Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 2, 373-383.

On Induced Modules over Strongly Group-graded Algebras

Th. Theohari-Apostolidi H. Vavatsoulas

Department of Mathematics Aristotle University of Thessaloniki Thessaloniki, 54006 Greece

1. Introduction

Let G be a group, Λ a strongly G-graded R-algebra for a commutative ring R with unity and H a subgroup of G of finite index. The aim of this paper is to examine the induced representations of the R-algebra Λ, whenever Λ is an artinian ring or an R-order and R is a complete discrete valuation ring, that is in such conditions that the Krull-Schmidt theorem holds.

In the second section Λ is a strongly G-graded ring without other assumptions. We firstly examine the relative H-projectivity on Λ under a condition on H which generalizes the condition that [G : H] is a unit in Λ (Theorem 2.2). So we generalize the relevant theorems ([15], Theorem 1.1), ([17], Proposition 7), ([12], Theorem 7.26) and ([2], Lemma 2.5) on strongly group graded rings or crossed products.

In the third section we assume that G is finite and H is a normal subgroup of G. Moreover Λ is an Artin R-algebra or an R-order with R a complete discrete valuation ring and modΛ is the category of finitely generated left Λ-modules or the category of left Λ-lattices. So the Krull-Schmidt theorem holds in modΛ. Let Λ = ⨁_{σ∈G} Λ_{σ} and Λ_{H} = ⨁_{σ∈H} Λ_{σ}. In this section given an indecomposable module V in modΛ_{H} and an indecomposable module W in modΛ we examine the decomposition of (V^{G})_{H} and (W_{H})^{G} into indecomposables. The Clifford theorem for the decomposition of induced modules from a normal subgroup H over group algebras ([3], Theorem 11.1) has been generalized by Tucker, Conlon and Ward for Clifford Systems and for H = {1} ([3], Proposition 11.16). F. Van Oystaeyen ([18], Proposition 2.5) and Z.B. Rong ([16], Proposition 1.3) considering strongly group-graded rings generalized the Clifford theorem for H = {1}. Here we generalize this theorem for strongly group graded rings and for a non-trivial normal subgroup H correcting the result of S. Hussein and F. Van Oystayen ([11], Theorem 2.7).

In the fourth section we prove the corresponding theorem for invariant modules of group rings ([3], Proposition 10.31) for crossed-products.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1999 Heldermann Verlag

374 Th. Theohari-Apostolidi, H. Vavatsoulas: On Induced Modules …

We cite [1] and [3] for the representation theory, [4], [5], [6], [7], [13] for the theory of group-graded rings, and [8], [9] for the theory of graded Artin algebras.

2. Basic concepts and relative projectivity

By a ring Λ we mean an associative ring with identity 1 = 1_{Λ}. All Λ-modules are left unitary modules unless otherwise stated. Let G be a group with identity 1 = 1_{G}. A G-graded ring Λ is a ring together with a direct sum decomposition

Λ = ⨁_{σ∈G} Λ_{σ}

as additive groups, where the additive subgroups Λ_{σ} of Λ satisfy

Λ_{σ}Λ_{τ} ⊆ Λ_{στ} for all σ, τ ∈ G

and Λ_{σ}Λ_{τ} is the usual module structure of additive subgroups of Λ. If the equality holds, that is

Λ_{σ}Λ_{τ} = Λ_{στ} for all σ, τ ∈ G

then Λ is called strongly (or fully) G-graded ring. A consequence of the definition of a G-graded ring is that 1 ∈ Λ_{1}.

A special type of strongly G-graded ring is the crossed product in which every component contains at least one unit of the ring Λ. In this case there exist units \overline{σ} in Λ_{σ}, for σ ∈ G, a map

α : G × G → U(Λ_{1}),

where by U(Λ_{1}) we mean the group of units of a ring Λ_{1}, and an action

φ : G → Aut(Λ_{1}), φ(σ)(r) = ^{σ}r

for all σ ∈ G and r ∈ Λ_{1}, such that

and  \begin{matrix}Λ = & ⨁_{σ∈G} Λ_{1}\overline{σ} \\ \overline{σ}r = ^{σ}r \overline{σ}, & \overline{σ} \overline{τ} = α(σ, τ)\overline{στ} \end{matrix}

for σ, τ in G and r ∈ Λ_{1}, that is α is a factor set of G on U(Λ_{1}).

