FROM STABLE EQUIVALENCES TO RICKARD EQUIVALENCES FOR BLOCKS WITH CYCLIC DEFECT

RAPHAËL ROUQUIER

1. Introduction

Let G and H be two finite groups, p a prime number. Let O be a complete discrete valuation ring with residue field k of characteristic p and with field of fractions K of characteristic 0, “big enough” for G and H. Let A and B be two blocks of G and H over O.

Let M be a (A⊗B^{∘})-module, projective as A-module and as B^{∘}-module, where B^{∘} denotes the opposite algebra of B. We denote by M^{∗} the (B ⊗ A^{∘})-module Hom_{O}(M, O).

We say that M induces a stable equivalence between A and B if

M ⊗_{B} M^{∗} ≃ A ⊕ projectives as (A ⊗ A^{∘}) − modules and
M^{∗} ⊗_{A} M ≃ B ⊕ projectives as (B ⊗ B^{∘}) − modules.

Let C be a complex of (A ⊗ B^{∘})-modules, all of which are projective as Amodules and as B^{∘}-modules.

Denoting by C^{∗} the O-dual of C, we say that C induces a Rickard equivalence between A and B if

C ⊗_{B} C^{∗} is homotopy equivalent to A as complexes of (A ⊗ A^{∘}) − modules and

C^{∗} ⊗_{A} C is homotopy equivalent to B as complexes of (B ⊗ B^{∘}) − modules.

By [Ri4, 5.5], from a complex C inducing a Rickard equivalence between A and B, one can construct a module M inducing a stable equivalence between A and B as follows : In the derived bounded category of A ⊗ B^{∘}, the complex C is isomorphic to a complex with only one term which is not projective as (A ⊗ B^{∘})- module, V in degree −n and then the n-th Heller translate (syzygy) M = Ω^{n}(V) induces a stable equivalence between A and B.

The main result of this note is a partial converse under very special assumptions (theorem 6). Since there are well-known situations where a module M induces a stable equivalence between two blocks (remark 9), for example when the Sylow p-subgroups of G are TI, H is the normalizer of a Sylow p-subgroup of G and A, B are principal blocks, it is tempting to try to construct a complex with two terms, M in degree 0 and a projective module in degree −1, inducing a Rickard equivalence between A and B. Using theorem 6, we prove that it is indeed possible when the Sylow p-subgroups of G are cyclic or when G = A_{5} or SL_{2}(8) and p = 2.

2. A criterion for derived equivalences between blocks

2.1. Some lemmas. Let A′ be an O-free O-algebra, finitely generated as an O-module.

If V is an A′-module, let P_{V} be an A′-module which is a projective cover of V. We will denote by Rad(V) the radical of V and by hd(V) the head V/Rad(V) of V, i.e., its largest semi-simple quotient.

If M and N are two A′-modules, we say that M and N are disjoint if they have no non-zero isomorphic direct summands. If M and N are projective, they are disjoint if and only if Hom_{A′}(M, hd(N)) = 0 or equivalently, Hom_{A′}(N, hd(M)) = 0.

If X is an O-module, we define X̄ = X ⊗ k.

Lemma 1. Let P, Q and R be three projective A′-modules and φ : P ⊕ Q ↠ R a surjective morphism. Assume that Q and R are disjoint. Then, the restriction φ_{|P} of φ to P is surjective.

Let U, V and W be three injective Ā′-modules and φ : W ↪ U ⊕ V an injective morphism. Assume that V and W are disjoint. Then, denoting by p_{U} the projection of U ⊕ V onto U, the map p_{U}φ is injective.

Proof. Let h : R → hd(R) be the canonical projection. Since hφ : P ⊕Q ↠ hd(R) is surjective and Hom_{A′}(Q, hd(R)) = 0 by assumption, hφ_{|P} is surjective. Hence, φ(P)+Rad(R) = R and by Nakayama’s lemma, φ(P) = R. The second assertion follows immediately by duality since V and W are disjoint implies that V^{∗} and W^{∗} are disjoint.

Lemma 2. Let M be an (A ⊗ B^{∘})-module, projective as A-module and as B^{∘}- module. A projective cover of M is

⨁_{W} P_{M⊗_{B}W} ⊗ P_{W}^{∗}

where W runs over a complete set of representatives of isomorphism classes of simple B-modules. This module is isomorphic to

⨁_{V} P_{V} ⊗ P_{M^{∗}⊗_{A}V}^{∗}

where V runs over a complete set of representatives of isomorphism classes of simple A-modules.

