SOURCE ALGEBRAS AND SOURCE MODULES

J. L. Alperin, Markus Linckelmann, Raphaël Rouquier

May 1998

The aim of this note is to give a self-contained approach in module theoretic terms to two fundamental results in block theory, due to L. Puig [6, 14.6]: first, there is an embedding of the source algebra of the Brauer correspondent of a block of some finite group into a source algebra of that block, and second, the source algebras of the Brauer correspondent can be described explicitly. Our proof of the first result - Theorem 5 and its Corollary 6 below - is essentially the translation to our terminology of the proof in [1, 4.10].

The second result, describing the source algebras of the Brauer correspondent, follows also from [3], but the proofs in [3] use both the main result on the structure of nilpotent blocks in [5] as well as techniques from [6, 4, 6], while our approach - from Proposition 9 onwards - requires only some classical results on the structure of blocks with a central defect group.

An account of the results in [6] on source algebras of blocks with a normal defect group can also be found in [7, 45], except for the explicit description of the central extension Ê (see Section 10 below) as being defined in terms of the multiplicity algebra of the block, which is quoted in [7, (45.10)(b)] without proof, referring back to [6] at the end of [7, 45].

1. Let p be a prime, O a complete local principal ideal domain having a residue field k = O/J(O) of characteristic p; that is, either O is a complete discrete valuation ring or O = k.

Throughout the paper, G is a finite group, b a block of OG and P a defect group of b; that is, b is a primitive idempotent of Z(OG) and P is a p−subgroup of G such that the subgroup ΔP = {(u, u^{−1})}_{u∈P} of G × G^{0} is a vertex of OGb as O(G × G^{0})−module, where G^{0} is the group opposite to G and OG is considered as O(G × G^{0})−module through (x, y).a = xay for any x, y ∈ G and a ∈ OG. Note that OGb has the trivial OΔP − module O as source, since we have an isomorphism of O(G × G^{0})−modules OG ≅ Ind_{ΔG}^{G×G^{0}}(O) mapping x ∈ G to (x, 1) ⊗ 1_{O}.

Definition 2. A source module of the block b of OG is an indecomposable direct summand M of Res_{G×P^{0}}^{G×G^{0}}(OGb) having ΔP as vertex.

Remark 3. Since G × P^{0} contains the vertex ΔP of the O(G × G^{0})−module OGb, the block b has a source module M. The algebra End_{O(G×1)}(M)^{0} is then a source algebra of the block b (cf. [4]), which explains our terminology. Indeed, any direct summand of OGb as left OG−module is of the form OGi for some idempotent i in OGb. Since M is a direct summand of OGb as O(G×P^{0})−module, we have in fact M = OGi for some P −stable idempotent i in OGb; moreover, as M is indecomposable, i is primitive in (OGb)^{P}, and the condition that ΔP be a vertex of M is then equivalent to requiring Br_{P} (i) ≠ 0, where Br_{P} : (OG)^{P} ⟶ kC_{G}(P) is the Brauer homomorphism (cf. [7, (37.5)(b)]). The map sending a ∈ iOGi to right multiplication by a on M is an isomorphism of interior P −algebras iOGi ≅ End_{O(G×1)}(M)^{0}. Note that M is projective as left OGb−module and as right OP −module. The block b can have more than one isomorphism class of source modules, but they are all conjugate under the action of N_{G}(P) in the sense, that if M′ is another source module of b, there is an automorphism φ of P induced by conjugation with an element of N_{G}(P) such that M′ ≅ Res_{Id_{G}×φ}(M). This follows from the interpretation of source modules as actual sources of a module in Remark 6 below.

4. There is a block e of OC_{G}(P) having Z(P) as defect group such that OC_{G}(P)e is isomorphic to a direct summand of Res_{C_{G}(P)×C_{G}(P)^{0}}^{G×G^{0}}(OGb); any other block of OC_{G}(P) with this property is then conjugate to e by some element of N_{G}(P). Equivalently, e is a block of OC_{G}(P) satisfying Br_{P} (be) ≠ 0; that is, (P, e) is a maximal b−Brauerpair.

