THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS

RAPHAËL ROUQUIER

1. Introduction

Let O be a complete discrete valuation ring with algebraically closed residue field k of characteristic p > 0 and field of fractions K of characteristic 0, “big enough” for the groups considered.

Let G be a finite group, OGe a block of G with a cyclic defect group D. Let N = N_{G}(D) and ONe′ the block of N corresponding to OGe.

In this work, we give a “self-contained” account of the theory of blocks with cyclic defect groups with aim the proof that the derived category of a block with cyclic defect is (in a strong sense) locally determined — a special case of Broué’s conjecture [Br1] :

Theorem 1.1. The blocks OGe and ONe′ are splendidly Rickard equivalent. If O_{p}(G) ≠ 1, then these blocks are in addition Morita equivalent.

Some remarks.

• That the blocks should be not merely Rickard equivalent, but also splendidly Rickard equivalent, is Rickard’s refinement of Broué’s conjecture.

• When O_{p}(G) ≠ 1, there need not be a splendid Morita equivalence !

• The block ONe′ has a very simple structure : it is Morita equivalent to OD ⋊ E, where E = N_{G}(D, e_{D})/C_{G}(D) and (D, e_{D}) is an e-subpair (Proposition 2.15).

Let us outline the organization of the proof of the theorem.

In §3, we prove that a stable equivalence between OGe and OD ⋊ E (the very local case) can always be lifted in a nice way to a Rickard equivalence. To elucidate part of the structure of OGe, we follow very closely Thompson’s original approach [Th], translated to the setting of an abstract stable equivalence of Morita type using Broué’s philosophy [Br3]. Ideas of Green [Gr], Alperin [Alp] and Linckelmann [Li4] enable us to understand well enough OGe and the stable equivalence to be left with a problem which is solved with (a simplification of) the ideas of [Rou].

Theorem 1.1 is then proved in §4 by induction on the order of G : it is enough to give a splendid Rickard equivalence between OGe and the corresponding block OHf for H = N_{G}(R), where O_{p}(G) < R ≤ D and [R : O_{p}(G)] = p, since by induction OHf is splendidly Rickard equivalent to ONe′.

First, we consider the case where O_{p}(G) = 1. By induction, we know that OHf is Morita equivalent to OD ⋊ E. Now, Green’s correspondence gives a stable equivalence between OGe and OHf, hence between OGe and OD ⋊ E, which can be lifted to a Rickard equivalence by the results of §3. We have then constructed a splendid Rickard equivalence between OGe and OHf.

Date: October 1997.

The author thanks for its hospitality the Isaac Newton Institute for Mathematical Sciences, Cambridge, where this paper has been written.

Assume now O_{p}(G) ≠ 1 but O_{p}(G) is central in G. Then, the blocks have a unique simple module and we have already a Rickard equivalence for the blocks of the groups modulo O_{p}(G). This equivalence turns out to be nice enough to be lifted to an equivalence between OGe and OHf.

Finally when O_{p}(G) ≠ 1 but O_{p}(G) is not central in G, we extend the known Rickard equivalence between the blocks of C_{G}(O_{p}(G)) and C_{H}(O_{p}(G)), following Marcus [Mar].

In §2, we give various general results (i.e., not specific to blocks with cyclic defect) pertaining to §3 and §4.

In the final part, we illustrate the constructions with an explicit study of P SL_{2}(p).

As a final remark, let us say that the structure of blocks with cyclic defect has been determined originally by Dade [Da, Fe]. The existence of a Rickard equivalence is due to Rickard [Ri1] and Linckelmann [Li2] (the Morita equivalence when O_{p}(G) ≠ 1 goes back to Linckelmann and Puig [LiPu, Li1]).

My thanks go to M. Broué for many a patient answer to my questions and to G. Robinson for useful discussions.

2. Miscellany : stable equivalences, Rickard equivalences and more

2.1. Notations. Let G be a finite group, e a block idempotent of OG (i.e., a primitive idempotent of the center of OG) and A = OGe the corresponding block of OG. By an A-module, we mean a finitely generated left A-module. The sign ⊗ means ⊗_{O}. For V an O-module, we put V^{∗} = Hom(V, O). We write KA and kA for the algebras K ⊗ A and k ⊗ A. We denote by A^{∘} the O-algebra opposite to A.

Let H be another finite group, f a block idempotent of OH and B = OHf.

2.2. Stable category, stable equivalences and invariants. We follow here [Br2, Br3] for the main concepts associated with the stable category. We denote by A − \overline{mod} the stable category of A, whose objects are the A-modules. Let us recall that a relatively O-projective A-module is an A-module of the form A ⊗_{O} U for some O-module U. Given two A-modules V and W, the set of morphisms \overline{Hom}(V, W) in A − \overline{mod} is Hom(V, W) modulo the submodule of O-projective morphisms, i.e., morphisms which factor through a relatively O-projective module.

Let Ω_{A⊗A^{∘}} be the kernel of the multiplication map A ⊗ A^{∘} → A. Let Ω_{A⊗A^{∘}}^{−1} be the cokernel of the map η_{A} : A → A ⊗ A^{∘} dual to the multiplication. These are (A ⊗ A^{∘})-modules whose restrictions to A and A^{∘} are projective. The functors Ω_{A⊗A^{∘}} ⊗_{A} − and Ω_{A⊗A^{∘}}^{−1} ⊗_{A} − induce inverse self-equivalences of A − \overline{mod}. Given an A-module V, we denote by ΩV an A-module without projective direct summand such that Ω_{A⊗A^{∘}} ⊗_{A} V ≃ ΩV in A − \overline{mod} — such a module is unique up to isomorphism.

Let f : V → W be an injection between A-modules. There exists a morphism ϕ : W → (A ⊗ A^{∘}) ⊗_{A} V such that we have a commutative diagram :

[figure: V; f; η_{A}⊗1; W; ϕ; (A ⊗ A^{∘}) ⊗_{A} V]

Let δ be the induced map W/V → Ω_{A⊗A^{∘}}^{−1} ⊗_{A} V. This gives rise to a “standard distinguished triangle” :

V →^{f} W → W/V ⇝^{δ} Ω_{A⊗A^{∘}}^{−1} ⊗_{A} V.

The category A−\overline{mod} becomes a triangulated category, with translation functor Ω_{A⊗A^{∘}} ⊗_{A} − and distinguished triangles the triangles isomorphic to the standard distinguished triangles.

We denote by R^{p}(A) the kernel of the decomposition map R(KA) → R(kA). We denote by R̄(A) the quotient of R(KA) by the submodule generated by the characters of the projective Amodules. Note that the restriction of the usual bilinear form on R(KA)×R(KA) to R^{p}(KA)× R(KA) factors through R^{p}(KA) × R̄(A) and gives a perfect pairing between R^{p}(KA) and ×R̄(A).

Let M be an (A ⊗ B^{∘})-module, projective as an A-module and as a B^{∘}-module.

The module M induces a stable equivalence of Morita type between A and B if

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

Since we are dealing with blocks, Rickard has noted that it is enough to check one of the properties :

Lemma 2.1. Let M be an (A ⊗ B^{∘})-module, projective as an A-module and as a B^{∘}-module. If A is not a matrix algebra over O and

M ⊗_{B} M^{∗} ≃ A ⊕ projective modules,

then M induces a stable equivalence between A and B.

Proof. Since A is not a matrix algebra over O, it follows that A is not a projective (A ⊗ A^{∘})- module, hence M is not projective.

The functors M ⊗_{B} − and M^{∗} ⊗_{A} − between B −\overline{mod} and A−\overline{mod} are left and right adjoint. Consider the associated natural maps η : B → M^{∗} ⊗_{A} M and ε : M^{∗} ⊗_{A} M → B. By [Mac, §IV.1, Theorem 1], the map

1 ⊗ η : M ⊗_{B} B → M ⊗_{B} M^{∗} ⊗_{A} M

is a split injection. As M ⊗_{B} M^{∗} ⊗_{A} M is isomorphic to M in (A ⊗ B^{∘}) − \overline{mod}, the map 1 ⊗ η is an isomorphism in (A ⊗ B^{∘}) − \overline{mod}. Similarly, the map

1 ⊗ ε : M ⊗_{B} M^{∗} ⊗_{A} M → M ⊗_{B} B

is an isomorphism in (A ⊗ B^{∘}) − \overline{mod}. It follows that the composite (1 ⊗ ε)(1 ⊗ η) = 1 ⊗ (εη) is an automorphism of M in (A ⊗ B^{∘}) − \overline{mod}. Since B is a block, the algebra End_{B⊗B^{∘}}(B) is local. If the composite εη is not invertible, it is in the radical of this algebra, hence gives an element in the radical of \overline{End}_{B⊗B^{∘}}(B). This can’t happen, since after applying M ⊗_{B} − we get a non-zero isomorphism ! Hence, εη is invertible and as a consequence, M^{∗} ⊗_{A} M ≃ B ⊕ P for some module P.