Proof. Let V be an Ā-module and W a B̄-module. We have

HomB̄^{∘}(M̄, V ⊗ W^{∗}) ≃ HomB̄^{∘}(M̄, V ⊗ W^{∗}) ≃ M̄^{∗} ⊗B̄^{∘} (V ⊗ W^{∗})

since M̄ is projective as B̄^{∘}-module. Hence,

HomB̄^{∘}(M̄, V ⊗ W^{∗}) ≃ (M̄ ⊗B̄ W)^{∗} ⊗ V ≃ Hom_{k}(M̄ ⊗B̄ W, V)

and finally

Hom_{Ā⊗B̄^{∘}}(M̄, V ⊗ W^{∗}) ≃ HomĀ(M̄ ⊗B̄ W, V).

Now, we have

hd(M) ≃ ⨁_{V,W} dim Hom_{A⊗B^{∘}}(M, V ⊗ W^{∗})(V ⊗ W^{∗})

≃ ⨁_{W} (⨁_{V} dim Hom_{A}(M ⊗_{B} W, V)V ) ⊗ W^{∗}

where V (resp. W) runs over the simple A-modules (resp. B-modules) up to isomorphism, hence

hd(M) ≃ ⨁_{W} hd(M ⊗_{B} W) ⊗ W^{∗},

so a projective cover of M is ⨁_{W} P_{M⊗_{B}W} ⊗ P_{W}^{∗} .

To get the second description, replace A by B^{∘} and B by A^{∘} in the first description : a projective cover of M as B^{∘} ⊗(A^{∘})^{∘}-module is ⊕_{V} P_{M⊗_{A^{∘}}V} ⊗P_{V^{∗}} where V runs over the simple A^{∘}-modules. This module is isomorphic to ⊕_{V} P_{M⊗_{A^{∘}}V^{∗}} ⊗P_{V} where V runs over the simple A-modules, hence a projective cover of M as (A⊗B^{∘})-module is ⊕_{V} P_{V} ⊗P_{V^{∗}⊗_{A}M} where V runs over the simple A-modules.

Lemma 3 (Linckelmann, [Li2, 6.8]). Let M be an (A⊗B^{∘})-module inducing a stable equivalence between A and B. Then, M has a unique non-projective direct summand, up to isomorphism.

Proof. Let M = M_{1} ⊕ M_{2}. Since M^{∗} ⊗_{A} M ≃ B ⊕ projectives, we have M^{∗} ⊗_{A} M_{1} ⊕M^{∗} ⊗_{A} M_{2} ≃ B ⊕ projectives as (B ⊗B^{∘})-modules. As B is indecomposable as (B ⊗ B^{∘})-module, there exists i ∈ {1, 2} such that M^{∗} ⊗_{A} M_{i} is projective as (B ⊗ B^{∘})-module, so M ⊗_{B} M^{∗} ⊗_{A} M_{i} is projective as (A ⊗ B^{∘})-module. Now, (M ⊗_{B}M^{∗})⊗_{A}M_{i} ≃ M_{i}⊕ projectives as (A⊗B^{∘})-modules, hence M_{i} is projective as (A ⊗ B^{∘})-module.

Remark 4. A similar proof shows that a complex of (A ⊗ B^{∘})-modules C inducing a Rickard equivalence between A and B has a unique non-homotopy equivalent to zero direct summand, up to isomorphism.

Lemma 5 (Linckelmann, [Li2, 6.3]). Let M be an indecomposable (A ⊗ B^{∘})- module inducing a stable equivalence between A and B. For any simple B-module V, the A-module M ⊗_{B} V is indecomposable.

Proof. (Linckelmann) Denote by soc(Ā) the largest semi-simple Ā-submodule of Ā. Recall that an Ā-module V has no projective direct summand if and only if soc(Ā)V = 0. We have soc(Ā ⊗ B̄^{∘}) = soc(Ā) ⊗ soc(B̄^{∘}). Since M has no projective direct summand, soc(Ā⊗B̄^{∘})M = 0, hence soc(Ā)(M ⊗_{B} soc(B̄)) = 0, which means that M ⊗_{B} soc(B̄) has no projective direct summand. But, if V is a simple B-module, it is a direct summand of soc(B̄), so M ⊗_{B} V has no projective direct summand : as M induces a stable equivalence, M ⊗_{B} V has a unique indecomposable non projective direct summand and the lemma follows.

