ON EQUIVALENCES BETWEEN BLOCKS OF GROUP

ALGEBRAS: REDUCTION TO THE SIMPLE COMPONENTS

Andrei Marcus

Abstract. A conjecture of Michel Broué states that if D is an abelian Sylow p-subgroup of a finite group G, and H = N_{G}(D), then the principal blocks of G and H are Rickard equivalent. The structure of groups with abelian Sylow p-subgroups, as determined by P. Fong and M.E. Harris, raises the following question: assuming that Broué’s conjecture holds for the simple components of G, under what conditions does it hold for G itself? Due to the structure of G, this problem requires mainly the lifting of Rickard complexes to p′ -extensions of the simple components and the construction of complexes over wreath products. We give here these reduction steps, which may be regarded as a “Clifford theory” of tilting complexes.

1. Introduction and preliminaries

Let G be a finite group, O a complete discrete valuation ring with quotient field K of characteristic 0, maximal ideal 𝔭 and residue field k = O/𝔭 of characteristic p > 0. We shall assume that K and k are “big enough” for all the finite groups considered here. An OG-module will be a unitary, finitely generated, O-free, and (unless otherwise stated) left OG-module. We denote by OG-mod (respectively mod-OG) the category of left (respectively right) OG-modules. If A is an additive category, then C^{b}(A) will be the category of bounded complexes in A.

In [Br2], [Br3] and [Br4], the following conjectures were stated:

1.1. Conjecture. If D is an abelian Sylow p-subgroup of G and H = N_{G}(D), then the principal blocks of OG and OH are isotypic.

1.2. Conjecture. If D is an abelian Sylow p-subgroup of G and H = N_{G}(D), then the principal blocks of OG and OH are Rickard equivalent.

Let e ∈ Z(OG) and f ∈ Z(OH) be the principal block idempotents. By a Rickard equivalence between OGe and OHf we mean that it is given a bounded complex (C, d) of (OGe, OHf)-bimodules such that: (1.3.a) For each integer i, C^{i} is projective in OGe-mod and in mod-OHf. (1.3.b) If we denote C^{∗} = Hom_{O}(C, O), then there are homotopy equivalences of complexes of bimodules C ⊗_{OHf} C^{∗} ∼ OGe and C^{∗} ⊗_{OGe} C ∼ OHf. (1.3.c) For each integer i, C^{i} is a p-permutation O[G × H^{op}]-module with vertex contained in δ(D) = {(x, x^{−1}) | x ∈ P } ⊆ G × H^{op}.

If these conditions hold, then C will be called a Rickard tilting complex for OGe-mod and OHf -mod. These complexes were called “splendid” in [R2], and their main feature is

Typeset by AMS-TEX

that they induce derived equivalences between the principal blocks of the local subgroups C_{G}(P) and C_{H}(P) for all subgroups P of D. The local tilting complexes are obtained in the following way (see [R2, Sections 4 and 5]): (1.4.a) Apply to C the Brauer functor

Br_{δ(P)}^{G×H^{op}}(−) = (−)(δ(P)): O[G × H^{op}]-mod → k[N_{G×H^{op}}(δ(P))]-mod.

(Recall that for an OG-module M,

Br_{P}^{G}(M) = M(P) = M^{P} /( ∑_{Q<P} Tr_{Q}^{P}(M^{Q}) + 𝔭M^{P}) = k ⊗_{O} (M^{P} / ∑_{Q<P} Tr_{Q}^{P}(M^{Q})),

where M^{P} is the set of points fixed by P, and Tr_{Q}^{P}: M^{Q} → M^{P} is the trace map.) (1.4.b) Regard C(δ(P)), by restriction, as a complex of p-permutation k[C_{G}(P) × C_{H}(P)^{op}]-modules. (1.4.c) Finally, lift C(δ(P)) to obtain a complex C_{P} of p-permutation O[C_{G}(P) × C_{H}(P)^{op}]-modules, which will be a Rickard tilting complex for the principal blocks of O[C_{G}(P)] and O[C_{H}(P)].

In [FH], Conjecture 1.1 was reduced, by using the classification of finite simple groups, and by developing a Clifford theory for isometries and isotypies, to the case of simple components of G. The full result was finally achieved for p = 2. In this case, three of the needed isotypies for the simple components were already provided by [Br2], and the fourth was constructed in [FH].

By [Br4] and [Ri, Section 6], Conjecture 1.2 implies Conjecture 1.1, and we show that it also can be reduced to the case of the simple components of G. The reduction steps are similar to those of [FH], and may be considered as a Clifford theory for tilting complexes. Unfortunately, we do not have yet tilting complexes for the simple components, but, as applications, we shall check our lifting conditions on some known examples.

Since we shall deal with a more general situation, let us recall some definitions. The Frobenius category 𝔉𝔯_{p}(G) of a finite group G has as objects the p-subgroups of G, and the morphisms between two p-subgroups are those group-homomorphisms which are induced by inner automorphisms of G. Then two finite groups G and H are said to have the same p-local structure if they have a common Sylow p-subgroup D (not necessarily abelian), and the embedding of D in G and H induce an equivalence of the categories 𝔉𝔯_{p}(G) and 𝔉𝔯_{p}(H). This means that: (1.5) For any subgroups P, Q of D and any isomorphism ϕ: P → Q, there is an element g ∈ G such that ϕ(x) = gxg^{−1} , for x ∈ P, if and only if there is an element h ∈ H such ϕ(x) = hxh^{−1} for x ∈ P.

It is well-known (see [Br4, Section 1]) that in our previous situation (when D is abelian and H = N_{G}(D)), the groups G and H do have the same p-local structure.

We shall also discuss the lifting of Morita equivalences from normal subgroups and we shall make use of the following result which was discovered independently by L. Puig and L. Scott.

1.6. Theorem. Assume that G and H have the same p-local structure, and let OGe and OHf be two blocks with defect group D. Then the following statements are equivalent:

(i) The blocks OGe and OHf are Puig equivalent (that is, the source algebras of OGe and OHf are isomorphic as interior D-algebras).

(ii) There is a Morita equivalence between OGe and OHf defined by an (OGe, OHf)- bimodule M (and by its O-dual M^{∗}) such that M is a relatively δ(D)-projective ppermutation O[G × H^{op}]-module.

In particular, if OGe and OHf are Puig equivalent, then they are Rickard equivalent and moreover, one obtains Morita equivalences between the principal blocks of the local subgroups C_{G}(P) and C_{H}(P), where P ≤ D, by applying to the bimodule M the same algorithm as in (1.4) (see [Br3, Section 6.C]).

For other definitions and results concerning Broué’s conjectures we refer to [Br2], [Br3], [Br4], [P] and [R2]. Some needed facts will be given in each section.

In Section 2 we follow closely [FH] in setting the notations and presenting the structure of groups with abelian Sylow p-subgroups. In Section 3 we give conditions under which Rickard and Puig equivalences can be lifted from normal subgroups, and in Section 4 we deal with wreath products. We show that a Rickard or a Puig equivalence between the principal blocks of the base groups of two wreath products can always be lifted. The problem is making the symmetric group to act on a tensor power of a complex. We solve this problem in a natural and elementary way. Finally, the conclusions which follow from these reduction steps are drawn in Section 5. We also give here some examples in which the lifting conditions are satisfied.

2. The structure of groups with abelian Sylow p-subgroups

Consider a finite group G and let D be a Sylow p-subgroup of G. We first point out that we may assume that O_{p′}(G) = 1, where O_{p′}(G) is the maximal normal p′ -subgroup of G.

Denote G_{0} = O_{p′}(G), Ḡ = G/G_{0} and let π: G → Ḡ, π(g) = ḡ be the canonical map. For a subgroup X of G denote X̄ = π(X) = XG_{0}/G_{0} . Then D ≃ D̄, so D may be viewed as a common Sylow p-subgroup of G and Ḡ. By [FH, Proposition 7.C], π induces an equivalence between 𝔉𝔯_{p}(G) and 𝔉𝔯_{p}(Ḡ). Moreover, if P is a subgroup of D, we have CḠ(P̄) = \overline{C_{G}(P)} ≃ C_{G}(P)/C_{G_{0}}(P).

Let OGe be the principal block of OG and OḠē the principal block of OḠ. Then Br_{P}^{G}(e) is the principal block idempotent of kC_{G}(P), BrḠ ; P̄(ē) is the principal block idempotent of kCḠ(P̄), and we have BrḠ ; P̄(ē) = \overline{Br_{P}^{G}(e)}.

Moreover, π induces the isomorphism of interior G-algebras OGe ≃ OḠē, and consider the (OGe, OḠē)-bimodule M = OḠē. Then M is a relatively δ(D)-projective p-permutation O[G × Ḡ^{op}]-bimodule, and M ⊗_{OḠē} −: OḠē-mod → OGe-mod is an equivalence, that is, the principal blocks of OG and OḠ are Puig equivalent. As a (kC_{G}(P), kḠ(P̄))-bimodule, M(δ(P)) is isomorphic to k[CḠ(P̄)] BrḠ ; P̄(ē), and induces a Morita equivalence between the principal blocks of kC_{G}(P) and kḠ(P̄). As in [R2, Section 5], this equivalence can be lifted to a Morita equivalence between the principal blocks of OC_{G}(P) and OCḠ(P̄).

Let now H be another group having D as a Sylow p-subgroup, OHf the principal block of OH, and denote H_{0} = O_{p′}(H), H̄ = H/H_{0} . By the remarks above we deduce: (2.1) G and H have the same p-local structure if and only if Ḡ and H̄ have the same p-local structure, and if OḠē and OH̄f̄ are Puig (respectively Rickard equivalent), then OGe and OHf are Puig (respectively Rickard) equivalent.

Assume now in addition that O_{p′}(G) = 1 and that D is abelian. By (2.1) we see that this is no loss in the context of Broué’s conjectures. By the results of [FH, Section 5] (obtained by using at certain stages the classification of finite simple groups), there is an embedding ι of G such that:

(2.2)  ∏_{i=0}^{s} X_{i}^{Ω_{i}} ≤ ι(G) ≤ ∏_{i=0}^{s}(X̃_{i} ≀ Σ_{i})

where:

– X_{0} = O_{p}(G), the maximal normal p-subgroup of G,

– X̃_{0} is a split extension of X_{0} by a p′ -subgroup of Aut(X_{0}),

– |Ω_{0}| = |Σ_{0}| = 1,

– X_{1}, …, X_{s} is a complete system of representatives for the isomorphism classes of simple components of G,

– X_{i} is a normal subgroup of X̃_{i} and X̃_{i}/X_{i} is a Hall p′ -subgroup of Aut(X_{i})/X_{i}, for 1 ≤ i ≤ s,

– Ω_{1}, …, Ω_{s} are disjoint finite sets,

– Σ_{i} is a p′ -subgroup of the symmetric group S(Ω_{i}), for 1 ≤ i ≤ s.

