Invent. math. 141, 365–397 (2000) Digital Object Identifier (DOI) 10.1007/s002220000072

The hyperfocal subalgebra of a block Lluis Puig CNRS, Universit´e de Paris 7, 6 Av Bizet, 94340 Joinville-le-Pont, France Oblatum 2-VII-1999 & 17-XII-1999 Published online: 29 March 2000 –  Springer-Verlag 2000

1. Introduction 1.1. It is known from Grün’s First Theorem (cf. Th. 4.2, Chap. 7 in [6]) that, in a finite group G, the intersection T ∩ [G, G] – usually called the focal subgroup – of a Sylow p-subgroup T with the so-called derived subgroup of G is determined by the G-conjugation of elements in T – namely it is generated by the elements [x, u] when u runs on T and x on the set of elements of G fulfilling ux ∈ T . More precisely, it follows from Alperin’s Fusion Theorem that this intersection is generated by the subgroups [NG (Q), Q] when Q runs on the set of subgroups of T (cf. [1]); as a matter of fact, the same theorem proves that, denoting by L(G) the last term of the lower central series of G, the intersection T ∩ L(G) is generated by the subgroups [x, Q] when Q runs on the set of subgroups of T , and x on the set of p -elements of NG (Q). In other terms, denoting by S the subgroup of T generated by these commutators – let us call it the hyperfocal subgroup of T – it can be proved that: there is a unique normal subgroup H of G such that 1.1.1

H∩T = S

and

G = H.T

.

1.2. It is obvious that this last statement implies Frobenius Criterion on the existence of a normal p-complement in G and, since our joint papers with Michel Brou´e (cf. [2], [3]), both have understood the existence of a focal subgroup – actually, of a focal Brauer pair – in any block b of G, namely, denoting by (P, e) a maximal (b, G)-Brauer pair (cf. §1 in [2]), the subgroup U of P generated by the elements [x, u] when (u, f ) runs on the set of (b, G)-Brauer elements such that (u, f ) ∈ (P, e), and x ∈ G on the set of elements such that (u, f )x ∈ (P, e) (cf. Def. 2.1 in [2]). It was already clear from our main result in [2] that the group of linear characters of P/U acts on the set of ordinary irreducible characters of G in b, and even acts freely on the subset of those of height zero (cf. Th. 1.5 in [3]); yet, it was not clear how to push it further.

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1.3. Certainly, there was no obstacle in considering the hyperfocal subgroup V of P generated by the subgroups [x, Q] when (Q, f ) runs on the set of (b, G)-Brauer pairs such that (Q, f ) ⊂ (P, e) and x runs on the set of p -elements of NG (Q, f ), and again the analogous condition to Frobenius Criterion – namely the condition of b being nilpotent (cf. Def. 1.1 in [3]) – implies V = {1}; but the kind of inductive argument reasonably employed to obtain 1.1.1 from Grün’s First Theorem seemed hopeless for blocks, and this remark had been of no help when determining in [12] the structure of the source algebra B = i(OG)i of a nilpotent block (recall that i is a primitive idempotent of (OG b)P such that BrP (i) = 0). However, the main idea in [12] – the existence of a section B → B ⊗O O P of the augmentation map – has been successfully extended by Fan Yun to any block, for some non-necessarily abelian quotients of P (cf. [4]). 1.4. Recently, a conjectural remark on Raphaël Rouquier’s lifting of splendid Rickard equivalences to central extensions, in the blocks with abelian defect groups (cf. [15]) – we seek that his result mainly come from the fact that when such a block is a central extension, both focal subgroups coincide – rose the question again and, this time, we have understood the significance of the hyperfocal subgroup V of P: there is a P-stable subalgebra D of the source algebra B, unique up to (BP )∗ -conjugation, such that 1.4.1

V ·i = D ∩ P·i

and

B = ⊕u D·u

where u runs on a set of representatives for P/V in P. 1.5. Let us restate it with more detail, fixing first our notation. Let p be a prime number, k an algebraically closed field of characteristic p, O a complete discrete valuation ring of characteristic zero – indeed, some uniqueness results are only proved in characteristic zero – with residue field k, G a finite group, b a block of G over O – namely b is a primitive idempotent of Z(OG) –, P a defect group of b and γ a local point
∗ of P on OG b – that is to say, up to NG (P)-conjugation, γ is the (OG) P -conjugacy class of the idempotent i above and, setting α = {b}, Pγ is a defect pointed group of Gα on OG (cf. §37 in [16]); we set B = (OG)γ , considered as an O P-interior algebra. Moreover, denoting by e the block of CG (P) determined by γ (cf. §41 in [16]), (P, e) is a maximal (b, G)-Brauer pair. 1.6. Recall that a self-centralizing pointed group Qδ on OG is a local pointed group on OG such that Z(Q) is a defect group of the block b(δ) of C G (Q) determined by δ – which, in particular, is nilpotent – or, equivalently, such that (OG)δ (Q) ∼ = kZ(Q) (cf. Cor 2.13 in [13]); an essential pointed group Rε on OG is a self-centralizing pointed group on OG such that the quotient N¯ = NG (Rε )/R·C G (R) contains a proper subgroup M¯ fulfilling the following condition (cf. Theorem A.9 in [13] or §48 in [16]). ¯ ¯ and does not divide |M¯ ∩ M¯ x¯ | for any x¯ ∈ N¯ − M, 1.6.1 p divides |M| and then, it is well-known that (cf. Proposition 5, Chap. II in [8], or A.10.2 in [13])

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1.6.2 the set of normal subgroups of N¯ which are not p -groups has a unique minimal element XG (Rε ). 1.7. Similarly to the so-called local control (cf. §3 in [12], or §49 in [16]), the hyperfocal subgroup of Pγ admits several equivalent definitions, namely it is generated by each one of the following families of subgroups:
 1.7.1 [O p NG (Pγ ) , P] and [O p (X), R]u where Rε runs on a set of representatives such that Rε ⊂ Pγ for the G-conjugacy classes of essential pointed groups on OG b, X is the converse image of XG (Rε ) in NG (Rε ), and u runs on P. 1.7.2 [x, Q] where Qδ runs on the set of local pointed groups on OG such that Q δ ⊂ Pγ , and x on the set of p -elements of NG (Q δ ). 1.7.3 [x, Q] where (Q, f ) runs on the set of (b, G)-Brauer pairs such that (Q, f ) ⊂ (P, e), and x on the set of p -elements of NG (Q, f ). In Lemma 7.9 below, we prove that these families generate indeed the same subgroup of P, but in the proof of our main result we only use the family 1.7.2. Actually, we are able to define the hyperfocal pointed subgroup of Pγ – which is quite satisfactory – since the hyperfocal subgroup of Pγ has a unique local point on B (cf. Prop. 4.2 below). Theorem 1.8. Let Q be a normal subgroup of P containing the hyperfocal subgroup of Pγ , and denote by U a set of representatives for P/Q in P. Then, there is a P-stable O-subalgebra D of B, unique up to (BP )∗ -conjugation, containing the image of Q and fulfilling 1.8.1

B = ⊕u∈U D·u

.

1.9. In particular, if we assume that the block b is nilpotent, we can apply this theorem choosing Q = {1} to get easily the main result of [12] (cf. Corollary 7.7); we emphasize that this application really provides a new proof, since the proof of Theorem 1.8 need not quote that result. As in [12], the main tool to prove this theorem is the existence of sections for suitable augmentation maps in an inductive setting (see Lemma 7.3 below), which overpasses the situation discussed in [4]; a posteriori, this existence and the main results in [4] – except that Fan Yun makes no hypothesis on k – are consequences of the theorem above (see 7.10 below). 1.10. Let us make a comment on some aspects of the proof. We argue by induction on |P/Q| and, assuming that Q = P, we consider a normal subgroup R of P such that Q ⊂ R and |R/Q| = p; then, we may assume that there is a P-stable O-subalgebra E of B containing the image of R and fulfilling B = ⊕v∈V E·v, where V is a set of representatives for P/R in P; but, to go on, we need a more precise uniqueness of E than which is obtained from the inductive hypothesis (cf. Th. 6.3 below), and some knowledge on the so-called local category of E (cf. Props. 4.2 and 4.3 below).

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1.11. Consequently, we prove first a stronger version of the uniqueness part of our theorem, and moreover we determine the local category of a P-stable O-subalgebra D as described in our theorem, which a fortiori has its own interest. Actually, D admits compatible structures of P-algebra and of O Q-interior algebra; that is to say, D becomes an O Q-interior P-algebra as it has been introduced in [14], and equality 1.8.1 becomes an O P-interior algebra isomorphism (cf. 2.1) B∼ = D ⊗Q P

1.11.1

;

thus, in order to speak about a local category of D, we have to consider the so-called fusions between pointed groups, whether or no the corresponding groups are contained in Q. Moreover, let δ be a local point of Q on OG such that Q δ ⊂ Pγ ; although Qδ need not be self-centralizing, we will prove that the block b(δ) of CG (Q) is nilpotent: we call nil-centralized such a local pointed group on B and will prove that it determines a unique local pointed group on D. 1.12. This paper is divided in seven sections. In Sect. 2, we introduce the fusions between the pointed groups on an O N-interior G-algebra, for a normal subgroup N of G, which allow us to consider the corresponding local category. In Sect. 3, we consider some features of the nil-centralized pointed groups on OG, which we apply in the description of the local category of D in Sect. 4: this description goes further than what we need in order to prove the theorem above, and the interested reader will observe that the main result [12] is not necessary when dealing with self-centralizing pointed groups. We prove the more precise version of the uniqueness of D in Sect. 6 and its existence in Sect. 7; in both proofs, we need some facts on p-adic analysis that, although certainly known, we recall in Sect. 5. We follow the general notation and terminology of [16] and [14, §2]; in particular, all the O-algebras and O-modules are considered to be O-free of finite O-rank. 2. Fusions in O N-interior G-algebras 2.1. In this section we need not assume that O has characteristic zero. Let G be a finite group and N a normal subgroup of G; following [14], an O N-interior G-algebra A is an O-algebra simultaneously endowed with a G-algebra and an O N-interior algebra structures such that, for any x ∈ G, any y ∈ N and any a ∈ B, we have 2.1.1

(y·a)x = y x ·ax

,

(a·y)x = ax ·y x

and

y−1 ·a·y = a y

.