2.2. The criterion. We denote by R_{K}(A) (resp. R_{K}(B)) the group of characters of KA = K ⊗ A (resp. KB).

Let us now state the main result :

Theorem 6. Let M be an (A ⊗ B^{∘})-module, projective as A-module and as B^{∘}- module. Let δ′ : P′ ↠ M be a projective cover of M. Let P be a direct summand of P′, δ = δ′_{|P} and C = (0 ⟶ P ⟶^{δ} M ⟶ 0) (M is in degree 0). Assume

(a_{1}) M^{∗} ⊗_{A} M ≃ B ⊕ Q where Q is a projective (B ⊗ B^{∘})-module, (a_{2}) M ⊗_{B} M^{∗} ≃ A ⊕ R where R is a projective (A ⊗ A^{∘})-module, (b_{1}) Res_{B^{∘}}^{A⊗B^{∘}} P̄ and Res_{B^{∘}}^{A⊗B^{∘}} P̄′/P̄ are disjoint, (b_{2}) Res_{A}^{A⊗B^{∘}} P̄ and Res_{A}^{A⊗B^{∘}} P̄′/P̄ are disjoint,

(c) KC induces an isometry between R_{K}(A) and R_{K}(B).

Then, C induces a Rickard equivalence between A and B.

Proof. ^{1} Remark first that (b_{1}) implies that

(b′) Res_{B}^{B⊗A^{∘}} P̄^{∗} and Res_{B}^{B⊗A^{∘}} (P̄′/P̄)^{∗} are disjoint.

We have

C^{∗}⊗_{A}C = (0 → M^{∗}⊗_{A}P ⟶^{(δ^{∗}⊗id,id⊗δ)} P^{∗}⊗_{A}P ⊕M^{∗}⊗_{A}M \underrightarrow{( \begin{matrix}id⊗δ \\ −δ^{∗}⊗id \end{matrix} )} P^{∗}⊗_{A}M → 0).

Since KC induces an isometry between R_{K}(A) and R_{K}(B), the character of K(C^{∗} ⊗_{A} C) as (B ⊗ B^{∘})-module is equal to the character of B. Hence,

K(M^{∗} ⊗_{A} P ⊕ P^{∗} ⊗_{A} M) ≃ K(P^{∗} ⊗_{A} P ⊕ Q).

We know that P is a projective (A⊗B^{∘})-module and Res_{B}^{B⊗A^{∘}} M^{∗} is projective, so M^{∗} ⊗_{A} P is projective as (B ⊗ B^{∘})-module. Similarly, P^{∗} ⊗_{A} M, P^{∗} ⊗_{A} P and Q are projective (B ⊗ B^{∘})-modules. Hence

M^{∗} ⊗_{A} P ⊕ P^{∗} ⊗_{A} M ≃ P^{∗} ⊗_{A} P ⊕ Q, and

M̄^{∗} ⊗_{A} P̄ ⊕ P̄^{∗} ⊗_{A} M̄ ≃ P̄^{∗} ⊗_{A} P̄ ⊕ Q̄. (1)

Let Q̄ = Q̄_{1} ⊕ Q̄_{2} where

Res_{B^{∘}}^{B⊗B^{∘}} Q̄_{2} and Res_{B^{∘}}^{A⊗B^{∘}} P̄ are disjoint,

(2)

(3) Res_{B^{∘}}^{B⊗B^{∘}} Q̄_{1} and Res_{B^{∘}}^{A⊗B^{∘}} P̄′/P̄ are disjoint.

1Using an unpublished result of J. Rickard, one can actually prove the theorem without the assumptions (a_{2}) and (b_{2}).

(Since the map pQ̄(id ⊗ δ̄′) : M̄^{∗} ⊗_{A} P̄′ → Q̄ is surjective, every indecomposable direct summand of Res_{B^{∘}}^{B⊗B^{∘}} Q̄ is isomorphic to a direct summand of Res_{B^{∘}}^{B⊗B^{∘}} P̄′, so Q̄_{1} and Q̄_{2} are unique up to isomorphism).

The map pQ̄_{1}(id ⊗ δ̄′) : M̄^{∗} ⊗_{A} P̄′ → Q̄_{1} is surjective and using (3),

Res_{B^{∘}}^{B⊗B^{∘}} Q̄_{1} and Res_{B^{∘}}^{B⊗B^{∘}} (M̄^{∗} ⊗_{A} P̄′) / (M̄^{∗} ⊗_{A} P̄) are disjoint,
hence Q̄_{1} and (M̄^{∗} ⊗_{A} P̄′) / (M̄^{∗} ⊗_{A} P̄) are disjoint

and it follows from lemma 1 that the map

pQ̄_{1}(id ⊗ δ̄) : M̄^{∗} ⊗_{A} P̄ → Q̄_{1}

is surjective.

