BLOCKS WITH CYCLIC DEFECT

GROUPS, AND CLIFFORD EXTENSIONS

ANDREI MARCUS

“Babeş-Bolyai” University

Faculty of Mathematics and Computer Science

Str. Mihail Kogălniceanu nr. 1

RO-3400 Cluj-Napoca, Romania

Abstract. We prove that the residual Clifford extension of any non-exceptional character belonging to a p-block with cyclic defect groups of a twisted group algebra is naturally isomorphic to the Clifford extension of any simple module belonging to that block. We also show that Dade’s inductive conjecture holds for blocks with cyclic defect groups.

1. Introduction

Let O be a complete discrete valuation ring such that the residue field k = O/J(O) is algebraically closed of characteristic p > 0, and the quotient field K has characteristic 0 and it is a splitting field for all the K-algebras considered in this paper.

Let K be a normal subgroup of the finite group H, G = H/K, and consider a twisted group algebra K_{α}H. Actually we may assume that α ∈ Z^{2}(H, O^{×}), and we denote by the same α the restriction of α to a subgroup, and also its image via the map O^{×} → k^{×}. 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.

Let b is a G-invariant block of O_{α}K having cyclic defect group D, and let b′ be the Brauer corresponding block of O_{α}K′, where K′ is the normalizer of the unique subgroup Ω(D) of order p of D. When G is a p′ -group, we proved in [10] that the splendid Rickard equivalence between bOK and b′OK′ obtained by Rouquier in [12] extends to a G-graded Rickard equivalence between bO_{α}H and b′O_{α}H′, where H′ = N_{H}(Ω(D)), and consequently, Dade’s inductive conjecture holds in this situation.

1991 Mathematics Subject Classification. 20C20, 20C05, 20C25, 16W50, 16S35.

Key words and phrases. twisted group algebras, group graded algebras, stable equivalence, Clifford extensions, cyclic defect groups.

If G is arbitrary, we do not have a derived equivalence between bO_{α}H and b′O_{α}H′, but we still have that the 1-component bO_{α}Kb′ of the G-graded (bO_{α}H, b′O_{α}H′)- bimodule bO_{α}Hb′ induces a stable Morita equivalence between bO_{α}K and b′O_{α}K′. In Section 2 we investigate the general properties of the functors defined by such graded bimodules, and we prove that they preserve the Clifford extensions of indecomposable modules and the residual Clifford extensions of irreducible characters. A useful observation is that the bimodule inducing the Heller translation also appears as the 1-component of a graded bimodule.

In Section 3 we give a twisted group algebra version of Külshammer’s theorem [6] on blocks with normal defect groups. Our proof is inspired by [1], but we also point out that we are in the presence of a graded Morita equivalence. We deduce that the block algebra b′O_{α}K′ is Morita equivalent to the ordinary group algebra O(D ⋊ E), where E is the inertial quotient of b. This allows us to conclude that the main results from the theory of blocks with cyclic defect groups generalize to twisted group algebras. We show in Section 4 that the residual Clifford extension of any non-exceptional character is naturally isomorphic to the Clifford extension of any simple module belonging to b. Finally, using these natural isomorphisms, we give some consequences towards Dade’s inductive conjecture for blocks with cyclic defect groups.

Our notations and general assumptions are standard and follow [10]. Various forms of Dade’s conjecture were explained in [3]. We refer the reader to [14] for results on block theory, to [8] for cyclic defect group theory, and to [9] for representations of group graded algebras.

2. Clifford extensions and stable equivalences

2.1. Let G be a finite group and R = ⨁_{g∈G} R_{g} a symmetric G-graded crossed product O-algebra, such that the restriction of the symmetrizing form of R is a symmetrizing form for R_{1} . We assume that KR := K ⊗_{O} R is a semisimple K-algebra, and that K is a splitting field for all the K-subalgebras of KR which will occur below. Denote also kR = k ⊗_{O} R.

2.2. The group G acts on the simple KR_{1} -modules, and let

G_{X} = {g ∈ G | KR_{g} ⊗_{KR_{1}} X ≃ X in KR_{1}-mod}

be the stabilizer of the simple KR_{1} -module X. 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 the group of homogeneous units of KE(X) is called the Clifford extension of X. Let E(X) denote the unique G_{X} -graded maximal suborder of KE(X). Then the group of homogeneous units of kE(X) = k ⊗_{O} E(X) is called the residual Clifford extension of X.

2.3. The group G also acts on indecomposable R_{1} -modules, and let G_{U} denote the stabilizer of U. The group of homogeneous units of the twisted group algebra

kE(U) := End_{R}(R ⊗_{R_{1}} U)^{op}/J_{gr}(End_{R}(R ⊗_{R_{1}} U)^{op})

of k and G_{U} is the Clifford extension of U. Note that if V is a simple kR_{1} -module, then kE(V) = End_{kR}(kR ⊗_{kR_{1}} V)^{op} .

We shall need the following result which is due to T. Ikeda [5, Lemma 3.1].

2.4. Lemma. Assume that U is a G-invariant indecomposable kR_{1} -module having a simple socle (or a simple head) V . Then the G-graded k -algebras kE(U) and kE(V) are isomorphic.

