MODULES GRADED BY G-SETS AND CLIFFORD THEORY

Andrei Marcus

September 1993

Introduction

Modules graded by G-sets occur in group-representation theory when dealing with induced modules. If K is a field, H is a subgroup of the group G, G/H = { gH | g ∈ G }, and M is a left KH -module, then the KG-module KG ⊗_{KH} M induced from M has a decomposition KG ⊗_{KH} M = ⨁_{x∈G/H}(Kx ⊗_{KH} M) into additive subgroups, where Kx = ⨁_{g∈x} Kg, such that for g ∈ G, Kg(Kx ⊗_{KH} M) = Kgx ⊗_{KH} M. Gradings by transitive G-sets were introduced by E.C. Dade in [5], and the general definition was given in [14]. We recall here some basic facts.

Assume that G is a group, A is a left G-set, R = ⨁_{x∈G} R_{g} is a G-graded ring and M is a left R-module. We say that M is A-graded if M = ⨁_{x∈A} M_{x} (as additive groups) and R_{g}M_{x} ⊆ M_{gx} for any g ∈ G and x ∈ A. We denote by (G, A, R)-gr the category of A-graded R-modules and grade preserving R-homomorphisms. It was shown in [14] that this is a Grothendieck category with a projective generator.

Several functors were constructed and studied in [5], [13], [14] and [16], and these constructions and their applications provided the inspiration for the present work. In order to set our context, in the first section of this paper we give more general definitions. We fix a homomorphism θ: G → H of groups and a homomorphism ϕ: A → B of G-sets, where B is an H -set, hence a G-set by restriction of scalars. We define first the “ungrading” functor for graded rings U(θ): G-grring → H-grring and its right adjoint, the “truncation” functor T (θ): H-grring → G-grring. Then we construct the ungrading functor for graded modules U(θ, ϕ): (G, A, R)-gr → (H, B, U(θ)(R))-gr and it’s right adjoint F(θ, ϕ), and the truncation functor for graded modules T (θ, ϕ): (H, B, S)-gr → (G, A, T (θ)(S))-gr and it’s left adjoint, the induction functor I(θ, ϕ). These functors behave well with respect to compositions of homomorphisms of groups and G-sets. Notice that another general presentation (in terms of bigraded bimodules) of functors between categories of modules graded by G-sets is given in [20].

In the second section we restrict to the case modules graded by transitive G-sets. If H is a subgroup of G, then (G/H, R)-gr is the category (G, G/H, R)-gr. We establish some properties of the functors defined above and we construct the coinduction functor Coind: (H/K, R_{H})-gr → (G/K, R)-gr (where K ≤ H ≤ G), which is a right adjoint of the

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truncation functor. The main result of this section is Theorem 2.16, from which we obtain correspondences between the simple objects of the categories (G/K, R)-gr, (H/K, R_{H})-gr and (G/H, R)-gr, generalizing results of [5] and [11]. The proofs rely on the properties of the induction and coinduction functors, and also on some standard facts about torsion theory.

A well-known theorem of E.C. Dade states that the categories R_{1} -mod and R-gr are equivalent via the induction and the truncation functors if and only if 1 ∈ R_{g}R_{g^{−1}} for all g ∈ G. We generalize this in Section 3 by showing that the categories (H/K, R_{H})-gr and (G/K, R)-gr are equivalent via the induction and the truncation functors if and only if 1 ∈ R_{gH}R_{Hg^{−1}} for all g ∈ G.

The main ingredient in the construction of the coinduction functor mentioned above is the following observation: if M is a finitely generated G-graded R-module and N is a G/H -graded R-module, then Hom_{R}(M, N) is in a natural way a G/H -graded left E - module, where E = End_{R}(M)^{op}. This observation is also used in Section 3, where we show that the equivalences (which we shall call Clifford correspondences) CD[M] ⇄ E-mod and (R|M)-mod ⇄ (E|E)-mod preserve the G/H -graded modules (and also the relative H - projective modules in the strongly graded case). Here M is again a finitely generated G-graded R-module, CD[M] is, as in [8], the full subcategory of R-mod consisting of R-modules N for which there is an exact sequence M^{(J)} → M^{(I)} → N → 0 in Rmod, (R|M)-mod consists of R-modules which divide a direct sum of copies of M, and (E|E)-mod is the category of projective E -modules.

As consequences of our results, we obtain a graded version of Morita’s Theorem, and in particular, some equivalences involving the smash product. We also deduce Clifford Correspondences for arbitrary subgroups for both simple and indecomposable modules.

In this paper, rings are always associative with unity, and modules are left, unless otherwise stated. We refer to [3], [4] and [17] for general results on graded rings, and to [22] for Grothendieck categories and torsion theory.

1. Functors associated to homomorphisms of groups and of G-sets

Let G be a group and A a (left) G-set. We shall denote by G-grring the category of G-graded rings (associative with unit element), and if R = ⨁_{g∈G} R_{g} is a G-graded ring, then (G, A, R)-gr will denote the category of A-graded R-modules. An R-module is Agraded, if it has a decomposition M = ⨁_{x∈A} M_{x} into a direct sum of abelian subgroups, such that R_{g}M_{x} ⊆ M_{gx}, for any g ∈ G and x ∈ A. An R-homomorphism f: M → M′ belongs to this category if f(M_{x}) ⊆ M′_{x} for any x ∈ A.

We shall fix a group-homomorphism θ: G → H and θ′: H → K, an H -set B and a K - set C, ϕ: A → B will be a homomorphism of G-sets and ϕ′: B → C a homomorphism of H -sets (by restriction of scalars). The aim of this section is to define define some functors naturally associated to these homomorphisms. The proofs of the most of the statements will be ommited.

1.1. The grade-forgetting functor U(θ). If R is a G-graded ring, let U(θ)(R) = R̄ = ⊕_{h∈H} R̄_{h}, where R̄_{h} = ⨁_{g∈θ^{−1}(h)} R_{g} (with the convention that the sum of an empty family is 0). Then R̄ is an H -graded ring, and R̄ = R as rings. If f: R → R′ is a morphism in G-grring, then U(θ)(f) = f̄: R̄ → R̄′, where f̄ = f as functions. We obtain in this way a functor U = U(θ): G-grring → H-grring, which will be called the “ungrading” functor.

Let α_{R} = α_{R}(θ): R → U(θ)(R), α_{R}(r_{g}) = r_{g} ∈ R̄_{θ(g)} for any r_{g} ∈ R_{g}. Then α_{R}(θ) is a ring homomorphism (α_{R}(θ) = id_{R} as functions) and α_{R}(R_{g}) ⊆ R̄_{θ(g)}. If f: R → R′ is a morphism of G-graded rings, then f̄ = U(θ)(f) is the unique morphism in H -grring such that f̄∘ α_{R}(θ) = α_{R′}(θ) ∘ f.

The functors U(θ′ ∘ θ) and U(θ′) ∘ U(θ): G-grring → K-grring are naturally isomorphic, and identifying U(θ′ ∘ θ)(R) with U(θ′)(U(θ)(R)), we have the equality αR̄(θ′) ∘ α_{R}(θ) = α_{R}(θ′ ∘ θ).

1.2. Examples. a) Let N be a normal subgroup of G and θ: G → H = G/N be the canonical projection. Then for any h ∈ G, R̄_{h} = ⨁_{g∈h} R_{g}, so R̄ is obtained by ”forgetting” partially the grading of R.

b) If θ: G ↪ H is the inclusion, then R̄_{h} = R_{h} if h ∈ G and R̄_{h} = 0 if h ∉ G.

c) Let θ: G → G, θ(g) = xgx^{−1}, where x ∈ G is fixed. Then R̄_{h} = R_{θ^{−1}(h)} = R_{x^{−1}hx} for any h ∈ G. Denote R̄ = ^{x}R. Then (^{x}R)_{^{x}g} = R_{g}, where ^{x}g = xgx^{−1}.

1.3. The right adjoint T (θ) of U(θ). Let S be an H -graded ring, and let T (θ)(S) = S̃ = ⨁_{g∈G} S̃_{g}, where S̃_{g} = S_{θ(g)}. Clearly S̃ is a G-graded ring. If f: S → S′ is a morphism of H -graded rings, then T (θ)(f) = f̃: S̃ → S̃′ is given by f̃ = ⨁_{g∈G}(f_{θ(g)} : S_{θ(g)} → S′_{θ(g)}).

Let β_{S} = β_{S}(θ): S̃ → S, β_{S}(s̃_{g}) = s̃_{g} ∈ S_{θ(g)} for any s̃_{g} ∈ S̃_{g}. Then β_{S}(θ) is a ring-homomorphism and β_{S}(s̃_{g}) ⊆ S_{θ(g)} (∀)g ∈ G. If f: S → S′ is a morphism of H - graded rings, then f̃ = T (θ)(f) is the unique morphism in G-grring such that f̃∘ β_{S}(θ) = β_{S′}(θ) ∘ f.

The functors T (θ′ ∘θ) and T (θ)∘T (θ′): K-grring → G-grring are naturally isomorphic, and identifying T (θ′ ∘θ)(T) with T (θ)(T (θ′)(T)), we have that β_{T} (θ′ ∘θ) = β_{T} (θ′)∘βT̃(θ) (here T ∈ K -grring and T̃ = T (θ′)(T)).

1.4. Examples. a) Let θ: G ↪ H be the inclusion map. In this case we denote S̃ = S_{G} = ⨁_{g∈G} S_{g}, and we see that S̃ is obtained by “truncation”, and β_{S}(θ) is the inclusion S̃ → S.

b) Let N be a normal subgroup of G, and θ: G → H = G/N be the canonical projection. Then S̃_{g} = S_{gN} (∀)g ∈ G, so S̃ is obtained by “inflation”.

c) Let θ: G → G, θ(g) = xgx^{−1}. Then S̃_{g} = (S^{x})_{g} = S_{xgx^{−1}}.

d) If θ: G → H is an isomorphism, then T (θ) = U(θ^{−1}) is the inverse of U(θ).

1.5. Lemma. T (θ) is a right adjoint of U(θ).

Proof. We define the functorial isomorphisms

Hom_{G-grring}(−, T (−)) ⇄_{β}^{α} Hom_{H-grring}(U(−), −)

as follows: if u ∈ Hom_{G-grring}(R, T (S)), then α_{R,S}(u)(r̄_{h}) = β_{S}(u(α_{R}^{−1}(r̄_{h}))); if v ∈ Hom_{H-grring}(U(R), S), then β_{R,S}(v)(r_{g}) = v(α_{R}(r_{g})) ∈ S̃_{g} = S_{θ(g)}.

Similar functors may be constructed in the case of modules graded by G-sets.

