Modular representations of Loewy length two

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MODULAR REPRESENTATIONS OF LOEWY LENGTH TWO

M. E. CHARKANI and S. BOUHAMIDI

Received 23 October 2002

LetGbe a finitep-group,Ka field of characteristicp, andJthe radical of the group algebraK[G]. We study modular representations using some new results of the theory of extensions of modules. More precisely, we describe theK[G]-modules Msuch thatJ2_{M=0 and give some properties and isomorphism invariants which} allow us to compute the number of isomorphism classes ofK[G]-modulesMsuch that dimK(M)=µ(M)+1, where µ(M)is the minimum number of generators of theK[G]-moduleM. We also compute the number of isomorphism classes of indecomposableK[G]-modulesMsuch that dimK(Rad(M))=1.

2000 Mathematics Subject Classification: 16G10, 20C05.

1. Introduction. LetKbe a field of characteristicp >0, andG a finitep -group. The aim of this paper is to studyK[G]-modules of Loewy length two; these areK[G]-modulesM such thatJ2_{M=0, where J is the radical of the} group algebraK[G](see [8]). Another characterization is the following: aK[G] -module is of Loewy length two if and only if its annihilatorIis not a maximal ideal containingJ2_{. In general, the Loewy length of a K[G]-moduleM is the} smallest integernsuch thatJn_{·M=0; observe thatnexists becauseK[G]is}

artinian, andJis a nilpotent ideal. The Loewy length is equal to the length of thekernel filtrationofM(see [5]), defined by

M(n)_=x∈M|Jn_{·x=0. (1.1)}

Modules of Loewy lengthnare described in Propositions2.1and2.2; as a con-sequence, we find that everyK[G]-module of Loewy length two is an extension ofKs _byKt_{, wheres,tare positive integers, andGacts trivially onK}s _andKt_.

Since the Krull-Schmidt-Azumaya theorem holds for K[G]-modules, it is more convenient to study indecomposableK[G]-modules. InProposition 3.1 andCorollary 3.2, we give necessary conditions for aK[G]-module of Loewy length two to be indecomposable. IfMis indecomposable of Loewy length two, thenJ·M=MG_{, the subspace consisting of elements ofMinvariant under the} G-action (see Theorem 3.4). In Proposition 3.7 we will see that this result is no longer valid if we consider modules of higher Loewy length. As another consequence ofTheorem 3.4, we find that the minimum number of generators of an indecomposableK[G]-moduleM of Loewy length two, extension ofKs

Propositions4.1and4.2give a classification of indecomposableK[G]-modules of Loewy length two. InTheorem 5.5, we determine the number η(s,m)of isomorphism classes of indecomposableK[G]-modulesMof dimensions+1 and of Loewy length two such that dimK(Rad(M))=1, andm=µ(G)is the minimum number of generators of G. We distinguish two different cases: if

s > µ(G), then all modules are decomposable, and ifs≤µ(G), then the num-ber of isomorphism classes of indecomposable modules is equal to the numnum-ber ofs-dimensional subspaces ofKm_.

Throughout this paper,Kwill be a commutative field of characteristicp >0,

G a finitep-group, andK[G]the group algebra. We denote byJ the Jacob-son radical of K[G] and µ(G) the minimum number of generators of G. A

K[G]-moduleMwill be assumed to be finitely generated;µ(M)is the minimal number of generators ofM, and we will denote

MG_{= {m∈M|g·m=m∀g∈G},}

InvGM= {g∈G|g·m=m∀m∈M}.

(1.2)

Also recall that the sequence(Jk)kis called the principal filtration ofK[G]. The

descending chain(Jk_M)

kof submodules ofMis called the Loewy sequence of M(see [8, page 284]). The sequence(Jk_{M)kis also called theJ-filtration ofM.}

2. Preliminaries on extensions of modules. LetΛbe a commutative ring and L and N twoΛ-modules. We recall (see [1, 6, 10]) that an extension of

Lby N is a triple(f ,M,g), whereM is aΛ-module and f, gareΛ-module homomorphisms such that the sequence

0→N f→M g→L→0 (2.1)

is exact. Two extensions(f ,M,g)and(f_{,M,g)ofLbyNare equivalent if} there exists an isomorphism ofΛ-modulesφ:M→M_{such thatφ◦f=fand}

g_{◦φ=g. Let ¯E}

Λ(L,N)be the set of equivalence classes of extensions ofLby

N. IfΛ=K[G], then it is well known (see [9], [3, Section 75], and [4, Proposition 25.10]) that

¯

EΛ(L,N)H1(G,T ), (2.2)

whereT =HomK(L,N)is aK[G]-module, by takinghσ =σ◦h◦σ−1for all h∈T andσ∈G.