Proof. Let V be the socle of U, so V is G-invariant too. The restriction of homomorphism from kR ⊗_{kR_{1}} U to kR ⊗_{kR_{1}} V is well defined and gives a grade-preserving ring homomorphism ρ from End_{kR}(kR ⊗_{kR_{1}} U)^{op} to End_{kR}(kR ⊗_{kR_{1}} V)^{op} . In degree 1, ρ induces the isomorphism

End_{kR_{1}}(U)^{op}/J(End_{kR_{1}}(U)^{op}) ≃ End_{kR_{1}}(V)^{op} ≃ k,

hence Ker ρ = J_{gr}(End_{kR}(kR ⊗_{kR_{1}} U)^{op}). A dual argument applies if U has a simple head. □

2.5. The residual Clifford extension of a simple KR_{1} -module X can also be computed in the following way. Assume that the R_{1} -lattice U is a G_{X} -invariant O-form of X. (Note that in general, G_{U} is a subgroup of G_{X}.) Then the residual Clifford extension of X is naturally isomorphic to the Clifford extension of U.

Indeed, denoting E = End_{R}(R ⊗_{R_{1}} U)^{op} , we have that E_{G_{X}} is a crossed product of E_{1} and G_{X}, and kE(U) = E/J_{gr}(E) is a twisted group algebra of k and G_{X}. Moreover, we have that

K ⊗_{O} E_{G_{X}} ≃ End_{KR_{G_{X}}} (KR_{G_{X}} ⊗_{KR_{1}} X)^{op} = KE(X),

so E_{G_{X}} is the unique maximal G_{X} -graded suborder of KE(X).

2.6. We shall apply the results of this section mostly in the following situation, when invariant O-forms exist.

Let K be a normal subgroup of the finite group H, let G = H/K, and let K_{α}H be a twisted group algebra, where α ∈ Z^{2}(H, K^{×}). Assuming that K is big enough, K_{α}H has a unique maximal H -graded suborder, so we may assume that α ∈ Z^{2}(H, O^{×}), and then the maximal suborder is O_{α}H. By [4, Section 6], we may assume that the group Ĥ = hU(O_{α}H) of (H -graded) homogeneous units of O_{α}H has a finite subgroup H^{∗} covering H, such that exp H^{∗} = exp Z, where Z = H^{∗} ∩ O^{×}. Let ζ: Z → O^{×} be the inclusion, and let V be the OZ -module of O-rank 1 determined by ζ. By Clifford theory, the functor

(K ⊗_{O} V) ⊗_{K} −: K_{α}H-mod ⟶ (KH^{∗}|K ⊗_{O} V)-mod

is an equivalence, where (KH^{∗}|K ⊗_{O} V)-mod denotes the category of KH^{∗} -modules lying over K ⊗_{O} V .

Let W be a simple K_{α}K -module, and let W^{∗} = (K⊗_{O} V)⊗_{K} W be the KK^{∗} -module corresponding to W. Then W and W^{∗} have the same stabilizer G_{W} in G. Since K contains enough roots of unity, by the main result of [12], W^{∗} has a G-invariant O-form U^{∗}, which clearly lies over V . Applying to U^{∗} the inverse of the equivalence

V ⊗_{O} −: O_{α}K-mod ⟶ (OK^{∗}|V)-mod,

we obtain a G_{W} -invariant O_{α}K -module U with the property K ⊗_{O} U ≃ W.

2.7. Returning to our general situation 2.1, let b be a block idempotent of R_{1} , and let A = bRb, which is a crossed product of bR_{1} and the stabilizer G_{b} of b. We have that C := C_{A}(A_{1}) = bC_{R}(R_{1}) is a G_{b} -graded G_{b} -algebra, while kE(b) := C/J_{gr}(C) is a twisted group algebra of k and the normal subgroup

G[b] = {g ∈ G | bR_{g} ≃ bR_{1} as Δ_{1} -modules}

of G_{b}. The group of homogeneous units of kE(b) is the Clifford extension of b.

2.8. Lemma. Let U be an indecomposable R_{1} -module belonging to the block b. Then G[b] ⊆ G_{U} ⊆ G_{b}, and there is a monomorphism

μ = μ(b, U): kE(b) ⟶ kE(U)

of G_{U} -graded G_{U} -algebras.

Proof. If g ∈ G[b] and a ∈ C_{A}(A_{1})_{g}, let

ϕ_{a}: R ⊗_{R_{1}} U → R ⊗_{R_{1}} U, r ⊗ u ↦ ra ⊗ u.

This is well defined, and ϕ_{a} belongs to End_{R}(R ⊗_{R_{1}} U)_{g}^{op} . Moreover, if a belongs to C_{A}(A_{1})_{g} ∩ J_{gr}(C_{A}(A_{1})), that is, 1 − a_{g} has an inverse b ∈ C_{A}(A_{1})_{g^{−1}} , then 1 − ϕ_{a} has as inverse the map corresponding to b, hence ϕ_{a} ∈ J_{gr}(End_{R}(R ⊗_{R_{1}} U)^{op}). Therefore we may define μ by sending the class of a to the class of ϕ_{a}.