We set H = N_{G}(P, e), the subgroup of all elements in N_{G}(P) which stabilize e, and E = H/P C_{G}(P), the inertial quotient of the block b (associated with e). Moreover, e is still a block of both OP C_{G}(P) and OH having P as defect group. If k is large enough for OC_{G}(P)e then E is a p′−group (see e.g. [7, (37.12)]).

Observe that N_{G×P^{0}}(ΔP) = ΔP (C_{G}(P)×1) ⊂ H ×P^{0}. Therefore, if we take a source module N of OHe, it follows from the Green correspondence [7, (20.8)], that Ind_{H×P^{0}}^{G×P^{0}}(N) has, up to isomorphism, a unique indecomposable direct summmand M with vertex ΔP and trivial source, and then M is a source module for OGb. The next result makes this more precise:

Theorem 5.

(i) Up to isomorphism, there is a unique indecomposable direct summand X of Res_{G×H^{0}}^{G×G^{0}}(OGb) with vertex ΔP which is isomorphic to a direct summand of

Ind_{H×H^{0}}^{G×H^{0}}(OHe).

(ii) If N is a source module for the block e of OH, then M = X ⊗_{OH} N is a

source module for the block b of OG.

Proof. The existence and uniqueness of X follows from the Green correspondence; similarly, since M is a direct summand of Ind_{H×P^{0}}^{G×P^{0}}(N), the Green correspondence implies that M = M′ ⊕ M′′, where M′ is indecomposable with vertex ΔP and all indecomposable direct summands of M′′ have vertices strictly smaller than ΔP.

But X is also isomorphic to a direct summand of Ind_{G×P^{0}}^{G×H^{0}}(Z) for some indecomposable O(G×P^{0})−module Z with vertex ΔP. Thus, using Mackey’s formula, the O(G × P^{0})−module M = X ⊗_{OH} N is isomorphic to a direct summand of

Res_{G×P^{0}}^{G×H^{0}}Ind_{G×P^{0}}^{G×H^{0}}(Z) = ⊕_{x∈[H^{0}/P^{0}]}^{(1,x)}Z .

Since H normalizes P, the module ^{(1,x)}Z has vertex ^{(1,x)}ΔP for any x ∈ H^{0}. Together with the fact that all indecomposable direct summands of M′′ have vertex strictly smaller than ΔP follows that M = M′, which concludes the proof. □

Remark 6. The Green correspondence does not require the ground ring to be commutative; one might therefore as well consider the group algebra AG^{0} of G^{0} over the ring A = OG. We have an obvious isomorphism AG^{0} ≅ O(G × G^{0}), which allows us to consider OGb as AG^{0}−module. With the notation of Theorem 5, as AG^{0}−module, OGb has vertex P^{0}, the source module M, considered as AP^{0}−module is a source of OGb and the AH^{0}−module X is the Green correspondent of OGb having M as source.

Theorem 5 is a particular case of [1, 4.10], just translated to a module theoretic language: with the notation of the Theorem, let f be a primitive idempotent in (OGb)^{H} such that Br_{P} (ef) ≠ 0 and let j be a primitive idempotent in (OHe)^{P} such that Br_{P} (j) ≠ 0. Then X ≅ OGf and N ≅ OHj. Thus M ≅ OGjf, and the assertion that M is indecomposable is equivalent to the assertion that the idempotent jf is primitive in (OGb)^{P}.

Note that, again by the Green correspondence, the unique indecomposable direct summand of Res_{H×H^{0}}^{G×H^{0}}(X) with vertex ΔP is isomorphic to OHe. From this and Theorem 5 we obtain the embedding of source algebras, due to L. Puig:

Corollary 7. ([6, 14.6], [1, 4.10]) Set A = End_{O(G×1)}(M)^{0} and B = End_{O(H×1)}(N)^{0}. The functor X ⊗_{OH} − induces an injective homomorphism of

interior P −algebras

B ⟶ A

which is split as homomorphism of B − B−bimodules; in particular, we have

rk_{O}(A)/|P | ≡ rk_{O}(B)/|P | (mod p) .

