Furuta Yoshiomi

Yamashita Hiroshi

古田 孝臣

山下 浩

Kanazawa University Repository for Academic Resources

ScLRep・KanazawaUniv・Vol、２７，Ｎｏｌｐｐｌ－３Ｊｕｎｅｌ９８２RepresentationMoClulesamltheAugumentationldealofaFinitegroupYoshiomiFuRuTAandHiroshiYAMAsHITA＊此ｐａｍｌｚｃ"ｔｑ／Ｍｚ伽"ｍｊｉＣＳ,Ftzc"/jbﾉｑ／Sbjb"ｃＥ,KZz"αzaz(ﾉαＵ)zjzﾉcぷ１３ノ（ReceivedApril26,1982）Abstract・ＦｏｒａｆｉｎｉｔｅｇｒｏｕｐＧａｎｄａＧ－ｍｏｄｕｌｅＡ,thestructureofI'ＷＩ臓十lAisstudied,whereIistheaugumentationidealofthegroupringZ[Ｇ］．1．ＬｅｔＧｂｅａｆｉｎｉｔｅｇｒｏｕｐａｎｄＡｂｅａＧ－ｍｏｄuleDenotebyItheaugumentationidealofthegroupringZ[Ｇ］ofGwithrespecttotheringofintegers、ＰｕｔＧ｡＝Ｇ/[ＧＧ］，where[ＧＧ]standsforthecommutatorsubgroupofGDenotebyG害蝿thetensorproductQ⑭…⑭Goof"-copiesofGb､Then川/I洞十lAisisomorphictoafactorgroupofGﾂ⑭(MZ1)．Thisisshownin［1］togetherwithothercohomologicalrelationshipbymeansofclassfieldtheory，ｗｈｅｎＧｉｓｔｈｅＧａｌｏｉｓｇｒｏｕｐｏｆａＧａｌｏｉｓｅｘｔｅnsionKofanalgebraicnumberfield虎andAisacongruentidealclassgrouｐｏｆＫＴｈｅｐｕｒｐｏｓｅｏｆｔｈｅｐｒｅｓｅｎｔｐａｐｅｒｉｓｔｏｇｅｔａｓｉｍｐｌｅｐｒｏｏｆｏｆｔｈｅｐｕｒｅｌygrouptheoreticalpartoftheaboveresult．2．LetIbeasabovetheaugumentationidealofthegroupringZ[Ｇ］ＬＥＭＭＡ１・WMCzzﾉｃ仇c/Ｍｂｚ(ﾉ蝿ｃＭｇｍｅ"CCSノ〃Ｚ[Ｇ］ノbγα〃ぴα"‘でCl/Ｃ：（ｉ）ぴで－１＝（ぴ－１)＋(で－１）ｍｏｄＩ２（ii）ず（ぴ－１）＝ぴ－１ ｍｏｄＩ２（iii）ぴでぴ-1で－１－１＝O modI2PRooF。（ｉ）ａｎｄ（ii）areimpliedfrom(ｄ－１)(て－１）＝（ぴで－１)＿(ぴ＿,)＿(で－，）ａｎｄで(ぴ－１)－(ぴ－１）＝に－１）（ぴ－１).Ｍｏｒｅｏｖｅｒ（ｉ）implies(ぴ＿,)＋(ぴ-,-,）＝ぴぴ－，－，＝ＯｍｏｄＩ２・Henceぴ~'－１＝－(ぴ－１)ｍｏｄＩ２・Thisimplies（iii）ｂｙusing（ｉ）again．LEMMA2・Ｌｅｔのｂｅα''０`Zppj'２９０／Ｇｂ⑭(A/、)わ、/12ﾉMb1/ii"Ｂａｂｙの(5②α）＝(ぴ￣ｌｍｍｏｄＩ２Ａ，掌Presentaddress：KanazawaWoman，sJuniorCollege-1２ YoshiomiFuRuTAandHiroshiYAMAsHITA＊z(ノル”ケα"‘αα”ｅノセ"e伽ｑ/ＵＭ"‘Ａ/ＬＡ””Sc"〃byぴq/Ｇα"‘αCl/Ａ花Ｗｃ"zﾉelly・ＺＭｚＰｊ"伽ccsczs"沈c"zﾉｅ〃o〃o"oゅﾉﾉＭｚ、PROOF・Let＆＝品.Thenthereexistsanelementpof［Ｇ,Ｇ］suchthaM＝oip,andweseethatoh-1＝（oi-1)＋(p－１）＝ｏｉ－１ｍｏｄＩ２ｂｙ（ｉ）ａｎｄ(iii)ｏｆＬｅｍｍａＬＭｏｒｅｏｖｅｒ(ｉ）ofLemmalimpliesthat(ぴ"－１)α＝〃(ぴ－１)czmodI2Aforanyinteger〃・Hence①iswelldefined,anditisclearthatのinducesasurjectivehomomorphism．