A special type of crossed product is the skew group ring in which the factor set is trivial.

If the G-graded ring Λ is an algebra over a commutative ring R and if each Λ_{σ}, σ ∈ G, is an R-submodule of Λ, then we call Λ a G-graded R-algebra. So if C is the center of Λ_{1}, then Λ is a G-graded C-algebra.

If H is a subset of G and Λ is a G-graded ring, we define

Λ_{H} = ⨁_{σ∈H} Λ_{σ}.

Thus if H is a subgroup of G, Λ_{H} is an H-graded ring and if Λ is strongly graded, then so is Λ_{H}.

If V is a left Λ_{H}-module we denote by V^{G} the induced module of V in G, that is

V^{G} = Λ ⨂_{Λ_{H}} V

and for the left Λ-module W we write W_{H} to indicate that W is regarded as left Λ_{H}- module, W_{H} is called the restriction of W on H.

In the sequel by Λ we will mean a strongly G-graded ring for a group G and H a subgroup of G of finite index in G. Moreover by T we will indicate a left transversal of H in G containing 1 ∈ G.

Given the left Λ_{H}-module V and an element σ ∈ G, we define the conjugate module

Λ_{σ} ⨂_{Λ_{H}} V ≅ Λ_{σH} ⨂_{Λ_{H}} V,

which is a left Λ_{σHσ^{−1}}-module, and we denote it by ^{σ}V. Hence, if H \underline{◁} G, (V^{G})_{H} = ⨁_{t∈T} ^{t}V. I. Reiten and C. Riedtmann defined the conjugate module for crossed products ([15], p. 235) and they showed for H \underline{◁} G that ^{σ}V is Λ_{H}-isomorphic to the left Λ_{H}-module whose underlying set and additive structure is the same as for V and where r ∗ v = ^{σ}rv and \overline{h} ∗ v = \overline{σ}^{−1}\overline{h}\overline{σ}v for r ∈ Λ_{1}, v ∈ V and h ∈ H.

Since Λ_{σ}Λ_{σ^{−1}} = Λ_{1} for all σ ∈ G, we can find elements α_{σ}^{(i)} ∈ Λ_{σ} and α_{σ^{−1}}^{(i)} ∈ Λ_{σ^{−1}} such that

∑_{i=1}^{n_{σ}} α_{σ}^{(i)}α_{σ^{−1}}^{(i)} = 1  (2.1)

for some natural number n_{σ}.

The next proposition describes the elementary properties of the conjugate modules. The proof is analogue of that in group ring case. See also ([2], Lemma 2.1).

Proposition 2.1. Let V, Z be left Λ_{H}-modules and σ ∈ T.

i) ^{g}(^{σ}V) ≅ ^{gσ}V as Λ_{gσH(gσ)^{−1}}-modules. ii) Let H \underline{◁} G. (^{σ}V)^{G} ≅ V^{G} as left Λ-modules. iii) Let H \underline{◁} G. ^{σ}V is a simple (resp. indecomposable) Λ_{H}-module if and only if V is a simple (resp. indecomposable) Λ_{H}-module.

We recall that a left Λ-module W is called H-projective if the exact sequence of left Λmodules 0 → X → Y → W → 0, for which the associated sequence of Λ_{H}-modules 0 → X_{H} → Y_{H} → W_{H} → 0 splits, is also a splitting sequence of Λ-modules. Equivalently, W is H-projective if and only if W |(W_{H})^{G}, where the notation X | Y means that the module X is isomorphic to a direct summand of the module Y.

In this paper by (s) we indicate the following

Condition. There exists an element r ∈ C_{Λ_{1}}(Λ_{H}) = {λ ∈ Λ_{1}: λx = xλ, ∀x ∈ Λ_{H}}, the centralizer of Λ_{H} in Λ_{1}, such that

∑_{t∈T} ∑_{i=1}^{n_{t}} α_{t}^{(i)}rα_{t^{−1}}^{(i)} = 1,

where α_{σ}^{(i)} for σ ∈ G are as in the relation (2.1) and for some integer n_{t}.

