TWISTED GROUP ALGEBRAS, NORMAL SUBGROUPS

AND DERIVED EQUIVALENCES

ANDREI MARCUS

“Babeş-Bolyai” University

Faculty of Mathematics and Computer Science

Str. Mihail Kogălniceanu nr. 1

RO-3400 Cluj-Napoca, Romania

13.04.2000

To Professor Klaus Roggenkamp on his 60th birthday

Abstract. We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade’s conjectures. We prove that an extended form of Broué’s conjecture implies Dade’s Inductive Conjecture in the abelian defect group case; this is a consequence of the fact that Rickard equivalences induced by complexes of graded bimodules preserve the relevant Clifford theoretical invariants. As an application, we show that these conjectures hold for p′ -extensions of blocks with with cyclic defect groups.

Introduction

Let K be a normal subgroup of a finite group H, G = H/K, and let O_{α}H be a twisted group algebra, where O is a complete discrete valuation ring with algebraically closed residue field k = O/J(O) of characteristic p > 0, quotient field K of characteristic 0 which is also a splitting field for all the K-algebras considered below, and α ∈ Z^{2}(H, O^{×}) is a cocycle.

E. Dade stated in [D4] several forms of a conjecture and one of them, the Inductive Conjecture, involves Clifford extensions associated to blocks of O_{α}K and to simple K_{α}K - modules.

1991 Mathematics Subject Classification. 20C20, 20C05, 20C25, 16W50, 16S35. Key words and phrases. Twisted group algebra, crossed product, Rickard equivalence, Clifford exten-

sions, linear source modules, cyclic defect groups.

This paper was written while the author was visiting the Mathematical Institute of the “Friedrich Schiller” University Jena. He thanks the Alexander von Humboldt Foundation for the financial support, and the people from the institute for their hospitality. The author is especially grateful to Burkhard Külshammer for numerous helpful suggestions. Thanks are also given to Lluís Puig for many useful descussions and comments.

The first aim of this paper is to provide a structural look to this conjecture. Let b be a block of O_{α}K with defect group D. Assume for the moment that b is G-invariant and let R = O_{α}Hb and R′ = O_{α}N_{H}(D)b′, where b′ ∈ Z(O_{α}N_{K}(D)) is the Brauer correspondent of b. Then R and R′ are strongly G-graded O-algebras with R_{1} = O_{α}Kb and R′_{1} = O_{α}N_{K}(D)b′. Moreover, by a theorem of M. Harris and R. Knörr, the Brauer correspondence induces a bijection between the blocks of R and the blocks of R′.

We say that the Rickard tilting complex C induces a G-graded derived equivalence between R and R′ (see [M1] and [M2]) if C is a complex of G-graded (R, R′)-bimodules, and the isomorphism

C^{∗} ⊗_{R} C ≃ R′, C ⊗_{R′} C^{∗} ≃ R

hold in the appropriate homotopy categories of graded bimodules.

The consideration of graded derived equivalences is also related to the following question. Let c be a block of R and let c′ be the Harris-Knörr correpondent of c. Assuming that the blocks b and b′ are derived equivalent, under what conditions does it follow that c and c′ are derived equivalent. We investigate graded Rickard equivalences in Section 1 in the more general setting of G-graded symmetric algebras, and we show that they preserve all the invariants appearing in Dade’s conjectures.

Our second aim is to study the local structure of graded derived equivalences. In sections 2 and 3 we show that a special type of graded derived equivalence, the so called splendid equivalences, between the blocks b and b′ of two twisted group algebras O_{α}K and O_{α′}K′ having the G-equivalent Brauer category induces, by applying the Brauer functor to the tilting complex, another graded derived equivalence between suitable subalgebras of the Brauer quotients (O_{α}H)(Q) and (O_{α′}H′)(Q), for any subgroup Q of D. In the case of principal blocks, this was done in [M1], and our result here is a common generalization of [M1, Corollary 3.9] and [Li2, Theorem 2.1]. Derived equivalences compatible with the local structure of blocks of group algebra were studied in full generality by L. Puig [Pu1]. Here we prefer to adapt the more particular and direct approach to splendid equivalences between arbitrary blocks of group algebra given by M. Linckelmann [Li1], [Li2] and M. Harris [Ha] (which in turn generalize the work of J. Rickard [Ri2]), since it will be enough for our final applications.

Dealing with twisted group algebras leads to the consideration of a generalization of p-permutation modules, the linear source modules introduced by R. Boltje [Bo]. As was shown in [BK], the Brauer construction works well when applied to such modules, which are involved in the definition of the splendid equivalence. We also need to develop some tools of local representation theory of twisted group algebras, in addition to what can be found in [D3] and [Pu2].

TWISTED GROUP ALGEBRAS, NORMAL SUBGROUPS AND DERIVED EQUIVALENCES 3

In section 4 we state an extended version of Broué’s abelian defect group conjecture, which (in the case of abelian defect groups), implies the Inductive Conjecture. Various forms of Dade’s conjecture were checked in many cases, one of them being the Projective Invariant conjecture for blocks with cyclic defect groups in [D5]. In sections 5 and 6 we prove that Rouquier’s construction in [Rou] of a splendid Rickard equivalence between OKb and ON_{K}(D)b′, where D is cyclic, generalizes to the twisted group algebra case, and if G is a p′ -group, it also yields a graded Rickard equivalence.

In this paper, rings will always be associative with unity element and modules are unitary, finitely generated and left, unless otherwise specified. Categories and functors are additive, and even linear over O, k or K, when this applies. The general reference for the topic of the paper are [D4], [KZ] and [Th]. We also refer to [Br2], [Li1] and [Li2] for general facts on various types of equivalences between blocks, and to [M4] for Clifford theory and representations of strongly graded algebras, and to [D3] and [Pu2] for results on twisted group algebras (O^{×} -groups). Our notations are standard, and we shall recall the needed definitions in each section.

1. Symmetric algebras and graded derived equivalences

1.1. Let O be a complete discrete valuation ring and assume that the residue field k = O/J(O) is algebraically closed of characteristic p > 0, the quotient field K has characteristic 0 and it is also a splitting field for all the K-algebras considered below. We shall assume that all O-modules are free of finite rank.

Let also a finite group G and let R = ⨁_{g∈G} R_{g} and S = ⨁_{g∈G} S_{g} be two G-graded crossed product O-algebras. Let hU(R) = ⋃_{g∈G}(R_{g}∩U(R)) be the group of homogeneous (graded) units of R, so we have the group extension

1 → U(R_{1}) → hU(R) → G → 1.

The group hU(R) acts on R by conjugation, such that for u_{g} ∈ hU(R) ∩ R_{g}, ^{u_{g}}R_{h} = R_{ghg^{−1}} . This induces an action of G on the centralizer C_{R}(R_{1}) of R_{1} in R, which is a G-graded subalgebra of R.

We assume that R and S are symmetric O-algebras, such that the symmetrizing forms of R and S are G-invariant symmetrizing forms for R_{1} and S_{1} respectively.

Denote KR = K ⊗_{O} R, kR = k ⊗_{O} R and assume that KR and KS (or equivalently KR_{1} and KS_{1} ) are semisimple K-algebras, and that K is a splitting field for all the K-subalgebras of KR and KS.

1.2. It is well-known that the above assumptions hold if R = bO_{α}H and R_{1} = bO_{α}K, where K is a normal subgroup of the finite group H, G = H/K, α ∈ Z^{2}(H, O^{×}) and b is a G-invariant central idempotent of O_{α}K.

Actually, if we start with an (H -graded) twisted group algebra K_{α}H, where α ∈ Z^{2}(H, K^{×}) and K is big enough, then by the results of [D3, Sections 6, 7], K_{α}H has a maximal H -graded suborder which is then a twisted group algebra of H over O. This means that we may choose α ∈ Z^{2}(H, O^{∗}). Moreover, there is a finite subgroup H′ of hU(O_{α}H) and an epimorphism OH′ → O_{α}H of H -graded algebras. As in [Pu2], we denote Ĥ = hU(O_{α}H). It follows that H′ is a finite subgroup of Ĥ covering H, and the kernel Z of the epimorphism H′ → H is a central subgroup of H′.

Recall also that the short exact sequence

1 → 1 + J(O) → O^{×} → k^{×} → 1

splits naturally, so any twisted group algebra k_{α}H can be lifted to a twisted group algebra over O.

1.3. By an (R, S)-bimodule we mean a module over R ⊗_{O} S^{op} . Observe that R ⊗_{O} S^{op} is a strongly G × G-graded O-algebra with components (R ⊗_{O} S^{op})_{(g,h)} = R_{g} ⊗_{O} S_{h}^{op} , and it has a strongly G-graded diagonal subalgebra

Δ = Δ(R ⊗_{O} S^{op}) = ⨁_{g∈G}(R_{g} ⊗_{O} S_{g}^{op}),

where S_{g}^{op} = S_{g^{−1}}. Remark that an (R_{1}, S_{1})-bimodule is the same as a Δ_{1} -module.

Recall that if O_{α}H and O_{β}H are twisted group algebras, then Δ(O_{α}H ⊗_{O} O_{β}H) ≃ O_{αβ}H (as H -graded algebras), and since (O_{α}H)^{op} is naturally isomorphic to O_{α^{−1}}H, we have that Δ(O_{α}H ⊗_{O} (O_{α}H)^{op}) ≃ OH.

A G-graded (R, S)-bimodule is, by definition, an R ⊗_{O} S^{op} -module with a decomposition M = ⨁_{x∈G} M_{x} such that R_{g}M_{x}S_{h} = M_{gxh}. By [M2, Section 2], we have that M_{1} is a Δ-module and there are natural isomorphisms

M ≃ (R ⊗_{O} S^{op}) ⊗_{Δ} M_{1} ≃ R ⊗_{R_{1}} M_{1} ≃ M ⊗_{S_{1}} S.

In particular, O_{α}H ≃ (O_{α}H ⊗_{O} (O_{α}H)^{op}) ⊗_{OH} O as H -graded (OH, OH)-bimodules.

1.4. We say that R and S are graded derived equivalent if there are bounded complexes C of G-graded (R, S)-bimodules and D of G-graded (S, R)-bimodules inducing a derived equivalence between R and S such that the canonical maps C⊗^{L}_{S}D → R and D⊗^{L}_{R}C → S are isomorphisms in the respective derived categories of G-graded bimodules.

As was pointed out by J. Rickard in [KZ, Section 9.2.2], in the context of symmetric algebras the definition can be simplified. A (G-graded) two-sided tilting complex C can be chosen such that C is projective as left and as right module, and all except one term is projective as a bimodule. The inverse equivalence is induced by the O-dual C^{∗} of C, which is a complex G-graded (S, R)-bimodules. Actually, we have the isomorphisms C⊗_{S} C^{∗} ≃ R and C^{∗}⊗_{S} C ≃ S in the appropriate chain homotopy categories of G-graded bimodules. Therefore, the functors C ⊗_{S} − and C^{∗} ⊗_{R} − are equivalences between the chain homotopy categories K^{b}(R-mod) and K^{b}(S-mod).

1.5. Let H be a subgroup of G, and let D^{b}(KR_{H}) be the bounded derived category of the category KR_{H}-mod.

It is well-known that the canonical embedding of KR_{H}-mod into D^{b}(KR_{H}) induces an isomorphism

K_{0}(KR_{H}) ≃ K_{0}(D^{b}(KR_{H})).

We identify these two groups and we denote by [X] ∈ K_{0}(KR_{H}) the class of an object X ∈ D^{b}(KR_{H}). The group K_{0}(KR_{H}) is endowed with a scalar product defined by

⟨[X], [X′]⟩ = ∑_{i} dim_{K} Hom_{D^{b}(KR_{1})}(X[i], X′).

For any g ∈ G, the functor

KR_{g} ⊗_{KR_{H}} −: KR_{1}-mod → KR_{^{g}H}-mod

is an equivalence with inverse KR_{g^{−1}} ⊗_{KR_{1}} −. In particular, if H is a normal subgroup of G, we have that G/H acts on KR_{H}-mod, on D^{b}(KR_{H}) and also on the corresponding Grothendieck groups K_{0}(KR_{H}) and K_{0}(D^{b}(KR_{H})).

