A note on the basic Morita equivalences

SCIENCE CHINA Mathematics. ARTICLES. March 204 Vol.57 No.3: 483 490 doi: 0.007/s425-03-4763- A note on the basic Morita equivalences HU XueQin School of Mathematical Sciences, Capital Normal University, Beijing 00048, China Email: sky9708@63.com Received November, 202; accepted January 23, 203; published online December 9, 203 Abstract Let G and G be two finite groups, and p be a prime number. k is an algebraically closed field of characteristic p. We denote by b and b the block idempotents of G and G over k, respectively. We assume that the block algebras kgb and kg b are basically Morita equivalent. Puig and Zhou (2007) proved that the corresponding block algebras of some special subgroups of G and G are also basically Morita equivalent. We investigate the relationships between the basic Morita equivalences of two kinds of subgroups of G and G : We find a module such that its induced module and its restricted module induce the basic Morita equivalences respectively, hence give a precise description of these basic Morita equivalences. Keywords block algebra, basic Morita equivalence, graded algebra, source algebra MSC(200) 20C05, 20C5 Citation: Hu X Q. A note on the basic Morita equivalences. Sci China Math, 204, 57: 483 490, doi: 0.007/ s425-03-4763- Introduction Let p be a prime number and k be an algebraically closed field of characteristic p. In [4, Section 7], Puig introduced the so-called basic Morita equivalences between two Brauer k-blocks of finite groups G and G. In this situation, the corresponding defect groups are isomorphic and the suitable Brauer correspondent blocks of the centralizers of the subgroups of the defect groups are also basically Morita equivalent. In [5], the last result has been extended to the blocks of the normalizers (see [5, Theorem.4]). Our main purpose here is to investigate these two basic Morita equivalences and give an explicit relationship between them, which may be regarded as a Clifford theory of basic Morita equivalences. Let G be a finite group and b be a block idempotent of kg, i.e., a primitive central idempotent of kg. A pair (L,τ) is called a pointed group on kg, where L is a subgroup of G and τ is a point of (kg) L, i.e., a conjugacy class of primitive idempotents of (kg) L. Here, (kg) L is the set of L-fixed elements of kg and a pointed group (L,τ) will always be written L τ. By the definition, G {b} is a pointed group on kg. Let P γ be a defect pointed group of G {b}. We denote by LP(P γ ) the set of all local pointed groups on kg contained in P γ. For any Q δ LP(P γ ), there exists a unique block idempotent of kc G (Q) determined by δ (see [6, Section 40]). We denote it by b δ. It is well known that for any subgroup H such that C G (Q) H N G (Q δ ), b δ is also a block idempotent of kh. Let G be another finite group and b be a block idempotent of kg. Assume that kgb and kg b are basically Morita equivalent, namely, there exists an indecomposable k(g G )-module M, which induces a Morita equivalence between them and the dimension of its source modules is prime to p. In this c Science China Press and Springer-Verlag Berlin Heidelberg 203 math.scichina.com link.springer.com

484 Hu X Q Sci China Math March 204 Vol. 57 No. 3 situation, by [4, Corollary 7.4], we can identity a vertex P of M with defect groups of b and b. Choosing suitable defect pointed groups P γ of G {b} and P γ of G {b }, Puig and Zhou [5] got a bijection and for any Q δ LP(P γ ), they get an equation F : LP(P γ ) LP(P γ ) F kg (Q δ ) = F kg (Q δ ). Here, Q δ is the image of Q δ under F and the definitions of F kg (Q δ ) and F kg (Q δ ) can be referred to the next section. By [3, Theorem 3.], F kg (Q δ ) = E G (Q δ ) = N G (Q δ )/C G (Q) and F kg (Q δ ) = E G (Q δ ) = N G (Q δ )/C G (Q), so from the bijection F, we can obtain an isomorphism N G (Q δ )/C G (Q) = N G (Q δ )/C G (Q). (.) We denote by K the inverse image of the diagonal subgroup of (N G (Q δ )/C G (Q)) (N G (Q δ )/C G (Q)) through the canonical group homomorphism N G (Q δ ) N G (Q δ ) (N G (Q δ )/C G (Q)) (N G (Q δ )/C G (Q)). For any subgroup H satisfying C G (Q) H N G (Q δ ), let H be the subgroup of N G (Q δ ) such that H /C G (Q) is the image of H/C G (Q) under the isomorphism (.). In [5], it has been proved that khb δ and kh b δ are basically Morita equivalent. Then we are ready to state our main theorem in this paper. Theorem.. Keep the hypothesis and the notations as above. For any Q δ LP(P γ ), there exist a k(n G (Q δ ) N G (Q δ ))-module U which induces a basic Morita equivalence between kn G (Q δ )b δ and kn G (Q δ )b δ and a k(c G (Q)) C G (Q))-module W, which induces a basic Morita equivalence between kc G (Q)b δ and kc G (Q)b δ such that W can be extended to a kk-module V and U is the induced module from V. As an application of this theorem, we can also get a similar result for khb δ and kh b δ. Corollary.2. Notations as above. There exists a k(h H )-module U H, which induces a basic Morita equivalence between khb δ and kh b δ such that U H is the induced module from the restricted module Res K K (H H )(V). 2 Notation and terminology In this section, we introduce some necessary notations which will be used in the next section. Through this section, G is a finite group. Let A be a k-algebra. A is called a G-graded algebra provided there exists a set {A g g G} of k-vector subspaces of A indexed by G such that the following two conditions hold: A = g GA g, as k-vector spaces, (2.) and We refer to A g as the g-component of A. When (2.2) is replaced by the stronger condition A is called a strongly G-graded algebra. A x A y A xy, for all x,y G. (2.2) A x A y = A xy, for all x,y G,