Now,

M ⊗_{B} (M^{∗} ⊗_{A} M) ≃ M ⊕ M ⊗_{B} P
and (M ⊗_{B} M^{∗}) ⊗_{A} M ≃ M ⊕ projective modules,

hence M ⊗_{B} P is projective. As P is isomorphic to a direct summand of M^{∗} ⊗_{A} M ⊗_{B} P, we finally deduce that P is projective.

Assume we have a module M inducing a stable equivalence between A and B.

Then, the functors M⊗_{B}− and M^{∗}⊗_{A}− induce inverse equivalences of triangulated categories between the stable categories B − \overline{mod} and A − \overline{mod}. The maps R(KA) → R(KB) and R(KB) → R(KA) induced by M ⊗_{B} − and M^{∗} ⊗_{A} − give rise to

• inverse isometries between R^{p}(A) and R^{p}(B) and

• inverse isomorphisms between R̄(A) and R̄(B).

Lemma 2.2. Let V be an indecomposable non relatively O-projective B-module. Let W be an A-module without projective direct summands such that

M ⊗_{B} V ≃ W ⊕ relatively O-projective modules.

Then, W is indecomposable.

If V is free over O and V ⊗ k is indecomposable, then, W ⊗ k is indecomposable.

Proof. Assume W = W_{1} ⊕ W_{2}, with W_{1}, W_{2} ≠ 0. Then,

V ⊕ relatively O-projective modules ≃ M^{∗} ⊗_{A} W_{1} ⊕ M^{∗} ⊗_{A} W_{2}.

As M^{∗} ⊗_{A} W_{1} and M^{∗} ⊗_{A} W_{2} contain both a non relatively O-projective direct summand, we get a contradiction.

Assume now V ⊗ k is indecomposable and W ⊗ k has a non-zero projective direct summand. So, there is a projective A-module P and a surjective map W ↠ P ⊗ k. As W is free over O, we have Ext^{1}(W, P) = 0, hence the surjection lifts to a map W → P, which is forced to be surjective by Nakayama’s lemma, hence which splits, contradicting the property of W to have no projective direct summand.

The following very useful result comes from [Li4] :

Lemma 2.3. The (A ⊗ B^{∘})-module M is the direct sum of a non-projective indecomposable module and of a projective module.

If M is indecomposable, then for any simple B-module V, the A-module M ⊗_{B} V is indecomposable.

Proof. Let M = M_{1} ⊕ M_{2}. Since M^{∗} ⊗_{A} M ≃ B ⊕ projective modules, we have

M^{∗} ⊗_{A} M_{1} ⊕ M^{∗} ⊗_{A} M_{2} ≃ B ⊕ projective modules.

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

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

2.3. Derived category and Rickard equivalences. The definition of a splendid Rickard equivalence follows [Ri4]. We denote by D^{b}(A) the bounded derived category of A and by K^{b}(A) the homotopy category of bounded complex of A-modules. Viewing a module as a complex concentrated in degree 0, we have a fully faithful functor from the category of A-modules to D^{b}(A) and to K^{b}(A).

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

The complex C induces a Rickard equivalence between A and B (or is a Rickard complex) if

C ⊗_{B} C^{∗} ≃ A ⊕ complex homotopy equivalent to 0
C^{∗} ⊗_{A} C ≃ B ⊕ complex homotopy equivalent to 0.

Assume we have such a complex C. Then, the functors C ⊗_{B} − and C^{∗} ⊗_{A} − induce inverse equivalences of triangulated categories between the categories D^{b}(B) and D^{b}(A) and between the categories K^{b}(A) and K^{b}(B).

They give rise to inverse “perfect” isometries between R(KA) and R(KB), to inverse isomorphisms between the centers of A and B,... [Br1].

Suppose the complex C has homology only in one degree, isomorphic to M. If C is a Rickard complex, then M induces a Morita equivalence between A and B. The converse doesn’t hold : if M induces a Morita equivalence between A and B, we deduce that C ⊗_{B} C^{∗} ≃ A ⊕ R, where R has zero homology, but might be non homotopy equivalent to 0. Nevertheless, if C has only one non-zero term, M, then C induces a Rickard equivalence if and only if M induces a Morita equivalence.

As for stable equivalences, Rickard [Ri4] has proved the following :

Lemma 2.4. Let C be a bounded complex of (A ⊗ B^{∘})-modules, all of which are projective as A-modules and as B^{∘}-modules. If

C ⊗_{B} C^{∗} is homotopy equivalent to A,

then C induces a Rickard equivalence between A and B.

(The proof is similar to the proof of Lemma 2.1)

In D^{b}(A ⊗ B^{∘}), the complex C is isomorphic to a bounded complex of modules which are all projective except the degree n term N, for some integer n. The module M = Ω^{n}N induces a stable equivalence between A and B [Ri3, Corollary 5.5].

Lemma 2.5. Let C be a bounded complex of (A⊗B^{∘})-modules, all of whose terms are projective but the degree 0 term M, which is projective as an A-module and as a B^{∘}-module. Assume M induces a stable equivalence between A and B. Then, A is a direct summand of C ⊗_{B} C^{∗} and B is a direct summand of C^{∗} ⊗_{A} C.

Proof. The functors C ⊗_{B} − and C^{∗} ⊗_{A} − between K^{b}(B) and K^{b}(A) are left and right adjoint and induce natural maps η : B → C^{∗} ⊗_{A} C and ε : C^{∗} ⊗_{A} C → B. Let η′ : B → M^{∗} ⊗_{A} M and ε′ : M^{∗} ⊗_{A} M → B be the natural maps induced by the functors M ⊗_{B} − and M^{∗} ⊗_{A} −. The composite maps εη and ε′η′ have the same image in \overline{End}_{B⊗B^{∘}}(B). As M induces a stable equivalence between A and B, the map ε′η′ is invertible, hence εη is invertible. It follows that B is isomorphic to a direct summand of C^{∗} ⊗_{A} C. The second property is obtained by a similar proof.

Assume A and B have a common defect group D. When the modules occurring in C are direct summands of permutation modules induced from ΔD = {(x, x)|x ∈ D} ⊂ G×H^{∘}, the complex C is called splendid and the equivalence of derived categories it induces is called a splendid Rickard equivalence. Such an equivalence gives rise to derived equivalences for corresponding blocks of centralizers of p-subgroups of D in G and H ([Ri4], [Ha] ; cf also [Pu],[Li3]).

Following a suggestion of M. Harris, we have

Lemma 2.6. Assume H′ ⊆ H are two subgroups of G. Let f′ be a block idempotent of H′. Assume e, f and f′ have defect D. Let C be a splendid Rickard complex for OGe⊗(OHf)^{∘} and C′ a splendid Rickard complex for OHf ⊗ (OH′f′)^{∘}. Then, C ⊗_{OHf} C′ is a splendid Rickard complex for OGe ⊗ (OH′f′)^{∘}.

Proof. Composing Rickard equivalences gives a Rickard equivalence. The point is to check that the modules in C ⊗_{OHf} C′ have the required properties.

Let M = Ind_{ΔQ}^{G×H^{∘}} O ≃ OG⊗_{OQ} OH be the trivial module induced from a diagonal subgroup ΔQ to G × H^{∘}, where Q ≤ D. Let N = Ind_{ΔR}^{H×H′^{∘}} O, where R ≤ D. We have

M ⊗_{OH} N ≃ OG ⊗_{OQ} OH ⊗_{OR} OH′ ≃ Ind_{Q×R^{∘}}^{G×H′^{∘}} Res_{Q×R^{∘}}^{H×H^{∘}} Ind_{ΔH}^{H×H^{∘}} O.

By Mackey’s formula, we obtain

M ⊗_{OH} N ≃ ⨁_{(h_{1},h_{2})∈Q×R^{∘}\H×H^{∘}/ΔH} Ind_{(ΔH)^{(h_{1},h_{2})}∩(Q×R^{∘})}^{G×H′^{∘}} O

≃ ⨁_{h∈Q\H/R} Ind_{Δ(Q^{h}∩R)}^{G×H′^{∘}} O

It follows that M ⊗_{OH} N is a direct sum of direct summands of permutations modules induced from ΔD to G × H′^{∘}.

Homotopy equivalence is detected by a Sylow p-subgroup :

Lemma 2.7. Let G′ be a subgroup of G of index prime to p. Let g : X → Y be a morphism between two bounded complexes of OG-modules. Then, g is a homotopy equivalence if and only if Res_{G′}^{G} g is a homotopy equivalence.