For a cocycleα∈Z1_{(G,T ), the corresponding extension isM=N×αL, which} is equal toN×Las a vector space, withG-action

Proposition2.1. Letnbe a positive integer. AnyK[G]-moduleMof Loewy lengthnis isomorphic toKt_{×αL, wheret=dimK(J}n−1_{M),Lis aK[G]-module}

of Loewy lengthn−1, andα∈Z1_{(G,T ), withT=Hom}

K(L,Kt).

Proof. The moduleJn−1_{Mis aK[G]-module of Loewy length 1 and is} iso-morphic toKt_{, wheretis a positive integer. The moduleMis an extension of} L=M/Jn−1_{Mwhich is aK[G]-module of Loewy lengthn−1 byJ}n−1_Mwhich is a trivialK[G]-module.

Proposition2.2. Letnbe a positive integer. AnyK[G]-module of Loewy length nis isomorphic to N×αKs_{, where s=µ(M), N is a K[G]-module of} Loewy lengthn−1, andα∈Z1_{(G,T ), withT=HomK(K}s_,N).

Proof. The moduleMis an extension ofM/JMbyJM, and it is clear that

JMis aK[G]-module of Loewy lengthn−1 andM/JMis a trivialK[G]-module.

Recall from [11] thatH1_{(G,T )=Hom(G,T )ifT=HomK(K}s_,Kt_).

Corollary2.3. AK[G]-module of Loewy length two is isomorphic toKt_×α Ks_{, where 0≠α∈Hom(G,T )andT=Hom}

K(Ks,Kt).

LetM=N×αLbe aK[G]-module and consider theK[G]-modules

KerG(α)=

σ∈G

Kerα(σ ), ImG(α)=

σ∈G

α(σ )(L). (2.4)

We put Rg(α)=dimK(ImG(α)). Let {σ1,σ2,...,σm}be a system of genera-tors ofG, wherem=µ(G)is the minimal number of generators ofG, and let

(e1,...,es)be a basis ofL. We defineαj(σi)=α(σi)(ej). It is then easy to see that{αj(σi)|i∈ {1,...,m}, j∈ {1,...,s}}is a system of generators of ImG(α).

Therefore,

Rg(α)≤InfdimK(N),µ(G)dimK(L). (2.5)

For aK[G]-moduleM, we define

InvG(M)= {σ∈G|σ·x=x∀x∈M}. (2.6)

Proposition2.4. LetM=N×αKs_{be aK[G]-module extension ofK}s_byN. IfHis a subgroup of InvG(N), then

MH_{=N×αKerH(α). (2.7)}

Proof. Take (m,n)∈ MH_{. For every h∈ H, we have that h·(m,n) =} (m,n). Buth·(m,n)=(m+α(h)(h·n),h·n), soh·n=nandα(h)(n)=0. Son∈KerH(α), and thereforeMH⊆N×α(KerH(α)). The converse inclusion

3. Decomposability of modules of Loewy length two. In this section, we establish some characterizations of the decomposability ofK[G]-modules of Loewy length two.

Proposition3.1. LetM=N×αLbe aK[G]-module andα∈Z1_{(G,T ).} (1)Each direct summand ofLincluded inKerG(α)is a direct summand ofM.

(2)IfImG(α)is a direct summand ofN, thenImG(α)×αLis a direct summand ofM.

Proof. (1) IfL=L_{⊕L, whereLand Lare twoK[G]-submodules ofL} such thatL_{⊆KerG(α), thenM=N×αL⊕0×αL.}

(2) If there exists aK[G]-submoduleN _{ofN such thatN=ImG(α)⊕N,} thenM=N_{×α0⊕ImG(α)×αL.}

Corollary 3.2. If M =Kt_×αKs _{is an indecomposable K[G]-module of}

Loewy length two, then Rg(α)=tand KerG(α)= {0}.

Proof. This is an immediate consequence ofProposition 3.1and the fact that Rg(α)=dimK(ImG(α)).

Remark3.3. We deduce fromCorollary 3.2thatM=Kt_×αKs _{is a}

decom-posableK[G]-module ifsµ(G) < t.

Theorem3.4. LetGbe a finitep-group. IfM is an indecomposable K[G]-module of Loewy length two, thenMG_=JM.

Proof. We can writeM=JM×αKs_{and, byProposition 2.4, we haveM}G JM×KerG(α). The moduleMis an indecomposableK[G]-module, so it follows fromCorollary 3.2that KerG(α)= {0}. Therefore, dimK(MG)=dimK(JM)and MG_{=JMsinceJM⊆M}G_.

