PROJECTIVE MODULES OVER TWISTED

GROUP ALGEBRAS OF p-SOLVABLE GROUPS

Andrei Marcus

Abstract. In this paper we generalize some results of P. Fong and I.M. Isaacs on modular representation theory of p-solvable groups to the case of twisted group algebras and the further to the case of strongly group graded algebras. For this, we use a categorical approach to Clifford theory.

1. Introduction

Let O be a complete discrete valuation ring with algebraically closed residue field k of characteristic p > 0, and let G be a finite group. If V is an O-module, then either V is O-free of finite rank or J(O)V = 0 and V has finite dimension over k.

Let further R = ⨁_{g∈G} R_{g} be a strongly G-graded O-algebra (that is,R_{g} is an Osummand of R, and R_{g}R_{h} = R_{gh} for all g, h ∈ G). For a subset X of G we denote R_{X} = ⨁_{x∈G} R_{x}. If V is a (left) R-module, then (R|V)-mod is the full subcategory of R-mod consisting of direct summands of finite direct sums of copies of V . We say that V is isotypic if all the indecomposable direct summands of V are isomorphic.

The main result of this paper states as follows. Assume that G is p-solvable and let 1 = N_{0} < N_{1} < ⋯ < N_{r} = G be a chain of normal subgroups of G such N_{i}/N_{i−1} is either a p-group or a p′ -group, for i = 1, …, r. Let also H be a Hall p′ -subgroup of G. (This means that the index of H in G is a power of p; it is well-known that such an H exists and that any p′ -subgroup of G is contained in a G-conjugate of H.) Let finally M be an indecomposable R_{1} -module.

1.1. Theorem. a) Let V ∈ (R|R ⊗_{R_{1}} M)-mod be an indecomposable R-module. Then there is an indecomposable R_{H} -module W ∈ (R_{H}|R_{H} ⊗_{R_{1}} M)-mod such that V ≃ R ⊗_{R_{H}} W.

b) Let W, W′ ∈ (R_{H}|R_{H} ⊗_{R_{1}} M)-mod be indecomposable R_{H} -modules such that the restrictions Res_{R_{H∩N_{i}}}^{R_{H}} W, Res_{R_{H∩N_{i}}}^{R_{H}} W′ are isotypic for i = 0, …, r − 1, and R ⊗_{R_{H}} W ≃ R⊗_{R_{H}} W′. Then R⊗_{R_{H}} W is an indecomposable R-module, and there is g ∈ N_{G}(H) such that W′ ≃ R_{gH} ⊗_{R_{H}} W (as R_{H} -modules).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20, 20C25, 20C05, 16W50, 16D90.

Key words and phrases. p-solvable groups, twisted group algebras, strongly graded algebras, Clifford theory.

These statements are graded versions of the main results of Hubert Fottner and Burkhardt Külshammer [FK]. Their proof rely on the theory of G-algebras and on some technical results on lifting idempotents with group actions involved. We shall give here a short proof using the categorical approach to the Clifford theory of indecomposable modules of [M1] and [M2], and induction, of course. The main results of the above papers will be used in Section 2 to show that it is enough to deal with projective modules over twisted group algebras, and the same results are needed in Section 3 to prove the theorem in this case by induction. It should be noted that in the case of ordinary group algebras 1.1.a) is due to P. Fong [P], while b) to I.M. Isaacs [I].

We refer the reader to [NV] for general facts on graded rings and to [K] for results on twisted group algebras and projective representations. Our notations tend to follow those of [M1] and [M2], and we shall recall the needed facts in the next section.

2. Clifford theory for strongly graded algebras

Let O, k, G, R and M be as in the introduction. The purpose of this section is to show that in Theorem 1.1, R can be replaced by a twisted group algebra of G and k.

2.1. If α ∈ Z^{2}(G, k^{∗}) then denote, as in [K], by k^{α}G the twisted group algebra having k -basis \{ḡ | g ∈ G\} and multiplication ḡh̄ = α(g, h)\overline{gh}. Then k^{α}G is a G-graded algebra in an obvious way. If β ∈ Z^{2}(G, k^{∗}), then k^{α}G ≃ k^{β}G as G-graded k -algebras if and only if α and β are cohomologous.

If G is a p-group, then it is well-known that Z^{2}(G, k^{∗}) = 1 and k^{α}G ≃ kG is a local ring with k^{α}G/J(k^{α}G) ≃ k.