By definition, the stabilizer of X ∈ D^{b}(KR_{1}) is the subgroup

G_{X} = {g ∈ G | KR_{g} ⊗_{KR_{1}} X ≃ X in D^{b}(KR_{1})}

of G. The above scalar product on K_{0}(KR_{1}) is clearly G-invariant, and the G-set Irr(KR_{1}) of isomorphism classes of simple KR_{1} -modules is an orthonormal ℤ-basis of K_{0}(KR_{1}).

1.6. Let X be a simple KR_{1} -module. Since K is a splitting field for KR_{1} , we have that

KE(X) = End_{KR}(KR ⊗_{KR_{1}} X)^{op}

is a twisted group algebra of K and G_{X}, and KR ⊗_{KR_{1}} X is a G-graded (KR, KE(X))- bimodule. The group of homogeneous units of KE(X) is a central extension of K^{∗} by G_{X}, and it is often called the Clifford extension of X.

Assume that X belongs to the p-block b of R_{1} . The twisted group algebra KE(X) has a unique G_{X} -graded suborder E(X), and let kE(X) = k ⊗_{O} E(X). Then kE(X) is a twisted group algebra of G_{X} over k, and its group of homogeneous units is called the residual Clifford extension of X.

If V is a simple KE(X)-module belonging to the p-block β of E(X), then X ⊗_{K} V ≃ (bKR ⊗_{bKR_{1}} X) ⊗_{KE(X)} V is a simple KR-module lying over X, and the block Bc of R to which X ⊗_{K} V belongs lies over b; then c is said to be linked to β by the Clifford theory of X (see [D3, Section 13]).

1.7. Let further D^{b}(kR_{H}) be the bounded derived category of kR_{H}-mod and let D_{perf}^{b}(kR_{H}) be the full subcategory of D^{b}(kR_{H}) consisting of perfect complexes (that is, complexes isomorphic to complexes of projective kR_{1} -modules), and denote by kR_{H}-proj the subcategory of finitely generated projective kR_{H} -modules.

Let K_{0}(D^{b}(kR_{H})) be the Grothendieck group of D^{b}(kR_{H}), and K_{0}(kR_{H}-proj) the Grothendieck group of the subcategory kR_{H}-proj of finitely generated projective kR_{H} - modules. The set Irr(kR_{1}) of isomorphism classes of simple kR_{1} -modules is a ℤ-basis of K_{0}(D^{b}(kR_{H})), while the set Pim(kR_{H}) of isomorphism classes of projective indecomposable projective kR_{H} -modules is a ℤ-basis of K_{0}(kR_{H}-proj).

By [KZ, Proposition 6.3.3], the canonical embedding induces an isomorphism between the Grothendieck groups

K_{0}(kR_{H}) ≃ K_{0}(D^{b}(kR_{H}))

and

K_{0}(kR_{H}-proj) ≃ K_{0}(D_{perf}^{b}(kR_{H})).

There is a pairing

K_{0}(kR_{H}-proj) × K_{0}(kR_{H}) → ℤ,

given by

⟨[P], [X]⟩ = ∑_{i}(−1)^{i} dim_{k} Hom_{D^{b}(kR_{H})}(P [i], X),

defining a duality between K_{0}(kR_{H}-proj) and K_{0}(kR_{H}).

For any g ∈ G, the functor

kR_{g} ⊗_{kR_{H}} −: kR_{H}-mod → kR_{^{g}H}-mod

induces an isomorphism between K_{0}(kR_{H}) and K_{0}(kR_{^{g}H}), and between K_{0}(kR_{H}-proj) and K_{0}(kR_{^{g}H}-proj).

In particular, G acts on K_{0}(kR_{1}) and K_{0}(kR_{1}-proj), and we denote by G_{X} and G_{P} the stabilizers of X ∈ Irr(kR_{1}) and P ∈ kR_{1}-proj. Also, the above duality between K_{0}(kR_{1}) and K_{0}(kR_{1}-proj) is G-invariant.

1.8. If X is an indecomposable R_{1} -module, then, since k is algebraically closed, we have that

kE(X) = End_{kR}(kR ⊗_{kR_{1}} X)^{op}/J_{gr}(End_{kR}(kR ⊗_{kR_{1}} X)^{op})

is a twisted group algebra of k and G_{X}, where J_{gr} denotes the graded Jacobson radical, and as above, its group of homogeneous units is a central extension of k^{∗} and G_{X} called the Clifford extension of X.

1.9. The Cartan-Decomposition triangle T (R_{H}) of R_{H} is the commutative diagram

\xymatrix{K_{0}(KR_{H}) \ar[rr]^{Dec} &  & K_{0}(kR_{H}) \\  & K_{0}(kR_{H}-proj) \ar@{-->}[ul]^{_{_{t} Dec}} \ar@{-->}[ur]_{Car}}

where the maps are defined as follows.

If P is a perfect complex of kR_{H} -modules, then the Cartan map Car: K_{0}(kR_{H}-proj) → K_{0}(kR_{H}) sends the class [P] of P to its class in K_{0}(kR_{1}).

The decomposition map Dec: K_{0}(KR_{H}) → K_{0}(kR_{H}) sends the class [X] of a KR_{H} - module X to the class of k ⊗_{O} X_{0} ∈ kR_{1}-mod, where X_{0} is an R_{H} -lattice such that X ≃ K ⊗_{O} X_{0} .

Reduction modulo J(O) is an isomorphism between K_{0}(R_{H}-proj) and K_{0}(kR_{H}-proj). Using this, the adjoint

t Dec: K_{0}(R_{H}-proj) → K_{0}(KR_{H})

sends [P] to [K ⊗_{O} P].

These maps are compatible with the metric structure of T (R_{H}), and notice that if H is a normal subgroup of G, then Dec, ^{t} Dec and Car are G/H -maps, that is, Car^{g}[P] = gCar[P], Dec(^{g}[X]) and ^{t} Dec^{g}[X] = ^{g}(^{t} Dec[X]) for any g ∈ G.

1.10. The enveloping algebra R^{en} = R ⊗_{O} R^{op} of R is a strongly G × G-graded algebra with components R_{(g,h)}^{en} = R_{g} ⊗_{O} R_{h}^{op} , where R_{h}^{op} = R_{h^{−1}} . The group G × G acts on the category of (R_{1}, R_{1})-bimodules by X ↦ ^{(g,h)}X = R_{g} ⊗_{R_{1}} X ⊗_{R_{1}} R_{h^{−1}}, and G acts on the set of ideals of R_{1} by I ↦ ^{g}I = R_{g}IR_{g^{−1}}.

If b is a block idempotent of R_{1} , then let

G_{b} = {g ∈ G | R_{g}bR_{g^{−1}} = bR_{1}}

denote the stabilizer of b. Then bRb is a strongly G_{b} -graded algebra. Moreover,

Tr_{G_{b}}^{G}(b) = ∑_{g∈[G/G_{b}]}^{g}b

is a G-invariant central idempotent of R_{1}. In order to simplify the notation, we may assume that b is G-invariant, that is, G_{b} = G, and denote A = bR = Rb. Regard A = ⨁_{g∈G} bR_{g} as a G-graded (R_{1}, R)-bimodule (or as a module over the strongly Ggraded algebra R_{1} ⊗_{O} R^{op}), and consider the endomorphism ring

C = End_{R_{1}⊗_{O}R^{op}}(A)^{op}.

Then C is a G-graded ring and C may be identified with the opposite G-graded subring bC_{R}(R_{1}) of R. Since b is a block of R_{1} , we have that C_{1} ≃ End_{R_{1}⊗_{O}R_{1}^{op}}(A_{1}) ≃ bZ(R_{1}) is a commutative local ring.

We are interested in the stabilizer G[b] of bR_{1} regarded as a Δ_{1} -module. Clearly, G[b] is the largest subgroup H of G for which C_{H} is a crossed product, and we have that

G[b] = {g ∈ G | bR_{g} ≃ bR_{1} as Δ_{1}-modules}
= {g ∈ G | C_{g}C_{g^{−1}} = C_{1}}
= {g ∈ G | bC_{R_{g}}(R_{1})bC_{R_{g}^{−1}}(R_{1}) = bZ(R_{1})}

is a normal subgroup of G_{b} = G.

Since k is algebraically closed and A_{1} is an indecomposable R_{1} ⊗_{O} R_{1}^{op} -module, we have as in 1.8 that C/J_{gr}(C) is a twisted group algebra of k and G[b], which will be denoted by kE(b). Again, the group of homogeneous units of kE(b) is called the Clifford extension of the block b of R_{1} .

1.11. Let X be a simple KR_{1} -module belonging to the p-block b of R_{1} . We have that G[b] ⊆ G_{X} ⊆ G_{b}, and with the notations of 1.6 and 1.10, there is a natural grade preserving monomorphism between kE(b) and kE(X), which is also compatible with the action of G_{X} on these algebras. Since this was stated in [D1, 13.10] and [D4, 5.3] for twisted group algebras, let us give a short argument in our situation.

The inclusion G_{X} ⊆ G_{b} is clear, and if g ∈ G[b], then bR_{g} ≃ bR_{1} as (R_{1}, R_{1})- bimodules, hence KR_{g}b ⊗_{KR_{1}b} X ≃ X as KR_{1} -modules, so g ∈ G_{X}. Moreover, if r ∈ C_{g} = bC_{R}(R_{1})_{g}^{op} , then multiplication with r on KR_{G[b]} ⊗_{KR_{1}} X gives an endomorphism μ_{r} of degree g; thus we have the monomorphism kE(b) → kE(X). Finally, if h ∈ G_{X}, u ∈ hU(R)∩R_{h} and r ∈ C_{g}, then it is easy to see that the endomorphism μ_{uru^{−1}} induced by multiplication with uru^{−1} is ^{h}μ_{r} (where by definition ^{h}ϕ_{r} = ψ_{h}μ_{r}ψ_{h}^{−1} , with ψ_{h} an invertible element of degree h in kE(X)), that is, μ_{^{h}r} = ^{h}μ_{r}.

We may now conclude the above discussion.

1.12. Theorem. Assume that the complex C of G-graded (R, S)-bimodules and its Odual induces a graded derived equivalence between R and S. Then the following statements hold:

a) For each subgroup H of G, there is an isomorphism between the Cartan - Decomposition triangles T (R_{H}) and T (S_{H}), preserving the canonical bases.

These isomorphisms are compatible with the restriction, induction, truncation and conjugation functors.

b) If b is a block idempotent of R_{1} , then b′S_{1} = C_{1}^{∗} ⊗_{R_{1}} bR_{1} ⊗_{R_{1}} C_{1} is a block of S_{1}, G_{b} = G_{b′} and bCb′ induces a G_{b} -graded derived equivalence between bRb and b′Sb′, while C induces a G-graded derived equivalence between R Tr_{G_{b}}^{G} b = RbR and S Tr_{G_{b}}^{G} b′ = Sb′S.

c) Assume that the simple KR_{1} -module X belonging to the p-block b corresponds to the simple KS_{1} -module X′ belonging to the p-block b′. Then we have the natural isomorphisms KE(X) ≃ KE(X′) of G_{X} -graded G_{X} -algebras, kE(b) ≃ kE(b′) of G[b]-graded G_{b} -algebras, and these isomorphisms are compatible with the monomorphism kE(b) → kE(X) of 1.11 and with the linkage of blocks defined in 1.6.