Proof. Let Z be the cone of g. Then, g is a homotopy equivalence if and only Z is homotopy equivalent to 0. Assume Z is not homotopy equivalent to 0. Up to homotopy, we can assume Z = ⋯ Z_{r−1} ⟶^{α} Z_{r} → 0, where α is not a split surjection. So, α gives a non-zero element in Ext_{OG}^{1}(Z_{r}, ker α). As [G : G′] is prime to p, the restriction map Ext_{OG}^{1}(Z_{r}, ker α) → Ext_{OG′}^{1}(Res_{G′}^{G} Z_{r}, Res_{G′}^{G} ker α) is injective, hence the restriction to G′ of α doesn’t split, i.e., the restriction to G′ of Z is not homotopy equivalent to 0 and the restriction to G′ of g is not a homotopy equivalence.

The following lemma due to A. Marcus [Mar] shows how to solve (certain) extension problems for Rickard complexes :

Lemma 2.8. Let G′ and H′ be normal subgroups of G and H with G/G′ = H/H′ = E. Assume e and f are block idempotents of OG′ and OH′. Let C′ be a Rickard complex for eOG′ and fOH′. Assume C′ extends to a complex C̃ of OL-modules, where L = {(g, h) ∈ G × H^{∘}|(gG′, hH′^{∘}) ∈ ΔE}.

If E has order prime to p or if C has only one non-zero term, then, C = Ind_{L}^{G×H^{∘}} C̃ is a Rickard complex for eOG and fOH.

Proof. We have

Res_{G×H′^{∘}}^{G×H^{∘}} C = Res_{G×H′^{∘}}^{G×H^{∘}} Ind_{L}^{G×H^{∘}} C̃ ≃ Ind_{G′×H′^{∘}}^{G×H′^{∘}} C′

by Mackey’s formula. Similarly,

Res_{G′×H^{∘}}^{G×H^{∘}} C ≃ Ind_{G′×H′^{∘}}^{G′×H^{∘}} C′.

Consequently,

Res_{G′×G^{∘}}^{G×G^{∘}} C ⊗_{OH} C^{∗} ≃ C′ ⊗_{OH′} OH ⊗_{OH} C^{∗} ≃ C′ ⊗_{OH′} C′^{∗} ⊗_{OG′} OG

is homotopy equivalent to eOG′ ⊗_{OG′} OG ≃ eOG.

In particular, C ⊗_{OH} C^{∗} has homology only in degree 0, M, with Res_{G′×G^{∘}}^{G×G^{∘}} M ≃ eOG. The natural map g : eOG → C ⊗_{OH} C^{∗} induces a map g_{0} = H_{0}(g) : eOG → M. As the restriction of M to OG^{∘} is isomorphic to eOG, the OG^{∘}-submodule of M generated by g_{0}(e), the unit of M = End_{K^{b}(OH^{∘})}(C), must be isomorphic to eOG, hence g_{0} is an isomorphism.

If the index of G′ × G^{∘} in G × G^{∘} is prime to p, it follows that g is a homotopy equivalence, since its restriction to G′ × G^{∘} is a homotopy equivalence (Lemma 2.7). Similarly, one proves that C^{∗} ⊗_{OG} C is homotopy equivalent to fOH and the lemma follows.

Let us finally consider normal p-subgroups.

Let R be a normal p-subgroup of G. We put Ḡ = G/R. For M an OG-module, we put M̄ = OḠ ⊗_{OG} M and for ϕ : M → N a morphism of OG-modules, we put ϕ̄ = 1 ⊗ ϕ : M̄ → N̄.

Lemma 2.9. Let ϕ : M → N be a morphism between OG-modules. Then, ϕ is surjective if and only if ϕ̄ is surjective.

Let N be an OG-module whose restriction to OR is projective. Then,

• every projective direct summand of N̄ lifts to a projective direct summand of N,

• if M is a projective cover of N, then M̄ is a projective cover of N̄.

Proof. Assume ϕ̄ is surjective. Then, N = ϕ(M)+rad(OR)N, hence N = ϕ(M) by Nakayama’s lemma and ϕ is surjective.

Note that given a projective indecomposable OG-module N, the projective OḠ-module N̄ is indecomposable, and we obtain a bijection between the set of isomorphism classes of projective indecomposable OG-modules and the set of isomorphism classes of projective indecomposable OḠ-modules.

Let M be a projective OG-module and f : N → M̄ a surjection. Let M_{0} be the kernel of the canonical surjective map M → M̄. Then, M_{0} is a direct summand of Ind_{R}^{G} Res_{R}^{G} M_{0}. In particular,

Ext_{OG}^{1}(N, M_{0}) ≤ Ext_{OG}^{1}(N, Ind_{R}^{G} Res_{R}^{G} M_{0}) ≃ Ext_{OR}^{1}(Res_{R}^{G} N, Res_{R}^{G} M_{0}).

Since Res_{R}^{G} N is projective, this last Ext-group is zero, hence the morphism f lifts to a morphism ϕ : N → M. By the first part of the lemma, this morphism is surjective, hence split surjective, since M is projective.

If ϕ : M → N is a projective cover of N, then ϕ̄ is a surjective map. Assume there is a direct summand M′ of M̄ such that the restriction of ϕ̄ to M′ is surjective. Using the second part of the lemma, we can lift M′ to a direct summand M′′ of M, and the first part of the lemma shows that the restriction of ϕ to M′′ is surjective. So, M = M′′ and M̄ = M′, i.e., M̄ is a projective cover of N̄.

Homotopy equivalence between OG-modules can sometimes be controlled by OḠ :

Lemma 2.10. Let n be an integer and ϕ : X → Y be a morphism between two bounded complexes of OG-modules whose components are direct sums of indecomposable modules with trivial source and vertices Q such that |Q ∩ R| = n.

Then, ϕ is a homotopy equivalence if and only if ϕ̄ is a homotopy equivalence.

Proof. Let P be a Sylow p-subgroup of G. Let Q be a subgroup of P. We have

Res_{P}^{G} Ind_{Q}^{G} O ≃ ⨁_{g∈P \G/Q} Ind_{P ∩Q^{g}}^{P} O

and |(P ∩ Q^{g}) ∩ R| = |Q ∩ R|. Since Ind_{P ∩Q^{g}}^{P} O is indecomposable (it has a unique simple quotient), every direct summand of Res_{P}^{G} Ind_{Q}^{G} O has a vertex Q′ such that |Q′ ∩ R| = |Q ∩ R|. Hence, the assumptions of the lemma hold if we restrict the modules from G to P and replace G by P. By Lemma 2.7, we can then assume that G = P.

As in the proof of Lemma 2.7, we are reduced to the following problem. Let ϕ : M → N be a morphism between OG-modules whose components are direct sums of indecomposable modules with trivial source and vertices Q such that |Q ∩ R| = n. Assume ϕ̄ is a split surjection. Then, we have to show that ϕ is a split surjection. By Lemma 2.9, we know already that ϕ is a surjection.

We may of course assume that N is indecomposable. But, this implies that N has a unique simple quotient, hence that N̄ has a unique simple quotient and is therefore indecomposable. If M = M_{1} ⊕ M_{2}, then the restriction of ϕ̄ to M̄_{1} or to M̄_{2} is a split surjection : so, we can assume M, hence M̄, are indecomposable as well. Then, ϕ is a split surjection if and only if it is an isomorphism, i.e., if and only if M and N have the same rank. Let Q and Q′ be subgroups of G such that M ≃ Ind_{Q}^{G} O and N ≃ Ind_{Q′}^{G} O. We have rank M̄ = [G : RQ] and rank N̄ = [G : RQ′]. Since ϕ̄ is an isomorphism, we have |RQ| = |RQ′|. But, by assumption, |R ∩ Q| = |R ∩ Q′|, hence |Q| = |Q′| and ϕ is an isomorphism.

Under good circumstances, normal p-subgroups can be factored out, in order to check that a complex induces a Rickard equivalence :

Lemma 2.11. Let R be a common normal p-subgroup of G and H and C a bounded complex of (OGe ⊗ (OHf)^{∘})-modules, each of which is a direct sum of indecomposable modules with trivial source and vertices Q such that Q ∩ (1 × H^{∘}) = Q ∩ (G × 1) = 1 and R × R^{∘} ≤ (R × 1)Q = (1 × R^{∘})Q. Let ē and f̄ be the images of e and f through the canonical morphisms OG → OḠ and OH → OH̄ and C̄ = OḠē ⊗_{OG} C ⊗_{OH} OH̄f̄.

Then, C is a Rickard complex for OGe and OHf if and only C̄ is a Rickard complex for OḠē and OH̄f̄.