Notice that an OG-interior algebra has an evident O N-interior G-algebra structure; more generally, if N is a normal subgroup of G contained in N, A has an O N  -interior G-algebra structure that we denote by ResNN ,G ,G (A)

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or, simply, by ResNN  (A) . Following [5], we denote by A ⊗N G the corresponding crossed product considered as an OG-interior algebra, namely the O-module A ⊗ON OG endowed with the distributive product defined by 2.1.2

(a ⊗ x)(a ⊗ x  ) = a(a )x

−1

⊗ xx 

for any a , a ∈ A and any x , x ∈ G, and with the evident group homomorphism 2.1.3

G −→ (A ⊗N G)∗

,

and the main purpose of this section is to relate A and A ⊗N G from the pointed group structure point of view whenever G/N is a p-group. 2.2. Let Hβ and Kγ be pointed groups on A; analogously to Definition 2.5 in [10] (but we have not found a similar formulation), we say that the group exomorphism determined by an injective group homomorphism ϕ : K → H fulfilling ϕ(y)N = yN, for any y ∈ K, is an A-fusion from Kγ to Hβ if, for some i ∈ β and some j ∈ γ , there is a ∈ A∗ such that jA j ⊂ (i Ai)a and, for any y ∈ K, we have
 2.2.1 (a j)y = y−1 ϕ(y) ·a j or, equivalently, (a j ⊗ 1)·y = ϕ(y)·(a j ⊗ 1) in A ⊗N G, which agrees with Proposition 2.12 in [10] (except that j ⊗ 1 need not be primitive in (A ⊗ N G) K !); in particular, if N contains H and K then ϕ˜ is an ordinary A-fusion. As in this case, this condition does not depend on the choices of i and j, and we may replace ϕ by any N ∩ H-conjugate, but we do not claim that it is fulfilled by any representative ϕ of ϕ˜ such that ϕ (y)N = yN for any y ∈ K : we call a representative fulfilling the above condition A-representative of ϕ. ˜ 2.3. We denote by FA (K γ , Hβ ) the set of A-fusions from Kγ to Hβ and write FA (K γ ) instead of FA (K γ , K γ ); recall that, for any subgroup L of G, E L (K γ , Hβ ) is the set of group exomorphisms induced by the elements x ∈ L such that Kγ ⊂ (Hβ )x , and, as in the usual case (cf. 2.10 in [10]), 2.3.1

E N (K γ , Hβ ) ⊂ FA (K γ , Hβ ) ;

we will denote by κxK : K ∼ = K x the isomorphism mapping y ∈ K on yx .   H) Moreover, if N is a normal subgroup of G contained in N, ϕ˜ ∈ Hom(K, is a ResNN  (A)-fusion from Kγ to Hβ if and only if it is an A-fusion and an A-representative ϕ of it fulfills ϕ(y)N = yN  for any y ∈ K. On the other hand, if A is an O N-interior G-algebra and f : A → A is an O N-interior G-algebra embedding, it is quite clear that, as in the ordinary case (cf. Prop. 2.14 in [10]), we have 2.3.2

FA (K γ , Hβ ) = FA (K γ , Hβ ) ,

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L. Puig

where, as usual, we identify β and γ with the corresponding points on A (cf. §15 in [16]). 2.4. If L δ is another pointed group on A and ψ˜ ∈ FA (L δ , K γ ), we claim that ϕ˜ ◦ ψ˜ is an A-fusion from Lδ to Hβ ; indeed, in that case L ⊂ K·N ⊂ ˜  a representative of δ, and c H·N; let ψ be an A-representative of ψ, an invertible element of A fulfilling  A ⊂ ( jA j)c and the corresponding ac equalities
 2.2.1; hence, we have  A ⊂ (i Ai) and, for any y ∈ L, we get ϕ ψ(y) N = ψ(y)N = yN and 2.4.1

(ac ⊗ 1)·y = (a j ⊗ 1)(c ⊗ 1)·y = (a j ⊗ 1)·ψ(y)·(c ⊗ 1)

 . = ϕ ψ(y) ·(a j ⊗ 1)(c ⊗ 1) = ϕ ψ(y) ·(ac ⊗ 1)

 In particular, FA (K γ ) is a subgroup of A ut(K) and, as in the ordinary case (cf. 6.7 in [11]), equalities 2.2.1 induce a group homomorphism
 2.4.2 FA (K γ ) −→ N( jA j)∗ ⊗ K j ⊗ K ( jA K j)∗ ⊗ K which, for any ϕ˜ ∈ FA (K γ ), any A-representative ϕ, and any a ∈ A∗ fulfilling j a = j and the corresponding equalities 2.2.1, maps ϕ˜ on on the class of a j ⊗ 1; when Kγ is local, this homomorphism allows us to consider the corresponding k∗ -group 

2.4.3 Fˆ A (K γ ) −→ N( jA j)∗ ⊗K j ⊗ K j + J( jA K j) ⊗ K . Moreover notice that, for any x ∈ G, from equalities 2.2.1 we immediately get

−1 2.4.4 FA (K γ )x , (Hβ )x = κ˜ xH ◦ FA (K γ , Hβ ) ◦ κ˜ xK . As in [10], we call local category of A the category formed by the local pointed groups on A and the A-fusions between them. 2.5. As in the ordinary case (cf. 2.11 in [10]), any A-fusion can be obtained as a composition of a bijective one and an A-fusion determined by an inclusion −1 of pointed groups; indeed, with the notation above, ja determines a point γ  of K  = ϕ(K) on A such that Kγ  ⊂ Hβ , since, for any y ∈ K and any c ∈ A, we have
ϕ(y) −1 ( jc j)a ⊗ 1 = ϕ(y)−1 ·(a j ⊗ 1)(c ⊗ 1)( ja−1 ⊗ 1)·ϕ(y) 2.5.1

= (a j ⊗ 1)·y−1 ·(c ⊗ 1)·y·( ja−1 ⊗ 1) a−1

= ( jc y j) −1

,

⊗1 −1



−1

−1

so that K  centralizes ja and we get j a A K j a = ( jA K j)a . Now assume that |K| = |H|; thus we have ϕ(K) = H and it is easily checked

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that ϕ˜ −1 is an A-fusion from Hβ to K γ ; more precisely, setting G¯ = G/N, consider the followong subgroup of H × K 2.5.2

¯

H ×G K = {(x, y) | x ∈ H , y ∈ K and xN = yN }

and the O(H×G¯ K)-module structure of i A j defined by (x, y)·c = xy−1 ·c y for any (x, y) ∈ H ×G¯ K and any c ∈ i A j; then, any group isomorphism ψ: K ∼ = H fulfilling ψ(y)N = yN, for any y ∈ K, determines the following subgroup of H ×G¯ K −1

2.5.3

∆ψ (K) = {(ψ(y), y)} y∈K

and the A-fusions from Kγ to Hβ are described by the following statement. Lemma 2.6. With the notation and the hypothesis above, a group isomorphism ϕ : K ∼ = H fulfilling ϕ(y)N = yN, for any y ∈ K, is an A-representative of an A-fusion from Kγ to Hβ if and only if we have 2.6.1

( jAi)∆ϕ−1 (H) ·(i A j)∆ϕ (K) = jA K j

.

Proof: In any case, it is easily checked that the left member is contained in the right one and that, actually, it is a two-sided ideal there; moreover, if ϕ˜ ∈ FA (K γ , Hβ ), ϕ is an A-representative and a is an invertible element of A fulfilling ia = j and equalities 2.2.1, then a j and ja−1 respectively belong to (i A j)∆ϕ (K) and to ( jAi)∆ϕ−1 (H) , which proves equality 2.6.1. Conversely, since jAK j/J( jA K j) ∼ = k, equality 2.6.1 forces the existence of ∆ϕ−1 (H) c ∈ ( jAi) and d ∈ (i A j)∆ϕ (K) such that cd belongs to ( jAK j)∗ ; actually, up to a new choice of d, we may assume that cd = j and then, since dcdc = d jc = dc and j = cdcd, dc is a nonzero idempotent of i AH i, so that dc = i. In particular, this proves that i and j are conjugate in A and, choosing b ∈ A∗ such that ib = j, we claim that the element a = d + (1 − i) b (1 − j) is inversible in A and fulfills equalities 2.2.1; indeed, we have 

a c + (1 − j) b−1 (1 − j) = d + (1 − i) b (1 − j)
 2.6.2 × c + (1 − j) b−1 (1 − i) = dc + (1 − i) b (1 − j) b−1 (1 − i) = 1 and equalities 2.2.1 follow from the fact that ∆ϕ (K) fixes a j = d. 2.7. Let A be an O N-interior G-algebra and f : A → A an O N-interior G-algebra homomorphism; considering only the G-algebra structures, recall that f is a
semicovering if f is unitary and, for any p-subgroup P of G,    we have f J(A P ) ⊂ J(A P ) and f(i) is primitive in A P for any primitive idempotent i of AP (cf. §3 in [7] and §2 in [13]); in this case, we still say that f is a strict semicovering if moreover Ker( f ) contains no nonzero

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idempotents or, equivalently, Ker( f ) ⊂ J(A). In particular, if Pγ and Q δ are local pointed groups on A such that Q ⊂ P·N and f is a semicovering, then f(γ) and f(δ) are respectively contained in local points γ and δ of P and Q on A (cf. Prop. 3.15 in [7]), and we clearly have FA (Q δ , Pγ ) ⊂ FA (Q δ , Pγ  ) .

2.7.1

Next proposition is the main result of this section; as Fan Yun pointed out to us, some of the arguments on A ⊗N G-fusions need not assume that G/N is a p-group, which allows the possibility of seeking a unified proof with the main result of [10]; but we have not found an adequate formulation. Proposition 2.8. If G/N is a p-group, the canonical map A −→ A ⊗N G= Aˆ is a strict semicovering G-algebra homomorphism and moreover, for any local pointed groups Pγ and Q δ on A, denoting by Pγˆ and Q δˆ the corresˆ an A-fusion ˆ ponding local pointed groups on A, ϕ˜ from Q δˆ to Pγˆ admits Q ˜ ◦ κ˜ x for suitable x ∈ G such that Qx ⊂ N· P a decomposition ϕ = ψ ˜
 and ψ˜ ∈ FA (Q δ )x , Pγ ; in particular, FA (Pγ ) is normal in FAˆ (Pγˆ ) and the quotient FAˆ (Pγˆ )/FA (Pγ ) is a p-group. Proof: If R is a p-subgroup of G, it is quite clear that  ˆ 2.8.1 A(R) = (A ⊗ t)(R) t∈T R

where T R is a set of representatives for (G/N)R in G; since we are assuming that (G/N) R is a p-group and, for any t , t ∈ T R , we clearly have 2.8.2

(A ⊗ t)(R)·(A ⊗ t  )(R) ⊂ (A ⊗ tt  )(R)

,

the first statement follows from Proposition 2.2 in [13] and Theorem 3.16 in [7]. ˆ Since any A-fusion can be obtained as a composition of a bijective one ˆ and an A-fusion determined by an inclusion of pointed groups, we may asˆ sume without loss that ϕ˜ is an exoisomophism; choosing an A-representative ϕ (i.e. any representative) of ϕ, ˜ i ∈ γ and j ∈ δ and setting ıˆ = i ⊗ 1 and ˆ = j ⊗ 1, it follows from Lemma 2.6 that 2.8.3

ˆ ı )∆ϕ−1 ( P) ·(ˆı Aˆ ) (ˆ Aˆ ˆ ∆ϕ (Q) = ˆ Aˆ Q ˆ ;

but, since Qδˆ is local, this equality implies that the k-linear map

 ˆ ı ) ∆ϕ−1 (P) ⊗k (ˆı Aˆ ) 2.8.4 (ˆ Aˆ ˆ ∆ϕ (Q) −→ (ˆ Aˆ )(Q) ˆ induced by the multiplication in Aˆ is surjective; moreover, it is clear that   ˆı = 2.8.5 ıˆ Aˆ ˆ = i A j t ⊗ t −1 and ˆ Aˆ jAi t ⊗ t −1 t∈T

t∈T

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where T is a set of representatives for G/N in G, and therefore, denoting by U the (possibly empty) set of t ∈ T such that y−1 tϕ(y) ∈ tN for any y ∈ Q (recall that the existence of ϕ˜ supposes that Q·N = P·N), we get
 
 (ˆı Aˆ ) (i A j u ⊗ u −1 ) ∆ϕ (Q) ˆ ∆ϕ (Q) = 2.8.6

u∈U

 
 ;
−1 ˆ ı ) ∆ϕ−1 (P) = ( jAi u ⊗ u) ∆ϕ−1 (P) (ˆ Aˆ u∈U −1

consequently, there are u , v ∈ U, c ∈ i A ju and d ∈ jAi v such that ∆ϕ (Q) fixes c ⊗ u−1 , ∆ϕ−1 (P) fixes d ⊗ v and the product (d ⊗ v)(c ⊗ u−1 ) is invertible in ˆ Aˆ Q ; ˆ thus, up to a new choice, we may assume that v = u −1 and dcu = j. In particular, we get yu N = ϕ(y)N for any y ∈ Q, and we claim that u ψ = ϕ ◦ κuQ−1 determines an A-fusion from (Qδ )u to Pγ ; indeed, since ∆ϕ (Q) fixes c ⊗ u−1 and ∆ϕ−1 (P) fixes d ⊗ u, it is easily checked that c and u d u respectively belong to (i A ju )∆ψ (Q ) and to ( j u Ai)∆ψ −1 ( P) ; then, since u u d c = j , the claim follows from Lemma 2.6. Finally, assume that Qδ = Pγ . For any η˜ ∈ FA (Pγ ), we have u ϕ˜ −1 ◦ η˜ ◦ ϕ˜ = κ˜ uP−1 ◦ ψ˜ −1 ◦ η˜ ◦ ψ˜ ◦ κ˜ uP
 and, since ψ˜ −1 ◦ η˜ ◦ ψ˜ ∈ FA (Pγ )u , it follows from equality 2.4.4 that ϕ˜ −1 ◦ η˜ ◦ ϕ˜ belongs to FA (Pγ ). On the other hand, if ϕ˜ is a p -element of FAˆ (Pγˆ ), from the above decomposition and from equality 2.4.4 again, we get
 ˜ P = ψ˜ ◦ κ˜ uP n = ξ˜ ◦ κ˜ uPn 2.8.8 id
n for some integer n
prime to p and a suitable ξ˜ ∈ FA (Pγ )u , Pγ ), so that κ˜ uPn  n still belongs to FA Pγ , (Pγ )u ; consequently, setting w = un and always
wi i i+1  applying equality 2.4.4, it follows that κ˜ wP belongs to FA (Pγ )w , (Pγ )w for any i ≥ 0; but, since n is prime to p and we are assuming that G/N is a p-group, we have wm = u y ufor some integer m and a suitable y ∈ N; in
conclusion, since κ˜ wPm and κ˜ yP respectively belong to FA Pγ , (Pγ )u y and

 to FA (Pγ )u , (Pγ )u y , κ˜ uP belongs to FA Pγ , (Pγ )u and then ϕ˜ belongs to FA (Pγ ).