Proof. a) By [M2, Corollary 3.5], we have that for each subgroup H of G, C_{H} and C_{H}^{∗} ≃ (C_{H})^{∗} induce a H -graded derived equivalence between R_{H} and S_{H}, and that these equivalences commute with restriction, induction, truncation and conjugation. Now use [Br, Proposition 3.3], which says that K ⊗_{O} C_{H} and K ⊗_{O} C_{H}^{∗} induce a derived equivalence (clearly graded) between KR_{H} and KS_{H}, while k ⊗_{O} C_{H} and k ⊗_{O} C_{H}^{∗} induce a derived equivalence between kR_{H} and kS_{H}.

b) follows by a) and from the fact that C ⊗_{O} C^{∗} induces a G × G-graded derived equivalence between R^{en} and S^{en} , sending R to S.

c) Regard X as a complex concentrated in degree 0. Then the image of X in the derived category of KS_{1}-mod is X′ in degree 0 or a shift of it. Consequently, ϕ ↦ id_{KC^{∗}} ⊗_{KR} ϕ gives the natural grade-preserving isomorphism

End_{KR}(KR ⊗_{KR_{1}} X)^{op} ≃ End_{D^{b}(KR)}(KR ⊗_{KR_{1}} X)^{op}
≃ End_{D^{b}(KS)}(KC^{∗} ⊗_{KR} (KR ⊗_{KR_{1}} X))^{op},

where we denoted KC^{∗} = K ⊗_{O} C^{∗}. By [M2, Lemma 2.9], we have that

S ⊗_{S_{1}} C_{1}^{∗} ≃ C_{1}^{∗} ⊗_{R_{1}} R

as complexes of G-graded (S, R)-bimodules, hence

KS ⊗_{KS_{1}} (KC_{1}^{∗} ⊗_{KR_{1}} X) ≃ C_{1}^{∗} ⊗_{KR_{1}} (KR ⊗_{KR_{1}} X)

as complexes of G-graded S -modules. It follows that we have a natural isomorphism

End_{KR}(KR ⊗_{KR_{1}} X)^{op} ≃ End_{KS}(KS ⊗_{KS_{1}} (KC_{1}^{∗} ⊗_{KR_{1}} X))^{op}

of G_{X} -graded K-algebras. The action of G_{X} on KE(X) is defined as follows: if g ∈ G and ϕ ∈ KE(X), then ^{g}ϕ = u_{g}ϕu_{g^{−1}} , where u_{g} is an invertible element of degree g in KE(X), so the above isomorphism is a G_{X} -isomorphism by naturality.

By b), we may assume that G_{b} = G, so bRb = bR and b′Sb′ = b′S. For any subgroups H and K of G, we have that the given derived equivalence induce equivalences between the chain homotopy categories K^{b}(R_{H}-mod-R_{K}) and K^{b}(S_{H}-mod-S_{K}) of (R_{H}, R_{K})-bimodules and (S_{H}, S_{K})-bimodules respectively. We apply this observation and b) to the G-graded (R_{1}, R)-bimodule A = bR and we conclude that there are the isomorphisms

C^{op} = End_{R_{1}⊗_{O}R^{op}}(A) ≃ End_{D^{b}(R_{1}⊗_{O}R^{op})}(A)
≃ End_{D^{b}(S_{1}⊗_{O}S^{op})}(C_{1}^{∗} ⊗_{R_{1}} A ⊗_{R} C)
≃ End_{D^{b}(S_{1}⊗_{O}S^{op})}(C_{1}^{∗} ⊗_{R_{1}} bR_{1} ⊗_{R_{1}} C_{1} ⊗_{S_{1}} S)
≃ End_{D^{b}(S_{1}⊗_{O}S^{op})}(b′S_{1} ⊗_{S_{1}} S)
≃ End_{S_{1}⊗_{O}S^{op}}(b′S) = C′^{op}

of G-graded O-algebras. Therefore, by 1.10, we have the isomorphism kE(b) ≃ kE(b′) of G[b]-graded k -algebras. The fact that this is also an isomorphism of G-algebras follows from the following more general fact. Let Y , Z be objects of K^{b}(R-mod) and let Y′ = C^{∗} ⊗_{R} Y ≃ C_{1}^{∗} ⊗_{R_{1}} Y , Z′ = C^{∗} ⊗_{R} Z ≃ C_{1}^{∗} ⊗_{R_{1}} Z be the objects of K^{b}(S-mod) corresponding to Y , Z respectively. We may regard Y , Z as objects of K^{b}(R_{1}-mod) by scalar restriction; then Hom_{K^{b}(R_{1}-mod)}(Y, Z) is a KG-module, and the natural map

Hom_{K^{b}(R_{1}-mod)}(Y, Z) → Hom_{K^{b}(S_{1}-mod)}(Y′, Z′)

induced by C^{∗} is kG-linear. This applies to our situation since bR is a G-graded R ⊗_{O} R^{op} -module, and the G-action arises by regarding bR as an R_{1} ⊗_{O} R^{op} -module.

With the notations of 1.11 observe that the following diagram of graded G_{X} -algebras is commutative.

kE(b) ⟶^{kC^{∗}⊗_{kR}(−)⊗_{kR}kC} kE(b′)
μ↓  ↓^{μ′}
KE(X)  ⟶_{kC^{∗}⊗_{kR}−}  KE(X′)-mod

Finally for the compatibility with the linkage of blocks observe that the diagram

(KRb | X)-mod ⟵^{(KRb⊗_{KR_{1}b}X)⊗_{KE(X)}−} KE(X)-mod
C^{∗}⊗_{KRb}↓  ↓^{≃}
(KSb′ | X′)-mod ⟵_{b′KS⊗_{KS_{1}b′} X′)⊗_{KE(X′)}−} KE(X′)-mod

is commutative, where (KRb | X)-mod denotes the category of bKR-modules lying over X. □

1.13. We have that the 1-component C_{1} of C induces a derived equivalence between R_{1} and S_{1} . In fact, C_{1} is a complex of Δ-modules, where Δ = Δ(R ⊗_{O} S^{op}), so in particular, C_{1} satisfies

R_{g} ⊗_{R_{1}} C_{1} ⊗_{S_{1}} S_{g^{−1}} ≃ C_{1} in D^{b}(Δ_{1}).

This shows that graded derived equivalences are stronger than the equivariant derived equivalences considered in [M3].

On the other hand, let H be a normal p′ -subgroup of G and C_{1} a complex of (R_{1}, S_{1})- bimodules inducing a G-equivariant derived equivalence between R_{1} and S_{1} . Assume also that C_{1} extends to Δ_{H} = ⨁_{h∈H} Δ_{h}, and that the isomorphism between Δ_{g} ⊗_{Δ_{1}} C_{1} and C_{1} holds in D^{b}(Δ_{H}). If D = (R_{H} ⊗_{O} S_{H}^{op})⊗_{Δ_{H}} C_{1} , then by [M3, Lemma 2.10], D induces a G/H -equivariant derived equivalence between R_{H} and S_{H}.

2. p-Monomial modules and the Brauer construction

2.1. Recall that an O-lattice V is called a permutation module if it is a direct sum of modules of the form Ind_{H}^{G} O, where H is a subgroup of G, and V is called a trivial source module if Res_{P}^{G} V is a permutation module, where P is a Sylow p-subgroup of G. This amounts to say that that V has a P -stable basis.

A generalization of this concept was introduced in [Bo]. The OG-lattice V is called monomial if there is a G-set X and a decomposition V = ⨁_{x∈X} V_{x} of V into Osubmodules which are free of O-rank 1, or equivalently, if V is a direct sum of modules of the form Ind_{H}^{G} O_{ϕ}, where O_{ϕ} is an OH -module of O-rank 1 (so it is determined by a group homomorphism ϕ: H → O^{×}. Finally, V is called a linear source module if for a Sylow p-subgroup P of G, Res_{P}^{G} V is a monomial module.

Trivial source modules are also called p-permutation modules, and by analogy, linear source modules will be called here p-monomial modules.

2.2. As it was shown by R. Boltje and B. Külshammer, the Brauer construction works well with respect to p-monomial modules, [BK] containing generalizations of many results from [Br1]. Actually, they even considered the so called “generalized fixed points” V^{(H,ϕ)} , but here we shall not need this level of generalization. So recall that for any OG-module V and any P subgroup P of G, the Brauer construction (quotient) of V with respect to P is the kN_{G}(P)-module

V (P) = Br_{P}^{OG}(V) = V^{P} /( ∑_{Q<P} Tr_{Q}^{P}(V^{Q}) + J(O)V^{P})

= k ⊗_{O} (V^{P} / ∑_{Q<P} Tr_{Q}^{P}(V^{Q})).

Let also Br_{P}^{OG}: V^{P} → V (P) be the canonical epimorphism of ON_{G}(P)-modules.

By [Br1, (1.3)], if V is relatively H -projective and P is not conjugate to a subgroup of H, the V (P) = 0.

2.3. Let V be a p-monomial OG-module, P a p-subgroup of G, and let V = ⨁_{x∈X} V_{x}, where V_{x} is O-free of rank 1 and X is a set on which a Sylow p-subgroup of N_{G}(P) acts.

We denote by P \X the set of orbits P x of P on X, by [P \X] a complete set of representatives for these orbits, and P_{x} is the stabilizer in P of an element x ∈ X. (2.3.1) We have the equality

V^{P} = ⨁_{x} Tr_{P_{x}}^{P} V_{x},

where x runs over the set of elements of [P \X] with the property that V_{x} is a trivial OP_{x} -module. In particular, V^{P} is a p-monomial ON_{G}(P)-module. (2.3.2) Let further Q be a subgroup of P. If J(O) = 0, then

Tr_{Q}^{P}(V^{Q}) = ⨁_{x} Tr_{P_{x}}^{P} V_{x},

where x runs over the set of elements of [P \X] with the property that V_{x} is a trivial OP_{x} -module, and there exists y ∈ P x such that P_{y} ⊆ Q.

(2.3.3) There is a decomposition

Br_{Q}(V^{P}) = ⨁_{x} Br_{Q}(Tr_{Q_{x}}^{Q} V_{x}),

where x runs over the set of elements of [P \X] with the property that V_{x} is a trivial OP_{x} -module and P x ∩ V^{Q} ≠ ∅. (2.3.4) If J(O)V = 0, then

V^{P} ∩ ∑_{R<Q} Tr_{R}^{Q}(V^{R}) = ⨁_{x} Tr_{P_{x}}^{P} V_{x},

where x runs over the set of elements of [P \X] with the property that V_{x} is a trivial OP_{x} -module and P x ∩ V^{Q} ≠ ∅. In particular, V (P) is a p-permutation kN_{G}(P)-module with decomposition

V (P) = ⨁_{x} Br_{P} (V_{x}),

where x runs over the set of elements of [P \X] with the property that P_{x} = P and V_{x} is a trivial OP_{x} -module.

These are generalizations of [BrPu, Lemmas 1.3 and 1.4], and see also [BK, Lemma 1.5 and Proposition 2.4] for even more general versions of (2.3.1) and of the last statement. (2.3.5) As a consequence of (2.3.4), we have that if V = ⨁_{x∈X} V_{x} and W = ⨁_{y∈Y} W_{y} are p-monomial OG-modules, then there is an isomorphism

(V ⊗_{O} W)(P) ≃ V (P) ⊗_{k} W (P),

of kN_{G}(P)-modules, sending Br_{P} (V_{x} ⊗_{O} W_{y}) onto Br_{P} (V_{x}) ⊗_{k} Br_{P} (W_{y}), for all components V_{x} and W_{y} fixed by P (see also [Ha, Lemma 2.1]).

2.4. Using [BK, Proposition 2.4] we get the following generalization of [Br1, (1.4)] (which can also be obtained using (2.2) and Mackey’s formula).

Let H be a subgroup of G, ϕ ∈ Hom(H, O^{×}), P a p-subgroup of G,

T_{G}(P, H) = {g ∈ G | P ≤ ^{g}H = gHg^{−1}},

and let T be a system of representatives for the double cosets in N_{G}(P)\T_{G}(P, H)/H. Then

(Ind_{H}^{G} O_{ϕ})(P) = ⨁_{\begin{matrix}g∈T \\ Res_{P}^{^{g}H}(^{g}O_{ϕ})≃O \end{matrix}} Ind_{N_{^{g}H}(P)}^{N_{G}(P)}(^{g}k_{ϕ}).

