that, for the algebra homomorphism ␰ˆ: D#G E⬘ we have X ˆ. s D⬘ ⭈ ␰ˆ⬘ Ž Gˆ. s iDi ⭈ i ␰ˆŽ Gˆ. i s i ␰ˆŽ D#Gˆ. i; ␰ˆ⬘ Ž D#G

from which one can deduce that the following two k-homomorphisms ␳ ˆ and ␫ are both D#G-homomorphisms. The left multiplication by i gives a surjective k-homomorphism:

␳ : Aj s V ª V ⬘ s iAj,

aj ¬ iaj;

STABLE DECOMPOSITIONS

729

and the inclusion map ␫ : iAj s V⬘ ª V s Aj is of course injective; and ˆ ␳␫ s id V ⬘. Therefore ␳ is a split D#G-homomorphism. 4. POINTED GROUPS AND TWISTED GROUP ALGEBRAS In this section we give the proofs of the results 1.5᎐1.9 stated in the Introduction. We begin with necessary preparations. 4.1. LEMMA. Let A be a G-algebra and G␣ be a pointed group on A; let A␣ s iAi for i g ␣ ; let H be a subgroup of G. For all pointed groups H␤ ; G␣ , set ␤ ⬘ s ␤ l A␣ s  j < j g ␤ and ij s j s ji4 . 4.1.1. H␤ ⬘ are just the all pointed groups of H on A␣ . 4.1.2. NG Ž H␤ ⬘ . s NG Ž H␤ ., and A␣ Ž H␤ ⬘ . is embedded into AŽ H␤ . as NG Ž H␤ .-algebras. 4.1.3. there are an isomorphism DŽ H␤ ⬘ . ( DŽ H␤ ., and a compatible isomorphism DŽ H␤ ⬘ .# NˆG Ž H␤ ⬘ . ( DŽ H␤ .# NˆG Ž H␤ ., and ¨ ia this isomorphism, V Ž H␤ ⬘ . is a direct summand of V Ž H␤ . as DŽ H␤ .# NˆG Ž H␤ .-modules. Ž G. Ž G . denotes 4.1.4. DŽ G␣ . is anti-isomorphic to AG ␣ rJ A␣ , where J A␣ the radical. Proof. For any pointed group H␤ on A it is clear that H␤ ; G␣ if and only if ␤ ⬘ s ␤ l A␣ / ⭋. Thus 4.1.1 is obviously true. Hence NG Ž H␤ ⬘ . s NG Ž H␤ .. For the radicals we have J Ž iA H i . s iJ Ž A H . i; so the composition NG Ž H .-algebra homomorphism A␣H ª A H ª A H rJ Ž A H . induces an NG Ž H .-algebra embedding A␣HrJ Ž A␣H . ª A HrJ Ž A H . ; so all of 4.1.2 are proved. Consequently, 4.1.3 follows from Lemma 3.8. For 4.1.4, taking H␤ s G␣ in 4.1.3, we get DŽ G␣ . ( DŽ G␣ ⬘ ., where ␣ ⬘ s  i4 : AG␣ s iAG i. Hence the conclusion follows from the Definition 3.6.1. 4.2. LEMMA. Notations as in 4.1 abo¨ e and further assume that H is normal in G and G␣ is projecti¨ e relati¨ e to H. Ž

Ž H Ž H .. G is surjecti¨ e, hence 4.2.1. The natural map AG ␣ ª A␣ rJ A␣ H .. G A␣ has a unique idempotent i s 1 A ␣. 4.2.2. G permutes all the pointed groups H␤ ; G␣ transiti¨ ely.

A␣H rJ Ž

FAN AND KULSHAMMER ¨

730

4.2.3. Gi¨ en H␤ ; G␣ , the following trace map is an algebra isomorphism Ž where ␤ ⬘ s ␤ l A␣ as in 4.1.: TrHG : A␣ Ž H␤ ⬘ .

N G Ž H␤ .

G

ª Ž A␣H rJ Ž A␣H . . .