If h ∈ G_{U}, t ∈ hU(R) ∩ R_{h}, and the maps ψ_{h}: R ⊗_{R_{1}} U → R ⊗_{R_{1}} (R_{h} ⊗_{R_{1}} U) and ψ_{h^{−1}} : R ⊗_{R_{1}} U → R ⊗_{R_{1}} (R_{h^{−1}} ⊗_{R_{1}} U) are isomorphisms, then the class of the map ϕ_{hah^{−1}} : r ⊗ u ↦ rtat^{−1} ⊗ u is the class of ψ_{h}ϕ_{a}ψ_{h^{−1}} . □

2.9. The Grothendieck group K_{0}(KR_{1}) of KR_{1}-mod has as ℤ-basis the G-set Irr(KR_{1}) of isomorphism classes of simple KR_{1} -modules, and it is endowed with the G-invariant scalar product

⟨[X], [X′]⟩ = dim_{K} Hom_{KR_{1}}(X, X′).

Similarly, K_{0}(kR_{1}) has as ℤ-basis the G-set Irr(kR_{1}) of simple kR_{1} -modules, and K_{0}(kR_{1}-proj) has as ℤ-basis the G-set Pim(kR_{1}) of indecomposable projective kR_{1} - modules. There is a G-invariant duality between K_{0}(kR_{1}-proj) and K_{0}(kR_{1}) defined by

⟨[P], [U]⟩ = dim_{k} Hom_{kR_{1}}(P, U).

The Cartan-Decomposition triangle T (R_{1}) of R_{1} is the commutative diagram

\xymatrix{K_{0}(KR_{1}) \ar[rr]^{Dec} &  & K_{0}(kR_{1}) \\  & K_{0}(kR_{1}-proj) \ar[ul]^{_{_{t} Dec}} \ar[ur]_{Car}}

where Dec, Car and ^{t} Dec are G-maps preserving the metric structure of T (R_{1}).

Let Pr(KR_{1}) = Im(^{t} Dec) be the ℤ-submodule of K_{0}(KR_{1}) generated by the characters of projective R_{1} -modules, and denote K_{0}^{p}(KR_{1}) = Ker Dec, which is the orthogonal complement of Pr(KR_{1}).

Finally, by [2], the stable Cartan-Decomposition triangle T^{st}(R_{1}) of R_{1} is the diagram

\xymatrix{K_{0}^{st}(KR_{1}) \ar[r]^{^{Dec^{st}}} & K_{0}^{st}(kR_{1}),}

where K_{0}^{st}(KR_{1}) = K_{0}(KR_{1})/ Pr(KR_{1}) and K_{0}^{st}(kR_{1}) = K_{0}(kR_{1})/ Im Car.

2.10. Let R′ be another G-graded crossed product satisfying the same conditions as R in 2.1. Then R ⊗_{O} R′^{op} is a G × G-graded algebra, and its diagonal subalgebra

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

is a G-graded crossed product.

Let M = ⨁_{x∈G} M_{x} be a G-graded (R, R′)-bimodule. Then M_{1} is a Δ-module, and there are natural isomorphisms

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

M is called exact if _{R}M and M_{S} are projective modules.

2.11. Theorem. Let M be a G-graded exact (R, R′)-bimodule, and assume that M_{1} and its O-dual M_{1}^{∨} induce a Morita stable equivalence between R_{1} and R′_{1} . Then the following statements hold.

a) There is a G-isomorphism between K_{0}^{p}(KR_{1}) and K_{0}^{p}(KR′_{1}), and a G-isomorphism of stable Cartan-Decomposition triangles

\xymatrix{K_{0}^{st}(KR_{1}) \ar[d]_{≃} \ar[r]^{^{Dec^{st}}} & K_{0}^{st}(kR_{1}) \ar[d]^{≃} \\ K_{0}^{st}(KR′_{1}) \ar[r]^{_{Dec_{st}}} & K_{0}^{st}(kR′_{1}).}

b) Let U be an indecomposable nonprojective R_{1} -module in the block b of R_{1} corresponding to the indecomposable nonprojective R′_{1} -module U′ belonging to the block b′ of R′_{1} . Then there is a commutative diagram of G_{U} -graded G_{U} -algebras

\xymatrix{kE(b) \ar[d]_{≃} \ar[r]^{μ(b,U)} & kE(U) \ar[d]^{≃} \\ kE(b′) \ar[r]^{μ(b′,U′)} & kE(U′).}

Proof. a) The existence of the isomorphism between K_{0}^{p}(KR_{1}) and K_{0}^{p}(KR′_{1}), and the isomorphism between T^{st}(R_{1}) and T^{st}(R′_{1}) is well-known (see [2, Proposition 5.3]). Additionally, observe that the isomorphisms in 2.10 imply that R_{g} ⊗_{R_{1}} M_{1} ≃ M_{1} ⊗_{R′_{1}} R′_{g} as (R_{1}, R′_{1})-bimodules for all g ∈ G. Consequently, if U is an R_{1} -module, then for every g ∈ G we have the isomorphism

M_{1}^{∨} ⊗_{R_{1}} (R_{g} ⊗_{R_{1}} U) ≃ R′_{g} ⊗_{R′_{1}} (M_{1}^{∨} ⊗_{R_{1}} U)

of R′_{1} -modules. The graded bimodules M and M^{∨} induce similar isomorphisms over k and over K, hence a G-morphism between T (R_{1}) and T (R′_{1}), and finally, a Gisomorphism between T^{st}(R_{1}) and T^{st}(R′_{1}).

b) Consider the G-graded O-algebras E = End_{R}(R⊗_{R_{1}} U)^{op} and E′ = End_{R′}(R′⊗_{R′_{1}} U′)^{op} .