2.8.7

2.9. If M is a normal subgroup of G contained in N, the map sending a ⊗M x to a ⊗N x, for any a ∈ A and any x ∈ G, defines an OG-interior algebra homomorphism 2.9.1

f : ResNM (A) ⊗ M G −→ A ⊗ N G

;

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−1 moreover, if [G,
N] ⊂ M then the ∗ set Z = {n ·1 A ⊗ M n | n ∈ N} is N a subgroup of Z Res M (A) ⊗ M G isomorphic to N/M and, considering O Z as an O N-interior algebra, we have evident O N- and OG-interior algebra isomorphisms

2.9.2

O ⊗O Z

Res NM (A) ⊗ M N ∼ = A ⊗O O Z 
Res NM (A) ⊗ M G ∼ = A ⊗N G

;

indeed, it is quite clear that Z centralizes A ⊗M 1 (cf. 2.1.1) and, for any x ∈ G and any n ∈ N, we have (1 ⊗ x)
−1 n ·1 A ⊗ M n A M = (n −1 )x ·1 A ⊗ M n x = (n −1 )x ·1 A ⊗ M [x, n −1 ]n = n −1 ·1 A ⊗ M n

,

whereas, for any m ∈ N, we get
−1 
 n ·1 A ⊗ M n m −1 ·1 A ⊗ M m = n −1 (nm −1 n −1 )·1 A ⊗ M nm = (nm)−1 ·1 A ⊗ M nm

.

Corollary 2.10. With the notation above, assume that G/M is a p-group and that M contains [G, N]. For any local pointed group Pγˆˆ on ResNM (A)⊗ M G, f(γˆˆ ) is contained in a local point γˆ of P on A ⊗N G. Proof: For any local pointed group Pγˆˆ on Aˆˆ = Res NM (A) ⊗ M G, it follows from Proposition 2.8 that we can find a primitive idempotent i in AP such that i ⊗M 1 ∈ γˆˆ and then, according to Proposition 3.15 in [7], we have Br P (i) = 0; thus, according to the same propositions, i ⊗N 1 = f(i ⊗M 1) ˆ still belongs to a local point γˆ of P on A. 3. Nil-centralized pointed groups on OG 3.1. In this section we need not assume that O has characteristic zero. We say that a local pointed group Qδ on OG is nil-centralized if the block b(δ) of CG (Q) determined by δ is nilpotent; in particular, any self-centralizing pointed group on OG is nil-centralized (cf. 1.6). As a matter of fact, many arguments on self-centralizing pointed groups can be extended to the nilcentralized ones, and we are interested on them since, as we announce in the introduction, the hyperfocal pointed subgroup is nil-centralized; however, the interested reader will notice that, in the proof of our main result, it suffices to consider the self-centralizing ones. 3.2. Let Q δ be a local pointed group on O, respectively denote by ν and ζ the points of NG (Q δ ) and Q·CG (Q) on OG such that (cf (37.7) in [16]) 3.2.1

Q δ ⊂ Q·C G (Q)ζ ⊂ NG (Q δ )ν

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375

and consider a local pointed group Rε on OG contained in NG (Q δ )ν and containing a defect pointed group of Q·CG (Q)ζ , so that Qδ ⊂ Rε . If Q δ is nil-centralized, it follows from Proposition 6.5 and Theorem 6.6 in [7] that b(δ) is a nilpotent block of R·CG (Q) too, that we have a kCR (Q)-interior R-algebra isomorphism (OG)ε (Q) ∼ = (OG)ε (Q δ ) ⊗k kC R (Q)
 and that (OG)ε (Q δ ) is a Dade R/Q-algebra such that (OG)ε (Q δ ) (R/Q) ∼ = k. Conversely, since a source algebra of the block b(δ) of CG (Q) is clearly embedded in the kCR (Q)-interior algebra (OG)ε (Q), it follows from the main result in [12] that isomorphism 3.2.2 implies that this block is nilpotent, so that Qδ is nil-centralized. 3.2.2

Proposition 3.3. Let Pγ and Q δ be local pointed groups on OG such that Q δ ⊂ Pγ . If Q δ is nil-centralized, for any subgroup R of P containing Q, there is a unique local point ε of R on OG such that Rε ⊂ Pγ and then Rε is nil-centralized and contains Qδ . In particular, Pγ is nil-centralized. Proof: It is well-known (cf. (40.4) in [16]) that there are a local point ε of R on OG such that Rε ⊂ Pγ and then a local point δ of Q on OG such that Qδ ⊂ Rε ; in particular, Pγ contains Qδ and Q δ which implies b(δ) = b(δ ) (cf. (40.4) in [16]) and therefore, since this block is nilpotent, we get δ = δ (cf. Th. 1.2 in [3]). Now, it suffices to prove that Rε is nil-centralized and, arguing by induction on |P/Q|, we may assume that |R/Q| = p; in that case, R normalizes Qδ (cf. Cor. 1.5 in [9]) and therefore b(δ) is also a nilpotent block of
R·CG (Q) (cf. Prop. 6.5 in [7]), so that their Brauer correspondants in BrR b(δ) are nilpotent blocks of the centralizer C R·C G (Q) (R) =
C G (R)  (cf. Th. 1.2 in [3]); but, the inclusion Qδ ⊂ Rε implies b(ε)BrR b(δ) = b(ε) (cf. (40.4) in [16]), where b(ε) denotes the corresponding block over k. 3.4. Recall that if A is a G-algebra, for any pointed group Hβ on A we call the simple quotient A(Hβ ) of A
H determined by β multiplicity algebra of Hβ , and we set mβ (A) = dimk A(Hβ )sβ (i) where sβ (i) is the image of i ∈ β in A(Hβ ) (cf. Def. 2.2 in [9]); more generally, if Kγ is another pointed group on A such that K ⊂ H, we set
 3.4.1 m βγ = dimk sγ (i)A(K γ )sγ ( j) where sγ (i) and sγ ( j) are the images of i ∈ β and j ∈ γ in A(Kγ ); that is to say, if Kγ ⊂ Hβ then γ can be identified with a point of K on Aβ and we have m βγ = m γ (Aβ ). Proposition 3.5. For any nil-centralized pointed groups Qδ and Rε on OG such that Rε ⊂ Q δ , we have (mδε )2 ≡ 1 (mod p).

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Proof: We argue by induction on |Q : R| and may assume that R = Q; by Proposition 3.3, for any subgroup T of NQ (Rε ) = N Q (R) there is a unique local point θ of T such that Rε ⊂ Tθ ⊂ Q δ ; but it is clear that (cf. Def. 2.2 in [9])  3.5.1 m δε = m δη m ηε η

where η runs on the set of points of T on OG, and according to Lemma 5.4 in [9], p divides mηε whenever η is not local; consequently we get 3.5.2

m δε ≡ m δθ m θε

(mod p)

and therefore, we may assume that Rε  Q δ . Let µ be the point of NG (Rε ) on OG fulfilling Qδ ⊂ NG (Rε )µ (cf. (37.7) in [16]) and Pγ a maximal local pointed group on OG which contains a defect pointed group of NG (Rε )µ γ γ containing Qδ ; since we have already proved that mε ≡ m δ m δε (mod p) (cf. 5.5.2), now it suffices to prove that
γ 2
γ 2 3.5.3 mε ≡ 1 ≡ mδ (mod p) . That is to say, if λ is the point of R·CG (R) on OG fulfilling Rε ⊂ R·C G (R)λ (cf. (37.7) in [16]), it suffices to prove the congruence above whenever Qδ is a maximal local pointed group on OG containing a defect pointed group of R·CG (R)λ ; then, choosing T = NQ (R) above, it is clear that Tθ is self-centralizing (cf. (41.3) in [16]) and, by the induction hypothesis, we may assume that 1 ≡ (mδθ )2 (mod p); moreover, from 3.2 we get
θ 2 (m ε ) = dimk (OG)θ (Rε ) ≡ 1 (mod p), so that the statement follows from congruence 3.5.2 again. Remark 3.6. Notice that, if in this proof we assume that Rε is selfcentralizing, we have 3.6.1

(OG)(Rε ) ∼ = kC¯ G (R) b¯ (ε) ,

where C¯ G (R) = C G (R)/Z(R) and b¯ (ε) is the image in kCG (R) of the block determined by ε; in particular, setting T¯ = T/R, it
is clear that  (OG)(Rε ) is ¯ ¯ ¯ ¯ a Dade T -algebra (since T is a defect group of k T ·C G (R) b¯ (ε)) and that the image in kCC¯ G (R) (T¯ ) of a primitive idempotent of kCG (T ) is primitive too (cf. Prop. 2.2 in [13]); consequently, we need not quote 3.2 to get 
3.6.2 (OG)θ (Rε ) (T¯ ) ∼ =k . Corollary 3.7. Let Qδ be a local pointed group on OG, α et β the respective points of G and Q·CG (Q) on OG such that Qδ ⊂ Q·C G (Q)β ⊂ G α , and Pγ

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a defect pointed group of Gα containing a defect pointed
 group Q·CP (Q)ε of Q·CG (Q)β . Then, |C P (Q)| divides dimk (OG)γ (Q) and we have
 
 dimk (OG)γ (Q) ≡  E C G (Q) Q·C P (Q)ε  (mod p) . 3.7.1 |C P (Q)| Proof: Since (OG)(Q) ∼ = kCG (Q), (OG)γ (Q) is a projective kCP (Q)-module by left multiplication. Let Rθ be a defect pointed group of Q·CG (Q)β such that Rθ ⊂ Pγ ; it is quite clear that Rθ is self-centralizing (cf. (41.3) in [16]) and therefore, by Proposition 3.3, there is a unique local point ε of Q·C P (Q) on OG fulfilling 3.7.2