The condition (s) holds if the order of G is a unit in Λ_{1}, since we can take in this case r = [G : H]^{−1}. Moreover in the crossed product case the condition (s) means the existence of an element r ∈ C_{Λ_{1}}(Λ_{H}) such that

∑_{t∈T} \overline{t}r\overline{t}^{−1} = 1.

Now if we consider the local classical crossed product case with L/K a Galois extension of local fields and G the corresponding finite Galois group, then the condition (s) means that the field F corresponding to the normal subgroup H is a tamely ramified extension over K. In this case Λ_{1} is a Dedekind domain with quotient field L.

The next theorem generalizes ([15] , Theorem 1.1) in which |G| is a unit in Λ and Λ is a crossed product, ([17], Proposition 7) in which Λ is a local classical crossed product, and ([2], Lemma 2.5) in which H contains a Sylow p-subgroup of G and Λ is a finitely dimensional strongly G-graded algebra over a field of characteristic p.

Theorem 2.2. Let H be a subgroup of G of finite index and suppose that condition (s) holds. Then every left Λ-module M is relative H-projective.

Proof. We shall prove under the conditions of the theorem that M |(M_{H})^{G}. For this we define the maps

f : (M_{H})^{G} → M, λ ⊗ m ↦ λm,

for λ ∈ Λ and m ∈ M, which is obviously a Λ-homomorphism, and

Ψ : M → (M_{H})^{G}, m ↦ ∑_{t∈T} ∑_{i=1}^{n_{t}} α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}m

for m ∈ M, and r, α_{t}^{(i)}, α_{t^{−1}}^{(i)}, n_{t} as in the statement of the condition (s). We shall prove that Ψ is also a Λ-homomorphism and f ∘ Ψ = 1_{M}.

At first we shall prove that Ψ does not depend on the chosen decomposition of 1. For this let β_{t}^{(j)} and β_{t^{−1}}^{(j)} be elements of Λ_{t} and Λ_{t^{−1}} respectively, 1 ≤ j ≤ m_{t}, such that

1 = ∑_{j=1}^{m_{t}} β_{t}^{(j)}β_{t−1}^{(j)}.

Then

∑_{i=1}^{n_{t}} α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}m = ∑_{i=1}^{n_{t}} (∑_{j=1}^{m_{t}} β_{t}^{(j)}β_{t^{−1}}^{(j)} ) α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}m =

= ∑_{i=1}^{n_{t}} ∑_{j=1}^{m_{t}} (β_{t}^{(j)} ⊗ rβ_{t^{−1}}^{(j)}α_{t}^{(i)}α_{t^{−1}}^{(i)}m) =

= ∑_{j=1}^{m_{t}} β_{t}^{(j)} ⊗ rβ_{t^{−1}}^{(j)} (∑_{i=1}^{n_{t}} α_{t}^{(i)}α_{t^{−1}}^{(i)}) m =

= ∑_{j=1}^{m_{t}} β_{t}^{(j)} ⊗ rβ_{t^{−1}}^{(j)}m.

Moreover Ψ is independent of the choice of T. Indeed, let S be another left transversal of H in G. Then for every t ∈ T there exists exactly one h_{t} ∈ H such that th_{t} = σ_{t}. Therefore if we choose elements α_{h_{t}}^{(j)} and α_{h_{t}^{−1}}^{(j)} in Λ_{h_{t}} and Λ_{h_{t}^{−1}} respectively such that

1 = ∑_{j∈J} α_{h_{t}}^{(j)}α_{h_{t}^{−1}}^{(j)}

for some finite index set J, then we define

α_{σ_{t}}^{(i)} = ∑_{j∈J} α_{t}^{(i)}α_{h_{t}}^{(j)} and α_{σ_{t}^{−1}}^{(i)} = ∑_{j∈J} α_{h_{t}^{−1}}^{(j)}α_{t^{−1}}^{(i)}, 1 ≤ i ≤ n_{t},

and we get

∑_{i=1}^{n_{t}} α_{σ_{t}}^{(i)}α_{σ_{t}^{−1}}^{(i)} = ∑_{i=1}^{n_{t}} ∑_{j∈J} α_{t}^{(i)}α_{h_{t^{−1}}}^{(j)} α_{h_{t}^{−1}}^{(j)}α_{t^{−1}}^{(i)} = ∑_{i=1}^{n_{t}} α_{t}^{(i)}α_{t^{−1}}^{(i)} = 1.