1.6. The grade-forgetting functor U(θ, ϕ). If M ∈ (G, A, R)-gr, let U(M) = U(θ, ϕ)(M) = M̄ = ⨁_{y∈B} M_{y}, where M̄_{y} = ⨁_{x∈ϕ^{−1}(y)} M_{x}. Let r̄_{h} ∈ R̄_{h} and m̄_{y} ∈ M̄_{y}. We may consider that r̄_{h} = r_{g} ∈ R_{g} with θ(g) = h and m̄_{y} = m_{x} ∈ M_{x} with ϕ(x) = y, so the element r_{g}m_{x} ∈ M_{gx} is defined. Define now the multiplication by r̄_{h}m̄_{y} = r_{g}m_{y} ∈ M̄_{hϕ(x)}. It follows that M̄ is a B -graded R̄-module and M̄ = M as abelian groups. If f: M → M′ is a morphism in (G, A, R)-gr, then U(f) = f̄: M̄ → M̄′ is a morphism in (H, B, R̄)-gr, where f̄ = f as functions, hence we have obtained an exact functor U = U(θ, ϕ): (G, A, R)-gr → (H, B, U(θ)(R))-gr.

Let α_{M} = α_{M}(θ, ϕ): M → U(M), α_{M}(m_{x}) = m_{x} ∈ M̄_{ϕ(x)}. Then α_{M}(θ, ϕ) is a homomorphism of abelian groups (α_{M} = id_{M} as functions) and α_{M}(r_{g}m_{x}) = α_{R}(r_{g})α_{M}(m_{x}) ∈ M̄_{ϕ(gx)} for every r_{g} ∈ R_{g}, m_{x} ∈ M_{x}. If f: M → M′ is a morphism in (G, A, R)-gr, then f̄ = U(f) is the unique morphism in (H, B, R̄)-gr such that f̄∘ α_{M} = α_{M′} ∘ f.

The functors U(θ′ ∘ θ, ϕ′ ∘ ϕ) and U(θ′, ϕ′) ∘ U(θ, ϕ): (G, A, R)-gr → (K, C, U(θ′ ∘ θ)(R)) are naturally isomorphic, and identifying the objects U(θ′ ∘ θ, ϕ′ ∘ ϕ)(M) and U(θ′, ϕ′)(U(θ, ϕ)(M)), we have that α_{M}(θ′ ∘ θ, ϕ′ ∘ ϕ) = αM̄(θ′, ϕ′) ∘ α_{M}(θ, ϕ).

1.7. Examples. a) Let θ = id_{G}, I ≤ K ≤ G and ϕ: G/I → G/K the natural homomorphism of G-sets. Then M̄_{y} = ⨁_{x∈y} M_{x}, so M̄ is obtained by “forgetting” partially the gradation of M.

b) Let θ = id_{G}, A = B = G and ϕ: G → G defined by ϕ(x) = xg where g ∈ G is fixed. Then for y ∈ G, M̄_{y} = M_{ϕ^{−1}(y)} = M_{yg^{−1}}, so M̄ is the g^{−1} -suspension M(g^{−1}) of M.

1.8 The right adjoint F(θ, ϕ) of U(θ, ϕ). If N ∈ (H, B, R̄)-gr, then let F(N) = F(θ, ϕ)(N) = N̂ = ⨁_{x∈A} N̂_{x}, where N̂_{x} = N_{ϕ(x)}, and the multiplication is defined by r_{g}n̂_{x} = α_{R}(r_{g})n_{ϕ(x)} ∈ N̂_{gx} = N_{ϕ(gx)}, for r_{g} ∈ R_{g} and n_{ϕ(x)} = n̂_{x}. If f: N → N′ is a morphism in (H, B, R̄)-gr, then f̂ = F(θ, ϕ)(f): N̂ → N̂′, where f̂_{x} = f_{ϕ(x)} : N_{ϕ(x)} → N′_{ϕ(x)}.

We obtain in this way an exact functor F = F(θ, ϕ): (H, B, R̄)-gr → (G, A, R)-gr.

Let β_{N} = β_{N}(θ, ϕ): N̂ → N, β_{N}(n̂_{x}) = n̂_{x} = n_{ϕ(x)} ∈ N_{ϕ(x)}. We have that β_{N} is a homomorphism of additive groups and β_{N}(r_{g}n̂_{x}) = α_{R}(r_{g})β_{N}(n̂_{x}) ∈ N_{ϕ(gx)}. If f: N → N′ is a morphism in (H, B, R̄)-gr, then f̂ is the unique morphism in (G, A, R)-gr such that f ∘ β_{N} = β_{N′} ∘ f̂.

The functors F(θ′ ∘ θ, ϕ′ ∘ ϕ) and F(θ, ϕ) ∘ F(θ′, ϕ′): (K, C, U(θ′ ∘ θ)(R))-gr → (G, A, R)-gr are naturally isomorphic, and identifying the objects F(θ′ ∘ θ, ϕ′ ∘ ϕ)(P) and F(θ, ϕ)(F(θ′, ϕ′)(P)) (where P ∈ (K, C, U(θ′ ∘ θ)(R))-gr), we have that β_{P} (θ′ ∘ θ, ϕ′ ∘ ϕ) = βP̂(θ, ϕ) ∘ β_{P} (θ′, ϕ′).

1.9. Lemma. F(θ, ϕ) is a right adjoint of U(θ, ϕ).

Proof. We have the functorial isomorphisms

Hom_{(G,A,R)-gr}(−, F(−)) ⇄_{β}^{α} Hom_{(H,B,R̄)-gr}(U(−), −)

as follows: if u ∈ Hom_{(G,A,R)-gr}(M, N̂) and m̄_{y} ∈ M̄_{y}, then define α_{M,N}(u)(m̄_{y}) = β_{N}(u(α_{M}^{−1}(m̄_{y}))) ∈ N_{y}; if v ∈ Hom_{(H,B,R̄)-gr}(M̄, N) and m_{x} ∈ M_{x}, then β_{M,N}(v)(m_{x}) = v(α_{M}(m_{x})) ∈ N̂_{x} = N_{ϕ(x)}.

Let now S ∈ H-grring. We define the “truncation” functor for graded modules T (θ, ϕ): (H, B, S)-gr → (G, A, S̃)-gr, where S̃ = T (θ)(S) ∈ G-grring. In some particular cases T (θ, ϕ) coincides with F(θ, ϕ).

1.10. The truncation functor T (θ, ϕ). If N ∈ (H, B, S)-gr, let T (N) = T (θ, ϕ)(N) = Ñ = ⨁_{x∈A} Ñ_{x}, where Ñ_{x} = N_{ϕ(x)}. For s̃_{g} ∈ S̃_{g} and ñ_{x} = n_{ϕ(x)} ∈ Ñ_{x}, the multiplication is given by s̃_{g}ñ_{x} = β_{S}(s̃_{g})n_{ϕ(x)} ∈ Ñ_{gx} = N_{ϕ(gx)}. If f: N → N′ is a morphism in (H, B, S)-gr, then f̃ = T (f): Ñ → Ñ′, f̃ = ⨁_{x∈A} f̃_{x}, where f̃_{x} = f_{ϕ(x)} : N_{ϕ(x)} → N′_{ϕ(x)}.

It can be easily seen that T : (H, B, S)-gr → (G, A, S̃)-gr is an exact functor.

Let γ_{N} = γ_{N}(θ, ϕ): Ñ → N, γ_{N}(ñ_{x}) = ñ_{x} = n_{ϕ(x)} ∈ N_{ϕ(x)}. Then γ_{N} is a homomorphism of additive groups and γ_{N}(s̃_{g}ñ_{x}) = β_{S}(s̃_{g})γ_{N}(ñ_{x}) ∈ S_{θ(g)}N_{ϕ(x)} ∈ N_{ϕ(gx)}. If f: N → N′ is a morphism in (H, B, S)-gr, then f̃ = T (f) is the unique morphism in (G, A, S̃)-gr such that f ∘ γ_{N}(θ, ϕ) = γ_{N′}(θ, ϕ) ∘ f̃.

The functors T (θ′ ∘ θ, ϕ′ ∘ ϕ) and T (θ, ϕ) ∘ T (θ′, ϕ′): (K, C, T)-gr → (G, A, T (θ′ ∘ θ)(T))-gr are naturally isomorphic, and identifying the objects T (θ′ ∘ θ, ϕ′ ∘ ϕ)(P) and T (θ, ϕ)(T (θ′, ϕ′)(P)), we have that γ_{P} (θ′ ∘ θ, ϕ′ ∘ ϕ) = γ_{P} (θ′, ϕ′) ∘ γP̃(θ, ϕ). Here T ∈ K-grring and P ∈ (K, C, T)-gr.

1.11 Examples. a) Let I ≤ G ≤ H, θ: G → H and ϕ: G/I → H/I the inclusions. Then S̃ = S_{G} and Ñ = N_{G/I} are obtained by “truncating” the H -graded ring S and and the H/I -graded S -module N.

b) Let θ = id_{G}, ϕ: G → G, ϕ(x) = xg, where g ∈ G is fixed. Then Ñ is the G-suspension N(g) of N, where N(g)_{x} = N_{xg}.

We continue this section with the construction of the induction functor, which will be a left adjoint of T (θ, ϕ) 1.12. The induction functor I(θ, ϕ). Let M ∈ (G, A, S̃)-gr. Consider first the left S - module S⊗_{ℤ}M with the multiplication given by s(s′⊗m) = ss′⊗m. We have the decomposition S ⊗_{ℤ} M = ⨁_{h∈H} ⨁_{x∈A} S_{h} ⊗_{ℤ} M_{x}. For y ∈ B let (S ⊗_{ℤ} M)_{y} = ⨁_{hϕ(x)=y} S_{h} ⊗_{ℤ} M_{x}. It follows that S ⊗_{ℤ}M = ⨁_{y∈B}(S ⊗_{ℤ}M)_{y} as abelian groups. If s_{h}⊗m_{x} ∈ (S ⊗_{ℤ}M)_{y}, then hϕ(x) = y. Let s_{h′} ∈ S_{h′}. We have that s_{h′}(s_{h}⊗m_{x}) = s_{h′}s_{h}⊗m_{x} ∈ (S_{h′h}⊗_{ℤ}M)_{h′y}, hence S ⊗_{ℤ} M ∈ (H, B, S)-gr. Using the ring homomorphism β_{S}: S̃ → S, we may consider the S -module S ⊗S̃ M, and the natural epimorphism π: S ⊗_{ℤ} M → S ⊗S̃ M. The kernel of π is the S -submodule of S⊗_{ℤ}M generated by the set {ss̃⊗m−s⊗s̃m | s ∈ S, s̃ ∈ S̃, m ∈ M}. Each element ss̃ ⊗ m − s ⊗ s̃m is a sum of elements of the same form, with s, s̃ and m homogenous, hence Kerπ is a B -graded S -submodule of S ⊗_{ℤ} M. We define now I(M) = I(θ, ϕ)(M) = S ⊗S̃ M ∈ (H, B, S)-gr. If f: M → M′ is a morphism in (G, A, S̃)-gr, we consider first the S -morphism S ⊗_{ℤ} f: S ⊗_{ℤ} M → S ⊗_{ℤ} M′. Since S ⊗_{ℤ} f(ss̃ ⊗ m − s ⊗ s̃m) = ss′ ⊗ f(m) − s ⊗ s̃f(m), there is a homomorphism I(θ, ϕ)(f) = S ⊗S̃ f: S ⊗S̃ M → S ⊗S̃ M′ such that S ⊗S̃ f(s ⊗S̃ m) = s ⊗S̃ f(m), and the induction functor I(θ, ϕ) = S ⊗S̃ −: (G, A, S̃)-gr → (H, B, S)-gr is defined.