Proof. Since M = X ⊗_{OH} N, tensoring by X induces a homomorphism of interior

P −algebras from B to A. This homomorphism is split injective as homomorphism of B − B−bimodules, because N is isomorphic to a direct summand of Res_{H×P^{0}}^{G×P^{0}}(M) by the usual Green correspondence. Since moreover any other direct summand has a vertex of order strictly smaller than |P |, we get the statement on the O−ranks. □

If k is large enough, it is possible to be much more precise about source modules of OHe and the corresponding source algebras. For this, we recall some standard facts from block theory (see e.g. [7, 39]).

8. Assume that k is large enough for OC_{G}(P)e; that is, the semi-simple quotient of OC_{G}(P)e is isomorphic to a matrix algebra over k. Set \overline{C} = C_{G}(P)/Z(P) and denote by \overline{e} the image of e in O\overline{C} under the canonical surjection OC_{G}(P) ⟶ O\overline{C}.

8.1. The idempotent \overline{e} is a block of O\overline{C} having the trivial subgroup as defect group, and the block algebra O\overline{C}\overline{e} is isomorphic to a matrix algebra M_{n}(O) over O for some positive integer n;

8.2. as block of OC_{G}(P), e has Z(P) as defect group and the block algebra OC_{G}(P)e is isomorphic to the matrix algebra M_{n}(OZ(P)) over OZ(P);

8.3. as block of OP C_{G}(P), e has P as defect group and the block algebra OP C_{G}(P)e is isomorphic to the matrix algebra M_{n}(OP) over OP.

The next proposition is a translation of the statements in 8 to our terminology involving source modules, from which we obtain the source algebras in each of the cases in 8 (as originally determined by L. Puig in [6, 14.5, 14.6]):

Proposition 9. Let \overline{V} be a projective indecomposable O\overline{C}\overline{e}−module, and let V be the O(ΔP (C_{G}(P) × 1))−module obtained by taking a projective cover of \overline{V} as OC_{G}(P)e−module and restricting it through the group homomorphism ΔP (C_{G}(P) × 1) ⟶ C_{G}(P) which maps (uz, u^{−1}) to z for any u ∈ P and any z ∈ C_{G}(P). Set

U = Ind_{ΔP (C_{G}(P)×1)}^{P C_{G}(P)×P^{0}}(V) .

(i) Up to isomorphism, \overline{V} is the unique source module of O\overline{C}\overline{e}, and we have an algebra isomorphism End_{O(\overline{C}×1)}(\overline{V})^{0} ≅ O.

(ii) Up to isomorphism, Res_{C_{G}(P)×Z(P)^{0}}^{ΔP (C_{G}(P)×1))}(V) is the unique source module of OC_{G}(P)e, and the natural map

OZ(P) ⟶ End_{O(C_{G}(P)×1)}(V)^{0}

is an algebra isomorphism.

(iii) Up to isomorphism, U is the unique source module of OP C_{G}(P)e, and the natural map

OP ⟶ End_{O(P C_{G}(P)×1)}(U)^{0}

is an algebra isomorphism.

Proof. Statement (i) is a trivial consequence of the fact that by 8.1 the block algebra O\overline{C}\overline{e} is a matrix algebra over O and has defect group \{1\}.

The restriction to C_{G}(P) of V is indecomposable, since it is a projective cover of the simple module k ⊗_{O} \overline{V}. Thus V and its restriction to C_{G}(P) × Z(P)^{0} are

indecomposable. Moreover, ΔP acts trivially on V. Thus ΔP and ΔZ(P) are vertices of V and its restriction to C_{G}(P)×Z(P)^{0}, respectively. This shows already, that Res_{C_{G}(P)×Z(P)^{0}}^{ΔP (C_{G}(P)×1}(V) is a source module of OC_{G}(P)e. As the normalizer of the defect group Z(P) acts transitively on the isomorphism classes of source modules (cf. Remark 3), this shows also the uniqueness, up to isomorphism in this situation. The natural algebra homomorphism in (ii) is injective, since V is projective as

OZ(P)−module. Since O ⊗_{OZ(P)} End_{O(C_{G}(P)×1)}(V) ≅ End_{O(\overline{C}×1)}(\overline{V}) ≅ O, it

follows from Nakayama’s lemma, that the algebra homomorphism in (ii) is also surjective. This shows (ii).

