TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS

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

0. Introduction

If G is a finite group, k a field of characteristic p > 0, A a block of kG with defect group D, and B is the Brauer corresponding block of kN_{G}(D), then a conjecture of M.Broué [1] states that if D is abelian, then the bounded derived categories D^{b}(A) and D^{b}(B) are equivalent as triangulated categories. Moreover, it is conjectured that this equivalence is compatible with p′ -extensions of the group G.

Equivalences between D^{b}(A) and D^{b}(B) were characterized by J. Rickard in terms of one-sided tilting complexes [14], and in terms of two-sided tilting complexes [15]. Recall that in the case of the Morita equivalence, if P is a progenerator of A-Mod, the P becomes an (A, End_{A}(P)^{op})-module. One of the main difficulties in the case of derived equivalences is that if T is an one-sided tilting complex of A-modules, then End_{K(A)}(T)^{op} act on T only up to homotopy.

For two-sided tilting complexes, compatibility with p′ -extensions was studied in [7] and [8]. This compatibility is expressed in terms of the existence of certain tilting complexes of graded bimodules. The present paper is motivated by recent constructions due to T. Okuyama [11], [22], [13]. His work was completed by S. Koshitani, N. Kunugi and H. Miyachi (see [6] and the references given there), who verified Broué’s conjecture when D is elementary abelian of order 9, and Okuyama himself verified the conjecture for the principal p-block of SL(2, p^{n}) in [13]. Although it ends up with two-sided tilting complexes, Okuyama’s method is based on constructions of one-sided tilting complexes, one of them originating from J. Rickard (see [5, Proposition 5.1.1]).

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

Key words and phrases. graded algebras, crossed product, symmetric algebras, tilting complex, Rickard equivalence, abelian defect groups.

2 ANDREI MARCUS

This paper is organized as follows. Section 1 contains general results on categories of modules graded by G-sets and their derived categories. The main result is a graded version of Rickard’s fundamental theorem, stated in Theorem 2.4 below. Our arguments use Keller’s proof of Rickard’s theorem [3]. This is then applied in the last section where we give easy conditions implying that Okuyama’s equivalences are compatible with p′ - extensions. In particular, this discussion gives an alternative proof of certain parts of [6], but our method has wider applicability. Another consequence is that Broué’s conjecture holds for the principal p-block of GL(2, p^{n}).

In this paper rings are associative with unity, module will mean finitely generated left and unital module, and complex will mean cochain complex. All the needed background material is contained in [5] and [9]. Finally, one should mention that other methods or variants of the above methods of constructing derived equivalences are presented in [16], [14], [15], [4] and [19]. We hope to discuss them and their applications in a subsequent paper.

1. Derived categories of graded modules

1.1. Let k be a commutative ring, G a group and R = ⨁_{g∈G} R_{g} a G-graded k -algebra. For a subgroup H of G, we denote by (G/H, R)-Gr the category of G/H -graded Rmodules M = ⨁_{x∈G/H} M_{x} and grade-preserving R-homomorphisms. In particular, (G/G, R)-Gr is the category R-Mod of R-modules, and (G/1, R)-Gr is the category R-Gr of G-graded R-modules. Another example is provided by G-graded bimodules. If S is anothe G-graded k -algebra, a G-graded (R, S)-bimodule is an object of the category (G × G/δ(G), R ⊗_{k} S^{op})-Gr, where δ(G) is the diagonal subgroup of G × G.

Recall that (G/H, R)-Gr is a Grothendieck category having a projective generator.

1.2. Let K ≤ H be subgroups of G. We shall make use of the ungrading (gradeforgetting) functor

U = U_{G/H}^{G/K}: (G/K, R)-Gr → (G/H, R)-Gr

sending a G/K graded R-module M to U(M) = M with components U(M)_{x} = ⨁_{y∈G/K, y⊆x} M_{y} and a morphism f to f.

This is an exact functor, and has a right adjoint

F = F_{G/H}^{G/K}: (G/H, R)-Gr → (G/K, R)-Gr

sending a G/H -graded R-module M to F(M) = ⨁_{y∈G/K} F(M)_{y}, where F(M)_{y} = M_{x} for y ⊆ x. Then F is exact, and if G is a finite group, then F is also a left adjoint of U.

We shall also consider the g -suspension (conjugation)

S_{g} = S_{g}^{G/H}: (G/H, R)-Gr → (G/gHg^{−1}, R)-Gr

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 3

sending M to S_{g}(M) = M(g), where M(g)_{y} = M_{yg} for all y ∈ G/gHg^{−1} .

1.3. The ungrading functor U is separable. This means that for any G/K -graded Rmodules M and M′, the trace map

Tr: Hom_{(G/H,R)-Gr}(M, M′) → Hom_{(G/K,R)-Gr}(M, M′)

defined by f ↦ Tr(f), Tr(f)(m_{x}) = f(m_{x})_{x} for all m_{x} ∈ M_{x}, x ∈ G/K, satisfies Tr(U(f)) = f for all f ∈ Hom_{(G/K,R)-Gr}(M, M′). The following statements are easy consequences of the separability of U.

a) Let P be a G-graded R-module. If U(P) is a projective R-module, then P is projective in R-Gr.

b) Let I be a G-graded R-module. If U(I) is an injective R-module, then I is injective in R-Gr.

Since the direct sum ⨁_{g∈G} R(g) of the conjugates of R is a projective generator of R-Gr which is projective in R-Mod, the converse of a) is true. By [10, Corollary 2.2], if G is finite, then the converse of b) also holds.

1.4. Clearly, the above functors may be extended to functors between categories of complexes. For instance, U induces the functors

U = U_{G/H}^{G/K}: C^{∗}((G/K, R)-Gr) → C^{∗}((G/H, R)-Gr),
U = U_{G/H}^{G/K}: H^{∗}((G/K, R)-Gr) → H^{∗}((G/H, R)-Gr),
U = U_{G/H}^{G/K}: D^{∗}((G/K, R)-Gr) → D^{∗}((G/H, R)-Gr),

where ∗ = ∅, −, +, b, which are again separable.

Another application of the separability of U is the following

1.5. Lemma. Let f, g: C → D be morphisms in C((G/K, R)-Gr). If U_{G/K}^{G/K}(f) and U_{G/K}^{G/K}(g) are homotopic, then f and g are homotopic.

Indeed, if s: C → D[1] is a homotopy between U(f) and U(g), then Tr(s) is a homotopy between f and g.