Let δ_{M} = δ_{M}(θ, ϕ): M → S ⊗S̃ M, δ_{M}(m) = 1 ⊗ m. Then δ_{M} is a homomorphism of abelian groups and δ_{M}(s̃_{g}m_{x}) = β_{S}(s̃_{g})δ_{M}(m_{x}) ∈ (S ⊗S̃ M)_{ϕ(gx)}. If f: M → M′ is a morphism in (G, A, S̃)-gr, then S ⊗S̃ f = I(f) is the unique morphism in (H, B, S)-gr such that (S ⊗S̃ f) ∘ δ_{M} = δ_{M′} ∘ f.

The functors I(θ′ ∘ θ, ϕ′ ∘ ϕ) and I(θ′, ϕ′) ∘ I(θ, ϕ): (G, A, T (θ′ ∘ θ)(T))-gr → (K, C, T)-gr are naturally isomorphic, and identifying the objects I(θ′ ∘ θ, ϕ′ ∘ ϕ)(M) and I(θ′, ϕ′)(I(θ, ϕ)(M)), we have that δ_{M}(θ′ ∘ θ, ϕ′ ∘ ϕ) = δ_{S⊗S̃M}(θ′, ϕ′) ∘ δ_{M}(θ, ϕ). Here T ∈ K-grring and this property is the transitivity of induction.

1.13. Lemma. The functor I(θ, ϕ) is a left adjoint of T (θ, ϕ).

Proof. Let M ∈ (G, A, S̃)-gr and N ∈ (H, B, S)-gr. We define the natural homomorphisms

Hom_{(H,B,S)-gr}(S ⊗S̃ −, −) ⇄_{γ}^{δ} Hom_{(G,A,S̃)-gr}(−, T (−))

as follows: if u ∈ Hom_{(H,B,S)-gr}(S ⊗S̃ M, N), let δ_{M,N}(u)(m) = u(1 ⊗S̃ m); if v ∈ Hom_{(G,A,S̃)-gr}(M, Ñ), let γ_{M,N}(v)(s⊗S̃m) = sγ_{N}(v(m)). Let s̃ ∈ S̃. Then δ_{M,N}(u)(s̃m) = u(1 ⊗ s̃m) = u(s̃ ⊗ m) = u(s̃(1 ⊗ m)) = s̃u(1 ⊗ m) = s̃δ_{M,N}(u)(m). If m_{x} ∈ M_{x}, x ∈ A, then 1 ⊗ m_{x} ∈ (S ⊗S̃ M)_{ϕ(x)} and δ_{M,N}(u)(m_{x}) ∈ N_{ϕ(x)}, that is δ_{M,N}(M_{x}) ⊆ Ñ_{x}, hence δ_{M,N} is well defined and it is natural. Let y ∈ B; if hϕ(x) ≠ y for any x ∈ A and any h ∈ H, then (S ⊗S̃ M)_{y} = 0. Otherwise, let h ∈ H and x ∈ A be such that hϕ(x) = y, s_{h} ∈ S_{h} and m_{x} ∈ M_{x}. Then v(m_{x}) ∈ Ñ_{x} = N_{ϕ(x)} and s_{h}v(m_{x}) ∈ S_{h}N_{ϕ(x)} ⊆ N_{y}. It follows that γ_{M,N}(v)((S ⊗S̃ M)_{y}) ⊆ N_{y}, so δ_{M,N} is well defined and it is also natural.

Further, γ_{M,N}(δ_{M,N}(u))(s⊗m) = sδ_{M,N}(u)(m) = su(1⊗m) = u(s⊗m), hence γ∘δ = id. Also, δ_{M,N}(γ_{M,N}(v))(m) = γ_{M,N}(v)(1 ⊗ m) = v(m), hence δ ∘ γ = id and the lemma is proved.

1.14. Let M ∈ (G, A, S̃)-gr and N ∈ (H, B, S)-gr. The functorial isomorphisms δ and γ define the functorial morphisms ξ: I ∘ T → id_{(H,B,S)-gr} and ζ: id_{(G,A,S̃)-gr} → T ∘ I as follows: ξ_{N} = γ_{Ñ,N}(idÑ): S ⊗S̃ Ñ → N, ξ_{N}(s ⊗ ñ) = sγ_{N}(ñ) and ζ_{M} =

δ_{M,I(M)}(id_{I(M)}): M → { \atop S ⊗S̃ M}, ζ_{M}(m_{x}) = 1 ⊗ m_{x} ∈ ({ \atop S ⊗S̃ M})_{x} = (S ⊗S̃ M)_{ϕ(x)}.

2. Modules graded by transitive G-sets

We shall use the constructions of the preceding section in the case of transitive G-sets, being especially interested in the investigation of the structure of graded simple modules, and of their behaviour with respect to our constructions.

We fix the group G and the subgroups I ≤ K ≤ H ≤ G. Then G/H is a (left) G-set and for σ ∈ G, ^{σ}H/^{σ}K is a ^{σ}H -set, where ^{σ}H = σHσ^{−1}. We denote by (G/H, R)-gr the category (G, G/H, R)-gr.

The homomorphisms θ = id_{G} and ϕ: G/H → G/K, ϕ(gK) = gH (or ϕ(x) = y where x ⊆ y) induce the pair of adjoint functors U: (G/K, R)-gr → (G/H, R)-gr and F: (G/H, R)-gr → (G/K, R)-gr, where for M ∈ (G/K, R)-gr, U(M) = M̄ = ⨁_{y∈G/H} M̄_{y}, M̄_{y} = ⨁_{x⊆y} M_{x} and for N ∈ (G/H, R)-gr, F(N) = Ñ = ⨁_{x∈G/K} Ñ_{x}, where Ñ_{x} = N_{y} for x ⊆ y. Remark that F(N) = T (θ, ϕ)(N).

There are some natural R-homomorphisms which can be defined: 2.1.a. If M ∈ (G/K, R)-gr, let ζ_{M}: M → F(U(M)), ζ_{M}(m_{x}) = m_{x} ∈ { \atop U(M)}_{x}. It follows that ζ_{M} is a morphism in (G/K, R)-gr. 2.1.b. Let N ∈ (G/H, R)-gr. Then we have a morphism ξ_{N}: U(F(N)) → N in (G/H, R)-gr, defined by ξ_{N}(ñ_{x}) = ñ_{x} ∈ N_{ϕ(x)}, (∀)x ∈ G/K.

If H/K is a finite set, define η_{N}: N → U(F(N)) by η_{N}(n_{y}) = ñ = ∑_{x∈G/K} ñ_{x}, where ñ_{x} = n_{y} if x ⊆ y and ñ_{x} = 0 otherwise. We see that η_{N} is also a morphism in (G/H, R)-gr, η_{N} is injective, ξ_{N} is surjective, and (ξ_{N} ∘η_{N})(n) = |H/K|n for any n ∈ N.

As immediate consequences we record: 2.2. If |H/K| is invertible in R, then N is a direct summand of U(F(N)) in (G/H, R)-gr. 2.3. If M is a generator of the category (G/K, R)-gr, then U(M) is a generator of the category (G/H, R)-gr.

Indeed, if N ∈ (G/H, R)-gr, then there is a set J and an epimorphism U(M^{(J)}) →

U(F(N)) ⟶^{ξ_{N}} N.

We prove first two basic and useful results, which are known for G-graded R-modules. For the definition of separability of functors we refer to [16].

2.4. Proposition. U is a separable functor. Consequently, if U(M) is projective (respectively injective) in (G/H, R)-gr, then M is projective (respectively injective) in (G/K, R)-gr.

Proof. Let M, M′, N, N′ ∈ (G/K, R)-gr. We have the homomorphisms

Hom_{(G/K,R)-gr}(M, M′) ⇄_{∼}^{U} Hom_{(G/H,R)-gr}(U(M), U(M′))

where for v ∈ Hom_{(G/H,R)-gr}(U(M), U(M′)), ṽ(m_{x}) = v(m_{x})_{x}, (∀)m_{x} ∈ M_{x}. It is easy to see that if u ∈ Hom_{(G/K,R)-gr}(M, M′), then { \atop U(u)} = u, and if the diagram

U(M) ⟶^{v} U(M′)
U(f)↓  ↓^{U(f′)}
U(N) ⟶^{w} U(N′)

commutes, then the diagram

M ⟶^{ṽ} M′
f↓  ↓^{f′}
N ⟶^{w̃} N′

also commutes, for any f ∈ Hom_{(G/K,R)-gr}(M, N) and f′ ∈ Hom_{(G/K,R)-gr}(M′, N′). The remaining statements follow easily from here, using that U is an exact functor.

2.5. Proposition. If M ∈ (G/K, R)-gr and N is an essential subobject of M in (G/K, R)-gr, then N̄ is an essential subobject of M̄ in (G/H, R)-gr.

Proof. N is essential in M if and only if for any m_{x} ≠ 0 there is r ∈ h(R) such that rm_{x} ∈ N \ {0}. Let m̄_{y} = m_{x_{1}} + ⋯ + m_{x_{t}} ∈ N̄_{y}, where x_{i} ⊆ y and m_{x_{i}} ≠ 0. We show by induction on t that there is r ∈ h(R) such that rm̄_{y} ∈ N \ {0}. For t = 1 the assertion follows by hypothesis. Let m̄′ = m_{x_{1}} + ⋯ + m_{x_{t−1}}. Since m_{x_{t}} ≠ 0, there is r ∈ h(R) such that rm_{x_{t}} ∈ N \ {0}. The element rm̄_{y} − rm̄_{x_{t}} = rm̄′ has at most t − 1 nonzero components. If rm̄′ = 0 then rm̄_{y} ∈ N \ {0}. If rm̄′ ≠ 0, then by the induction hypothesis, there is s ∈ h(R) such that srm̄′ ∈ N \ {0}. We have srm̄_{y} = srm_{x_{t}} + srm̄′ and the assumption srm̄_{y} ≠ 0 gives srm_{x_{t}} = srm′ = 0, which contradicts the choice of s. It follows that srm̄_{y} ∈ N \ {0} and the proposition is proved.

Our first result on graded simple modules is the following generalization of [5], Theorem 12.10.

2.6. Proposition. Assume that H/K is finite. If N ∈ (G/H, R)-gr is a simple object, then there exists a simple object M ∈ (G/K, R)-gr and a monomorphism N → U(M) in (G/H, R)-gr.

Proof. We have the homomorphism η_{N}: N → U(Ñ), η_{N}(n_{y}) = ∑_{x∈G/K} ñ_{x} as in (2.1.b). There is 0 ≠ n_{y} ∈ N_{y} such that Rn_{y} = N. Then the finite set {ñ_{x} | x ⊆ y} generates Ñ in (G/K, R)-gr. Indeed, let ñ_{gK} ∈ Ñ_{gK}, so ñ_{gK} = n_{gH} ∈ N_{gH}, and assume that y = σH, σ ∈ G. Since n_{gH} ∈ Rn_{σH}, we have that n_{gH} = ∑_{h∈H} r_{ghσ^{−1}}n_{σH}, where only a finite number of r_{ghσ^{−1}} ∈ R_{ghσ^{−1}} are nonzero. It follows that ñ_{gK} = ∑_{h∈H} r_{ghσ^{−1}}ñ_{σh^{−1}}K = n_{σH}.

