STATIC MODULES AND CLIFFORD

THEORY FOR STRONGLY GRADED RINGS

Andrei Marcus

Abstract. We use the concept of static module to obtain the direct (non-stable) Clifford correspondence for strongly graded rings in the case of simple and of indecomposable modules. These correspondences are compatible with induction and the main results of Clifford theory are easily obtained from here.

1. Introduction

Let G be a finite group, R = ⨁_{x∈G} R_{x} a strongly G-graded ring, and consider a G-graded left R-module M = ⨁_{x∈G} M_{x} which is finitely generated in R-mod. By definition, the x-suspension M(x) of M is the G-graded R-module with M(x)_{y} = M_{yx}. Further, E = End_{R}(M)^{op} has a G-grading E = ⨁_{x∈G} E_{x} where E_{x} = Hom_{R-gr}(M, M(x)) ≃ Hom_{R_{1}}(M_{y}, M_{yx}) for every x, y ∈ G. In this way M becomes a G-graded R, E -bimodule.

By a well-known result of E.C. Dade [5], [6], if M is gr-simple, the category (RgM)-mod of R-modules generated by M is equivalent with E -mod, via the functors Hom_{R}(M, −) and M ⊗_{E} −. Our objective is to obtain a direct Clifford correspondence in the case when M is gr-indecomposable. Let J_{gr}(E) be the graded Jacobson radical of E and let D = E/J_{gr}(E). Using the notion of static module and a graded version of Fitting’s Lemma, we obtain the following theorem, which is the main result of the paper:

Theorem. Assume that M is gr-indecomposable of finite length in R-gr. Then the composite functor D ⊗_{E} Hom_{R}(M, −) induces an isomorphism between the Grothendieck groups associated to the category (R|M)-mod of R-modules which divide a finite direct sum of copies of M and to the category (D|D)-mod of finitely generated projective D-modules. This isomorphism is compatible with induction from subgroups, that is, for any subgroup H of G, the following diagram commutes to within natural equivalences of functors:

(R|M)-mod  ⟶^{D⊗_{E}Hom_{R}(M,−)}  (D|D)-mod
R⊗_{R_{H}} −↑  ↑^{D⊗_{D_{H}} −}
(R_{H}|M_{H})-mod ⟶^{D_{H}⊗_{E_{H}} Hom_{R_{H}} (M_{H},−)} (D_{H}|D_{H})-mod

Typeset by AMS-TEX

1991 Mathematics Subject Classification 20C20 , 16D90 , 16S35 , 16W50

where R_{H} = ⊕_{x∈H}R_{x} (and M_{H}, D_{H} are defined similarly).

Other Clifford type theorems are easy consequences of this theorem and the case of simple modules is also discussed.

2. Static modules and induction

Let A be a ring (associative with unit element), M a left A-module, and let D = End_{A}(M)^{op}, so M is an A, D-bimodule.

1.1. Definition. a) An A-module V is M -static if the natural A-homomorphism M ⊗_{D} Hom_{A}(M, V) → V , m ⊗ f ↦ mf = f(m) is an isomorphism.

b) A D-module W is called M -static if the natural D-homomorphism W → Hom_{A}(M, M ⊗_{D} W), w ↦ f_{w}, f_{w}(m) = m ⊗ w is an isomorphism.

Let C_{0} = (AsM)-mod be the full subcategory of A-mod consisting of M -static A-modules, and (DsM)-mod the full subcategory of D-mod consisting of M -static D-modules.

For the following result see Alperin [1] and Nauman [15].

2.2. Theorem. a) If M is finitely generated, then the A-module V is M -static iff there is an exact sequence M^{(J)} → M^{(I)} → V → 0 such that the sequence Hom_{A}(M, M^{(J)}) → Hom_{A}(M, M^{(I)}) → Hom_{A}(M, V) → 0 is also exact.

b) There is an equivalence (AsM)-mod ⇄ (DsM)-mod, given by the functors H = Hom_{A}(M, −) and T = M ⊗_{D} −.

c) If C is a full subcategory of (AsM)-modand H(C) is the image of C under H, then the restrictions of H and T give the equivalence C ⇄ H(C).