2.5. Let H be a subgroup of G and let P ≤ D be p-subgroups of H such that C_{G}(P)T_{H}(P, D) = T_{G}(P, D). If V is a relatively D-projective p-monomial OH -module, then

Ind_{C_{H}(P)}^{C_{G}(P)}(Res_{C_{H}(P)}^{H_{H}(P)} V (P)) ≃ Res_{C_{G}(P)}^{N_{G}(P)}((Ind_{H}^{G} V)(P))

as kC_{G}(P)-modules.

This follows without difficulty by 2.4 and by adapting the proof of [M1, Lemma 3.7].

2.6. Our motivation for the above study come from the fact that, by 1.3, the twisted group algebra A = O_{α}H is a p-monomial OH -module, that is, a p-monomial H -algebra. More generally, if K is a normal subgroup of H, then O_{α}K is a p-monomial H -algebra.

Let \{h̄ | h ∈ H\} be and O-basis of A such that h̄h̄′ = α(h, h′)\overline{hh′} for all h, h′ ∈ H. Then by 2.3 we have that A(P) is the twisted group algebra

A(P) = ∑_{h∈C_{H}(P), P fixes h̄} k Br_{P} (h̄) = kᾱC_{H}(P̂),

where C_{H}(P̂) is the subgroup of H fixing the elements of P̂ ≤ Ĥ = hU(A), and ᾱ ∈ Z^{2}(C_{H}(P̂), k^{×}) is the image of α under reduction modulo J(O) and restriction to C_{H}(P̂). Usually, we shall denote this cocycle by the same α.

Let G = H/K and consider the subgroup G(P̂) = C_{H}(P̂)/C_{K}(P̂) of G. It follows that A(P) is a G(P̂)-graded N_{G}(P)-algebra over k.

2.7. If L is a subgroup of H, the the L-graded subalgebra A_{L} = O_{α}L of A can be regarded as a G-graded algebra, with components equal to zero outside LK/K. A G-graded O-algebra R is called a G-graded A_{L} -interior algebra, if there is a gradepreserving O-algebra map A_{L} → R.

Let D be a p-subgroup of H and let R be a G-graded A_{D} -interior algebra. By definition, a G-graded monomial (A_{D}, A_{D})-bimodule is a G-graded bimodule M with a decomposition M = ⨁_{x∈X} M_{x} into free O-submodules of rank 1, where is a set acted by the diagonal subgroup δ(D) = {(h, h) | h ∈ D}, and for any x ∈ X, the elements of M_{x} are homogeneous (with respect to the G-grading).

In particular, let i be an idempotent in A^{D}; then we have the decomposition

A = iAi + iA(1 − i) + (1 − i)Ai + (1 − i)A(1 − i)

of A into a direct sum of G-graded (A_{D}, A_{D})-bimodules.

2.8. If P, Q are subgroups of D, then

A = ⨁_{x∈[P \H/Q]} A_{P xQ}

is a decomposition into a direct sum of indecomposable (A_{P} , A_{Q})-bimodules; therefore, any indecomposable (A_{P} , A_{Q})-bimodule summand of iAi is of the form A_{P xQ}, with x ∈ H. These bimodules are related to the local structure of blocks (see 3.5 below) so let us discuss them in more detail.

Since A_{P xQ} = ⨁_{y∈P xQ} A_{y} and P xQ ⊆ H, we may regard A_{P xQ} as an H -graded (and also G-graded) (A_{P} , A_{Q})-module in am obvious way.

From another point of view, A_{P xQ} is an A_{P} ⊗_{O} A_{Q}^{op} -module graded by the transitive P × Q-set P xQ, and A_{P} ⊗_{O} A_{Q}^{op} is a P × Q-graded algebra. The stabilizer of x is the subgroup S = {(y, x^{−1}yx) | y ∈ P ∩^{x}Q} of P × Q, so the map δ_{x}: P ∩^{x}Q → S, y ↦ (y, x^{−1}yx) is a group isomorphism. It follows that we have the natural isomorphism

A_{P xQ} ≃ (A_{P} ⊗_{O} A_{Q}^{op}) ⊗_{(A_{P} ⊗_{O}A_{Q}^{op})_{S}} A_{x},

and (A_{P} ⊗_{O} A_{Q}^{op})_{S} is isomorphic to a twisted group algebra O_{α_{x}}(P ∩^{x}Q). In the particular case when x = 1, we have that α_{x} is the trivial cocycle.

Alternatively, denoting T = P ∩ ^{x}Q, the group homomorphism T → Q, y ↦ y^{x} induces an O-algebra homomorphism ϕ: A_{T} → A_{Q}; then A_{Q} has a structure of an (A_{T} , A_{Q})-bimodule, denoted by _{ϕ}A_{Q}, and the map

A_{P xQ} → A_{P} ⊗_{A_{T}} _{ϕ}A_{Q}, ūx̄v̄ ↦ ū ⊗ v̄

is an isomorphism of (A_{P} , A_{Q})-bimodules.

3. The Brauer category and local equivalences

3.1. Let K be a normal subgroup of a finite group H, let G = H/K, and consider the twisted group algebra A = O_{α}H, regarded here as a G-graded algebra. Let b be a block of A_{1} = O_{α}K, N = H_{b} = {x ∈ H | ^{x}b = b} and G_{b} = N/K. Then β = {b} is a point of both N and K on A_{1} , and ω = {Tr_{N}^{H} b} is a point of H on A_{1} . Observe that A_{ω} = A Tr_{N}^{H} b is a G-graded subalgebra of A, and A_{β} = bAb = bO_{α}N is a G_{b} -graded subalgebra.

A defect pointed group D_{δ} of N_{β} is also a defect pointed group of H_{ω}. Recall thet D is a minimal p-subgroup of N such that b ∈ Tr_{D}^{N}(A_{1})^{D}, or equivalently, a maximal p-subgroup of N such that Br_{D}^{A_{1}}(b) ≠ 0. Moreover, by [KP, Proposition 5.3] (generalized to twisted group algebras) we have that D̄ = DK/K is a Sylow p-subgroup of G_{b}, and there is a defect pointed group D̃δ̃ such that D̃δ̃ ≤ D_{δ}, and we also have that D̃ = D∩K.

Let Aδ̃ = jAj, where j ∈ (bA_{1})D̃ is a primitive idempotent such that BrA_{1} ; D̃(j) ≠ 0. Then Aδ̃ is a G-graded subalgebra of A, and the interior O_{α}D̃-algebra jA_{1}j is a source algebra of the interior O_{α}K -algebra bA_{1} . With these notations we have the following.

3.2. Proposition. a) A_{ω} ≃ Ind_{O_{α}N}^{O_{α}H} A_{β} as G-graded interior O_{α}H -algebras, so A_{ω} and A_{β} are Morita equivalent.

b) Aδ̃ is a crossed product of jA_{1}j and G_{b}, and it is graded Morita equivalent to A_{β}.

c) For any local pointed group D̃δ̃ ≤ P_{γ} ≤ D_{δ}, A_{γ} is a strongly G_{b} -graded interior O_{α}P -algebra, graded Morita equivalent to A_{β}.

Proof. a) We have that A_{β} is a G_{b} -graded interior O_{α}N -algebra, so we may consider the induced interior O_{α}H -algebra

Ind_{O_{α}N}^{O_{α}H} A_{β} = O_{α}H ⊗_{O_{α}N} A_{β} ⊗_{O_{α}N} O_{α}H,

with structural map x̄ ↦ ∑_{y∈[H/N]} x̄ȳ⊗a⊗ȳ^{−1} , where {x̄ | x ∈ H} is an O-basis of O_{α}H as in 2.6, and [H/N] is a complete set of representatives for the left cosets {xK | x ∈ H}. Then Ind_{O_{α}N}^{O_{α}H} A_{β} is a G-graded O-algebra, where for any g ∈ G,

(Ind_{O_{α}N}^{O_{α}H} A_{β})_{g} = ⨁_{\begin{matrix}xyzK=g \\ x,z∈H, y∈N \end{matrix}}(x̄ ⊗ (A_{β})_{yK} ⊗ z̄).

Moreover, since ^{g}b · b = 0 for g ∈ G \ G_{b}, the map

f: Ind_{O_{α}N}^{O_{α}H} A_{β} → A_{ω},  x̄ ⊗ a ⊗ ȳ ↦ x̄aȳ

is an isomorphism of interior O_{α}H -algebras. It follows that the bimodules bA and Ab induce a Morita equivalence between A_{ω} and A_{β}.

b) Since jb = bj = j, we have that jAj_{j}A_{β}j. By [KP, 1.6] we have that N = H(D̃δ̃) ≤ N, and moreover N = KN_{H}(D̃δ̃) by the Frattini argument. Hence G_{b} = N_{H}(D̃δ̃)K/K, so for each g ∈ G_{b} there is a homogeneous unit u_{g} ∈ hU(A_{β}) of degree g such that u_{g}ju_{g}^{−1} ∈ δ̃. It follows that there is v_{g} ∈ U(A_{1}D̃) such that u_{g}ju_{g}^{−1} = v_{g}jv_{g}^{−1} . Consequently, v_{g}^{−1}u_{g}j = jv_{g}^{−1}u_{g} is a unit of degree g in jAj.

It is well-known that the bimodules bA_{1}j = A_{1}j and jA_{1}b = jA_{1} induce a Morita equivalence between bA_{1} and jA_{1}j. The above shows that bA_{1}j is a Δ(A_{β} ⊗_{O} Aop ; δ̃)- module, hence by the results of [M2, Section 3], the bimodules bAj and jAb induce a G_{b} -graded Morita equivalence between A_{β} and jAj.

c) Let i ∈ (bA_{1})^{P} be a primitive idempotent such that Br_{P}^{A_{1}}(i) ≠ 0, so ib = bi = i and Aγ = iAi = iA_{β}i is a G_{b} -graded interior O_{α}P -algebra. There is j ∈ δ̃ such that ij = ji = j so we have A_{β} ⊇ A_{γ} ⊇ Aδ̃, and by b),

A_{β} ⊇ A_{β}iA_{β} = bAiAb ⊇ bAjAb = A_{β}.

Consequently, the G_{b} -graded bimodules A_{β}i = bAi and iA_{β} = iAb give a G_{b} -graded Morita equivalence between A_{β} and A_{γ}.

The above argument also shows that (bA_{1})i(bA_{1}) = bA_{1} , so iA_{1} and A_{1}i induce a Morita equivalence between bA_{1} and iA_{1}i. It follows that for any g ∈ G_{b} and any unit u_{g} ∈ bA_{g} we have

(iA_{g}i)(iA_{g^{−1}}i) = iu_{g}bA_{1}ibA_{1}u_{g}^{−1}i = iu_{g}(bA_{1})u_{g}^{−1}i = ibA_{1}i = iA_{1}i,

that is, A_{γ} is strongly G_{b} -graded. □

3.3. Since A = O_{α}H and A_{1} = O_{α}K are p-monomial H -algebras, the observations made in 2.3 show that the results of [BP, Section 1], [Li1, Section 3] and [Li2, Sections 6,7] generalize to our situation. We only state here those results in the form form we need.

Recall that a (b, H)-Brauer pair is a pair (P, e) where P is a P -subgroup of H and e is a block of A_{1}(P) such that Br_{P} (b)e ≠ 0. If we fix a primitive idempotent i ∈ δ, where D_{δ} is a defect pointed group of b, then for any subgroup P of D there is a unique block e_{P} of A_{1}(P) such that Br_{P} (i)e_{P} ≠ 0.

The set of (b, H)-Brauer pairs form a partially ordered set on which the group H acts by conjugation.

3.4. Let (P, e) and (Q, f) be (b, H)-Brauer pairs. An (H, G)-fusion from P to Q is an algebra homomorphism O_{α}P → O_{α}Q induced by conjugation with an element h ∈ H, modulo automorphisms of O_{α}Q induced by conjugation with elements q ∈ Q; moreover, we take into account the action of H on G by left translation, that is, H -fusions induced by g, h ∈ H such that gK ≠ hK are different. We denote by E_{H,G}(P, Q) the set of (H, G)-fusions from P to Q.