If F(N) is a simple object in (G/K, R)-gr, then the proposition is proved. Otherwise, let N = {L ≤ F(N) | L ≠ F(N)}. Obviously, N is nonempty, and since F(N) is finitely generated, we have that N is inductively ordered by inclusion. Let L be a maximal element of N , so Ñ/L = M is a simple object in (G/K, R)-gr. Let η′ be the composite homomorphism N → U(Ñ) → U(M). We have only to show that η′ ≠ 0. If η′ = 0, then η_{N}(n_{gH}) ∈ L for any n_{gH} ∈ N_{gH} and gH ∈ G/H. It follows that Ñ_{x} ⊆ L_{x} for any x ∈ G/K, hence Ñ ⊆ L, which is a contradiction.

For further need, we give now the definition of the suspension (conjugation) functor (see also [5], Section 3.)

2.7. Let θ = id_{G} and ϕ: G/^{σ}K → G/K, ϕ(y) = yσ, where σ ∈ G is fixed. Then ϕ is an isomorphism of G-sets and it’s inverse is ϕ^{−1} : G/K → G/^{σ}K, ϕ^{−1}(x) = xσ^{−1}. It follows that ϕ induces the equivalence of categories S_{G/K}^{σ} = T (θ, ϕ) = F(θ, ϕ): (G/K, R)-gr → (G/^{σ}K, R)-gr. For M ∈ (G/K, R)-gr, S_{G/K}^{σ}(M) = M(σ) = ⨁_{y∈G/^{σ}K} M(σ)_{y}, where M(σ)_{y} = M_{yσ}, and if f: M → M′ is a morphism in (G/K, R)-gr, then S_{G/K}^{σ}(f) = f as functions. The inverse of this equivalence is (S_{G/K}^{σ})^{−1} = S_{G/^{σ}K}^{σ^{−1}}. For N ∈ (G/^{σ}K, R)-gr, S_{G/^{σ}K}^{σ^{−1}}(N) = N(σ^{−1}) and N(σ^{−1})_{x} = N_{xσ^{−1}} for every x ∈ G/K. Remark that if τ ∈ σK, then S_{G/K}^{τ} = S_{G/K}^{σ}.

2.8. The inclusions θ: H → G and ϕ: H/K → G/K define the truncation functor (−)_{H/K} = T (θ, ϕ): (G/K, R)-gr → (H/K, R_{H})-gr. This is a right adjoint of the induction functor R ⊗_{R_{H}} −: (H/K, R_{H})-gr → (G/K, R)-gr. We denote by (−)_{σH} the composite functor (−)_{^{σ}H/^{σ}K} ∘ S_{G/K}^{σ}: (G/K, R)-gr → (^{σ}H/^{σ}K, R_{^{σ}H})-gr, which is a right adjoint of (S_{G/K}^{σ})^{−1} ∘ (R ⊗_{R_{H}} −).

Using the suspension and the truncation, we may define G/K -graded groups of homomorphisms.

2.9. Denote by U_{σ} the functor U(θ, ϕ): R-gr → (G/^{σ}K)-gr defined by θ = id_{G} and ϕ: G → G/^{σ}K. Let M ∈ R-gr and N ∈ (G/K, R)-gr, so N(σ) ∈ (G/^{σ}K, R)-gr. Let HOM_{G/K,R}(M, N)_{σK} = Hom_{(G/^{σ}K,R)-gr}(U_{σ}(M), N(σ)) ⊆ Hom_{R}(M, N). If f ∈ HOM_{G/K,R}(M, N)_{σK} ⋂ HOM_{G/K,R}(M, N)_{σ′K}, then for every τ ∈ G, we have that f(M_{τ}) ⊆ N(σ)_{τ^{σ}K} = N_{τσK} on the one hand, and f(M_{τ}) ⊆ N_{τσ′K} on the other hand. It follows that σK = σ′K or f = 0. We may therefore define HOM_{G/K,R}(M, N) = ⨁_{σK∈G/K} HOM_{G/K,R}(M, N)_{σK} ⊆ Hom_{R}(M, N). We have that HOM_{G/K,R}(M, N) is a G/K -graded left END_{R}(M)^{op} -module. Indeed, let f ∈ HOM_{G/K,R}(M, N)_{σK} and α ∈ E_{ρ}. Then (αf)(M_{τ}) = f(α(M_{τ})) ⊆ f(M_{τρ)} ⊆ N_{τρσK}, hence αf ∈ HOM_{G/K,R}(M, N)_{ρσK}.

In order to have HOM_{G/K,R}(M, N) = Hom_{R}(M, N), we may find conditions similar to those of [3], Section 3. For convenience, we give here the whole proof.

2.10 Proposition. The subgroup HOM_{G/K,R}(M, N) consists of all R-homomorphisms f: M → N for which there is a finite set F ⊆ G/K such that for every τ ∈ G, f(M_{τ}) ⊆ ∑_{σK∈F} N(σ)_{τ^{σ}K} = ∑_{σK∈F} N_{τσK}.

Proof. If f ∈ HOM_{G/K,R}(M, N), then f(M_{τ}) = ∑_{σK∈F} f_{σK}(M_{τ}) ⊆ ∑_{σK∈F} N_{τσK}, hence f satisfies the condition.

Assume now that f ∈ Hom_{R}(M, N) satisfies the condition for a finite set F ⊆ G/K. If τ ∈ G, then N = ⨁_{σK∈G/K} N_{τσK}. It follows that there are unique f_{τ,τσK} ∈ Hom_{R_{1}}(M_{τ} , N_{τσK}) such that f(m_{τ}) = ∑_{σK∈G/K} f_{τ,τσK}(m_{τ}) for every τ ∈ G and m_{τ} ∈ M_{τ} . (All the sums considered here have a finite number of nonzero terms.) We have that f_{τ,τσK}(m_{τ}) = f(m_{τ})_{τσK} ∈ N_{τσK}. For σK ∈ G/K, let f_{σK} ∈ Hom_{R_{1}}(M, N), f_{σK}(∑_{τ∈G} m_{τ}) = ∑_{τ∈G} f_{τ,τσK}(m_{τ}). Clearly, f_{σK} is well-defined, and f_{σK}(M_{τ}) = f_{τ,τσK}(M_{τ}) ⊆ N_{τσK}. If σK ∉ F, then f_{τ,τσK} = 0 for any τ ∈ G, and hence f_{σK} = 0.

Consequently, f = ∑_{σK∈F} f_{σK} in Hom_{R_{1}}(M, N). We need only to prove that each f_{σK} is R-linear, that is, f_{σK}(r_{ρ}m_{τ}) = r_{ρ}f_{σK}(m_{τ}) for every r_{ρ} ∈ R_{ρ} and m_{τ} ∈ M_{τ} . We have that r_{ρ}f(m_{τ}) = ∑_{σK∈F} r_{ρ}f_{σK}(m_{τ}) = f(r_{ρ}m_{τ}) = ∑_{σK∈F} f_{σK}(r_{ρ}m_{τ}) and both r_{ρ}f_{σK}(m_{τ}) and f_{σK}(r_{ρ}m_{τ}) belong to N_{ρτσK}. Since N = ⨁_{σK∈G/K} N_{ρτσK}, it follows that r_{ρ}f_{σK}(m_{τ}) = f_{σK}(r_{ρ}m_{τ}).

2.11. Corollary. HOM_{G/K,R}(M, N) = Hom_{R}(M, N) if either of the following condition is satisfied:

a) G/K is a finite set.

b) M is a finitely generated R-module.

c) Supp(M) = {g ∈ G | M_{g} ≠ 0} and Supp(N) = {x ∈ G/K | N_{x} ≠ 0} are finite sets.

Proof. a) is obvious by the Proposition.

b) Let {m_{i} | i = 1, …, t} be a set of generators of M. Since each m_{i} is a finite sum of homogenous elements, we may assume that m_{i} is homogenous of degree σ_{i}. We have that M = ∑_{i=1}^{t} ∑_{ρ∈G} R_{ρ}m_{i}, where R_{ρ}m_{i} ⊆ M_{ρσ_{i}} and f(m_{i}) ⊆ ∑_{σK⊆F_{i}} N_{σ_{i}σK}, where F_{i} ⊆ G/K is finite. Let F = ⋃_{i=1}^{t} F_{i}. It follows that M_{τ} = ∑_{i=1}^{t} R_{τσ_{i}^{−1}}m_{i} and f(M_{τ}) = ∑_{i=1}^{t} R_{τσ_{i}^{−1}}f(m_{i}) ⊆ ∑_{i=1}^{t} ∑_{σK∈F_{i}} R_{τσ_{i}^{−1}}N_{σ_{i}σK} ⊆ ∑_{i=1}^{t} ∑_{σK∈F_{i}} N_{τσK} = ∑_{σK∈F} N_{τσK}.

c) Let Supp(M) = {τ_{1}, …, τ_{s}}, Supp(N) = {σ_{1}K, …, σ_{t}K} and F = {τ_{i}^{−1}σ_{j}K | i = 1, …, s, j = 1, …, t}. If f ∈ Hom_{R}(M, N) and τ ∈ G, then for τ ∉ Supp(M), f(M_{τ}) = 0. If τ ∈ Supp(M), then f(M_{τ}) ⊆ N = ∑_{σK∈F} N_{τσK}.

We are now able to construct the coinduction functor Coind(−): (H/K, R_{H})-gr → (G/K, R)-gr. If M ∈ (H/K, R_{H})-gr, then Coind(M) is a subset of Hom_{R_{H}} (R, M), where the multiplication is given by af(r) = f(ra) for all a, r ∈ R. Let ρ ∈ H and σ ∈ G. Then Coind_{σρK} is a subset of {f ∈ Hom_{R_{H}} (R, M) | f(R_{Hτ^{−1}}) = 0 for σH ≠ τH} = Hom_{R_{H}} (R_{Hσ^{−1}}, M). We have further that R_{Hσ^{−1}} is an H - graded R_{H} -module in a natural way, and consider the ungrading functor U_{ρ}^{H}: R_{H}-gr → (H/^{ρ}K, R_{H})-gr. By definition, Coind(M)_{σρK} = Hom_{(H/^{ρ}K,R_{H})-gr}(U_{ρ}^{H}(R_{Hσ^{−1}}), M(ρ)) and Coind(M) = ⨁_{x∈G/H,y∈H/K} Coind(M)_{σ_{x}ρ_{y}K}, where {σ_{x} | x ∈ G/H} and {ρ_{y} | y ∈ H/K} are transversals for the left cosets. Let a_{g} ∈ R_{g} and f_{σρK} ∈ Coind(M)_{σρK}. If r ∉ R_{Hσ^{−1}ρ^{−1}}, then ra_{g} ∉ R_{Hσ^{−1}} and a_{g}f_{σρK}(r) = f_{σρK}(ra_{g}) = 0. If r ∈ R_{hρKρ^{−1}σ^{−1}g^{−1}} then a_{g}f_{σρK}(r) ∈ M_{hρK}, hence a_{g}f_{σρK} ∈ Coind(M)_{gσρK}. We have used that f_{σρK}(R_{hρKρ^{−1}σ^{−1}}) ⊆ M(ρ)_{hρKρ^{−1}} = M_{hρK} for all h ∈ H. It follows that Coind(M) ∈ (G/K, R)-gr. If ψ: M → M′ is a morphism in (H/K, R)-gr, then Coind(ψ)(f_{σρK}) = ψ ∘ f_{σρK}, so the functor Coind(−) is defined.