If H is a subgroup of G, we shall denote k^{α}H = k^{res_{H}^{G}α}H, where res_{H}^{G}α ∈ H^{2}(H, k^{∗}) is the restriction of α to H. If W is a k^{α}H -module and V is a k^{α}G-module, then Ind_{k^{α}H}^{k^{α}G} W = k^{α}G ⊗_{k^{α}H} W is the induced module, and Res_{k^{α}H}^{k^{α}G} V is the k^{α}G-module obtained by scalar restriction via the inclusion k^{α}H ↪ k^{α}G.

If N is a normal subgroup of G, it will be useful to regard k^{α}G as a G/N -graded algebra, where for x = gN ∈ G/N, (k^{α}G)_{x} = ḡk^{α}N.

2.2. The graded Jacobson radical of R is, by definition, the intersection of maximal graded left ideals of R, and it coincides with the two-sided graded ideal J_{gr}(R) = J(R_{1})R = RJ(R_{1}). Then R/J_{gr}(R) is a strongly graded k -algebra with (R/J_{gr}(R)_{1} ≃ R_{1}/J(R_{1}). It is well-known that J_{gr}(R) ⊆ J(R), and if G is a p′ -group, then J_{gr}(R) = J(R).

TWISTED GROUP ALGEBRAS OF p-SOLVABLE GROUPS 3

2.3. Remark. The following observation will be crucial in the next section. Let N be a normal p-subgroup of G and H a p′ -subgroup of G such that G = HN. Let α ∈ Z^{2}(G, k^{∗}) and regard k^{α}G as a G/N -graded k -algebra. Let f denote the composition k^{α}H ↪ k^{α}G → k^{α}G/J_{gr}(k^{α}G) (where, using the G/N -grading, J_{gr}(k^{α}G) = J(k^{α}N)J(k^{α}G), and by the last statement of (2.2) we have J_{gr}(k^{α}G) = J(k^{α}G) since G/N is a p′ -group). Clearly, f is a homomorphism of G/N -graded k -algebras, and since k^{α}N is a local ring, we have that k^{α}G/J_{gr}(k^{α}G) is a twisted group algebra, so there is β ∈ Z^{2}(H, k^{∗}) such that k^{α}G/J_{gr}(k^{α}G) ≃ k^{β}H. It follows immediately that f is an isomorphism of H -graded k -algebras. Consequently, we have the injective map k^{α}H → k^{α}G and the surjective map k^{α}G → k^{α}H given by the composition k^{α}G → k^{β}H ⟶^{f^{−1}} k^{α}H.

2.4. Consider now the indecomposable R_{1} -module M and let

G_{M} = I_{G}(M) = {g ∈ G | R_{g} ⊗_{R_{1}} M ≃ M in R_{1}-mod}

be the stabilizer (inertia group) of M. Let further E = End_{R_{1}}(R ⊗_{R_{1}} M)^{op}. Then it is well-known that E is a G-graded O-algebra (not necessarily strongly graded) with

E_{g} ≃ {f ∈ E | f(R_{x} ⊗_{R_{1}} M) ⊆ R_{xg} ⊗_{R_{1}} M for all x ∈ G}.

In particular, E_{1} ≃ End_{R_{1}}(M)^{op}, and moreover, for any subgroup H of G, E_{H} ≃ End_{R_{H}} (R_{H} ⊗_{R_{1}} M)^{op}.

2.5. Let D = E/J_{gr}(E). Then, since E_{1} is a local ring and k is algebraically closed, we have that D is a twisted group algebra of G_{M} and k, so there is α ∈ H^{2}(G_{M}, k^{∗}) such that D = k^{α}G_{M} (see [M1, Section 3] for details).

2.6. By [M1, Theorem 4.1], we have that the additive functor

D ⊗_{E} Hom_{R}(R ⊗_{R_{1}} M, −): (R|R ⊗_{R_{1}} M)-mod → (D|D)-mod

induces an isomorphism between the Grothendieck groups associated to these categories, where (D|D)-mod is the category of (finitely generated) projective D-modules. Moreover, this functor commutes with induction from subgroups, restriction, truncation and conjugation (see [M1, Theorem 4.1 and Remark 4.5.b), and the functors

(−)_{G_{M}} : (R|R ⊗_{R_{1}} M)-mod → (R_{G_{M}} |R_{G_{M}} ⊗_{R_{1}} M)-mod

and

R ⊗_{R_{H}} −: (R_{G_{M}} |R_{G_{M}} ⊗_{R_{1}} M)-mod → (R|R ⊗_{R_{1}} M)-mod

induce isomorphisms between the Grothendieck groups of these categories ([M1, Theorem 4.1.c)]).