Similarly, an (H, G)-fusion from (P, e) to (Q, f) is an (H, G)-fusion from P to Q as above, with the additional property that ^{h}(P, e) = (^{h}P,^{h}e) ≤ (Q, f). (Notice that with the notation of 3.3, h ∈ H induces a fusion between (P, e) and (Q, f) if and only if hP ≤ Q and ^{h}e_{P} = e_{^{h}P} .) We denote by E_{H,G}((P, e), (Q, f)) the set of (H, G)-fusions from (P, e) to (Q, f). (See also [KP, 2.8] and [Pu2, 1.6] for related definitions.)

3.5. (H, G)-fusions can be described in terms of bimodule direct summands of A_{δ} = iAi (see 2.8). Linckelman’s [Li1, Lemma 3.3.v] and [Li2, 7.7 and 7.8] generalize as follows.

Let P, Q be subgroups of D, T a subgroup of P, and ϕ: T → Q a group homomorphism. If the (O_{α}P, O_{α}Q)-bimodule O_{α}P ⊗_{O_{α}T} _{ϕ}O_{α}Q is isomorphic to a summand of iAi, the ϕ is injective and the class ϕ̃ of ϕ belongs to E_{H,G}((T, e_{T}), (Q, e_{Q})).

Conversely, if C_{D}(Q̂) is a defect group of e_{Q} and ϕ: T → Q is a group homomorphism such that ϕ̃ ∈ E_{H,G}((T, e_{T}), (Q, e_{Q})), then the (O_{α}T, O_{α}Q)-bimodule _{ϕ}O_{α}Q is isomorphic to a summand of iAi.

Notice also that by [Li1, Lemma 3.3 i, ii], C_{D}(Q̂) is always contained in a defect group of e_{P} , and there is h ∈ H such that ^{h}(P, e_{P}) ≤ (D, e_{D}) and C_{D}(^{h}P̂) is a defect group of he_{P} .

3.6. For a (b, H)-Brauer pair (P, e_{P}) define as in 3.1 the stabilizer

G_{(P,e_{P})} = {h̄C_{K}(P̂) | h ∈ C_{H}(Ô), ^{h}e_{P} = e_{P} },

so e_{P} A(P)e_{P} = e_{P} A(P)_{G_{(P,e_{P} )}} is a G_{(P,e_{P})} -graded crossed product k -algebra. Always with i ∈ δ fixed, in order to simplify the notation, we denote

R = bAb, G_{P} = G_{(P,e_{P})} and R^{(P)} = e_{P} A(P)e_{P} .

3.7. Next we consider bimodules. Let K′ be a normal subgroup of the finite group H′ such that there is an isomorphism

ι′: G′ = H′/K′ → G = H/K.

Assume that D is a common Sylow p-subgroup of H and H′ such that D̃ = D ∩ K = D ∩ K′.

Consider the twisted group algebra A′ = O_{α′}H′ satisfying Res_{D}^{H} α = Res_{D}^{H′} α′, and observe that for any subgroup P of D, the (A, A′)-bimodule

A ⊗_{O_{α}P} A′ ≃ (A ⊗_{O} A′^{op}) ⊗_{OδP} O

is p-monomial.

Let b′ be a block of A′_{1} = O_{α′}K′, so β′ = {b′} is a point of K′ on A′_{1} and also of N′ = N_{H′}(K′_{β′}) on A′_{1} ; let D_{δ′} be a defect pointed group of N′_{β′} and D̃δ̃′ a defect pointed group of K′_{β′} as in 3.1.

Let further i′ ∈ δ′ be fixed, and for each subgroup P of D let e′_{P} be the block of A′(P) = k_{α′}C_{H′}(P̂) such that Br_{P}^{A′}(i′)e′_{P} ≠ 0.

3.8. We shall make the following assumptions. (3.8.1) For each subgroup P of D, ι′ induces the isomorphism

C_{H}(P̂)K′/K′ ≃ C_{H}(P̂)K/K.

(3.8.2) For any subgroups P, Q of D we have the equality

E_{H,G}((P, e_{P}), Q, e_{Q})) = E_{H′,G}((P, e′_{P}), Q, e′_{Q})),

that is, given an algebra homomorphism ϕ: O_{α}P → O_{α}Q, there is h ∈ H and a homogeneous unit h̄ ∈ A of degree h such that ϕ(u) = h̄uh̄^{−1} for all u ∈ O_{α}P if and only if there is h′ ∈ H′ and a homogeneous unit h̄′ ∈ A′ of degree h such that ϕ(u) = h̄′uh̄′^{−1} for all u ∈ O_{α}P, and ι(h′K) = hK.

We say in this case that A_{β} and A′_{β′} have G_{b} -equivalent Brauer categories. By [M3, Proposition 3.4.a)], these assumptions imply that G_{b} = G_{b′} and G_{(P,e_{P})} = G_{(P,e′_{P})} for any subgroup P of D.

For the result we also need a third twisted group algebra A′′ = O_{α′′}H′′ and an isomorphism ι′: G′′ = H′′/K′′ → G = H/K satisfying the same assumptions as A′ is 3.7 and 3.8. To simplify notation, let

R = A_{β} = bAb, R′ = A′_{β′} = b′Ab′, R′′ = A′′_{β′′} = b′′Ab′′; R^{(P)} = e_{P} A(P)e_{P} , R^{′(P)} = e′_{P} A′(P)e′_{P} , R^{′′(P)} = e′′_{P} A′′(P)e′′_{P} ; Δ = Δ(R ⊗_{O} R′^{op}), Δ′′ = Δ(R ⊗_{O} R′′^{op}); Δ^{(P)} = Δ(R^{(P)} ⊗_{k} R′^{(P)}^{op}), Δ^{(P)} = Δ(R^{(P)} ⊗_{k} R′′^{(P)}^{op}).

3.9. Theorem. Let M be a G_{b} -graded direct summand of the (A_{β}, A′_{β′})-bimodule bAi ⊗_{O_{α}P} i′A′b′, and let M′ be a G_{b} -graded direct summand of the (A′_{β}, A′′_{β′′})-bimodule b′A′i′ ⊗_{O_{α}Q} i′′A′′b′′, where P and Q are subgroups of D. Then a) For any subgroup P of D there is an isomorphism

e_{P} M(δP)e′_{P} ≃ (R^{(P)} ⊗_{k} R′^{(P) op}) ⊗_{Δ^{(P)}} M_{1}(δP)

of G_{P} -graded (R^{(P)}, R^{′(P)})-bimodules.

b) For any subgroup P of D there are an isomorphisms

e_{P} (M ⊗_{R′} M′)(δP)e′′_{P} ≃ e_{P} M(δP)e′_{P} ≃ ⊗_{R^{(P)}}e′_{P} M′(δP)e′′_{P}
≃ (R^{(P)} ⊗_{k} R′^{(P)}^{op}) ⊗_{Δ_{′′}_{(P)}} (M_{1} ⊗_{R′_{1}} M′_{1})(δP)

of G_{P} -graded (R^{(P)}, R^{′′(P)})-bimodules.

c) M and its O-dual M^{∗} induce a G_{b} -graded Morita equivalence between R and R′ if and only if there is an isomorphism ϕ: iAi → i′A′i′ of G_{b} -graded algebras satisfying ϕ(ui) = ui′ for all u ∈ O_{α}D, such that M ≃ Ai ⊗_{iAi} _{ϕ}i′A.

Proof. a) Notice that R^{(P)} and R^{′(P)} are G_{P} -graded (O_{α}H_{b})(P)-interior, respctively O_{α′}H′_{b′} -interior N_{G}(P)-algebras, and for any g ∈ G_{P} ≤ G_{b}, R_{g}, R′_{g} and M_{g} are monomial OδP -modules. It follows that

R(P) ⊗_{k} R′(P)^{op} ≃ (R ⊗_{O} R′^{op})(P)

and

Δ(R(P) ⊗_{k} R′(P)^{op}) ≃ (Δ(R ⊗_{O} R′^{op}))(P)

Further, e_{P} M(δP)e′_{P} is an (R^{(P)}, R^{′(P)})-bimodule graded by a subgroup of G containing G_{P} , and clearly, it has no nonzero components outside G_{P} . The 1-component M_{1} of M is a Δ-module, so M_{1}(δP) is a Δ(δP)-module.

The isomorphism stated in a), which is equivalent to the isomorphism

e_{P} M_{1}(δP)e′_{P} ≃ e_{P} M(δP)_{1}e′_{P}

of Δ^{(P)} -modules, will follow once we verify in this situation the condition in 2.5. Similarly to the proof of [M1, Corollary 3.9], this amounts to the equality

(C_{H}(P̂)_{e_{P}} × C_{H′}(P̂)_{e′_{P}})T_{δG}((δP, e_{P} ⊗_{k} e′_{P}), (δD, e_{D} ⊗_{k} e′_{D}))
= T_{H×H′}((δP, e_{P} ⊗_{k} e′_{P}), (δD, e_{D} ⊗_{k} e′_{D})),

where δG denotes the subgroup {(h, h′) ∈ H × H′ | hK = h′K′} of H × H′, and (h, h′) ∈ T_{H×H′}((δP, e_{P} ⊗_{k} e′_{P}), (δD, e_{D} ⊗_{k} e′_{D})) means that ^{h}(P, e_{P}) ≤ (D, e_{D}), h′(P, e′_{P}) ≤ (D, e′_{D}), and conjugation with h̄ and h̄′ induce the same algebra homomorphism from O_{α}P to O_{α}D. The above equality easily follows from our assumptions 3.8, so a) holds.

b), c) Again, the proof of [Li1, Theorem 1.1], [Li2, Lemma 2.3 and Theorem 4.1] can be adapted without difficulty. Regarding the statement about the gradings, observe that the natural map

M(δP) ⊗_{A′(P)} M′(δP) → (M ⊗_{A′} M′)(δP),

induced by the map M^{δP} ⊗_{A′(P)} M′^{δP} → (M ⊗_{A′} M′)^{δP} , is graded preserving. Next, if C_{D}(P̂) is a defect group of e_{P} , then the G_{b} -graded Morita equivalence between R and iAi of Proposition 3.2 c) induce a G_{P} -graded Morita equivalence between R^{(P)} and Br_{P} (i)A(P) Br_{P} (i) = (iAi)(P). Actually, by [Li2, 7.6 and 7.9], one can assume that C_{D}(P̂) is a common defect group of e_{P} , e′_{P} and e′′_{P} . Then we have the homomorphism

(iMi′)(δP) ⊗_{(i′A′i′)(P)} (iM′i′′)(δP) → (iMi′ ⊗_{i′A′i′} i′M′i′′)(δP)

of G_{P} -graded ((iAi)(P), (i′A′i′)(P))-bimodules, which is an isomorphism by the proof of [Li2, Lemma 2.3 (ii)] and the remarks made in 2.6–2.8. □

3.10. Corollary. Let C be a G_{b} -graded Rickard tilting complex of (R, R′)-bimodules all whose terms are direct summands of the bimodules bAi ⊗_{O_{α}P} i′A′b′, where P runs over the subgroups of D.

Then for any subgroup P of D, e_{P} C(δP)e′_{P} and its k -dual induce a G_{P} -graded derived equivalence between R^{(P)} and R^{′(P)} , and this equivalence is compatible with truncation, and induction from the diagonal subalgebra Δ^{(P)} .

3.11. Following [Li1, 1.10] and [Li2, Definition 1.1], the complex C in Corollary 3.10 will be called a splendid tilting complex.

M. Harris provided in [Ha] a direct generalization to arbitrary blocks of Rickard’s [Ri] concept of splendid equivalence. An examination of Harris’ proofs shows that the main result of that paper also generalizes when considering twisted group algebras and the presence of normal subgroups; moreover, for any subgroup P of D, there is a close relationship between C(δP) and C_{1}(δP) as in Theorem 3.9 above. Let us state this properly, cf. [Ha, Theorems 1.6 and 1.7].