1.6. A complex T ∈ H(R-Gr) is called G-invariant if T (g) ≃ T (in the category H(R-Gr)) for all g ∈ G. T is called weakly G-invariant if T (g) ∈ add(T) for all g ∈ G.

1.7. Lemma. Let T ∈ H(R-Gr), E = End_{H(R)}(T)^{op} , and assume that G is a finite group.

a) E is a G-graded algebra with components

E_{g} ≃ Hom_{H(R-Gr)}(T, T (g)).

4 ANDREI MARCUS

b) E is strongly graded (crossed product) if and only if T is weakly G-invariant (Ginvariant).

Proof. a) As in the case of modules, it is easy to see that E′ = End_{C(R)}(T)^{op} is a Ggraded algebra with components E′_{g} = (Hom_{C(R-Gr)}(T, T (g)). We have to show that the ideal containing the morphisms which are homotopic to zero is a G-graded ideal, that is, if f = ∑_{g∈G} f_{g} ∈ E′ is homotopic to zero, then f_{g} is homotopic to zero for all g ∈ G. Indeed, observe that f_{g} is the image of f through the trace map

Tr: Hom_{C(R)}(U(T), U(T (g)) → Hom_{C(R-Gr)}(T, T (g)).

Then, if s: U(T) → U(T)[−1] satisfies f = sd+ds, then as in Lemma 1.5, f_{g} = s_{g}d+ds_{g}.

b) is clear, since E is a crossed product means that each component E_{g} contains a unit of E, while E is strongly graded means that each component E_{g} belongs to add(E_{1}).

1.8. A complex P ∈ H((G/K, R)-Gr) is homotopically projective if

Hom_{H((G/K,R)-Gr)}(P, C) = 0

for each acyclic complex C ∈ H((G/K, R)-Gr). The complex I ∈ H((G/K, R)-Gr) is homotopically injective if

Hom_{H((G/K,R)-Gr)}(C, I) = 0

for each acyclic complex C ∈ H((G/K, R)-Gr).

1.9. Lemma. Let P ∈ H((G/K, R)-Gr). Then P is homotopically projective if and only if H_{G/H}^{G/K}(P) is homotopically projective.

If G is finite and I ∈ H((G/K, R)-Gr), then P is homotopically injective if and only if H_{G/H}^{G/K}(P) is homotopically injective.

Proof. Let C ∈ H((G/H, R)-Gr) be acyclic. Then F_{G/H}^{G/K}(C) is also acyclic, and since F is a right adjoint of U, we have

Hom_{H((G/H,R)-Gr)}(U(P), C) ≃ Hom_{H((G/K,R)-Gr)}(P, F(C)) = 0.

Conversely, let C ∈ H((G/K, R)-Gr) be acyclic. Then Hom_{H((G/K,R)-Gr)}(P, C) = 0, since the trace map

Hom_{H((G/H,R)-Gr)}(P, C) → Hom_{H((G/K,R)-Gr)}(P, C)

is surjective.

In the case of injectivity, the proof is similar, but we have to use that if G is finite, then F is also a left adjoint of U.

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 5

1.10. A complex P of G/K -graded R-modules is called perfect if it is bounded, and its terms are finitely generated projective R-modules. We denote by (G/K, R)-Grperf the full subcategory of D((G/K, R)-Gr) consisting of complexes quasi-isomorphic to a perfect complex.

Rickard’s characterization of R-perf [14, Proposition 6.3] also holds in this situation: a complex P ∈ D((G/K, R)-Gr) belongs to (G/K, R)-Grperf if and only if it is compact, that is, the functor

Hom_{D((G/K,R)-Gr)}(P, −) → Ab

commutes with infinite direct sums.

It can be easily seen that if P belongs to (G/K, R)-Grperf, then U_{G/H}^{G/K}(P) belongs to (G/H, R)-Grperf.

1.11. Let H_{p}((G/K, R)-Gr) and H_{i}((G/K, R)-Gr) denote the full subcategories of homotopically projective, respectively homotopically injective, objects of H((G/K, R)-Gr). Then [3, Theorems 8.1.1 and 8.1.2] generalize to these categories: there are equivalences of categories p: D((G/K, R)-Gr) → H_{p}((G/K, R)-Gr) and i: D((G/K, R)-Gr) → H_{i}((G/K, R)-Gr), with quasi-inverses induced by the quotient functor H((G/K, R)-Gr) → D((G/K, R)-Gr), where for each complex C of G/K -graded R-modules, p(C) is a homotopically projective resolution of C, and i(C) is a homotopically injective resolution of C.

1.12. Proposition. A full triangulated subcategory of D((G/K, R)-Gr) equals D((G/K, R)-Gr) if and only if it contains R(g), g ∈ G and it is closed under infinite direct sums.

Proof. The statement follows by infinite dévissage (see [3, 8.1.5]), since D((G/K, R)-Gr) is a triangulated category with infinite direct sums, and the conjugates R(g) of R are compact objects generating D((G/H, R)-Gr).

2. Graded derived equivalences

2.1. Let R and S be G-graded k -algebras, and let D_{1} and D_{2} be full triangulated subcategories of D(R-Gr) and D(S-Gr) respectively, both closed under the conjugation functors S_{g} = (−)(g), g ∈ G. Then a functor F: D_{1} → D_{2} is said to be G-graded if F ∘ S_{g} = S_{g} ∘ F for all g ∈ G.

Starting with a complex X = ⨁_{g∈G} X_{g} of G-graded bimodules, one obtains an adjoint pair of G-graded functors as follows. If C is a complex of G/K -graded S -modules, then X ⊗_{S} C is a complex of G/K -graded R-modules, where for all x ∈ G/K, (X ⊗_{S} C)_{x} = ⨁_{hy=x} X_{h} ⊗_{S} C_{y}. Then for all g ∈ G, (X ⊗_{S} C)(g) = X ⊗_{S} C(g) as objects of C((G/^{g}K, R)-Gr). We obtain the functor

X ⊗_{S} −: H((G/K, S)-gr) → H((G/K, R)-Gr)

and its derived functor X ⊗^{L}_{S} −: D((G/K, S)-gr) → D((G/K, R)-Gr), where X ⊗^{L}_{S} C = X ⊗_{S} pC.

Now assume that G is finite and let C be a complex of G/K -graded R-modules. Then Hom_{R}(X, C) is a complex of G/K -graded S -modules, where for all x ∈ G/K,

Hom_{R}(X, C)_{x} = {f ∈ Hom_{R}(X, C) | fX_{g} ⊆ C_{gx} for all g ∈ G}.