First, we claim that G_{U} = G_{U′}. Indeed, if g ∈ G_{U}, then R_{g} ⊗_{R_{1}} U ≃ U, and

R′_{g} ⊗_{R′_{1}} (M^{∨} ⊗_{R_{1}} U) ≃ M_{1}^{∨} ⊗_{R_{1}} (R_{g} ⊗_{R_{1}} U) ≃ M^{∨} ⊗_{R_{1}} U.

Since U′ is the unique non-projective summand of M^{∨} ⊗_{R_{1}} U, it folows that g ∈ G_{U′}. Conversely, we also have that G_{U′} ⊆ G_{U}.

The functor M^{∨}⊗_{R}− induces a homomorphism E → End_{R′}(R′⊗_{R′_{1}} (M^{∨}⊗_{R_{1}} U))^{op} of G-graded algebras. This homomorphism is compatible with conjugation, since if g ∈ G_{U} and ϕ ∈ E_{h} is an isomorphism, then M^{∨} ⊗_{R} ϕ is also an isomorphism of degree g, and the image of ϕαϕ^{−1} under M^{∨} ⊗_{R} − is (M^{∨} ⊗_{R} ϕ)(M^{∨} ⊗_{R} α)(M^{∨} ⊗_{R} ϕ^{−1}), for all α ∈ E.

The projective endomorphisms of U form an ideal of E_{1} , which is clearly G_{U} - invariant, so it is the 1-component of a G_{U} -graded ideal of E_{G_{U}} contained in J_{gr}(E_{G_{U}}). Similar statements hold for End_{R′}(R′ ⊗_{R′_{1}} (M^{∨} ⊗_{R_{1}} U))^{op} and E′. Since M_{1} induces a stable Morita equivalence, we have that \overline{End}_{R_{1}}(U) ≃ \overline{End}_{R′_{1}}(U′). It follows that

kE(U) = E/J_{gr}(E) ≃ E′/J_{gr}(E′) = kE(U′)

as G_{U} -graded G_{U} -algebras.

Let A = bRb and A′ = b′Rb′. Multiplication on the right gives an isomorphism C_{A}(A_{1}) → End_{R⊗_{R_{1}} R_{1}^{op}}(A)^{op} of G_{b} -graded algebras. Then by arguments as above we deduce that the functor M^{∨} ⊗_{R} (−) ⊗_{R_{1}} M_{1} induces an isomorphism kE(b) ≃ kE(b′) of G_{U} -graded G_{U} -algebras.

Let ϕ_{c} be the endomorphism of _{R}A_{R_{1}} corresponding to the element c ∈ C_{A}(A_{1}). Then by Lemma 2.8, the image in kE(U) of the class of ϕ_{c} is the class of ϕ_{c} ⊗_{R_{1}} M_{1} , and the image in kE(b′) of the class of ϕ_{c} is the class of M^{∨} ⊗_{R} ϕ_{c} ⊗_{R_{1}} M_{1} . Sice the map μ(b, U): kE(b) → kE(U) is induced by multiplication with scalars, it follows immediately that the diagram is commutative. □

2.12. The previous result applies in the case of the Heller translation. Let Ω_{R_{1}⊗_{O}R_{1}^{op}} be the kernel of the (R_{1}, R_{1})-bimodule map R_{1} ⊗_{O} R_{1}^{op} → R_{1} given by the multiplication. It is well-known that Ω_{R_{1}⊗_{O}R_{1}^{op}} and its O-dual induce a stable Morita autoequivalence of R_{1} . Unfortunately, it does not extend to a Δ(R⊗_{O} R^{op})-module unless G is a p′ -group. We can however replace Ω_{R_{1}⊗_{O}R_{1}^{op}} with the kernel Ω_{Δ} of the multiplication map

Δ(R ⊗_{O} R^{op}) = ⨁_{g∈G}(R_{g} ⊗_{O} R_{g^{−1}}) ⟶ R_{1} → 0.

This map is Δ(R⊗_{O} R^{op})-linear, and since Δ(R⊗_{O} R^{op}) is still a projective R_{1} ⊗_{O} R_{1}^{op} - module, we have that Ω_{R_{1}⊗_{O}R_{1}^{op}} and Ω_{Δ} induce the same functor on the stable category of R_{1}-mod. The above theorem applies with the G-graded (R, R)-bimodule

M = (R ⊗_{O} R^{op}) ⊗_{Δ(R⊗_{O}R^{op})} Ω_{Δ} ≃ Ω_{R⊗_{O}R^{op}}.

3. Blocks with normal defect groups

3.1. Let b be a block with defect group D of the twisted group algebra A = O_{α}G, and let (D, e) be a maximal b-Brauer pair, so e is a block of A(D) = k_{α}C_{G}(D̂) with defect group C_{D}(D̂). The semisimple quotient of eA(D) is a simple k -algebra which will be denoted S ≃ M_{n}(k), and let V̄ be the unique simple S -module.

Let H = N_{G}(D), N = N_{G}(D, e) = {h ∈ H | ^{h}e = e}, and let c be the Brauer correspondent of b in O_{α}H. Let also f be the block of O_{α}C_{G}(D̂) with defect group C_{D}(D̂) satisfying f̄ = e. Then, since c = Tr_{N}^{H} f, the algebra cO_{α}H is Morita equivalent to fO_{α}H. Note that f is also a block with defect group D of O_{α}DC_{G}(D̂) and of O_{α}N.