From (1), Q̄ is isomorphic to a direct summand of M̄^{∗} ⊗_{A} P̄ ⊕ P̄^{∗} ⊗_{A} M̄, hence Q̄_{2} is isomorphic to a direct summand of P̄^{∗} ⊗_{A} M̄ using (2). By (b’),

Res_{B}^{B⊗B^{∘}} (P̄^{∗} ⊗ M̄) and Res_{B}^{B⊗B^{∘}} (P̄^{′∗} ⊗ M̄/P̄^{∗} ⊗ M̄) are disjoint
hence Res_{B}^{B⊗B^{∘}} Q̄_{2} and Res_{B}^{B⊗B^{∘}} (P̄^{′∗} ⊗ M̄/P̄^{∗} ⊗ M̄) are disjoint.

Now, since (δ̄′^{∗} ⊗ id)_{|Q̄_{2}} : Q̄_{2} → P̄^{′∗} ⊗_{A} M̄ is injective, lemma 1 implies that

(δ̄^{∗} ⊗ id)_{|Q̄_{2}} : Q̄_{2} → P̄^{∗} ⊗_{A} M̄

is injective.

Let R̄_{2} be a submodule of P̄^{∗} ⊗_{A} M̄ such that P̄^{∗} ⊗_{A} M̄ = R̄_{2} ⊕ Im(δ̄^{∗} ⊗ id)_{|Q̄_{2}}. We introduce

f_{2} = pR̄_{2}(id ⊗ δ̄) : P̄^{∗} ⊗_{A} P̄ → R̄_{2}
and f′_{2} = pR̄_{2}(id ⊗ δ̄′) : P̄^{∗} ⊗_{A} P̄′ → R̄_{2}

We have P̄^{∗} ⊗_{A} M̄ ≃ Q̄_{2} ⊕ R̄_{2}, so, by (1), as Q̄_{1} is isomorphic to a direct summand of M̄^{∗} ⊗_{A} P̄, the module R̄_{1} is a direct summand of P̄^{∗} ⊗_{A} P̄, hence, by (b_{1}),

Res_{B^{∘}}^{B⊗B^{∘}} R̄_{2} and Res_{B^{∘}}^{B⊗B^{∘}} ((P̄^{∗} ⊗_{A} P̄′)/(P̄^{∗} ⊗_{A} P̄)) are disjoint.

Since f′_{2} is surjective, lemma 1 implies that f_{2} is surjective.

It follows that the map id ⊗ δ̄ − δ̄^{∗} ⊗ id is surjective. By Nakayama’s lemma, the map id ⊗ δ − δ^{∗} ⊗ id is also surjective and hence splits. By duality, the map δ^{∗} ⊗ id + id ⊗ δ is injective and splits. Hence, the complex C^{∗} ⊗_{A} C is homotopy equivalent to B.

Similarly, the complex C ⊗_{B} C^{∗} is homotopy equivalent to A. Hence, the complex C induces a Rickard equivalence between A and B.

2.3. An application. Let M be an indecomposable (A ⊗ B)^{∘}-module inducing a stable equivalence between A and B.

Assume that for every simple A-module V, the head of M^{∗} ⊗_{A} V is simple.

Theorem 7. If there exists a direct summand P of

⨁_{V} P_{V} ⊗ P_{M^{∗}⊗_{A}V}^{∗} ≃ ⨁_{W} P_{M⊗_{B}W} ⊗ P_{W}^{∗}

(V runs over the simple A-modules and W over the simple B-modules) such that 0 → P ⟶^{0} M → 0 induces an isometry between R_{K}(KA) and R_{K}(KB), then there is a complex C = 0 → P → M → 0 inducing a Rickard equivalence between A and B.

Proof. The modules P and M being projective as A-modules and as B^{∘}-modules, the isometry induced by 0 → P ⟶^{0} M → 0 is perfect [Br3, 1.2] and it follows that the algebras A and B have the same number s of isomorphism classes of simple modules [Br3, 1.5].

If V is a simple A-module, the modules P_{V} and P_{M^{∗}⊗_{A}V}^{∗} are indecomposable. Hence, a projective cover of M is a sum of s indecomposable (A ⊗ B^{∘})-modules, which are mutually non-isomorphic when restricted to A or when restricted to B^{∘}. Hence, if P is a direct summand of P′ then Res_{B^{∘}}^{A⊗B^{∘}} P and Res_{B^{∘}}^{A⊗B^{∘}} P′/P are disjoint and Res_{A}^{A⊗B^{∘}} P and Res_{A}^{A⊗B^{∘}} P′/P are disjoint. Now, theorem 6 gives the conclusion.