Moreover, for all g ∈ G, Hom_{R}(X, C(g)) = Hom_{R}(X, C)(g) as objects of the category C((G/^{g}K, S)-Gr). We obtain the functor

Hom_{R}(X, −): H((G/K, R)-Gr) → H((G/K, S)-Gr)

and its derived functor

RHom_{R}(X, −): D((G/K, R)-Gr) → D((G/K, S)-Gr),

where RHom_{R}(X, C) = Hom_{R}(X, iC).

2.2. An object T ∈ D(R-Gr) is called a G-graded tilting complex over R if it satisfies the following conditions:

a) T ∈ R-Grperf (that is, T, regarded as a complex of R-modules, belongs to R−perf).

b) Hom_{D(R)}(T, T [n]) = 0 for all n ≠ 0.

c) add(T) generate R-perf as a triangulated category.

2.3. For the next theorem, we need the generalizations of the above results to the case of G-graded DG algebras and G/K -graded DG modules. We say that R is a G-graded DG algebra if there is a decomposition R = ⨁_{g∈G} ⨁_{p∈ℤ} R_{g}^{p} into k -submodules such that denoting R_{g} = ⨁_{p∈ℤ} R_{g}^{p} and R^{p} = ⨁_{g∈G} R_{g}^{p}, R = ⨁_{g∈G} R_{g} is a G-graded algebra, R = ⨁_{p∈ℤ} R^{p} is a ℤ-graded algebra, and the differential d: R → R satisfies d(R_{g}^{P}) ⊆ R_{g}^{p+1} and d(ab) = (da)b + (−1)^{p}ad(b) for all a ∈ R^{p}, b ∈ R.

A G/K -graded DG R-module is an R-module endowed with a decomposition M = ⨁_{x∈G/K} ⨁_{p∈BbbZ} M_{x}^{p} into k -submodules and a differential d: M → M satisfying d(M_{x}^{p}) ⊆ M_{x}^{p+1} and d(am) = d(a)m + (−1)^{p}ad(m) for all a ∈ R^{p} and m ∈ M.

For example, if R is a G-graded algebra, then it becomes a G-graded DG algebra by defining R^{p} = R for p = 0 and R^{p} = R for p ≠ 0. Moreover, if T is a complex of G-graded R-modules, then the complex E = End_{R}(T)^{op} is a G-graded DG algebra, and

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 7

if M is a complex of G/K -graded R-modules, then Hom_{R}(T, M) is a G/K -graded DG module.

If R and S are G-graded DG algebras, then a G-graded DG (R, S)-bimodule X is defined in a similar way, and note that X may be regarded as a G × G/δ(G)-graded DG R ⊗_{k} S^{op} -module. The homotopy category H((G/G, R)-Gr is defined as in [3, 8.2.3] and the derived category D((G/G, R)-Gr is defined as in [3, 8.2.5], Moreover, Lemma 1.9, 1.10, 1.11, Proposition 1.12 and the construction of the functors X ⊗^{L}_{R} −, RHom_{R}(X, −) carry over to G-graded DG algebras.

2.4. Theorem. Let k be a commutative ring, G a finite group and R, S two G-graded k -algebras. The following statements are equivalent.

(i) There is a G-graded tilting complex T ∈ D(R-Gr) and an isomorphism S → End_{D(R)}(T)^{op} of G-graded algebras.

(ii) There is a complex X of G-graded (R, S)-bimodules such that the functor

X ⊗^{L}_{S} −: D(S) → D(R)

is an equivalence.

(iii) There are equivalences F : D(S) → D(R) and F^{gr} : D(S-Gr) → D(S-Gr) of triangulated categories such that F^{gr} is a G-graded functor and the diagram

D(S) ⟶^{F} D(R)
U↑ ↑^{U}
D(S-Gr) ⟶_{F^{gr}} D(R-Gr)

is commutative.

(iv) There are equivalences F_{perf} : D(S−perf) → D(R−perf) and F_{perf}^{gr} : D(S-Grperf) → D(S-Grperf) of triangulated categories such that F_{perf}^{gr} is a G-graded functor and U ∘ F_{perf}^{gr} = F_{perf} ∘ U.

Proof. Assume that (i) holds. Then E = End_{B}(T)^{op} is a G-graded DG algebra, and

E_{−} = (⋯ → E^{−2} → E^{−1} → Z^{0}E → ⋯)

is a G-graded DG subalgebra of E. By 2.3, T is a G-graded DG (R, E)-bimodule, so T is a G-graded DG (R, E_{−})-bimodule. Let pT ∈ H((G × G, R ⊗ E_{−}^{op})-Gr) be the homotopically projective resolution of T, and let X = pT ⊗_{E_{−}} S, which is a Ggraded (R, S)-bimodule. Then by the argument of [3, 8.3.1], there is a quasi-isomorphism ϕ: T → X is D(S-Gr), and by [3, Proposition 8.1.4], the functor X⊗^{L}_{S}− is an equivalence.

By 2.1 we have that F^{gr} = X ⊗^{L}_{S} − is a graded functor and it is compatible with the grade-forgetting functor. It follows that for all g ∈ G, the adjunction morphism

ϕ_{S(g)} : S(g) → RHom_{R}(X, X ⊗^{L}_{S} S(g))

is invertible in D(S-Gr), so again the argument of [3, 8.1.4] implies that F^{gr} is an equivalence too. This shows that (ii) implies (iii).

The implication (iii) to (iv) follows by Rickard’s characterization of R-perf and the remarks made in 1.10.

To prove that (iv) implies (i) let T = F_{perf}^{gr}(S). Then T belongs to R-grperf, and by [3, Theorem 8.3.2] and 1.10 it follows that T is a G-graded tilting complex. Moreover, since F_{perf}^{gr} is a graded functor and compatible with the ungrading functor, it follows by Lemma 1.7 that the ring homomorphism

S ≃ End_{D(S)}(S)^{op} → End_{D(R)}(T)^{op}

induced by F_{perf} is grade-preserving.