Since k is algebraically closed, the inertial quotient E = N/DC_{G}(D̂) of b with respect to e can be regarded as a p′ -subgroup of Out(D), and consequently of Aut(D). We may consider the semidirect product D ⋊ E having C_{D}(D̂) as a normal subgroup, so any twisted group algebra O_{γ}(D ⋊ E) may be regarded and an E -graded algebra and also as an algebra graded by (D ⋊ E)/C_{D}(D̂) ≃ N/C_{G}(D̂).

3.2. The group N acts on eA(D) and also on the simple quotient S, and moreover, S is a k_{α}(DC_{G}(D̂))-interior algebra. It follows that there is a unique E -graded crossed product S ∗_{α} E with 1-component S, endowed with a homomorphism k_{α}H → S ∗_{α} E of E -graded k -algebras.

The simple S -module V̄ is E -invariant, hence there is an isomorphism

End_{S∗_{α}E}((S ∗_{α} E) ⊗_{S} V̄)^{op} ≃ k_{β}E,

where β ∈ Z^{2}(E, k^{×}). Consequently, (S ∗_{α} E) ⊗_{S} V̄ is an E -graded (S ∗_{α} E, k_{β}E)- bimodule, and V̄ is a module over the diagonal subalgebra Δ̄ = Δ(S ∗_{α} E ⊗_{k} k_{β}E^{op}), which is isomorphic to S ∗_{αβ^{−1}} E.

(One may elaborate a bit more on the structure of Δ̄. We have the homomorphisms O_{α}H → S ∗_{α} E and O_{β}H → k_{β}E of E -graded algebras, where we denote by the same β the 2-cocycle of H with values in O^{×} obtained from β by inflation from E to H. It follows that Δ̄ is an E -graded O_{αβ^{−1}}H -interior k -algebra.)

Let O_{α}D ∗_{β} E be the crossed product of O_{α}D and E determined by β and the action of E on D. Notice that this is actually a twisted group algebra O_{γ}(D ⋊ E), where γ ∈ Z^{2}(D ⋊ E, O^{×}) is the product between α extended trivially to D ⋊ E and the inflation of β from E to D ⋊ E.

Finally, in this section we denote R = fO_{α}N = fA_{N}, regarded as an N/C_{G}(D̂)- graded O-algebra, with 1-component R_{1} = fA_{C_{G}(D̂)} .

3.3. Theorem. There is an N/C_{G}(D̂)-graded Morita equivalence between R and O_{γ}(D ⋊ E).

Proof. Regard V̄ as an fA_{C_{G}(D̂)} -module via the epimorphism fA_{C_{G}(D̂)} → S, and let V ∈ fA_{C_{G}(D̂)}-mod be the projective cover of V̄ . We may regard V as an (fA_{C_{G}(D̂)}, A_{C_{D}(D̂)})-bimodule (because A_{C_{D}(D̂)} is commutative). Since V is a projective fA_{C_{G}(D̂)} -module, the O-algebra map A_{C_{D}(D̂)} → End_{eA_{C_{G}(D̂)}}(V)^{op} given by multiplication on the right is injective. Since End_{R_{1}}(V)/J(End_{R_{1}}(V)) ≃ End_{S}(V̄) ≃ k, it follows by Nakayama’s lemma the this map is also surjective. Consequently, the (R_{1}, A_{C_{D}(D̂)})- bimodule V induces a Morita equivalence between R_{1} and A_{C_{D}(D̂)} = R′_{1} .

Now let D̄ = D/C_{D}(D̂) ≃ DC_{G}(D̂)/D, and consider the D̄-graded algebras RD̄ = fA_{DC_{G}(D̂)} and R′ ; D̄ = A_{D}. One can easily see that the diagonal subalgebra of RD̄ ⊗_{O} A_{D}^{op} is (fA_{C_{G}(D̂)}⊗_{O}1)O(δ(D)), so we have a surjective O-algebra map Δ(RD̄⊗_{O}A_{D}^{op}) → R_{1}. It follows that V extends to a Δ(RD̄ ⊗_{O} A_{D}^{op})-module with vertex δ(D), by letting δ(D) act trivially on V .

By [9, Theorem 5.1.2], the D̄-graded (RD̄, A_{D})-bimodule

U := (RD̄ ⊗_{O} A_{D}^{op}) ⊗_{Δ(RD̄⊗_{O}A_{D}^{op})} V ≃ RD̄ ⊗_{R_{1}} V ≃ V ⊗_{A_{C_{D} (D̂)}} A_{D}

induces a D̄-graded Morita equivalence between RD̄ and A_{D}. By Green’s indecomposability theorem, U is indecomposable even as an RD̄ -module.

Now regard R and A_{D} ∗_{β} E as E -graded algebras, and consider the diagonal subalgebra Δ′ = Δ(R ⊗_{O} (A_{D} ∗_{β} E)^{op}), which is also E -graded with 1-component Δ′_{1} = RD̄ ⊗_{O} A_{D}^{op} . Observe that Δ′_{1}/J(Δ′_{1}) ≃ S and U/J(Δ′_{1})U ≃ V . By Lemma 2.4 it follows that the Clifford extension of the Δ′_{1} -module U is isomorphic to the Clifford extension of V . We have that Δ′/J(Δ′_{1})Δ′ is isomorphic to Δ̄ = Δ((S ∗_{α} E)⊗_{k} (k_{β}E)^{op}) as E -graded algebras.