Hu X Q Sci China Math March 204 Vol. 57 No. 3 485 Fixing a G-graded algebra A, a left A-module M is called a graded A-module provided there exists a set {M g g G} of A -submodules of M indexed by G such that the following two conditions hold: M = g GM g, direct sum of A -modules (2.3) and We refer to M g as the g-component of M. When (2.4) is replaced by the stronger condition A x M y M xy, for all x,y G. (2.4) A x M y = M xy, for all x,y G, M is called a strongly graded A-module. Let A be an interior G-algebra and H α be a pointed group on A. Fix i α; an automorphism ϕ of H is an A-fusion of H α if there exists a A such that i a = i and (ai) y = ϕ(y) ai for any y H, where A denotes the multiplicative group of all its invertible elements. We denote by F A (H δ ) the set of all the A-fusions of H α, which becomes a group with the composition of maps. When A is the group algebra kg and P γ is a local pointed group on A, F A (P γ ) = E G (P γ ) (see [3, Theorem 3.]). Let Q δ be a local pointed group on A. For any ϕ F A (Q δ ), the interior G-algebra structure induces an action of ϕ (Q) on A such that A becomes a k ϕ (Q)-module, where ϕ (Q) = {(ϕ(u),u) u Q}. Denote all the ϕ (Q)-fixed elements of A by N ϕ A (Q), i.e., N ϕ A (Q) = {a A u Q, ϕ(u) a u = a}. In [5], it defines a distributive product on the external direct sum N A (Q δ ) = N ϕ A (Q) ϕ F A(Q δ ) such that it becomes an interior N G (Q δ )-algebra. Setting /( N A (Q δ ) = N A (Q δ ) ϕ F A(Q δ ) Ker(Br A ϕ(q)) ) = ϕ F A(Q δ ) A( ϕ (Q)), where Br A ϕ(q) is the usual Brauer homomorphism associated with ϕ(q) and the k ϕ (Q)-module A, A( ϕ (Q)) is the quotient of this homomorphism, it is called extended Brauer quotient (see [5]). And the interior N G (Q δ )-algebra structure of N A (Q δ ) induces an interior N G (Q δ )-algebra structure of NA (Q δ ). 3 Proof of Theorem. First we will give the proof of the main theorem. Here, we still use notations and some results in [5]. Readers can refer to it for details. We set and N = N G (Q δ ), C = C G (Q) N = N G (Q δ ), C = C G (Q). For any s N and N/C-graded algebra B and graded B-module E, we will simply denote the sccomponent of B and E by B s and E s throughout this section. Let Q δ be a local pointed group on kg contained in P γ. Then there is a unique pointed group N α such that Q δ N α (see [6, Proposition 37.7]). Let T ν be its defect pointed group. From [5, Section 4],