2.3. Remarks. a) M is an M -static A-module and D is an M -static D-module.

b) (AsM)-modand (DsM)-modare closed under direct summands and finite direct sums.

c) If M is finitely generated, then (AsM)-modand (DsM)-modare closed under arbitrary direct sums.

d) Let C_{1} = (A|M)-mod(respectively C_{2} = (A∥M)-mod) be the full subcategory of A-modconsisting of modules which divide a finite (respectively arbitrary) direct sum of copies of M. Then C_{1} ⊆ (AsM)-modand H(C_{1}) is the category of finitely generated projective D-modules. If M is finitely generated then C_{2} ⊆ (AsM)-mod and H(C_{2}) is the category of projective D-modules.

e) Let C_{3} = (AgM)-mod= σ(M) be the full subcategory of A-modgenerated by M. Assume that M is a finitely generated, self generator and projective object in (AgM)-mod. Then (AgM)-mod= (AsM)-modand H(C_{3}) = D-mod.

This applies especially in the case when A is a G-graded ring and M is a gr-simple A-module (see [5],[6],[9]).

Consider now a unit-preserving homomorphism A → B of rings and let N = B ⊗_{A} M, E = End_{B}(N)^{op}. Then the map φ: D → E, defined by φ(d)(b ⊗ m) = b ⊗ md = b ⊗ d(m) is also a unit-preserving homomorphism of rings. Let C′_{0} = (BsN)-mod, C′_{1} = (B|N)-mod, C′_{2} = (B∥N)-mod and C_{3} = (BgN)-mod, and consider the additive functors B ⊗_{A} −: A-mod → B -mod and E ⊗_{D} −: D-mod→ E -mod.

The following lemma should be compared with [7], Proposition 3.10.

2.4. Lemma. Suppose that if V ∈ C_{i} and W ∈ H(C_{i}), then B ⊗ V ∈ C′_{i} and E ⊗_{A} W ∈ H(C′_{i}) (i = 1, 2, 3). Then the following diagram commutes to within natural equivalences of functors:

C′_{i} ⟶^{Hom_{B}(N,−)} H(C′_{i})
B⊗_{A}−↑  ↑^{E⊗_{D}−}
C_{i} ⟶^{Hom_{A}(M,−)} H(C_{i})

The assumptions are fulfilled in the following situations:

a) i = 1;

b) i = 2 and M is a finitely generated A-module;

c) i = 3 and M and N are finitely generated, self generator and projective in (AgM)-mod(respectively in (BgN)-mod).

Proof. If W ∈ H(C_{i}) then clearly B ⊗_{A} (M ⊗_{D} W) and N ⊗_{E} (E ⊗_{D} W) are naturally isomorphic B -modules, so the first assertion follows. The rest is also clear since the functors B ⊗_{A} − and E ⊗_{D} − are right exact and commute with direct sums.

2.5. Remarks. a) Assume that N = B ⊗_{A} M is an M -static A-module. Then C′_{0} = (BsN)-modis the category of B -modules which are M -static as A-modules, C′_{1} (respectively C′_{2}) is the category of B -modules which divide in A-moda finite (respectively arbitrary) direct sum of copies of N.

Consequently, H(C′_{0}) is the category of E -modules which are M -static when regarded as D-modules, H(C′_{1}) (respectively H(C′_{2})) is the category of E -modules which are finitely generated projective (respectively projective) as D-modules.

b) Let finally C′_{4} = (BrM)-modbe the category of B -modules U which have a presentation B ⊗_{A} M^{(J)} → B ⊗_{A} M^{(I)} → U → 0 which splits in A-mod. Let F = End_{A}(N) so E ⊆ F. Then C′_{4} ⊆ (BsN)-modand H(C′_{4}) is the category of E -modules W for which F ⊗_{E} W is projective in F -mod.

These facts are proved in [15] and in [1] and are related to the stable Clifford theory of [2].