Proof. By the assumption, A␣ is projective relative to H, see Lemma H 2.2; hence Ž A␣ .HG s AG ␣ . Applying Lemma 2.6 to the GrH-algebra A␣ , we get the first conclusion of 4.2.1; the second conclusion of 4.2.1 is clear since AG ␣ is local and has the unique idempotent i s 1 A ␣. For 4.2.2, it is clear that A␣HrJ Ž A␣H . s

Ł

H␤;G␣

A␣ Ž H␤ ⬘ . ,

where H␤ runs over H␤ ; G␣ and ␤ ⬘ s ␤ l A␣ as in 4.1. It is clear that G permutes the simple factors, corresponding to the G-permutation on the set of all H␤ ; G␣ . Suppose 4.2.2 is false, then there is more than one G-orbit of the simple factors. Every such G-orbit forms a G-subalgebra; hence A␣H rJ Ž A␣H . is a direct product of more than one G-algebra; consequently, Ž A␣H rJ Ž A␣H .. G has more than one idempotent; this contradicts to 4.2.1. So 4.2.2 is proved. At last, 4.2.3 follows from 4.2.2 clearly, just as the argument for Ž3.7.4.. 4.3. COROLLARY. Notations as in 4.2. Then G␣ is projecti¨ e relati¨ e to a pointed group H␤ on A if and only if H␤ ; G␣ . Proof. Assume H␤ ; G␣ . Since ␣ ; AGH , by Lemma 2.2 we have a pointed group H␤ 0 ; G␣ such that G␣ is projective relative to H␤ 0 . On the other hand, by 4.2.2, H␤ is G-conjugate to H␤ 0 . So G␣ is also projective relative to H␤ . For the necessity, since only the G-algebra A H is considered, in order to shorten the symbols we assume H s 1. Let i g ␣ and i g Tr1G Ž A ␤ A. for a pointed group 1␤ , our aim is to show that 1␤ ; G␣ , or equivalently, to show that iAŽ1␤ . i / 0; see the sketch before Lemma 2.2. But i g Tr1G Ž A ␤ A. implies that i g Tr1G Ž iA ␤ Ai . since i is an idempotent and centralized by G. Mapping it to A s ArJ Ž A., we get that i g Tr1G ŽiA ␤ Ai.. However, by definition of AŽ1␤ ., we have A ␤ As AŽ1␤ .; thus iAŽ1␤ . i / 0. We remark that the above results 4.2.2 and 4.3 are related to w3, 4.1 and 4.2x; in fact w3x worked in a more general case where the case that H is normal in G is included, though under the hypothesis that k is algebraically closed. 4.4. COROLLARY. l A␣ as in 4.1.

Notations as in 4.2 abo¨ e. Let H␤ ; G␣ and ␤ ⬘ s ␤

731

STABLE DECOMPOSITIONS

4.4.1. On the G-algebra A␣ the multiplicity module V Ž H␤ ⬘ . is an indecomposable projecti¨ e DŽ H␤ .# NˆG Ž H␤ .-module. 4.4.2. The di¨ ision algebra DŽ G␣ . is anti-isomorphic to the head of the endomorphism algebra End DŽ H .# Nˆ Ž H .Ž V Ž H␤ ⬘ ... ␤

␤

G

Proof. Consider the isomorphism 4.2.3 for the H␤ ; G␣ of this statement. Since Ž A␣H rJ Ž A␣H .. G is local by 4.2.1, so is the algebra A␣ Ž H␤ ⬘ .

N G Ž H␤ .

s End DŽ H␤ .# NˆG Ž H␤ . Ž V Ž H␤ ⬘ . . ,

Ž 4.4.3.