Q δ ⊂ Rθ ⊂ Q·C P (Q)ε ⊂ Pγ

,

which implies Q·C P (Q)ε ⊂ Q·C G (Q)β by the uniqueness of
β (cf. (37.7)  in [16]), so that Rθ = Q·C P (Q)ε ; moreover, since BrQ (OG) Q·X = (OG)(Q) X for any subgroup X of CG (Q), it is easily checked that C P (Q)Br Q (ε) is a defect pointed group of CG (Q)Br Q (β) on (OG)(Q), and therefore (OG)ε (Q) is a source kC P (Q)-interior algebra of CG (Q)Br Q (β) , so that (cf. Prop. 14.6 in [11])
 
 dimk (OG)ε (Q) 3.7.3 ≡  E C G (Q) Q·C P (Q)ε  (mod p) . |C P (Q)| On the other hand, any other point η of Q·CP (Q) on (OG)γ is not local (cf. (40.4) and (41.1) in [16]) and therefore, BrQ (η) is either {0} or a nonlocal point of CP (Q) on (OG)γ (Q); in the last case, for any j ∈ η, we have BrQ ( j) = TrCT P (Q)() for a suitable proper subgroup T of CP (Q) and some idempotent  of (O G)γ (Q)T such that  u = 0 for any u ∈ C P (Q)−T (cf. (23.1) in [16]); hence, for any
point η of Q·C γ and any  P (Q) on (OG)
 j  ∈ η , p|C P (Q)| divides dimk j(OG) j  (Q) and dimk j  (OG) j(Q) . γ Since Q·C P (Q)ε is self-centralizing, we have (mε )2 ≡ 1 (mod p) by Proposition 3.5, which completes the proof. 4. On the local category of a hyperfocal subalgebra 4.1. In this section we need not assume that O has characteristic zero. As in 1.5 above, let b be a block of G and Pγ a defect pointed group of b. Set B = (OG)γ = i(OG)i for some i ∈ γ , and assume that there is a P-stable O-subalgebra D of B containing the image of a subgroup Q of P and fulfilling B = ⊕u∈U D·u where U is a set of representatives for P¯ = P/Q in P; thus, Q is normal, D has an evident O Q-interior P-algebra structure and we have an O P-interior algebra isomorphism B ∼ = D ⊗ Q P. According to our main result, a hyperfocal subalgebra of B is one of these

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O-subalgebras – actually, it will be clear from the next proposition and from our main result that it is a minimal one – and the purpose of this section is to describe the relationship between the local categories of B and D. Notice that D is a direct summand of B as O(P ×P¯ P)-modules and therefore it has a P × P¯ P-stable O-basis. Proposition 4.2. The inclusion D ⊂ B is a strict semicovering P-algebra homomorphism and, for any local pointed group Rε on B and any local point εˆ of R on D contained in ε, we have 4.2.1

FD (Rεˆ , Pγˆ ) = FB (Rε , Pγ ) .

In particular, Q contains the hyperfocal subgroup of Pγ , it has a unique local point δ on B, and Qδ is nil-centralized. Proof: The first statement follows from Proposition 2.8 above; moreover, for from any local pointed group Rε on B and any ϕ˜ ∈ FB (Rε ,
Pγ ), it follows  the same proposition that there are u ∈ P and ψ˜ ∈ FD (Rεˆ )u , Pγˆ such that ϕ˜ = ψ˜ ◦ κ˜ uR , and therefore we have ϕ˜ = (˜κuP )−1 ◦ ϕ˜ = (˜κuP )−1 ◦ ψ˜ ◦ κ˜ uR , which belongs to FD (Rεˆ , Pγˆ ) by equality 2.4.4. In particular, if x is a p -element  ut(R) is contained in FD (Rεˆ ) (cf. 2.3.1) and of NG (Rε ), the image of x in A therefore, choosing j ∈ εˆ , [x, R]· j is contained in jD j ∩ R· j = (R ∩ Q)· j (cf. 2.2.1), so that [x, R] ⊂ R ∩ Q. Notice that, if moreover x centralizes
R ∩ Q, it centralizes R (cf. Th. 3.2, Chap. 5 in [6]); but, since BrQ (OG) Q·X = (OG)(Q) X for any subgroup X of CG (Q), C R (Q)Br Q (ε) runs on the whole set of local pointed groups on B(Q) when Rε runs on the set of all the local pointed groups on B such that Q ⊂ R ⊂ Q·C P (Q); consequently, if δ is a local point of Q on OG such that Qδ ⊂ Pγ , the
block b(δ) of CG (Q) is nilpotent, since P normalizes the Brauer pair Q, b(δ) (cf. (40.4) in [16]) and therefore it contains a defect group of b(δ); in particular, δ is unique (cf. Th. 1.2 in [3]). Proposition 4.3. Let Rε be a local pointed group on B and ζ the point of R·C G (R) on OG such that Rε ⊂ R·C G (R)ζ , and assume that Pγ contains a defect pointed group of R·CG (R)ζ . Then, C P (R) acts transitively on the set of local points of R on D contained in ε and Q·CP (R) contains any u ∈ P fulfilling 4.3.1

[u, R] ⊂ Q

and

(D·u)R ∩ B ∗ = ∅ ;

in particular, if Rε is nil-centralized then R has a unique local pointˆε on D and, respectively denoting by θ and θˆ the local points of NP (R) on B and D, we have kC P (R)- and kC Q (R)-interior NP (R)-algebra isomorphisms 4.3.2

Bθ (R) ∼ = Dθˆ (R) ⊗C Q (R) C P (R) Dθˆ (R) ∼ = Dθˆ (Rεˆ ) ⊗O kC Q (R)

.

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Proof: Let T be the set of u ∈ P fulfilling condition 4.3.1; it is easily checked that T is a subgroup of P containing Q·CP (R); moreover, for any v ∈ P such that [v, R] ⊂ Q, it is clear that R stabilizes D·v and that we have BrR (D·v) = (D·v)(R); thus, for any u ∈ T , we get

 4.3.3 dimk (D·v)(R) = dimk (D·vu)(R) ; hence, since (D·v)(R) is a direct summand of (OG)(R) ∼ = kCG
(R), as kC Q (R)-modules by left multiplication, |CQ (R)||T/Q| divides dimk B(R) and therefore, by 3.7.1, we necessarily have T = Q·CP (R). Let εˆ and δˆ local points of R on D contained in ε and choose i ∈ εˆ and j ∈ δˆ ; since i and j are conjugate in BR , we have Bi ∼ = B j as B ⊗O O R-modules by multiplication by B on the left and by R on the right; consequently, there is u ∈ P such that D i ∼ = (D·u) j as D ⊗O O R-modules (since both members are indecomposable) or, equivalently, such that D i ∼ = −1 u −1 u −1 Resg (D j u ) where g : D ⊗O O R ∼ D ⊗ O R maps d ⊗ v on d ⊗ v = O
−1  for any d ∈ D and any v ∈ R; thus, since D(1 − i) and Resg D(1 − j u ) −1 are isomorphic too, there is d ∈ D∗ fulfilling Did = D ju , D(1 − i)d = −1 −1 D(1 − j u ) and vd = dvu for any v ∈ R, so that du ∈ (D·u)R ∩ B ∗ and j = i du ; in particular, up to a new choice, we may assume that u ∈ CP (R) and then we get d ∈ (DR )∗ and δˆ = εˆ u . Assume that Rε is nil-centralized and let θ be the local point of NP (R) = N P (Rε ) on B fulfilling Rε ⊂ N P (R)θ ⊂ Pγ (cf. Prop. 3.3) and θˆ a local point of N P (R) on D contained in θ (cf. Prop. 3.15 in [7]); then, the multiplicity of the inclusion DN P (R) ⊂ B R at ε and θˆ is not zero, and therefore there is a point εˆ of R on D such that Rεˆ ⊂ N P (R)θˆ and that the mutiplicity of the inclusion DR ⊂ B R at ε and εˆ is not zero too (cf. Def. 2.2 in [9]), so that εˆ ⊂ ε and εˆ is local (cf. Prop. 3.15 in [7]); by 3.2, setting S = Bθ (Rε ), we have a kC P (R)-interior NP (R)-algebra isomorphism 4.3.4

Bθ (R) ∼ = S ⊗k kC P (R)

and therefore, respectively denoting by θˆ S and θ S the unique local points of N P (R) on S◦ ⊗O Dθˆ and on S◦ ⊗O Bθ (cf. Th. 5.3 in [12]), we have (S◦ ⊗O Bθ )θ S (R) ∼ = kC P (R) (cf. Prop. 5.6 in [12]); hence, since (S◦ ⊗O Dθˆ ·u)θˆ S (R) is a projective direct summand of (S◦ ⊗O Bθ )θ S (R) as kC Q (R)-modules for any u ∈ CP (R), we necessarily get 4.3.5

(S◦ ⊗O Dθˆ )θˆ S (R) ∼ = kC Q (R) .

Now, the evident OCQ (R)-interior NP (R)-algebra embeddings 4.3.6

S ⊗O kC Q (R) −→ S ⊗O S◦ ⊗O Dθˆ (R) ←− Dθˆ (R)

determine a kCQ (R)-interior NP (R)-algebra embedding Dθˆ (R) → S ⊗O kC Q (R)

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which actually is an isomorphism since the unity element is primitive in S N¯ P (R) . In particular, εˆ is the unique local point of R on Dθˆ and therefore it is fixed by NP (R), so that it is the unique local point of R on D contained in ε. Corollary 4.4. Any O Q-interior P-algebra automorphism of D which acts trivially on FˆD (Pγˆ ) is induced by an element of Z(P)·(DP )∗ . Proof: An O Q-interior P-algebra automorphism h of D determines an O P-interior algebra automorphism of B and if h acts trivially onFˆD (Pγˆ ) = FˆB (Pγ ) (cf. Prop. 4.3), it is induced by some a ∈ (BP )∗ (cf. Prop. 14.9 in [11]); in particular, Da is an indecomposable direct summand of B as (D ⊗O D◦ ) ⊗ Q P-modules and therefore, since 4.4.1

(Da)(P) = D(P)Br P (a) = {0} ,

it is isomorphic to D·u for some u ∈ P such that P normalizes u Q or, equivalently, Q contains [u, P]; moreover, if d·u is the image of a by such an isomorphism, for some d ∈ D, it is clear that P fixes d·u and that there is d ∈ D such that d d·u = u; thus, u belongs to Q·Z(P) and, up to a new choice, we actually may assume that u ∈ Z(P); in that case it is easily checked that d belongs to (DP )∗ and induces the same automorphism of D as a·u−1 . 5. Some p-adic analysis 5.1. In this section we collect some elementery facts on p-adic analysis; they certainly are well-known, but play such an important role in the proof of our main result that, for the reader’s convenience, we give complete proofs. Lemma 5.2. Let B be an O-algebra, D an O-subalgebra of B, M an O-submodule of B such that B = M + D and π an element of J(O). Then, we have 5.2.1

1 + π·B = (1 + π·M)·(1 + π·D) .

Moreover, if D ∩ M = {0}, the map (1 + π·M) × (1 + π·D) → 1 + π·B determined by the product is bijective. Proof: It is clear that (1 + π·M)·(1 + π·D) ⊂ 1 + π·B; moreover, for any b ∈ B and any  ≥ 1, if we assume that we have already obtained the decomposition (which is trivial for  = 1 ) 5.2.2

−1 −1     1 + π·b = 1 + π h ·m h + π  ·b 1 + π h ·dh h=1

h=1

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for suitable b ∈ B, {m h }1≤h< in M and {dh }1≤h< in D, then we obviously can choose m ∈ M and d ∈ D such that b = m  + d , and therefore, setting −1   d = d 1 + π h ·dh

5.2.3

and

h=1 
−1  π h−1 ·m h b+1 = −d 1 + π  ·d h=1

which respectively belong to D and B, we get −1 
 −1 1 + π·b = 1 + π h ·m h + π  ·b 1 + π  ·d h=1

5.2.4

−1 
 × 1 + π  ·d 1 + π h ·dh

.

h=1       π h ·m h + π +1 ·b+1 1 + π h ·dh = 1+ h=1

h=1



Now, setting m = h≥1 π h ·m h in M and d = h≥1 π h ·dh in D (we mean the corresponding p-adic limits), we have 5.2.5

1 + π·b ≡ (1 + π·m)(1 + π·d)

(mod π ·B)

for any  ≥ 1, so that the equality holds in B. Moreover, if we have (1 + π·m)(1 + π·d) = (1 + π·m )(1 + π·d  ) for some m  ∈ M and some d ∈ D such that d = d, we get 1 + π·m = (1 + π·m  )(1 + π  ·c) for some  ≥ 1 and a suitable c ∈ D − π·D, and therefore, if m c = n + c for suitable n ∈ M and c ∈ D, we obtain m − m  − π  ·n = π −1 ·c + π  ·c , so that c + π·c = 0; this contradiction proves the last statement. Lemma 5.3. Assume that O contains a primitive p-th root of unity ξ. Let D be a commutative O-algebra, set J = J(D) and consider the groups (ξ − 1)·J endowed with the sum and 1 + (ξ − 1)· J endowed with the product. The maps exp : (ξ − 1)·J −→ 1 + (ξ − 1)· J log : 1 + (ξ − 1)· J −→ (ξ − 1)· J

respectively sending m ∈ (ξ − 1)· J to 1 + h≥1 m h /h! and 1 − m to

h h≥1 m /h are group isomorphisms, inverse of each other, which preserve 5.3.1

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the intersections with any O-subalgebra D . In particular, the p-th power determines a group isomorphism 5.3.2

1 + (ξ − 1)·(J ∩ D ) ∼ = 1 + (ξ − 1) p ·(J ∩ D ) .