3. Endomorphism rings of gr-indecomposable modules

We fix now a finite group G, a G-graded ring R = ⨁_{x∈G} R_{x}, and a finitely generated G-graded R-module M = ⨁_{x∈G} M_{x}. The x-suspension M(x) of M is the G-graded R-module with M(x)_{y} = M_{yx}. Then we have that E = End_{R}(M)^{op} = ⨁_{x∈G} E_{x} is a G-graded ring with E_{x} = Hom_{R-gr}(M, M(x)). Let

G(M) = { x ∈ G | M ≃ M(x) in R-gr }

be the stabilizer of M. It is well-known that if M is gr-simple then E = E_{G(M)} = ⨁_{x∈G(M)} E_{x} is a crossed-product of the skew-field E_{1} = End_{R-gr}(M) with G(M).

We shall consider the graded Jacobson radical J_{gr}(E) of E. It is also well-known that J_{gr}(E) ⊆ J(E).

We need the construction of the functor R ¯⊗_{R_{H}} − introduced in [5]. Let H be a subgroup of G and N an R_{H} -module. Then R ⊗_{R_{H}} N is a G/H -graded Rmodule. Let Soc_{H}(R ⊗_{R_{H}} N) be the largest G/H -graded R-submodule of R ⊗ N with trivial H -component. This is called “the H -null socle” of R ⊗ N. Then, by definition, R ¯⊗_{R_{H}} N = (R ⊗ N)/ Soc_{H}(R ⊗ N), which is also a G/H -graded R-module, and the functor R ¯⊗_{R_{H}} − : R_{H} -mod→ (G/H, R)-gr can be defined and it is transitive.

3.1. Lemma. If H is a subgroup of G, then J_{gr}(E_{H}) = J_{gr}(E) ∩ E_{H}.

Proof. Let e ∈ J_{gr}(E) and let S = ⊕_{x∈G}S_{x} be a gr-simple left E -module. By the results of [7], for any x ∈ G, S(x)_{H} = 0 or S(x)_{H} is a gr-simple R_{H} -module so eS(x)_{H} = 0. It follows that eS = 0 and consequently e ∈ J_{gr}(E).

Let now e ∈ J_{gr}(E) ∩ E_{H} and let S be a gr-simple E_{H} -module. Again by [7], S′ = E ¯⊗_{E_{H}} S is a gr-simple E -module with S′_{H} ≃ S. We have that eS′ = 0 so eS = 0 and e ∈ J_{gr}(E_{H}).

We shall also need a graded version of Fitting’s Lemma. Recall that if the Ggraded R-module M has finite length in R-gr, then, by the structure of gr-simple modules, M has finite length in R-mod too.

3.2. Lemma. Assume that M is gr-indecomposable and has finite length in R-gr. Then the following assertions hold:

a) Every homogenous element of E is a unit or is nilpotent.

b) E_{1} = End_{R-gr}(M) is a local ring and D = E/J_{gr}(E) is a G(M)-gr-field, that is, it is a crossed product of the skew-field E_{1}/J(E_{1}) = (E/J_{gr}(E))_{1} with G(M).

c) We have that J(E_{1})E ⊆ J_{gr}(E) = J(E_{1})E_{G(M)} ⊕ (⨁_{x∉G(M)} E_{x}) and D is naturally isomorphic to D_{1} ⊗_{E_{1}} E_{G(M)} .

Proof. a) Let f ∈ E_{x}, so f: M → M(x) is a grade-preserving R-homomorphism. It follows that Ker f and f(N) are graded submodules of M for every graded submodule N of M. We prove that f is surjective if and only if it is injective.

Assume first that f is injective. We have the descending chain

M ⊃ f(M) ⊃ f^{2}(M) ⊃ ⋯ ⊃ f^{n}(M) = f^{n+1}(M)

of graded submodules of M, so for every u ∈ M there exists v ∈ M such that f^{n}(u) = f^{n+1}(v). It follows that f(v) = u ∈ f(M) hence f is surjective.

Assume now that f is surjective. There exists n ≥ 1 such that

0 ⊂ Ker f ⊂ ⋯ ⊂ Ker f^{n} = Ker f^{n+1}.

Let u ∈ Ker f. Then there is v ∈ M such that u = f^{n}(v). But f(u) = 0 so v ∈ Ker f^{n+1} = Ker f^{n}, hence u = 0 and f is injective.