2.5. As in the proof of (i)⇒(ii), one can show that for each subgroup H ≤ K of G, the functor X ⊗^{L}_{S} − induces an equivalence between D((G/K, S)-Gr) and D((G/K, R)-Gr). Moreover, for all subgroups K ≤ H of G, the diagram

D((G/K, S)-Gr) ⟶^{X⊗^{L}−} D((G/K, R)-Gr)
U_{G/H}^{G/K} ↓ ↓^{U_{G/H}^{G/K}}
D((G/K, S)-Gr) ⟶_{X⊗^{L}−} D((G/H, R)-Gr)

2.6. In the next section we shall consider the situation when R and S are G-graded symmetric crossed products over a field k, where G is a finite group. Assume that X is a complex of G-graded (R, S)-bimodules such that the functor X ⊗^{L}_{S} −: D(S) → D(R) is an equivalence. Then, using the procedure described in [5, 9.2.2], we obtain that X is quasi-isomorphic to a bounded complex of finitely generated G-graded (R, S)-bimodules, projective as left R-modules and as right S -modules, and all but one term is projective as a bimodule. This implies that the functor

X ⊗_{S} −: H^{b}(S) → H^{b}(R)

is an equivalence. Then X is called a Rickard tilting complex or a split endomorphism tilting complex. The inverse equivalence is induced by the k -dual of X.

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 9

3. On Okuyama’s tilting complexes

3.1. In this section R is a symmetric crossed product graded by the finite group G, and we shall denote R_{1} = A.

Let P_{0} be a projective R-module. Recall that a map f: R → X of R-modules is called a right P_{0} -approximation of X if P ∈ add(P_{0}), and the map

Hom_{R}(P′, f): Hom_{R}(P′, P) → Hom_{R}(P′, X)

is surjective for all P′ ∈ add(P_{0}). Note that every R-module X has, up to isomorphism, a unique minimal right P_{0} -approximation, and any right P_{0} -approximation of X contains a minimal one as a direct summand.

For later use, we need to study the behaviour of minimal right P_{0} -approximations with respect to the restriction functor R-mod → A-mod.

3.2. Lemma. Let Q_{0} be a G-invariant projective A-module, and let P_{0} = R ⊗_{A} Q_{0} . Assume that G is a p′ -group.

If f: P → X is a minimal right P_{0} -approximation of the R-module X, then f: _{A}P → AX is a minimal right Q_{0} -approximation of _{A}X.

Proof. The G-invariance of Q_{0} implies the add(Q_{0}) = add(_{A}P_{0}), hence _{A}P belongs to add(Q_{0}).

Let Q′ ∈ add(Q_{0}) and consider the map

Hom_{A}(Q′, f): Hom_{A}(Q′,_{A}P) → Hom_{R}(Q′,_{A}X).

Since R ⊗_{A} Q′ belongs to add(P_{0}), by the adjunction isomorphism

Hom_{R}(R ⊗_{A} −, −) ≃ Hom_{A}(−,_{A}(−))

it follows that Hom_{A}(Q′, f) is surjective. Therefore, f: _{A}P → _{A}X is a minimal right Q_{0} -approximation of _{A}X. In order to prove that it is minimal, observe that one obtains a minimal right Q_{0} -approximation of _{A}X by taking a projective cover Q of the Asubmodule

X_{0} = ∑_{f∈Hom_{A}(Q_{0},X)} Im f

of X, and considering the composition Q → X_{0} → X. Let ϕ: Q_{0} → X be an A-map, and let g ∈ G. Since X is an R-module and Q_{0} is G-invariant, we have the composite A-map

Q_{0} ⟶^{≃} R_{g} ⊗_{A} Q_{0} ⟶^{R_{g}⊗_{A}Q_{0}} R_{g} ⊗_{A} Im ϕ ⟶^{mult} X.

10 ANDREI MARCUS

It follows that R_{g} Im f is the image of an A-map Q_{0} → X, hence X_{0} is an Rsubmodules of X. Using again the above adjunction isomorphism, we deduce that X_{0} = ∑_{ϕ∈Hom_{R}(P_{0},X)} Im ϕ. The lemma now follows by the well-known fact that a projective cover Q of _{A}X_{0} is actually an R-module, and also a projective cover of _{R}X_{0} . 3.3. Let {S_{i} | i ∈ I} be a set of representatives for the isomorphism classes of simple A-modules, and let P_{i} be a projective cover of S_{i}. The index set I becomes a G-set via the action of G on simple A-modules.

Let I_{0} be a subset of I, and define, as in [11], the cochain complex

P^{•}(I_{0}) = ⨁_{i∈I} P_{i}^{•} = (⋯ → 0 → P^{−1} ⟶^{δ_{0}} P^{0} → 0 → ⋯),

where, δ_{0} = ⨁_{i∈I} δ_{i}, for i ∈ I_{0}

P_{i}^{•} = (⋯ → 0 → R_{i} ⟶^{δ_{i}} P_{i} → 0 → ⋯),

with R_{i} in degree −1, P_{i} in degree 0, and δ_{i}: R_{i} → P_{i} is a minimal right ⨁_{i∈I_{0}} P_{i} - approximation of P_{i}, and for i ∉ I_{0} ,

P_{i}^{•} = (⋯ → 0 → P_{i} ⟶^{δ_{i}} 0 → ⋯),

with P_{i} in degree −1.

Denote also C = End_{H(A)}(P^{•}(I_{0}))^{op} and E = End_{H(R)}(R ⊗_{A} P^{•}(I_{0}))^{op} . For i ∈ I, let P̂_{i} be the indecomposable projective C -module corresponding to the indecomposable direct summand P_{i}^{•} of P^{•}(I_{0}), so Ŝ_{i} = P̂_{i}/ rad P̂_{i} is a simple C -module. 3.4. Remark. Before stating the next result, we need an observation on stable Morita equivalences. Assume for the moment that G is a p′ -group, R and S are two strongly G-graded k -algebras, and let T^{•} be an one-sided tilting complex of G-graded R-module with endomorphism ring End_{H(R)}(T^{•})^{op} ≃ S. Denote Δ = Δ(R ⊗_{k} S^{op}). By the preceding section, there is a two-sided tilting complex X^{•} of G-graded (R, S)-bimodules. Then X_{1}^{•} is a complex of Δ-modules, and also a two-sided tilting complex of (R_{1}, S_{1})- bimodules. Let Y_{1}^{•} be a projective resolution of X_{1}^{•} as Δ-modules. It is possible to truncate Y_{1}^{•} to obtain a complex

(⋯ → 0 → Ker d^{n} → Y_{1}^{n} → Y_{1}^{n+1} → ⋯),

of Δ-modules quasi-isomorphic to X_{1}^{•}, such that all the terms but Ker d^{n} are projective Δ-modules, and Ker d^{n} is projective as an R_{1} -module and as a right S_{1} -module. Let M_{1} = Ω^{n}(Ker d^{n}) and N_{1} = Ω^{−n}(Hom_{R_{1}}(Ker d^{n}, R_{1})). Then by [15, Corollary 5.5] or [5, Proposition 6.3.8], M_{1} and N_{1} induce a stable Morita equivalence between R_{1} and S_{1} . Furthermore, M is a Δ-module, and by [9, Proposition 5.3.7] it follows that (R ⊗_{k} S^{op}) ⊗_{Δ} M_{1} induces a graded stable Morita equivalence between R and S.