Proof. Let Q be a p-subgroup of G × H^{∘} such that Q ∩ (1 × H^{∘}) = Q ∩ (G × 1) = 1 and R × R^{∘} ≤ (R × 1)Q = (1 × R^{∘})Q. By Mackey’s formula, we have

Res_{R×R^{∘}}^{G×H^{∘}} Ind_{Q}^{G×H^{∘}} O ≃ ⨁_{g∈R×R^{∘}\G×H^{∘}/Q} Ind_{(R×R^{∘})∩Q^{g}}^{R×R^{∘}} O

and Q′ = (R × R^{∘}) ∩ Q^{g} satisfies Q′ ∩ (1 × H^{∘}) = Q′ ∩ (G × 1) = 1 and R × R^{∘} = (R × 1)Q′ = (1 × R^{∘})Q′. Let N = Ind_{Q′}^{R×R^{∘}} O. We have

O ⊗_{OR} N ⊗_{OR} O ≃ Hom_{OR⊗(OR)^{∘}}(N, O) ≃ O.

We have

Res_{R×1}^{R×R^{∘}} N ≃ Ind_{1}^{R×1} O

since R×R^{∘} = (R×1)Q′ and (R×1)∩Q′ = 1. It follows that O ⊗_{OR} N ≃ O. So, the canonical surjective map

O ⊗_{OR} N → O ⊗_{OR} N ⊗_{OR} O

is an isomorphism. Similarly, the canonical surjective map

N ⊗_{OR} O → O ⊗_{OR} N ⊗_{OR} O

is an isomorphism. So, if M is a direct summand of Ind_{Q}^{G×H^{∘}} O, the canonical maps give isomorphisms of O(G × H^{∘})-modules

M ⊗_{OR} O ≃ O ⊗_{OR} M ≃ O ⊗_{OR} M ⊗_{OR} O.

Consequently, we have

OḠ ⊗_{OG} C ⊗_{OH} C^{∗} ⊗_{OG} OḠ ≃ OḠ ⊗_{OG} C ⊗_{OH} OH̄ ⊗_{OH} C^{∗} ⊗_{OG} OḠ ≃ C̄ ⊗_{OH̄} C̄^{∗}

and

OH̄ ⊗_{OH} C^{∗} ⊗_{OG} C ⊗_{OH} OH̄ ≃ C̄^{∗} ⊗_{OḠ} C̄.

The components of C ⊗_{OH} C^{∗} are direct sums of direct summands of modules

Ind_{Q_{1}}^{G×H^{∘}} O ⊗_{OH} (Ind_{Q_{2}}^{G×H^{∘}} O)^{∘} ≃ OG ⊗_{OQ_{1}} OH ⊗_{OQ_{2}} OG

where Q_{i} ∩ (G × 1) = Q_{i} ∩ (1 × H^{∘}) = 1 and R × R^{∘} ≤ (R × 1)Q_{i} = (1 × R^{∘})Q_{i} for i ∈ {1, 2}.

Let ϕ : G × H^{∘} → G and ψ : G × H^{∘} → H^{∘} be the canonical projections. Then, we have isomorphisms Q_{i} → ϕ(Q_{i}) and Q_{i} → ψ(Q_{i}) and R is contained in ϕ(Q_{i}) and ψ(Q_{i}).

We have

Res_{ψ(Q_{1})×ψ(Q_{2})^{∘}}^{H^{∘}×H} OH ≃ ⨁_{g∈ψ(Q_{1})×ψ(Q_{2})^{∘}\H×H^{∘}/ΔH} Ind_{(ψ(Q_{1})×ψ(Q_{2})^{∘})∩(ΔH)^{g}}^{ψ(Q_{1})×ψ(Q_{2})^{∘}} O.

Note that, for Q′ = (ψ(Q_{1}) × ψ(Q_{2})^{∘}) ∩ (ΔH)^{g}, we have |Q′ ∩ (R × R^{∘})| = |ΔR ∩ (ψ(Q_{1}) × ψ(Q_{2})^{∘})^{g^{−1}}| = |R|. Hence,

Ind_{Q_{1}}^{G×H^{∘}} O ⊗_{OH} (Ind_{Q_{2}}^{G×H^{∘}} O)^{∘} ≃ Ind_{Q_{1}×Q_{2}^{∘}}^{G×G^{∘}} Res_{ψ(Q_{1})×ψ(Q_{2})^{∘}}^{H^{∘}×H} OH

is a direct sum of modules Ind_{Q′′}^{G×G^{∘}} O, where |Q′′ ∩ (R × R^{∘})| = |R|. Similarly, the components of C^{∗} ⊗_{OG} C have trivial source and vertices Q′′ such that |Q′′ ∩ (R × R^{∘})| = |R|.

Let η : OGe → C ⊗_{OH} C^{∗} and η′ : OHf → C^{∗} ⊗_{OG} C be the natural maps induced by the functors C ⊗_{OHf} − and C^{∗} ⊗_{OGe} −. Then, η̄ = 1 ⊗_{OR} η ⊗_{OR} 1 : OḠē → C̄ ⊗_{OH̄} C̄^{∗} and η̄′ = 1⊗_{OR}η′⊗_{OR}1 : OH̄f̄ → C̄^{∗}⊗_{OḠ} C̄ are the natural maps induced by the functors C̄ ⊗_{OH̄f̄}− and C̄^{∗} ⊗_{OḠē} −. By Lemma 2.10, η (resp. η′) is a homotopy equivalence if and only if η̄ (resp. η̄′) is a homotopy equivalence.

2.4. Some more lemmas. For a module V, we denote by P_{V} a projective cover of V.

Lemma 2.12. Let M be an (A ⊗ 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 Bmodules.

Proof. Let V be a simple A-module and W a simple B-module. We have an isomorphism of (A ⊗ A^{∘})-modules

Hom_{B^{∘}}(M, V ⊗ W^{∗}) ≃ Hom_{O}(M ⊗_{B} W, V)

given by f ↦ (m ⊗_{B} w ↦ f(m)(w)) and the lemma follows from the isomorphism

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

The following well-known lemma solves the problem of lifting modules through cyclic p′- extensions.

Lemma 2.13. Let H be a normal subgroup of G with E = G/H a cyclic p′-group. Let M be a G-stable OH-module. Then, there exists an OG-module M̃ extending M and for any such module, we have

Ind_{H}^{G} M ≃ Res_{ΔG}^{G×G^{∘}}(M̃ ⊗ OE).

Proof. Let g ∈ G generating G/H. Since M is G-stable, there exists ϕ ∈ End_{O}(M) such that

ϕ(g^{−1}hg(m)) = hϕ(m) for all h ∈ H and m ∈ M.

Let γ = ϕ^{e}g^{−e}, where e = |E|, and R be the subring of End_{OH}(M) generated by γ.

Suppose there is α ∈ R such that α^{e} = γ. Let ψ = α^{−1}ϕ. Then, ψ^{e} acts on M as g^{e} and ψ(g^{−1}hg(m)) = hψ(m) for h ∈ H and m ∈ M. It follows that we can extend the action of H on M to an action of G by letting g act as ψ.

The existence of α follows from the fact that R is a finite algebra over the strictly henselian ring O, hence the etale extension R[X]/(X^{e} − γ) of R must be trivial : first, replacing R by one of its blocks, we can assume it is local. Then, the equation α^{e} = γ has e distinct roots in the residue field of R. By Hensel’s lemma, these solutions can be lifted to R and we are done.

Let M̃ be an OG-module extending M. Then,

Res_{ΔG}^{G×G^{∘}} Ind_{G×H^{∘}}^{G×G^{∘}}(M̃ ⊗ O) ≃ Ind_{H}^{G} Res_{H}^{G} M̃

by Mackey’s formula.

Let us recall some basic definitions of local block theory and some properties related to Brauer’s first main theorem (see for example [Alp, §IV]).

Assume H is a subgroup of G. We say that the block OGe corresponds to OHf if OGe is a direct summand of Ind_{H×H^{∘}}^{G×G^{∘}} OHf.

Let D be a defect group of OGe. When N_{G}(D) ≤ H, there is a unique block idempotent f of OH such that OGe corresponds to OHf : OHf is called a Brauer correspondent of OGe.

If R is a p-subgroup of D, e_{R} is a block of ORC_{G}(R) (equivalently, a block of OC_{G}(R)) and OGe corresponds to ORC_{G}(R)e_{R}, then (R, e_{R}) is called an e-subpair.

Assume furthermore R is normal in G and e_{R} is G-stable. Then, e_{R} = e, D ∩ RC_{G}(R) is G-conjugate to a defect group of ORC_{G}(R)e_{R} and p |̸[G : DC_{G}(R)].

The next two lemmas deal with blocks of groups having a normal p-subgroup.

Lemma 2.14. Let R be a normal p-subgroup of G, (R, f) an e-subpair and H its normalizer. Then, OGf induces a Morita equivalence between OGe and OHf.

Proof. We have ef = f and (g^{−1}fg)f = 0 for g ∈ G − H. The multiplication map fOG ⊗_{OGe} OGf → fOGf is an isomorphism and

fOGf = ∑_{g∈G/H} OHgg^{−1}fgf = OHf.

The lemma is then a consequence of Lemma 2.4.

Proposition 2.15. Assume a defect group D of OGe is normal in G and e is a block of OC_{G}(D). Let E = G/DC_{G}(D). Assume E is cyclic. Then, OGe is Morita equivalent to OD ⋊ E.

