関西学院大学リポジトリ
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(2) Contents 1 Introduction. 2Modu1e derivations 2．1. Basic properties of1oca11y ni1potent derivations. 2．2. δ一mO（1u1eS. 2．3. The case where B＝λ［V］. 2．4 2．5. The homoIogica1property ofδ一modu1es The re1ation betweenδ∈LND（B）andδ一mod−uIes. 2．6. Projective trianguIabi1ity. 2．7. InvertibIeδ一modu1es. 3 The14th prob1em of Hi1bert 3．1. Suf巳。ient conditions for丘nite generation. 3．2 Symmetric tensor a1gebras ofδ一modu1es．. 3．3 Di丘erentia1modu1es．. 4 In丘nite1y many generators 4−1 A simp1i丘。ation of Kuroda，s counterexamp1e 4．2 A genera1ization of Theorem4．1．3. 5 Further prob1ems and−comments 5．1. Tensor products. 5．2 Dua1mod−u1es. 1 5. 5 8. 16 26. 38 40 43. 46 46 50 52. 58 58 63. 69 69 71.
(3) Chapter1 Introduction Atheoryofa伍nea1gebraicva■ietieshasnotbeenwe11−deve1oped−inhigher dimensiona1case and we often use a1gebraic group actions on a伍ne a1gebraic varieties in order to1ower the dimension，e．g．，by considering the quotient space，orbits，etc．However，an a1gebraic group action is not treated in the case ofunipotent a1gebraic groups．One examp1e is an a1gebraic action ofthe. additive group scheme Gα．An a1gebraic action ofthe additive group scheme Gαon an a伍ne scheme Spec B is described equiva1ent1y in terms of a1oca11y. ni1potent derivationδon the coord−inate ring B．The Gα一invariant subring. λin B and−the quotient morphismπ：Spec B→Specλare given by Kerδ and the inc1usionλ→B，Contrary to this simp1e d−escription，λandπ have properties which are not contro11ed within the frameworks of ord−inary. a1gebraic geometry，e．g．，the existence of counterexamp1es to the fourteenth. prob1em ofHi1bert，the non−surjectivity ofthe quotient morphismπ，etc．（see Nagata［17］，Freud−enburg［91a皿d Bonnet［11）． Letたbe a丘e1d of characteristic zero．The fourteenth prob1em of Hi1bert asks if R＝K∩引”1ゾ＿，”η］is丘nite1y generated overん，whereん［”1ゾ．．，”η］. is a po1ynomia1ring and K is a sub丘e1d ofん（”1，＿，叫）containingた．There. have been constructed−many counterexamp16s inc1uding the丘rst one due t0. M．Nagata［171．In most cases，the subring R is the invariant subaIgebra of a1oca11y ni1potentん一derivationδonん［”1，．．べ，”肌］（Roberts［19］，Kojima−. Miyanishi［101，Freudenburg［81，Daig1e−Freudenburg［31，Kuroda［111etc．）． UnsoIved important probIems such as Jacobian conjecture and Canceua− tion prob1em can be reinterpreted as prob1ems concerning1oca11y ni1potent d−erivations．Further，1oca11y niIpotent derivations themseIves have many un− soIved prob1ems．For examp1e，there is a prob1em which asks if it is possib1e 1.