Now by the well-known argument we obtain that there is n ≥ 1 such that M ≃ f^{n}(M) ⊕ Ker f^{n} in R-gr. Since M is gr-indecomposable, we conclude that f^{n} = 0 or f is an isomorphism.

b) Let I be a maximal left ideal of E_{1} and let f ∈ I so f^{n} = 0 for some n ≥ 1. Let g ∈ E \ I. Then Eg + I = E and hg + f = 1 for some h ∈ E_{1} . It follows that hg(1 + f + ⋯ + f^{n−1}) = (1 − f)(1 + f + ⋯ + f^{n−1}) = 1 − f^{n} = 1 hence g is a unit. This means that I is the set of all nonunits of E_{1} so J(E_{1}) = I and E_{1} is local.

Let now I be a maximal graded left ideal of E and f ∈ I a homogenous element so f^{n} = 0 for some n ≥ 1. Let g ∈ E \ I be another homogenous element. By the above argument we obtain that g is a unit. It follows that I is the graded ideal generated by the set of all homogenous units of E, hence J_{gr}(E) = I.

Clearly, if x ∈ G \ G(M) then E_{x} ⊆ J_{gr}(E). Therefore D is a crossed product of E_{1}/J(E_{1}) with G(M) since for x ∈ G(M), E_{x} contains invertible elements.

c) We have that J_{gr}(E)∩E_{1} = J(E_{1}) and J(E_{1})E is a graded ideal of E, hence J_{gr}(E) ⊇ J(E_{1})E. Also, since E_{G(M)} is a strongly graded ring,

J_{gr}(E_{G(M)}) = J_{gr}(E) ∩ E_{G(M)} = J(E_{1})E_{G(M)} = E_{G(M)}J(E_{1}),

and if x ∈ G \ G(M) then E_{x} ⊆ J_{gr}(E), so

J_{gr}(E) = J_{gr}(E_{G(M)}) ⊕ (⨁_{x∉G(M)} E_{x}) = J(E_{1})E_{G(M)} ⊕ (⨁_{x∉G(M)} E_{x}).

Consider now the exact sequence 0 → J(E_{1}) → E_{1} → D_{1} → 0 and apply the functor − ⊗_{E_{1}} E_{G(M)} . It follows that

D_{1} ⊗_{E_{1}} E_{G(M)} ≃ E_{1} ⊗_{E_{1}} E_{G(M)}/J(E_{1}) ⊗_{E_{1}} E_{G(M)} ≃ E_{G(M)}/J(E_{1})E_{G(M)} ≃ D.□

As in the first section, let (E|E)-mod (respectively (D|D)-mod) be the category of finitely generated projective E -modules (respectively D-modules). Under the hypothesis of the above lemma, J_{gr}(E) is a nilpotent ideal. Since J_{gr}(E) ⊆ J(E), the idempotents of D can be lifted modulo J_{gr}(E). Also, if P ∈ (E|E)-mod then P/J_{gr}(E)P is naturally isomorphic to D ⊗_{E} P.

If C is a category of modules, we denote by K(C) the Grothendieck group of C.

3.3. Proposition. Assume that M is gr-indecomposable and has finite length. Then we have:

a) The additive functor D ⊗_{E} − : (E|E)-mod → (D|D)-mod induces an isomorphism K((E|E)-mod) ≃ K((D|D)-mod) of groups.

b) If H is a subgroup of G then the following diagram commutes to within natural equivalences of functors:

(E|E)-mod ⟶^{D⊗_{E}−} (D|D)-mod
E⊗_{E_{H}} −↑  ↑^{D⊗_{D_{H}} −}
(E_{H}|E_{H})-mod ⟶^{D_{H}⊗_{E_{H}} −} (D_{H}|D_{H})-mod

c) If H = G(M) then the additive functor E ⊗_{E_{H}} − induces the isomorphism K((E_{H}|E_{H})-mod) ≃ K((E|E)-mod).

Proof. a) follows from [8], Proposition 22.15.

b) By Lemma 3.1, D_{H} = (E/J_{gr}(E))_{H} = E_{H}/J_{gr}(E)_{H} = E_{H}/J_{gr}(E_{H}). Therefore, if P ∈ (E_{H}|E_{H})-mod then we have the natural isomorphism of D-modules D ⊗_{D_{H}} (D_{H} ⊗_{E_{H}} P) ≃ D ⊗_{E} (E ⊗_{E_{H}} P).

c) By Lemma 3.3, D = D_{G(M)} , so the assertion follows immediately from a) and b).