Proof. Let ē be the image of e in O(C_{G}(D)/Z(D)). The canonical map from the center of OC_{G}(D)e to the center of O(C_{G}(D)/Z(D))ē is onto. Since O is complete, this forces ē to be a block of O(C_{G}(D)/Z(D)). This block has defect zero, hence has a unique simple module. So, OC_{G}(D)e has a unique simple module.

Let H = D ⋊ E and L = N_{G×H^{∘}}(ΔD). The map C_{G}(D) → G × H^{∘}, x ↦ (x, 1) factors through L and gives an injection φ : C_{G}(D) → L/ΔD with cokernel isomorphic to E.

The G-stable block e of OC_{G}(D) has a unique simple module V, which is consequently G-stable. The action of C_{G}(D) on V lifts to an action of L/ΔD on V (Lemma 2.13). Let P_{V} be a projective cover of V as an O(L/ΔD)-module and Q the restriction of P_{V} to L. Let M = Ind_{L}^{G×H^{∘}} Q and P = Res_{C_{G}(D)×Z(D)^{∘}}^{L} Q. Then, P has vertex ΔZ(D) and P̄ = O(C_{G}(D)/Z(D)) ⊗_{C_{G}(D)} P ⊗_{Z(D)} O is a projective indecomposable O(C_{G}(D)/Z(D))ē ⊗ Omodule, hence it induces a Morita equivalence between O(C_{G}(D)/Z(D))ē and O. It follows from Lemma 2.11 that P induces a Morita equivalence between OC_{G}(D)e and OZ(D) and from Lemma 2.8 that M induces a Morita equivalence between OGe and OH.

What this proposition actually determines is a source algebra of the block. Note that instead of assuming E cyclic, one can assume the blocks are principal to get the same result. In general, a similar proof shows that OGe is Morita equivalent to a twisted group algebra O_{∗}D ⋊ Ê when D is normal in G, as proven by L. Puig [Thé, §45, theorem 12].

3. Blocks stably equivalent to OD ⋊ E

Let G be a finite group, e a block of G with positive defect. Let E be a p′-subgroup of the group of automorphisms of a non-trivial cyclic p-group D. Let A = OGe and B = OD ⋊ E. Let M be an indecomposable (A ⊗ B^{∘})-module which is projective as an A-module and as a B^{∘}-module.

This section is devoted to the proof of

Theorem 3.1. Assume M induces a stable equivalence between A and B. Then, there exists a direct summand N of a projective cover p : P_{M} ↠ M of M such that the complex 0 → N ⟶^{p_{|N}} M → 0 induces a Rickard equivalence between A and B.

If E = 1, then N = 0 or N = P_{M}, i.e., M or ΩM induces a Morita equivalence between A and B.

Let Ψ : R(KB) → R(KA) and Θ : R(KA) → R(KB) be the maps induced by the functors M ⊗_{B} − and M^{∗} ⊗_{A} −.

3.1. Exceptional characters. An exceptional character of B is defined to be the character of an irreducible KB-module with a non-trivial D-action. A non-exceptional character of B is an irreducible character which is not exceptional. Let θ_{x} be the sum of the exceptional characters of B. The simple B-modules are the simple OE-modules. They lift uniquely to O-free B-modules, with projective covers the indecomposable projective B-modules. If V is a one-dimensional O-free OE-module, then its projective cover as a B-module is Ind_{E}^{H} V, whose character is the character of V plus θ_{x}. Recall also that the indecomposable kB-modules are uniserial.

The set {θ̄}_{θ exceptional} is a basis of R̄(B) (we denote by x̄ the image of x ∈ R(KB) under the canonical morphism R(KB) → R̄(B)).

Assume B has at least two exceptional characters. An irreducible character ψ of A is called exceptional if there exists two exceptional characters θ, θ′ of B such that ⟨ψ, Ψ(θ) − Ψ(θ′)⟩ ≠ 0. Note that θ − θ′ ∈ R^{p}(B), hence ||Ψ(θ) − Ψ(θ′)|| = 2. So, the number of exceptional characters of A and B are the same. The rank of R^{p}(B) is this number and R^{p}(A) and R^{p}(B) have the same rank. It follows that R^{p}(A) is not generated by linear combinations of exceptional characters, hence A has at least one non-exceptional character.

We denote by ψ_{x} the sum of the exceptional characters of A.

When B has a unique exceptional character θ_{x}, then we pick an irreducible character ψ_{x} = ψ_{θ_{x}} of A such that ψ̄_{x} ≠ 0 and we call it exceptional (actually, any irreducible character of A will do).

For the non-exceptional characters, we have

Lemma 3.2. Let ψ be a non-exceptional irreducible character of A. Then, \overline{Θ(ψ)} is a multiple of θ̄_{x}.

Proof. Let θ, θ′ be two exceptional characters of B. We have ⟨Θ(ψ), θ −θ′⟩ = ⟨ψ, Ψ(θ −θ′)⟩ = 0. We are now done since θ̄ = −θ̄_{x}.

3.2. Decomposition numbers.

Lemma 3.3. Let V be a non-projective O-free A-module such that V ⊗k is indecomposable. Let W be a non-projective indecomposable B-module such that M^{∗}⊗_{A}V ≃ W ⊕ projective modules. Let ζ be the character of W. Then,

(i) ζ is multiplicity free (ii) ζ contains at most one non-exceptional character (iii) if ζ contains a non-exceptional character, then it doesn’t contain all exceptional characters.

Proof. By Lemma 2.2, W ⊗k is indecomposable and non projective. Since W ⊗k is indecomposable, it is the quotient of an indecomposable projective kB-module. Hence, W is the quotient of an indecomposable projective B-module. It follows that the character ζ of W is multiplicity free and contains at most one non-exceptional character. Since W ⊗ k is not projective, we know in addition that if ζ contains a non-exceptional character, then it doesn’t contain all exceptional characters.

Lemma 3.4. Let χ be a character of A and P an indecomposable projective A-module such that χ is contained in the character of P.

Then, there exists an O-free A-module V with character χ and projective cover P. In particular, V ⊗ k is indecomposable.

Proof. Let L′ be a KA-submodule of K ⊗P such that K ⊗P/L′ has character χ. Let L = L′∩P and V = P/L. Then, V has the required properties.

As a consequence of Lemmas 3.2, 3.3 and 3.4, we obtain :

Corollary 3.5. Let ψ be a non-exceptional character of A. Then, \overline{Θ(ψ)} = ±θ̄_{x}.

If χ and χ′ are two characters, we say that χ is contained in χ′ if χ′ − χ is a character.

Proposition 3.6. Let P be an indecomposable projective A-module. Then, its character is the sum of two distinct non-exceptional characters or the sum of a non-exceptional character and of all exceptional characters.

Proof. Let η be the character of P. Assume there are non-exceptional characters ψ and ψ′ such that ψ + ψ′ is strictly contained in η. By Lemma 3.4, there is an O-free A-module V with character ψ + ψ′, such that V ⊗ k is indecomposable. By Lemma 3.5, \overline{Θ(ψ)} + \overline{Θ(ψ′)} is zero or ±2θ̄_{x}. By Lemma 3.3, the second possibility can’t arise. Now, assume \overline{Θ(ψ)} + \overline{Θ(ψ′)} = 0. Let W be a B-module such that M^{∗} ⊗_{A} V ≃ W ⊕ projective modules, with W ⊗k indecomposable. Then, the character of W is 0 in R̄(B). By Lemma 3.3, this is impossible.

We assume now that η contains at most one non-exceptional character. Let θ, θ′ be two distinct exceptional characters of B. We have

⟨η, Ψ(θ) − Ψ(θ′)⟩ = ⟨Θ(η), θ − θ′⟩ = 0.

Let ψ and ψ′ be the two distinct exceptional characters of A such that Ψ(θ) − Ψ(θ′) = ±ψ ± ψ′. Then, ⟨η, ψ⟩ = ⟨η, ψ′⟩, so ψ and ψ′ arise with the same multiplicity in η. It follows that there is a positive integer α such that χ = η − αψ_{x} is zero or a non-exceptional character.

Assume χ = 0. Then the block A has a projective indecomposable module with character η′ different from η such that ⟨η, η′⟩ ≠ 0. The character η′ is a non-zero multiple of ψ_{x} plus a non-exceptional character ψ. But this implies that ψ̄ = 0, which is impossible ! Hence, χ is a non-exceptional character.

Now, \overline{Θ(η)} = 0 = ±θ̄_{x} + α\overline{Θ(ψ_{x})} by Lemma 3.5. This implies α = 1 and \overline{Θ(ψ_{x})} = ±θ̄_{x} and we are done.