2.12. Proposition. a) Coind(−) is a right adjoint of (−)_{H/K}.

b) (−)_{H/K} ∘ Coind(−) ≃ id_{(H/K,R_{H})-gr}.

Proof. a) Let M ∈ (H/K, R_{H})-gr and N ∈ (G/K, R)-gr. We define the functorial morphisms

Hom_{(H/K,R_{H})-gr}((−)_{H/K}, −) ⇄_{δ}^{γ} Hom_{(G/K,R)-gr}(−, Coind(−))

as follows: if u belongs to Hom_{(H/K,R_{H})-gr}(N_{H/K}, M), then define γ_{N,M}(u)(n_{σρK})(r) = ∑_{h_{y}∈H/^{ρ}K} u(r_{h_{y}ρKρ^{−1}σ^{−1}}n_{σρK}) = u(r_{Hρ^{−1}σ^{−1}}n_{σρK}); if v ∈ Hom_{(G/K,R)-gr}(N, Coind(M)), then δ_{N,M}(v)(n_{ρK}) = v(n_{ρK})(1) ∈ M_{ρK}. Denote f_{σρK} = γ_{N,M}(u)(n_{σρK}). Then f_{σρK}(r) = 0 if r ∉ R_{Hσ^{−1}} and f_{σρK}(r) ∈ N_{hρK} if r ∈ R_{hρKρ^{−1}σ^{−1}}, hence f_{σρK} ∈ Coind(M)_{σρK} and γ_{N,M} is well-defined. Denote now f_{ρK} = v(n_{ρK}). Then f_{ρK} ∈ Coind(M)_{ρK}, hence f_{ρK}(r) = 0 if r ∉ H and f_{ρK}(r) ∈ M_{hρK} if r ∈ R_{hρKρ^{−1}}. It follows that f_{ρK}(1) = δ_{N,M}(v)(n_{ρK}) ∈ M_{ρK}, so δ_{N,M} is well defined too.

Further, we have that δ_{N,M}(γ_{N,M}(u))(n_{ρK}) = γ_{N,M}(u)(n_{ρK})(1) = u(n_{ρK}), that is δ ∘ γ = id; we also have that γ_{N,M}(δ_{N,M}(v))(n_{σρK})(r) = δ_{N,M}(v)(r_{Hρ^{−1}σ^{−1}}n_{σρK}) = v(n_{σρK})(r), since v(n_{σρK}) = f_{σρK} ∈ Coind(M)_{σρK} and f_{σρK}(r) = 0 if r ∉ R_{Hσ^{−1}}. It follows that γ ∘ δ = id.

b) Let η: (−)_{H/K} ∘ Coind(−) → id_{(H/K,R_{H})-gr} be the functorial morphism given by η_{M}(f_{ρK}) = f_{ρK}(1). It is easy to see that η_{M} is an isomorphism in (H/K, R_{H})-gr.

2.13. Proposition. Let ν: id_{(G/K,R)}-gr → Coind(−) ∘ (−)_{H/K} be the functorial morphism given by ν_{N}(n_{σρK})(r) = r_{Hρ^{−1}σ^{−1}}n_{σρK}. Then Im(ν_{N}) is essential in Coind(N_{H/K}).

Proof. Let 0 ≠ f ∈ Coind(N_{H/K})_{σρK}. There is 0 ≠ a_{ρkρ^{−1}σ^{−1}} such that f(a_{ρkρ^{−1}σ^{−1}}) = n_{ρK} ≠ 0. Let b ∈ R. We have that ν_{N}(n_{ρK})(b) = b_{Hρ^{−1}}n_{ρK} = b_{Hρ^{−1}}f(a_{ρkρ^{−1}σ^{−1}}) = f(b_{Hρ^{−1}}a_{ρkρ^{−1}σ^{−1}}). On the other hand, we have (a_{ρkρ^{−1}σ^{−1}}f)(b) = f(ba_{ρkρ^{−1}σ^{−1}}) = f(b_{Hρ^{−1}}a_{ρkρ^{−1}σ^{−1}}) since f(b_{τ} a_{ρkρ^{−1}σ^{−1}}) = 0 for τ ∉ Hρ^{−1}. It follows that a_{ρkρ^{−1}σ^{−1}}f = ν_{N}(n_{ρK}) ∈Imν_{N}, and this is nonzero since ν_{N}(n_{ρK})(1) = n_{ρK} ≠ 0.

Using the truncation, induction and coinduction functors, we shall establish correspondences between the simple objects of the categories (G/K, R)-gr and (H/K, R_{H})-gr. We need first some notations.

Let C_{H/K} be the full subcategory of (G/K, R)-gr consisting of objects N for which N_{H/K} = 0. Since (−)_{H/K} is an exact functor, C_{H/K} is a localizing subcategory of (G/K, R)-gr and is closed under direct products. It follows that C_{H/K} is a hereditary torsion class, and the corresponding torsion free class is F_{H/K} = {N ∈ (G/K, R)-gr | Hom_{(G/K,R)-gr}(X, N) = 0 (∀)X ∈ C_{H/K}}. Denote by t_{C_{H/K}} the idempotent radical associated to C_{H/K}, so t_{C_{H/K}} (N) is the largest subobject of N ∈ (G/K, R)-gr belonging to C_{H/K} and F_{H/K} = {N ∈ (G/K, R)-gr | t_{C_{H/K}} (N) = 0}.

Remark that Hom_{(G/K,R)-gr}(U(R), N) = N_{H/K}, so C_{H/K} = {N ∈ (G/K, R)-gr | Hom_{(G/K,R)-gr}(R, N) = 0}. Consequently, C_{H/K} is the torsion free class generated by U(R) and C_{H/K} is a TTF-class. Let T_{H/K} = {N ∈ (G/K, R)-gr | Hom_{(G/K,R)-gr}(N, X) = 0 (∀)X ∈ C_{H/K}} be the corresponding torsion class and let t_{T_{H/K}} be the idempotent radical associated to T_{H/K}. We have that t_{T_{H/K}} (N) = RN_{H/K} and T_{H/K} = {N ∈ (G/K, R)-gr | RN_{H/K} = N} is the class of H/K -generated objects. Clearly, C_{H/K} = {N ∈ (G/K, R)-gr | RN_{H/K} = 0}. In other words, N/RN_{H/K} ∈ C_{H/K} and RN_{H/K} is the smallest subobject L of N such that N/L ∈ C_{H/K}.

If N ∈ (G/K, R)-gr is a simple object and N_{H/K} ≠ 0 then clearly N ∈ F_{H/K} ∩ T_{H/K}. We shall say in this case that N belongs to (G/K, R)-sim_{H/K}.

Recall that for N ∈ (G/K, R)-gr, there are the natural G/K -graded homomorphisms μ_{N}: R ⊗_{R_{H}} N_{H/K} → N, μ_{N}(r_{σ} ⊗ n_{ρK}) = r_{σ}n_{ρK}, and ν_{N}: N → Coind(N_{H/K}), ν_{N}(n_{σρK})(r) = r_{Hρ^{−1}σ^{−1}}n_{σρK}, for σ ∈ G, ρ ∈ H.

2.14. Proposition. a) Im μ_{N} = t_{T_{H/K}} (N) = RN_{H/K}.

b) Ker μ_{N} and Coker μ_{N} belong to C_{H/K}.

c) μ is a functorial isomorphism if and only if C = {0}.

d) The categories (H/K, R_{H})-gr and (G/K, R)-gr/C_{H/K} are equivalent via the composite functors

\xymatrix{(H/K, R_{H})-gr \ar[r]^{R⊗_{R_{H}} −} & (G/K, R)-gr \ar[l]^{(−)_{H/K}} \ar[r]^{F_{H/K}} & (G/K, R)-gr/C_{H/K} \ar[l]^{S_{H/K}}}

where F_{H/K} and S_{H/K} are the canonical functors.

Proof. a) is obvious.

b) Let L = Ker(μ_{N}). Then L_{H/K} = Ker(μ_{N})_{H/K}: R_{H/K} ⊗_{R_{H}} N_{H/K} → N_{H/K} is an isomorphism, hence L_{H/K} = 0. Further, Coker(μ_{N}) = N/ Im μ_{N} = N/RN_{H/K} ∈ C_{H/K}.

c) Assume that μ is an isomorphism and let N ∈ C_{H/K}. We have that R ⊗_{R_{H}} N_{H/K} and N are isomorphic via μ_{N}, so N = 0. Conversely, assume that C_{H/K} = {0} and let N ∈ (G/K, R)-gr. Then Ker μ_{N} = Coker μ_{N} = 0 by b), so μ_{N} is an isomorphism.

d) is a consequence of b). The details are as in [13], Theorem 3.1.

2.15. Proposition. a) Ker ν_{N} = t_{C_{H/K}} (N) and Coker ν_{N} belong to C_{H/K}.

b) ν_{N} is a monomorphism if and only if for any 0 ≠ n_{σρK} ∈ N_{σρK}, R_{Hρ^{−1}σ^{−1}}n_{σρK} ≠ 0. In other words, N ∈ F_{H/K} if and only if N is H/K -faithful.

c) N belongs to F_{H/K} ∩ T_{H/K} if and only if ν_{N} induces an isomorphism between N and the submodule of Coind(N_{H/K}) generated by Coind(N_{H/K})_{H/K} ≃ N_{H/K}.

Proof. a) Let P = Ker ν_{N} and p_{ρK} ∈ P_{ρK}. We have that p_{ρK} = ν_{N}(p_{ρK})(1) = 0 for any ρ ∈ H, hence P ⊆ t_{C_{H/K}} (N). Let now n_{σρK} ∈ t_{C_{H/K}} (N)_{σρK}. We have that (Rn_{σρK})_{H/K} = 0, so R_{Hρ^{−1}σ^{−1}}n_{σρK} = 0. We obtain that ν_{N}(n_{σρK})(r) = r_{Hρ^{−1}σ^{−1}}n_{σρK} = 0 for any r ∈ R, hence n_{σρK} ∈ Ker ν_{N}.

Let f ∈ Coind(N_{H/K})_{H/K}, n_{H/K} = f(1). Then ν_{N}(n_{H/K}) = f, so Coker(ν_{N})_{H/K} = 0.

b) follows immediately from a) and the definition of ν.

c) If N ∈ F_{H/K}, then ν_{N} is a monomorphism. If N ∈ T_{H/K}, then RN_{H/K} = N, and therefore Imν_{N} is the subobject of Coind(N_{H/K}) generated by Coind(N_{H/K})_{H/K}. The converse is obvious.