We change now the notations setting G′ = ι(G), G = ∏_{i=0}^{s} X_{i}^{Ω_{i}}, G̃ = ∏_{i=0}^{s}(X̃_{i} ≀ Σ_{i}), H = N_{G}(D), H′ = N_{G′}(D) and H̃ = NG̃(D). If D_{i} = X_{i} ∩ D, then D_{i} is a Sylow p-subgroup of X_{i} and D = ∏_{i=0}^{s} D_{i}. Denote Y_{i} = N_{X_{i}}(D_{i}) and Ỹ_{i} = NX̃_{i}(D_{i}), so H = ∏_{i=0}^{s} Y_{i}^{Ω_{i}} and H̃ = ∏_{i=0}^{s}(Ỹ_{i} ≀ Σ_{i}). (Remark that for i = 0 we have D_{0} = O_{p}(G) = X_{0} = Y_{0} and X̃_{0} = Ỹ_{0} .) By the Frattini argument we have X̃_{i} = Ỹ_{i}X_{i}, hence the map α_{i}: X̃_{i}/X_{i} → Ỹ_{i}/Y_{i}, α_{i}(y_{i}X_{i}) = y_{i}Y_{i} is an isomorphism. Moreover, α_{i}, 1 ≤ i ≤ s, induce the isomorphism

α = ∏_{i=0}^{s} α_{i}^{≀Σ_{i}}: ∏_{i=0}^{s}(X̃_{i}/X_{i}) ≀ Σ_{i} → ∏_{i=1}^{s}(Ỹ_{i}/Y_{i}) ≀ Σ_{i}, ((y_{i}X_{i}^{Ω_{i}}, σ_{i})) ↦ ((y_{i}Y_{i}^{Ω_{i}}, σ_{i}))

such that α(G′/G) = H′/H.

In the next sections we shall give conditions under which Broué’s Conjecture 1.2 holds for G′ provided that it holds for the simple components X_{i} of G′. The reduction steps are motivated by the structure theorem above. For each i, an equivalence between the principal blocks of OX_{i} and OY_{i} has to be lifted to an equivalence between the principal blocks of OX̃_{i} and OỸ_{i}, then to an equivalence between the principal blocks of O[X̃_{i} ≀ Σ_{i}] and O[Ỹ_{i} ≀ Σ_{i}], then to an equivalence for the direct products ∏_{i=0}^{s}(X̃_{i} ≀ Σ_{i}) and ∏_{i=0}^{s}(Ỹ_{i} ≀ Σ_{i}), and finally to an equivalence for subgroups of these direct products (corresponding under the isomorphism α). We shall see that restrictive conditions have to be imposed only at the first step.

3. Block equivalences and normal subgroups

Let X̃, Ỹ be finite groups and let X ≤ X̃, Y ≤ Ỹ be normal subgroups such that there is an isomorphism α: X̃/X → Ỹ /Y . For simplicity of the notation, we identify via α these two isomorphic groups and denote G = X̃/X = Ỹ /Y. So an element g of G will mean a class xX with x ∈ X̃ and also a class yY with y ∈ Ỹ such that α(xX) = yY.

Let e ∈ Z(OX) and f ∈ Z(OY) be G-invariant block idempotents, that is, e is invariant under X̃ and f is invariant under Ỹ , so e ∈ Z(OX̃) and f ∈ Z(OỸ). Since we deal with normal subgroups, it will be convenient to express our results in the language of group-graded algebras. We refer to [D2] for definitions and basic facts. Denote then R = eOX̃ and S = fOỸ . It follows that R and S are fully G-graded O-algebras with R_{1} = eOX and S_{1} = fOY , and the blocks of R, respectively S, are precisely the blocks of OX̃, respectively OỸ , covering e, respectively f (see for instance [Al, Section 15]).

The opposite ring S^{op} is a fully G^{op} -graded O-algebra with components S_{g}^{op} = S_{g}, g ∈ G^{op}. Consider the fully G × G^{op} -graded O-algebra A = R ⊗_{O} S^{op} and the diagonal morphism δ: G → G × G^{op}, δ(g) = (g, g^{−1}). Let Δ = A_{δ(G)} = ⨁_{g∈G}(R_{g} ⊗_{O} S_{g^{−1}}^{op}), which is a fully δ(G)-graded subalgebra of A with Δ_{δ(1)} = R_{1} ⊗_{O} S_{1}^{op}. Since G × G^{op} = ⋃_{g∈G}(g, 1)δ(g) = ⋃_{g∈G}(1, g)δ(G), it also follows that we have a natural isomorphism of R ⊗_{O} S_{1}^{op} -modules

(3.1.a)  A = ⨁_{g,h∈G} R_{gh} ⊗_{O} S_{h^{−1}}^{op} ≃ ⨁_{g,h∈G} R_{g} ⊗_{R_{1}} R_{h} ⊗_{O} S_{h^{−1}}^{op}
≃ ⨁_{g∈G} R_{g} ⊗_{R_{1}} Δ = R ⊗_{R_{1}} Δ

and similarly, isomorphisms of R_{1} ⊗_{O} S^{op} -modules

(3.1.b)  A = ⨁_{g,h∈G} R_{h} ⊗_{O} S_{h^{−1}g}^{op} ≃ ⨁_{g,h∈G}(1 ⊗_{O} S_{g}^{op}) ⊗_{S_{1}^{op}} (R_{h} ⊗_{O} S_{h^{−1}}^{op})
≃ ⨁_{g∈G} S_{g}^{op} ⊗_{S_{1}^{op}} Δ = S^{op} ⊗_{S_{1}^{op}} Δ

In the same way, we obtain the isomorphisms

(3.1.c)  A ≃ Δ ⊗_{R_{1}} R  (as right R ⊗_{O} S_{1}^{op}-modules)
(3.1.d)  A ≃ Δ ⊗_{S_{1}^{op}} S^{op}  (as right R_{1} ⊗_{O} S^{op}-modules)

Clearly, all these isomorphisms preserve the gradings.

3.2. Remarks. In what will follow, we shall mainly use that R and S are crossed-products of G with R_{1} and S_{1} respectively. Using representations of these crossed products we may give explicitely the isomorphisms (3.1) and we shall discuss other needed facts. Denote R = R_{1,ρ}^{α}[G] = {rḡ | r ∈ R_{1}, g ∈ G} and S = S_{1,σ}^{β}[G] = {sg̃ | s ∈ S_{1}, g ∈ G}, where ḡ ∈ R_{g} and g̃ ∈ S_{g} are homogenous units of degree g, and ρ: G → Aut(R_{1}), σ: G → Aut(S_{1}) are defined by ρ(g)(r) = ḡrḡ^{−1}, σ(g)(s) = g̃sg̃^{−1} . The functions ρ and σ induce grouphomomorphisms ρ̄: G → Out(R_{1}) and σ̄: G → Out(S_{1}) which do not depend on the choice of ḡ and g̃. The multiplications are defined by (rḡ)(r′h̄) = r · ρ(g)(r′) · α(g, h)\overline{gh} and (sg̃)(s′h̃) = s·σ(g)(s′)·β(g, h)g̃h, where α: G×G → R_{1}^{∗}, β: G×G → S_{1}^{∗} are the factor sets given by ḡh̄ = α(g, h)\overline{gh} and g̃h̃ = β(g, h)g̃h. Then A = R ⊗_{O} S^{op} = \{rḡ ⊗_{O} h̃s | r ∈ R_{1}, s ∈ S_{1}, g, h ∈ G} and Δ = A_{δ(G)} = {rḡ ⊗_{O} g̃^{−1}s | r ∈ R_{1}, s ∈ S_{1}, g ∈ G}. The element rḡ ⊗_{O} h̃s of A corresponds to rḡh̄ ⊗_{R_{1}} (h̄^{−1} ⊗_{O} h̃s) ∈ R ⊗_{R_{1}} Δ by the isomorphism (3.1.a), and to g̃h̃s ⊗_{S_{1}^{op}} (rḡ ⊗_{O} g̃^{−1}) ∈ S^{op} ⊗_{S_{1}^{op}} Δ by (3.1.b). Conversely, rḡ ⊗_{R_{1}} (r′h̄ ⊗_{O} h̃^{−1}s′) ∈ R ⊗_{R_{1}} Δ corresponds to rḡr′h̄ ⊗_{O} h̃^{−1}s′ ∈ A in (3.1.a), and h̃s ⊗_{S_{1}^{op}} (r′ḡ ⊗_{O} g̃^{−1}s′) ∈ S^{op} ⊗_{S_{1}^{op}} Δ corresponds to r′ḡ ⊗_{O} g̃^{−1}s′h̃s ∈ A in (3.1.b).

b) Let M be a Δ-module. We have that R⊗_{R_{1}} M is a G-graded (R, S_{1})-bimodule and M ⊗_{S_{1}} S is a G-graded (R_{1}, S)-bimodule. We may use (3.1.a) to obtain isomorphisms

(3.3.a)  A ⊗_{Δ} M ≃ (R ⊗_{R_{1}} Δ) ⊗_{Δ} M ≃ R ⊗_{R_{1}} M
(3.3.b)  A ⊗_{Δ} M ≃ (S^{op} ⊗_{S_{1}^{op}} Δ) ⊗_{Δ} M ≃ M ⊗_{S_{1}} S

as (R, S_{1})-bimodules, respectively as (R_{1}, S)-bimodules. These isomorphisms preserve G-gradings, so we may regard R ⊗_{R_{1}} M and M ⊗_{S_{1}} S as isomorphic G-graded (R, S)- bimodules, by transport of structure. Then, under (3.3.a), (rḡ⊗_{O} h̃s)⊗_{Δ}m ∈ A⊗_{Δ}M maps to rḡh̄⊗_{R_{1}} (h̄^{−1}⊗_{O} h̃)ms ∈ R⊗_{R_{1}} M, rḡ⊗_{R_{1}} m ∈ R⊗_{R_{1}} M maps to (rḡ⊗_{O} 1)⊗_{Δ}m ∈ A⊗_{Δ} M, and under (3.3.b), (rḡ⊗_{O} h̃s)⊗_{Δ} m ∈ A⊗_{Δ} M corresponds to (rḡ⊗_{O} g̃^{−1})m⊗_{S_{1}} sh̃g̃ ∈ M ⊗_{S_{1}} S, and m⊗_{S_{1}} sh̃ ∈ M ⊗_{S_{1}} S to (1⊗_{O} h̃s)⊗_{Δ} m ∈ A⊗_{Δ} M. Consequently, the right S -module structure of R⊗_{R_{1}} M is defined by (rḡ⊗_{R_{1}} m)sh̃ = rḡh̄⊗_{R_{1}} (h̄^{−1} ⊗_{O} h̃)(ms) and the left module structure of M ⊗_{S_{1}} S is defined by rḡ(m ⊗_{S_{1}} sh̃) = (rḡ⊗_{O} g̃^{−1})m ⊗_{S_{1}} sh̃g̃.