where the equality follows from 3.1.5 or Ž3.6.2.. Thus V Ž H␤ ⬘ . on A␣ is an indecomposable DŽ H␤ .# NˆG Ž H␤ .-module since its endomorphism algebra is local; and it is projective by Proposition 3.7 since A␣ is projective relative to H. For the statement 4.4.2, note that, by 4.1.3, we have DŽ G␣ . ( DŽ G␣ ⬘ .; further, by 4.1.4, DŽ G␣ ⬘ . is anti-isomorphic to the head of AG ␣ ; but, by H H .. G Ž Ž 4.2.1, AG is mapped onto A rJ A . Therefore, the second assertion ␣ ␣ ␣ is derived from the isomorphism 4.2.3 and the equality Ž4.4.3.. 4.5. LEMMA. Let A be a G-algebra and H be a subgroup of G; let i g AG be an idempotent. Assume that i s TrHG Ž j . for j g A H is an orthogonal decomposition of idempotents. 4.5.1. For any H F K F G, TrHK Ž j . is an idempotent on A K ; and it is primiti¨ e on A K only if j is primiti¨ e on A H. 4.5.2. Assume that i is primiti¨ e on AG and i s TrHG Ž j . s TrHG Ž j⬘. both are orthogonal decompositions of idempotents for j g ␤ and j⬘ g ␥ where H␤ and H␥ are pointed groups on A. If H␤ and H␥ are conjugate by G, y1 then there are an a g Ž A*. G and y g NG Ž H . such that j a y s j⬘. Proof. The statement 4.5.1 is clearly true. For 4.5.2, we have y g G y1 y1 y1 such that Ž H␤ . y s H␥ . That is, H y s H, i.e., y g NG Ž H .; and ␤ y s ␥ , yy1 yy1 i.e., j g ␥ . Thus both j , j⬘ g ␥ . By definition, ␥ is a conjugacy class of primitive idempotents of A H ; so, there is a b g Ž A*. H such that y1 bj y by1 s j⬘. On the other hand, we have: TrHG Ž j y

y1

. s TrNG Ž H . TrHN G

GŽ H .

y1

Ž j y . s TrNG Ž H .TrHN G

GŽ H .

Ž j . s TrHG Ž j . ;

thus 4.5.2 follows from the following fact which appears in w9x. 4.5.3. Let i s TrHG Ž j . s TrHG Ž j⬘. with idempotents j, j⬘ g A H be two orthogonal decompositions. If there is b g Ž A*. H such that bjby1 s j⬘, then there is an a g Ž A*. G such that ajay1 s j⬘. To see it, we set u s TrHG Ž bj . and ¨ s TrHG Ž jby1 .; then both u, ¨ g iAi, and a direct calculation shows that u¨ s i, hence ¨ u s i because iAi is a

FAN AND KULSHAMMER ¨

732

unitary O-algebra of finite O-rank. Let a s Ž1 y i . q u and c s Ž1 y i . q

¨ , noting that 1 y i annihilates both u and ¨ , we can check that ac s 1

and ca s 1 and ajc s j⬘.

4.6. A Proof of Proposition 1.5. Assume that G␣ is free relative to H. Let i g ␣ . Then A␣ s iAi is free relative to H, see Lemma 2.2. For any pointed group H␤ ; G␣ , setting ␤ ⬘ s ␤ l A␣ as in 4.1, by Proposition 3.7 and cf. 4.2.3 we have that V Ž H␤ ⬘ . on A␣ is a free DŽ H␤ .# NˆG Ž H␤ .-module; however, V Ž H␤ ⬘ . is indecomposable by Corollary 4.4.1; in other words, DŽ H␤ .# NˆG Ž H␤ . has an indecomposable free module, hence it must be a local algebra. Conversely, assume DŽ H␤ .# NˆG Ž H␤ . is a local algebra for a pointed group H␤ ; G␣ . Let A␣ s iAi for i g ␣ and ␤ ⬘ s ␤ l A␣ be the same as above. By Proposition 3.7 and cf. Lemma 4.2, V Ž H␤ ⬘ . is a projective DŽ H␤ .# NˆG Ž H␤ .-module because A␣ is projective relative to H. However, DŽ H␤ .# NˆG Ž H␤ . is local, this forces that V Ž H␤ ⬘ . is a free DŽ H␤ .# NˆG Ž H␤ .module. So A␣ is free relative to H by Proposition 3.7; this also implies that G␣ is free relative to H, cf. Lemma 2.2. The conclusion 1.5.1 follows from 4.2.2 and 4.5.2 directly. At last we prove 1.5.2. By the assumption, there is an H␥ ; G␣ and j⬘ g ␥ such that i s TrHG Ž j⬘. g ␣ is an orthogonal decomposition. From y1 4.2.2 we have y g G such that Ž H␥ . y s H␤ . Thus, cf. 4.5.3 above, there is y1 an a g Ž A*. G such that j a y s j⬘. Then i s TrHG Ž j⬘ . s TrHG Ž j a y

y1

y1

. s ž TrHG Ž j y . /

a

a

s Ž TrHG Ž j . . .