4. Direct Clifford theory for strongly graded rings

Assume that G is a finite group and R = ⊕_{x∈G}R_{x} is a strongly G-graded ring. Further, let M = ⊕_{x∈G}M_{x} be a gr-indecomposable R-module of finite length, so M ≃ R ⊗_{R_{1}} M_{x} where M_{1} is an R_{1} -module of finite length. Also G(M) = { x ∈ G | M_{1} ≃ R_{x} ⊗_{R_{1}} M_{1} in R_{1}-mod }. As in the previous section let E = End_{R}(M)^{op} and D = D_{G(M)} = E/J_{gr}(E). If M_{1} is simple then D = E = E_{G(M)} .

If M is not gr-simple let C_{G} = (R|M)-mod and D_{G} = (D|D)-mod and if M is gr-simple let C_{G} = (RgM)-mod and D_{G} = D-mod = E -mod. Then clearly C_{H} = (R_{H}|M_{H})-mod and D_{H} = (D_{H}|D_{H})-mod, or C_{H} = (R_{H}gM_{H})-mod and D_{H} = E_{H} -mod = D_{H} -mod, respectively.

We can now state the direct and two-step Clifford Theorem.

4.1. Theorem. With the above notations, the following assertions hold:

a) The functor D ⊗_{E} Hom_{R}(M, −) induces an isomorphism K(C_{G}) ≃ K(D_{G}) of Grothendieck groups.

b) For every subgroup H of G, the following diagram commutes to within natural equivalences of functors :

C_{G} ⟶^{H_{G}} D_{G}
R⊗_{R_{H}} −↑  ↑^{D⊗_{D_{H}} −}
C_{H} ⟶^{H_{H}} D_{H}

c) The functor R ⊗_{R_{G(M)}} − : C_{G(M)} → C_{G} induces an isomorphism of the Grothendieck groups K(C_{G(M)}) and K(C_{G}); the inverse of this isomorphism is induced by the truncation functor (−)_{G(M)} : (G/G(M), R)-gr → R_{H}-mod. In particular, every object of C_{G} is G/G(M)-gradable. Also, the following diagram commutes:

K(C_{G}) ⟶^{H_{G}} K(D_{G(M)})
R⊗_{R_{G(M)}} −↑  H_{G(M)}
K(C_{G(M)})

d) If H is a subgroup of G then H(M) = H ∩ G(M) and the following diagram commutes:

K(C_{G})  K(C_{G(M)}) ⟶^{H_{G(M)}} K(D_{G(M)})
↑  ↑  ↑
K(C_{H})  K(C_{H(M)}) ⟶^{H_{H(M)}} K(D_{H(M)})

Proof. a) follows from Theorem 2.2, Remarks 2.3.d) and e) and Proposition 3.3.a). b) follows from Lemma 2.4 and Proposition 3.3.b).

c) is a consequence of a), b) and Lemma 3.2 and d) follows easily from a), b) and c).

4.2. Remarks. a) If M is gr-simple we do not need to pass to Grothendieck groups; in this case H_{G} , H_{H} and R ⊗_{R_{G(M)}} − are equivalences of categories.

b) If M is not gr-simple and P ∈ (D|D)-mod, then the corresponding object N ∈ (R|M)-mod may be obtained as follows: let d ∈ D be a primitive idempotent such that P ≃ Dd and let e ∈ E be a primitive idempotent such that d = e + J_{gr}(E). Then N ≃ M ⊗_{E} Ee ≃ R ⊗_{R_{G(M)}} M_{G(M)}e since we clearly have that e ∈ E_{G(M)} .

The following result is an immediate consequence of Theorem 4.1.c).

4.3. Corollary. Let M_{G(M)} = ⨁_{i=1}^{s} N_{i} be a decomposition of the induced module M_{G(M)} = R_{G(M)} ⊗_{R_{1}} M_{1} into indecomposable R_{G(M)} -modules corresponding to a decomposition D = ⨁_{i=1}^{s} P_{i} of D into principal indecomposable modules. Then:

a) M ≃ ⨁_{i=1}^{s} R ⊗_{R_{G(M)}} N_{i} is a decomposition of M into indecomposable R-modules and R ⊗ N_{i} ≃ R ⊗ N_{j} implies N_{i} ≃ N_{j}.

b) If M is gr-simple then M is a semisimple R-module if and only if E = E_{G(M)} is a semisimple ring.