We define now the functor X : (H/K, R_{H})-gr → (G/K, R)-gr by X (M) = R ⊗_{R_{H}} M/t_{C_{H/K}} (R ⊗_{R_{H}} M) ≃ Im(ν_{R⊗M}) ⊆ Coind(M).

2.16. Theorem. a) X (M) ≃ t_{T_{H/K}} (Coind(M)_{H/K}) = R Coind(M)_{H/K}.

b) X and (−)_{H/K} give an equivalence between the categories (H/K, R_{H})-gr and F_{H/K}∩ T_{H/K}.

Proof. Consider the commutative diagram

R ⊗_{R_{H}} M  ⟶^{ν_{R⊗M}} Coind(M)
↓  ↑
R ⊗_{R_{H}} M/ Ker ν_{R⊗M} ⟶ Im ν_{R⊗M}

Obviously, R ⊗_{R_{H}} M and R ⊗_{R_{H}} M/ Ker ν are R_{H} -generated (belong to T_{H/K}) and the composition

R ⊗_{R_{H}} M/ Ker ν ⟶ Im ν_{R⊗M} ⟶ Coind(M)

is the homomorphism ν_{R⊗M/ Ker ν}. Both statements follow from Proposition 2.15.

2.17. Lemma. a) If L is an essential subobject of M ∈ (H/K, R_{H})-gr, then X (L) is an essential subobject of Coind(M) and of X (M) ∈ (G/K, R)-gr.

b) If M ∈ (H/K, R_{H})-grsim, then X (M) is a simple object and is the socle of Coind(M) in (G/K, R)-gr.

Proof. a) Let 0 ≠ f_{ρK} ∈ Coind(M)_{H/K}. We have that R_{Hρ^{−1}}f_{ρK} ≠ 0 since ν: Coind(M) → Coind(Coind(M)_{H/K}) ≃ Coind(M) is an isomorphism, so Coind(M) is C_{H/K} -torsion free. It follows that X (L)_{H/K} ∩ R_{Hρ^{−1}}f_{ρK} ≠ 0.

b) Let again 0 ≠ f_{ρK} ∈ Coind(M)_{H/K}. Then 0 ≠ R_{Hρ^{−1}}f_{ρK} ⊆ X (M)_{H/K} ≃ M, hence R_{Hρ^{−1}}f_{ρK} = X (M)_{H/K} and therefore X (M) = Rf_{ρK}. If L is a nonzero subobject of X (M), then L_{H/K} ≠ 0, so there is 0 ≠ f_{ρK} ∈ L_{H/K}. Since X (M) is essential in Coind(M) and X (M) is a simple object, it follows that X (M) is the socle of Coind(M).

2.18. Corollary. Let (G/K, R)-grsim_{H/K} be the category of the simple objects N of (G/K, R)-gr, for which N_{H/K} ≠ 0. Then this category is equivalent, via the functors X and (−)_{H/K}, with the category (H/K, R_{H})-grsim.

Combining (2.7), (2.16) and (2.18), we obtain:

2.19. Corollary. a) The functors S_{G/^{σ}K}^{σ^{−1}} ∘ X_{H/K} = X_{σH/K} and (−)_{σH/K} give an equivalence between (^{σ}H/^{σ}K, R_{^{σ}H})-gr and F_{σH/K} ∩ T_{σH/K}, where F_{σH/K} and T_{σH/K} are the classes corresponding to the subcategory C_{σH/K} = {N ∈ (G/K, R)-gr | N_{σH/K} = 0}.

b) X_{σH/K} induces an equivalence between the categories (^{σ}H/^{σ}K, R_{^{σ}H})-grsim and (G/^{σ}K, R)-grsim_{σH/K}.

c) Let N ∈ (G/K, R)-grsim. There is a nonzero component N_{σH/K}. Then N_{σH/K} ∈ (^{σ}H/^{σ}K, R_{^{σ}H})-grsim and N ≃ X_{σH/K}(N_{σH/K}) = R ⊗_{R_{^{σ}H}} N_{σH/K}/t_{C_{σH/K}} (R ⊗_{R_{^{σ}H/^{σ}K}} N_{σH/K}).

3. Equivalences of categories

We consider the abelian categories A and B and the additive covariant functors F: A → B and G: B → A such that F is a left adjoint of G. It follows that F is right exact and commutes with direct sums. For any M ∈ A and N ∈ B, we have the natural isomorphisms

Hom_{A}(M, G(N)) ⇄_{β_{M,N}}^{α_{M,N}} Hom_{B}(F(M), N),

which determine the functorial isomorphisms ξ: F ∘ G → id_{B} and ζ: id_{A} → G ∘ F.

For M ∈ A we define the following full subcategories of A (see also [8] and [9] for the first two definitions):

3.1.a. σ[M] consisting of objects subgenerated by M. 3.1.b. CD[M] consisting of objects U for which there is an exact sequence M^{(J)} → M^{(I)} → U → 0 in A. 3.1.c. Add[M] consisting of objects U which divide a direct sum of copies of M. 3.1.d. add[M] consisting of objects U which divide a finite direct sum of copies of M.

Analogously, for N ∈ B we have the full subcategories σ[N], CD[N], Add[N] and add[N] of B.

We fix an object N of B and let M = G(N) ∈ A. Consider the following statements:

3.2.i. G(N^{(I)}) is naturally isomorphic to M^{(I)} for any set I. 3.2.ii. The restriction of G to σ[N] is right exact. 3.2.iii. ξ_{N}: F(G(N)) → N is an isomorphism.

3.2.a. F and G induce an equivalence between add[M] and add[N]. 3.2.a. F and G induce an equivalence between Add[M] and Add[N]. 3.2.a. F and G induce an equivalence between CD[M] and CD[N].

The following obvious lemma will be the general pattern for our next results:

3.3. Lemma. (i) Statements (3.2.iii) and (3.2.a) are equivalent.

(ii) If (3.2.i) holds, then (3.2.iii) and (3.2.b) are equivalent.

(iii) If (3.2.i) and (3.2.ii) hold, then (3.2.iii) and (3.2.c) are equivalent.

We shall use first this lemma to generalize Dade’s theorem which asserts that R_{1} - mod is equivalent with R-gr via R ⊗_{R_{1}} − and (−)_{1} if and only if 1 ∈ R_{g}R_{g^{−1}} for any g ∈ G. Consider the functors I = R ⊗_{R_{H}} −: (H/K, R_{H})-gr → (G/K, R)-gr and T = (−)_{H/K}: (G/K, R)-gr → (H/K, R_{H})-gr. A projective generator of (G/K, R)-gr is N = ⨁_{g∈G} R(g^{−1}) and a projective generator of (H/K, R_{H})-gr is M = ⨁_{h∈H} R_{H}(h^{−1}).

3.4. Proposition. The following statements are equivalent:

(i) The functors R ⊗_{R_{H}} − and (−)_{H/K} are equivalences of categories.

(ii) R ⊗_{R_{H}} M ≃ ⨁_{h∈H} R(h^{−1}) is a generator of (G/K, R)-gr.

(iii) R ⊗_{R_{H}} R(g^{−1})_{H/K} ≃ R(g^{−1}) in (G/K, R)-gr for any g ∈ G.

(iv) The functors R ⊗_{R_{H}} −: (H/I, R_{H})-gr → (G/I, R)-gr and (−)_{H/I}: (G/I, R)-gr → (H/I, R_{H})-gr are equivalences, for any subgroup I of H.

(v) 1 ∈ R_{gH}R_{Hg^{−1}} for any g ∈ G.

Proof. By Lemma 3.3, the functors I and T give an equivalence between (H/K, R_{H})-gr and CD[R ⊗_{R_{H}} M], hence (i) and (ii) are equivalent. Statement (iii) holds if and only if I(T (N)) ≃ N. Since T (N) is a generator of (H/K, R_{H})-gr, again Lemma 3.3 implies that (i) and (iii) are equivalent. Statements (i) and (iv) are also equivalent, since μ_{R(g^{−1})} : R⊗_{R_{H}} R(g^{−1})_{H} → R(g^{−1}) is a morphism in (G/I, R)-gr for any subgroup I of H.

Assume that (v) holds and let g ∈ G. It follows that there are a finite set J, the elements h_{j} ∈ H, a_{j} ∈ R_{gh_{j}}, b_{j} ∈ R_{h_{j}^{−1}g_{j}^{−1}}, j ∈ J, such that ∑_{j∈J} a_{j}b_{j} = 1. Define f: (⨁_{h∈H} R(h^{−1}))^{(J)} → R(g^{−1}) by f(1^{(j)})_{h_{j}} = b_{j} if h = h_{j} and 0 if h ≠ h_{j}, where 1_{h}^{(j)} ∈ R(h^{−1})^{(j)} ⊆ (⨁_{h∈H} R(h^{−1}))^{(J)}. Clearly, f is an epimorphism in (G/K, R)-gr. Since ⨁_{g∈G} R(g^{−1}) is a generator of this category, it follows that (ii) holds.

Assume finally that (iii) holds and let g ∈ G. It follows that there exist a finite set J, and elements h_{j} ∈ H, a_{j} ∈ R_{gh_{j}}, b_{j} ∈ R_{h_{j}^{−1}g^{−1}}, j ∈ J, such that μ_{R(g^{−1})}(∑_{j∈J} a_{j}b_{j}) = ∑_{j∈J} a_{j}b_{j} = 1, hence 1 ∈ R_{gH}R_{Hg^{−1}} and (v) holds.

The next applications of Lemma 3.3 will be in the context of Hom and tensor functors. As remarked in [12] and [1], some results on stable Clifford theory [3] hold under very general hypotheses. We develop here some of these ideas, indicating how they can be used to generalize the main result of [23], which is a stable Clifford theorem which does not assume any finiteness conditions.

Let then A be a ring, A an abelian subcategory of A-mod, M an object of A and D = End_{A}(M)^{op}. It follows that M is an A, D-bimodule. Consider the functors H = Hom_{A}(M, −): A → D-mod and G = M ⊗_{D} −: D-mod → A. Then H is a right adjoint of G and we have the natural homomorphisms ξ_{N}: M ⊗_{D} Hom_{A}(M, N) → N, ξ_{N}(m ⊗ f) = mf = f(m) and ζ_{U}: U → H(G(U)), ζ_{U}(u)(m) = m⊗_{D} u, where N ∈ A and U ∈ D-mod.

3.5. Definition. a) N ∈ A is called M -static if ξ_{N} is an isomorphism.

b) U ∈ D-mod is called M -static if ζ_{U} is an isomorphism.

Denote by A_{M} the full subcategory of A consisting of M -static objects and by (DsM)-mod the full subcategory of D-mod consisting of M -static D-modules.

The argument of [1], Lemma 2, can be used to obtain the following:

3.6. Proposition. a) If Hom_{A}(M, −) commutes with direct sums (in A) of copies of M, then N ∈ A is M -static if and only if there is an exact sequence M^{(J)} → M^{(I)} → N → 0 in A such that the sequence H(M^{(J)}) → H(M^{(I)}) → H(N) → 0 is exact in D-mod.

b) The restrictions of H and G give an equivalence between A_{M} and (DsM)-mod.