Now we remark that

\begin{matrix}∑_{σ_{t}∈S} ∑_{i=1}^{n_{t}} α_{σ_{t}}^{(i)} ⊗ rα_{σ_{t}^{−1}}^{(i)}m = ∑_{σ_{t}∈S} ∑_{i=1}^{n_{t}} ∑_{j∈J} α_{t}^{(i)}α_{h_{t}}^{(j)} ⊗ rα_{h_{t}^{−1}}^{(j)}α_{t^{−1}}^{(i)}m = \\ = ∑_{t∈T} ∑_{i=1}^{n_{t}} ∑_{j∈J} α_{t}^{(i)} ⊗ rα_{h_{t}}^{(j)}α_{h_{t}^{−1}}^{(j)}α_{t^{−1}}^{(i)}m = ∑_{t∈T} ∑_{i=1}^{n_{t}} α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}m = Ψ(m), \end{matrix}

so Ψ does not depend on the chosen set T. We will show that Ψ is a Λ-homomorphism. For this it is enough to show that

Ψ(λ_{σ}m) = λ_{σ}Ψ(m) for λ_{σ} ∈ Λ_{σ}, σ ∈ G, m ∈ M.

Let σ be a fixed element of G, obviously σ^{−1}t ranges over the left transversal of H in G as t ranges over the elements of T. We choose a decomposition of 1 in Λ_{σ^{−1}t} · Λ_{t^{−1}σ},

1 = ∑_{j∈J} α_{σ^{−1}t}^{(j)}α_{t^{−1}σ}^{(j)},

for a finite index set J. Then

Ψ(λ_{σ}m) = ∑_{t∈T} ∑_{i=1}^{n_{t}} α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}λ_{σ}m =

= ∑_{t∈T} ∑_{i=1}^{n_{t}} ∑_{j∈J} α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}λ_{σ}α_{σ^{−1}t}^{(j)}α_{t^{−1}σ}^{(j)}m =

= ∑_{t∈T} ∑_{i=1}^{n_{t}} ∑_{j∈J} α_{t}^{(i)}α_{t^{−1}}^{(i)}λ_{σ}α_{σ^{−1}t}^{(j)} ⊗ rα_{t^{−1}σ}^{(j)}m =

= ∑_{t∈T} ∑_{j∈J} λ_{σ}α_{σ^{−1}t}^{(j)} ⊗ rα_{t^{−1}σ}^{(j)}m =

= λ_{σ}Ψ(m).

Hence Ψ(λ_{σ}m) = λ_{σ}Ψ(m) for all σ ∈ G and m ∈ M. Finally for all m ∈ M we have

fΨ(m) = f (∑_{t∈T} ∑_{i=1}^{n_{t}} α_{t}^{(i)} ⊗ rα_{t^{−1}}^{(i)}m) = ∑_{t∈T} ∑_{i=1}^{n_{t}} α_{t}^{(i)}rα_{t^{−1}}^{(i)}m = m.

because of the condition (s), so f is a splitting epimorphism and the theorem is proved.

Corollary 2.4. Let G be a finite group and H a subgroup of G for which the condition (s) holds. Then Λ is of finite representation type if and only if Λ_{H} is of finite representation type.