Let A = O_{α}H, A′ = O_{α′}H′ be twisted group algebras with isomorphism ι′: G′ → G as in 3.7. Assume that b, b′ are G-invariant blocks of A_{1} , A′_{1} respectively, and that for any subgroups P, Q of D, ι′ induce the equality

E_{H,G}(P, Q) = E_{H′,G′}(P, Q).

Let C be a Rickard tilting complex of G-graded (R, R′)-bimodules such that all the terms of C are relatively δD-projective p-monomial A ⊗_{O} A′^{op} -modules. (3.11.1) If P ≤ D and (P, e_{P}) is a (b, H)-subpair, then there is a unique (b′, H′)- subpair (P, e′_{P}) such that e_{P} Ce′_{P} is a Rickard tilting complex of G_{P} -graded (R^{(P)}, R^{′(P)})- bimodules, and all its terms are p-permutation A(P) ⊗_{k} A^{′(P)}^{op} -modules. Moreover, e_{P} C_{1}(δP)e′_{P} = 0 for any (b′, H′)-subpair (P, e′) ≠ (P, e′_{P}).

(3.11.2) Fix a maximal (b, H)-subpair (D, e_{D}), let (P, e_{P}) ≤ (D, e_{D}), and let (D, e′_{D}), (P, e′_{P}) be the (b′, H′)-subpairs defined by (3.11.1). Then (P, e′_{P}) ≤ (D, e′_{D}). (3.11.3) If (P, e_{P}), (Q, e_{Q}) ≤ (D, e_{D}), then we have the equality

E_{H,G}((P, e_{P} , (Q, e_{Q})) = E_{H′,G′}((P, e′_{P}), (Q, e′_{Q})).

4. Abelian defect groups and Dade’s conjectures

4.1. Let A = O_{α}H, R = bAb, D_{δ} and D̃δ̃ be as in the preceding section, and let H′ = N_{H}(D̃), K′ = N_{K}(D̃), G′ = H′/K′ and b′ ∈ Z(O_{α}K′) be the Brauer correspondent of b. Let also ι′: G′ → G be the map induced by the inclusion of H′ in H. Condition (3.8.1) clearly holds in this case, and by [KP, 1.6] we know that

G_{b} = N_{H}(D̃, eD̃)K/K ≃ N_{H′}(D̃, eD̃)K′/K′ = G′_{b′}.

Recall also from [HK] that the Brauer correspondence induces a bijection, called the Harris-Knörr correspondence, between the blocks of N_{H}(D̃) lying over b and the blocks of N_{H′}(D̃) lying over b′. (The module theoretic proof given by Alperin [Al] also works for twisted group algebras.

When D̂ = hU(O_{α}D) is abelian, then the following generalization of [Th, Proposition 4.9.6], shows that condition (3.8.2) also holds, that is, the Brauer categories of A_{β} and A_{β′} are G_{b} -equivalent.

4.2. Proposition. Assume that D̂ is abelian, and let P_{γ} be a local pointed group on R_{1} = A_{1}b contained in D_{δ}. If h ∈ H satisfies ^{h}P_{γ} ≤ D_{δ}, then h = xz, with x ∈ N_{H}(D̃δ̃) and z ∈ C_{K}(P̂).

Proof. By hypothesis we have that P_{γ} ≤ D_{δ}^{h}, and in particular P ≤ D and P ≤ D^{h}.

Since P̂ is abelian, we have that C_{H}(P̂) ⊇ P C_{K}(P̂), so by [Th, Proposition 37.7], there is a unique point ϵ of C_{H}(P̂) on R_{1} such that P_{γ} ≤ C_{H}(P̂)_{ϵ}. There always exist a point ϵ′ of C_{H}(P̂) on R_{1} such that D_{δ} ≤ C_{H}(P̂)_{ϵ′}; but then P_{γ} ≤ D_{δ} ≤ C_{H}(P̂)_{ϵ′}, hence ϵ = ϵ′.

There is also a point ϵ̃ of C_{K}(P̂) on R_{1} such that C_{K}(P̂)ϵ̃ ≤ C_{H}(P̂)_{ϵ}, and then D̃δ̃ ≤ C_{K}(P̂)ϵ̃.

By the same argument we obtain that D_{δ}^{h} ≤ C_{H}(P̂)_{ϵ} and D̃δ̃ and D̃h ; δ̃ are maximal local pointed groups contained in K_{β}, so there is z ∈ C_{K}(P̂) such that ^{z}D̃δ̃ = D̃h ; δ̃. It follows that hz ∈ N_{H}(D̃δ̃) and h ∈ (hz)z^{−1} ∈ N_{H}(D̃δ̃)C_{K}(P̂). □

4.3. We shall consider the following extended version of Broué’s conjecture:

Let b be a block of O_{α}K and D_{δ} a defect pointed group of N_{H}(K_{{b}}). If D̂ is abelian, then there is a G_{b} -graded Rickard equivalence between bO_{α}Hb and b′O_{α}N_{H}(D̃)b′, where b′ is the Brauer correspondent of b.

4.4. Denote by

C : (P_{0} < P_{1} < ⋯ < P_{n})

a p-chain of K and by |C| = n the length of C. The groups K and H act on the set of p-chains of K, and let N_{K}(C) and N_{H}(C) be the normalizers of C in K and H respectively. Denote also N_{G}(C) = N_{H}(C)/N_{K}(C). We have that

C_{K}(P_{n}) ≤ N_{K}(C) ≤ N_{K}(P_{n}),

and if c is a block of O_{α}N_{K}(C), then by [KR, Lemma 3.2] and [D3, 10.14], the induced block c^{K} of OK is defined.

The group N_{G}(C) acts on the set Irr(K_{α}N_{K}(C)) of simple K_{α}N_{K}(C)-modules, so if X ∈ Irr(K_{α}N_{K}(C)), then the stabilizer N_{G}(C)_{X} and the Clifford extension KE(X) are defined as in 1.6. If c = c(X) denotes the block of O_{α}N_{K}(C) to which X belongs, then we shall consider the Clifford extension kE(c(X)) of c defined in 1.10.

Assume that c^{K} = b, where c = c(X). Then by [D4, (5.2)] or [D1, 8.1], we have that G[b] is a subgroup of N_{G}(C)[c], and the Brauer hommorphism associated to the final subgroup P = P_{n} of C induces a grade preserving monomorphism

Br_{C}: kE(b) → kE(c).

Also, by [D4, (5.3)] of 1.11 above there is a grade preserving monomorphism

μ[C, X]: kE(c) → kE(X),

and denote

μ[b, C, X] = μ[C, X] ∘ Br_{C}: kE(b) → kE(X).

Recall finally that the defect def(X) and the height ht(X) are defined by

p^{def(X)} = (\frac{|N_{K}(C)|}{dim_{K} X})_{p} , ht(X) = def(c(X)) − def(X).

4.5. Let d ≤ 0 be an integer, F is a subgroup of N_{G}(C), O_{β}F a twisted group algebra, and κ: kE(b) → k_{β}F a grade preserving monomorphism, or equivalently, a monomorphism

1 ⟶ k^{×} ⟶ hU(kE(b)) ⟶ G[b] ⟶ 1
∥  ↓^{κ}  ↓^{⊆}
1 ⟶ k^{×} ⟶ hU(k_{β}F) ⟶ F ⟶ 1

of central extensions; then denote by k(C, b, d, F, κ) the number of simple modules X ∈ Irr(K_{α}N_{K}(C)) satisfying

def(X) = d, c(X)^{K} = b, N_{G}(C)_{X} = F and (kE(X), μ[b, c, X]) ≃ (k_{β}F, κ)

(that is, there is an isomorphism γ: kE(X) → k_{β}F such that κ = γ ∘ μ[b, c, X]), and by k(β, d, F, κ) the number of of simple modules X ∈ Irr(K_{α}N_{K}(C)) satisfying

def(X) = d, N_{G}(C)_{X} = F and (kE(X), μ[b, c, X]) ≃ (k_{β}F, κ).

Then k(C, b, d, F, κ) depends only on the K -conjugacy class of C.

4.6. With the above notations, Dade’s Inductive Conjecture [D4, (5.8)] states as follows. If O_{p}(K) = 1 and def(b) > 0 then

∑_{C∈F/K}(−1)^{|C|}k(C, b, d, F, κ) = 0,

where F is one of the families P, E, N or U of p-chains of K introduced in [KR, Section 2].

4.7. Proposition. The extended abelian defect group conjecture 4.3 implies Dade’s Inductive Conjecture 4.6.

Proof. The argument of [KR, Proposition 5.5] also works here. We consider the family N of chains C = (P_{0} < ⋯ < P_{n}) such that P_{i} ⊴ P_{n} for 0 ≤ i ≤ n. So let b be a G-invariant block of O_{α}K such that D̂ is abelian, where D > 1 is a defect group of b in H, and let c be a block of O_{α}N_{G}(C) such that c^{K} = b.

Assume that P_{n} is a defect group of c, and let C′ = (P_{0} < ⋯ < P_{n−1}). Then the induced block c′ = c^{N_{K}(C′)} is defined, it has defect group P_{n}, and (c′)^{K} = b. By hypothesis, there is an N_{G}(C)_{c} -graded Rickard equivalence between c and c′ = c^{N_{K}(C′)} , and by [Br2, Proposition 4.5], defects of simple modules are preserved. It follows that

k(β, d, F, κ) = k(c′, d, F, κ).

Indeed, if X is a simple K_{α}N_{G}(C)-module belonging to the block c, corresponding to the simple K_{α}N_{G}(C′)-module X′ belonging to c′, then the following diagram is commutative.

kE(b) ⟶^{Br_{C}} kE(c) ⟶^{μ[C,X]} kE(X)
∥  ↓^{≃}  ↓^{≃}
kE(b) ⟶^{Br_{C′}} kE(c′) ⟶^{μ[C′,X′]} kE(X′)

Now assume that P_{n} is not a defect group of c. Then the defect group P_{n+1} of c satisfies P_{n} < P_{n+1} , and let C′ = (1 < P_{1} < ⋯ < P_{n} < P_{n+1}) ∈ N . Then N_{K}(C′) = N_{N_{K}(C)}(P_{n+1}) and there is a unique block c′ of N_{K}(C′) inducing c. As above we have the equality

k(c, d, F, κ) = k(c′, d, F, κ).

It follows that in the alternating sum 4.6 we can pair off the contributions coming from the various blocks, and the proposition is proved.

5. Cyclic defect groups: the ordinary group algebra case

5.1. Let b be a G-invariant block of OK, and assume that G = G_{b} is a p′ -group and that the defect group D = D̃ of b is cyclic.

Let H′ = N_{H}(D), K′ = N_{K}(D), b′ the Brauer correspondent of b, R = bOH, R′ = b′OH′, and let A′_{δ′} be the source algebra of R′. By Proposition 3.2 b), we have that A′_{δ′} is a G-graded crossed product. In this case, it follows by the main result of [KP] that there is a matrix OP -interior algebra B with P -stable O-basis and a twisted group algebra S = O_{β}L such that A′_{δ′} ≃ B ⊗_{O} O_{β}L; moreover, S = O_{β}L is also a G-graded crossed product, so it is G-graded Morita equivalent to A′_{δ′} and to R′, and by [Rou, Proposition 10.2.5], S_{1} is the group algebra O(D ⋊ E), where E = N_{K}(D, e_{D})/C_{K}(D).

Denote further Q = O_{p}(K), and let D′′ be the (unique) subgroup of D containing Q such that [D′′ : Q] = p. Let H′′ = N_{H}(D′′), K′′ = N_{K}(D′′), b′′ the block of OK′′ coresponding to b (notice that K′ ≤ K′′ and H′ ≤ H′′), and let R′′ = OH′′b′′.

The aim of this section is to prove the following theorems.

5.2. Theorem. There is a G-graded splendid Rickard equivalence between R and R′.

Moreover, if Q = O_{p}(K) ≠ 1, then R and R′ are graded Morita equivalent (but this equvalence is not necessarily splendid).