Remark that M is an M -static object of A, D is an M -static left D-module, and A_{M} and (DsM)-mod are closed under direct summands and finite direct sums. If H commutes with direct sums of copies of M, then A_{M} and (DsM)-mod are closed under arbitrary direct sums.

We shall regard add[M], Add[M], CD[M] and σ[M] as full subcategories of A. Then Lemma 3.3 immediately gives:

3.7. Proposition. a) add[M] is equivalent to the category of finitely generated projective D-modules.

b) If H commutes with direct sums of copies of M, then Add[M] is equivalent to the category of projective D-modules.

c) If H commutes with direct sums of copies of M and M is projective in σ[M], then CD[M] is equivalent to D-mod.

Assume further that A = A-mod and let α: A → B be a unital homomorphism of rings. Denote N = B ⊗_{A} M, E = End_{B}(N)^{op}, hence there is a unital homomorphism D → E of rings. By [12], Lemma 4.5, we have:

3.8. If N is M -static as an A-module, then B ⊗_{A} M and M ⊗_{D} E are isomorphic A, D-bimodules.

It follows that the functor Hom_{B}(N, −): B-mod → E-mod is naturally equivalent with Hom_{A}(M, −) and the functor N ⊗_{E} −: E-mod → B-mod is naturally equivalent with M ⊗_{D} −. Using these facts and Theorem 1.3 of García Hernández and Gómez Pardo [8], one easily obtains the following generalization of [23], Theorem 2.5.

3.9. Proposition. Assume that (3.8) holds and that M is projective in σ[M]. Let J be the two-sided ideal of D consisting of endomorphisms which factors through a finitely generated submodule of M, and let F = {I ≤_{D} D | J ⊆ I} = {I ≤_{D} D | MI = M}. Then J is an idempotent ideal, F is a left Gabriel topology on D, and the functors H and G give an equivalence between CD[N] and the category (E|F)-mod consisting of E -modules which are F -closed as D-modules.

We return to graded modules. Let again R be a G-graded ring and assume that M ∈ R-gr is finitely generated in R-mod. The correspondence given by the functors Hom_{R}(M, −): R-mod → E-mod and M ⊗_{E} −: E-mod → R-mod, where E = End_{R}(M)^{op}, will be called “Clifford Correspondence”. We shall establish some properties of this correspondence, beeing concerned especially with the preservation of G/H -gradings. We fix the subgroups K ≤ H of G.

Consider the functors U: R-gr → R-mod and F: R-mod → R-gr of the first section and denote by M̃ the G-graded R-module F(U(M)), so M̃ ≃ ⨁_{g∈G} M(g^{−1}) in R-gr, by [13], Lemma 3.1. Using the ungrading functor, M̃ may be viewed as an object of (G/K, R)-gr. In this way we consider the full subcategories C_{1}^{G/K} = Add[M̃] and C_{2}^{G/K} = CD[M̃].

We have that E is a G-graded ring, with g -component E_{g} = Hom_{R-gr}(M, M(g)) = Hom_{R-gr}(M(g^{−1}), M), and denote E_{K} = ⨁_{g∈K} E_{g}. It is well-known that the G/K -graded E -module Ẽ = ⨁_{g∈G} E(g^{−1}) is a projective generator of (G/K, E)-gr. It follows that the category D_{1}^{G/K} = Add[Ẽ] consists of projective G/K -graded E -modules, and D_{2}^{G/K} = CD[Ẽ] = (G/K, E)-gr.

We shall also consider the the full category σ^{G/K}[M] of (G/K, R)-gr consisting of objects subgenerated by M̃, and let finally E_{1}^{G/K} be the category of projective E_{K} - modules, and E_{2}^{G/K} = E_{K}-mod.

3.10. Lemma. If M̃ is projective in σ^{G/H}[M], then M̃ is projective in σ^{G/K}[M].

Proof. Let M′ and M′′ be objects of σ^{G/K}[M]. A diagram of the form

M̃
↓
M′ ⟶ M′′ ⟶ 0

in σ^{G/K}[M], may be viewed as a diagram in σ^{G/H}[M]. If there is a morphism M̃ → M′ in (G/H, R)-gr such that the diagram is commutative, then as in Proposition 2.4, there is a morphism M̃ → M′ in (G/K, R)-gr such that the diagram is commutative.

This lemma and [9], Corollary 2.5.(i) yield:

3.11. Proposition. The following statements are equivalent:

(i) M is projective in σ[M] = σ^{G/G}[M] ⊆ R-mod.

(ii) M̃ is projective in σ^{G/K}[M].

(iii) M is projective in σ^{G/1}[M] = σ^{gr}[M] ⊆ R-gr.

Assume now that N ∈ (G/K, R)-gr. Then by (2.9), (2.10) and (2.11), Hom_{R}(M, N) = HOM_{G/K,R}(M, N) ∈ (G/K, E)-gr. Also, if U ∈ (G/K, E)-gr, then M ⊗_{E} U ∈ (G/K, R)-gr, where the G/K -grading of M ⊗_{E}U is given by M_{g}⊗_{E}U_{g′K} ⊆ (M ⊗_{E}U)_{gg′K}.

We may now conclude all our observations. In what follows, i ∈ {1, 2} and if i = 2, we assume that M is projective in σ[M].

3.12. Theorem. a) The functors Hom_{R}(M, −) and M ⊗_{E} − induce an equivalence between the categories C_{i}^{G/K} and D_{i}^{G/K}.

b) Assume that M is G-invariant. Then the functors Hom_{(G/K,R)-gr}(M, −) and M⊗_{E_{K}} − induce an equivalence between the categories C_{i}^{G/K} and E_{i}^{G/K}, and the following diagram commutes to within natural equivalences of functors:

[figure: C_{i}^{G/K}; Hom_{R}(M,−); Hom_{(G/K,R)-gr}(M,−); D_{i}^{G/K}; M⊗_{E}−; (−)_{K}; E_{i}^{G/K}; M⊗_{E_{K}} −; E_{i}^{G/K}; E⊗_{E_{K}} −]

c) The equivalences of a) commute with the suspension and the ungrading functors. Moreover, if in addition M is G-invariant, then the diagram

[figure: C_{i}^{G/K}; H_{G/K}; U; E_{i}^{G/K}; M⊗_{E_{K}} −; E⊗_{E_{K}} −; E_{H}⊗_{E_{K}} −; D_{i}^{G/K}; (−)_{K}; D_{i}^{H/K}; (−)_{K}; U; E⊗_{E_{H}} −; D_{i}^{G/K}; (−)_{H/K}; U; C_{i}^{G/H}; H_{G/H}; E_{i}^{G/H}; M⊗_{E_{H}} −; E⊗_{E_{H}} −; D_{i}^{G/H}; (−)_{H}]

is commutative, where we have denoted by H_{G/K} the functor Hom_{(G/K,R)-gr}(M, −).

Proof. We have that Hom_{R}(M, M(g)) ≃ E(g) in E-gr and M ⊗_{E} E(g) ≃ M(g) in R-gr, for any g ∈ G. It follows that Hom_{R}(M, M̃) ≃ Ẽ in E-gr and M ⊗_{E} Ẽ ≃ M̃ in R-gr. Further, if N ∈ R-gr, then, since M is finitely generated, Hom_{(G/K,R)-gr}(M, N) = ⨁_{g∈K}(M, N(g)) = Hom_{R}(M, N)_{K}, so Hom_{(G/K,R)-gr}(M, M(g)) = E(g)_{K}. Also, if M is G-invariant, we have that E is a crossed-product of E_{1} with G, and M̃ divides in R-gr a direct sum of copies of M ⊗_{E_{K}} (⨁_{g∈K} E_{K}(g)) ≃ ⨁_{g∈K} M(g). It is easy to see that for N ∈ (G/K, R)-gr and U ∈ (G/K, E)-gr, the canonical morphisms M ⊗_{E} Hom_{R}(M, N) → N and U → Hom_{R}(M, M ⊗_{E} U) preserve G/K -gradings. Consequently, a) and b) follow at once from lemma 3.3. The rest of the statements are straightforward verifications.

Assume that R is strongly graded and let N, N′ ∈ R-mod. Then Hom_{R_{1}}(N, N′) has a G-module structure given by the Miyashita action (see [4], Section 2), such that Hom_{R_{H}} (N, N′) is the set of H -fixed elements. By Higman’s Criterion for strongly graded rings [4], N is relatively R_{H} -projective (divides in R-mod a G/H -graded R-module) if and only if N is H -regulate (there is f ∈ End_{R_{H}} (N) such that Tr_{H}^{G}(f) = id_{N}).

3.13. Corollary. Assume that R is strongly graded.

a) The Clifford Correspondence commutes with induction from subgroups, that is, the following diagram is commutative:

\xymatrix{C_{1}^{G/K} \ar[r]^{Hom_{R}(M,−)} & D_{i}^{G/K} \ar[l]^{M⊗_{E}−} \\ C_{i}^{H/K} \ar[u]^{R⊗_{R_{H}} −} \ar[r]^{Hom_{R_{H}} (M_{H},−)} & D_{i}^{H/K} \ar[u]_{E⊗_{E_{H}} −} \ar[l]^{M_{H}⊗_{E_{H}} −}}

b) If in addition M is G-invariant, then the Clifford Correspondence preserves relatively R_{H} -projective modules and vertices of R-modules. Moreover, if U, U′ ∈ E-mod, then the map Hom_{E_{1}}(U, U′) → Hom_{R_{1}}(M ⊗_{R} U, M ⊗_{R} U′) ≃ Hom_{R_{1}}(M_{1} ⊗_{R_{1}} U, M_{1} ⊗_{R_{1}} U′), f ↦ M_{1} ⊗_{R_{1}} f is an isomorphism of G-modules.

3.14. Remark. These results generalize (for finite G) the results of [15]. Take M = ⨁_{g∈G} R(g^{−1}), so M is finitely generated, projective and G-invariant. We have that Hom_{R-gr}(M, M(g)) = R{g} = {α ∈ M_{G}(R) | α(x, y) ∈ R_{xgy^{−1}}} = R{1}ḡ, where ḡ(x, y) = 1 if y = xg and ḡ(x, y) = 0 if y ≠ xg; R{1} is the smash-product of R and G, M_{G}(R) = R{G} = R{1}Ḡ is a skew group-ring of R and G, and R-gr is equivalent with R{1}-mod. Further, End_{(G/K,R)-gr}(M) = R{K} = ⨁_{g∈K} R{g} and for N ∈ (G/K, R)-gr, Hom_{(G/K,R)-gr}(M, N) = Col_{G,K}(N) = {α ∈ Col_{G}(N) | α(g) ∈ N_{gK} for any g ∈ G}. Moreover, if K ≤ H, Col_{G,H}(N) = R{H} ⊗_{R{K}} Col_{G,K}(N). If R is strongly graded, then ⨁_{g∈G} R(g^{−1}) is isomorphic in R-gr to R ⊗_{R_{1}} R = ⨁_{g∈G}(R_{g} ⊗_{R_{1}} R) (see [2], Section 5).

We show now that a graded version of Morita I follows from our results. For another treatment see [2], Theorem 3.2.