486 Hu X Q Sci China Math March 204 Vol. 57 No. 3 we only need to prove the theorem under the situation T ν P γ. Here, we assume that T ν P γ and in this situation T = N P (Q δ ). Similarly, for kg there is a unique pointed group N α such that Q δ N α. Set T ν = F(T ν ) and Q δ = F(Q δ ). Then we have Q δ T ν P γ and T ν is a defect pointed group of N α. We choose j ν and set A = Br Q (j)knbr Q (j). By [5, Proposition 3.3 and Corollary 3.7], it is proved that A is the source algebra of knb δ. Similarly, we can choose j ν and set A = Br Q (j )kn Br Q (j ). Then A is the source algebra of kn δ. On the other hand, we denote by ˆν the point of T on kn containing Br Q (j). Then ˆν can be viewed as a local point of T on kc and C T (Q) is a defect group of b δ viewed as a block idempotent of kc. Then there is a defect pointed group C T (Q) ε of C {bδ } such that C T (Q) ε Tˆν. Moreover, we have that kcbr Q (j)c = kcb δ from the fact that kcεc = kcb δ, where kcεc is the ideal of kcb δ generated by ε. Similarly, we can also get kc Br Q (j )C = kc b δ. From this observation, it is easy to see that the k-algebra A is a strongly N/C-graded algebra, and for any s N, its sc-component is in particular, its -component is A s = kbr Q (j)csbr Q (j), A = kbr Q (j)cbr Q (j). For the k-algebra A, we have the similar result, and denote by A the corresponding -component. Assuming that k(g G )-module M induces the basic Morita equivalence between kgb and kg b, we can view the p-subgroup P as a vertex of M; let N be a kp-module source of M. Then by [6, Corollary 28.7], there exists a kn P (Q δ )-module N Q such that (End k (N))(Q) = End k (N Q ), where (End k (N))(Q) is the Brauer quotient at Q over the interior P-algebra End k (N). And in [5, Section 4], it gets an interior T-algebra embedding f : A End k (N Q ) k A. Precisely speaking, the embedding f is obtained from the embedding and the isomorphisms For any N jkgj (Q δ ) End k (N Q ) k Nj kg j (Q δ ) (3.) µ : NjkGj (Q δ ) = A, µ : Nj kg j (Q δ ) = A. ψ F jkgj (Q δ )=F kg (Q δ ) = F kg (Q δ ), thereexists aunique x ψ C N/C suchthat for anyu Q, we haveψ(u)=x ψ ux ψ (see[3, Proposition2.4 and Theorem 3.]). Then by [5, Corollary3.7], µ( N ψ jkgj (Q δ)) = A x. It is similar for µ and N ψ j kg j (Q δ ). At the same time, the embedding (3.) maps N ψ jkgj (Q δ) to End k (N Q ) k Nψ j kg j (Q δ ). So we have the embedding f preserves the graded structures. In particular, the embedding f induces an interior C T (Q)-algebra embedding A End k (N Q ) k A, for convenience, we still denote it by f. From the fact kcbr Q (j)c = kcb δ and kc Br Q (j )C = kc b δ, by [6, Theorem 9.9] we can get that the module kcbr Q (j) induces the Morita equivalence between kcb δ and A, and the module Br Q (j )kc

Hu X Q Sci China Math March 204 Vol. 57 No. 3 487 induces the Morita equivalence between A and kc b δ. From the two embeddings above, we can get a k(n N )-module knbr Q (j) A (f(br Q (j)) (N Q k A )) A Br Q (j )kn, which induces a Morita equivalence between knb δ and kn b δ and a k(c C )-module kcbr Q (j) A (f(br Q (j)) (N Q k A )) A Br Q (j )kc, which induces a Morita equivalence between kcb δ and kc b δ. We set X = knbr Q (j), Y = f(br Q (j)) (N Q k A ), X = Br Q (j )kn. It is easily to see that X is a graded module both as a left kn-module and a right A-module. It is similar for X. For (A,A )-bimodule Y, we have Y = f(br Q (j)) (N Q k A x ), x C N /C for any x N,(y,y ) K, f(br Q (j)) (N Q k A x ) is an (A,A )-bimodule and A y (f(br Q (j)) (N Q k A x )) f(br Q(j)) (N Q k A y x ), (f(br Q (j)) (N Q k A x )) A y f(br Q(j)) (N Q k A x y ), so Y is a graded module both as a left A-module and a right A -module. Since the k-algebras A and A are strongly graded algebras, the modules X,X and Y are also strongly graded modules (see [, Theorem 2.8]). And for any x N,(x,x ) K, f(br Q (j)) (N Q k A x ) is either xc-component as a graded left A-module or x C -component as a graded right A -module. In particular, we have the corresponding -components as follows: X = kcbr Q (j), Y = f(br Q (j)) (N Q k A ), X = Br Q(j )kc. Proposition 3.. Notations as above. Then we have (i) the k(n N )-module X A Y A X induces a basic Morita equivalence between knb δ and kn b δ ; (ii) the k(c C )-module X A Y A X induces a basic Morita equivalence between kcb δ and kc b δ. Proof. (i) By [7, Lemma 2] and [6, Lemma 44.6], we have X A Y A X Ind N N T T (Y), and as k(t T)-modules, Y N Q k A. On the other hand, as a k(t T)-module any one of the direct summands of A is isomorphic to ktg T for some g N (see [6, Lemma 44.]). So there exists n N such that X A Y A X is relatively T n -projective where T n = {(u,n un ) u T n T}. At the same time, it is well known that there is a surjective homomorphism between the vertex of X A Y A X and the defect of b δ viewed as a block idempotent of kn (see [4, Theorem 6.9]). So we have that T = T n and T n is a vertex of X A Y A X. Then by [4, Corollary 7.4] X A Y A X induces a basic Morita equivalence between knb δ and kn b δ. (ii) We notice that in the proof of [7, Lemma 2], it is only used the fact that b δ Tr N T (NT Br Q (j) N T ),