Let us denote by CF(G, K) the space of class functions G → K, by CF(A, K) the subspace generated by R_{K}(A). We denote by CF_{p}(G, K) (resp. CF_{p′}(G, K)) the subspace of CF(G, K) consisting of class functions which vanish on p-regular (resp. p-singular) elements and CF_{p}(A, K) (resp. CF_{p′}(A, K)) the intersection CF_{p}(G, K) ∩ CF(A, K) (resp. CF_{p′}(G, K) ∩ CF(A, K)).

As the next lemma shows, in the situation of theorem 7, if the map induced by 0 → P ⟶^{0} M → 0 is an isometry on a subspace of CF(A, K) which contains a complement of CF_{p}(A, K), then it is an isometry :

Lemma 8. Let P_{1}, P_{2} be two projective (A ⊗ B^{∘})-modules and C = 0 → P_{1} ⟶^{0} M ⊕ P_{2} → 0. Let I be the map between R_{K}(A) and R_{K}(B) induced by C. Let X be a subspace of CF(A, K) such that CF(A, K) = X + CF_{p}(A, K).

If the restriction of I to X is an isometry, then I is an isometry.

Proof. Let f, g ∈ CF(A, K). We decompose f and g as f = f_{p}+f_{p′} and g = g_{p}+g_{p′} where f_{p}, g_{p} ∈ CF_{p}(A, K) and f_{p′}, g_{p′} ∈ CF_{p′}(A, K). Since I is perfect [Br3, 1.2], I(f_{p}), I(g_{p}) ∈ CF_{p}(B, K) and I(f_{p′}), I(g_{p′}) ∈ CF_{p′}(B, K). Hence, the scalar product of I(f) and I(g) is <I(f), I(g)>=<I(f_{p}), I(g_{p})> + <I(f_{p′}), I(g_{p′})>.

Furthemore, the restriction of I to CF_{p}(A, K) is an isometry because M induces a stable equivalence between A and B and as P_{1} and P_{2} are projective, the map induced by M between R_{K}(A) and R_{K}(B) is equal to I on CF_{p}(A, K) [Br2, 5.3]. It follows that <I(f_{p}), I(g_{p})>=<f_{p}, g_{p}> and we have now to prove that < I(f_{p′}), I(g_{p′}) >=< f_{p′}, g_{p′} >. But, as CF(A, K) = X + CF_{p}(A, K), we can decompose f_{p′} and g_{p′} as f_{p′} = f_{1} + f_{2} and g_{p′} = g_{1} + g_{2} where f_{1}, g_{1} ∈ X and f_{2}, g_{2} ∈ CF_{p}(A, K). Now, <f_{1}, g_{1}>=<f_{p′}, g_{p′}> − <f_{2}, g_{2}> and <I(f_{1}), I(g_{1})>=< I(f_{p′}), I(g_{p′})> − <I(f_{2}), I(g_{2})>. Finally, we know that <I(f_{1}), I(g_{1})>=<f_{1}, g_{1}> and <I(f_{2}), I(g_{2})>=<f_{2}, g_{2}>, hence <I(f_{p′}), I(g_{p′})>=<f_{p′}, g_{p′}>.

Remark 9. Stable equivalences induced by bimodules arise for example in the following situation [Br2, 6.4] :

Assume that H is a subgroup of G with index prime to p and e, f are the units of A and B. Following Broué, let us assume that for every non trivial p-subgroup P of H, we have N_{G}(P) = N_{H}(P)O_{p′}C_{G}(P). Then, the (A ⊗ B^{∘})-module eOGf induces a stable equivalence between A and B. Let M be an indecomposable non-projective direct summand of eOGf ; by lemma 3, such a module is unique up to isomorphism ; we have eOGf = M⊕ projectives, so M induces a stable equivalence between A and B.

Example 1. Let G = SL_{2}(4) = A_{5} and H = A_{4} = 2^{2} ⋊ 3 a Borel subgroup, p = 2. The principal block ekG of G has three simple modules : k, S_{1} and S_{2} of dimension 2. The module Res_{H}^{G}(S_{1}) is a non-split extension of V_{2} by V_{1}, where V_{1} and V_{2} are the two non-trivial non-isomorphic simple kH-modules and Res_{H}^{G}(S_{2}) is a non-split extension of V_{1} by V_{2}. An immediate character calculation shows that

0 → P_{S_{1}} ⊗ P_{V_{1}}^{∗} ⊕ P_{S_{2}} ⊗ P_{V_{2}}^{∗} ⟶^{0} eOG → 0

induces an isometry between R_{K}(eKG) and R_{K}(KH). Hence, by remark 9 and theorem 7, there exists a complex 0 → P_{S_{1}} ⊗ P_{V_{1}} ⊕ P_{S_{2}} ⊗ P_{V_{2}} → eOG → 0 inducing a Rickard equivalence between the principal blocks of G and H, result due to J. Rickard [Ri3].