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 11

3.5. Proposition. a) P^{•}(I_{0}) is a tilting complex for A.

b) If I_{0} is a G-subset of I, then E is a crossed product of C and G.

c) There is an isomorphism ^{ ĝ}S_{i} ≃ ^{g}Ŝ_{i} of C -modules. Proof. a) is due to J. Rickard (see [5, Proposition 5.1.1]) and

b) follows from the preceding section, since P^{•}(I_{0}) is G-invariant. For c), we may argue using [12, Proposition 1], or [11, Lemma 2.1].

Let i ∈ I \ I_{0} , so for all g ∈ G, ^{g}i ∈ I \ I_{0} . By [12, Proposition 1] we have

Ŝ_{i} ≃ Hom_{A}(P^{0}, S_{i}) ≃ Hom_{R}(R ⊗_{A} P^{0}, R ⊗_{A} S_{i})_{1},

where P^{0} = ⨁_{j∈I\I_{0}} P_{j}. Let E′ = End_{C(R)}(R ⊗_{A} P^{0})^{op} . Then E′ is a crossed product, since P^{•}(I_{0}) is G-invariant as a complex. Moreover, E′ operates naturally on Hom_{R}(R ⊗_{A} P^{0}, R ⊗_{A} S_{i})_{1} , such that null-homotopic maps operate trivially. It follows that Hom_{R}(R ⊗_{A} P^{0}, R ⊗_{A} S_{i})_{1} is a G-graded E -module. Consequently, we have

ĝS_{i} ≃ Hom_{A}(P^{0}, R_{g} ⊗_{A} S_{i})
≃ Hom_{R}(R ⊗_{A} P^{0}, R ⊗_{A} S_{i})_{g}
≃ E_{g} ⊗_{C} Hom_{R}(R ⊗_{A} P^{0}, R ⊗_{A} S_{i})_{1}
≃ E_{g} ⊗_{C} Ŝ_{i} = ^{g}Ŝ_{i}.

For i ∈ I, let U_{i}/S_{i} ≤ P_{i}/S_{i} be that largest submodule of P_{i}/S_{i} each whose composition factors are in I \ I_{0} . Then again by [12, Proposition 1],

Ŝ_{i} ≃ Coker(ρ_{i}: Hom_{A}(P^{0}, U_{i}) → Hom_{A}(P^{1}, U_{i}),

where P^{1} = (⨁_{j∈I\I_{0}} R_{j}) ⊕ (⨁_{i∈I_{0}} P_{i}), and ρ_{i} is induced by δ_{0} . Observe that since I_{0} is a G-subset of I, both P^{0} and P^{1} are G-invariant A-modules, and moreover, U_{^{g}i}/S_{^{g}i} ≤ P_{^{g}i}/S_{^{g}i} ≃ R_{g} ⊗_{A} (P_{i}/S_{i}), so U_{^{g}i} ≃ R_{g} ⊗_{A} U_{i} = ^{g}U_{i}. We have that the natural map

ρ̄_{i}: Hom_{R}(R ⊗_{A} P^{0}, R ⊗_{A} U_{i}) → Hom_{R}(R ⊗_{A} P^{1}, R ⊗_{A} U_{i})

is a homomorphism of G-graded E -modules, and the argument of the preceding paragraph shows that ρ̄_{i} = E ⊗_{C} ρ_{i}. We conclude that

Ŝ_{^{g}i} ≃ Coker ρ_{^{g}i} ≃ Coker(E_{g} ⊗_{C} ρ_{i})
≃ E_{g} ⊗_{C} Coker ρ_{i} ≃ ^{g}Ŝ_{i}.

Alternatively, there is a (C, A)-bimodule L inducing a stable equivalence between A and C. By Remark 3.4, L is actually a Δ(E ⊗ R^{op})-module. By [11, Lemma 2.1], we have that for i ∈ I \ I_{0} , Ŝ_{i} ≃ L_{A} ⊗ S_{i}, and therefore

Ŝ_{^{g}i} ≃ L ⊗_{A} (R_{g} ⊗_{A} S_{i}) ≃ E_{g} ⊗_{C} (L ⊗_{A} S_{i}) ≃ ^{g}Ŝ_{i}.

For i ∈ I_{0} , denote W_{i} = P_{i}/U_{i}. By the above remarks, U_{^{g}i} ≃ ^{g}U_{i}, hence W_{^{g}i} ≃ ^{g}W_{i} for all g ∈ G, and again by [11, Lemma 2.1],

Ŝ_{^{g}i} ≃ L ⊗_{A} W_{^{g}i} ≃ L ⊗_{A} (R_{g} ⊗_{A} W_{i})
≃ E_{g} ⊗_{C} (L ⊗_{A} W_{i}) ≃ ^{g}Ŝ_{i}.

3.6. Let S be another symmetric crossed product graded by the finite group G. Denote S_{1} = B, Δ = Δ(R ⊗_{k} S^{op}), and Γ = R ⊗_{k} S^{op} . In adition, assume that A and B are connected non-semisimple symmetric algebras. Let {T_{i} | i ∈ I} be a set of representatives for the isomorphism classes of simple B -modules. As before, we regard I as a G-set via the action of G on simple A-modules, and let I_{0} be a G-subset of I.

Assume that the (A, B)-bimodule M induces a stable Morita equivalence between A and B, and as in [12], consider the complex of (A, B)-bimodules

P^{•}(M, I_{0}) = (⋯ → 0 → P ⟶^{δ} M → 0 → ⋯),

with P in degree −1 and M in degree 0, where δ: P → M is a right minimal ((⨁_{i∈I} P_{i}) ⊗_{k} B^{op})-approximation of M.

Let A^{(1)} = End_{H(A)}(P^{•}(M, I_{0}))^{op} and R^{(1)} = End_{H(R)}(R⊗_{A} P^{•}(M, I_{0}))^{op} . Regarded as a complex of A-modules, P^{•}(M, I_{0})) is a direct sum of complexes isomorphic to P_{i}^{•}, i ∈ I. Let S_{i}^{(1)} be a simple A^{(1)} -modules corresponding to an indecomposable summand isomorphic to P_{i}^{•} of _{A}P^{•}(M, I_{0}).