5.3. Theorem. Let M_{1} be an indecomposable direct summand of the (R_{1}, R′′_{1})-bimodule bRb′′ with vertex δD.

a) There is a direct summand N_{1} of the (R_{1}, R′′_{1})-bimodule R_{1} ⊗_{OQ} R′′_{1} such that

C_{1} = (⋯ → 0 → N_{1} ⟶^{m} M_{1} → 0 → …)

is a splendid Rickard tilting complex of (R_{1}, R′′_{1})-bimodules. Moreover, if Q = O_{p}(K) ≠ 1, then C_{1} has homology only in one degree.

b) C_{1} extends to a complex of Δ(R ⊗_{O} R′′^{op}) so the complex

C = (R ⊗_{O} R′′^{op}) ⊗_{Δ(R⊗_{O}R′′^{op})} C_{1}

induces a G-graded Rickard equivalence between R and R′′.

5.4. We rely heavily on the case G = 1 of the above theorems, which were proven by R. Rouquier [Rou] (or [KZ, Theorems 10.1 and 10.3]). Notice that he used the term “splendid tilting complex” in the sense of Rickard [Ri2] and Harris [Ha] (see also 3.11). However by [Li2, Section 5], we have that Rouquier’s complex is splendid in the sense of the more restrictive definition [Li1, 1.10] and [Li2, Definition 1.1], which is adopted in this paper.

Since Rouquier’s proof was by induction on the order of K, we shall need to follow the steps of his proofs, and use a similar inductive argument. An important point is that composition of splendid (respectively G-graded) equivalences is a splendid (respectively G-graded) equivalence. We give here a rather complete proof, for convenience.

First of all, Theorem 5.2 follows from Theorem 5.3. Indeed, by induction, there is a graded splendid Rickard equivalence between R′′ and R′ given by a complex with homology in only one degree. By Theorem 5.3 there is a graded splendid Rickard equivalence between R and R′′. Composing these equivalences, we obtain a graded splendid Rickard equivalence between R and R′; moreover, of Q ≠ 1, then the Rickard tilting complex has homology in only one degree.

Before going further, we need to extend the complex constructed in [Rou, Section 10.3]. More precisely, we can prove the following.

5.5. Proposition. Assume that M is a G-graded (R, S)-bimodule such that M_{1} induces a stable Morita equivalence between R_{1} and S_{1} . Then there exist a complex

(⋯ → 0 → N ⟶^{π} M → 0 → …)

inducing a G-graded Rickard equivalence between R and S, where N_{1} is a direct summand of the projective cover of M_{1} , and π_{1} is the restriction of the epimorphism P (M_{1}) → M_{1} to N_{1} .

Moreover, if E = 1, then N = 0 or N = P (M), so M or ΩM induce a graded Morita equivalence between R and S.

Proof. Denote Δ = Δ(R ⊗_{O} S^{op} , and let P_{1} be a projective cover of M_{1} in Δ-mod; then P/J(Δ_{1})P = P_{1}/J(Δ)P_{1} ≃ M_{1}/J(Δ)M_{1} = M_{1}/J(Δ_{1})M_{1} as Δ-modules, hence Res_{Δ_{1}}^{Δ} P_{1} is also a projective cover P (M_{1}) of M_{1} in Δ_{1}-mod. Recall from [Rou, 10.3.4] that

P (M_{1}) ≃ ⨁_{0≤i≤e−1} P (Ω^{2i}Σ) ⊗_{O} P (Ω^{2i}O)^{∗},

where e = |E| and Σ = M_{1} ⊗_{S_{1}} O. Since for all g ∈ G, S_{s} ⊗_{S_{1}} Ω^{2i}O ≃ Ω^{2i}O and

R_{g} ⊗_{R_{1}} Σ = R_{g} ⊗_{R_{1}} (M_{1} ⊗_{S_{1}} O) ≃ M_{1} ⊗_{S_{1}} (S_{g} ⊗_{S_{1}} O) ≃ M_{1} ⊗_{S_{1}} O ≃ Σ,

we have that each term of P (M_{1}) is G-invariant. There is a subset I of {0, …, e − 1} such that taking

N_{1} = ⨁_{i∈I} P (Ω^{2i}Σ) ⊗_{O} P (Ω^{2i}O)^{∗},

we have that 0 → N_{1} → M_{1} → 0 is a Rickard tilting complex of (R_{1}, S_{1})-bimodules. But N_{1} is a G-invariant Δ_{1} -summand of M_{1} , and M_{1} , M_{1}/N_{1} have no common indecomposable direct summands. It follws by a well-known argument of Dade (see [M4, Theorem 3.1.16]) that N_{1} is a Δ-summand of M_{1} . Let N = Δ ⊗_{Δ_{1}} N_{1} and C = (⋯ → 0 → N → M → 0 → …). By [Rou, Theorem 10.2], C_{1} and its O-dual C_{1}^{∗} induce a Rickard equivalence between R_{1} and S_{1} , (and, if in addition E = 1, then N_{1} = 0 or N_{1} = P (M_{1}), and by [M2, Theorem 4.8] C induce a graded Rickard equivalence between R and S. □

5.6. Next we assume that Theorem 5.3 holds for all finite groups K_{1} ⊴ H_{1} with |K_{1}| < |K|. First we consider the case Q = O_{p}(K) = 1, so |D′′| = p.

We have that bOKb′′ induces a stable equivalence between R_{1} and R′′_{1} , and by a result of Linckelmann [KZ, Lemma 10.2.3], bOKb′′ has a unique indecomposable non-projective bimodule summand M_{1} , which still induces a stable equivalence between R_{1} and R′′_{1} . Since bOKb′′ extends to a Δ(R ⊗_{O} R′′^{op})-module, and bOKb′′/M_{1} have no common indecomposable direct summands, it follows as above that that M_{1} is a Δ(R ⊗_{O} R′′^{op})- summand of bOKb′′, and by [M2, Proposition 5.7] M = (R ⊗_{O} R′′^{op}) ⊗_{Δ(R⊗_{O}R′′^{op})} M_{1} induces a graded stable equivalence between R and R′′.

Now, by induction, Theorem 5.2 holds for R′′, so there is an (R′′, S)-bimodule M′ inducing a G-graded Morita equivalence between R′′ and S, so the indecomposable Ggraded bimodule M′′ = M ⊗_{R′′} M′ induces a stable equivalence between R and S. Using Proposition 5.5, one can easily deduce (similarly to [Rou, 10.4.1]), that Theorem 5.3 holds when Q = 1.

5.7. Assume that Q = O_{p}(K) ≠ 1, and denote by “¯ ” the canonical map H → H/Q = H̄. Then b̄ and b̄′′ are blocks with defect group D̄ of K and K̄ respectively, and by induction, R̄ = OH̄b̄ is graded Rickard equivalent to S̄, where S̄_{1} ≃ O(D̄ ⋊ E). One may actually assume that b = e_{Q}, where (Q, e_{Q}) is a b-subpair, since the (OHb, ON_{H}(Q, e_{Q})e_{Q})-bimodule bOHe_{Q} induces a Morita equivalence between (OHb and ON_{H}(Q, e_{Q})e_{Q}). Then by [Rou, 10.4.2], we have that the canonical maps N_{K}(D, e_{D})/C_{K}(D) → K/C_{K}(Q) and N_{K′′}(D, e_{D})/C_{K′′}(D) → K′′/C_{K′′}(Q) are isomorphisms, and O_{p}(C_{K}(Q)) = O_{p}(C_{K′′}()) = Q.

(5.7.1) Assume that 1 ≠ Q = O_{p}(K) ≤ Z(K). By induction, R̄ and R̄′′ are graded derived equivalent, and since E = 1, R̄_{1} and R̄′′_{1} have a unique simple module. Let M′_{1} be an indecomposable direct summand of the Δ(R ⊗_{O} R′′^{op})/O(δQ)-module bOKb′′ with vertex δD/δQ, and let f′_{1} : N′_{1} → M′_{1} be the projective cover of M′_{1} . We have that M′′_{1} = M′_{1} ⊗_{OQ} O is an indecomposable direct summand of b̄OK̄b̄′′ (as a Δ(R̄ ⊗_{O} R̄′′^{op})- module and as an (R̄_{1}, R̄′′_{1})-bimodule), and by [Rou, Lemma 10.2.9], N′′_{1} = N′_{1} ⊗_{OQ} O is a projective cover of M′′_{1} . Let M_{1} = M′_{1} and N_{1} = N′_{1} regarded as Δ(R⊗_{O} R′′^{op})-modules, and let M′′, N′′, M and N be the G-graded bimodules obtained from M′′_{1} , N′′_{1} , M_{1} and N_{1} , respectively.

If M′′ induces a graded Morita equivalence between R̄ and R̄′′, then let C be the complex with M in degree 0. If Ker(f′′: N′′ → M′′) induces a graded Morita equivalence between R̄ and R̄′′, then let

C = (⋯ → 0 → N ⟶^{f′} M → 0 → ⋯)

with M in degree 0 and N in degree 1. Using [Rou, Lemma 10.2.11], one deduces that C induces a graded Rickard equivalence between R and R′′.

(5.7.2) Assume finaly that Q = O_{p}(K) is not contained in Z(K). Then the proof in [Rou, p. 218] produces a Rickard complex

C_{1} = (⋯ → M′ → ⋯) or C_{1} = (⋯ → N′ → M′ → ⋯)

of E -graded (R_{1}, R′′_{1})-bimodules, where M′ is an E -graded summand of bOKb′′, and N′ is an E -graded summand of the bimodule R ⊗_{OQ} R′′, isomorphic to a projective cover of M. The 1-component M′_{1} of M′ is the unique indecomposable (OC_{K}(Q)b, OC_{K′′}(Q)b′′)- bimodule summand of bOC_{K}(Q)b′′, and M′ is a G-invariant bimodule. Since bOKb′′ extends to a Δ(R ⊗_{O} R′′^{op})-module, it follows that M′ extends to a Δ(R ⊗_{O} R′′^{op})- module, and therefore C_{1} also extends to a complex of Δ(R ⊗_{O} R′′^{op})-modules. We obtain a Rickard complex C of G-graded (R, R′′)-bimodules, and this concludes the proof of Theorem 5.3.

6. Cyclic defect groups: the twisted group algebra case

6.1. We return to our twisted group algebra A = O_{α}H, and in this section we shall assume that the grading group G has p′ -order.

Let b be a block of A_{1} = O_{α}K having cyclic defect group D = D̃, and let H′, K′, G′ and b′ be as in 4.1. Let further D′′ be the subgroup of order p of D, H′′ = N_{H}(D′′), K′′ = N_{K}(D′′), A′′ = O_{α}H′′, and let b′′ be the block of A′′_{1} = O_{α}K′′ corresponding to b.

Denote also R = bAb, R′ = b′A′b′ and R′′ = b′′A′′b′′, so all these algebras are strongly G_{b} -graded. The main result of this section says that there are G_{b} -graded Rickard equivalences between R, R′ and R′′. We also derive consequences concerning Dade’s conjectures.

6.2. By the results of [D3, Section 6] and [D5, Section 6], we may assume that Ĥ has a finite subgroup H^{∗} covering H, such that, denoting Z = H^{∗} ∩ O^{×}, we have that exp(H^{∗}) = exp(Z). Then Z is a cyclic central subgroup of H^{∗}, and it decomposes into the direct product of its Sylow p-subgroup Z_{p} and its Hall p′ -subgroup Z_{p′}. Morever, the inverse image D^{∗} = D̂ ∩ H^{∗} of D in H^{∗} is abelian, and D^{∗} = Z_{p} × D.

Let ζ: Z → O^{×} be the inclusion, and let V be the OZ -module of O-rank 1 determined by ζ. Then V ≃ V_{p} ⊗_{O} V_{p′}, where V_{p} is the restriction of V to Z_{p}, and V_{p′} is the restriction of V to Z_{p′}. Note that V belongs to the block e^{∗} of OZ, where

e^{∗} = |Z_{p′}|^{−1} ∑_{z∈Z_{p′}} z^{−1}ζ(z).