Example 2. Let G = SL_{2}(8) and H = 2^{3} ⋊7 a Borel subgroup, p = 2. Then, theorem 7 applies also to construct a complex inducing a Rickard equivalence between the principal blocks of G and H : The (A ⊗ B^{∘})-bimodule eOG is indecomposable. We leave to the reader to check that a projective cover of this module is :

P_{1} ⊗ Q_{1}^{∗} ⊕ P_{2_{1}} ⊗ Q_{2_{1}}^{∗} ⊕ P_{2_{2}} ⊗ Q_{2_{2}}^{∗} ⊕ P_{2_{3}} ⊗ Q_{2_{3}}^{∗} ⊕ P_{4_{1}} ⊗ Q_{4_{1}}^{∗} ⊕ P_{4_{2}} ⊗ Q_{4_{2}}^{∗} ⊕ P_{4_{3}} ⊗ Q_{4_{3}}^{∗}

(where P_{1} (resp. Q_{1}) is a projective cover of the trivial A-module (resp. Bmodule), P_{2_{1}}, P_{2_{2}} and P_{2_{3}} (resp. P_{4_{1}}, P_{4_{2}} and P_{4_{3}}) are projective covers of the three non-isomorphic 2-dimensional (resp. 4-dimensional) simple A-modules and Q_{2_{1}}, Q_{2_{2}}, Q_{2_{3}}, Q_{4_{1}}, Q_{4_{2}}, Q_{4_{3}} are projective covers of the six non-isomorphic nontrivial simple B-modules) and that the complex

0 → ⊕P_{4_{1}} ⊗ Q_{4_{1}}^{∗} ⊕ P_{4_{2}} ⊗ Q_{4_{2}}^{∗} ⊕ P_{4_{3}} ⊗ Q_{4_{3}}^{∗} ⟶^{0} eOG → 0

induces an isometry between R_{K}(A) and R_{K}(B), so that by remark 9 and theorem 7, there exists a complex 0 → ⊕P_{4_{1}} ⊗ Q_{4_{1}}^{∗} ⊕ P_{4_{2}} ⊗ Q_{4_{2}}^{∗} ⊕ P_{4_{3}} ⊗ Q_{4_{3}}^{∗} → eOG → 0 inducing a Rickard equivalence between the principal blocks of G and H.

3. Application to principal blocks with cyclic defect

Let G be a finite group with a cyclic Sylow p-subgroup P and let H = N_{G}(P). As before, A = OGe and B = OHf are the principal blocks of G and H, where e and f are primitive idempotents of the centers of OG and OH.

The functor e Ind_{H}^{G} induces a stable equivalence between A and B with inverse functor f Res_{H}^{G} (remark 9).

As conjectured by J. Rickard (cf [Ri2]), a slight modification of these functors leads to a derived equivalence, and this proves in particular the conjecture of Broué and Rickard on abelian defect, for principal blocks with cyclic defect (cf [Br1]) :

Theorem 10. There exists a projective (A ⊗ B^{∘})-module Y and a map ϕ : Y → eOGf such that, if C = 0 ⟶ Y ⟶^{ϕ} eOGf ⟶ 0, then C induces a Rickard equivalence between A and B. In particular, C is a Rickard tilting complex of p-permutation modules.

Note that the fact that A and B are derived-equivalent was already known by the work of Rickard and Linckelman (cf [Ri1] and [Li1]).

3.1. Construction of C.

Let us quote some classical results about A (cf [Gr]).

The set of irreducible characters of KA is Irr(A) = {χ_{1}, …, χ_{e}} ∪ {χ_{λ}}_{λ∈Λ} where χ_{1}, …, χ_{e} are the non-exceptional characters and the χ_{λ}, λ ∈ Λ, are the exceptional characters. (In the case there is only one exceptional character, one can choose it different from the character 1_{G}.)

Define χ_{e+1} = ∑_{λ∈Λ} χ_{λ} and Γ = {χ_{1}, …, χ_{e+1}}.

The Brauer tree T_{A} is then defined as follows :

• the set of its vertices is Γ,

• two vertices v and v′ are incident if and only if v + v′ is the character of a projective indecomposable A-module. We denote by {v, v′} the corresponding edge.