Corollary3.5. LetGbe a finitep-group. LetMbe aK[G]-module of Loewy length two such that dimK(Rad(M))=1. ThenMis an indecomposableK[G] -module if and only ifMG=JM.

Proof. It suffices to prove one direction. The ringK[G]is a local ring (see [8]), so Rad(M)=JM. But dimK(Rad(M))=1 andMG_{=JM, so dimK(M}G_)=1,

and thereforeMis an indecomposableK[G]-module.

Corollary3.6. IfM is an indecomposableK[G]-module of Loewy length two, thenMis isomorphic toKt_×αKs_witht=dim

K(MG)ands=µ(M).

Proof. This is a consequence ofTheorem 3.4and the fact thatK[G]is a local ring, and thereforeµ(M)=dimK(M/JM).

Proposition3.7. LetGbe a finitep-group, withpan odd prime andnan integer such thatdimK(Jn−2/Jn−1)≥2. Then there exists an indecomposable cyclicK[G]-moduleM of Loewy lengthnsuch thatJn−1_{M is strictly included}

inMG_.

Proof. The ringK[G]being an artinian and noetherian,Jhas a composi-tion series. Hence, there exists an idealI ofK[G]such thatJn−1_⊂I⊂Jn−2 and such thatJn−2_{/I is an irreducible K[G]-module (isomorphic toK). The} idealIJis strictly included inJn−1_{; otherwiseJ}n−2_{is included inI, and} there-foreJn−2_{=I. So, J}n _{⊂IJ ⊂ J}n−1 _{and then the Loewy length of the K[G]} -moduleK[G]/IJisn. We haveMG_{=(IJ:J)/IJ. SinceJ}n−1_{⊂I⊆(IJ:J)and}

Jn−1_M=Jn−1_/IJ,Jn−1_{Mis strictly included inM}G_.

4. Isomorphism classes of modules of Loewy length two. When we study the modules of Loewy length two, we need to know when two modulesM= Kt_×αKs _{and M=K}t_×βKs _{are isomorphic. This is what we do in the next}

proposition.

Proposition4.1. LetM=Kt×αKs andM=Kt×βKs be indecomposable modules of Loewy length two. Then M andM _{are isomorphic if and only if}

there exist an automorphismϕofKt_{and an automorphismψofK}s _{such that} ϕ·α·ψ=β, where(ϕ·α·ψ)(g)=ϕ◦α(g)◦ψ.

Proof. The proof is an adaption of the proof of [3, Lemma 81.8]: if M andM_{are indecomposable, then, byTheorem 3.4,M}G_=MG_=Kt_{×{0}, and}

therefore any homomorphism fromM toM_{is a triangular matrix of} homo-morphisms and keepsKt_{globally invariant. For more details, the reader may}

consult [2].

Proposition4.2. LetM=Kt_×αKs_andM=Kt_×βKs _{be two} indecompos-able extensions ofKs _byKt_{. IfMandMare isomorphic, thenRg(α)=Rg(β).}

Proof. IfMandM_{are isomorphic, then there exist automorphismsϕand}

ψofKt_andKs_{such thatϕ·α·ψ=β. Then ImG(β)is generated byβ(σj)(ei),}

and therefore dimK(ImG(β))=dimKα(σj)(ψ(ei))=dimK(ImG(α)).

The converse ofProposition 4.2 is not true in general; under a certain hy-pothesis, we have equivalence.

Proposition4.3. LetGbe a finitep-group ands,ttwo integers such that t=sµ(G). LetM=Kt_×αKs _andM=Kt_×βKs _{be two indecomposable} exten-sions ofKs_byKt_{. ThenMMif and only if Rg(α)=Rg(β).}

Proof. IfM=Kt_×αKs _andM=Kt_×βKs _{are indecomposable such that}

Rg(α)=Rg(β), then Rg(α)=Rg(β)=t=sm. The system (α(σj)ei)has t

elements and generatesKt_{, so it is a basis ofK}t_{. Similarly,(β(σ}

j)ei)is a basis

ofKt_{. Then we define an automorphismϕofK}t_{byϕ(α(σj)(ei))=β(σj)(ei).}

Corollary4.4. LetGbe a finitep-group such thatm=µ(G).

(1) There exists only one isomorphism class of cyclicK[G]-modulesM of Loewy length two such that dimK(M)=m+1.