488 Hu X Q Sci China Math March 204 Vol. 57 No. 3 and But it is easily checked that b δ Tr N T (N T BrQ (j ) N T ). b δ Tr C C T(Q)(C CT(Q) Br Q (j) C CT(Q) ), b δ Tr C C T(Q)(C C T(Q) BrQ (j ) C C T(Q) ), and the k(c T (Q) C T (Q))-module structure of A is similar with the k(t T)-module structure of A. Similarly we can prove (ii). Next, we will analysethe relationshipbetween the k(n N )-module X A Y A X and the k(c C )- module X A Y A X. We have the following result. Proposition 3.2. The k(c C )-module X A Y A X can be extended to a kk-module V, and the k(n N )-module X A Y A X is the induced module from V. Proof. Since A is a strongly N/C-graded algebra, by [, Theorem 2.8], we have as a (kc, A)-bimodule, X = X A A and as an (A,A )-bimodule, Y = A A Y. It is similar for X. Then Res N N C C (X A Y A X ) = Res N N C C (X A Y A (A A X )) = Res N N C C (X A Y A X ) = Res N N C C (X A (A A Y ) A X ) = Res N N C C (X A Y A X ) = (gx A Y A X ). (3.2) gc N/C Denote by ϕ the isomorphism between Res N N C C (X A Y A X ) and gc N/C (gx A Y A X ). For any g N,(s,s ),(t,t ) K,x g X g,y s Y s,x t X t, since X,Y and X are strongly graded modules, there exist x X,y Y,x X and a t A t,a st A st such that then x t = a t x, y s a t = a sty, x g a st = gstx, ϕ(x g A y s A x t ) = gstx A y A y. In particular, for any (h,h ) K,r X,y h Y h,r X, Denote ϕ(r A y h A r ) hx A Y A X. (3.3) {r A y g A r r X,y g Y g,r X } by Z g where (g,g ) K. From this observation, as vector spaces, we have X A Y A X = Z g. g C N /C Next, we will show that the subspace Z is invariant under the action of K and is the required kkmodule V. Since f(a g ) End(N Q ) k A g, for any (g,g ) K,x A y A x Z, (g,g ) (x A y A x ) = gx A y A x g

Hu X Q Sci China Math March 204 Vol. 57 No. 3 489 and there exist l X,l X,a g A g,a g A g such that gx = l a g,x g = a g l, so (g,g ) (x A y A x ) = gx A y A x g = l A f(a g )y a g A l Z. Hence, Z is invariant under the action of K, which means that Z is a kk-module. So we have Then X A Y A X = Ind N N K (Z ). Res N N C C (X A Y A X ) = Res N N C C (IndN N K (Z )). (3.4) From (3.2) (3.4), ϕ : Res K C C (Z ) = X A Y A X. This completes the proof. Proof of Theorem.. Set U = X A Y A X, W = X A Y A X and V = Z. By the two propositions above, these three modules are satisfied the conditions of the theorem. So the theorem is completed. Next, we will get the proof of Corollary.2 by Theorem.. We will still use the notations as above. Recall that for any subgroup H satisfying C H N, we denote by H the subgroup of N such that H /C is the image of H/C under the isomorphism (.). For any gc N/C, denote its image under the isomorphism (.) by g C. By Theorem. above, it s easily seen that the k(c C )-module W can be extended to a k((h H ) K)-module. So by [2, Theorem 3.4], the k(h H )-module h C H /C Z h induces a Morita equivalence between khb δ and kh b δ. Furthermore, we can get the following result. Lemma 3.3. Notations as above. The k(h H )-module h C H /C Z h induces a basic Morita equivalence between khb δ and kh b δ. Proof. Let R be the vertex of h C H /C Z h. Denote respectively by ρ : R H and ρ : R H the restriction of the first and the second projection maps, by R and R the images of ρ and ρ, and by σ : R R and σ : R R the corresponding group homomorphisms. Then R is a defect group of b δ viewed as the block idempotent of kh. By Theorem., we have Then, there exists (g,g ) K such that Z h = Ind H H K (H H ) (ResK K (H H ) (V)). h C H /C R (H H ) ( (g,g ) (T)). For any (,s ) Kerσ, we can get that (,s ) (g,g ) (T), i.e., (,g s g ) (T). So the map σ is injective. By [4, Corollary 7.4], this completes the proof. Proof of Corollary.2. Set U H := h C H /C Z h. Then by the lemma above, the k(h H )-module U H is satisfied the condition of the corollary. We can get the proof of the corollary. Acknowledgements The author is very grateful to the anonymous referees for their careful reading of the paper, and for suggestions which helped the author to improve it.

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