Therefore 1.5.2 is derived. 4.7. A Proof of Lemma 1.6. Assume G␣ is projective relative to H␤ . By Lemma 2.2, ␣ : AGH . Let H s H0 F H1 F ⭈⭈⭈ F Hn s G be a subnormal series of subgroups. If n s 1; i.e., H is normal in G, then 4.3 shows that H␤ ; G␣ , as required. Now assume n ) 1. Write i g ␣ as an orthogonal sum of primitive idempotents on A H ny 1 : i s i1 q i 2 q ⭈⭈⭈ , and assume i1 g ␣ 1 for a pointed group Ž Hny1 .␣ 1. Since i g TrHG Ž A H␤ A H ., by Mackey decomposition and the normality of Hny 1 in G, we have ig

x

x

TrHHxny 1 Ž A H ␤ xA H . .

Ý xgGrH

Noting that i1 s ii1 g A H ny 1 and the right-hand side of the above is an ideal of A H ny 1 , we see that i1 belongs to this ideal. By Rosenberg’s lemma, e.g., see w12, 4.9x, there is an x such that x

i1 g TrHHxny 1 Ž A H ␤ xA H

x

.

hence i1x

y1

g TrHH ny 1 Ž A H␤ A H . .

STABLE DECOMPOSITIONS

733

y1

Setting ␤ny 1 s Ž ␣ 1 . x , we have Ž Hny1 .␤ ny 1 projective relative to H␤ , and Ž Hny 1 .␤ ; G␣ since ŽŽ Hny1 .␤ . x s Ž Hny1 .␣ ; G␣ ; hence G␣ is projecny 1 ny 1 1 tive relative to Ž Hny 1 .␤ ny 1 by Corollary 4.3. Then by induction the desired subnormal projective series of pointed groups exists. The converse part is clearly true. 4.8. COROLLARY. Let G␣ and H␤ be pointed groups on a G-algebra A, and assume that H is subnormal in G of p-primary index. If G␣ is projecti¨ e relati¨ e to H␤ , then < D Ž G␣ . s : k < < DŽ H␤ .s : k <. Proof. If H is normal in G, then, applying the notation in Corollary 4.4.1, V Ž H␤ ⬘ . is the unique indecomposable projective DŽ H␤ .# NˆG Ž H␤ .module, see Lemma 2.11; and the conclusion follows from Corollary 4.4.2 and Lemma 2.11. In general case, from Lemma 1.6 proved just now, we have a subnormal projective series of pointed groups from H␤ to G␣ , thus the result is obtained by induction on the length of the subnormal projective series. 4.9. A Proof of Theorem 1.7. We prove the equivalences of the three statements by induction on the length n of the subnormal projective series of pointed groups. Note that the case of n s 1 is just Proposition 1.5; thus we assume n ) 1. 1.7.1 « 1.7.2. Let i g ␣ and j g ␤ such that i s TrHG Ž j . is an orthogonal decomposition of idempotents. From Lemma 4.5.1 we see that i⬘ s TrHH ny 1 Ž j . is a primitive idempotent on A H ny 1 Žsince i s TrHGny 1Ž i⬘. is primitive on AG .; hence we have a pointed group Ž Hny 1 .␤ ny 1 on A such that i⬘ g ␤ny 1; consequently, H␤ ; Ž Hny1 .␤ ny 1 ; G␣ and both G␣ to Ž Hny 1 .␤ and Ž Hny1 .␤ ny 1 to H␤ are relative free. Then, by induction, ny 1 1.7.2 is proven. 1.7.2 « 1.7.3. Trivial. 1.7.3 « 1.7.1. By induction, for i g ␣ there is an h g ␤ny 1 such that i s TrHGny 1Ž h. is an orthogonal decomposition of idempotents, and for h g ␤ny 1 there is a j g ␤ such that h s TrHH ny 1 Ž j . is an orthogonal decomposition of idempotents; thus i s TrHG Ž j . is an orthogonal decomposition of idempotents. The ‘‘conjugation’’ part follows from 4.5.3 directly. 4.10. A Proof of Corollary 1.8. This is clear by Lemma 2.3.1 and 4.5.2. 4.11. A Proof of Corollary 1.9. The 1.9.1 follows from Corollary 4.8 straightforwardly. For 1.9.2, by Lemma 2.2, G␣ is projective relative to a pointed group H␤ ; so by Lemma 1.6 there is a subnormal projective series H␤ s Ž H0 .␤ ; Ž H1 .␤ ; ⭈⭈⭈ ; Ž Hn .␤ s G␣ ; and it follows from 1.9.1 that p ¦ 0 1 n < DŽŽ Ht .␤ .s : k < for every t. Hence DŽŽ Ht .␤ .# NˆH ŽŽ Ht .␤ . is a local algebra t t tq 1 t