3.15. Corollary. Let R and S be G-graded rings, _{R}M_{S} and _{S}N_{R} G-graded bimodules and (R, S, M, N, α, β) a graded Morita context, that is α: M ⊗_{S}N → R and β: N ⊗_{R}M → S are homomorphisms of G-graded bimodules (α(M_{g} ⊗ N_{g′}) ⊆ R_{gg′} and β(N_{g} ⊗ N_{g′}) ⊆ S_{gg′}). Let α_{H}: M_{H} ⊗_{S_{H}} N_{H} → R_{H}, α_{H}(m ⊗_{S_{H}} n) = α(m ⊗_{S} n) and β_{H}: N_{H} ⊗_{R_{H}} M_{H} → S_{H}, β_{H}(n⊗_{R_{H}} m) = β(n⊗_{R} m). Assume that α and β are surjective. Then the assertions of Morita I hold, and in addition we have:

a) S ≃ End_{R}(M)^{op} ≃ End_{R}(N) and R ≃ End_{S}(N)^{op} ≃ End_{S}(M) as G-graded rings.

b) N ≃ Hom_{S}(M, S) ≃ Hom_{R}(M, R) and M ≃ Hom_{R}(N, R) ≃ Hom_{S}(M, S) as Ggraded bimodules.

c) If P ∈ (G/K, R)-gr, then the natural map ϕ_{P} : N ⊗_{R} P → Hom_{R}(M, P), ϕ_{P} (n ⊗ p)(m) = α(m ⊗ n)p is an isomorphism of G/H -graded S -modules.

d) For any K ≤ G, the categories (G/K, R)-gr and (G/K, S)-gr are equivalent via the functors N ⊗_{R} − and M ⊗_{S} −. These equivalences commute with the suspension and the ungrading functor.

e) Assume that R and S are strongly graded rings. Then α_{1} and β_{1} are surjective, (R_{K}, S_{K}, M_{K}, N_{K}, α_{K}, β_{K}) is a surjective Morita context for any K ≤ G, and the equivalences N_{K} ⊗_{R_{K}} − and M_{K} ⊗_{S_{K}} − commute with the induction and the truncation functors, preserve relatively projective modules and are compatible with the Miyashita action.

Proof. a) and b) are already verified in [2]. We know that ϕ_{P} is an isomorphism of S -modules, and let m_{g} ∈ M_{g}, n_{g′} ∈ N_{g′} and p_{hK} ∈ P_{hK}, where g, g′, h ∈ G. Then α(m_{g} ⊗ n_{g′})(p_{hK}) ∈ R_{gg′}P_{hK} ⊆ P_{gg′hK}, hence ϕ_{P} (n_{g′} ⊗ p_{hK}) ⊆ Hom_{R}(M, P)_{ghK}. It follows that ϕ_{P} is G/K -graded, so c) holds, and d) is a consequence of Theorem 3.12.

Assume now that R is strongly graded and let n_{g} ∈ N_{g}, m_{g^{−1}} ∈ M_{g^{−1}}. There are n_{1}^{i} ∈ N_{1} and r_{g}^{i} ∈ R_{g}, i = 1, …, a and m_{1}^{j} ∈ M_{1}, r_{g^{−1}}^{j} ∈ R_{g^{−1}}, j = 1, …, b such that n_{g} = ∑_{i=1}^{a} n_{1}^{i}r_{g}^{i} and m_{g^{−1}} = ∑_{j=1}^{b} r_{g^{−1}}^{j}m_{1}^{j}. Then β(n_{g} ⊗_{R} m_{g^{−1}}) = ∑_{i,j} β(n_{1}^{i}r_{g}^{i} ⊗_{R} r_{g^{−1}}^{j}m_{1}^{j}) = ∑_{i,j} β(n_{1}^{i}r_{g}^{i}r_{g−1}^{j} ⊗_{R} m_{1}^{j}) = ∑_{i,j} β_{1}(n_{1}^{i}r_{g}^{i}r_{g^{−1}}^{j} ⊗_{R_{1}} m_{1}^{j}), hence β_{1} is surjective. The rest of e) follows from this and from Theorem 3.12.

We will show that our results provide an alternative proof of [18], Theorem 2.1, avoiding the use of the smash product, but using the equivalences given by Theorem 3.12. We shall also obtain a variant of [18], Theorem 3.1, which may be considered as a Clifford Correpondence for arbitrary subgroups. If M ∈ R-gr, then, as in [10], we denote by G(M) the stabilizer with respect to suspension(conjugation) of M in G.

3.16. Theorem. Assume that R has finite support. Let N be a finite subgroup of G and N a simple object of (G/H, R)-gr.

a) There is M ∈ R-gr, gr-simple, uniquely determined up to a suspension such that N ∈ C_{2}^{G/H}.

b) There is a set I, a monomorphism α: N → (⨁_{h∈H} M(h^{−1}))^{(I)} and an epimorphism β: (⨁_{h∈H} M(h^{−1}))^{(I)} → N in (G/H, R)-gr such that β ∘ α = |H(M)|id_{N}.

c) Assume that N is |H(M)|-torsion free. Then N divides M in (G/H, R)-gr, CD[N] ⊆ CD[M], and there is an idempotent e_{H} ∈ E_{H} such that CD[N] is equivalent with e_{H}Ee_{H}-mod, where E = End_{R}(M)^{op}.

Proof. a) By Proposition 2.6, taking K = 1, it follows that there is M ∈ R-gr and a monomorphism N → M in (G/H, R)-gr, so clearly N ∈ C_{2}^{G/H}. The unicity follows from [9], Proposition 3.1.

b) Theorem 3.12 tells that the functors Hom_{R}(M, −) and M ⊗_{E} − give an equivalence between C_{2}^{G/H} and (G/H, E)-gr, which coincides with (G(M)/H(M), E_{G(M)})-gr, where H(M) = G(M) ∩ H and E = E_{G(M)}. Let U = Hom_{R}(M, N) ∈ (G(M)/H(M), E)-gr. This category is equivalent with E_{H(M)}-mod, so U ≃ E ⊗_{E_{H(M)}} V , where V = U_{H(M)}. By (2.1.b) we have the R_{H(M)} -homomorphisms η_{V} : V → U(F(V)) and ξ: U(F(V)) → V such that ξ_{V} ∘ η_{V} = |H(M)|id_{V} . But F(V) ∈ E_{H(M)}-gr, and E_{H(M)} is a H(M)-gr-simple left E_{H(M)} -module. It follows that there is a set I such F(V) divides W = (⨁_{h∈H(M)} E_{H(M)}(h^{−1}))^{(I)}. We have then the monomorphism α′: V → W and the epimorphism β′: W → V in E_{H(M)}-mod such β′ ∘ α′ = |H(M)|id_{V} . Using the equivalence E ⊗_{E_{H(M)}} −, we obtain the homomorphisms ᾱ: U → (⨁_{h∈H(M)} E(h^{−1}))^{(I)} and β̄: (⨁_{h∈H(M)} E(h^{−1}))^{(I)} → U in (G(M)/H(M), E)-gr such that β̄ ∘ ᾱ = |H(M)|id_{U}. Finally, using the equivalence M ⊗_{E} − we obtain the monomorphism α: N → (⨁_{h∈H(M)} M(h^{−1}))^{(I)} and the epimorphism β: (⨁_{h∈H(M)} M(h^{−1}))^{(I)} → N in (G/H, R)-gr, such that β ∘ α = |H(M)|id_{N}.

c) If N has no |H(M)|-torsion, then |H(M)|N ≠ 0 and we immediately have that |H(M)| is invertible in M and in E. The monomorphism N → U(M) gives a monomorphism U = Hom_{R}(M, N) → Hom_{R}(M, U(M)) = U(E) in (G/H, E)-gr, hence we have a monomorphism V → E_{H} in E_{H}-mod. Since E_{H} = E_{H(M)} is semisimple artinian, it follows that V divides E_{H}, U divides U(E) in (G/H, E)-gr, and N divides U(M) in (G/H, R)-gr. Further, since σ[M] = CD[M] is a localizing subcategory of R-mod, it follows that CD[N] ⊆ CD[M] and σ[N] ⊆ CD[M]. Moreover, N is projective in σ[M], hence it is projective in σ[N] also. It follows that CD[N] is equivalent with End_{R}(N)^{op}-mod. By [18], Theorem 3.2.2), we have an idempotent e_{H} ∈ E_{H} such that End_{R}(N)^{op} ≃ e_{H}Ee_{H}.

Finally we establish a similar result for gr-indecomposable modules, which were investigated in [10]. For this, we assume that G is finite, R is strongly graded such that R_{1} is left artinian. Denote D = E/J_{gr}(E), where J_{gr}(E) is the graded Jacobson radical of E, and let n = |H|.

3.17. Theorem. Assume that N ∈ (G/H, R)-gr is an indecomposable object, finitely generated in R-mod, such that n is invertible in N. Then there is M ∈ R-gr, grindecomposable and finitely generated, such N divides M in (G/H, R)-gr. Further, there is an idempotent d_{H} ∈ D_{H} such that the Grothendieck group of add[N] is isomorphic with the Grothendieck group of the category of finitely generated projective d_{H}Dd_{H} -modules.

Proof. By (2.1.b), we have the homomorphisms η_{N}: N → U(F(N)) and ξ_{N}: U(F(N)) → N in (G/H, R)-gr, such that ξ_{N} ∘ η_{N} = n.id_{N}, where F ∈ R-gr is a finitely generated R-module. Since N is invertible in N, and N is an indecomposable object, it follows that N divides in (G/H, R)-gr a G-graded gr-indecomposable R-module. We obtain that N ∈ add[M] and add[N] ⊆ add[M] (regarded as full subcategories of R-mod). We have that add[M] is equivalent with the category of finitely generated projective End_{R}(N)^{op} - modules. As above, there is an idempotent e_{H} ∈ E_{H} such that End_{R}(N)^{op} ≃ e_{H}Ee_{H}. Let d_{H} = e_{h} + J_{gr}(E) ∈ D. By [10], Lemma 3.1, the idempotent d_{H} may be identified with element e_{H} + J_{gr}(E_{H}) ∈ D_{H}. Let I be the kernel of the canonical ring epimorphism e_{H}Ee_{H} → d_{H}Dd_{H}. Since J_{gr}(E) ⊆ J(E), we may consider the epimorphism d_{H}Dd_{H} → d′_{H}D/J(D)d′_{H}, where d′_{H} = d_{H} + J(D), and D/J(D) ≃ E/J(E). The kernel of the composition of these two epimorphisms is e_{H}J(E)e_{H}J(e_{H}Ee_{H}), hence I ⊆ J(e_{H}Ee_{H}). The last statement follows as in [10], Proposition 3.3.a).

Notice that by [10], Lemma 3.2, D = D_{G(M)} is a crossed product of D_{1} with G(M) and as in [18], d_{H}Dd_{H} plays the role of a generalized Hecke algebra. Rings of this kind have been studied in [7].

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“Babes-Bolyai” University, Faculty of Mathematics and Computer Science, Str. M. Kogălniceanu nr. 1, 3400 Cluj-Napoca, Romania