3.3. The Brauer tree and its walk.

Lemma 3.7. Let L be a simple B-module. Then, the module M ⊗_{B} L has a unique simple quotient V and the correspondence L ↦ V induces a bijection between the sets of isomorphism classes of simple B-modules and simple A-modules.

Proof. Let V be a simple A-module and W = M^{∗} ⊗_{A} V. Then, W is indecomposable (Lemma 2.3), hence it has a unique simple submodule L_{V}. So, we have a map h from the set of isomorphism classes of simple A-modules to the set of isomorphisms classes of simple B-modules given by V ↦ L_{V}.

Let now L be a simple B-module and U = M ⊗_{B} L. Let V be a simple A-module which is a quotient of the indecomposable module U. Then,

\overline{Hom}(L, M^{∗} ⊗_{A} V) ≃ \overline{Hom}(U, V) ≠ 0.

It follows that the map h is surjective. Let now V_{1} and V_{2} be two simple A-modules such that W_{1} = M^{∗} ⊗_{A} V_{1} and W_{2} = M^{∗} ⊗_{A} V_{2} have the same simple submodule L. Since an injective hull of L is uniserial, there is an injection W_{i} → W_{j} for some i, j with {i, j} = {1, 2}. Such an injection between modules with no projective direct summand is not an O-projective morphism, hence \overline{Hom}(W_{i}, W_{j}) ≠ 0, so \overline{Hom}(V_{i}, V_{j}) ≠ 0 and V_{i} and V_{j} are isomorphic. This proves the injectivity of h.

Since h is bijective, given a simple B-module L, the module M ⊗_{B} L has a unique simple quotient.

The set {Ω^{2i}k}_{0≤i≤e−1} is a complete set of simple kB-modules (up to isomorphism). By Lemma 2.3, the module M ⊗_{B} Ω^{2i}k is indecomposable. Hence, M ⊗_{B} Ω^{2i}O is indecomposable as well. On the other hand, M ⊗_{B} − commutes with Heller translation up to projective modules, i.e., M ⊗_{B} Ω^{2i}O ≃ Ω^{2i}S ⊕ projective modules, where S = M ⊗_{B} O. It follows that M ⊗_{B} Ω^{2i}O ≃ Ω^{2i}S.

Proposition 3.8. The character of Ω^{i}S is a non-exceptional character or the sum of the exceptional characters.

Proof. Let χ be the character of Ω^{i}S.

When i is even, Ω^{i}S has a unique simple quotient, hence Ω^{i}S is a quotient of a projective indecomposable A-module P_{i}. In particular, χ is contained in the character of P_{i}.

Assume now i is odd. Then, we have an exact sequence

0 → Ω^{i}S → P_{i−1} → Ω^{i−1}S → 0.  (1)

Again, we see that χ is contained in the character of a projective indecomposable module.

We have χ̄ = ±\overline{Ψ(θ_{x})} since the character of Ω^{i}O is non-exceptional or equal to θ_{x}. Hence, χ̄ = ±ψ̄_{x} and we get the conclusion from Proposition 3.6.

Let us now define the Brauer tree T of A. The set of vertices is {ψ}_{ψ non exceptional} ∪ {ψ_{x}}. The vertices ψ and ψ′ are incident if ψ + ψ′ is the character of an indecomposable projective module. This defines a graph whose number of edges is the number of isomorphism classes of simple A-modules. By Proposition 3.6, this is a tree. The vertex corresponding to ψ_{x} is called exceptional.

Let v_{i} be the vertex corresponding to the character of Ω^{i}S. Then, there is an edge l_{i} connecting v_{i} and v_{i+1}, due to the exact sequence (1). The set {l_{2i}} is the set of all edges of T .

Note that Ω^{2e}O ≃ O, where e is the order of |E|. It follows that v_{2e+i} = v_{i} and l_{2e+i} = l_{i}.

For 0 ≤ i ≤ e − 1, let v′_{i} ∈ {v_{2i}, v_{2i+1}} be the further vertex of l_{2i} from the exceptional vertex. Let I be the set of non-negative integers i ≤ e − 1 such that v′_{i} = v_{2i+1}.

Note that {v′_{i}} is the set of non-exceptional vertices of T .

3.4. Construction of the complex. By Lemma 2.12, a projective cover of the (A ⊗ B^{∘})- module M is

⨁_{0≤i≤e−1} P_{Ω^{2i}S} ⊗ P_{Ω^{2i}O}^{∗}.

Let

N = ⨁_{i∈I} P_{Ω^{2i}S} ⊗ P_{Ω^{2i}O}^{∗}

and let

C = 0 → N ⟶^{ϕ} M → 0

where M is in degree 0 and ϕ is the restriction to N of a surjection ⨁_{0≤i≤e−1} P_{Ω^{2i}S}⊗P_{Ω^{2i}O}^{∗} → M. For 0 ≤ i, j ≤ e − 1, we have

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

Hence, we have an isomorphism in D^{b}(A)

C ⊗_{B} Ω^{2i}O ≃ \begin{cases}0 → Ω^{2i+1}S → 0 → 0 & if i ∈ I, \\ 0 → 0 → Ω^{2i}S → 0 & otherwise \end{cases}

where Ω^{2i}S is in degree 0 and Ω^{2i+1}S in degree −1.

In particular, the (Lefschetz) character of C ⊗_{B} Ω^{2i}O is ε_{i}v′_{i} where ε_{i} = −1 if i ∈ I and ε_{i} = 1 otherwise.

Lemma 3.9. We have Hom_{D^{b}(A)}(C ⊗_{B} Ω^{2i}k, C ⊗_{B} Ω^{2j}k[−1]) = 0 for all i, j.

Proof. Let us recall first that \overline{Hom}_{B}(Ω^{n}k, k) = 0 unless n ≡ 0(mod 2e). Put now T = Hom_{D^{b}(A)}(C ⊗_{B} Ω^{2i}k, C ⊗_{B} Ω^{2j}k[−1]).

Since \overline{Hom}(Ω^{2i}S ⊗ k, Ω^{2j+1}S ⊗ k) ≃ \overline{Hom}(Ω^{2i}k, Ω^{2j+1}k) = 0, we deduce that T = 0 unless i ∉ I and j ∈ I, in which case T ≃ Hom(Ω^{2i}S ⊗ k, Ω^{2j+1}S ⊗ k). Then, we have to prove that there are no k-projective morphisms from Ω^{2i}S ⊗ k to Ω^{2j+1}S ⊗ k. As k-projective morphisms Ω^{2i}S ⊗ k → Ω^{2j+1}S ⊗ k lift to O-projective morphisms Ω^{2i}S → Ω^{2j+1}S, we are done, since Hom(Ω^{2i}S, Ω^{2j+1}S) = 0 (the character of Ω^{2i}S is v′_{i} and the character of Ω^{2j+1}S is v′_{j}, hence, these are distinct since i ≠ j).

Corollary 3.10. The complex C^{∗} ⊗_{A} C is homotopy equivalent to its 0-homology.

Proof. Let L, L′ be two simple B-modules. We have

Hom_{D^{b}(B⊗B^{∘})}(C^{∗} ⊗_{A} C, L′ ⊗ L^{∗}[−1]) ≃ Hom_{D^{b}(A)}(C ⊗_{B} L, C ⊗_{B} L′[−1]) = 0

by Lemma 3.9. Hence, C^{∗} ⊗_{A} C has no homology in degree 1. Since the degree 1 component of C^{∗} ⊗_{A} C is projective, C^{∗} ⊗_{A} C is homotopy equivalent to a complex with no component in degree 1. Since C^{∗} ⊗_{A} C is self-dual, it is homotopy equivalent to its 0-homology.

By Lemma 2.5, we have H_{0}(C^{∗} ⊗_{A} C) ≃ B ⊕ Q, where Q is a projective (B ⊗ B^{∘})-module. Now,

Hom_{D^{b}(KB⊗(KB)^{∘})}(K ⊗ (C^{∗} ⊗_{A} C), K ⊗ (Ω^{2i}O ⊗ (Ω^{2j}O)^{∗})) ≃
Hom_{D^{b}(KA)}(K ⊗ (C ⊗_{B} Ω^{2i}O), K ⊗ (C ⊗_{B} Ω^{2j}O)) = δ_{ij}K,
hence Hom(K ⊗ Q, K ⊗ (Ω^{2i}O ⊗ (Ω^{2j}O)^{∗})) = 0

for all i, j. This implies Q = 0. So, we have proven that C^{∗} ⊗_{A} C is homotopy equivalent to B (seen as a complex concentrated in degree 0). By Lemma 2.4, we conclude that C is a Rickard complex. This completes the proof of the first part of Theorem 3.1.

Note that, if E = 1, then a projective cover of M is indecomposable : it follows that C has homology only in one degree and it is isomorphic to M or ΩM[1], from which we derive the second part of Theorem 3.1.

4. Local study

Let G be a finite group, e a block of G with a non-normal cyclic defect group D.