(4) CHAPTERユ．工NTRρDU’CT工ON to describe a11Ioca11y ni1potent d−erivations on po1ynomia1rings，Hence it is signi丘。ant to study1oca11y ni1potent d−erivations，. In this artic1e，we extend a1oca11y ni1potent derivationδon a ring B t0. B−modu1es M and−investigate the properties of such B−mod−u1es M「and the. extended d−erivations．By using modu1e theory，we expect to so1ve unso1ved. prob1ems conceming1oca11y ni1potent derivations on rings and．to know new properties of the associated a1gebraic varieties．. We introducethe notionofa（B，δ）一modu1e（abbreviated as aδ一modu1e），. which is a B−modu1e with a modu1e derivationδM：M→M「such thatδM is compatib1e withδand is1oca11y ni1potent．Prototype examp1es areδ一 id−ea1s of B and the residue rings of B with respect toδ一id−ea1s．It seems that. this no1：ion has not been considered except for scattered introduction of the. notions simi1ar七〇七his but con丘ned−in the1imi七ed−circums七ances．Hence we begin with deve1oping some standard−theory ofδ一moduIes and then shift to. the study of aδM−invariant submod−u1e Mo二KerδM，which is anルmoduIe． Some of centraI prob1ems conceming Mo inc1ude the fo11owing．. PR0BLEM1．1．1．∫8M geηθm古θdわV Moα5αB−moあZθ9 PR0BLEM1．1．2．∫5Moα力η伽佃9εηθm加dλ一mo∂ωθ〃。仇加d M45αガη伽佃 g㎝θm右θdB−m0〃θ9 We can give on1y partia1so1utions to these prob1ems．Prob1em1．1．1is a伍rmative1y answered．if one of the fo11owing conditions is satis丘ed一． 。（1）ル7is a free B−modu1e of rank one（see Proposition2．2．14）． （2）δon B has a s1ice（see1Lemma2．2，3，（2））、 （3）Some geometric con（iitions are sa七is丘ed一（see Proposition2，2．18）．. Meanwhi1e，Prob1em1．1．1is negative1y answered in the case where M「is a projective B−modu1e of rank one（Examp1e2．7．1）．Prob1em1．1．2is a伍rma一 七ive1y answered−if one of the fo11owing conditions is satis丘ed。. （1）M is a free B−moduIe of rank one（see Proposition2．2−14）． （2）δhas a s1ice（see Lemma2．2，3，（2））．. （3）〃is torsion−free as a B−modu1e and−A is a noetherian domain（see Theorem3．1．3）、.
(5) 3. （4）M「is torsion−free as a B−modu1e and B is an a伍neん一d−omain of d−imen−. sion≦3（see Coro11ary3．1．4）．. （5）Mo is a projectiveんmod−u1e（see Lemma3．1・8）・. （6）The3−modu1e Bλんgenerated by Mo is a free B−modu1e with a basis ｛e1ゾ，．，eη｝such that e｛∈Mo（see Lemma3．1．9）．. （7）B＝刈μ1is a po1ynomia1ring in one variab1e over a noetherian do皿ain λ，α：＝δ（ひ）is a nonzerq e1ement ofλandαhas no torsion in M（see Lemma2．3．1）．. If M has torsion as a B−modu1e，it is rather easy to construct a coun−. terexamp1g（Lemma2．3．3）to Prob1em1，1．2．When we try to construct a counterexamp1e in the case where M is a torsion−free B−modu1e and−B is. an a伍neん一domain，λhas tobe non一五nite1y generated overた．We construct counterexamp1es in the血ee case by making use ofthe counterexamp1es to the fourteenth prob1em of Hi1bert given by Roberts［19］，Kojima−Miyanishi［10］，. Freudenburg［81，and−Daig1e−Freudenburg［31．In such examp1es，we take B to be a po1ynomia1ring and M to be the d−i授erentia1modu1eΩB／ん。n whichδ. gives a natura1mod−u1e derivation．In the case where dimB≧5there exists a. counterexamp1e，but Prob1em1．1．2is openinthe casewhere dimB。。4．We note that there is no examp1e obtained yet in the case dimB＝4for which λis not丘nitely generated ove工ん．In order to prove the in丘nite generation of M；o，we need exp1icit forms of generators ofλas aん一a1gebra．. Prob1em1．1．2is c1ose1y re1ated−to the fourteenth prob1em of Hi1bert．If. Prob1em1．1．一2has acounterexamp1ewith afree B−mod−u1e M，we can consid−er. the symmetric tensor a1gebra R＝巧（M「）on which the modu1e derivation δM extend−s natura．11y as a1oca11y ni1potent d−erivation．Then the invariant. subring of R und−er this derivation gives rise to a comterexamp1e to the fourteenth prob1em of Hi1bert，where infinite1y many gener包tors of M；o give. in丘ni七e1ymanygeneratorsoftheinvariantsubringof月（Lemma3．2．1）．With the same setting as above but without a§suming that M is a counterexamp1e. to Prob1em1．1，2，we may ask if M is a counterexamp1e to Prob1em1．1．2 provid−ed理（M）is a counterexamp1e to the fourteenth prob1em of Hi1bert． The answer is negative（Theorem3，2．2）． Di田。u1ty onδ一mod−u1es1ies a1so in the comp1exity ofδitse1f when the. dimension of B is higher．But it seems reasonab1e that we assumeδto be.