3. Induced representation modules

In this section we assume that Λ_{1} is an artinian ring or a complete discrete valuation ring and G a finite group. Thus the Krull-Schmidt Theorem holds for the finitely generated left Λ-modules. We denote by R the centralizer C_{Λ_{1}}(Λ) of Λ in Λ_{1} and by modΛ the category of finitely generated left Λ-modules, if Λ_{1} is artinian ring, or the category of finitely generated left Λ-modules which are R-torsion-free, that is the Λ-lattices, if Λ_{1} is a complete discrete valuation ring. In this last case Λ is an R-order and by ([10], Proposition 6.4) Λ is a crossed product order. We refer to [3] for the representation theory of orders.

If W ≅ ⨁_{i=1}^{m} W_{i} is the decomposition of W in the category modΛ into indecomposable Λ-modules, then W_{i}, 1 ≤ i ≤ m, are called the indecomposable components of W.

If V is an indecomposable module in modΛ_{H} and V^{G} ≅ ⨁_{i=1}^{n} Y_{i}, where Y_{i}, 1 ≤ i ≤ n, are the indecomposable components of V^{G} in modΛ, then

(V^{G})_{H} ≅ ⨁_{t∈T} ^{t}V ≅ ⨁_{i=1}^{n}(Y_{i})_{H}.

In this section we shall study the decomposition of (Y_{i})_{H} into indecomposable modules in modΛ_{H} and for a module W in modΛ the decomposition of (W_{H})^{G} into indecomposable modules in modΛ.

In the whole of this section H will be a normal subgroup of G. For a module V in modΛ_{H} we denote by N_{V} the set of all elements g ∈ G such that ^{g}V ≅ V as left Λ_{H}- modules. N_{V} is a subgroup of G and is called the inertia group of V. For convenience we include into the next proposition the elementary properties of N_{V}. The proof is analogous to that of the corresponding result for group rings ([13], [3]).

Proposition 3.1. Let V be a left Λ_{H}-module and W a left Λ-module.

i) ^{g}V ≅ ^{gσ}V as Λ_{H}-modules for g ∈ G and σ ∈ N_{V}. ii) H ≤ N_{V}, that is ^{h}V ≅ V as Λ_{H}-modules for h ∈ H. iii) N_{W_{H}} = G. Moreover if Λ is a crossed product, then N_{Λ_{H}} = G. iv) N_{^{g}V} = gN_{V} g^{−1} for g ∈ G.

v) (V^{G})_{H} ≅ ⨁_{g∈K}(^{g}V)^{([N_{V} :H])}, where K is a left transversal of N_{V} in G.

The next result reduces the induction on G-invariant Λ_{H}-modules, that is Λ_{H}-modules V such that ^{g}V ≅ V, ∀g ∈ G as Λ_{H}-modules, and generalizes the corresponding result for group rings and for strongly graded algebras but for trivial H ([12], ch. 4, Theorem 5.2). We omit its proof since it is analogous to those of the above cases.

Proposition 3.2. Let V be an indecomposable module in modΛ_{H} with inertia group N and let

V^{N} ≅ U_{1} ⊕ ⋯ ⊕ U_{s}  (3.1)

be the decomposition of V^{N} into indecomposable modules in modΛ_{N}. Then s ≤ [N : H], (U_{i})_{H} ≅ V^{(m_{i})} for some m_{i} ≤ [N : H], and U_{i}^{G}, 1 ≤ i ≤ s, are the indecomposable components of V^{G}. Moreover if U_{i}^{G} ≅ U_{j}^{G}, then U_{i} ≅ U_{j} for 1 ≤ i, j ≤ s.

In [2] P. R. Boisen gives the graded version of the theory of vertices and sources for group rings and among other things he proves that there exists an indecomposable Λ_{H}-module V such that W | V^{G} for a given indecomposable H-projective Λ-module W ([2], Theorem 2.5, (2)). The next result gives more information in this direction.

Theorem 3.3. Let W be an indecomposable module in modΛ and suppose that W is H-projective.

i) There exists an indecomposable Λ_{H}-module V such that

W_{H} ≅ ⨁_{g∈K} (^{g}V )^{(m)} and m ≤ [N : H],

where K is a left transversal of the inertia group N of V in G.

ii) If

V^{N} ≅ ⨁_{i=1}^{s} U_{i}^{(t_{i})}

is the decomposition of V^{N} into indecomposable Λ_{N}-modules, where V is that of i), then there exists ρ ∈ {1, 2, …, s} such that W ≅ U_{ρ}^{G}. Moreover if α_{w} is the multiplicity of W in (W_{H})^{G}, then α_{w} = t_{ρ}m[G : N].