3.7. Proposition. Assume that G is a p′ -group, I_{0} is a G-invariant subset of I, and that M is a Δ(R ⊗_{k} S^{op})-module.

a) R^{(1)} is a G-graded crossed product, graded Morita equivalent to E.

b) There is a G-graded algebra map S → R^{(1)} , and the (S, R^{(1)})-bimodule R^{(1)} induces a graded stable equivalence of Morita type between S and R^{(1)} .

c) There is an isomorphism of B -modules _{B}S_{^{g}i}^{(1)} ≃ ^{g}S_{i}^{(1)} .

Proof. a) Since M is a Δ-module, by Lemma 3.2 we may assume that δ: P → M is a right minimal (Δ ⊗_{Δ_{1}} ((⨁_{i∈I} P_{i}) ⊗_{k} B^{op}))-approximation of M, that is, we may assume that P^{•}(M, I_{0}) is a complex of Δ-modules. Then R⊗_{A} P^{•}(M, I_{0}) ≃ Γ⊗_{Δ} P^{•}(M, I_{0}) is a complex of G-graded (R, S)-bimodules, so right multiplication induces a map S → R^{(1)} of G-graded algebras. Since S is a crossed product, it follows that R^{(1)} is a crossed product. This implies that P^{•}(M, I_{0}) is G-invariant as a complex of A-modules, and since _{A}P^{•}(M, I_{0}) is a direct sum of complexes isomorphic to P_{i}^{•}, i ∈ I, each occurring with non-zero multiplicity, it follows easily that there is a Morita equivalence between A^{(1)} and C, which extends to a G-graded Morita equivalence between R^{(1)} and E.

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 13

b) We already have seen that R^{(1)} may be regarded as a G-graded (S, R^{(1)})-bimodule. By [12, Proposition 2], we know that A^{(1)} and B are stably Morita equivalent. More precisely, the proof shows that there is an isomorphism of Δ(S ⊗_{k} S^{op})-modules

Hom_{A}(M, M) ≃ B ⊕ U,

where U is a projective B ⊗_{k} B^{op} -module. This implies that _{B}A_{B}^{(1)} ≃ B ⊕ V as Δ(S ⊗_{k} S^{op})-modules, where V is a projective B ⊗_{k} B^{op} -module. The statement now follows by [8, Proposition 5.7], using also the fact that R^{(1)} and S are symmetric.

c) By [12, Corollary 1], for i ∈ I \ I_{0} there is an isomorphism of B -modules _{B}S_{i}^{(1)} ≃ Hom_{A}(M, S_{i}), hence we also have the following isomorphisms:

BS_{^{g}i}^{(1)} ≃ Hom_{A}(M, R_{g} ⊗_{A} S_{i})
≃ Hom_{R}(R ⊗_{A} M, R ⊗_{A} S_{i})_{g}
≃ S_{g} ⊗_{B} Hom_{A}(M, S_{i})
≃ ^{g}_{B}S_{i}^{(1)}.

For i ∈ I_{0} , _{B}S_{i}^{(1)} is isomorphic to the unique non-projective indecomposable summand of Hom_{A}(M, W_{i}), so the same argument applies in this case too.

3.8. Let L^{(1)} be the projective-free part of the (A ⊗_{k} A^{(1)op})-module M ⊗_{B} A^{(1)} , and consider the complex of (A, A^{(1)})-bimodules

P^{•}(L^{(1)}, I_{0}) = (⋯ → 0 → P^{(1)} ⟶^{δ^{(1)}} L^{(1)} → 0 → ⋯),

with P^{(1)} in degree −1 and L^{(1)} in degree 0, where δ^{(1)} : P^{(1)} → L^{(1)} is a right minimal ((⨁_{i∈I} P_{i}⊗_{k}A^{(1)op})-approximation of L^{(1)} . By [12, Theorem 3], P^{•}(L^{(1)}, I_{0}) is a Rickard tilting complex of (A, A^{(1)})-bimodules, and moreover, _{A}P^{•}(L^{(1)}, I_{0})_{B} ≃ P^{•}(M, I_{0}).

3.9. Proposition. P^{•}(L^{(1)}, I_{0}) extends to a complex of Δ(R ⊗_{k} R^{(1)op}))-modules, and it is isomorphic to P^{•}(M, I_{0}) as complexes of Δ(R ⊗_{k} S^{op}))-modules.

Proof. Since M is a Δ(R ⊗_{k} S^{op}))-module and A^{(1)} is a Δ(S ⊗_{k} R^{(1)op})), we have that M ⊗_{B} A^{(1)} is a Δ(R ⊗_{k} R^{(1)op}))-module (see [8, Section 2]). Since G is a p′ -group, it follows by [2] that R^{(1)} also extends to a Δ(R ⊗_{k} R^{(1)op}))-module. Now using Lemma 3.2 we conclude that P^{•}(L^{(1)}, I_{0}) is a complex of Δ(R ⊗_{k} R^{(1)op}))-modules.

We may also regard L^{(1)} as a Δ-module, and by Proposition 3.7 b) we have that

Γ ⊗_{Δ} L^{(1)} ≃ Γ ⊗_{Δ} M ⊕ (projective G-graded (R, S)-bimodules),

that is, we have the isomorphism of Δ-modules

L^{(1)} ≃ M ⊕ (projective Δ-modules).

14 ANDREI MARCUS

Moreover, P^{•}(L^{(1)}, I_{0}) is a right Δ ⊗_{Δ_{1}} (⨁_{i∈I} P_{i} ⊗_{k} B)-approximation of L^{(1)} , so P^{•}(M, I_{0}) is a direct summand of P^{•}(L^{(1)}, I_{0}) in H^{b}(Δ-proj). It follows by [12, Theorem 3] that these complexes are actually isomorphic.

3.10. Remark. In [11] and [12], Okuyama gave a method of constructing Rickard tilting complexes of (A, B)-bimodules (eve splendid tilting complexes in the case of blocks of group algebras), starting with a stable Morita equivalence between A and B. Let us point out what happens in the graded situation. We continue to use the notation of 3.6.

Consider the A-modules X_{i} = M ⊗_{B} T_{i}, i ∈ I. If all X_{i} are are simple A-modules, then by Linckelmann’s theorem [5, Theorem 11.2.3], A and B are Morita equivalent, so if M is a Δ-module, then R and S are graded Morita equivalent.