3．WedenotebyG'nthedirectproductof〃-copiesofG,anddenotebyOz(Ｇ,Ａ)thegroupofcochainsofGofdimensiｏｎ〃withAasitscoeffientmoduleForアＥＣ"(Ｇ,Ａ),let△ゲーニ(Ti-1)…(Th-1〃(でi,…,でh)，wherethesumistakenoverall（Ti,…,Th）ｏｆＧ"・ＤｅｎｏｔｅｂｙＲ"(ＧＡ）ｔｈｅｓｕｂｇｒｏｕｐｏｆＣ”（ＧＡ）consistedof′ｓｕｃｈｔｈａｔ△Ｖ＝０.LetノリｂｅａｈｏｍｏｍｏｒｐｈｉｓｍｏｆＣ"(Ｇ,Ａ)toG亨蝿⑭似/ＬＡ)definedbyﾉ区(ｆ）＝二肩⑭…⑭面⑭八でi,…川)，where′(Ti,…川)standsfortheelementofA/L4representedby′(Ti,…川)andthesumistakenoverall(Ti,…,Ｚｎ)ｏｆＧ"．THEoREM・jVbm加卯bcj咽Czsa60zﾉc，ｚ(ﾉｅノiczzﾉｅG房"⑭(Ｍｎ）I"Ａ/'''十1Ａ= ﾉｕＲ"(ＧＡ）PROOF、ApplyingLemma2repeatedly,wehaveasurjectivehomomorphismの(")ｏｆＧ害鰯⑭(6A/ＬＡ)toIWI'Ｍ,whichisdefinedby①(")(団⑭…⑭5,m⑭α）＝(oi-1)…(oh-1)αｍｏｄＩ"+1Ａ，where＆ａｎｄａａｒｅｅｌｅｍｅｎｔｓｏｆＧｂａｎｄＡ/ZArepresentedbyaeGandczeArespectively(ｉ＝1,…,〃)．LetusdetermineKerP(")．Put(可＝(＆,…,瓦),whichrunsoverallelementsofG:,andletrepresentativesainGofeachclassofQbefixedonceforalLＮｏｗｌｅｔα(可beelementsofA/L4suchthatの(三扇⑧…⑭斑⑧α(5))＝0．Ｔｈｅｎ二(oi-1)…(ｏｈ－１）α何＝０腕o〃'Ｍ,whereα(副arere缶bsentativesofα(す)inAThere＆istelements6⑭）oｆＡｓｕｃｈｔｈａｔ二(oi-1)…(oh-1)α何＝二(p1-1ﾉ…(p"+1-1ノ６(pﾉ，（前 （pノwhereい)=ＷＡｌ)｡G剛Put(雷)＝(〃…)ｃｃ伽dlet611:|=,三ルー1)6ⅡRepresntationModulesandtheAugumentationIdealofaFinitegroup ３where６(p)runsoverall(p)ｓｕｃｈthat(p,,…,仏）＝（Ti,…，助)．Ｐｕｔ′(Ti,…川）＝α(功一6tIor-611ilaccordingas(Ｔｌ,…川)＝(｡i,…川)forsome(可ornotThen八R"(Ｍ）MoreoverwehavqM＝三肩⑭…剛⑭'(両,…川)＝二面⑭…⑭5瀬⑭α(動,because6I:iｌ（５）Ｅn．ＮｏｗｗｅｈａｖｅＫｅｒ①(")亡ﾉｕＲ(")(Ｃ,Ａ)．Theoppositeinclusionisclearbydefinition．References[1］ＹFuruta,OnnilpotentfactorsofcongruentidealclassgroupsofGaloisextensions,NagoyaMathJ,６２(1976),13-28.

Representation Modules and the Augumentation Ideal of a Finite group

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Representation Modules and the Augumentation Ideal of a Finite group

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