Proof. i) W |(W_{H})^{G} since W is H-projective. Therefore W | V^{G} for some indecomposable Λ_{H}-module by the Krull-Schmidt theorem. Thus W_{H} |(V^{G})_{H}. Now from v) of Proposition 3.1 and the Krull-Schmidt theorem we get that W_{H} is a direct sum of conjugates of V. However ^{g}W_{H} ≅ W_{H} for all g ∈ G as follows from iii) of Proposition 3.1 and so all the multiplicities of the conjugates of V in the decomosition of W_{H} are equal, this proves i).

ii) We get the relation (3.1). Since V^{G} ≅ (V^{N}) ≅ ⨁_{i=1}^{s} U_{i}^{G}, from the Krull-Schmidt theorem and Proposition 3.2 we get that W ≅ U_{i}^{G} for some i ∈ {1, …, s}. The second statement of ii) follows from i).

Corollary 3.4. If W is an indecomposable Λ-module and α_{w} = 1, then W_{H} is an indecomposable Λ_{H}-module.

In the sequel we assume that R is an artinian ring. The next theorem generalizes the Clifford’s theorem for strongly group graded algebras.

Theorem 3.5. Let W be a simple Λ-module, H \underline{◁} G and V a simple Λ_{H}-module contained in W_{H}. Then the following holds:

i) W_{H} is a semisimple Λ_{H}-module, where the simple summands are isomorphic to conjugates {^{σ}V: σ ∈ G} of V.

ii) Let N be the inertia group of V and U the homogeneous component of W_{H} containing V. Then W_{H} ≅ ⨁_{σ∈K}(^{σ}V)^{(e)}, where K is a left transversal of N in G and for some natural number e.

iii) W ≅ U^{G} as Λ-modules and U is a simple submodule of V^{N}.

Proof. i) Let V be a simple Λ_{H}-submodule of W_{H}. Then the conjugates ^{σ}V, σ ∈ G, are also simple Λ_{H}-modules. We remark that Λ_{σ}V ≅ ^{σ}V as Λ_{H}-modules and ∑_{σ∈G} Λ_{σ}V is a Λ-submodule of W. Hence W = ∑_{σ∈G} Λ_{σ}V, since W is simple, and so W_{H} is a semisimple Λ_{H}-module as a sum of simple Λ_{H}-modules. This proves i). ii) Let U be the homogeneous component of W_{H} containing V. It is clear that Λ_{σ}V, σ ∈ K , are the non-isomorphic conjugates of V. We remark that Λ_{σ}U, σ ∈ G, is also a homogeneous component of W_{H} and since ∑_{σ∈G} Λ_{σ}U ⊂ W we get that W = ΣΛ_{σ}U. Hence the group G acts transitively on the set of the homogeneous components of W_{H} which is in bijective correspondence with a left transversal of the stabilizer Ñ of U in G. Hence W_{H} = ⨁_{σ∈X} Λ_{σ}U for a left transversal X of Ñ in G. Moreover since W = Λ_{σ}W, for σ ∈ Γ, we remark that the multiplicity of Λ_{σ}V in Λ_{σ}U is the multiplicity e of V in U, so

W_{H} = (⨁_{σ∈X} Λ_{σ}V )^{(e)} .

It is obvious that N ≤ Ñ and we claim that N = Ñ. Indeed, if x ∈ Ñ, then Λ_{x}U = U and so ⨁_{σ∈Y} Λ_{xσ}V = ⨁_{σ∈Y} Λ_{σ}V, for some subset Y of a left transversal of H in N. This means that there exists σ ∈ N such that Λ_{xσ}V ≅ V as Λ_{N}-modules and so xσ ∈ N, that is x ∈ N. Hence N = Ñ and we get ii). iii) From the proof of ii) we conclude that W = ⨁_{σ∈K} Λ_{σ}V and hence W ≅ V^{G} as Λmodules. If U is not a simple Λ_{N}-module, then there exists a non-zero proper Λ_{N}-module U′ of U. Thus ∑_{σ∈K} Λ_{σ}U′ ⊂ ∑_{σ∈K} Λ_{σ}U = W which is a contradiction, since W is simple. This proves iii).