Proof:

Set m = (ξ − 1)·n for some n ∈ J ∩ D ; since pO = (ξ − 1) p−1 O, if h = ∈N r p where 0 ≤ r ≤ p − 1 for any  ∈ N, it is easily checked that

5.3.3

h!O = p

j≥1

≥ j

r p− j



O = (ξ − 1)h−

Nr O

∈

and therefore mh /h! and a fortiori mh /h belong to (ξ − 1)·(J ∩ D )h ; consequently, the maps exp and log are well-defined, and it is straightforward to see that they are group homomorphisms fulfilling 5.3.4

log ◦ exp = id(ξ−1)·J

and

exp ◦ log = id1+(ξ−1)·J

.

Thus, the p-th power in 1+(ξ −1)·(J ∩ D ) becomes the multiplication by p in the additive group (ξ − 1)·(J ∩ D ), which determines an injective group homomorphism of image equal to p(ξ − 1)·(J ∩ D ) = (ξ − 1) p ·(J ∩ D ) and, setting D = O·1 + p·D we already know that exp determines a group isomorphism 5.3.5

(ξ − 1)·(J ∩ D ) ∼ = 1 + (ξ − 1)·(J ∩ D ) .

5.4. Moreover, we need the following easy consequence of Lemma 5.3; an equivalent version of this consequence already appears as Lemma A1.4 in [14] with a direct but less clear proof. Lemma 5.5. Assume that O contains a primitive p-th root of unity ξ. For any commutative O-algebra D, the torsion subgroup of 1 + (ξ − 1)·D is  the product i∈I < ξ·i >, where I is the set of primitive idempotents of D. In particular, if M is an indecomposable OG-module and the image of G is contained in idM + (ξ − 1)·EndO (M), then M ∼ = O and this image is contained in < ξ >. Proof: Set J = J(D); the group isomorphism (ξ − 1)· J ∼ = 1 + (ξ − 1)· J shows that 1+(ξ −1)· J has a trivial torsion subgroup and then the existence of the evident group isomorphism 5.5.1

D/J ∼ = (1 + (ξ − 1)·D)/(1 + (ξ − 1)· J )

mapping the class of d ∈ D on the class of 1 + (ξ − 1)·d proves that the torsion subgroup of 1 + (ξ − 1)·D has exponent p; moreover, we may assume that
I = {1}. If t is an element of order p in 1 + (ξ − 1)·D, we have t = λ· 1 + (ξ − 1)·n for suitable λ ∈ 1 + (ξ − 1)O and n ∈ J, and

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383


p therefore we get 1 = λp · 1 + (ξ − 1)·n ; that is to say, λp ·1 belongs to the intersection

 5.5.2 O·1 ∩ 1 + (ξ − 1) p ·J = 1 + (ξ − 1) p J(O) ·1 p p −1 and
therefore there  p is µ ∈ 1 + (ξ − 1)· J(O) such that µ ·1 = (λ ·1) = 1 + (ξ − 1)·n , which implies that µ·1 = 1 + (ξ − 1)·n by isomorphism 5.3.2, so that t = λµ·1. In order to prove the last statement, we may assume that M is faithful and identify G with its image; then, it is clear that G is a p-group and, for any f ∈ Z(G), setting f = idM +(ξ−1)·g, for a suitable g ∈ EndO (M), and considering the O-subalgebra D generated by g, it follows that f ∈< ξ·idM > since G centralizes D; in particular, we get 
5.5.3 G ∩ id M + (ξ − 1)2 ·EndO (M) = {id M } .

6. Uniqueness of the hyperfocal O-subalgebra 6.1. We borrow the notation from 1.5 above and denote by K the quotient field of O. Let Q be a normal subgroup of P and U a set of representatives for P/Q in P; in this section we prove the uniqueness part of Theorem 1.8; explicitely, we show that a P-stable O-subalgebra D of B = (OG)γ = i(OG)i fulfilling  6.1.1 D ∩ Pi = Qi and B= D·u u∈U

is unique up to (B P )∗ -conjugation. Notice that, according to Proposition 4.2, the existence of D forces Q to contain the hyperfocal subgroup of Pγ . First of all, we need a more precise form of a particular case of Corollary 4.4; then, in Theorem 6.3 we prove a more precise form of the uniqueness – the ordinary form corresponds to the case π ∈ O∗ – that we need in the proof of the existence. Proposition 6.2. For any π ∈ J(O), an O Q-interior P-algebra automorphism of D inducing the identity on D/π·D is induced by some element d of i + π· J(D P ). Proof: Assume that an O Q-interior P-algebra automorphism h of D induces the identity on D/π·D. We argue by induction on |P/Q|; if Q = P, we have D = B and, since Fˆk⊗O B (P1⊗γ ) = FˆB (Pγ ), h acts trivially on FˆB (Pγ ), so that it is induced by a suitable a ∈ i + J(BP ) (cf Proposition
14.9 in [11])  having a central image in B/π·B; but, since Z(OG) covers Z (O/πO)G , Z(B) covers Z(B/π·B); hence, up to a new choice of a, we may assume that a ∈ i + π·J(B P ). If Q = P, let R be a normal subgroup of P such that Q ⊂ R and |R/Q| = p , and, denoting by W a set of representatives for R/Q

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in R, set E = ⊕w∈W D·w; it is quite clear that h determines an O R-interior P-algebra automorphism hE of E inducing the identity on E/π·E and, by the induction hypothesis, we may assume that there is e ∈ i + π·J(EP ) such that h E = int E (e). Moreover, since P acts trivially on R/Q, we have 6.2.1

Z(E) P = ⊕w∈W Z(E) P ∩ D·w

and thus, choosing an O-submodule Mw of (D·w) P which is a complement of Z(E) P ∩ D·w for any w ∈ W, it follows from Lemma 5.2 that we may choose e in i + π·(⊕w∈W Mw ) and then, this choice is unique. ˆ be an extension of O containing a primitive p-th root of Finally, let O ∗ ˆ a group homomorphism such that Ker(ϕ) = Q, unity ξ and ϕ : R → O ˆ ⊗O E ∼ ˆ ⊗O E inducing ˆ and consider the O-algebra automorphism tϕ : O =O ˆ ⊗O D and mapping 1 ⊗ v·i on ϕ(v) ⊗ v·i for any v ∈ R; the identity on O since idOˆ ⊗ h E = intO⊗ ˆ O E (1 ⊗ e) commutes with tϕ , and tϕ stabilizes ˆ ⊗O Mw , it follows from the uniqueness part of Lemma 5.2 that ⊕w∈W O tϕ (1 ⊗ e) = 1 ⊗ e, so that e belongs to D. Theorem 6.3. Assume that there is another P-stable O-subalgebra D of B fulfilling condition 6.1.1, such that D + π·B = D + π·B for some π ∈ O. There is a ∈ i + π·J(B P ) such that D = Da . Proof: We may assume that Q = P; let R be a normal subgroup of P such that Q ⊂ R and |R/Q| = p, and, denoting by W a set of representatives containing 1 for R/Q in R, consider the O R-interior P-algebras E = ⊕w∈W D·w and E  = ⊕w∈W D ·w; it is clear that, arguing by induction on |P/Q|, we may assume that E = E c for some c ∈ i + π·J(B P ) and −1 then, replacing D by (D )c , we still may assume that E = ⊕w∈W D ·w. In that case, it is clear that the inclusions D ⊂ E and D ⊂ E induce O P-interior algebra homomorhisms 6.3.1

t:B∼ = D ⊗ Q P −→ Res QR (E) ⊗ Q P t : B ∼ = D ⊗ Q P −→ Res QR (E) ⊗ Q P

and that both are sections of the canonical map (cf. 2.9.1) 6.3.2

f : B˜ = Res QR (E) ⊗ Q P −→ E ⊗ R P ∼ =B

;

but recall that Z = {w−1 ·i ⊗ Q w}w∈W is central in B˜ (cf. 2.9), which implies that B˜ has an O Z-algebra structure, and that we have (cf. 2.9.2) B˜ = ⊕w∈W t(B)(w−1 ·i ⊗ Q w) 6.3.3

= ⊕w∈W t  (B)(w−1·i ⊗ Q w) ;  ˜ −1 ·i ⊗ Q w − i ⊗ Q 1) B(w Ker( f ) = w∈W

The hyperfocal subalgebra of a block

385

hence, according to Lemma 6.5 below, there is a ∈ i ⊗Q 1 + J( B˜ P ) such that t  = int B˜ (a) ◦ t and, in particular, f(a) is central in B. Moreover, setting ¯ = O/π·O and B¯ = B/π·B, and denoting by D, ¯ D¯  , E, ¯ t¯, t¯ and a¯ O  the corresponding items, the hypothesis D¯ = D¯ implies t¯ = t¯ and then, ¯ ⊗ Q P. But we have by the top equality in 6.3.3, a¯ is central in ResQR ( E)
 ˜ = Z(B) and, since Z(OG) covers Z(OG) ¯ ¯ f Z( B) , Z(B) covers Z( B), R ˜ covers Z(ResQ ( E) ¯ ⊗ Q P) (cf. 6.3.3). Consequently, we may so that Z( B) choose 6.3.4

a ∈ i ⊗Q 1 + π·Ker( f ) ∩ J( B˜ P ) .

ˆ be the Galois extension of K containing On the other hand, let K ˆ the ring of integers of K, ˆ and a primitive p-th root of unity ξ and O   ˆ ⊗O B, Dˆ = O ˆ ⊗O D, Dˆ = O ˆ ⊗O D and Eˆ = O ˆ ⊗O E; set Bˆ = O ∗ ˆ consider a group homomorphism ϕ : R → O fulfilling Ker(ϕ) = Q, so ˆ mapping that Im(ϕ) =< ξ >, and the O-algebra homomorphism O Z → O −1 ˆ ˆ v ·i ⊗ Q v on ϕ(v) for any v ∈ R, denote by O ϕ the evident O R-interior ˆ notice that we have an O ˆ R-interior algebra ˆ ϕ ⊗Oˆ E; algebra, and set Eˆ ϕ = O isomorphism (cf. 2.9.2)
 ˆ ⊗O Z Res QR (E) ⊗ Q R . 6.3.5 Eˆ ϕ ∼ =O It is easily checked that we have an O P-interior algebra homomorphism 6.3.6

ˆ ⊗O Z B˜ ∼ qϕ : B˜ −→ O = Eˆ ϕ ⊗ R P

mapping e⊗Q u on e⊗R u ∈ Eˆ ϕ ⊗ R P, for any e ∈ E and any u ∈ P; moreover notice that, for any v ∈ R, we have qϕ (v·i ⊗ Q v−1 ) = ϕ(v)−1 ·i ⊗ R 1. ˆ P-interior algebra isomorConsequently, qϕ ◦ t and qϕ ◦ t  determine O phisms (cf. 6.3.3) and t ϕ : Bˆ ∼ tϕ : Bˆ ∼ = Eˆ ϕ ⊗ R P = Eˆ ϕ ⊗ R P 
and we have tϕ = int Eˆ ϕ ⊗ R P qϕ (a) ◦ tϕ ; actually, according to the condition 6.3.4 and to the bottom equality in 6.3.3, qϕ (a) belongs to
 i ⊗ R 1 + π(ξ − 1)· J ( Eˆ ϕ ⊗ R P) P ; but, it is quite clear that 6.3.7