4.4. Corollary. Let N be an R-module such that either N ∈ (R|M)-mod is an indecomposable R-module if M is not gr-simple, or N ∈ (RgM)-mod is a simple R-module if M is gr-simple. Then the following assertions hold:

a) _{R_{1}}N is a direct sum of R_{1} -modules conjugate to M_{1} such that each type of isomorphism appears with the same multiplicity e.

b) There exists an indecomposable R_{G(M)} -submodule N′ of N such that _{R_{1}}N′ is the sum of R_{1} -submodules of N isomorphic to M_{1} and N ≃ R ⊗_{R_{G(M)}} N′.

c) The multiplicity e equals dim_{D_{1}}(D ⊗_{E_{G(M)}} Hom_{R_{G(M)}}(M_{G(M)}, N′)).

Proof. a),b) In the first case N has finite length in R_{1} -mod and the assumption of indecomposability implies that N divides M in R-mod, so by the Krull-Schmidt Theorem _{R_{1}}N is a direct sum of conjugates of M_{1} . In the second case, since N is a simple R-module, it is a factor module of M = R ⊗_{R_{1}} M_{1} which is a semisimple R_{1} -module having all the components conjugated to M_{1} .

By Theorem 4.1.c), there is an indecomposable in the first case, respectively simple in the second case R_{G(M)} -module U such that N ≃ R ⊗_{R_{G(M)}} U. It follows that _{R_{1}}N ≃ ⨁_{i=1}^{t} R_{x_{i}G(M)} ⊗_{R_{G(M)}} U ≃ ⨁_{i=1}^{t} R_{x_{i}} ⊗_{R_{1}} U where { x_{i} | i = 1, ..., t } is a transversal for the left cosets of G(M) in G. Clearly _{R_{1}}U is homogenous and by the uniqueness asserted in the Krull-Schmidt Theorem it is isomorphic to the sum of all R_{1} -submodules of N isomorphic to M_{1} , so U ≃ N′. Since R_{x_{i}} ⊗_{R_{1}} − is an autoequivalence of R_{1} -mod, R_{x_{i}} ⊗_{R_{1}} N′ has the same number of components as N′.

c) We have that N′ and M_{G(M)} are finite direct sums of copies of M_{1} and

Hom_{R_{H}} (M_{H}, N′) ≃ Hom_{R_{H}} (R_{H} ⊗_{R_{1}} M_{1}, N′) ≃ Hom_{R_{1}}(M_{1}, N′)
≃ Hom_{R_{1}}(M_{1}, eM_{1}) ≃ e Hom_{R_{1}}(M_{1}, M_{1}) ≃ E_{1}^{e}

in E_{1} -mod, where H = G(M).

We also have that D = D_{H} and

D ⊗_{E_{H}} Hom_{R_{H}} (M_{H}, N′) ≃ (E_{1}/J(E_{1}) ⊗_{E_{1}} E) ⊗_{E} Hom_{R_{H}} (M_{H}, N′)
≃ E_{1}/J(E_{1}) ⊗_{E_{1}} Hom_{R_{1}}(M_{1}, eM_{1}) ≃ D_{1}^{e}

in D_{1}-mod. 4.5. Remarks. a) Any simple R-module N contains a simple R_{1} -submodule S, hence N ∈ (RgR ⊗_{R_{1}} S)-mod. Also, a simple (respectively indecomposable) Rmodule N ∈ (RgM)-mod (respectively (R|M)-mod) is just a simple (respectively indecomposable) object of (RgM)-mod (respectively (R|M)-mod).