These arguments show that we have three naturally isomorphic O-linear functors

A ⊗_{Δ} −, R ⊗_{R_{1}} −, − ⊗_{S_{1}} S: Δ-mod → A-mod.

c) Let T = T_{1,τ}^{γ} [G] = {tĝ | t ∈ T_{1}, g ∈ G} be a third crossed product O-algebra. We shall need to denote A(R, S) = R ⊗_{O} S^{op} and Δ(R, S) = A(R, S)_{δ(G)} . Let also A(R) = A(R, R) and Δ(R) = Δ(R, R). The proofs of the following statements are routine, and are left to the reader.

Assume that M is a Δ(R, S)-module and N is a Δ(S, T)-module. Then M ⊗_{S_{1}} N is a Δ(R, T)-module with

(rḡ ⊗_{O} ĝ^{−1}t)(m ⊗_{S_{1}} n) = (rḡ ⊗_{O} g̃^{−1})m ⊗_{S_{1}} (g̃ ⊗_{O} ĝ^{−1}t)n.

Assume that M is a Δ(S, R)-module and N is a Δ(S, T)-module. Then Hom_{S_{1}}(M, N) is a Δ(R, T)-module with

(rḡ ⊗_{O} ĝ^{−1}t)(f)(m) = (g̃ ⊗_{O} ĝ^{−1}t)f((g̃^{−1} ⊗_{O} rḡ)m).

In particular, if M is a Δ(R, S)-module, then Hom_{R_{1}}(M, R_{1}) and Hom_{S_{1}}(M_{S_{1}}, S_{1}) are Δ(S, R)-modules.

d) Recall that if M and N are R-modules, then Hom_{R_{1}}(M, N) is a G-module with (^{g}f)(m) = ḡf(ḡ^{−1}m), and there is a trace map Tr_{1}^{G}: Hom_{R_{1}}(M, N) → Hom_{R}(M, N), Tr_{1}^{G}(f) = ∑_{g∈G} ^{g}f.

e) We have that R is a symmetric crossed product algebra with symmetrizing form λ: R → O (where λ(∑_{α∈X̃} α_{x}x) = α_{1} ), and the restriction λ_{1} : R_{1} → O of λ to R_{1} is a symmetrizing form for R_{1} . Moreover, for all r ∈ R_{1} and g ∈ G, we have that λ_{1}(ḡrḡ^{−1}) = λ_{1}(r). Similar statements hold for S too, where we denote by μ: S → O the symmetrizing form. As we shall immediately see, the proof of the following theorem works under these more general assumptions. □

Let M be an (R_{1}, S_{1})-bimodule. Then its O-dual M^{∗} is an (S_{1}, R_{1})-bimodule, R⊗_{R_{1}} M is a G-graded (R, S_{1})-bimodule and Hom_{O}(R⊗_{R_{1}} M, O) is a G-graded (S_{1}, R)- bimodule (with components Hom_{O}(R ⊗_{R_{1}} M, O)_{g} = Hom_{O}(R_{g^{−1}} ⊗_{R_{1}} M, O), for g ∈ G), which is naturally isomorphic to M^{∗} ⊗_{R_{1}} R. Notice also that M is a projective R_{1} -module if and only if R ⊗_{R_{1}} M is a projective R-module.

Let finally (C, d) be a bounded complex of (R_{1}, S_{1})-bimodules. Then C^{∗} is a complex of (S_{1}, R_{1})-bimodules, (R ⊗_{R_{1}} C, R ⊗_{R_{1}} d) is a complex of G-graded (R, S_{1})-bimodules, and (R ⊗_{R_{1}} C)^{∗} ≃ C^{∗} ⊗_{R_{1}} R as complexes of G-graded (S_{1}, R)-bimodules.

We treat first the lifting of Morita equivalences and of derived equivalences.

3.4. Theorem. a) Assume that (M, M^{∗}) defines a Morita equivalence between R_{1} and S_{1} . If M is δ(G)-invariant and extends to a Δ-module, then R ⊗_{R_{1}} M (respectively S ⊗_{S_{1}} M^{∗}) is naturally a G-graded (R, S)-bimodule (respectively (S, R)-bimodule) and (R ⊗_{R_{1}} M, S ⊗_{S_{1}} M^{∗}) defines a Morita equivalence between R and S.

This equivalence induces Morita equivalences between the corresponding blocks of OX̃ and OỸ covering e and f.

b) Assume that G is a p′ -group and that (C, C^{∗}) defines a derived equivalence between R_{1} and S_{1} . If (C, d) extends to a complex of Δ-modules, then (R⊗_{R_{1}} C, S ⊗_{S_{1}} C^{∗}) defines a derived equivalence between R and S.

This equivalence induces derived equivalences between the blocks of OX̃ and OỸ covering e and f.

Proof. a) Since M extends to a Δ-module, M^{∗} becomes a right Δ-module. Using now (3.1.c,d) and the remarks preceding the theorem, we get:

M^{∗} ⊗_{Δ} A ≃ M^{∗} ⊗_{R_{1}} R ≃ (R ⊗_{R_{1}} M)^{∗} ≃ M^{∗} ⊗_{S_{1}^{op}} S^{op} ≃ S ⊗_{S_{1}} M^{∗} ≃ (M ⊗_{S_{1}} S)^{∗}

as G-graded (S, R)-bimodules, and it also follows that M^{∗} ⊗_{Δ} A ≃ (A ⊗_{Δ} M)^{∗} as right A-modules.

Next we need to show that (R ⊗_{R_{1}} M) ⊗_{S} (R ⊗_{R_{1}} M)^{∗} ≃ R as (R, R)-bimodules and that (R ⊗_{R_{1}} M)^{∗} ⊗_{R} (R ⊗_{R_{1}} M) ≃ S as (S, S)-bimodules. By Remark 3.2.c) we have that M ⊗_{S_{1}} M^{∗} is a Δ(R)-module and M^{∗} ⊗_{R_{1}} M is a Δ(S)-module. Clearly, R_{1} is a Δ(R)-module and S_{1} is a Δ(R)-module. Using the definitions given in Remark 3.2.c), it is a routine to check that the adjunctions between M ⊗_{S_{1}} M^{∗} and R_{1} are Δ(R)-linear, and the adjunctions between M^{∗} ⊗_{R_{1}} M and S_{1} are Δ(S)-linear. The isomorphism M ⊗_{S_{1}} M^{∗} ≃ R_{1} of Δ(R) modules gives, by (3.3) the isomorphism

A(R) ⊗_{Δ(R)} (M ⊗_{S_{1}} M^{∗}) ≃ A(R) ⊗_{Δ(R)} R_{1} ≃ R

of G-graded (R, R)-bimodules, and also the (R, R)-bimodule isomorphisms

A(R) ⊗_{Δ(R)} (M ⊗_{S_{1}} M^{∗}) ≃ R ⊗_{R_{1}} (M ⊗_{S_{1}} M^{∗})
≃ (R ⊗_{R_{1}} M) ⊗_{S_{1}} M^{∗}
≃ (R ⊗_{R_{1}} M) ⊗_{S} (S ⊗_{S_{1}} M^{∗})
≃ (R ⊗_{R_{1}} M) ⊗_{S} (R ⊗_{R_{1}} M)^{∗}

We may similarly prove the second isomorphism, and notice also that these isomorphisms preserve G-gradings.

Finally, in order to see that Hom_{R}(A ⊗_{Δ} M, R), Hom_{S}(A ⊗_{Δ} M, S) and (A ⊗_{Δ} M)^{∗} are isomorphic G-graded (S, R)-bimodules, it is enough to check that the isomorphisms of (S_{1}, R_{1})-bimodules

ϕ: Hom_{R_{1}}(M, R_{1}) → M^{∗}, ϕ(f)(m) = λ_{1}(f(m)),

and

ψ: Hom_{S_{1}}(M, S_{1}) → M^{∗}, ψ(f)(m) = μ_{1}(f(m))

are Δ(S, R)-linear. But this checking is a routine, using Remark 3.2.c) and e).

b) By Remark 3.2.b) we have three naturally isomorphic O-linear functors between the categories of complexes

A ⊗_{Δ} −, R ⊗_{R_{1}} −, − ⊗_{S_{1}} S: C^{b}(Δ-mod) → C^{b}(A-mod).

It follows that (R ⊗_{R_{1}} C, R ⊗_{R_{1}} d) may be regarded as a complex of G-graded (R, S)- bimodules, isomorphic to (A ⊗_{Δ} C, A ⊗_{Δ} d), and S ⊗_{S_{1}} C^{∗} may be wiewed as a complex of (S, R)-bimodules, isomorphic to (A ⊗_{Δ} C)^{∗}. Similarly, the naturally isomorphic O-linear functors

(− ⊗_{S_{1}} S) ⊗_{S} −, − ⊗_{S_{1}} −: mod − S_{1} × S-mod → Ab

give rise to naturally isomorphic O-linear functors

(− ⊗_{S_{1}} S) ⊗_{S} −, − ⊗_{S_{1}} −: C^{b}(mod − S_{1}) × C^{b}(S-mod) → C^{b}(Ab),

(where we take the total complex of the double complex obtained by the construction of tensor products).

As in the Morita case, we have the Δ(R)-linear chain maps ϵ: C ⊗_{S_{1}} C^{∗} → R_{1} and ϵ′: R_{1} → C ⊗_{S_{1}} C^{∗}, and the Δ(S)-linear chain maps η: C^{∗} ⊗_{R_{1}} C → S_{1} and η′: S_{1} → C^{∗}⊗_{R_{1}} C. By assumption, there are Δ(R)_{1} -linear homotopies between ϵ′∘ϵ and id_{C⊗_{S_{1}} C^{∗}}, and between ϵ ∘ ϵ′ and id_{R_{1}} , and Δ(S)_{1} -linear homotopies between η′ ∘ η and id_{C^{∗}⊗_{R_{1}} C}, and between η ∘ η′ and id_{S_{1}} . Since G is a p′ -group, we can apply the map |G|^{−1} Tr_{1}^{G} defined by the crossed products Δ(R), respectively Δ(S) to these homotopies to obtain that C ⊗_{S_{1}} C^{∗} and R_{1} are homotopic complexes of Δ(R)-modules, and that C^{∗} ⊗_{R_{1}} C and S_{1} are homotopic complexes of Δ(S)-modules.