6.3.8

ˆ = Eˆ ϕ ⊗ R 1 = t ϕ ( E) ˆ ; tϕ ( E)

thus, qϕ (a) stabilizes Eˆ ϕ ⊗ R 1 inducing the identity on the quotient Eˆ ϕ /π(ξ − 1)· Eˆ ϕ ; hence, according to Proposition 6.2, suitably modifying qϕ (a) we get
 6.3.9 aˇ ∈ i ⊗ R 1 + π(ξ − 1)·J ( Eˆ ϕ ⊗ R 1) P

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ˆ Q-interior still fulfilling tϕ = int Eˆ ϕ ⊗ R P (a)◦t ˇ ϕ . Moreover, considering the O R ˆ ∼ ˆ P-algebra isomorphism r : ResQ ( E) = E ϕ ⊗ R 1 which maps eˆ ∈ Eˆ on eˆ ⊗ R 1, the element aˆ = r −1 (a) ˇ belongs to i + π(ξ − 1)·J(Eˆ P ) and the compositions ˆ Q-interior P-algebra automorphisms sϕ = r −1 ◦ tϕ and sϕ = r −1 ◦ tϕ are O R ˆ of Res Q ( E), respectively inducing the identity on Dˆ and Dˆ  , and fulfilling 6.3.10

and sϕ = int Eˆ (a) ˆ ◦ sϕ .  sϕ (v·i) = ϕ(v)·(v·i) = s ϕ (v·i) for any v ∈ R

ˆ At that point, since P stabilizes the decomposition Eˆ = ⊕w∈W D·w, P P for any subgroup T of P we have ET = ⊕w∈W (D·w)T , so that ˆ ˆ moreover, according to Proposition 4.3, we have E(P) = ⊕w∈W ( D·w)(P); ∼ ˆ ˆ D(P) = kC Q (P) and E(P) ∼ = kC R (P); that is to say, considering the evident homomorphism 6.3.11


Br P ∼ ˆ Eˆ P −−−→ k ⊗k(C Q ( P)) E(P) = k C R (P)/C Q (P) ,


 P ˆ either C Q (P) = C R (P) and J( Eˆ P ) = Ker(Br P ) = J( Dˆ P )⊕ ⊕w∈W  ( D·w) where W  = W − {1}, or we have C R (P)/C Q (P) ∼ = R/Q and may assume that W ⊂ Z(P). In any case, we get 6.3.12

ˆ Z(P) + Ker(Br P ) Eˆ P = O ˆ J( Eˆ P ) ⊃ Ker(Br P ) = ⊕w∈W Ker(Br P ) ∩ D·w

and, arguing by induction, for any h ≥ 1 we get easily 6.3.13

ˆ . Ker(BrP )h = ⊕w∈W Ker(Br P )h ∩ D·w

In particular, according to Lemma 5.2 applied to the top equality in 6.3.12, in the first equality of 6.3.10 we may replace aˆ by an element – still noted aˆ – ˆ = (ϕ(v) − 1)O ˆ for any of i + π(ξ − 1)·Ker(Br P ) and then, since (ξ − 1)O v ∈ R − Q, applying the bottom equality in 6.3.12 we get  6.3.14 aˆ = i + (ξ − 1)· dˆ + (ϕ(w) − 1)·( dˆw ·w) , w∈W 

ˆ for any w ∈ W  . where dˆ ∈ π·Ker(Br P )∩ Dˆ and dˆw ·w ∈ π·Ker(Br P )∩ D·w Now, consider the recursive families {ˆch }h∈N and {ˆgh }h∈N in Eˆ ∗ fulfilling and, if h ∈ N, cˆ h+1 = gˆ h−1 cˆ h sϕ (ˆgh )  gˆ h = i − dˆ w,h ·w cˆ 0 = aˆ

6.3.15

w∈W 

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for a choice of dˆh ∈ π·Ker(Br P ) ∩ Dˆ and, for any w ∈ W  , a choice ˆ such that of dˆw,h w ∈ π·Ker(Br P )h+1 ∩ D·w  6.3.16 cˆ h = i + (ξ − 1)· dˆh + (ϕ(w) − 1)·( dˆw,h ·w) ; w∈W 

the existence (and the uniqueness) of such families follows from equality 6.3.14 (that is to say, we have dˆ◦ = dˆ and dˆw,◦ = dˆw for any w ∈ W  ) and from the following one  −1    cˆ h+1 = i − dˆw,h ·w cˆ h i − ϕ(w)·( dˆw,h ·w) 6.3.17 , w∈W  w∈W  = i + (ξ − 1)· dˆh − xˆh+1 + yˆh+1 − zˆh+1 where we put 6.3.18 xˆh+1 = yˆh+1 = zˆh+1 =

  ∈N w∈W 

 

∈N w∈W 

 

∈N w∈W 

+1   dˆw,h ·w (ϕ(w) − 1)·( dˆw,h ·w) w∈W 

+1  dˆw,h ·w (ˆch − i) − (ˆch − i) ϕ(w)·( dˆw,h ·w) w∈W 

;

+1  (ˆch − i) ϕ(w)·( dˆw,h ·w) dˆw,h ·w w∈W 

indeed, it is clear that xˆh+1 , yˆh+1 and zˆh+1 belong to π(ξ − 1)·Ker(BrP )h+2 and therefore we can apply equality 6.3.13.

h Consequently, the sequence of ordered products ˆ  h∈N has =0 g a limit in Eˆ 6.3.19

gˆ = lim

h→∞

h 

gˆ 



=0

which belongs to i +π·Ker(BrP ); moreover, the element
 cˆ = limh→∞ {ˆch } = gˆ −1 aˆ sϕ (ˆg) belongs to i + π(ξ − 1)· Ker(Br P ) ∩ Dˆ . In conclusion, we get (cf. 6.3.10) 
6.3.20 sϕ = int Eˆ (ˆg) ◦ int Eˆ (ˆc) ◦ sϕ ◦ int Eˆ (ˆg)−1
p and therefore, we have idEˆ = int Eˆ (ˆc)◦sϕ (cf. 6.3.10); but, since sϕ (ˆc) = cˆ , sϕ commutes with intEˆ (ˆc) and we get idEˆ = int Eˆ (ˆc p ); hence, cˆ p belongs to the intersection 

 ˆ P ∩ i+(ξ −1) p · Ker(Br P )∩ Dˆ = i+(ξ −1) p ·J Z( Dˆ P ) 6.3.21 Z( E)

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 ˆ P , so and thus, according to Lemma 5.3, cˆ belongs to i + (ξ − 1)· J Z( D) that we have intEˆ (ˆc) = idEˆ ; then, we get sϕ = int Eˆ (ˆg) ◦ sϕ ◦ int Eˆ (ˆg)−1 and therefore Dˆ  = gˆ Dˆ gˆ −1 . ˆ over Finally, let us consider the action of the Galois group Σ of K  −1 ˆ for any σ ∈ Σ, we still have Dˆ = σ(ˆg) Dσ(ˆ ˆ g) and therefore K on B; −1 ˆ gˆ σ(ˆg) determines an O Q-interior P-algebra automorphism of Dˆ inducˆ ˆ now, according to Proposition 6.2, there is ing the identity on D/π· D; P ˆ eˆ ∈ i + π·J( D ) inducing the same and therefore we have 
automorphism ˆ P . That is to say, the group σ(ˆg) = gˆ eˆ zˆ for some zˆ ∈ i + π·J Z( E) 6.3.22

 

ˆ P Σ X = i + π·J( Dˆ P ) i + π·J Z( E)

ˆ g−1 and the stabilizer acts on the set of gˆ ∈ i + π·J( Eˆ P ) such that Dˆ  = gˆ Dˆ of gˆ in X covers the quotient Σ and therefore, since Σ is a p -group, it contains a conjugate of Σ (cf. Prop. 4.6 in [11]); hence, up to a new choice, ˆ P. we may assume that Σ fixes gˆ or, equivalently, that gˆ belongs to i +π·J(E) Remark 6.4. Notice that, in this proof, we only apply Proposition 4.3 to P which need not quote neither [7] nor [12]. Lemma 6.5. Let C be an O P-interior algebra and g : C → B an O P-interior algebra homomorphism such that Ker(g) ⊂ J(C). If s : B → C and s : B → C are O P-interior algebra homomorphisms fulfilling g ◦ s = id B = g ◦ s

,

then they determine the same O P-interior algebra exomorphism.

 Proof: Since Ker IndGP (g) ⊂ J IndGP (C) and 6.5.1

IndGP (g) ◦ IndGP (s) = idIndG (B) = IndGP (g) ◦ IndGP (s ) , P

it is quite clear that, for any subgroup H of G, all these homomorphisms induce O-algebra isomorphisms between BH /J(B H ) and C H /J(C H ), so that they induce bijections between the sets of pointed groups on B and on C, and these bijections preserve the corresponding inclusions; in particular, IndGP (s) and IndGP (s ) induce the same bijection, namely the inverse map of the bijection induced by IndGP (g). Thus, P has a unique point γˆ on C whereas, denoting by α the point of G on IndGP (B) determined by the Higman embedding OG b → IndGP (B) (cf. (18.9) in [16]), IndGP (s)(α) and IndGP (s )(α) are contained in the same point αˆ of G on IndGP (C) and we have Pγˆ ⊂ G αˆ ; now, it is clear that IndGP (s)

The hyperfocal subalgebra of a block

389

and IndGP (s ) induce the unique nonzero OG-interior algebra homomorphism (cf. (25.7) in [16]) 6.5.2

OG b ∼ = IndGP (B)α −→ IndGP (C)αˆ

and therefore they induce the same O P-interior algebra exomorphism (cf. (25.7) in [16]) 6.5.3

B∼ =C = IndGP (B)γ −→ IndGP (C)γˆ ∼

;

on the other hand, it immediately follows from their definition (cf. p. 129 in [16]) that IndGP (s) and IndGP (s ) respectively induce s and s from B to C; consequently, we finally get s˜ = s˜ . 7. Existence of the hyperfocal O-subalgebra 7.1. We borrow the notation from 1.5 above, namely setting B = i(OG)i for i ∈ γ , and denote by K the quotient field of O. Let Q be a normal proper subgroup of P; in this section we prove the existence part of Theorem 1.8. If Q contains the hyperfocal subgroup of Pγ , arguing by induction on |P/Q| we consider a normal subgroup R of P such that Q ⊂ R and |R/Q| = p, and assume that there is a P-stable O-subalgebra E of B containing the image of R and fulfilling B = ⊕v∈V E·v, where V is a set of representatives containing 1 for P¯ = P/R in P; recall that E has a P ×P¯ P-stable O-basis Y (cf. 4.1). 7.2. First of all, considering the evident O R-interior P-algebra structure of E, we have to prove the existence of a section of the canonical O P-interior algebra homomorphism (cf. 2.9.1) 7.2.1

f : B˜ = Res QR (E) ⊗ Q P −→ E ⊗ R P ∼ =B

which, a posteriori, is an evident consequence of Theorem 1.8 and determines a unique O P-interior algebra exomorphism B → B˜ by Lemma 6.5. We also fix a set of representatives W containing 1 for R/Q in R, and set U = W·V and Z = {w−1 ·i ⊗ Q w}w∈W , so that U is a set of representatives ˜ ∗ (cf. 2.9). In particular, B˜ has for P/Q in P, and Z a subgroup of Z(B) an evident structure of O(P × Z)-interior algebra, it is easily checked that the set Y ⊗ P is a (P × Z) × (P × Z)-stable O-basis, and f induces an O P-interior algebra isomorphism (cf. 2.9.2) 7.2.2

O ⊗O Z B˜ ∼ =B

.

Lemma 7.3. With the notation above, assume that Q contains the hyperfocal subgroup of P. Then f : B˜ → B admits an O P-interior algebra section ˜ s : B −→ B.