Let Q be the subgroup of D containing R = O_{p}(G) as a subgroup of index p. Let H = N_{G}(Q) and f the block of H corresponding to e.

Theorem 1.1 will follow from the following more precise result :

Theorem 4.1. Let M be an indecomposable direct summand of the (OGe ⊗ (OHf)^{∘})-module eOGf with vertex ΔD. Then, there is a direct summand N of the (OGe ⊗ (OHf)^{∘})-module OGe ⊗_{OR} fOH such that the complex C = 0 → N ⟶^{m} M → 0 induces a splendid Rickard equivalence between OGe and OHf. Here, m is the restriction of the multiplication map OGe ⊗_{OR} fOH → eOGf.

If R ≠ 1, then C has homology only in one degree.

Note that the (OGe ⊗ (OHf)^{∘})-module eOGf has, up to isomorphism, a unique indecomposable direct summand with vertex ΔD (this can easily be deduced from the forthcoming proof of the theorem).

Let us check that a complex C as defined in the theorem is splendid. Note that eOGf is a direct summand of Ind_{ΔD}^{G⊗H^{∘}} O, hence is isomorphic to a direct sum of modules with trivial source and vertex contained contained in ΔD. Since OGe ⊗_{OR} fOH is isomorphic to a direct summand of Ind_{ΔR}^{G⊗H^{∘}} O, it follows that N is isomorphic to a direct sum of modules with trivial source and vertex contained in ΔR. Hence, C is splendid.

Let us prove by induction on the order of G that Theorem 1.1 follows from Theorem 4.1. By induction, we know that Theorem 1.1 holds for OHf : there is a splendid Rickard

equivalence between OHf and ONe′ given by a complex having homology only in one degree. Now, Theorem 4.1 gives a splendid Rickard equivalence between OGe and OHf. Hence, composing the two equivalences, we get a splendid Rickard equivalence between OGe and ONe′, by Lemma 2.6. Furthermore, when O_{p}(G) ≠ 1, the Rickard complex has homology only in one degree. Hence, Theorem 1.1 holds for G and, by induction, the proof of Theorem 1.1 is complete.

We assume now that Theorem 4.1 holds for all finite groups of order strictly less than the order of G. The rest of this section is devoted to proving that the theorem holds then for G.

4.1. O_{p}(G) = 1. Let us first consider the case where R = 1, i.e., Q has order p.

Following Alperin, we have :

Lemma 4.2. The module eOGf induces a stable equivalence between OGe and OHf.

Proof. We have

Res_{H⊗H^{∘}}^{G×G^{∘}} OG ≃ Res_{H⊗H^{∘}}^{G×G^{∘}} Ind_{ΔG}^{G×G^{∘}} O ≃ ⨁_{(g_{1},g_{2})∈H×H^{∘}\G×G^{∘}/ΔG} Ind_{(ΔG)^{(g_{1},g_{2})}∩(H×H^{∘})}^{H×H^{∘}} O.

Now, (Q × Q^{∘}) ∩ (ΔG)^{(g_{1},g_{2})} ≠ 1 if and only if g_{1}g_{2}^{−1} ∈ H. Since Q × Q^{∘} is the maximal elementary abelian subgroup of D × D^{∘} and is normal in H × H^{∘}, it follows that the OHf ⊗ (OHf)^{∘}-module f (Ind_{(ΔG)^{(g_{1},g_{2})}∩(H×H^{∘})}^{H×H^{∘}} O) f is projective when g_{1}g_{2}^{−1} ∉ H.

Hence,

OHf ⊗_{OH} OG ⊗_{OH} fOH ≃ OHf ⊕ projective modules.

Since f is the Brauer correspondent of e, OHf is a direct summand of Res_{H×H^{∘}}^{G×G^{∘}} OGe. So,

fOGe ⊗_{OG} eOGf ≃ fOGef ≃ OHf ⊗_{OH} OGe ⊗_{OH} fOH ≃ OHf ⊕ projective modules.

The result is now a consequence of Lemma 2.1.

Let M be an indecomposable non-projective direct summand of eOGf : by Lemma 2.3, it still induces a stable equivalence between OGe and OHf.

By induction, Theorem 1.1 holds for OHf. Hence, there is an (OHf ⊗ (OD ⋊ E)^{∘})-module M′ inducing a Morita equivalence between OHf and OD ⋊ E. So, the indecomposable module M_{0} = M ⊗_{OHf} M′ induces a stable equivalence between OGe and OD ⋊ E. By Theorem 3.1, there exists a direct summand N_{0} of a projective cover P_{0} of M_{0}, such that the complex C_{0} = 0 → N_{0} → M_{0} → 0 induces a Rickard equivalence between OGe and OD ⋊ E. It follows that

C_{1} = C_{0} ⊗_{OD⋊E} M′^{∗} ≃ 0 → N_{0} ⊗_{OD⋊E} M′^{∗} → M → 0

induces a Rickard equivalence between OGe and OHf. The module N_{0} ⊗_{OD⋊E} M′^{∗} is a direct summand of P_{M} = P_{0} ⊗_{OD⋊E} M′^{∗}, a projective cover of M, and the map N_{0} ⊗_{OD⋊E} M′^{∗} → M is the restriction of a surjective morphism P_{M} → M. Now, the multiplication map

m : OGe ⊗ fOH → eOGf

is surjective and eOGf = M ⊕ projective modules, hence there is a direct summand N of OGe ⊗ fOH such that the complex C = 0 → N ⟶^{m} M → 0 is isomorphic to C_{1}.

So, the theorem holds when R = 1.

4.2. O_{p}(G) ≠ 1. Let us now consider the case where R is non-trivial. Let Ḡ = G/R and ē be the image of e through the canonical morphism OG → OḠ. Similarly, let H̄ = H/R and f̄ be the image of f through the canonical morphism OH → OH̄. Note that the canonical map E = N_{G}(D, e_{D})/C_{G}(D) → Aut(D/R) is injective, since it factors through the group of p′-automorphisms of D. Hence, OH̄f̄ and OḠē are blocks with defect D/R.

Let (R, e_{R}) be an e-subpair. By Lemma 2.14, the (OGe ⊗ ON_{G}(R, e_{R})e_{R})-module eOGe_{R} induces a Morita equivalence. Hence, we can assume that G stabilizes e_{R}, that is, that e = e_{R}.

Let C̄ = C_{G}(R)/R. Since Ḡ/C̄ ≃ G/C_{G}(R) is a cyclic p′-group, a simple OC̄ē-module extends in exactly [G : C_{G}(R)] non-isomorphic ways to Ḡ and every simple OḠē-module is obtained in this way (Lemma 2.13). By induction, OḠē is derived equivalent to (D/R) ⋊ E, hence has |E| simple modules. It follows that [G : C_{G}(R)] = |E|, hence the canonical map N_{G}(D, e_{D})/C_{G}(D) → G/C_{G}(R) is an isomorphism. Note that O_{p}(C_{G}(R)) = R. Similarly, the canonical map N_{H}(D, e_{D})/C_{H}(D) → H/C_{H}(R) is an isomorphism and O_{p}(C_{H}(R)) = R.

4.2.1. 1 ≠ O_{p}(G) ≤ Z(G). By assumption, Theorem 4.1 holds for OḠē and OH̄f̄. The observation above shows that these two blocks have a unique simple module. Let M′ be an indecomposable direct summand of the O(G × H^{∘})/ΔR-module eOGf with vertex ΔD/ΔR. Since ēOḠf̄ is the direct sum of an indecomposable non-projective module and of a projective module, it follows from Lemma 2.9 that the O(Ḡ × H̄^{∘})-module M′′ = M′ ⊗_{OR} O is an indecomposable direct summand of ēOḠf̄. Let f′ : N′ → M′ be a projective cover of M′. Then, N′′ = N′ ⊗_{OR} O is a projective cover of M′′ (Lemma 2.9). Let M (resp. N) be the restriction of M′ (resp. N′) to G × H^{∘}.

If M′′ induces a Morita equivalence between OḠē and OH̄f̄, then let C be the complex with only one non-zero term, M, in degree 0. If the kernel of a surjective map f′′ : N′′ → M′′ induces a Morita equivalence between OḠē and OH̄f̄, then let C be the complex with N in degree −1, M in degree 0 and differential f′.

Then, it follows from Lemma 2.11 that C is a Rickard complex. Note that C has the form required.