Using these observations, we obtain as above that

(R ⊗_{R_{1}} C) ⊗_{S} (R ⊗_{R_{1}} C)^{∗} ≃ A(R) ⊗_{Δ(R)} (C ⊗_{S_{1}} C^{∗})

is homotopy equivalent to R as complexes of (R, R)-bimodules, and

(R ⊗_{R_{1}} C)^{∗} ⊗_{S} (R ⊗_{R_{1}} C) ≃ A(S) ⊗_{Δ(S)} (C^{∗} ⊗_{R_{1}} C)

is homotopy equivalent to S as complexes of (S, S)-bimodules. □

We consider further Puig and Rickard equivalences. Working with p-permutation modules and complexes of p-permutation modules, we shall refine the conclusions of Theorem 3.4, showing the relationship between the different local equivalences which are obtained applying the Brauer functor. In view of [R2, Section 5], it is no loss to consider only modules over k, so the Brauer functor will have a simpler expression.

We need some properties of this functor. The notations introduced here will be in force only in this section. Let H be a subgroup of a finite group G and P a p-subgroup of G. We denote by 1_{G} the trivial kG-module. For the following facts, see [Br1, (1.3) and (1.4)]. (3.5) Let M be a relatively H -projective kG-module. If P is not conjugate to a subgroup of H, then Br_{P}^{G}(M) = 0. (3.6) Let T be a complete system of representatives for the double cosets in N_{G}(P) \ T_{G}(P, H)/H, where T_{G}(P, H) = {g ∈ G | P ≤ ^{g}H = gHg^{−1}}. Then there is an isomorphism of k[N_{G}(P)/P]-modules

Br_{P}^{G}(Ind_{H}^{G}(1_{H})) ≃ ⨁_{g∈T} Ind_{N_{^{g}H}(P)}^{N_{G}(P)}(1_{N_{^{g}H}(P)}).

Take now a look to the following diagram of categories and functors:

kG-mod ⟶^{Br_{P}^{G}(−)} k[N_{G}(P)]-mod ⟶^{Res_{C_{G}(P)}^{N_{G}(P)}(−)} k[C_{G}(P)]-mod
Ind_{H}^{G}(−)↑  ^{Ind_{N_{H} (P)}^{N_{G}(P)}(−)}↑  ↑^{Ind_{C_{H} (P)}^{C_{G}(P)}(−)}
kH-mod ⟶^{Br_{P}^{H}(−)} k[N_{H}(P)]-mod ⟶^{Res_{C_{H} (P)}^{N_{H} (P)}(−)} k[C_{H}(P)]-mod

3.7. Lemma. Assume that H ≤ G and that P ≤ D are p-subgroups of H. Let M be a relatively D-projective p-permutation kH -module. If C_{G}(P)T_{H}(P, D) = T_{G}(P, D), then there is a natural isomorphism of k[C_{G}(P)]-modules

Ind_{C_{H}(P)}^{C_{G}(P)}(Res_{C_{H}(P)}^{N_{H}(P)}(Br_{P}^{H}(M))) ≃ Res_{C_{G}(P)}^{N_{G}(P)}(Br_{P}^{G}(Ind_{H}^{G}(M))).

Proof. Since M is a relatively D-projective p-permutation module, it is a direct summand of a direct sum of kH -modules of the form Ind_{Q}^{H}(1_{Q}), where Q ≤ D. Remark that, by (3.5), Br_{P}^{H}(Ind_{Q}^{H}(1_{Q})) = Br_{P}^{G}(Ind_{Q}^{G}(1_{Q})) = 0 if Q < P. So assume that P ≤ Q ≤ D and M = Ind_{Q}^{H}(1_{Q}).

Using (3.6) and the Mackey Decomposition, we get

Ind_{C_{H}(P)}^{C_{G}(P)}(Res_{C_{H}(P)}^{N_{H}(P)}(Br_{P}^{H}(M))) ≃ ⨁_{h∈H̃}( ⨁_{y∈H̄_{h}} Ind_{C_{H}(P)∩^{y}N_{^{h}Q}(P)}^{C_{G}(P)}(1_{C_{H}(P)∩^{y}N_{^{h}Q}(P)})),

where we have denoted by H̃ (respectively H̄_{h}) a complete set of representatives of C_{H}(P)\ T_{H}(P, Q)/Q (respectively of C_{H}(P) \ N_{H}(P)/N_{^{h}Q}(P)).

In the same way we obtain

Res_{C_{G}(P)}^{N_{G}(P)}(Br_{P}^{G}(Ind_{H}^{G}(M))) ≃ ⨁_{g∈G̃}( ⨁_{x∈Ḡ_{g}} Ind_{C_{G}(P)∩^{x}N_{^{g}Q}(P)}^{C_{G}(P)}(1_{C_{G}(P)∩^{x}N_{^{g}Q}(P)})),

where G̃ (respectively Ḡ_{g}) is a complete set of representatives of C_{G}(P) \ T_{G}(P, Q)/Q (respectively of C_{G}(P) \ N_{G}(P)/N_{^{g}Q}(P)).

Observe that we have C_{G}(P)T_{H}(P, Q) = T_{G}(P, Q) for any P ≤ Q ≤ D, and in particular, C_{G}(P)N_{H}(P) = N_{G}(P). It follows that we may choose representatives such that G̃ = H̃ and Ḡ_{h} = H̄_{h} for every h ∈ H̃, hence the lemma is proved.

We return now to the situation introduced at the beginning of the section and assume that e ∈ Z(kX) and f ∈ Z(kY) are the principal block idempotents. Let R = ekX̃, R_{1} = ekX, S = fkỸ , S_{1} = fkY, A = R ⊗_{k} S^{op} and Δ = A_{δ(G)} . We also denote

δ(G) = {(x, y^{−1}) ∈ X̃ × Ỹ^{op} | α(xX) = yY }.

We shall make the following additional assumptions: (3.8.a) G is a p′ -group. (3.8.b) D is a common Sylow p-subgroup of X and Y , and X̃, Ỹ have the same p-local structure in a way compatible with the isomorphism α: X̃/X → Ỹ /Y , that is, (see(1.5)): for any subgroup P, Q of D and any σ ∈ Hom(P, Q), there is x ∈ X̃ such that σ(u) = xux^{−1} for u ∈ P if and only if there is y ∈ Ỹ such that σ(u) = yuy^{−1} for u ∈ P, and x, y are related by α(xX) = yY. (3.8.c) For any subgroup P of D, α(CX̃(P)X/X) = CỸ (P)Y/Y.

If P is a subgroup of D, we denote X_{P} = C_{X}(P), Y_{P} = C_{Y} (P),X̃_{P} = CX̃(P), Ỹ_{P} = CỸ (P). Then e_{P} = Br_{P}^{X}(e) ∈ Z(kX_{P}) and f_{P} = Br_{P}^{Y} (f) ∈ Z(kY_{P}) are the principal block idempotents. The condition (3.8.c) imply that α induces the isomorphism α_{P} : X̃_{P} /X_{P} → Ỹ_{P} /Y_{P} , α_{P} (xX) ∩ Ỹ_{P} for x ∈ X̃_{P} . Denote by G_{P} these two isomorphic groups, and consider the fully G_{P} -graded k -algebras R^{P} = e_{P} kX̃_{P} and S^{P} = f_{P} kỸ_{P} with R_{1}^{P} = e_{P} kX_{P} and S_{1}^{P} = f_{P} kY_{P} . Then A^{P} = R^{P} ⊗_{k} S^{P op} is a fully G_{P} × G_{P}^{op} -graded k -algebra, and let Δ^{P} = A_{δ(G_{P})}^{P}, where we also identify

G_{P} = {(x, y^{−1}) ∈ X̃_{P} × Ỹ_{P} | α_{P} (xX_{P}) = yY_{P} }.

Denote finally δ: D → X × Y^{op}, δ(u) = (u, u^{−1}), and notice that C_{X×Y^{op}}(δ(P)) = X_{P} × Y_{P}^{op}, C_{δ(G)}(δ(P)) = δ(G_{P}) and C_{X̃×Ỹ^{op}}(δ(P)) = X̃_{P} × Ỹ_{P}^{op}.

3.9. Corollary. With the above notations, assume that conditions (3.8) hold.

a) Let M be a relatively δ(D)-proiective p-permutation (R_{1}, S_{1})-bimodule such that (M, M^{∗}) defines a Puig equivalence between R_{1} and S_{1} . Assume that M extends to a Δ-module _{Δ}M = M. Then M̃ = A ⊗_{Δ} M is a relatively δ(D)-projective p-permutation A-module and (M̃, M̃^{∗}) defines a Puig equivalence between R and S.

Moreover, the modules inducing the local equivalences are related in the following way (where P ≤ D):

Br_{δ(P)}^{X̃×Ỹ^{op}}(M̃) ≃ A^{P} ⊗_{Δ^{P}} Br_{δ(P)}^{X×Y^{op}}(M) (as A^{P} -modules).

b) Let C be a Rickard tilting complex for R_{1} and S_{1} . Asume that C extends to a complex _{Δ}C = C of Δ-modules. Then C̃ = A ⊗_{Δ} C is a Rickard tilting complex for R and S.

Moreover, we have:

Br_{δ(P)}^{X̃×Ỹ^{op}}(C̃) ≃ A^{P} ⊗_{Δ^{P}} Br_{δ(P)}^{X×Y^{op}}(C) (as complexes of A^{P} -modules).

Proof. The fact that M̃ (respectively C̃) induces a Morita (respectively derived) equivalence is already proved in Theorem 3.4. For the rest of the statements, remark first that since G is a p′ -group, the extension module _{Δ}M is also a relatively δ(D)-projective p-permutation k[δ(G)]-module, and hence A ⊗_{Δ} M is a relatively δ(D)-projective ppermutation k[X̃ × Ỹ^{op}]-module.