(2) IfG=(Z/pZ)m, then any cyclic K[G]-moduleM of Loewy length two such that dimK(M)=m+1 is isomorphic toK[G]/J2_.

Proof. (1) Any cyclicK[G]-moduleMis indecomposable. Then, byCorollary 3.6, a cyclicK[G]-moduleMof Loewy length two is isomorphic toKt_×αKwith t=dimK(MG_{). Since dimK(M)=m+1,t=mand all such modules are}

iso-morphic toKm_×αK.

(2) If M is a cyclic K[G]-module such that dimK(M)=m+1, then M is

indecomposable and isomorphic toK[G]/I, whereIis a proper ideal ofK[G]

containingJ2_{since dimK(K[G]/J}2_{)=m+1 (see [8, page 119, Theorem 2.3]),} and thenI=J2_.

Remark4.5. Proposition 4.3asserts that ifs andtare two integers such that t =sµ(G), then there exists only one isomorphism class of indecom-posableK[G]-modulesM of Loewy length two such that dimK(MG_{)=t and} µ(M)=s.

5. Extensions ofKs _{byK. We have seen inCorollary 3.6that all}

indecom-posable modules of a Loewy length two extension ofKbyKs are cyclic, and therefore they are isomorphic if and only if they have the same annihilator. In this section, we will study the isomorphism classes of modules of a Loewy length two extension ofKs byK, withsbeing a nonzero integer.

Proposition5.1. LetM=K×αKs _{be aK[G]-module extension ofK}s_byK. ThenMis an indecomposableK[G]-module if and only if KerG(α)= {0}.

Proof. IfM=K×αKs _{and Ker}

G(α)= {0}, then we conclude from Prop-osition 2.4that dimK(MG_{)=1, and thereforeM is an indecomposableK[G]}

-module.

From now on, we assume thatKis the finite field ofq=pr_{elements. From}

[7], we recall the following lemma.

Lemma5.2. LetV be aK-vector space of dimensionn. The number of m-dimensional subspaces ofV is given by the formula

n m

= m−1

i=0 qn−qi

m−1

i=0

qm_−qi. (5.1)

Proposition5.3. Letsandmbe nonzero integers. The number of orbits of the left action ofAut(Ks_)onHomK(Km_,Ks_{)is equal to}

min(m,s)

k=0

m k

Proof. Consider two homomorphismsf ,g:Km_→Ks_{. Thenfandgare in}

the same orbit if and only if there exists an automorphismϕofKs _{such that} ϕ◦f=g. Nowϕ◦f=gif and only if Kerf=Kerg, so two homomorphisms are in the same orbit if and only if they have the same kernel. Therefore, the number of orbits of the left action of Aut(Ks_{)on Hom}

K(Km,Ks)is equal to the

number of subspaces ofKm _{of dimensionk, wherekis an integer such that} m−min(m,s)≤k≤m. Now

m k = m m−k

, (5.3)

hence,

m

k=m−min(m,s)

m k

=

min(m,s)

k=0

m k

. (5.4)

Lemma 5.4. If M =K×αKs _{is a K[G]-module of Loewy length two, then} J·M=K×{0}.

Proof. LetM=K×αKs be aK[G]-module of Loewy length two. The pro-jection map pr2is aK[G]-homomorphism fromK×αKs toKs so pr2(J·(K×α

Ks_))⊆J·Ks _{(see [8]). Since K}s _{is a completely reducibleK[G]-module, we}

have J·Ks =0, and therefore pr2(J·M)=0. We haveJ·M ⊆K×KerG(α)

becauseM is aK[G]-module of Loewy length two. It follows thatJ·M⊆MG

andMG_{=K×KerG(α)(see Propositions2.4and 3.1). Since pr}

2J·M=0, we haveJ·M=K×{0}orJ·M=0. IfJ·M=0, thenMis a completely reducible

K[G]-module, hence aK[G]-module of Loewy length one, which contradicts our hypothesis thatMis of Loewy length two. We conclude thatJ·M=K×{0}.

LetGbe a finite abelianp-group withµ(G)=m(µ(G)is the minimal number of generators ofG). Letη(s,m)be the number of isomorphism classes ofK[G] -modules of Loewy length two which are extensions ofKs _byK.

Theorem5.5. Letsbe a positive integer and letGandmbe as above. Then

η(s,m)=

min(m,s)

k=1

k−1

i=0qm−qi

k−1

i=0qk−qi

. (5.5)

Proof. LetT=HomK(Ks_{,K)and take a basis{e}

1,...,es}ofKs and a basis

{e

1,...,em}ofKm. Also consider the map

f:Z1_{(G,T ) →Hom}

where α(x) =mi=1α(σi)(x)ei. IfM =K×αKs is a K[G]-module of Loewy

length two, thenα≠0 and Kerα=KerG(α).