By 3.2, V extends to a Δ̄-module, and since E is a p′ -group, it follows by a theorem of Dade (see [9, Corollary 3.1.9]) that U extends to a Δ′ -module. By [9, Theorem 5.1.2], the R ⊗_{O} (A_{D} ∗_{β} E)^{op} -module

W := (R ⊗_{O} (A_{D} ∗_{β} E)) ⊗_{Δ′} U

induces an E -graded Morita equivalence between R and A_{D} ∗_{β} E. This is actually an N/C_{G}(D̂)-graded equivalence, since

W ≃ R ⊗_{RD̄} U ≃ R ⊗_{R_{1}} V
≃ U ⊗_{A_{D}} (A_{D} ∗_{β} E) ≃ V ⊗_{A_{C_{D} (D̂)}} (A_{D} ∗_{β} E)

as N/C_{G}(D̂)-graded (R, A_{D} ∗_{β} E)-bimodules. □

3.3. Corollary. If D is cyclic, then R is Morita equivalent to O(D ⋊ E).

Proof. We have in general that for any group G, H^{2}(G, O^{×}) embeds into H^{2}(G, K^{×}), and since K^{×} is uniquely n-divisible for any positive n, H^{2}(G, K^{×}) = 1 when G is cyclic.

Since D is cyclic, E is cyclic too, hence the cohomology classes of Res_{D}^{H} α and β are trivial. □

4. Blocks with cyclic defect groups

4.1. As in the introduction, let K be a normal subgroup of the finite group H, let G = H/K, and let b be a G-invariant block of the twisted group algebra O_{α}K having cyclic defect group D. Let Ω(D) the unique subgroup of order p of D, denote K′ = N_{K}(Ω(D)), H′ = N_{H}(Ω(D)), and let b′ be the Brauer corresponding block of O_{α}K′.

We regard R = bO_{α}H and R′ = b′O_{α}H′ as G-graded O-algebras, and then M = bO_{α}Hb′ is a G-graded (R, R′)-bimodule with 1-component M_{1} = bO_{α}Kb′.

The proof of [8, Theorem 11.1] can be easily generalized to obtain:

4.2. Theorem. M_{1} induces a Morita stable equivalence between R_{1} and R′_{1} .

4.3. Theorem. b′O_{α}K′ is Morita equivalent to O(D ⋊ E), where E is the inertial quotient of b.

Proof. Let K′′ = N_{K}(D) and let b′′ be the Brauer correspondent of b in O_{α}K′′. By Corollary 3.4 we have that b′′O_{α}K′′ is Morita equivalent to O(D ⋊ E), so it remains to show that b′O_{α}K′ and b′′O_{α}K′′ are Morita equivalent. We shall use that this is true for ordinary group algebras (see [13, Theorem 10.2]) and the method of [10, Section 6].

As in [3, Section 6], let K^{∗} be a finite subgroup of K̂ covering K, such that the exponent of K^{∗} equals the exponent of Z = K^{∗} ∩ O^{∗}. Then Z is the direct product of its Sylow p-subgroup Z_{p} and its Hall p′ -subgroup Z_{p′}, and denoting D^{∗} = D̂ ∩ K^{∗}, we have D^{∗} = Z_{p} × D. If V is the OZ -lattice of O-rank 1 determined by the inclusion ζ: Z → O^{×}, 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′}.

Let b^{∗} be the unique block of OK^{∗} covering b, let K̄ = K^{∗}/Z_{p}, and let b̄ be the image of b^{∗} under the epimorphism OK^{∗} → OK̄. Then by [7, Theorem 7], there is a natural isomorphism OZ_{p} ⊗_{O} b̄OK̄ ≃ b^{∗}OK^{∗}. Note that the categories (b^{∗}OK^{∗}|V_{p})-mod and (b^{∗}OK^{∗}|V)-mod of module lying over V_{p} and V respectively coincide, and by Clifford theory, the categories bO_{α}K-mod and (b^{∗}OK^{∗}|V)-mod are equivalent.

Similar statements hold for b′O_{α}K′, b′^{∗}OK′^{∗}, b̄′OK̄′, and b′′O_{α}K′′, b′′^{∗}OK′′^{∗}, b̄′′OK̄′′, where K′^{∗} = N_{K^{∗}}(Z_{p} × Ω(D)), K̄′ = K′^{∗}/Z_{p}, K′′^{∗} = N_{K^{∗}}(D^{∗}), K̄′′ = K′′^{∗}/Z_{p}, and b′^{∗}, b̄′, respectively b′′^{∗}, b̄′′ are the Brauer corresponding blocks. Since b̄ has cyclic defect groups, we have that b̄′OK̄′ is Morita equivalent to b̄′′OK̄′′, and by the above, we deduce that b̄′O_{α}K̄′ is Morita equivalent to b̄′′O_{α}K̄′′. □

4.4. The above theorems imply that the results of [8, Sections 13 and 15] generalize to twisted group algebras with exactly the same proof. We briefly present here the facts we need.