The vertex χ_{e+1} is called the exceptional vertex of T_{A}. Every character of a projective indecomposable A-module is an edge of T_{A} and we have a bijection beween the set of edges of T_{A} and the set of characters of projective indecomposable Amodules. If v and v′ are two vertices of T_{A}, we denote by d(v, v′) the distance between v and v′.

There is a “walk” on T_{A} starting from 1_{G}, the trivial character of G, i.e., a sequence v_{0} = 1_{G}, v_{1}, …, v_{2e} of vertices of T_{A} such that v_{i} is incident with v_{i+1} for 0 ≤ i ≤ 2e − 1, with the following properties :

• Each edge is traversed twice, i.e., denoting by l_{i} the edge {v_{i}, v_{i+1}}, then for every edge l of T_{A}, there exists i and j two distinct integers, 0 ≤ i, j ≤ 2e − 1, such that l = l_{i} = l_{j} ; • denote by P_{i} a projective indecomposable module with character l_{i}. Then, we have a minimal projective resolution of the A-module O, periodic of period 2e :

⋯ → P_{0} → P_{2e−1} → ⋯ → P_{1} → P_{0} → O → 0. (4)

We have v_{2e} = v_{0}. Given three vertices v, v′, v” of T_{A}, we have d(v, v′) + d(v′, v”) ≡ d(v, v”) (mod 2), hence d(v_{i}, v_{0}) ≡ i (mod 2). Suppose l_{i} = l_{j}. Since T_{A} is a tree, we have v_{i} = v_{j+1} and v_{j} = v_{i+1}, hence i ≡ j + 1 (mod 2). It follows that {l_{2i}}_{0≤i≤e−1} is the set of all edges of T_{A}.

If X is an A-module (resp. a B-module) and i an integer, we define Ω_{A}^{i}(X) (resp. Ω_{B}^{i}(X)) to be the i-th Heller translate of X.

The character of Ω_{A}^{i}O is v_{i}.

The block B has a similar description which is a particular case of the previous one :

The Brauer tree of B, T_{B}, is a star whose center is the exceptional vertex, i.e., every edge of T_{B} is of the form {w, w′} where w′ is the exceptional vertex. There is a walk w_{0} = 1_{H}, w_{1}, …, w_{2e} on T_{B} such that :

• Every edge is traversed twice ;

• denote by Q_{i} a projective indecomposable module with character w_{i}+w_{i+1}. Then, we have a minimal projective resolution of the B-module O, periodic of period 2e :

⋯ → Q_{0} → Q_{2e−1} → ⋯ → Q_{1} → Q_{0} → O → 0.

Note that for any i, 0 ≤ i ≤ e−1, w_{2i+1} is the exceptional vertex and {w_{2i}}_{0≤i≤e−1} is the set of all non-exceptional characters of KB. The module Ω_{B}^{2i}O remains irreducible modulo p and its character is w_{2i}.

Since eOGf induces a stable equivalence between A and B, we have eOGf = M ⊕ U as (A ⊗ B^{∘})-modules, where M is indecomposable – and then M̄ is also indecomposable since M is a p-permutation module – and U is projective (cf lemma 3). We still have

M ⊗_{B} M^{∗} ≃ A ⊕ projectives and M^{∗} ⊗_{A} M ≃ B ⊕ projectives.

Since M induces a stable equivalence between A and B, tensoring by M commutes with Heller translates, up to projectives, hence M ⊗_{B} Ω_{B}^{2i}O ≃ Ω_{A}^{2i}O ⊕ projectives. Since M̄ is indecomposable and Ω_{B}^{2i}k is simple, M̄ ⊗B̄ Ω_{B}^{2i}k is indecomposable (cf lemma 5), so that

M ⊗_{B} Ω_{B}^{2i}O ≃ Ω_{A}^{2i}O. (5)

Now, since a projective cover of Ω_{A}^{2i}O is P_{2i} and a projective cover of Ω_{B}^{2i}O is Q_{2i}, it follows from lemma 2 that a projective cover of M is :

⨁_{0≤i≤e−1} P_{2i} ⊗ Q_{2i}^{∗} ⟶^{ψ} M.

For l = {v′, v”} an edge and v a vertex of T_{A}, define δ(l, v) = inf(d(v′, v), d(v”, v)). Let x be an integer, 0 ≤ x ≤ 2e, such that v_{x} is the exceptional vertex of T_{A}.

Let

X = ⨁_{δ(l_{2i},v_{x})≡x (mod 2)} P_{2i} ⊗ Q_{2i}^{∗}

and ϕ be the restriction of ψ to X. We then define D to be 0 ⟶ X ⟶^{ϕ} M ⟶ 0 (where M is in degree 0).