b) We may also see the effect of conjugation on the functor H_{G}. Let y ∈ G, H = G(M), ^{y}H = yY y^{−1} , and let N be an R-module as in Corollary 4.4. Then End_{R}(M(y))^{op} = ^{y}E where (^{y}E)_{x} = E_{y^{−1}xy} for every x ∈ G, and the stabilizer of M(y) is ^{y}H so ^{y}E_{H} = (^{y}E)_{^{y}H}. With these notations, and using Lemma 3.2.c) and Fitting’s Lemma (or Schur’s Lemma if M_{1} is simple), we obtain the isomorphisms

yD ⊗_{^{y}(E_{H})} Hom_{R}(M(y), N) ≃ (^{y}D_{1} ⊗_{^{y}E_{1}} ^{y}(E_{H})) ⊗_{^{y}(E_{H})} Hom_{R_{H}} (M(y)_{^{y}H}, N)
≃ D_{1} ⊗_{E_{1}} Hom_{R_{1}}(M_{y}, N)
≃ D_{1} ⊗_{E_{1}} Hom_{R_{1}}(M_{y}, ⊕_{i=1}^{t}R_{x_{i}} ⊗_{R_{1}} N′)
≃ D_{1} ⊗_{E_{1}} Hom_{R_{1}}(M_{y}, R_{y} ⊗_{R_{1}} eM_{1})
≃ eD_{1} ⊗_{E_{1}} Hom_{R_{1}}(M_{y}, M_{y}) ≃ D_{1}^{e}.

4.6. Corollary. (Dade) The indecomposable R_{1} -module M_{1} can be extended to an R_{G(M)} -module M′_{1} ∈ C_{G(M)} if and only if

(i) there exists a D-module P with dim_{D_{1}}P = 1 in case M_{1} is simple, (ii) there exists a projective D-module P with dim_{D_{1}}P = 1 in case M_{1} is not

simple.

4.7. Remarks. a) D has a module P of dimension 1 over D_{1} if and only if the group-extension D_{1}^{∗} ↣ hU(D) ↠ G(M) splits, and two such D-modules are isomorphic if and only if the corresponding splittings are D_{1}^{∗} -conjugate. Since D_{1} is a skew-field, P is a projective D-module if and only if it is D_{1} -projective or equivalently if and only if J(D)P = 0. This holds when J(E_{G(M)}) = J_{gr}(E_{G(M)}) or when |G(M)| is invertible in E_{1} . In this case, if α: G(M) → hU(D) is a splitting, then P ≃ Dd where d is the idempotent of D given by d = |G(M)|^{−1} ∑_{x∈G(M)} α_{x} (see also [3], Corollary 2.14 and [4], Theorem 2.8).

b) Let K be a field of characteristic p > 0, N ↣ H ↠ G an extension of p-groups, and consider the trivial KN -module 1_{N}. This module is G-invariant and End_{KN}(1_{N})^{H})^{op} ≃ KG. The trivial KG-module has dimension 1 over K and it is not a projective KG-module. It follows that the extension 1_{H} of 1_{N} to KH is not an N -projective KH -module.

The following corollary generalises [13], Theorem 3 and [10], Proposition 1.

4.8. Corollary. Assume that G(M) is a p-group and charD_{1} = p. Then M is a homogenous (isotypic) R-module, and

(i) if M_{1} is simple then all the simple modules of (RgM)-mod are isomorphic; (ii) if M_{1} is not simple then all the indecomposable R-modules of (R|M)-mod

are isomorphic.

Proof. By [12], Theorem 2, D is an Artinian ring such that D/J(D) is simple, hence all the simple D-modules, respectively all projective indecomposable Dmodules, are isomorphic, and the corollary follows from Theorem 4.1.

4.9. Remark. Assume that R is a finite dimensional K -algebra where K ⊆ R_{1} is a perfect field of characteristic p ≥ 0 eand let M_{1} be absolutely indecomposable, that is D_{1} = K. Then there is α ∈ Z^{2}(G(M), K^{∗}) such that D ≃ K^{α}G(M) as G(M)-graded K -algebras. If G(M) is a p-group then D is a local ring and M and M_{G(M)} are absolutely indecomposable modules. Also, every simple R-module N ∈ (RgM)-mod is a simple R_{1} -module and corresponds to the trivial KG(M)- module. Conversely, if K is algebraically closed and K^{α}G(M) is a local ring, then G(M) is a p-group.

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University Babes-Bolyai of Cluj Department of Mathematics str. M. Kogalniceanu nr. 1 3400 Cluj-Napoca Romania