First recall that Irr_{K}(D ⋉ E) is the disjoint union of the set

M = {μ ∈ Irr_{K}(D ⋊ E) | Ker μ ⊇ D}

of non-exceptional characters (which is in bijection with Irr_{K} E), and the set

Λ = {λ ∈ Irr_{K}(D ⋊ E) | D ⊉ Ker λ}
= {Ind_{D}^{D⋊E} ζ | ζ ∈ Irr_{K} D, ζ ≠ 1_{D}}

of exceptional characters, and we have the equalities |M| = |E| and |Λ| = \frac{|D|−1}{|E|}.

We also have that (4.4.1) {∑_{μ∈M} μ − λ | λ ∈ Λ} is a basis of K_{0}^{p}(K(D ⋊ E)) = Ker Dec. (4.4.2) {μ + ∑_{λ∈Λ} λ | μ ∈ M} is a basis of Pr(K(D ⋊ E)) (4.4.3) A character of a projective indecomposable O(D ⋊ E)-module is of the form μ + ∑_{λ∈Λ} λ, where μ ∈ M.

There is a bijective isometry Δ: K_{0}(K(D ⋊ E)) → K_{0}(KR_{1}), which extends the isometry between K_{0}^{p}(K(D ⋊ E)) and K_{0}^{p}(KR_{1}) induced by the stable equivalence. In particular, for λ ∈ Irr_{K}(K(D ⋊ E)), we may write Δ(λ) = δ(λ) · χ_{λ}, where χ_{λ} ∈ Irr_{K}(KR_{1}) and δ(λ) ∈ {1, −1}. Moreover, δ(λ) = δ(λ′) for λ, λ′ ∈ Λ, and the set of exceptional characters of KR_{1} is {χ_{λ} | λ ∈ Λ}.

4.5. This isometry is actually the “shadow” of a splendid derived equivalence between O(D ⋊ E) and R_{1} (see [13, Theorem 10.3] and also [10, Section 6] for the twisted case).

We also have a splendid derived equivalence between R_{1} and R′_{1} induced by the complex

C^{•} = (⋯ → 0 → P → M_{1} → 0 → …)

of (R_{1}, R′_{1})-bimodules, where M_{1} = bO_{α}Kb′ in degree 0, and P is a direct summand of the projective cover of the R_{1} ⊗_{O} R′_{1}^{op} -module M_{1} . Note that by [10, 6.7], the complex C^{•} is G-invariant, that is, R_{g} ⊗_{R_{1}} C^{•} ⊗_{R_{1}} R′_{g^{−1}} ≃ C^{•} as complexes of (R_{1}, R′_{1})- bimodules.

Green’s exact sequence also generalizes to our situation.

4.6. The algebras kR_{1} and kR′_{1} have the same number |E| of simple modules. Let |E| = e and I = {0, 1, …, e − 1}. We have that {Ω^{2i}k | i ∈ I} is a complete set of representatives of simple k(D ⋉ E)-modules.

Let {S_{i} | i ∈ I} (respectively {S′_{i} | i ∈ I}) be complete sets of representatives for the simple kR_{1} -modules (respectively kR′_{1} -modules). We choose the notation such that S′_{i} = Ω^{2i}(S′_{0}) corresponds to Ω^{2i}k under the Morita equivalence between kR′_{1} and k(D ⋊ E), and S_{i} is the socle of M̄^{∨} ⊗_{kR′_{1}} S′_{i}.

Let T′_{i} be an R′_{1} -lattice such that k ⊗_{O} T′_{i} ≃ S′_{i}. Note that T′_{i} is unique up to isomorphism. Let also P_{i} be the projective cover of S_{i} regarded as an R_{1} -module. The equalities Soc(M̄ ⊗_{kR′_{1}} ΩS′_{i}) = S_{ρ(i)} , i ∈ I define a permutation ρ of I. Let π be the cycle (0 1 … e − 1), and let σ = π ∘ ρ^{−1} . Then by [8, Theorème 15.4], there are unique submodules U_{i} and V_{i} of P_{i} satisfying the following properties:

(4.6.1) For all i ∈ I there are exact sequences

0 ⟶ U_{i} ⟶ P_{i} ⟶ V_{σ^{−1}(i)} ⟶ 0,
0 ⟶ V_{i} ⟶ P_{i} ⟶ U_{ρ^{−1}(i)} ⟶ 0.

(4.6.2) U_{i} ≃ M ⊗_{R′_{1}} T′_{i} and V_{i} ≃ M ⊗_{R′_{1}} T′_{ρ^{−1}(i)} .

(4.5.3) The characters of U_{i}, V_{i} belong to the set {χ_{μ} | μ ∈ M} ∪ {∑_{λ∈Λ} λ}.

(4.6.4) Ū_{i}, V̄_{i} are uniserial submodules of P̄_{i}, Ū_{i}+V̄_{i} = Rad P̄_{i} and Ū_{i}∩V̄_{i} = Soc P̄_{i} = S_{i}.

With this information we can prove the main result of this paper.

4.7. Theorem. If X is a simple KR_{1} -module with non-exceptional character, and S is a simple kR_{1} -module, then the residual Clifford extension kE(X) of X is naturally isomorphic to the Clifford extension kE(S) of S.