6.3. Denote A^{∗} = OH^{∗} and R^{∗} = b^{∗}OH^{∗}b^{∗}, where b^{∗} is the unique block of A_{1}^{∗} = OK^{∗} whose image through the epimorphism A^{∗} → A is b; actually, b^{∗} is a block of e^{∗}A_{1}^{∗}, and b^{∗} is said to cover the block b. Moreover, e^{∗}A^{∗} is a strongly G-graded algebra, the stabilizer G_{b^{∗}} of b^{∗} coincides with G_{b}, and D^{∗} is a defect group of b^{∗}.

Let further H̄ = H^{∗}/Z_{p}, K̄ = K^{∗}/Z_{p}, Ā = OH (also regarded as a G-graded algebra, and let b̄ be the image of b^{∗} under the epimorphism A^{∗} → A. Then b̄ is a block with defect group D of Ā_{1} = OK̄, and it has the same stabilizer Gb̄ = G_{b}. Let also R̄ = b̄Āb̄.

With these notations and assumptions, we have:

6.4. Theorem. There is an isomorphism of G_{b} -graded OD-interior algebras

σ: OZ_{p} ⊗_{O} R̄ → R^{∗}

such that ν ∘ σ = aug ∘ idR̄, where ν: R^{∗} → R̄ is the canonical epimorphism, and aug: OZ_{p} → O is the augmentation.

Morever, any two such isomorphisms are conjugate by an element of 1 + J(R_{1}^{∗}).

This is just a graded version of [Fan, Theorem 1.2]. One can prove it by using the method of [KOW, Theorem 7]. Observe only that R^{∗} is a separable OP^{∗} -interior algebra extension of A_{δ^{∗}}^{∗}, where (D^{∗}, δ^{∗}) is a defetc pointed group of b^{∗}, and that all the maps which occur in the proof of [KOW, Theorem 7 and Theorem 3] are grade preserving.

Notice also that the isomorphism σ comes from an isomorphism σ_{δ}: OZ_{p}⊗_{O} Āδ̄ → A_{δ^{∗}}^{∗}, satisfying similar compatibility and uniqueness up to (1 + J(A_{δ^{∗}}^{∗})_{1})-conjugation, where Dδ̄ is a defect pointed group of b̄.

6.5. Similar statements hold with R′ and R′′ replacing R. We have that H′^{∗} = Ĥ′∩H^{∗} = N_{H^{∗}}(D^{∗}) is a finite subgroup of Ĥ′ covering H′, and H′′^{∗} = Ĥ′′ ∩ H^{∗} = N_{H^{∗}}(D′′^{∗}) is a finite subgroup of Ĥ′′ covering H′′. Let H̄′ = H′^{∗}/Z_{p}, H̄′′ = H′′^{∗}/Z_{p}, b′^{∗} and b′′^{∗} be the (unique) blocks of OK′^{∗} and OK′′^{∗} covering b′ and b′′ respectively, and b̄′ and b̄′′ the images of b′^{∗} and b′′^{∗} in OK̄′ and OK̄′′ respectively. By [D5, Sections 5, 6], b^{∗}, b′^{∗} and b′′^{∗} correspond to each other under the Brauer correspondence, and so do b̄, b̄′ and b̄′′. Moreover, all these blocks have the same stabilizer G_{b}. The notations R′^{∗}, R′′^{∗}, R̄′ and R̄′′ will have the obvious meaning; then Theorem 6.4 gives the isomorphism OZ_{p} ⊗_{O} R̄′′ ≃ R′′^{∗}.

Now we can state the twisted group algebra version of Theorems 5.2 and 5.3.

6.6. Theorem. Under the assumptions 6.1, there are G_{b} -graded splendid Rickard equivalences between R, R′ and R′′.

Consequently, Dade’s Inductive conjecture holds for p′ -extensions of blocks with cyclic defect groups.

Proof. Since the defect group D̄ ≃ D of b̄ is cyclic, by Theorem 5.2 there is a complex C̄ inducing a G_{b} -graded splendid Rickard equivalence between R and R′. The terms of C̄ are direct summands of the (R̄, R̄′)-bimodules R̄ī ⊗_{OP̄} ī′R̄′, where ī ∈ δ̄ ⊂ R̄^{D} and ī′ ∈ δ̄′ ⊂ R̄′^{D} are source idempotents of b̄ and bar b′ respectively, and P̄ runs over the subgroups of D̄.

Let i^{∗} ∈ δ^{∗} ⊂ R^{∗}^{D^{∗}} and i′^{∗} ∈ δ′^{∗} ⊂ R′^{∗}^{D^{∗}} be source idempotents of b^{∗} and b′^{∗}, whose images in R̄ and R̄′ are ī and ī′ respectively. By Theorem 6.4 there are isomorphisms

OZ_{p} ⊗_{O} R̄ ≃ R^{∗}, OZ_{p} ⊗_{O} R̄′ ≃ R′^{∗}, OZ_{p} ⊗_{O} R̄δ̄ ≃ R_{δ^{∗}}^{∗} and OZ_{p} ⊗_{O} R̄′ ; δ̄′ ≃ R′_{δ′^{∗}}^{∗}

of G_{b} -graded OD^{∗} -interior algebras. Composing these isomorphisms with the Rickard equivalence between R̄ and R̄′, we obtain a graded Rickard equivalence between R^{∗} and R′^{∗}, induced by a complex C^{∗} (not the O-dual of C) whose terms are summands of the (R^{∗}, R′^{∗})-bimodules of the form

R^{∗}i^{∗} ⊗_{OP^{∗}} i′^{∗}R′^{∗} ≃ OZ_{p} ⊗_{O} (R̄ī ⊗_{OP̄} ī′R̄′).

We denote by (R^{∗}|V_{p})-mod the full subcategory of R^{∗}-mod generated by R^{∗}⊗_{e^{∗}OZ_{p}} V_{p}; the objects of this (abelian) category are the R^{∗} -modules lying over V_{p}. Since b^{∗} is a block lying over e^{∗}, we have that the subcategories (R^{∗}|V_{p})-mod and (R^{∗}|V_{p})-mod coincide (see also [D5, Lemma 6.8]), where the latter category is generated by R^{∗}⊗_{e^{∗}OZ} V . Similar statements hold for R′^{∗}, and the Rickard equivalence induced by C restrict to an equivalence between the triangulated categories K^{b}((R^{∗}|V_{p})-mod) and K^{b}((R′^{∗}|V_{p})-mod), so between K^{b}((R^{∗}|V)-mod) and K^{b}((R′^{∗}|V)-mod). Now since End_{R^{∗}}(R^{∗} ⊗_{e^{∗}OZ} V)^{op} ≃ R as G_{b} -graded algebras, Clifford theory gives us an equivalence

V ⊗_{O} − ≃ (R^{∗} ⊗_{e^{∗}OZ} V) ⊗_{R} −: R mod → (R^{∗}|V)-mod

of abelian categories, compatible with G_{b} -gradings. Let i be the image of i^{∗} throug the epimorphism A^{∗} → A, and notice that by [Pu2, Corollary 2.3], this map induces a bijection, preserving inclusion and localness, between the pointed groups L_{λ^{∗}}^{∗} on e^{∗}A^{∗} and the pointed groups L_{λ} on A. Again, similar statements hold for R′, so finally, we obtain a G_{b} -graded Rickard equivalence between R and R′, induced by a complex C whose terms are direct summands of bimodules of the form Ri ⊗_{OP} i′R′. □

6.7. In our final remark we do not need to assume that G is a p′ -group. By [M3] we have that Rouquier’s derived equivalence between R̄_{1} and R̄′_{1} is G_{b} -invariant (see also 1.13). Then the arguments if this section imply: (6.7.1) There are G_{b} -equivariant splendid Rickard equivalences between R_{1} , R′_{1} and R′′_{1} .

Indeed, let C̄_{1} be a Rickard tilting complex of (R̄_{1}, R̄′_{1})-bimodules. The proof of Theorem 6.6 gives a Rickard tilting complex C_{1} of (R_{1}, R′_{1})-bimodules, and clearly, the complex corresponding to R̄_{g} ⊗R̄_{1} C̄_{1} ⊗R̄′_{1} R̄′_{g^{−1}} is R_{g} ⊗_{R_{1}} C_{1} ⊗_{R′_{1}} R′_{g^{−1}} .

This observation completes the study made in [M3] with the case of twisted group algebras: it follows that the main result of [D5], stating that the Projective Invariant conjecture hold for blocks with cyclic defect groups, is also a consequence of (6.7.1).

References

[Al] J.L. Alperin, The Green correspondence and normal subgroups, J. Algebra 104 (1986), 74–77.

[Bo] R. Boltje, Linear source modules and trivial modules, Proc. Symp. Pure Math. 63 (1998), 7–30.

[BK] R. Boltje and B. Külshammer, A generalized Brauer construction and linear source modules, Preprint, 1998.

[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, Equivalences of blocks of group algebras, Proc. of the NATO Advanced Research Workshop on Representation of Algebras and Related Topics, Kluwer Academic Publishers, 1991.

[BrPu] M. Broué and L. Puig, Characers and local structure in G-algebras, J. Algebra 63 (1980), 306–317.

[D1] E.C. Dade, Block extensions, Ill. J. Math. 17 (1973), 198–273.

[D2] E.C. Dade, Extending group-modules in a relatively prime case, Math. Z. 186 (1984), 81–98.

[D3] E.C. Dade, Counting characters in blocks, II, J. reine angew. Math. 448 (1994), 97–190.

[D4] E.C. Dade, Counting characters in blocks, 2.9, Representation Theory of finite groups (R. Solomon, editor), Ohio State University Math Research Inst. Publications 6, Walter de Gruyter, 1997.

[D5] E.C. Dade, Counting characters in blocks with cyclic defect groups, I, J. Algebra 186 (1996), 934–969.

[Fan] Y. Fan, Relative local control and the block source algebras, Science in China (Ser. A) 40 (1997), 785–798.

[Ha] M.E. Harris, Splendid derived equivalences for blocks of finite groups, J. London Math. Soc. (2) 60 (1999), 71–82.

[HaKn] M.E. Harris and R. Knörr, Brauer correspondences for covering blocks of finite groups, Comm. Algebra 13 (1985), 1213–1218.

[KnRo] B. Knörr and G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. 39 (1989), 48–60.

[KZ] S. König and A. Zimmermann, Derived Equivalences for Group Rings, Springer-Verlag, Berlin, 1998.

[KOW] B. Külshammer, T. Okuyama and A. Watanabe, A lifting theorem with applications to blocks and source algebras, Preprint 1999.

[KP] B. Külshammer and L. Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), 17–71.

[Li1] M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, Turkish J. Math. 22 (1998), 93–107.

[Li2] M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, Preprint, 1998.

[M1] A. Marcus, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), 372–396.

[M2] A. Marcus, Equivalences induced by graded bimodules, Comm. Algebra 26 (1998), 713-731.

[M3] A. Marcus, Derived equivalences and Dade’s invariant conjecture, J. Algebra 221 (1999), 513–527.

[M4] A. Marcus, Representation Theory of Group Graded Algebras, Nova Science Publishers, New-York, 1999.

[Pu1] L. Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks, Birkhäuser Verlag, Basel, 1999.

[Pu2] L. Puig, Notes on O^{∗} -groups, Preprint, 1998.

[Ri1] J. Rickard, Derived categories and stable equivalences, J. Pure Appl. Algebra 61 (1989), 303–317.

[Ri2] J. Rickard, Splendid equivalences: Derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331–358.

[Rou] R. Rouquier, The derived category of blocks with cyclic defect groups, in S. König and A. Zimmermann, Derived Equivalences for Group Rings, Springer-Verlag, Berlin, 1998, pp. 199– 220.

[Th] J Thévenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995.

Current address: Mathematisches Institut, Friedrich-Schiller-Universität Jena, D-07740 Jena, Deutschland

E-mail address: amarcus@minet.uni-jena.de