The natural map OP ⟶ End_{O(P C_{G}(P)×1)}(U)^{0} is O−split injective, since U is projective as O(1 × P^{0})−module. In order to show the surjectivity, it suffices to show that both sides have the same O−rank. By Mackey’s formula, we have Res_{P C_{G}(P)×1}^{P C_{G}(P)×P^{0}}(U) ≅ Ind_{C_{G}(P)×1}^{P C_{G}(P)×1}(V). Thus Frobenius’ reciprocity implies that

End_{O(P C_{G}(P)×1)}(U) ≅ Hom_{O(C_{G}(P)×1)}(V, ⊕_{x}^{(x,1)}V), whith x running over a right

transversal of C_{G}(P) in P C_{G}(P). Now ^{(x,1)}V ≅ V as left OC_{G}(P)−modules for any x ∈ P C_{G}(P), and whence the rank of End_{O(P C_{G}(P)×1)}(U) is [P C_{G}(P) : C_{G}(P)]rk_{O}(End_{O(C_{G}(P)×1)}(V)) = [P : Z(P)]|Z(P)| = |P |. This shows that U is indecomposable, even when restricted to P C_{G}(P) × 1. Moreover, ΔP is a vertex of V, and thus of U, too. Therefore U is a source module for OP C_{G}(P)e. Again by Remark 3, it is unique up to isomorphism. □

Before we go further, we recall some facts about central extensions of a finite group by the group O^{×} of invertible elements in O. The reason for central extensions to occur in this context is the following: the module \overline{V}, viewed as module for P C_{G}(P) via the canonical homomorphism P C_{G}(P) ⟶ P C_{G}(P)/P ≅ C_{G}(P)/Z(P), cannot, in general, be extended to H. It can, however, be extended to a twisted group algebra O_{∗}Ĥ for some central O^{∗}−extension Ĥ of H which arises naturally from the action of H on the matrix algebra O\overline{C}\overline{e}.

10. Assume that k is large enough for the block e of OC_{G}(P). The group H acts on the matrix algebra S = O\overline{C}\overline{e}, since H normalizes P and stabilizes e. Let Ĥ be the subgroup of H × S^{×} of all pairs (x, s) ∈ H × S^{×} satisfying ^{x}t = sts^{−1} for all t ∈ S; that is, such that x and s induce the same automorphism of S. In other words, Ĥ is the pullback of the obvious maps from H and S^{×} to Aut(S).

By the Skolem-Noether theorem, any automorphism of S is inner; in particular, the projection to the first component Ĥ ⟶ H mapping (x, s) to x is surjective. The kernel of this map is {(1_{H}, λ1_{S})}_{λ∈O^{×}} ≅ O^{×}; thus Ĥ is a central extension of H by O^{×}. We consider Ĥ as O^{×}−group endowed with the map O^{×} ⟶ Ĥ sending λ ∈ O^{×} to λ̂ = (1_{H}, λ1_{S}) and call it the O^{×}−group defined by the action of H on S. Its opposite O^{×}−group is the group Ȟ, which is equal to Ĥ as abstract group, endowed with the group homomorphism O^{×} ⟶ Ȟ sending λ ∈ O^{×} to λ̌ = (1_{H}, λ^{−1}1_{S}).

The twisted group algebra O_{∗}Ĥ is the quotient of the group algebra OĤ by the ideal generated by the set of elements λ.x̂ − 1_{O}.(λ̂x̂) with λ running over O^{×} and x̂ running over Ĥ. This is an O−free algebra of O−rank |H|; indeed, if for any x ∈ H we denote by x̂ an element of Ĥ lifting x, then the set {x̂}_{x∈H} is an O−basis of O_{∗}Ĥ, and multiplication is given by x̂ŷ = λ_{x,y}x̂y for any two x, y ∈ H and uniquely determined coefficients λ_{x,y} ∈ O^{×} (this is a two-cocycle determining the central extension Ĥ of H). If the extension Ĥ of H splits; that is, Ĥ ≅ H×O^{×}, any such isomorphism induces an isomorphism of O−algebras O_{∗}Ĥ ≅ OH.