Consider a secondK[G]-moduleM_=K×βKs _{of Loewy length two and}

as-sume that ϕ:M →M _{is an isomorphism. It follows fromLemma 5.4 that}

ϕ(K× {0})⊆(K× {0})(ϕ(J·M)⊆J·M_{, see [8]). Then there exists an} auto-morphismψofKs_{such that}

ασi◦ψ=βσi (5.7)

for alli∈ {1,...,m}, which is equivalent to

α◦ψ=_β,˜ _(5.8)

so the number of isomorphism classes ofK[G]-modules that are extensions ofKs _{byKis equal to the number of orbits of the right action of Aut(K}s_)on

HomK(Ks_,Km_{)−{0}, which is equal to the number of orbits of the left action of}

Aut(Ks_{)on HomK(K}m_,Ks_{)−{0}, and our result follows fromProposition 5.3.}

LetM be aK[G]-module of Loewy lengthn, thenJn−1_M⊆MG_{. In order to}

study the quotientMG_/Jn−1_{M, we need the following definition.}

Definition 5.6. Let M be aK[G]-module of Loewy length two. We call dimK(MG/JM)the index ofM.

By Theorem 3.4, the index of an indecomposable K[G]-module of Loewy length two is equal to zero.

Corollary5.7. Letnbe a nonzero integer andGa finite abelianp-group, withµ(G)=m. Then the number of isomorphism classes ofK[G]-modules of Loewy length two, indexr, minimal number of generatorsnand dimension

n+1 is equal to(nm−r ).

Proof. It follows fromProposition 2.4andTheorem 3.4that such a mod-ule is of the formK×αKn_{, withr =dimK(M}G_{/J·M)=dimK(KerG(α))and}

KerG(α)=Ker(α) . Therefore, the number of isomorphism classes is equal to

the number of orbits of the left action of Aut(Kn_{)on the set of}

homomor-phismsf:Km_→Kn_{such that dimK(Imf )=n−r (seeTheorem 5.5), which is}

equal to(nm−r ), byProposition 5.3.

Corollary5.8. LetGbe a finite abelianp-group andsandmtwo positive integers.

(1) Ifs≤m, then there existmsisomorphism classes of indecomposable K[G]-modules which are extensions ofKs _byK.

(2) Ifs > m, then allK[G]-modules which are extensions of Ks _{by Kare}

Proof. IfM is an indecomposableK[G]-module, then the index ofM is equal to zero. The result is therefore an immediate consequence ofCorollary 5.7.

Remark 5.9. If s =m, then there exists only one isomorphism class of an indecomposableK[G]-module which is an extension of Ks _{byK. Indeed,}

m

m=1. This is a particular case ofProposition 4.3.

Note5.10. (1) LetM=N×αLbe anR[G]-module extension ofLbyNand letL_{be a submodule of theR[G]-moduleL, andNa submodule of theR[G]} -moduleNcontained inσ∈Gα(σ )(L). Then we can considerαas a cocycle

in Z1_{(G,T), where T=HomK(L,N), and N×αL is a submodule of the}

R[G]-moduleM.

(2) Let᏿= {σ1,σ2,...,σm}be a system of generators ofG=(Z/pZ)m and consider anm×s matrixA=(aij)∈ᏹm,s(K). LetM be theK-vector space with basisᏮ= {u1,u2,...,us+1}. We define the followingG-action onM:

σi·uj=

  

uj+aijus+1, ifj≤s,

uj, ifj=s+1; (5.9)

Mtogether with theK[G]-module structure (5.9) will be denoted byKA. Ifs≤ mandAis of ranks, then it can be shown thatM=KAis an indecomposable

K[G]-module of Loewy length two such that dimK(M)=s+1 and µ(M)= s. More precisely, KA is isomorphic to K×αKs_{, whereα∈Z}1_{(G,T ), T =} HomK(Ks,K), andα(σi)(ej)=aij.

(3) EveryK[G]-module as inCorollary 5.8is isomorphic to a certainKA.

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M. E. Charkani: Department of Mathematics, Faculty of Sciences Dhar-Mahraz, Uni-versity of Sidi Mohammed Ben Abdellah, BP 1796, Fes, Morocco

E-mail address:[email protected]

S. Bouhamidi: Department of Mathematics, Faculty of Sciences Dhar-Mahraz, Univer-sity of Sidi Mohammed Ben Abdellah, BP 1796, Fes, Morocco