Let us depict the following diagram:

k[X̃ × Ỹ^{op}]-mod ⟶^{Br_{δ(P)}^{X̃×Ỹ ^{op}} (−)} k[N_{X̃×Ỹ^{op}}(δ(P))]-mod ⟶^{Res} k[X̃_{P} × Ỹ_{P}^{op}]-mod
Ind↑  ^{Ind}↑  ↑^{Ind}
k[δ(G)]-mod ⟶^{Br_{δ(P)}^{δ(G)}(−)} k[N_{δ(G)}(δ(P))]-mod ⟶^{Res} k[δ(G_{P})]-mod
Res↓  ^{Res}↓  ↓^{Res}
k[X × Y^{op}] ⟶^{Br_{δ(P)}^{X×Y ^{op}} (−)} k[N_{X×Y^{op}}(δ(P))]-mod ⟶^{Res} k[X_{P} × Y_{P}^{op}]-mod

The two bottom squares clearly commute to within natural equivalences of functors, so Br_{δ(P)}^{δ(G)}(_{Δ}M) and Br_{δ(P)}^{X×Y^{op}}(M) are naturally isomorphic (R_{1}^{P} , S_{1}^{P})-bimodules, that is, Br_{δ(P)}^{X×Y^{op}}(M) extends to a Δ^{P} -module. The corollary is proved if we show that the assumption of Lemma 3.7 holds if we replace G by X̃ × Ỹ^{op}, H by δ(G), D by δ(D) and P by δ(P). So we have to verify that

(CX̃(P) × CỸ (P)^{op})T_{δ(G)}(δ(P), δ(D)) = T_{X̃×Ỹ^{op}}(δ(P), δ(D)).

Indeed, let (x, y) ∈ T_{X̃×Ỹ^{op}}(δ(P), δ(D)). This means that for any u ∈ P, x^{−1}ux = yuy^{−1} ∈ D. By (3.8.b) there is x_{1} ∈ X̃ such that α(x_{1}X) = y^{−1}Y and x_{1}^{−1}ux_{1} = yuy^{−1} for u ∈ P. We see that (xx_{1}^{−1}, 1) ∈ CX̃(P) × CỸ (P)^{op}, (x_{1}, y) ∈ T_{δ(G)}(δ(P), δ(D)), and (x, y) = (xx_{1}^{−1}, 1)(x_{1}, y). It follows that a) is proved, and similar statements hold for complexes.

3.10. Remarks. a) We have by (3.3) that Br_{δ(P)}^{X̃×Ỹ^{op}}(M̃) is actually a G_{P} -graded (A^{P} , B^{P})- bimodule, and Br_{δ(P)}^{X̃×Ỹ^{op}}(C̃) is a complex of G_{P} -graded (A^{P} , B^{P})-bimodules.

b) Returning to the notations of Lemma 3.7 and taking P = D, we see that T_{G}(D, D) = N_{G}(D), T_{H}(D, D) = N_{H}(D), and the left square of the diagram preceding the lemma becomes commutative for relatively D-projective p-permutation modules without any assumptions on C_{G}(P), that is, Ind_{N_{H}(D)}^{N_{G}(D)}(Br_{D}^{H}(M)) and Br_{D}^{G}(Ind_{H}^{G}(M)) are isomorphic k[N_{G}(D)]-module.

We refer to [D3] for conditions equivalent to the extendibility to Δ of a Δ_{1} -module M in the case when the order of G is invertible in Δ. We end this section with some observations concerning the extendibility of complexes.

Denote by C^{b}(Δ_{1}) the category of bounded complexes of Δ_{1} -modules, and let (C, d) be an object of this category. We say that C is G-invariant if C and Δ_{g} ⊗_{Δ_{1}} C are isomorphic in C^{b}(Δ_{1}) for any g ∈ G. If C extends to a complex of Δ-modules, then the multiplication map Δ_{g} ⊗_{Δ_{1}} C → Δ_{g}C = C is an isomorphism in C^{b}(Δ_{1}) for any g ∈ G. So assume further that C is G-invariant.

Consider the complex (Δ ⊗_{Δ_{1}} C, Δ ⊗_{Δ_{1}} d) of G-graded Δ-modules. Denote E = End_{Δ}(Δ⊗_{Δ_{1}} C)^{op} and E′ = End_{C^{b}(Δ)}(Δ⊗_{Δ_{1}} C)^{op}. Then E has a natural G-grading (see [D2]), and it is easy to see that E′ is a G-graded subring of E′. Since C is G-invariant, E and E′ are crossed-products of E_{1} ≃ End_{Δ_{1}}(C)^{op}, respectively E′_{1} = End_{C^{b}(Δ_{1})}(C)^{op} with G. Let hU(E′) = ⋃_{g∈G}(U(E′) ∩ E_{g}) be the group of homogenous units of E′. Then Dade’s basic criterion for extendability of modules can be adapted without difficulty to obtain:

3.11. Proposition. a) There is a complex of Δ-modules extending C if and only if the group-extension U(E′_{1}) ↣ hU(E′) ↠ G splits. If γ is a splitting, then the corresponding complex of Δ-modules has multiplication defined by a_{g}∗c = (a_{g}⊗c)γ(g^{−1}), for g ∈ G, a_{g} ∈ Δ_{g} and c ∈ C.

b) There is a bijection between the set if isomorphism classes in C^{b}(Δ) of complexes extending C, and the set of U(E′_{1})-conjugation classes of splittings.

Remark finally that the assumption on C in Theorem 3.4.b) and in Corollary 3.9.b) can be replaced with the following weaker one: (3.12) There is a bounded complex C̄ of Δ-modules such that the complex Res_{Δ_{1}}^{Δ} C̄ is homotopic to C.

4. Equivalences for wreath products

Let Ω = {1, …, n} and Σ a subgroup of the symmetric group S(Ω). We will show in this section how equivalences between two blocks of OX and OY lift to an equivalence between blocks of O[X ≀ Σ] and O[Y ≀ Σ]. We shall use the results of the preceding section, so our notations here will fit as much as possible with the notations introduced there. But first, we establish a lemma solving the main difficulty which appears when one deals with complexes over wreath products.

If M is an (O-free) O-module, denote M^{⊗ Ω} = M ⊗ ⋯ ⊗ M (n times), and if (C, d) is a (bounded) complex of O-modules, let again C^{⊗ Ω} = C ⊗ … ⊗ C (n times) with the differential defined by:

d(c_{i_{1}} ⊗ … ⊗ c_{i_{n}}) = ∑_{l=1}^{n}(−1)^{i_{1}+···+i_{l−1}}c_{i_{1}} ⊗ … ⊗ dc_{i_{l}} ⊗ … ⊗ c_{i_{n}},

where c_{i_{l}} belongs to the l-th factor C of C^{⊗ Ω} and has degree i_{l} ∈ ℤ. We shall denote for short c_{i_{1}} ⊗ … ⊗ c_{i_{n}} = c_{i_{1}} … c_{i_{n}} ∈ (C^{⊗ Ω})^{i_{1}+···+i_{n}}.

The next lemma shows that S(Ω) acts naturally on M^{⊗ Ω} and on C^{⊗ Ω} .

4.1. Lemma. a) Let M be an O-module. Then M^{⊗ Ω} becomes a left O[S(Ω)]-module by defining, for σ ∈ S(Ω) and m = m_{1} ⊗ … ⊗ m_{n} ∈ M^{⊗ Ω} ,

σ · m = m_{σ^{−1}(1)} ⊗ … ⊗ m_{σ^{−1}(n)}.

b) Let (C, d) be a complex of O-modules. Then there is a function ϵ: S(Ω) × ℤ_{2}^{Ω} → ℤ_{2} such that by defining:

σ · (c_{i_{1}} … c_{i_{n}}) = (−1)^{ϵ_{σ}(i_{1},...,i_{n})}c_{i_{σ^{−1}(1)}} … c_{i_{σ^{−1}(n)}},

C^{⊗ Ω} becomes a complex of O[S(Ω)]-modules.

Proof. a) is well-known. For b), we first show the uniqueness of a function ϵ_{σ} satisfying a certain condition. Assume that ϵ_{σ} exists. The action of σ must commute with the differential d. We have:

dσ · (c_{i_{1}} … c_{i_{n}}) = (−1)^{ϵ_{σ}(i_{1},...,i_{n})}d(c_{i_{σ^{−1}(1)}} … c_{i_{σ^{−1}(n)}})

= (−1)^{ϵ_{σ}(i_{1},...,i_{n})} ∑_{l=1}^{n}(−1)^{i_{σ^{−1}(1)}+···+i_{σ^{−1}(l−1)}}c_{i_{σ^{−1}(1)}} … dc_{i_{σ^{−1}(l)}} … c_{i_{σ^{−1}(n)}},

(where the differential is in the l-th place), and

σ · d(c_{i_{1}} … c_{i_{n}}) = σ · (∑_{l=1}^{n}(−1)^{i_{1}+···+i_{l−1}}c_{i_{1}} … dc_{i_{l}} … c_{i_{n}})

= ∑_{l=1}^{n}(−1)^{i_{1}+···+i_{l}}(−1)^{ϵ_{σ}(i_{1},...,i_{l}+1,...,i_{n})}c_{i_{σ^{−1}(1)}} … dc_{i_{σ^{−1}(σ(l))}} … c_{i_{σ^{−1}(n)}}

(where the differential is in the σ(l)-th place). We have dσ = σd if the following equalities hold in ℤ_{2} for 1 ≤ l ≤ n:

(4.2.1)  ϵ_{σ}(i_{1}, …, i_{l} + 1, …, i_{n}) = ϵ_{σ}(i_{1}, …, i_{l}, …, i_{n}) + i_{1} + ⋯ + i_{l−1}

+ i_{σ^{−1}(1)} + ⋯ + i_{σ^{−1}(σ(l)−1)}.

Due to a), we choose ϵ_{σ}(0, …, 0) = 0. This choice implies immediately that if ϵ_{σ} satisfying (4.2.1) exists, then it is unique.

Next, S(Ω) must act on C^{⊗ Ω} , so we compute:

(τσ) · (c_{i_{1}} … c_{i_{n}}) = (−1)^{ϵ_{τσ}(i_{1},...,i_{n})}c_{i_{(τσ)^{−1}(1)}} … c_{i_{(τσ)^{−1}(n)}},

and

τ · (σ · (c_{i_{1}} … c_{i_{n}})) = (−1)^{ϵ_{τ} (i_{1},...,i_{n})}σ · (c_{i_{τ^{−1}(1)}} … c_{i_{τ^{−1}(n)}})
= (−1)^{ϵ_{τ} (i_{1},...,i_{n})+ϵ_{σ}(i_{τ^{−1}(1)},...,i_{τ^{−1}(n)})}c_{i_{(τσ)^{−1}(1)}} … c_{i_{(τσ)^{−1}(n)}}.

We require then the following “1-cocycle” equality in ℤ_{2} :

(4.2.2)  ϵ_{τσ}(i_{1}, …, i_{n}) = ϵ_{τ} (i_{1}, …, i_{n}) + ϵ_{σ}(i_{τ^{−1}(1)}, …, i_{τ^{−1}(n)}).