We identify P C_{G}(P) with its canonical image in Ĥ (via the inclusion P C_{G}(P) ⊂ H and the obvious homomorphism P C_{G}(P) ⟶ P C_{G}(P)/P ≅ C_{G}(P)/Z(P) ⟶ S^{×}; see [6, 6.5]) and set Ê = Ĥ/P C_{G}(P). Similarly, we set Ě = Ȟ/P C_{G}(P). The O^{×}−groups Ê and Ě are central O^{×}−extensions of E.

The projection to the second component Ĥ ⟶ S^{×} mapping (x, s) to s induces a surjective algebra homomorphism O_{∗}Ĥ ⟶ O\overline{C}\overline{e} which extends the canonical homomorphism OP C_{G}(P)e ⟶ O\overline{C}\overline{e}. Note that this homomorphism maps the elements of P to the unit element of O\overline{C}\overline{e}. In particular, any O\overline{C}\overline{e}−module can be viewed as O_{∗}Ĥ/P −module through this homomorphism. Note that k ⊗_{O} \overline{V} is,

up to isomorphism, the unique simple O\overline{C}\overline{e}−module, which necessarily remains simple when considered as O_{∗}Ĥ/P −module.

Since E is a p′−group, the inverse image of E in Aut(P) has Int(P) as normal Sylow-p−subgroup, thus has a complement isomorphic to E, unique up to conjugacy (cf. [7, (45.1)]), which we are going to denote again by E. We denote by P ⋊ E the corresponding semi-direct product. The O^{×}−group Ě acts still on P via the canonical surjection Ě → E, and we denote again by P ⋊ Ě the corresponding semi-direct product, viewed as O^{×}−group in the obvious way; that is, with the O^{×}−group structure coming from Ě. We use the sign ∘ to denote the opposite product of a group.

Proposition 11. Let τ : Ĥ ⟶ H and π : Ĥ ⟶ Ê be the canonical surjections.

(i) The product map (τ, π) : Ĥ ⟶ H × Ê is injective, and its image consists of all pairs (x, ê) ∈ H × Ê with the property that the natural images of x and ê in Aut(P)/Int(P) coincide; equivalently, the triple (Ĥ, τ, π) is the pullback of the natural maps from H and Ê to Aut(P)/Int(P).

(ii) Set L̂ = N_{H×(P ⋊Ě)^{0}}(ΔP). For any (x, u ∘ ě) ∈ L̂, where x ∈ H, u ∈ P, ě ∈ Ě, there is a unique x̂ = α(x, u ∘ ě) ∈ Ĥ such that τ(x̂) = x and π(x̂)^{−1} = ě, and this map is a surjective homomorphism of O^{×}−groups

α : L̂ ⟶ Ĥ

which maps ΔP to P and whose kernel is 1 × Z(P)^{0}.

Proof. (i) Since ker(τ) ≅ O^{×} and ker(π) = P C_{G}(P), the product map (τ, π) is injective, and for any x̂ ∈ Ĥ, the images of τ(x̂) and π(x̂) in Aut(P)/Int(P) coincide. Let x ∈ H and ê ∈ Ê such that the images of x and ê in Aut(P)/Int(P) coincide. This means that the class ê ∈ Ê = Ĥ/P C_{G}(P) can be represented by an element of the form x̂ = (x, s) of Ĥ for some s ∈ S^{×}, which proves (i).

(ii) An element (x, u ∘ ě) ∈ H × (P ⋊ Ě)^{0} normalizes ΔP if x and u^{−1}ě^{−1}, the latter now viewed as element of P ⋊ Ê, induce the same automorphism of P. In particular, the images of x and ě^{−1} in Aut(P)/Int(P) coincide, which by (i) implies that there is a unique x̂ ∈ Ĥ satisfying τ(x̂) = x and π(x̂)^{−1} = ě. ¿From this uniqueness follows also that the map α thus defined is a group homomorphism. Conversely, given x̂ ∈ Ĥ, the elements x = τ(x̂) ∈ H and ê = π(x̂) ∈ Ê have same image in Aut(P)/Int(P), which means that there is u ∈ P such that (x, u ∘ ê^{−1}) normalizes ΔP. This shows the surjectivity of α. Moreover, if in this case both x = 1 and ê = 1, then u ∈ Z(P), which implies that ker(α) = 1 × Z(P)^{0}. Finally, α maps (u, u^{−1}) ∈ ΔP to the unique element û of Ĥ satisfying τ(û) = u and π(û) = 1 which forces û = u. □