4.2.2. O_{p}(G) ≰ Z(G). Note that OC_{H}(R)f is the Brauer correspondent of the block OC_{G}(R)e.

Let M be an indecomposable direct summand of the O(C_{G}(R) × C_{H}(R)^{∘})/ΔR-module eOC_{G}(R)f with vertex Δ(D/R). Let L = N_{G×H^{∘}}(ΔR). The restriction to L of the action of G × H^{∘} on OG leaves eOC_{G}(R)f invariant. This gives a natural extension of the action of (C_{G}(R) × C_{H}(R))/ΔR on eOC_{G}(R)f to L/ΔR. Since (eOC_{G}(R)f)/M is a sum of modules with vertices strictly contained in Δ(D/R), the module M is L-stable. The isomorphisms G/C_{G}(R) ≃ E and H/C_{H}(R) ≃ E induce an isomorphism L/(C_{G}(R) × C_{H}(R)^{∘}) ≃ E. So, by Lemma 2.13, there is an indecomposable summand M̃ of Ind_{Δ(H/R)}^{L/ΔR} O ≃ OC_{G}(R) lifting M.

Let N be a projective cover of M. This is an L-stable indecomposable O(C_{G}(R)×C_{H}(R)^{∘})/ΔRmodule, hence it lifts to a projective L/ΔR-module Ñ (Lemma 2.13). Since Ind_{(C_{G}(R)×C_{H}(R)^{∘})/ΔR}^{L/ΔR} N

is a projective cover of Ind_{(C_{G}(R)×C_{H}(R)^{∘})/ΔR}^{L/ΔR} M and M̃ is isomorphic to a direct summand of the latter, one may choose Ñ to be a projective cover of M̃. We have a natural map

Ind_{1}^{H/R} Res_{1}^{H/R} O → O

giving rise to a surjective map

f : Ind_{1}^{L/ΔR} O → Ind_{Δ(H/R)}^{L/ΔR} O.

So, we may choose Ñ to be a direct summand of Ind_{1}^{L/ΔR} O with f(Ñ) = M̃.

Let N′ = Ind_{L}^{G×H^{∘}} Res_{L}^{L/ΔR} Ñ and M′ = Ind_{L}^{G×H^{∘}} Res_{L}^{L/ΔR} M̃. Then, N′ is a direct summand of OGe ⊗_{OR} fOH, M′ is a direct summand of eOGf and m′(N′) = M′, where m′ is the multiplication map OGe ⊗_{OR} fOH → eOGf.

If Res_{C_{G}(R)×C_{H}(R)^{∘}}^{(C_{G}(R)×C_{H}(R)^{∘})/ΔR} M induces a Morita equivalence between OC_{G}(R)e and OC_{H}(R)f, then we define C to be M′. Otherwise, let C be the complex with N′ in degree −1, M′ in degree 0 and differential m′.

Then, Lemma 2.8 says that C is a splendid Rickard complex between eOG and fOH. Note that C has homology only in one degree. Hence, the proof of Theorem 4.1 is complete.

5. An example : P SL_{2}(p)

We make the constructions of §3 explicit for the group P SL_{2}(p).

Let V_{1} be the natural 2-dimensional representation of SL_{2}(p). Then, the simple kSL_{2}(p) modules are the symmetric powers S^{i}(V_{1}), where 0 ≤ i ≤ p − 1 [Alp, pp 14–16]. The center of SL_{2}(p) acts trivially on S^{i}(V_{1}) when i is even and we denote by V_{i} the module S^{2i}(V_{1}) induced from SL_{2}(p) to P SL_{2}(p). Then, the simple modules in the principal block e of G = P SL_{2}(p) are the V_{i}, 0 ≤ i ≤ \frac{p−3}{2}. There is only one other block in G, it has defect 0 and its simple module is the Steinberg module V_{\frac{p−1}{2}} .

For i < \frac{p−1}{2}, the dimension of V_{i} is 2i+1 < p. Let B be the normalizer of a Sylow p-subgroup of G. By Proposition 4.2, the (OGe⊗(OB)^{∘})-module OGe induces a stable equivalence between OGe and OB. For V a simple OGe-module, the module OGe ⊗ V has a dimension strictly smaller than p, hence it cannot have a projective direct summand. By Proposition 2.3, this implies the indecomposabiblity of OGe as a (OGe ⊗ (OB)^{∘})-module.

Since indecomposable kB-modules are uniserial, the restriction of V_{i} to B has a unique simple quotient W_{i}, for 0 ≤ i ≤ \frac{p−3}{2} (this is actually a special case of the general property of simple modules for groups with a (B, N)-pair to have a unique simple quotient when restricted to B). Furthermore, W_{i} ≃ W_{1}^{i}.

Let U be the Sylow p-subgroup of B and T a complement to U in B. The group T is cyclic, with order \frac{p−1}{2} and there are two conjugacy classes of non-trivial p-elements in B (hence also in G). Let x_{1}, x_{2} be representatives for these classes. Let T′ be a “Coxeter torus” of G, i.e., the centralizer of an element of order \frac{p+1}{2}. This is a cyclic group of order \frac{p+1}{2}.

Let η be the irreducible character of T which gives the simple module W_{1}. Let δ be a nontrivial irreducible character of T′ which occurs in the character of the restriction of V_{1} to T′ (the restriction is then 1 + δ + δ^{−1}). Let ε = ±1 with ε ≡ p (mod 4).

The character table of the principal block of G is :

x_{1}  x_{2}  t ∈ T  t′ ∈ T′
1  1  1  1  1
(p − 1)_{j}  −1  −1  0  −(δ^{j}(t′) + δ^{−j}(t′))
(p + 1)^{l}  1  1  η^{l}(t) + η^{−l}(t)  0
(\frac{p+ε}{2})^{i}  \frac{1}{2}(ε + i\sqrt{εp}) \frac{1}{2}(ε − i\sqrt{εp}) \begin{cases}η^{\frac{p−1}{4}} (t) & if ε = 1 \\ 0 & if ε = −1 \end{cases} \begin{cases}0 & if ε = 1 \\ −δ^{\frac{p+1}{4}} (t′) & if ε = −1 \end{cases}

where j ∈ \{1, …, \frac{p+ε−2}{4}\}, l ∈ \{1, …, \frac{p−ε}{4} − 1\} and i ∈ \{−1, 1\}.

The character table of B is :

x_{1}  x_{2}  t ∈ T
1_{l}  1  1  η^{l}(t)
(\frac{p−1}{2})i \frac{1}{2}(−1 + i\sqrt{−p}) \frac{1}{2}(−1 − i\sqrt{−p}) 0

where l ∈ \{0, \frac{p−1}{2} − 1\} and i ∈ \{−1, 1\}.

The restrictions of the irreducible characters of the principal block of G to B are :

• Res_{B}^{G} 1 = 1,

• Res_{B}^{G} (p − 1)_{j} = (\frac{p−1}{2})_{0} + (\frac{p−1}{2})_{1},

• Res_{B}^{G} (p + 1)_{l} = (\frac{p−1}{2})_{0} + (\frac{p−1}{2})_{1} + 1_{l} + 1_{\frac{p−1}{2} −l}.

• Res_{B}^{G} (\frac{p+ε}{2})_{i} = \begin{cases}(\frac{p−1}{2})i & if ε = −1, \\ (\frac{p−1}{2})_{i} + 1_{\frac{q−1}{4}} & if ε = 1. \end{cases}

The Brauer tree of OGe is

[figure: 1; _{(p − 1)_{1}}; _{(p + 1)_{1}}; _{(p − 1)_{2}}; _{(p + 1)_{2}}; _{(p − ε)_{\frac{p−2+ε}{4}}}; ∗; \begin{aligned}( & \frac{p+ε}{2} & ) \\ & & ±1 \end{aligned}; P_{0}; P_{\frac{p−3}{2}}; P_{1}; P_{\frac{p−5}{2}}; P_{\frac{p−ε}{4}}]

The arrows describe Green’s walk on the tree (i.e., the sequence of vertices v_{0}, v_{1}, …).

A projective cover of the (OGe ⊗ (OB)^{∘})-module OGe is

⨁_{0≤λ≤ \frac{p−3}{2}} P_{λ} ⊗ Q_{λ}^{∗}

where P_{λ} is a projective cover of V_{λ} and Q_{λ} a projective cover of W_{λ}. The module N constructed in §3.4 is

N = ⨁_{\frac{p+2+ε}{4} ≤λ≤ \frac{p−3}{2}} P_{λ} ⊗ Q_{λ}^{∗}

Restricting a surjective map

⨁_{0≤λ≤ \frac{p−1}{2}} P_{λ} ⊗ Q_{λ}^{∗} ↠ OGe

to N gives a complex

C = 0 → N → OGe

which induces a splendid Rickard equivalence between OB and OGe.

Let I be the isometry R(KGe) → R(KB) induced by this equivalence of derived categories. We have

• I(1) = 1,

• I((p + 1)_{l}) = 1_{l},

• I((p − 1)_{j}) = −1_{\frac{p−1}{2} −j},

• I((\frac{p+ε}{2})_{i}) = −ε (\frac{p−1}{2})_{εi}.

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U.F.R. de Mathématiques, Université Paris 7–Denis Diderot et UMR 9994 du CNRS, 2 Place Jussieu, F–75251 Paris Cedex 05, France

  E-mail address: rouquier@math.jussieu.fr