This implies (see [M1, Corollary 4.4]) that if V ∈ (R|R ⊗_{R_{1}} M)-mod is indecomposable, then Res_{R_{1}}^{R} V is an isotypic R_{1} -module if and only if G_{M} = G.

The above discussion implies immediately that Theorem 1.1 is equivalent to the following theorem.

2.7. Theorem. Assume that G is a p-solvable group, H is a Hall p′ -subgroup of G, and α ∈ Z^{2}(G, k^{∗}). Let 1 = N_{0} < N_{1} < ⋯ < N_{r} = G be a chain of normal subgroups of G such that N_{i}/N_{i−1} is either a p-group or a p′ -group.

a) Let V be a projective indecomposable k^{α}G-module. Then there is a simple k^{α}H - module W such that V ≃ Ind_{k^{α}H}^{k^{α}G} W.

b) Let W, W′ be simple k^{α}H -modules such that Res_{k^{α}(H∩N_{i})}^{k^{α}H} W and Res_{k^{α}(H∩N_{i})}^{k^{α}H} W′ are isotypic for i = 0, …, r − 1, and Ind_{k^{α}H}^{k^{α}G} W ≃ Ind_{k^{α}H}^{k^{α}G} W′ as k^{α}G-modules. Then Ind_{k^{α}H}^{k^{α}G} W is an indecomposable (and projective) k^{α}G-module, and there is g ∈ N_{G}(H) such that k^{α}Hḡ ⊗_{k^{α}H} W as k^{α}H -modules.

The proof of this theorem will be given in the next section. It will require Clifford theory of projective modules, so we recall the basic result [M2, Theorem 2.3] in the general context of this section.

2.8. Assume in addition that M is a projective R_{1} -module, and let S = M/J(R_{1})M and R′ = R/J_{gr}(R). It follows that S is a simple R′_{1} -module and End_{R′}(R′ ⊗_{R′_{1}} S)^{op} ≃ D as G_{M} -graded k -algebras. Moreover, there is a commutative diagram of categories

(R|R ⊗_{R_{1}} M)-mod ⟶^{Hom_{R}(R⊗_{R_{1}} M,−)} (E|E)-mod
R′⊗_{R}−↓  ↓^{D⊗_{E}−}
(R′|R′ ⊗_{R′_{1}} S) ⟶^{Hom_{R′} (R′⊗_{R′_{1}} S,−)} (D|D)-mod

and this diagram is also compatible with induction from subgroups.

3. A proof of Theorem 2.7.

Assume that G is p-solvable, H a Hall p′ -subgroup of G, and let 1 = N_{0} < N_{1} < ⋯ < N_{r} = G be a chain of normal subgroups as in (2.7).

a) We prove by induction on G that if α ∈ Z^{2}(G, k^{∗}) and V is a projective k^{α}G-module, then there is a simple k^{α}H -module W such that V ≃ Ind_{k^{α}H}^{k^{α}G} W.

The statement is trivial if G is a p′ -group (since then H = G) or if G is a p-group (then H = 1 and V = k^{α}G is the unique projective indecomposable k^{α}G-module).

Denote N = N_{1} and assume first that N ≠ 1 is a p′ -group. Since N is a normal subgroup of G, we have that N ⊆ H. Since V is a projective indecomposable k^{α}Gmodule, by Clifford theory there is a projective (and simple) k^{α}N -module M such that V ∈ (k^{α}G| Ind_{k^{α}N}^{k^{α}G} M)-mod. Let I/N = (G/N)_{M} be the stabilizer of M. By (2.6) there is a projective k^{α}I -module V_{0} such that V ≃ Ind_{k^{α}I}^{k^{α}G} V_{0} , and moreover Res_{k^{α}N}^{k^{α}I} V_{0} is an isotypic k^{α}N -module. By (2.6) and (2.5) V_{0} corresponds to a projective indecomposable k^{β}(I/N)-module Ṽ_{0} , where β ∈ Z^{2}(I/N, k^{∗}). Denote H_{0} = I ∩ H, so H_{0}/N is a Hall p′ -subgroup of I/N. Since |H_{0}/N| < |G|, by induction there is a projective and simple k^{β}H_{0} -module W̃_{0} such that Ṽ_{0} ≃ Ind_{k^{β}(H_{0}/N)}^{k^{β}(I/N)} W̃_{0} . Again by (2.6), W̃_{0} corresponds to a projective k^{α}H_{0} -module W_{0} ∈ (k^{α}H_{0}| Ind_{k^{α}N}^{k^{α}H_{0}} M)-mod and V_{0} ≃ Ind_{k^{α}H_{0}}^{k^{α}I} W_{0} . Denoting W = Ind_{k^{α}H_{0}}^{k^{α}H} W_{0} , we finally have