Remark. Zhou Bo Rong proved Theorem 3.5 for H = {1} ([16], Proposition 1.3). S. Houssein and F. Van Oystaeyen in ([11], Theorem 2.7) claimed that W ≅ U^{G}, but this is not true even for group rings ([16], Counterexample).

4. G-invariant modules in crossed products

In this section we assume that Λ = R ∗ G = ⨁_{σ∈G} R\overline{σ} is a crossed-product, G is a finite group and H is a subgroup of G.

Let W be a left R ∗ G-module. We call an element w of W G-invariant, if uw = w for all graded units u of R ∗ G, that is for all u ∈ Gr U(R ∗ G) = \{u\overline{σ}: u ∈ U(R), σ ∈ G\}. Let Inv_{G}(W) be the set of all G-invariant elements of W. It is clear that Inv_{G}(W) is a C(R ∗ G)-submodule of W, for C(R ∗ G) the center of R ∗ G.

Theorem 4.1. Let V be a left R ∗ H-module. Then

Inv_{G}(V^{G}) ≅ Inv_{H}(V)

as C_{R∗H}(R ∗ G)-module, where C_{R∗H}(R ∗ G) = {x ∈ R ∗ H: xy = yx, y ∈ R ∗ G}.

Proof. Firstly we prove that

Inv_{G}(V^{G}) = \{∑_{t∈T} \overline{t} ⊕ v : v ∈ Inv_{H}(V)\} ,

where T is a left transversal of H in G. We remark that every element of the right part of the above relation is defined independently of the set T. Indeed if S is another one and th_{t} = σ_{t} , for t ∈ T, h_{t} ∈ H and σ_{t} ∈ S, then for a_{t} = α(t, h_{t})^{−1} we have

∑_{t∈T} \overline{σ}_{t} ⊗ v = ∑_{t∈T} a_{t}\overline{t} ⊗ \overline{h}_{t}v = ∑_{t∈T} \overline{t} ⊗ \overline{t}^{−1}a_{t}\overline{t}v = ∑_{t∈T} \overline{t} ⊗ v,

since \overline{t}a_{t}\overline{t} ∈ U(R). Moreover for g ∈ G, u ∈ U(R) and v ∈ Inv_{H}(V) we get that

\begin{aligned}\overline{g} ∑_{t∈T} \overline{t} ⊗ v & = ∑_{t∈T} α(g, t)\overline{gt} ⊗ v = ∑ \overline{gt} ⊗ \overline{gt}^{−1}α(g, t)\overline{gt} v = \\ & = ∑_{t∈T} \overline{gt} ⊗ v = ∑_{t∈T} \overline{t} ⊗ v, \end{aligned}

and

u ∑_{t∈T} \overline{t} ⊗ v = ∑_{t∈T} \overline{t} ⊗ \overline{t}^{−1}u\overline{t}v = ∑_{t∈T} \overline{t} ⊗ v.

Hence ∑_{t∈T} \overline{t} ⊗ v ∈ Inv_{G}(V^{G}).

Let now ∑_{t∈T} \overline{t}⊗v_{t} be an element of Inv_{G}(V^{G}). Then for σ ∈ T we get \overline{σ}^{−1} ∑_{t∈T} \overline{t}⊗v_{t} = ∑_{t∈T} \overline{t} ⊗ v_{t}. Hence v_{σ} = v_{1} for all σ ∈ G, and ∑_{t∈T} \overline{t} ⊗ v_{t} = ∑_{t∈T} \overline{t} ⊗ v for some v ∈ V. It remains to prove that v belongs to Inv_{H}(V). For this we remark that if h ∈ H, then \overline{h} ∑_{t∈T} \overline{t} ⊗ v = ∑_{t∈T} \overline{t} ⊗ v. Hence \overline{h} · \overline{1} ⊗ v = \overline{1} ⊗ v so \overline{h}v = v. Similarly uv = v for u ∈ U(R). Therefore we have the equality mentioned above.