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Remark 7.4. As a matter of fact, this statement follows from Theorem 1.8 in [13]. Indeed, B˜ has an evident structure of O(P × Z)-interior algebra and it is quite clear that f induces an O P-interior algebra isomorphism O ⊗O Z B˜ ∼ = B (cf. 2.9.2). On the other hand, b still is a block of G × Z, and B ⊗O O Z clearly is an O(P × Z)-interior source algebra of this block; moreover, it follows from 7.3.1 below that any G-conjugacy class of local pointed groups on OG b has a representative Tθ on B which respectively determines local points θ˜ and θˆ of T × Z on B˜ and on B ⊗O O Z such that

 . 7.4.1 FB˜ (T × Z)θ˜ = E G×Z (T × Z)θˆ Consequently, according to Theorem 1.8 and Remark 1.9 in [13], f can be lifted to an O(P × Z)-interior algebra isomorphism B˜ ∼ = B ⊗O O Z. Actually, our proof below adapts the proof of Theorem 3.13 in [13] to our easier situation. Proof: According to Corollary 2.10, for any local pointed group Uδ on B there is a local point δ˜ of U on B˜ fulfilling f(δ˜ ) ⊂ δ and then we clearly have FB˜ (Uδ˜ ) ⊂ FB (Uδ ); let ν be the point of NG (Uδ ) on OG b such that Uδ ⊂ NG (Uδ )ν (cf. (37.7) in [16]); we claim that, if Pγ contains a defect pointed group of NG (Uδ )ν , then such a δ˜ is unique and we actually have 7.3.1

FB˜ (Uδ˜ ) = FB (Uδ ) .

Indeed, in that case if η˜ is another local point of U on B˜ such that f(˜η) ⊂ δ and we choose local points δˆ and ηˆ of U on E such that δˆ ⊗ Q 1 ⊂ δ˜ and ηˆ ⊗ Q 1 ⊂ η˜ (cf. Prop. 4.2), we have ηˆ = δˆ z for a suitable z ∈ CP (U) (cf. Prop. 4.3), and therefore we get 7.3.2

δ˜ = δ˜ z ⊃ δˆ z ⊗ Q 1 = ηˆ ⊗ Q 1 ⊂ η˜ .

Moreover, NP (Uδ ) covers a Sylow p-subgroup of FB (Uδ ) = E G (Uδ ) (cf. Lemma 3.10 in [12]) and therefore we get FB (Uδ ) = O p (FB (Uδ ) ·E P (Uδ ); but, according to the last statement of Proposition 2.8, we have  O p (FB (Uδ ) ⊂ FE (Uδˆ ) and then, since for any p -element x of NG (Uδ ) we are assuming that ux Q = u Q for any u ∈ U, we get (cf. 2.2 and 2.7.1)  7.3.3 O p (FB (Uδ ) ⊂ FRes RQ (E) (Uδˆ ) ⊂ FB˜ (Uδ˜ ) ; hence, since E P (Uδ˜ ) = E P (Uδ ) by the uniqueness of δ˜ , we obtain equality 7.3.1. On the other hand, consider B˜ as an O(P × Z)-interior algebra and idenG ˜ ˜ ˜ tify IndG×Z P×Z ( B) with Ind P ( B); since B has a (P × Z) × (P × Z)-stable G ˜ O-basis (cf. 7.2), IndP ( B) has a (G × Z) × (G × Z)-stable O-basis, and isomorphism 2.9.2 determines an OG-interior algebra isomorphism

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391

˜ ∼ O ⊗O Z IndGP ( B) = IndGP (B); thus, it follows from Theorem 2.9 in [13] that the induced map 7.3.4

˜ −→ IndGP (B) = A h = IndGP ( f ) : A˜ = IndGP ( B)

is a strict semicovering of OG-interior algebras; notice that, identifying ˜ and Uδ and Uδ˜ to the corresponding local pointed groups on A and A, always assuming that Pγ contains a defect pointed group of NG (Uδ )ν , since E G (Uδ˜ ) = FA˜ (Uδ˜ ) ∩ E G (U) (cf. 2.10 in [10]) and FA˜ (Uδ˜ ) = FB˜ (Uδ˜ ) (cf. Prop. 2.14 in [10]), equality 7.3.1 implies NG (Uδ˜ ) = NG (Uδ ) and then h induces a k∗ -group isomorphism 7.3.5

ˆ Nˆ¯ G (Uδ˜ ) ∼ = N¯ G (Uδ ) .

In particular, we get a k-algebra isomorphism (cf. Lemma 9.12 in [11]) 7.3.6

ˆ¯

ˆ¯

N G ( Pγ˜ ) N G ( Pγ ) ˜ γ˜ ) ∼ (k) ∼ (k) ∼ A(P = Indk∗ = Indk∗ = A(Pγ )

and, denoting by α the point of G on A determined by the Higman OG-interior algebra embedding OG b → A (cf. (18.9) in [16]), there is a point α˜ of G on A˜ having the same image than α in A(Pγ ) via isomorphisms 7.3.6 (cf. (19.1) in [16]). Now, it suffices to prove that h(˜α) ⊂ α; indeed, in that case h induces an OG-interior algebra homomorphism 7.3.7

h αα˜ : A˜ α˜ −→ Aα ∼ = OG b

and the structural map s : OG b → A˜ α˜ provides an obvious section; in particular, this shows that hαα˜ ( A˜ αHˆ ) = (OG b) H for any subgroup H of G, so that hαα˜ is a covering (cf. §25 in [16]), which is strict since Ker(h αα˜ ) ⊂ J( A˜ α˜ ) (cf. 2.9), and therefore s is a strict covering too (cf. Prop. 4.15 and Rem. 4.16 in [12]); hence, this structural map induces a section of f : B˜ ∼ = B (cf. (25.7) in [16]). = A˜ γ˜ → Aγ ∼ Choose ˜ ∈ α˜ and  ∈ α such that h(˜ )  =  =  h(˜ ), which is possible by our choice of α˜ ( is a suitable lifting of the image of h(˜ ) in A(Pγ ) to a primitive idempotent
of AG ); since all the points of 1 on A are local, it suffices to prove that sδ h(˜ ) = sδ () for any local pointed group Uδ on A, where sδ : AU → A(Uδ ) denotes the canonical
 map. Arguing by contradiction, let Uδ be a maximal one such that sδ h(˜ ) = sδ (); eventually replacing Uδ by a G-conjugate, we may assume that Uδ ⊂ Pγ ⊂ G α (cf. (16.7) in [16]) and, more precisely, we still may assume that Pγ contains a defect pointed ˜
group of NG (Uδ )ν ; moreover, according to our choice of α, we have sγ h(˜ ) = sγ () and therefore U = P. On
the other idempotent of AG fulfilling sδ ( j) = 0  hand, if
j is a primitive  and j h(˜ )− = j = h(˜ )− j, and β is the point of G on A containing j,

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then Uδ is a defect pointed group of Gβ ; indeed, since sδ ( j) = 0, G β contains
Uδ and,if Tθ is a defect pointed group of Gβ containing Uδ , we have sθ h(˜ ) −  = 0 which forces Uδ = Tθ by maximality; consequently,
 ¯ the idempotent sδ h(˜ ) −  belongs to A(Uδ )1NG (Uδ ) (cf. (18.8) in [16]). As above, let δ˜ be the local point of U on B˜ such that f(δ˜ ) ⊂ δ and set ˜ δ˜ ) = Endk (Vδ˜ ) A(U

7.3.8

and

A(Uδ ) = Endk (Vδ ) ,

considering Vδ˜ and Vδ as k∗ Nˆ¯ G (Uδ )-modules (cf. 6.4 in [11] and 7.3.5 above); then, since h is a semicovering, it induces a k∗ Nˆ¯ G (Uδ )-interior algebra em˜ δ˜ ) → A(Uδ ) (cf. (7.5) in [16]), and thus Vδ˜ becomes a direct bedding A(U summand of Vδ (cf. (15.4) in [16]). Moreover, according to Higman’s cri
 terion (cf. (17.3) in [16]), the k∗ Nˆ¯ G (Uδ )-module sδ h(˜ ) −  (Vδ ) is projective; but, according to our choice of α, ˜ Pγ˜ is a defect pointed group of G α˜ and therefore, since U = P, the k∗ Nˆ¯ G (Uδ )-module sδ˜ (˜ )(Vδ˜ ) has no nonzero projective direct summands (cf. (19.3) in [16]): hence, sδ˜ (˜ )(Vδ˜ ) has to be a direct summand of sδ ()(Vδ ) which, since Aα ∼ = OG b, is simple (cf. (37.6) in [16]), so that they are isomorphic.
 However, we have sδ h(˜ ) −  (Vδ ) = {0} and therefore there is at least another local point ε˜ of U on A˜ α˜ such that f(˜ε) ⊂ δ; it is clear that NG (Uε˜ ) ⊂ NG (Uδ ) and, mutatis mutandis, h induces a k∗ -group homomorphism Nˆ¯ G (Uε˜ ) → Nˆ¯ G (Uδ ) and a k∗ Nˆ¯ G (Uε˜ )-interior algebra embedding ˜ ε˜ ) → A(Uδ ); this time, the corresponding k∗ Nˆ¯ G (Uε˜ )-module sδ˜ (˜ )(Vε˜ ) A(U
 is a direct summand of the restriction of sδ h(˜ ) −  (Vδ ), so it is projective which forces Uε˜ to be a defect group of Gα˜ (cf. (19.3) in [16]), a contradiction. Theorem 7.5. Assume that Q contains the hyperfocal subgroup of P. Then, there is a P-stable O-subalgebra D of B containing the image of Q and fulfilling B = ⊕u∈U D·u. Proof: Borrowing the notation from 7.1 and 7.2, by Lemma 7.3 we know the existence of an O P-interior algebra homomorphism s : B → B˜ such that f ◦ s = id B ; in particular, s(E) is a P-stable O-subalgebra of B˜ containing the image of R and fulfilling  s(E) (i ⊗ Q v) + Ker( f ) ; 7.5.1 B˜ = v∈V

˜ B˜ is an O Z-algebra and it is clear that but, since Z is central in B, ˜ Ker( f ) ⊂ J(O Z)· B; consequently, by Nakayama’s Lemma we get 

  7.5.2 Res QR (E) ⊗ Q R (i ⊗ Q v) = B˜ = s(E)·O Z (i ⊗ Q v) v∈V

v∈V

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393

and ResQR (E) ⊗ Q R is a P-stable O Z-subalgebra of B˜ which contains the image of R. ˆ be the Galois extension of K generated by a primitive p-th Let K ∗ ˆ the ring of integers of K ˆ and ϕ : R → O ˆ a group root of unity ξ, O ˆ ⊗O B and Eˆ = O ˆ ⊗O E, homomorphism fulfilling Ker(ϕ) = Q; set Bˆ = O ˆ R-interior algebra and consider the O-algebra ˆ ϕ the evident O denote by O ˆ homomorphism O Z → O mapping w−1 ·i⊗ Q w on ϕ(w) for any w ∈ W, and ˆ notice that we have an O ˆ R-interior algebra Eˆ ϕ = O ˆ R-interior ˆ ϕ ⊗Oˆ E; the O algebra isomorphism (cf. 2.9.2)
 ˆ ⊗O Z Res QR (E) ⊗ Q R . 7.5.3 Eˆ ϕ ∼ =O It follows from equalities 7.5.2 and isomorphism 7.5.3 that s induces an ˆ P-interior algebra isomorphism O 7.5.4

ˆ ⊗O Z B˜ ∼ sϕ : Bˆ ∼ = Eˆ ϕ ⊗ R P =O

ˆ and Eˆ ϕ ⊗ R 1 are P-stable O-subalgebras ˆ of Eˆ ϕ ⊗ R P and then, sϕ ( E) containing the image of R and fulfilling (cf. 7.5.2) ˆ ⊗ R v) = Eˆ ϕ ⊗ R P = ⊕v∈V ( Eˆ ϕ ⊗ R 1)(i ⊗ R v) ; ⊕v∈V sϕ ( E)(i

moreover, if e ∈ E and s(e) = u∈W·V eu ⊗ Q u for suitable eu ∈ E, setting eˆ  = 1 ⊗ e in Eˆ = Eˆ ϕ for any e ∈ E, we have   eˆ u ⊗ R u and e= eu ·u 7.5.6 sϕ (ˆe) = 7.5.5

u∈W·V

u∈W·V



and therefore we get e = w∈W ew ·w and 0 = v ∈ V − {1}, so that   e sϕ (ˆe) = eˆ ⊗ R 1 − wv ·w ⊗ R v + 7.5.7

v∈V

= eˆ ⊗ R 1 −

w∈W



w∈W



ewv ·w for any eˆ wv ⊗ R wv

v∈V, w∈W


 ϕ(w)−1 − 1 ·ˆewv ⊗ R wv

v∈V, w∈W

which implies that sϕ (ˆe) belongs to eˆ ⊗ R 1 + (ξ − 1). Eˆ ϕ ⊗ R P. ˆ of Bˆ containing the Thus, sϕ−1 ( Eˆ ϕ ⊗ R 1) is a P-stable O-subalgebra ˆ ˆ and fulfilling image of R, having the same image as Eˆ in B/(ξ − 1)· B, 7.5.8

Bˆ = ⊕v∈V sϕ−1 ( Eˆ ϕ ⊗ R 1)·v

;

hence, it follows from Theorem 6.3 that there is aˆ ∈ i + (ξ − 1)· J( Bˆ P ) such that 7.5.9

sϕ−1 ( Eˆ ϕ ⊗ R 1)aˆ = Eˆ

.