V ≃ Ind_{k^{α}I}^{k^{α}G} V_{0} ≃ Ind_{k^{α}I}^{k^{α}G} Ind_{k^{α}H_{0}}^{k^{α}I} W_{0}
≃ Ind_{k^{α}H}^{k^{α}G} Ind_{k^{α}H_{0}}^{k^{α}H} W_{0} ≃ Ind_{k^{α}H}^{k^{α}G} W.

Assume now that N ≠ 1 is a p-group. Since V is a projective indecomposable k^{α}G-module, there is a projective indecomposable k^{α}N -module M such that V ∈ (k^{α}G| Ind_{k^{α}N}^{k^{α}G} M)-mod. But N is a p-group, so M ≃ k^{α}N is the unique projective indecomposable k^{α}N -module, hence M is G/N -invariant. By (2.6) and (2.7) V corresponds to a projective indecomposable k^{β}(G/N)-module Ṽ , where β ∈ Z^{2}(G/N, k^{∗}). We have that |G/N| < |G| and HN/N is a Hall p′ -subgroup of G/N. By induction it follows that there is a projective simple k^{β}(HN/N)-module W̃ such that Ṽ ≃ Ind_{k^{β}(HN/N)}^{k^{β}(G/N)} W̃. By (2.6) again, W̃ corresponds to a projective indecomposable k^{α}(HN)-module P ∈ (k^{α}(HN)| Ind_{k^{α}N}^{k^{α}(HN)} M)-mod such that V ≃ Ind_{k^{α}(HN)}^{k^{α}G} P.

Since W̃ is a simple k^{β}(HN/N)-module, by (2.8) it corresponds to a simple module W over k^{α}(HN)/J_{gr}(k^{α}(HN)), where k^{α}(HN) is regarded as a HN/N -graded algebra. By Remark 2.3, W is a projective simple k^{α}H -module. Clearly, k^{α}(HN) ⊗_{k^{α}H} W is a projective k^{α}(HN)-module, and multiplication induces an epimorphism Ind_{k^{α}H}^{k^{α}(HN)} W → W. By the commutativity of the diagram (2.8) we have that P is the projective cover of W (where W is regarded as a simple k^{α}(HN)-module), so P is a direct summand of Ind_{k^{α}H}^{k^{α}(HN)} W. Since W ≃ P/J(k^{α}N)P, we have dim_{k} P = |N|dim_{k}W; by comparing dimensions, it follows that P ≃ Ind_{k^{α}H}^{k^{α}N} W, hence V ≃ Ind_{k^{α}H}^{k^{α}G} W.

b) We use again induction on G, and we see that the statements are trivial if G is a p′ -group or a p-group.

Suppose first that N = N_{1} ≠ 1 is a p′ -group. By the assumption on isotypy, there is a G/N -invariant k^{α}N -module M such that W and W′ belong to the same category (k^{α}H| Ind_{k^{α}N}^{k^{α}G} M)-mod. Denoting V = Ind_{k^{α}H}^{k^{α}G} W ≃ Ind_{k^{α}H}^{k^{α}G} W′, we have that V belongs to the category (k^{α}G| Ind_{k^{α}N}^{k^{α}G} M)-mod. By (2.6) and (2.4) V corresponds to a projective k^{β}(G/N)-module, W and W′ correspond to projective simple k^{β}(H/H)-modules W̃ and W̃′ respectively, such that

Ṽ ≃ Ind_{k^{β}(H/N)}^{k^{β}(G/N)} W̃ ≃ Ind_{k^{β}(H/N)}^{k^{β}(G/N)} W̃′,

and Res_{k^{β}(H∩N_{i}/N)}^{k^{β}(H/N)} W̃ , Res_{k^{β}(H∩N_{i}/N)}^{k^{β}(H/N)} W̃′ are isotypic for i = 1, …, r − 1.