Also, ϵ_{τσ} must satisfy the “recurrence relations” (4.2.1), that is:

(4.2.3)  ϵ_{τσ}(i_{1}, …, i_{l} + 1, …, i_{n}) = ϵ_{τσ}(i_{1}, …, i_{n}) + i_{1} + ⋯ + i_{l−1}
+ i_{(τσ)^{−1}(1)} + ⋯ + i_{(τσ)^{−1}((τσ(l)−1)}.

Assume that there are ϵ_{σ} and ϵ_{τ} satisfying (4.2.1). Then we show that there is ϵ_{τσ} satisfying (4.2.3). Indeed, define η: ℤ_{2}^{Ω} → ℤ_{2} by

(4.2.4)  η(i_{1}, …, i_{n}) = ϵ_{τ} (i_{1}, …, i_{n}) + ϵ_{σ}(i_{τ^{−1}(1)}, …, i_{τ^{−1}(n)}).

Then we have

η(i_{1}, …,i_{l} + 1, …, i_{n}) = ϵ_{τ} (…, i_{l} + 1, …) + ϵ_{σ}(i_{τ^{−1}(1)}, …, i_{τ^{−1}(τ(l))} + 1, …, i_{τ^{−1}(n)})
= ϵ_{τ} (i_{1}, …, i_{n}) + i_{1} + ⋯ + i_{l−1} + i_{τ^{−1}(1)} + ⋯ + i_{τ^{−1}(τ(l)−1)}
+ ϵ_{σ}(i_{τ^{−1}(1)}, …, i_{τ^{−1}(n)}) + i_{τ^{−1}(1)} + ⋯ + i_{τ^{−1}(τ(l)−1)}
+ i_{(τσ)^{−1}(1)} + ⋯ + i_{(τσ)^{−1}((τσ)(l)−1)}
= η(i_{1}, …, i_{n}) + i_{1} + ⋯ + i_{l−1} + i_{(τσ)^{−1}(1)} + ⋯ + i_{(τσ)^{−1}((τσ)(l)−1)}.

Consequently, it is enough to show that ϵ_{σ} satisfying (4.2.1) exists for all transpositions σ = (l l + 1), 1 ≤ l < n. Indeed, if these ϵ_{σ} are defined, then for any ρ ∈ S(Ω) choose a decomposition of ρ into a product of such transpositions. Using definition (4.2.4), we define inductively ϵ_{ρ}, which will satisfy (4.2.1) by the above calculations. The uniqueness of ϵ_{ρ} implies that the definition does not depend on the choice of the decomposition of ρ. Now given σ, τ ∈ S(Ω) and setting ρ = τσ, then, defining η = ϵ_{τ} +^{τ}ϵ_{σ} as in (4.2.4), it follows by the uniqueness that η = ϵ_{ρ}, and hence by (4.2.2) that S(Ω) acts on C^{⊗ Ω} .

So we finally show that ϵ_{σ} is defined for σ = (l l + 1). Equalities (4.2.1), written for l and l + 1 become:

ϵ_{σ}(i_{1}, …, i_{l} + 1, i_{l+1}, …, i_{n}) = ϵ_{σ}(i_{1}, …, i_{l}, i_{l+1}, …, i_{n}) + i_{l+1}
ϵ_{σ}(i_{1}, ….i_{l}, i_{l+1} + 1, …, i_{n}) = ϵ_{σ}(i_{1}, …, i_{l}, i_{l+1}, …, i_{n}) + i_{l}.

Clearly, the remaining equalities are ϵ_{σ}(i_{1}, …, i_{k}+1, …, i_{n}) = ϵ_{σ}(i_{1}, …, i_{n}) for k ≠ l, l+1, and the function which satisfies these equalities is ϵ_{σ}(i_{1}, …, i_{n}) = i_{l}i_{l+1} .

For m = m_{1} ⊗ … ⊗ m_{n} ∈ M^{⊗ Ω}, c = c_{i_{1}} ⊗ … ⊗ c_{i_{n}} ∈ C^{⊗ Ω} and σ ∈ Σ, we shall also denote ^{σ}m = σ · m, ^{σ}c = σ · c, m^{σ} = σ^{−1} · m and c^{σ} = σ^{−1} · c. If X is a group and A is an O-algebra such that are given group-homomorphisms Σ → Aut X and Σ → Aut A, then we denote by X ⋊ Σ = {xσ | x ∈ X, σ ∈ Σ}, A ∗ Σ = {aσ | a ∈ A, σ ∈ Σ} with multiplications (xσ)(yτ) = x ·^{σ}y · στ in X ⋊ Σ, and (aσ)(bτ) = a ·^{σ}b · στ in A ∗ Σ.

We return now to our basic situation. Let X, Y be two groups and e ∈ Z(OX), f ∈ Z(OY) be block idempotents. Then ẽ = e^{⊗ Ω} = e ⊗ ⋯ ⊗ e ∈ (OX)^{⊗ Ω} ≃ O[X^{Ω}] and f̃ = f^{⊗ Ω} ∈ (OY)^{⊗ Ω} ≃ O[Y^{Ω}] are Σ-invariant block idempotents. Denote R_{1} = ẽO[X^{Ω}] ≃ (eOX)^{⊗ Ω} , S_{1} = f̃O[Y^{Ω}] ≃ (fOY)^{⊗ Ω} , and let R = R_{1} ∗ Σ ≃ ẽO[X ≀ Σ] and S = S_{1} ∗ Σ ≃ f̃O[Y ≀ Σ]. Then R is a fully Σ-graded O-algebra, and its elements are of the form rσ = (r_{1} ⊗ … ⊗ r_{n})σ, where r_{i} ∈ eOX and the multiplication is given by

(rσ)(r′σ′) = r ·^{σ}r′ · σσ′ = (r_{1}r′_{σ^{−1}(1)} ⊗ … r_{n}r′_{σ^{−1}(n)})σσ′,

and similar statements hold for S.

We may apply the results of the preceding section with ẽ, f̃ in the place of e, f, with X^{Ω}, Y^{Ω} in place of X, Y , with X ≀ Σ, Y ≀ Σ in place of X̃, Ỹ and with Σ in place of G. Since Σ is naturaly isomorphic to (X ≀ Σ)/X^{Ω} and to (Y ≀ Σ)/Y^{Ω} , the isomorphism α may be regarded to be just the identity. Denote again A = R ⊗_{O} S^{op} and Δ = A_{δ(Σ)} , where δ(Σ) = {(σ, σ^{−1}) | σ ∈ Σ} ⊆ Σ × Σ^{op}. There are isomorphisms A ≃ (R_{1} ⊗_{O} S_{1}^{op}) ∗ (Σ × Σ^{op}) and Δ ≃ (R_{1} ⊗_{O} S_{1}^{op}) ∗ δ(Σ) ≃ Δ_{1} ∗ Σ, so we may write Δ = {(r ⊗ s)σ | r ∈ R_{1}, s ∈ S_{1}^{op}, σ ∈ Σ}, with multiplication defined by

(r ⊗ s)σ · (r′ ⊗ s′)σ′ = (r ·^{σ}r′ ⊗^{σ}s′ · s)σσ′.

Let now M be an (eOX, fOY)-bimodule. Then M^{⊗ Ω} is an R_{1} ⊗_{O} S_{1}^{op} -module. The point is that due to Lemma 4.1, M^{⊗ Ω} extends to a Δ-module by defining

(r ⊗ s)σ · m = (r ⊗ s) ·^{σ}m = r_{1}m_{σ^{−1}(1)}s_{1} ⊗ … ⊗ r_{n}m_{σ^{−1}(n)}s_{n},

hence we may talk about the (R, S)-bimodule M ≀ Σ = A ⊗_{Δ} M. As we have seen in Section 3, M ≀ Σ is naturally isomorphic to R ⊗_{R_{1}} M and to M ⊗_{S_{1}} S. Let us make these isomorphisms to be explicit, as in Remark 3.2.

We identify R ⊗_{R_{1}} M^{⊗ Ω} with OΣ ⊗_{O} M^{⊗ Ω} = {σ ⊗ m | σ ∈ Σ, m ∈ M^{⊗ Ω}} with R-module structure given by rτ(σ ⊗ m) = τσ ⊗ r^{τσ} · m. A right S -module structure can also be defined by (σ ⊗m)sτ = στ ⊗m^{τ} ·s^{τ}. In the same way we identify M^{⊗ Ω} ⊗_{S_{1}} S with M^{⊗ Ω} ⊗_{O} OΣ, with right R-module structure defined by (m⊗σ)sτ = ms^{σ} ⊗στ. A left Rmodule structure is given by rτ(m⊗σ) = r ·^{τ}m⊗τσ. Finally, the map ϕ: OΣ⊗_{O} M^{⊗ Ω} → M^{⊗ Ω} ⊗_{O} OΣ, (σ ⊗ m) ↦ ^{σ}m ⊗ σ is an isomorphism of (R, S)-bimodules.

If C is a complex of (eOX, fOY)-bimodules, we define similarly, using Lemma 4.1, the complex of (R, S)-bimodules C ≀ Σ = A ⊗_{Δ} C ≃ OΣ ⊗_{O} C ≃ C ⊗_{O} OΣ.

The following result is almost proved by now.

4.3. Theorem. a) Assume that (M, M^{∗}) defines a Morita equivalence between eOX and fOY . Then (M ≀ Σ, (M ≀ Σ)^{∗}) defines a Morita equivalence between R and S, inducing Morita equivalences between the corresponding blocks of O[X ≀ Σ] and O[Y ≀ Σ] covering ẽ and f̃.

b) Assume that Σ is a p′ -group and that (C, C^{∗}) defines a derived equivalence between eOX and fOY . Then (C ≀ Σ, (C ≀ Σ)^{∗}) defines a derived equivalence between R and S, inducing derived equivalences between the corresponding blocks of O[X ≀ Σ] and O[Y ≀ Σ] covering ẽ and f̃.

Proof. If (M, M^{∗}) defines a Morita equivalence between eOX and fOY , then obviously, (M^{⊗ Ω}, (M^{⊗ Ω})^{∗}) defines a Morita equivalence between R_{1} and S_{1} . Similarly, if (C, C^{∗}) defines a derived equivalence of eOX and fOY , then, by [R1, Lemma 4.3], (C^{⊗ Ω}, (C^{⊗ Ω})^{∗}) defines a derived equivalence of R_{1} and S_{1} . Since both M^{⊗ Ω} and C^{⊗ Ω} extend to Δ, the conclusions follow from Theorem 3.4.