3.2. Proof of theorem 10.

Let i and j be two integers, 0 ≤ i, j ≤ e − 1. We have Q_{2j}^{∗} ⊗_{B} Ω_{B}^{2i}O ≃ Hom_{B}(Q_{2j}, Ω_{B}^{2i}O). Since Q_{2j} is a projective cover of Ω_{B}^{2i}O if and only if i = j, we have

Q_{2j}^{∗} ⊗_{B} Ω_{B}^{2i}O ≃ \begin{cases}O & if i = j, \\ 0 & otherwise. \end{cases}

Hence, we have

X ⊗_{B} Ω_{B}^{2i}O ≃ \begin{cases}P_{2i} & if δ(l_{2i}, v_{x}) ≡ x (mod 2), \\ 0 & otherwise. \end{cases}

It follows from (5) that

D ⊗_{B} Ω_{B}^{2i}O ≃ \begin{cases}0 → 0 → Ω_{A}^{2i}O → 0 & if δ(l_{2i}, v_{x}) ≢ x (mod 2), \\ 0 → P_{2i} → Ω_{A}^{2i}O → 0 & if δ(l_{2i}, v_{x}) ≡ x (mod 2) \end{cases}

where in both cases, Ω_{A}^{2i}O is in degree 0. Let I be the map between the group of characters of B, R_{K}(B), and the ring of characters of A, R_{K}(A), induced by D. By (4), we have :

I(w_{2i}) = \begin{cases}v_{2i} & if δ(l_{2i}, v_{x}) ≢ x (mod 2), \\ −v_{2i+1} & if δ(l_{2i}, v_{x}) ≡ x (mod 2). \end{cases}

Lemma 11. The restriction of the map I to the submodule of R_{K}(B) with basis {w_{0}, w_{2}, …, w_{2(e−1)}} is an isometry.

Proof. We have δ(l_{2i}, v_{x}) ≡ x (mod 2) if and only if δ(l_{2i}, v_{x}) = d(v_{2i}, v_{x}), since d(v_{2i+1}, v_{x}) ≡ x + 1 (mod 2). Hence, δ(l_{2i}, v_{x}) ≡ x (mod 2) if and only if d(v_{2i}, v_{x}) < d(v_{2i+1}, v_{x}). So, I(w_{2i}) is, up to sign, the furthest vertex of l_{2i} from v_{x}. Since T_{A} is a tree, the vertices corresponding to I(w_{2i}) and I(w_{2j}) are equal if and only if w_{2i} = w_{2j}. Note furthermore that I(w_{2i}) is, up to sign, an irreducible character. Hence, the lemma follows.

Corollaire 12. The map I is an isometry.

Proof. Indeed, we have CF (B, K) = K < w_{0}, w_{2}, …, w_{2(e−1)} > ⊕CF_{p}(B, K) and the result is given by lemma 8 and lemma 11.

The following is now a direct consequence of theorem 7 :

Theorem 13. The complex D induces a Rickard equivalence between A and B.

We obtain the exact formulation of theorem 10 by replacing D by 0 ⟶ X ⊕ U ⟶^{δ+id} M ⊕ U ⟶ 0, which is homotopy equivalent to D.

References

[Br1] M. Broué, Rickard equivalences and block theory, this issue (1994).

[Br2] M. Broué, Equivalences of blocks of group algebras, preprint LMENS (1993), to appear in Proceedings of the International Conference on Representations of Algebras, Ottawa, August 1992.

[Br3] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181-182 (1990), 61-92.

[Gr] J.A. Green, Walking around the Brauer tree, J. Aust. Math. Soc. 17 (1974), 197-213.

[Li1] M. Linckelmann, Derived equivalences for cyclic blocks over a p-adic ring, Math. Z. 207 (1991), 293-304.

[Li2] M. Linckelmann, The isomorphism problem for blocks with cyclic defect groups, preprint (1993).

[Ri1] J. Rickard, Derived categories and stable equivalence, J. of Pure and Appl. Alg. 61 (1989), 303-317.

[Ri2] J. Rickard, Conjectural calculation of a two-sided tilting complex for cyclic blocks, preprint (1990).

[Ri3] J. Rickard, Derived equivalences for principal blocks of A_{4} and A_{5}, preprint (1990).

[Ri4] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37-48.

DMI-ENS (CNRS UA 762), 45 rue d’Ulm, 75005 PARIS, FRANCE

E-mail address: Raphael.Rouquier@ens.fr