Proof. Let G_{0} be the stabilizer of S′_{0} . Since S′_{i} = Ω^{2i}(S′_{0}), it follows by Theorem 2.11 b) and Example 2.12 that kE(S′_{i}) ≃ kE(S′_{0}), for i = 0, …, e − 1. By [4, Lemma 4.7], G_{0} is also the stablizer of S_{i}, i = 0, …, e − 1. Again by Theorem 2.11 b) we have that kE(M̄_{1} ⊗_{kR′_{1}} S′_{i}) ≃ kE(S′_{i}), and since S_{i} = Soc(M̄_{1} ⊗_{kR′_{1}} S′_{1}), by Lemma 2.4 we deduce that kE(S_{i}) ≃ kE(S′_{0}).

Now let X be a simple KR_{1} -module whose character is nonexceptional, and we may assume that X ≃ K ⊗_{O} U_{i}. By [12, Proposition 5] we have that G_{X} = G_{0} , and by 4.6.2, G_{0} is also the stabilizer of U_{i}. Consequently, U_{i} is a G_{X} -invariant O-form of X, so by 2.5, kE(X) ≃ kE(U). Since Soc Ū_{i} = S_{i}, it follows by Lemma 2.4 that kE(U_{i}) ≃ kE(S_{i}), hence kE(X) ≃ kE(S′_{i}). □

4.8. Theorem. Dade’s inductive conjecture holds for blocks with cyclic defect groups.

Proof. As in the proof of [4, Theorem 7.11], in the case of blocks with cyclic defect groups, Dade’s inductive conjecture reduces to the equality

(4.8.1)  k(b, F, κ) = k(b′, F, κ),

where F is a subgroup of G, O_{β}F is a twisted group algebra, and the “augmentation” κ: kE(b) → k_{β}F is a graded monomorphism. Here k(b, F, k) denotes the number of simple KR_{1} -modules X satisfying G_{X} = F and such that there is a graded monomorphism γ: kE(X) → k_{β}F with κ = γ ∘ μ(b, X), where μ(b, X): kE(b) → kE(X) is the graded monomorphism coming from 2.5 and Lemma 2.8; k(b′, F, k) is defined as the number of simple KR′_{1} -modules X′ satisfying similar properties, but with μ(b, X′) instead of μ(b, X), where μ(b, X′) is the composition

μ(b, X′): kE(b) ⟶^{≃} kE(b′) ⟶ kE(b, X′).

Let X be a simple KR_{1} -module with non-exceptional character corresponding to the simple KR′_{1} -module X′ under the isometry described in 4.4. Then by 2.5, Theorem 2.11 and Theorem 4.7 it follows that we have the commutative diagram

(4.8.2)  \xymatrix{kE(b) \ar[r]^{μ(b,X)} \ar[d]_{≃} & kE(X) \ar[d]^{≃} \\ kE(b′) \ar[r]^{μ(b′,X′)} & kE(X′)}

and this implies that an equality similar to (4.8.1) holds if we count only non-exceptional characters with given augmentation.

Finally, let X′ be a simple KR′_{1} -module whose character is exceptional, and let U′ be a G_{X′} -invariant O-form of X′. Let

U^{•} = (⋯ → 0 → P ⊗_{R′_{1}} U′ → M_{1} ⊗_{R′_{1}} U′ → 0 → …)

be the complex coresponding to U′ under the derived equivalence 4.5. Then the character of X′ corresponds to the character of K ⊗_{O} U^{•} under the isometry described in 4.4, and let X be a simple KR_{1} -module affording the character of K ⊗_{O} U^{•}. Denote also U = M_{1} ⊗_{R′_{1}} U′. Since the complex C^{•} is G-invariant, we have that G_{X} = G_{X′} = G_{U′} = G_{U^{•}} = G_{U}, and in what follows, we may assume that G_{X} = G. Recall that kE(X′) ≃ kE(U′) by 2.5, and we have the commutative diagram of Theorem 2.11 b) for our current U and U′.

Note that by [11, Lemma 1.7 a)], the endomorphism ring in the homotopy category of a complex of G-graded modules is G-graded. Moreover, it is easy to generalize 2.5 to the homotopy category of bounded complexes, so that for the Clifford extension of X we have the natural isomorphisms

KE(X) = End_{KR}(KR ⊗_{KR_{1}} X)^{op}
≃ End_{H(KR)}(KR ⊗_{KR_{1}} (K ⊗_{O} U^{•}))^{op}
≃ K ⊗_{O} End_{H(R)}(R ⊗_{R_{1}} U^{•})^{op}

of G-graded algebras, where H(R) denotes the homotopy category of complexes of Rmodules. It follows that the residual Clifford extension kE(X) of X is naturally isomorphic to k ⊗_{O} End_{H(R)}(R ⊗_{R_{1}} U^{•})^{op} , so in order to obtain the commutative diagram (4.8.2) in the exceptional case, it is enough to show that the latter algebra is naturally isomorphic to kE(U).

We have an obvious homomorphism of G-graded O-algebras

ϕ: End_{C(R)}(R ⊗_{R_{1}} U^{•}) → End_{R}(R ⊗_{R_{1}} U),

where C(R) denotes the category of complexes of R-modules. Since the complex R ⊗_{R_{1}} U^{•} has only one non-projective term, this homomorphism induces in degree 1 ∈ G a homomorphism of O-algebras from End_{H(R)}(U^{•}) to the endomorphism ring of U in the stable category of R_{1} -modules. This implies that ϕ induces an isomorphism of G-graded algebras from k ⊗_{O} End_{H(R)}(R ⊗_{R_{1}} U^{•})^{op} to kE(U). □

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