4.4. Remark. It turns out that Theorem 4.3 is just a particular case of Theorem 3.4. Notice that (4.3.a) was already proved in [S, Proposition 2], and under the assumptions of Theorem 5 of that paper, we have obtained a derived equivalence of R and S, not only a stable equivalence of Morita type.

4.5. Remark. Assume in addition that: Σ is a p′ -group, eOX and fOY are the principal blocks, D is a common Sylow p-subgroup of X and Y and X, Y have the same p-local structure, M is a relatively δ(D)-projective p-permutation O[X × Y^{op}]-module and C is a bounded complex of relatively δ(D)-projective p-permutation O[X × Y^{op}]-modules. Then, in Theorem 4.3, the expression “Morita equivalence” can be replaced by “Puig equivalence” and “derived equivalence” by “Rickard equivalence”. Indeed, it is easy to verify that the following statements are true: e^{⊗ Ω} ∈ (OX)^{⊗ Ω} and f^{⊗ Ω} ∈ (OY)^{⊗ Ω} are the principal block idempotents; D^{Ω} is a common Sylow p-subgroup of X^{Ω} and Y^{Ω} and X^{Ω}, Y^{Ω} , respectively X ≀ Σ, Y ≀ Σ have the same p-local structure as required in (3.8.b); for any subgroup P of D^{Ω} , the obvious isomorphism α: X ≀ Σ/X^{Ω} → Y ≀ Σ/T^{Ω} has the property α(C_{X≀Σ}(P)X^{Ω}/X^{Ω}) = C_{Y ≀Σ}(P)Y^{Ω}/Y^{Ω} (as required in (3.8.c)); M^{⊗ Ω} and C^{⊗ Ω} are relatively δ(D^{Ω})-projective and p-permutation over k[X^{Ω} × Y^{Ω op}], and apply Corollary 3.9.

5. Conclusions and examples

In this section we put the things together in order to obtain equivalences between blocks in the situation presented in Section 2.

For 0 ≤ i ≤ s let X_{i} be a normal subgroup of X̃_{i} and Y_{i} a normal subgroup of Ỹ_{i}, such that there is an isomorphism α_{i}: X̃_{i}/X_{i} → Ỹ_{i}/Y_{i}. We identify these two factor groups and we denote them by G_{i} as in Section 3. Let further Ω_{0}, …, Ω_{s} be disjoint finite sets, and let Σ_{i} ≤ S(Ω_{i}). Then α_{i} induces the isomorphism α_{i}^{Ω_{i}}: X̃_{i}^{Ω_{i}}/X_{i}^{Ω_{i}} → Ỹ_{i}^{Ω_{i}}/Y_{i}^{Ω_{i}}, and these factor groups may be identified with G_{i}^{Ω_{i}}; α_{i} also induces the isomorphism α_{i}^{≀Σ_{i}}: X̃_{i} ≀Σ_{i}/X_{i}^{Ω_{i}} → Ỹ ≀Σ_{i}/Y_{i}^{Ω_{i}}, and we identify these factor groups with G_{i} ≀Σ_{i}. Consider the isomorphism

α = ∏_{i=0}^{s} α_{i}: ∏_{i=0}^{s}(X̃_{i} ≀ Σ_{i})/ ∏_{i=0}^{s} X_{i}^{Ω_{i}} → ∏_{i=0}^{s}(Ỹ_{i} ≀ Σ_{i})/ ∏_{i=0}^{s} X_{i}^{Ω_{i}},

and let G = ∏_{i=0}^{s} G_{i} ≀ Σ_{i}. Denote X = ∏_{i=0}^{s} X_{i}^{Ω_{i}}, Y = ∏_{i=0}^{s} Y_{i}^{Ω_{i}}, X̃ = ∏_{i=0}^{s} X̃_{i} ≀ Σ_{i}, Ỹ = ∏_{i=0}^{s} Ỹ_{i} ≀ Σ_{i}, and consider the subgroups X ≤ X′ ≤ X̃, Y ≤ Y′ ≤ Ỹ such that α(X′/X) = Y′/Y . Finally, identify X′/X = Y′/Y = G′ via the restriction of α.

Let e_{i} ∈ OX_{i} and f_{i} ∈ OY_{i} be block idempotents, and denote R^{i} = e_{i}^{⊗Ω_{i}}O[X̃_{i} ≀ Σ_{i}], S^{i} = f_{i}^{⊗Ω_{i}}O[Ỹ_{i} ≀ Σ_{i}], R_{1} = eOX, S_{1} = fOY (where e = ⊗_{i=0}^{s}e_{i}^{⊗Ω_{i}} and f = ⊗_{i=0}^{s}f_{i}^{⊗Ω_{i}}), R = eOX̃ ≃ ∏_{i=0}^{s} R^{i} and S = fOỸ ≃ ∏_{i=0}^{s} S^{i} (where the isomorphisms are of G-graded rings).

Furthermore, let A = R ⊗_{O} S^{op} ≃ ∏_{i=0}^{s} A^{i} (where A^{i} = R^{i} ⊗_{O} S^{i op}), and let Δ = A_{δ(G)} ≃ ∏_{i=0}^{s}(Δ^{i})^{⊗ Ω} (where Δ^{i} = (e_{i}OX̃_{i} ⊗_{O} f_{i}OỸ_{i}^{op})_{δ(G_{i})} ). We shall also consider the G′ -components R′ = R_{G′} = ⨁_{g∈G′} R_{g}, S′ = S_{G′}, A′ = A_{G′×G′^{op}} ≃ R′ ⊗_{O} S′^{op} and Δ′ = A′_{δ(G′)} .

For 0 ≤ i ≤ s, let M_{i} be an (e_{i}OX_{i}, f_{i}OY_{i})-bimodule and C_{i}^{·} a (bounded) complex of (e_{i}OX_{i}, f_{i}OY_{i})-bimodules, and denote M = ⊗_{i=0}^{s}M_{i}^{⊗Ω_{i}} and C^{·} = ⊗_{i=0}^{s}C_{i}^{·}^{⊗Ω_{i}}.

5.1. Theorem. Assume that for 0 ≤ i ≤ s, M_{i} extends to a Δ^{i} -module and C_{i}^{·} extends to a complex of Δ^{i} -modules.

a) If for 0 ≤ i ≤ s, (M_{i}, M_{i}^{∗}) (respectively (C_{i}^{·}, C_{i}^{·∗}) ) defines a Morita (respectively derived) equivalence between e_{i}OX_{i} and f_{i}OY_{i}, then M (respectively C) extends to Δ, and (A′ ⊗_{Δ′} M, (A′ ⊗_{Δ′} M)^{∗}) (respectively (A′ ⊗_{Δ′} C^{·}, (A ⊗_{Δ′} C^{·})^{∗}), provided that G_{i} and Σ_{i} are p′ -groups ) defines a Morita (respectively derived) equivalence between R′ and S′.

b) Assume in addition that for each i, the conditions (3.8.a,b,c) are satisfied by X_{i}, Y_{i}, X̃_{i}, Ỹ_{i}, G_{i} and α_{i}, and that Σ_{i} is a p′ -group. If for 0 ≤ i ≤ s, M_{i} (respectively C_{i}^{·}) defines a Puig (respectively Rickard) equivalence between e_{i}OX_{i} and f_{i}OY_{i}, then A′⊗_{Δ′} M (respectively A′ ⊗_{Δ′} C^{·}) defines a Puig (respectively Rickard) equivalence between R′ and S′.

The proof is a straightforward application of the results of Sections 3 and 4, and of the nice behavior of block equivalences with respect the direct products and subgroups described above. We leave the details to the reader, since the facts are much more simple then the notations which they involve.

(5.2) Since Broué’s conjectures are statements about the principal blocks, we are interested to know when the principal blocks of OX̃ and OỸ correspond under the above equivalences. More exactly, we want the 1-characters to correspond. So we have to look at the characters associated to M and C. Since the trivial characters correspond when passing to direct products and subgroups, we need only to analyze our basic situation of Section 3. We denote in this paragraph by 1_{X} the 1-character of KX. Assume that the complex C induces a derived equivalence between eOX and fOY , and let μ ∈ R(K[X × Y]) be the virtual character associated to C, that is, μ(x, y) = ∑_{i} tr((x, y^{−1}), C^{i}). (This sum has only the term i = 0 if we have a Morita equivalence.) It is well-known that μ induces a perfect isometry between eOX and fOY , by defining I_{μ}: R(KY f) → R(KXe), I_{μ}(ζ)(x) = \frac{1}{|Y |} ∑_{y∈Y} μ(x, y^{−1})ζ(y), and R_{μ}: R(KXe) → R(KY f), R_{μ}(ξ)(y) = \frac{1}{|X|} ∑_{x∈X} μ(x^{−1}, y)ξ(x). Thus μ determines a bijection J_{μ}: Irr(KXe) → Irr(KY f) and a sign function ϵ_{μ}: Irr(KXe) → {±1} such that R_{μ}(χ) = ϵ_{μ}(χ)J_{μ}(χ) for all χ ∈ Irr(KXe). Also μ may be written as μ = ∑_{χ∈Irr(KXe)} ϵ_{μ}(χ)(χ × \overline{J_{μ}(χ)}), where ζ̄(y) = ζ(y^{−1}) for ζ ∈ Irr(KY) and y ∈ Y . If C extends to a complex of Δ-modules as in Theorem 3.4, then μ also extends to a virtual character of δ(G). Let μ̃ be the isometry associated to A ⊗_{Δ} C. Then the 1-characters of X̃ and Ỹ will correspond (and consequently, the principal blocks of OX̃ and OỸ will correspond), if \frac{1}{|X̃|} ∑_{x∈X̃} μ̃(x^{−1}, y) = 1 for any y ∈ Ỹ . Remark also that if 1_{X} and 1_{Y} correspond via μ, and if 1_{δ(G)} (which is an extension of 1_{X} × ¯1_{Y} to δ(G)) is a component of the extension of μ to δ(G), then 1X̃ and 1Ỹ correspond via μ̃ by Frobenius reciprocity.