Otherwise, choose a subset I_{0} of I and replace A by A^{(1)} = End_{A}(P^{•}(M, I_{0}))^{op} (which is Morita equivalent to C = End_{A}(P^{•}(I_{0}))^{op} ) and M by a (B, A^{(1)})-bimodule M^{(1)} inducing a stable equivalence between B and A^{(1)} . If I_{0} is a G-subset of I, then by Propositions 3.7 and 3.9 we have that R^{(1)} is G-graded derived equivalent to R, and M^{(1)} is a Δ(S ⊗_{k} R^{(1)}^{op} -module. This procedure continues until a stage t when simple B -modules will correspond to simple A^{(t)} -modules.

The point is that by Proposition 3.5 c) and Proposition 3.7 c), the G-invariance of a set I_{s} of simple A^{(s)} -modules can be established from the knowledge of the action of G on the simple A-modules.

3.11. Example. Let X be a normal subgroup of the finite group X̃, D a Sylow psubgroup of X, and assume that G = X̃/X is a p′ -group and that D is abelian. Let Y = N_{X}(D), Ỹ = NX̃(D), B the principal block of kX, A the principal block of kY , S = BkX̃ and R = AkỸ .

Let Y_{0} = Y CỸ (D) and let A_{0} be the principal block of of kY_{0} . Then Y_{0} is a normal subgroup of Ỹ , and A_{0} and A are isomorphic k -algebras via restriction, by a theorem of Alperin and Dade. Denote E(X) = N_{X}(D)/C_{X}(D). Then by [6, Lemma 1.2], we have that Ỹ /Y ≃ E(X̃)/E(X), and E(X̃) is a p′ -subgroup of Aut(D).

Now consider the case when char k = 3 and D = C_{3} × C_{3} . The Hall p′ -subgroup of Aut D is isomorphic to the semidihedral group SD_{16} . Then Y = N_{G}(P) can be isomorphic to D⋊SD_{16} , D⋊Q_{8} , D⋊D_{8} or D⋊C_{4} ; these groups, and the non-isomorphic projective indecomposable modules for kSD_{16} , kQ_{8} , kD_{8} and kC_{4} are described in [11, Appendix].

By Clifford theory and the above remarks, it is enough to look at the action of SD_{16} on these projective indecomposable modules. Using the information given in [11, Appendix], it is easy to calculate the SD_{16} -conjugacy classes of primitive idempotents. By [6, Lemma 3.11] there is a G-graded stable Morita equivalence between R and S.

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 15

In the case when X is simple and has Sylow 3-subgroup isomorphic to C_{3} × C_{3} , Broué’s conjecture was verified in [11], and [6]. The calculation of the action of SD_{16} on primitive idempotents shows that in all the cases, the subsets I_{s} of I were chosen to be G-subsets.

3.12. Another method of constructing Rickard tilting complexes from stable equivalences was given by Okuyama [13]. The idea has its origins in the work of Rouquier [17], [18].

Let A, R, B and S be as in 3.6. We shall regard I as a G-set via the action of G on {T_{i} | i ∈ I} of simple B -modules. Assume that the (A, B)-bimodule M induces a stable equivalence between A and B.

Let τ_{i}: Q_{i} → T_{i} be a projective cover of T_{i}, and π_{i}: P_{i} → M ⊗_{B} T_{i} a projective cover of the A-module M ⊗_{B} T_{i}, i ∈ I. By [17, Lemma 2], a projective cover of M has the form ⨁_{i∈I} δ_{i}: ⨁_{i∈I} P_{i} ⊗_{k} Q_{i}^{∗} → M.

Let I_{0} be a subset of I, P = P (I_{0}) = ⨁_{i∈I_{0}} P_{i} ⊗_{k} Q_{i}^{∗}, and let δ = δ(I_{0}) = ⨁_{i∈I_{0}} δ_{i}: P → M. Consider the complex of (A, B)-bimodules

M^{•} = M^{•}(I_{0}) = (⋯ → 0 → P ⟶^{δ} M → 0 → ⋯),

with M in degree 0 and P in degree −1. Then by [13, Theorem 1.1], _{A}M^{•} is an one-sided tilting complex for A if and only if a certain condition (c) in [13, (1.5)] is satisfied.

Denote C = End_{H(A)}(M^{•}(I_{0}))^{op} and E = End_{H(R)}(R ⊗_{A} M^{•}(I_{0}))^{op} .

3.13. Proposition. Assume that _{A}M^{•} is a tilting complex for A, G is a p′ -group, M is a Δ-module and I_{0} is a G-subset of I.

a) M^{•} extends to a complex of Δ-modules.

b) E is a crossed product, and there is a graded stable Morita equivalence between E and S.

c) There is a complex N^{•} = N^{•}(I_{0}) of Δ(R ⊗_{k} E^{op})-modules such that _{A}N_{C}^{•} is a two-sided tilting complex of (A, C)-bimodules, and N^{•} is homotopy equivalent to M^{•} as complexes of Δ-modules.

Proof. a) The argument is the same as in [8, Example 5.8]. Since M is a Δ-module, the projective cover of _{A}M_{B} also extends to a Δ-module.

We have that T_{^{g}i} ≃ S_{g} ⊗_{B} T_{i}, so Q_{^{g}i} ≃ S_{g} ⊗_{B} Q_{i}, and since M is a Δ-module, M ⊗_{B} T_{^{g}i} ≃ R_{g} ⊗_{A} (M ⊗_{B} T_{i}). Consequently, P_{^{g}i} ≃ R_{g} ⊗_{A} P_{i} and

Δ_{g} ⊗_{Δ} (P_{i} ⊗_{k} Q_{i}^{∗}) ≃ R_{g} ⊗_{A} (P_{i} ⊗_{k} Q_{i}^{∗}) ⊗_{B} S_{g^{−1}}
≃ (R_{g} ⊗_{A} P_{i}) ⊗_{k} (S_{g} ⊗_{B} Q_{i})^{∗}
≃ P_{^{g}i} ⊗_{k} Q_{^{g}i}^{∗}.