734

FAN AND KULSHAMMER ¨

for every t, see Lemma 2.11. Therefore, by Theorem 1.7, G␣ is free relative to H␤ . We prove 1.9.3. First we prove the characterization of maximal G-stable orthogonal sets of idempotents on A. Let I be any G-stable orthogonal set of idempotents on A. Any proper refinement of I is a composition of the following two kinds of proper refinements. a. A G-orbit of I is properly refined into a G-orbit, so the G-orbit is lengthened. In this case, a member i of the orbit is refined in an orthogonal sum i s TrRC G Ž i. Ž j ., i.e., i g ACRG Ž i. for a proper subgroup R of CG Ž i .. Conversely, if i g ACRG Ž i. for a proper subgroup R of CG Ž i ., since condition 1.9.1 holds for R, by 1.9.2, i really has a such proper refinement Žnote that there is a pointed group Ž CG Ž i ..␦ such that i g ␦ hence Ž CG Ž i ..␦ is projective relative to R .. b. The number of G-orbits of I is increased, or equivalently, irkŽŽ O I . G . is increased. In fact, there are two subcases: either I is not complete, or a member i of I is not primitive on AC G Ž i. Žcf. 4.5.1.; but the further analysis is not necessary for our proof. From the above observations, it is clear that I is maximal if and only if irkŽŽ O I . G . s irkŽ AG . and every i g I satisfies i f ACRG Ž i. for any proper subgroup R of CG Ž i .. Now we prove the conjugation part of 1.9.3. Let I and I⬘ be two maximal G-stable sets as in 1.9.3. Then irkŽŽ O I⬘. G . s irkŽ AG . s irkŽŽ O I . G .; i.e., the number of the G-orbits of I equals the number of the r G-orbits of I⬘, say, the number is r. Let I s Dts1 It be the decomposition r of G-orbits and let e t s Ý i t g I t i t ; and let I⬘ s Dts1 ItX and eXt s Ý iXt g I tX iXt X X similarly for I⬘. Then both  e1 , . . . , e r 4 and  e1 , . . . , e r 4 are maximal orthogonal sets of idempotents on AG , so that there is an u g Ž A*. G such that, y1 reindexing the subscripts if necessary, we have e tu s eXt for t s 1, . . . , r; X X cf. Lemma 2.4. Considering the algebra e t Ae t , its identity element has two decompositions: eXt s

Ý

X X i tgI t

iXt s

Ý

i tgI t

y1

i tu .

Since both I and I⬘ have the property in 1.9.3, all the subgroups CG Ž iXt . y1 and CG Ž i tu . are defect groups of the primitive idempotent eXt on AG , see Lemma 2.3.1; thus all these subgroups are conjugate by G. Noting that y1 y1 y1 both ItX and Itu are G-transitive sets, we can pick iXt g ItX and i tu g Itu y1 such that CG Ž iXt . s CG Ž i tu .. Then from Corollary 1.8 we have an inverty1 y1 ible bt g Ž eXt AeXt . G such that ŽŽ It . u . b t s ItX in eXt AeXt . Let a t s bt u and a s a1 q ⭈⭈⭈ qa r , noting that 1 s eX1 q ⭈⭈⭈ qeXr is an orthogonal decompoy1 sition of idempotents, we get that a g Ž A*. G and I a s I⬘.

STABLE DECOMPOSITIONS

735

ACKNOWLEDGMENTS This work was done while the first author was visiting the Universitat ¨ Jena, April᎐July 1998, supported by DAAD. He thanks DAAD very much for the financial support. He is also grateful to the Mathematisches Institut, Universitat ¨ Jena and to the second author for their hospitality. Thanks are also given to Laurence Barker and Lluis Puig for many helpful discussions and comments.

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