The hypothesis if Theorem 5.1 has to be verified for each 0 ≤ i ≤ s. We discuss here two situations when this checking is easy. (5.3) With the notations of Section 2, the case i = 0 is trivial. We make here slightly more general assumptions. Suppose that X = Y, X̃ = Ỹ , D is a Sylow p-subgroup of X and that G = X̃/X is a p′ -group. Let eOX be the principal block, and denote R_{1} = S_{1} = eOX, R = S = eOX̃, and let M = eOX regarded as an R_{1}, S_{1} -bimodule. Then M is in fact a Δ-module simply because X is normal in X̃, and it is clearly a relatively δ(D)-projective p-permutation Δ-module. It follows that M defines a Puig autoequivalence of eOX and M̃ = A⊗_{Δ}M ≃ R⊗_{R_{1}} M ≃ R defines a Puig autoequivalence of eOX̃. (5.4) Another nice situation which may occur for 1 ≤ i ≤ s is that of isomorphic blocks. Let X be a normal subgroup of X̃ such that X̃ = CX̃(D)X where D is a Sylow p-subgroup of X and G = X̃/X is a p′ -group. Let e and ẽ be the principal block idempotents of OX and OX̃ respectively. By [D1, Theorem], the map eOX → ẽOX̃, r ↦ ẽr is an isomorphism of O-algebras. It follows that the source algebras of eOX and ẽOX̃ are isomorphic as interior D-algebras. (See also [H] and the final remark of [K] for related results). Let V = eOX̃, regarded naturally as a right eOX̃ -module, and as a left R_{1} -module by scalar restriction. Then V is a relatively δ(D)-projective p-permutation O[X × X̃^{op}]-module and defines a Puig equivalence between eOX and ẽOX̃. We see that the functor V ⊗_{ẽOX̃} − is just ResX̃ ; X(−) on ẽOX̃ -mod.

Let now Y be a normal subgroup of another group Ỹ such that D is also a Sylow p-subgroup of Y , Ỹ = CỸ (D)Y and X̃/X ≃ Ỹ /Y , and let f, f̃ be the principal block idempotents of OY and OỸ , respectively. It follows that if eOX and fOY are Puig (respectively Rickard) equivalent, then ẽOX̃ and f̃OỸ are Puig (respectively Rickard) equivalent.

There is also another argument for this assertion. Remark that δ(G) = (X × Y^{op})C_{δ(G)}(D × D^{op}). If we denote by d the principal block idempotent of Δ, then the Oalgebras (e⊗f)O[X ×Y^{op}] and dΔ are isomorphic by [D1]. Assume that M (respectively C) defines a Puig (respectively Rickard) equivalence between eOX and fOY . Then M (respectively C) extends to Δ via this isomorphism, hence A⊗_{Δ} M (respectively A⊗_{Δ} C) defines a Puig (respectively Rickard) equivalence between R and S, such that ẽOX̃ and f̃OỸ are corresponding blocks.

Remark finally that if X, X̃ and D are as above, and Ỹ = NX̃(D), Y = N_{X}(D), then Ỹ = CỸ (D)Y , so these observations apply.

5.5. Example. Groups with cyclic Sylow p-subgroups. Assume that the finite group X has a cyclic Sylow p-subgroup D and denote Y = H_{X}(D). Let R_{1} = ekX and S_{1} = fkY be the principal blocks. In [Rou2, Theorem 10] a Rickard tilting complex C = (0 → C^{−1} →^{d} C^{0} → 0) for R_{1} and S_{1} was constructed. The term C^{0} was taken to be eOXf, and the term C^{−1} was constructed as follows. The (R_{1}, S_{1})-bimodule eOXf decomposes as eOXf = M ⊕ U, where M is a non-projective indecomposable R_{1} ⊗_{O} S_{1}^{op} -module, and U is a projective R_{1} ⊗_{O} S_{1}^{op} -module. The projective cover in R_{1} ⊗_{O} S_{1}^{op}-mod of M is V = ⨁_{0≤i≤ϵ}(P_{2i} ⊗_{O} Q_{2i}^{∗}) ↠^{ψ} M, where ϵ is the inertial index of the block eOX, while P_{2i} = P (Ω_{R_{1}}^{2i}O) and Q_{2i} = P (Ω_{S_{1}}^{2i}O) are the projective covers of the Heller translates. Denote W = ⨁_{j}(P_{2j} ⊗_{O} Q_{2j}^{∗}), where j runs over a certain subset of {0, …, ϵ − 1} (we do not need more details here), and let ϕ be the restriction of ψ to W. Let finally C^{−1} = W ⊕ U and d = ϕ ⊕ id.

Assume now that X is a normal subgroup of X̃ such that G = X̃/X is a p′ -group, and let Ỹ = NX̃(D). The notations A, δ(G) and Δ will have the same meaning as in Section 3. We show that C extends to a complex C̄ of Δ-modules. We point out that X̃ still has D as a cyclic Sylow p-subgroup, so by [Rou2, Theorem 10], there is a Rickard tilting complex for the principal blocks of OX̃ and OỸ , but the complex A ⊗_{Δ} C̄ given by Theorem 3.4 has G-graded modules as its components, so may offer additional information.

Remark first that C^{0} extends to a Δ-module by putting (x, y^{−1})c = xcy^{−1} , where x ∈ X̃, y ∈ Ỹ are such that xX = yX. Since M is the non-projective part and U is the projective part of C^{0} , it follows that M and U are G-invariant Δ_{1} -modules, and by [D3, Theorem 3.11], that M and U extend to Δ-modules.

Let ψ̄: V̄ → M be a projective cover of M in Δ-mod, hence V̄ /J(Δ)V̄ ≃ M/J(Δ)M. Since J(Δ) = J(Δ_{1})Δ = ΔJ(Δ_{1}), we have V̄ /J(Δ_{1})V̄ ≃ M/J(Δ_{1})M ≃ V/J(Δ_{1})V in Δ_{1} -mod, that is, Res_{Δ_{1}}^{Δ}(V̄) is a projective cover of M, so it is isomorphic to V . By the proof of [Rou2, Theorem 7], W and V/W do not have isomorphic direct summands in common. Then by [D3, Theorem 3.11], there is a Δ-direct summand W̄ of V̄ such that W̄ ≃ W in Δ_{1} -mod. Let ϕ̄ be the restriction of ψ̄ to W̄. Then

C̄ = (0 ⟶ W̄ ⊕ U ⟶^{ϕ̄⊕id} ekXf ⟶ 0)

is a complex of Δ-modules, and Res_{Δ_{1}}^{Δ}(C̄) is isomorphic to C as complexes of Δ_{1} -modules.

Notice also that by [D3, Corollary 5.7], Ω_{R}^{2i}(O) is an R-module extending Ω_{R_{1}}^{2i}(O), and Ω_{S}^{2i}(O) is an S -module extending Ω_{S_{1}}^{2i}(O). Then the above argument implies that P_{2i} extends to R, and Q_{2i} extends to S. It follows that V extends to an A-module, and hence to a Δ-module, but the Δ_{1} -linear map ψ may not be Δ-linear, so we have to look for another extension of V .

5.6. Example. The principal 3-block of A_{5} . Although this example fits into the situation of [P] (see also [Br3, p.14]), we shall deal with permutations instead of regarding A_{5} ≃ SL_{2}(4). So let X = A_{5} , D = ⟨(1 2 3)⟩ a Sylow 3-subgroup of X, Y = N_{X}(D) = D ⋊ ⟨y⟩, where y = (1 2)σ, σ = (4 5), and let U = {e, (1 2)(3 4), (1 4)(2 3), (1 3)(4 2)} be a Sylow 2-subgroup of X. If M = k[X/U] is the k -space with basis X/U, then X acts on M by left multiplication, and D acts on M by right multiplication. Moreover, M extends to a right kY -module by defining xUtσ = σxtU for t ∈ {(1 2), (2 3), (1 3)}. We have that M is a relatively δ(D)-projective 3-permutation k[X × Y^{op}]-module, and defines a Puig equivalence between the principal 3-blocks of kX and kY .

Let now X̃ = S_{5} = X ⋊ G, where G = ⟨g⟩, g = (1 2), and Ỹ = NX̃(D). We see that M extends to a Δ-module by defining an action of G on M: g · xU = x^{g}U, which is correct, since g normalizes U. Consequently, Corollary 3.9 applies.

5.7. Example. Symmetric groups. We end with an example which was considered in [Rou1]. Let X be the symmetric group S_{p}, and let w < p so the wreath-product X ≀S_{w} is a subgroup of S_{pw}. Let D be a (cyclic) Sylow p-subgroup of X and let e (respectively f) be the principal block idempotents of OX (respectively OY , where Y = N_{X}(D)). Then by [Rou2, Theorem 10] and the results of Section 4 above, we deduce that e^{⊗w}O[X ≀ S_{w}] and f^{⊗w}O[Y ≀ S_{w}] are Rickard equivalent.

Acknowledgements. This paper was written while I was enjoying the hospitality of Ecole Normale Supérieure at Paris, due to a grant offered by the French Ministry of Research and Technology. I am grateful to Michel Broué, Lluis Puig and Raphaël Rouquier for their kind support and helpful dicussions, and also to Morton Harris and Jeremy Rickard for pointing out the gaps in the first version of the paper and for many other useful observations.

References

[Al] J.L. Alperin, Local representation theory, Cambridge studies in advanced mathematics, vol. 11, Cambridge University Press, Cambridge, 1986.

[Br1] M. Broué, On Scott modules and p-permutation modules: an approach through the Brauer homomorphism, Proc. Amer. Math. Soc. 93 (1985), 401–408.

[Br2] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181–182 (1990), 61–92.

[Br3] M. Broué, Equivalences of blocks of group algebras, Proceedings of the International Conference on Representations of Algebras, Ottawa 1992, North-Holland, 1993.

[Br4] M. Broué, Rickard equivalences and block theory, Preprint LMENS (1994).

[D1] E.C. Dade, Remarks on isomorphic blocks, J. Algebra 45 (1977), 254–258.

[D2] E.C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), 241–262.

[D3] E.C. Dade, Extending group-modules in a relatively prime case, Math. Z. 186 (1984), 81–98.

[FH] P. Fong and M.E. Harris, On perfect isometries and isotypies in finite groups, Invent. Math. 114 (1993), 139–191.

[H] M.E. Harris, Categorically equivalent and isotypic blocks, J. Algebra 166 (1994), 232–244.

[K] B. Külshammer, Morita equivalent blocks in Clifford theory of finite groups, Astérisque 181–182 (1990), 209–215.

[P] L. Puig, Algèbres de source de certains blocs des groupes de Chevalley, Astérisque 181–182 (1990), 221–236.

[R1] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48.

[R2] J. Rickard, Splendid equivalences: derived categories and permutation modules, Preprint (1994).

[Rou1] R. Rouquier, Sur le blocs à groupe de défaut abélien dans les groupes symétriques et sporadiques, J. Algebra 168 (1994), 648–694.

[Rou2] R. Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, Preprint LMENS (1994).

[Str] M.A. Stricker, Morita equivalences and stable equivalences for wreath products of finite groups, Comm. Algebra 20 (1992), 3691–3701.

“Babeş–Bolyai” University , Department of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, RO-3400 Cluj–Napoca, Romania