16 ANDREI MARCUS

Since I_{0} is a G-subset of I, it follows that P = P (I_{0}) is a G-invariant Δ_{1} -module. Clearly, P and P (M)/P have no common Δ_{1} -summands, hence P extends to a Δsummand of the projective cover P (M) of M.

b) As in Proposition 3.7, R ⊗_{A} M^{•} is a complex of G-graded (R, S)-bimodules, so right multiplication gives a map S → E of G-graded algebras. Therefore, E is a crossed product, and [13, (1.6)] implies that _{E}E_{S} induces a graded stable equivalence between E and S, that is,

C ≃ B ⊕ (projectives) as Δ(S ⊗_{k} S^{op})-modules,
C ⊗_{B} C ≃ C ⊕ (projectives) as Δ(E ⊗_{k} E^{op})-modules.

c) We show that the complex _{A}N_{C}^{•} = _{A}N(I_{0})_{C}^{•} constructed in [13, p. 5] extends to a complex of Δ(R ⊗_{k} E^{op})-modules. As in the proof of Proposition 3.5 c), E acts on R ⊗_{A} Ker δ ≃ H^{1}(R ⊗_{A} M^{•}) and on R ⊗_{A} Coker δ ≃ H^{0}(M^{•}), such that R ⊗_{A} Ker δ and R⊗_{A}Coker δ are G-graded (R, E)-bimodules, hence Ker δ and Coker δ are Δ(R⊗_{k}E^{op})- modules. It follows by b) that we have the isomorphisms of Δ(R ⊗_{k} E^{op})-modules

Ker δ ⊗_{B} C ≃ Ker δ ⊕ (Ker δ ⊗_{C} U)
Coker δ ⊗_{B} C ≃ Coker δ ⊕ (Coker δ ⊗_{C} U).

Consider an injective hull 0 → Ker δ → Q ⟶^{λ} X → 0 of Ker δ regarded as an A ⊗_{k} C^{op} - module. Then the k -dual Q^{∗} of Q is a projective cover of (Ker δ)^{∗}, which is a Δ(E ⊗_{k} R^{op})-module. It follows that Q is an injective hull of Ker δ regarded as a Δ(E ⊗_{k} R^{op})- module, and λ is a Δ(E ⊗_{k} R^{op})-homomorphism.

By the above isomorphisms it follows that the sequence 0 → Ker δ → Q ⟶^{λ} X → 0 is a direct summand of

0 → Ker δ ⊗_{B} C → P ⊗_{B} C → Im δ ⊗_{B} C → 0.

letting Im δ ⊗_{B} C ≃ X ⊕ Y , the exact sequence

0 → X ⊕ Y → M ⊗_{B} C → Coker δ ⊗_{B} C → 0

has an injective hull 0 → Y → Q′ → Z → 0 of Y as a Δ(R ⊗_{k} E^{op})-summand. Letting M ⊗_{B} C = N ⊕ Q′, further arguments of [13] give an exact sequence

0 → X ⟶^{μ} N → Coker δ → 0

of Δ(R ⊗_{k} E^{op})-modules such that the complex

N^{•} = N(I_{0})^{•} = (⋯ → 0 → Q ⟶^{δ_{0}} N → 0 → …)

of Δ(R ⊗_{k} E^{op})-modules, where δ_{0} = μ ∘ λ is isomorphic to M^{•} in H^{b}(Δ). Moreover, Hom_{A}(N^{•}, N^{•}) ≃ C in H^{b}(Δ(E ⊗_{k} E^{op})). Since R and E are symmetric algebras, this implies that N^{•} is a Rickard tilting complex of G-graded (R, E)-bimodules.

TILTING COMPLEXES FOR GROUP GRADED ALGEBRAS 17

3.14. Example. Okuyama proved in [13] that Broué’s conjecture holds for the principal p-block of SL(2, p^{n}) by repeatedly using his constructions of Rickard tilting complexes. We show here that this equivalence is compatible with p′ -outer automorphisms.

Let X = SL(2, p^{n}), D a Sylow p-subgroup of X, Y = N_{X}(D), A the principal block of kX and B the principal of kY .

Assume that X is a normal subgroup of X̃ and that G = X̃/X is a p′ -group. Let Ỹ = NX̃(D), R = AkX̃ and S = BkỸ .

Since X has trivial intersection Sylow p-subgroups, the (A, B)-bimodule M = _{A}A_{B} induces a stable Morita equivalence between A and B, and since G is a p′ -group, the (R, S)-bimodule _{R}R_{S} induces a graded stable equivalence between R and S.

The group X̃ acts on X by conjugation. There are two types of automorphisms of X which have to be considered.

Firstly, let σ be the Frobenius automorphism of k, and consider in this case the group X̃ = X ⋊ ⟨σ⟩ and Ỹ = Y ⋊ ⟨σ⟩.

Recall that the simple kX -modules are

S_{λ} = S_{λ_{0}}^{(0)} ⊗ S_{λ_{1}}^{(1)} ⊗ ⋯ ⊗ S_{λ_{n−1}}^{(n−1)}, 0 ≤ λ ≤ q − 1,

where λ = ∑_{i=0}^{n−1} λ_{i}p^{n}, with 0 ≤ λ_{i} ≤ p − 1, S_{λ_{i}} is the set of homogeneous polynomials of degree λ_{i} in k[X, Y], and S_{λ_{i}}^{(i)} is obtained by twisting i times with the Frobenius automorphism σ. The element x = ( \begin{matrix}a & b \\ c & d \end{matrix} ) of SL(2, p^{n}) acts on S_{λ} by sending the indeterminate X to aX + bY , and Y to cX + dY .

The set of simple A-modules is indexed by I = {λ | 0 ≤ λ ≤ q − 1, λ is even} when p is odd, and by I = {λ | 0 ≤ λ ≤ q − 1} if p = 2. Then I becomes a ⟨σ⟩-set.

The simple B -modules are T_{λ}, λ ∈ I, where T_{λ} is the socle of M^{∗} ⊗_{A} S_{λ} = Res_{Y}^{X} S_{λ}. These are one-dimensional modules, and we clearly have T_{^{σ}λ} = Soc Res_{Y}^{X} S^{σ}λ.

The application of Okuyama’s method requires at stage t the choice of a subset I_{t} of I. By [13, p. 11], I_{t} is the ⟨σ⟩-orbit of an element λ in I, so the assumptions of Proposition 3.13 are satisfied in this case.

The other situation which has to be considered occurs when taking X̃ = GL(2, p^{n}) and p ≠ 2. In this case the image of = X̃/X in Out X is a subgroup of order 2. Then all the simple A-modules S_{λ} are G-invariant since they extend to X̃, and the similar statement holds for the B -modules T_{λ}. In particular, [13] and this observation implies that Broué’s conjecture holds for the principal block of GL(2, p^{n}).

Acknowledgements

The research for this paper was done at the “Friedrich Schiller” University Jena,

where the author was supported by a Humboldt Fellowship. He is grateful to Burkhard

Külshammer for all his help and advise.

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