Since |G/N| < |G| and H/N is a Hall p′ -subgroup of G/N, by the induction hypothesis we have that Ṽ is an indecomposable k^{β}(G/N)-module, and there is gN ∈ N_{G/N}(H/N) such that

W̃′ ≃ k^{β}(H/N)\overline{gN} ⊗_{k^{β}(H/N)} W̃ .

By (2.6) it follows that V is an indecomposable k^{α}G-module and W′ ≃ k^{α}Hḡ ⊗_{k^{α}H} W.

Finally, assume that N ≠ 1 is a p-group. By Remark 2.3 we have an epimorphism k^{α}(HN) → k^{α}H, and regard W, W′ as simple k^{α}(HN)-modules by scalar restriction. Let P = P (W) and P′ = P (W′) be the projective covers of W and W′ respectively, so P and P′ are projective indecomposable k^{α}(HN)-modules. By the last part of the proof of a) we have that P ≃ Ind_{k^{α}H}^{k^{α}(HN)} W and P′ ≃ Ind_{k^{α}H}^{k^{α}(HN)} W′. Moreover, by assumption

V ≃ Ind_{k^{α}(HN)}^{k^{α}G} P ≃ Ind_{k^{α}(HN)}^{k^{α}G} P′.

Clearly, V belongs to (k^{α}G| Ind_{k^{α}N}^{k^{α}G} M)-mod while P and P′ belong to the category (k^{α}(HN)| Ind_{k^{α}N}^{k^{α}(HN)} M)-mod, where M ≃ k^{α}N is the unique projective indecomposable k^{α}N -module. (Hence M is G-invariant and we do not have to use the assumption on isotypy when we restrict to normal p-groups.) By (2.6) and (2.4) V corresponds to a projective k^{β}(G/N)-module Ṽ , while W and W′ correspond to projective k^{β}(HN/N)- modules W and W′ respectively, where β ∈ Z^{2}(G/N, k^{∗}). Moreover, by assumption it follows that

Ṽ ≃ Ind_{k^{β}(HN/N)}^{k^{β}(G/N)} W̃ ≃ Ind_{k^{β}(HN/N)}^{k^{β}(G/N)} W̃′.

Since |G/N| < G and HN/N is a Hall p′ -subgroup of G/N, the induction hypothesis can be applied, hence Ṽ is an indecomposable k^{β}(G/N)-module, and there is gN ∈ N_{G/N}(HN/N) ( so g ∈ N_{G}(H)) such that W̃′ ≃ k^{β}(H/N)\overline{gN} ⊗_{k^{β}(H/N)} W̃. By (2.6) it follows that V is an indecomposable k^{α}G-module and P′ ≃ k^{α}(HN)ḡ ⊗_{k^{α}(HN)} P as k^{α}(HN)-modules. Consequently, by Remark 2.3,

W′ ≃ P′/J(k^{α}N)P′ ≃ k^{α}Hḡ ⊗_{k^{α}H} P/J(k^{α}H)P ≃ k^{α}Hḡ ⊗_{k^{α}H} W

as k^{α}H -modules.

Acknowledgements. The author thanks the referee for his observations which improved the presentation of the paper.

References

[F] P. Fong, Solvable groups and modular representation theory, Trans. Amer. Math. Soc. 103 (1962), 484–494.

[FK] H. Fottner and B. Külshammer, On indecomposable and imprimitive modules for finite groups – a G-algebra approach, J. London Math. Soc. (2) 59 (1999), 828–844.

[I] I.M. Isaacs, Fong Characters in π -separable groups, J. Algebra 99 (1986), 83–107.

[K] G. Karpilovski, Projective representations of finite groups, Marcel Dekker, New-York, 1985.

[M1] A. Marcus, Static modules and Clifford theory for strongly graded rings, Publicationes Math. 42 (1993), 303–314.

[M2] A. Marcus, Clifford theory for projective modules over strongly graded rings, Comm. Algebra 23 (1995), 4393–4404.

[NV] C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North Holland, Amsterdam, 1982.

“Babeş-Bolyai” University Faculty of Mathematics and Computer Science str. M. Kogalniceanu nr. 1 3400 Cluj-Napoca Romania

   E-mail address: marcus@math.ubbcluj.ro