Now it is easy to see that the map

f : Inv_{G}(V^{G}) → Inv_{H}(V), ∑_{t∈T} \overline{t} ⊗ v ↦ v

is a C_{R∗H}(R ∗ G)-isomorphism.

Corollary 4.2. Let W be a left R ∗ G-module, then

Tr_{H}^{G} : Inv_{H}(W_{H}) → Inv_{G}(W), w ↦ ∑_{t∈T} \overline{t}w

is a C_{R∗H}(R ∗ G)-homomorphism. Particularly Tr_{H}^{G} is a splitting epimorphism if the condition (s) holds.

Proof. It is clear that Tr_{H}^{G} is a C_{R∗H}(R ∗ G)-homomorphism. If the condition (s) holds, we define

f : Inv_{G}(W) → Inv_{H}(W_{H}), w ↦ rw.

Then

Tr_{H}^{G} f(w) = ∑_{t∈T} \overline{t}rw = ∑_{t∈T} \overline{t}r\overline{t}^{−1} = w.

So Tr_{H}^{G} is a splitting epimorphism.

Acknowledgement. The authors would like to thank the referee for his (or her) very helpful remarks.

References

[1] Auslander, M.; Reiten, I.; Smalø, S. O.: Representation Theory of Artin Algebras. Cambridge Stud. Adv. Math. 36, Cambridge University Press, Cambridge 1995.

[2] Boisen, P. R.: The Representation Theory of Fully Group-Graded Algebras. J. Algebra 151 (1992), 160–179.

[3] Curtis, C.; Reiner, I.: Methods of Representation Theory with Applications to Finite Groups and Orders. Vol.I, Wiley, New York 1981.

[4] Dade, E. C.: Group Graded Rings and Modules. Math. Z. 174 (1980), 241–262.

[5] Dade, E. C.: The Equivalence of Various Generalizations of Group Rings and Modules. Math. Z. 181 (1982), 335–344.

[6] Dade, E. C.: Clifford Theory for Group Graded Rings. J. Reine Angew. Math. 369 (1986), 40–86.

[7] Dade, E. C.: Clifford Theory for Group Graded Rings II. J. Reine Angew. Math. 387 (1988), 148–181.

[8] Gordon, R.; Green, E.: Graded Artin Algebras. J. Algebra 76 (1982), 111–137.

[9] Gordon, R.; Green, E.: Representation Theory of Graded Artin Algebras. J. Algebra 76 (1982), 138–152.

[10] Haefner, J.: Graded Equivalence Theory with Applications. J. Algebra 172 (1995), 385–424.

[11] Hussein, S.; Van Oystaeyen, F.: Control Subgroups for Group Graded Rings. Comm. Algebra 20(8) (1992), 2219–2237.

[12] Karpilovsky, G.: The Algebraic Structure of Crossed-Products. North-Holland Math. Stud. 142, Amsterdam 1987.

[13] Karpilovsky, G.: Group representations. Vol.4, North-Holland Math. Stud. 182, Amsterdam 1995.

[14] Năstăsescu, C.; Van Oystaeyen, F.: Graded Ring Theory. Noth Holland Math. Library, 28, Amsterdam 1982.

[15] Reiten, I.; Riedtmann, C.: Skew Group Algebras in the Representation Theory of Artin Algebras. J. Algebra 92 (1985), 224–282.

[16] Zhou, Borong: Two Clifford’s Theorems for strongly Group Graded Rings. J. Algebra 139 (1991), 172–189.

[17] Theohari-Apostolidi, Th.: Local Crossed-Product Orders of Finite Representation Type. J. Pure Appl. Algebra 41 (1986), 87–98.

[18] Van Oystaeyen, F.: On Clifford Systems and Generalized Crossed-Products. J. Algebra, 87(2) (1984), 396–415.

Received April 16, 1998