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ˆ At that point, the O-algebra automorphism tϕ of Eˆ mapping eˆ ∈ Eˆ on aˆ −1 u sϕ (ˆe ⊗ R 1) fulfills tϕ (ˆe ) = tϕ (ˆe)u for any u ∈ P and tϕ (v·i) = ϕ(v)−1 ·(v·i) ˆ ˆ consequently, (tϕ ) p for any v ∈ R, and it acts trivially on E/(ξ − 1)· E; ˆ R-interior P-algebra automorphism of Eˆ and, setting tϕ = induces an O ˆ we have id Eˆ + (ξ − 1)·qϕ for a suitable qϕ ∈ EndOˆ ( E),

7.5.10

(tϕ ) p = id Eˆ +

p    p h=1

h

(ξ − 1)h ·(qϕ )h

;

ˆ ∈ id Eˆ + (ξ − 1) p · EndOˆ ( E) then, it follows from Proposition 6.2 that we have (tϕ ) p = int Eˆ (ˆe) for some element eˆ in i + (ξ − 1) p ·J( Eˆ P ). Moreover, since tϕ obviously commutes with (tϕ ) p , we have tϕ (ˆe) = eˆ zˆ 
ˆ P ; thus, setting for a suitable element zˆ in i + (ξ − 1) p · J Z( E) eˆ = i + (ξ − 1) p ·m ˆ and zˆ = i + (ξ − 1) p ·nˆ
 ˆ P , we get tϕ (m) where m ˆ =m ˆ + nˆ +(ξ −1) p ·mˆ nˆ ˆ ∈ J( Eˆ P ) and nˆ ∈ J Z( E) ˆ ˆ nˆ belongs to (ξ − 1)·Z(E) ˆ P; and, since tϕ acts trivially on E/(ξ − 1)· E, ˆ hence, the O-subalgebra  ˆ P+ ˆ m O· 7.5.11 Cˆ = Z( E) ˆh h∈N ˆ −1)·C, ˆ and (tϕ ) p acts is tϕ -stable, tϕ induces the identity on the quotientC/(ξ ˆ ˆ consequently, according to Lemma 5.5, tϕ stabilizes O-rank trivially on C; P ˆ P ˆ ˆ one direct sum decompositions in C and in Z( E) , and therefore Z(E) ˆ then, according to Lemma 5.2, we can has a tϕ -stable complement Nˆ in C; p ˆ choose eˆ ∈ i + (ξ − 1) · N and we have a unique choice, so that in that case we get tϕ (ˆe) = eˆ by the uniqueness of eˆ . ˆ fixed by tϕ and then, In conclusion, we can choose eˆ ∈ i + (ξ − 1) p · J(C) ˆ according to Lemma 5.3, there is a unique cˆ ∈ i + (ξ
− 1)· J(C) such  p that cˆ p = eˆ , so that cˆ is fixed by tϕ too; in particular, we get int Eˆ (ˆc−1 ) ◦ tϕ = id Eˆ ˆ ˆ Conseand moreover intEˆ (ˆc−1 ) ◦ tϕ induces the identity on E/(ξ − 1)· E. −1 ˆ quently, according to Lemma 5.5, intEˆ (ˆc ) ◦ tϕ stabilizes an O-rank one ˆ thus, since P fixes cˆ and tϕ is an O Q-interior direct sum decomposition in E; ˆ fulfilling tϕ (v·i) = ϕ(v)·(v·i) for P-algebra automorphism of ResQR ( E) any v ∈ R, setting 7.5.12

−1 Dˆ = Eˆ

,

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395

Dˆ contains the image of Q, P stabilizes Dˆ and, since Q = Ker(ϕ), we get ˆ and therefore we get also Eˆ = ⊕w∈W D·w 7.5.13

ˆ Bˆ = ⊕u∈U D·u

.

ˆ over K Finally, let us consider the action of the Galois group Σ of K ˆ ˆ for any σ ∈ Σ, we still have Bˆ = ⊕u∈U σ( D)·u and therefore, on B; P ˆ = Dˆ bˆ ; ˆ ˆ according to Proposition 6.2, there is b ∈ i + J( B ) such that σ( D) that is to say, the group
 7.5.14 X = i + J( Bˆ P )  Σ ˆ has an evident action on the set of P-stable O-subalgebras of Bˆ fulˆ filling equality 7.5.13 and the stabilizer of D in X covers the quotient Σ; hence, since Σ is a p -group, this stabilizer contains a conjugate of Σ (cf. Prop. 4.6 in [11]) and, up to a new choice of Dˆ on that set, we may assume that it is Σ-stable; in that case, since Σ fixes P, we get 7.5.15

B = ⊕u∈U Dˆ Σ ·u

.

7.6. When b is a nilpotent block of G, Q = {1} fulfills the hypothesis of Theorem 7.5 and, as we prove below, the main result in [12] is an easy consequence of it. That is to say, the next corollary provides a new proof of this main result since, strictly speaking, we never quote it in the proof of Theorem 7.5 (see Remarks 3.6 and 6.4). Corollary 7.7. If b is a nilpotent block then there is a unique isomorphism class of Dade P-algebras S such that we have an O P-interior algebra isomorphism B ∼ = SP. Proof: Choosing Q = {1} and applying Theorem 7.5, it suffices to prove that the O-subalgebra D is isomorphic to a full matrix algebra over O or, equivalently, that k ⊗O D is simple (cf. (7.3) in [16]). We know that B has a simple quotient S of dimension prime to p (cf. (44.9) in [16]); since B∼ = DP, D covers S and therefore S becomes a P-stable quotient of D; that is to say, SP becomes a quotient of k ⊗O B. But, since the structural group homomorphism P → Aut(S) can be lifted to a group homomorphism ρ : P → S∗ and we have SP ∼ = S ⊗k k P (cf. 1.8.1 in [12]), SP has a unique isomorphism class of projective indecomposable modules N and moreover N still is a projective indecomposable k ⊗O B-module (cf. (38.4) in [16]), with a unique isomorphism class of simple k ⊗O B-modules in a JordanHölder sequence. Consequently, N still is the unique projective indecomposable k ⊗O B-module up to isomorphism, and therefore the surjective map k ⊗O B → SP is injective too, so that k ⊗O D ∼ = S.

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7.8. It remains to prove that the families 1.7.1, 1.7.2 and 1.7.3 generate the same subgroup of P; it is already clear that the subgroup generated by one of these families is contained in the subgroup generated by following one, and therefore it suffices to prove the next lemma; in its proof, we make free use of the formalism introduced in the Appendix of [13]. Lemma 7.9. The subgroup of P generated by the union of the commutators
[O p NG (Pγ ) , P] and [O p (X), T ]u , where Tθ runs on a set of representatives E such that Tθ ⊂ Pγ for the G-conjugacy classes of essential pointed groups on OG b, X is the converse image of XG (Tθ ) in NG (Tθ ) and u runs on P, contains [x, U] for any (b, G)-Brauer pair (U, h) such that (U, h) ⊂ (P, e), and any p -element x of NG (U, h). Proof: We may assume that this subgroup is different from P and then consider a normal proper subgroup Q of P containing it; let R be a normal subgroup of P such that Q ⊂ R and |R/Q| = p, and set K =< x >; now, arguing by induction on |P/Q|, we may assume that R already contains [K, U]. Let δ be a local point of U on OG such that b(δ) = h and Uδ ⊂ Pγ ; we claim that, for any ϕ˜ ∈ E G (Uδ , Pγ ), any representative ϕ of ϕ˜ and any v ∈ [K, U], we have ϕ(v)Q = vQ. Indeed, applying Corollary A.12 in [13] to ϕ˜ − ˜ι, where ι : U → P is the inclusion map, and arguing by induction on the number of terms of the corresponding sum, it follows from Lemma A.5 in [13] that we may assume that there is ψ˜ ∈ E G (Uδ , Pγ ) such that ψ(v)Q = vQ for some representaive ψ of ψ˜ and that 7.9.1

˜ T ) ◦ ν˜ = 0 ϕ˜ − ψ˜ = µ ˜ ◦ (σ˜ − id

for some Tθ ∈ E ∪ {Pγ }, some µ ˜ ∈ E NG ( Pγ ) (Tθ , Pγ ), some p -element σ˜ either in EG (Pγ ) if Tθ = Pγ or in X G (Tθ )−M(µ), ˜ and some ν˜ ∈ E G (Uδ , Tθ ); that is to say, we have ϕ(v) = ν(v)ym

and ψ(v) = ν(v)mu ,
 for suitable m ∈ NG (Pγ ), y ∈ O p NG (Tθ ) and u ∈ P, and
some repre sentative ν of ν˜ ; hence, since ψ(v) ∈ R and NG (Pγ ) = O p NG (Pγ ) .P normalizes R and Q, and acts trivially on R/Q, we have ν(v) ∈ R and ν(v)Q = vQ, and therefore, since we are assuming that [y, T ] ⊂ Q, we get 7.9.2

7.9.3

ϕ(v)Q = ν(v)m Q = ν(v)Q = vQ

.

But, according to 2.15.3 in [14], there is z ∈ G such that, for any local point δ of U on OG fulfilling b(δ ) = h, we have (Uδ )z ⊂ Pγ . Hence, for any x ∈ K and any v ∈ [K, U], it follows from equality 7.9.3 that  vx z Q = vQ = vz Q and therefore, since [v, x ] ∈ [K, U], we still  get that [v, x  ]Q = [v, x  ]z Q = Q; that is to say, Q contains K[K, U] = [K, U] (cf. Chap. 5, Th. 3.6 in [6]).

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7.10. Finally, we explain the link with [4]. In that paper, Fan Yun says that G α is locally controlled by Pγ relatively to a normal subgroup Q of P if, for any local pointed group Uδ on B and any element x of G such that (Uδ )x ⊂ Pγ , there are u ∈ P and z ∈ G fulfilling x = zu and [z, U] ⊂ Q (cf. Def. 1.1 in [4]); in that case, for any p -subgroup K of NG (Uδ ), we claim that [K, U] ⊂ Q. Indded, arguing by induction on |P/Q|, we still may assume that a normal subgroup R of P fulfilling Q ⊂ R and |R/Q| = p contains [K, U]; then, if x ∈ K and we have x = zu for some u ∈ P and z ∈ G such that [z, U] ⊂ Q, for any v ∈ [K, U] we obtain [x, v] = u−1 z −1 v−1 zuv = [z, v]u [u, v]   and, since [P, R] ⊂ Q, we get [K, U] = K, [K, U] ⊂ Q (cf. Chap. 5, Th. 3.6 in [6]). Consequently, up to our assumption on k, Theorem 1.1 in [4] follows from Lemma 7.3 above. Similarly, the third statement in Theorem 1.2 follows from Theorem 7.5 above.

7.10.1

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