The group homomorphism α in the previous proposition allows us to consider any O_{∗}Ĥ/P −module as O_{∗}L̂/ΔP −module. In particular, the source module \overline{V} for O\overline{C}\overline{e} from Proposition 9 becomes an O_{∗}L̂/ΔP −module in this way. The next Proposition shows, that the OΔ(C_{G}(P) × 1))−module V from Proposition 9 extends to L̂.

Proposition 12. Let W be a projective cover of \overline{V} viewed as O_{∗}L̂/ΔP −module, and consider then W as O_{∗}L̂−module through the canonical map L̂ ⟶ L̂/ΔP. Let V be the OΔP (C_{G}(P) × 1)−module as in Proposition 9. We have

ResL̂ ; ΔP (C_{G}(P)×1)(W) ≅ V .

Proof. As left OC_{G}(P)−module, V is a projective cover of the simple module k ⊗_{O} \overline{V}; in particular, V has a unique maximal submodule rad(V). Since k ⊗_{O} \overline{V} is

still simple as O_{∗}L̂/ΔP −module, W has a unique maximal submodule rad(W), too, ResL̂ ; ΔP (C_{G}(P)⊗1)(W) has a direct summand isomorphic to V and we have

ResL̂ ; ΔP (C_{G}(P)×1)(W/rad(W)) ≅ V/rad(V) ≅ k ⊗_{O} \overline{V} .

This isomorphism implies that rad(W) is also a maximal submodule of

ResL̂ ; ΔP (C_{G}(P)×1)(W).

The group homomorphism α : L̂ ⟶ Ĥ from Proposition 11.(ii) induces an isomorphism L̂/ΔP (C_{G}(P) × 1) ≅ Ĥ/P C_{G}(P) ≅ Ê. As E is a p′−group, we have therefore J(O_{∗}L̂) = O_{∗}L̂.J(OΔP (C_{G}(P) × 1)). This means that rad(W) is also the radical and thus the unique maximal submodule of ResL̂ ; ΔP (C_{G}(P)×1)(W); in particular, the latter is indecomposable and whence isomorphic to V. □

We determine now the source modules of OHe and obtain from this description the structure of the source algebras of OHe as given by Puig in [6, 14.6] (extending earlier work of B. Külshammer on blocks with normal defect groups [2]). We keep the notation introduced above.

Theorem 13. Set N = Ind_{P C_{G}(P)×P^{0}}^{H×P^{0}}(U), where U is a source module for OP C_{G}(P)e.

(i) Up to isomorphism, N is the unique source module for OHe.

(ii) We have N ≅ Res_{H×P^{0}}^{H×(P ⋊Ě)^{0}}IndH×(P ⋊Ě)^{0} ; L̂(W).

(iii) As right O_{∗}(P ⋊ Ě)−module, IndH×(P ⋊Ě)^{0} ; L̂(W) is free of rank n.

(iv) (Puig [6, 14.6]) We have an isomorphism of interior P −algebras

End_{O(H×1)}(N)^{0} ≅ O_{∗}(P ⋊ Ě) ;

in particular, as an O−algebra, OHe is isomorphic to the matrix algebra M_{n}(O_{∗}(P ⋊ Ě)).

Proof. Since U is an indecomposable direct summand of OP C_{G}(P)e as O(P C_{G}(P) × P^{0})−module having ΔP as vertex, the module N is isomorphic to a direct summand of OHe as O(H × P^{0})−module having any source module of OHe as direct summand.

By Proposition 9 we have N ≅ Ind_{ΔP (C_{G}(P)×1)}^{H×P^{0}}(V), and by Proposition 12 together with Mackey’s formula, we get

N ≅ Ind_{ΔP (C_{G}(P)×1)}^{H×P^{0}}ResL̂ ; ΔP (C_{G}(P)×1)(W) ≅ Res_{H×P^{0}}^{H×(P ⋊Ě)^{0}}IndH×(P ⋊Ě)^{0} ; L̂(W) .

In particular, any such isomorphism shows that N has a right O_{∗}(P ⋊ Ě)− module structure, or equivalently, any such isomorphism induces a homomorphism of interior P −algebras

13.1. O_{∗}(P ⋊ Ě) ⟶ End_{O(H×1)}(N)^{0}

mapping a ∈ O_{∗}(P ⋊ Ě) to the endomorphism of N given by the action of a on N. Another application of Mackey’s formula shows that

Res_{1×(P ⋊Ě)^{0}}^{H×(P ⋊Ě)^{0}}IndH×(P ⋊Ě)^{0} ; L̂(W) ≅ Ind_{1×Z(P)^{0}}^{1×(P ⋊Ě)^{0}}ResL̂ ; 1×Z(P)^{0}(W)

is free as right O_{∗}(P ⋊ Ě)−module by 12 and has therefore rank n by 8.2; in particular, the homomorphism 13.1 is O−split injective. In order to show that 13.1 is an isomorphism it suffices to show that both sides have the same O−rank.

By an appropriate version of Frobenius’ reciprocity, we have an O−linear isomorphism End_{O(H×1)}(N) ≅ Hom_{O(P C_{G}(P)×1)}(U, Res_{P C_{G}(P)×1}^{H×P^{0}}(N)), and Mackey’s

formula implies that Res_{P C_{G}(P)×1}^{H×P^{0}}(N) ≅ ⊕_{x}^{(x,1)}U, where x runs over a set of

representatives in H of H/P C_{G}(P). The direct summands ^{(x,1)}U in this sum are all isomorphic to U as left OP C_{G}(P)−modules since the elements of H stabilize e and since OP C_{G}(P)e has only one isomorphism class of projective indecomposable modules (cf. 8.3). This shows that rk_{O}(End_{O(H×1)}(N)) = |E|rk_{O}(End_{O(P C_{G}(P)×1)}(U)) = |E||P |, where the last equality comes from 9(iii). Therefore 13.1 is an isomorphism. Since we know already that every source module of OHe occurs as direct summand, up to isomorphism, of N, all we have to show is that N is indecomposable as O(H ×P^{0})−module, or equivalently, that the unit element of O_{∗}(P × Ě) is primitive in (O_{∗}(P ⋊ Ě))^{P}. This is well-known, and there are various possibilities to show this: one way is to observe that the rank of any source algebra of OHe is at least |P ||E| because for any x ∈ H, any source algebra has a direct summand isomorphic to O[P x] as O(P × P^{0})−module. A more direct proof goes as follows, involving the Brauer construction [7, 11]: first, (O_{∗}(P ⋊ Ě))(P) ≅ OZ(P) and so O_{∗}(P ⋊ Ě) has a unique block and a unique local point of P whose multiplicity is then necessarily 1; second, E is a p′−group acting on the semi-simple quotient of O_{∗}(P ⋊ Ě)^{P} permuting transitively the indecomposable direct algebra summands (because any E−orbit yields an E−stable idempotent which lifts to a block of O_{∗}(P ⋊ Ě), whence must be the unit element). Combining these observations shows that the unit element is primitive in O_{∗}(P ⋊Ě)^{P}. Yet another proof of this fact follows from decomposing O_{∗}(P ⋊Ě) ≅ ⊕_{e∈E}OP e as O(P × P^{0})−module; the direct summands in this decomposition are

indecomposable, mutually non isomorphic and transitively permuted by the action of E. Thus O_{∗}(P ⋊ Ě) is indecomposable as O_{∗}((P ⋊ Ě) × P^{0})−module, showing again that the unit element is primitive in (O_{∗}(P ⋊Ě))^{P}. This concludes the proof of Theorem 13. □

References

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J. L. Alperin University of Chicago Department of Mathematics 5734 University Avenue Chicago, Illinois 60637

U.S.A.

Markus Linckelmann

and Raphaël Rouquier

CNRS, Université Paris 7 UFR de Mathématiques 2, place Jussieu 75251 Paris Cedex 05

FRANCE