(6) 4. CHAPTERユ．工NTRODσσnON. 丘xed−point free in ord−er to state七he properties which are1inked to the ge−. ometric structure of B and7r．Suppose B is one of the fo11owing rings：（i） a po1ynomia1ring in one，two or three variab1es over a丘e1d of characteristic zero，（ii）a PID，（iii）an a伍ne a1gebra ofd．imension one over a丘e1d of charac−. teristic zero．Then the丘xe（1−point freeness ofδimp1ies thatδhas a s1ice and. hence both Prob1ems1．1，1and−1．1．2have positive answers（Coro11ary2．2．4， Coro11ary2．2，6，and Coro11ary2．2，9）． If the d−imension of B is1ower，we can stud−y properties ofδ一modu1es in. detai1．Weinvestigatethecasewhere B is apo1ynomia1ringintwovariab1es and more genera1cases in§2．3．Noteworthy is the resu1t that if B is a norma1a舐ne C−d．omain of dimension two with a丘xed一一point free Gα一action， then any torsion−free，丘ni七e1y generated一（B，δ）一modu1e is a projective mod−u1e. and projective1y triangu1ab1e（Theorem2．6．7）．. We expect七〇make good use ofδ一modu1es七〇investigate properties of. rings and1oca11y ni1potent derivations on rings．We expect to construct homo1ogica1theory ofδ一mod−u1es and to so1ve unsoIved prob1ems concerning 16ca11y ni1potent derivations on rings with a new method．In§2．4，we give. partia1homo1ogica1properties．In§2．5，we give the re1ation betweenδand一 δ一mod−u1es．. In Chapter4，we give a s七ra．ightforward−way of constructing in丘nite1y. many generators ofKerδin the case ofsome counterexa皿p1es（Kuroda［111） to the fourteenth prob1em of Hi1bert．The generators are given in a more precise form（Theorem4，1．3）．We a1so give its app1ication．. In the丘na1chapter，we1ists unso1ved−prob1ems and give some examp1es. conceming them． Wさhere ad−d that the primary decomposition of aδ一modu1e is treated in. Takata［201．. Throughout this paper，we assume that a11rings are commutative and−. have七he identity e1ement．We a1so assume that a11rings and丘e1ds have characteristic zero．Ifλis an integra1d−omain，we d−enote the quotient丘e1d− ofλby Q（λ）．Given any ring B，we denote the set of units of3by B＊．.
(7) Chapter2 Modu1e derivations 2．1 Basic−properties of1oca11y ni1potent deriva− tiOnS In this section，we summarize the basic properties of1oca1Iy ni1potent deriva−. tions on rings that we岬ke useofbe1ow．Werefer the readers to［131，［91as basic references．. DEFINITI0N2．1．1．Let B be a ring and1etλo be a subring of B．Anλo− derivationδof−8is said to be1oca11y ni1potent if，for each6 ∈ B，there exists an integer／V such thatδN（わ）＝0．The set of a11λo−derivations of B is denoted by Derλo（B），and the set of a1110ca11y ni1potentλo−derivations of. B is denoted by LNDλ。（B）．When we say simp1y thatδis a derivation of. B，we meanthatδis a Z−derivation．We write Der（B）and LND（B）instead of Derz（B）and−LNDz（B）respective1y． Givenδ∈LNDλoB，the keme1λofδis a subring of B such thatλ⊃λo and satis丘es七he fo11owing proper七ies．. Lemma2．1．2．〃B6e㎝舳egm用。mα伽。㎝￠α伽伽g Q，δ∈LND（B）， αηdλ：Kerδ．τんeηωθんαηθj. ω川・α・伽伽g・プB・㎝古α伽伽gB＊，㎝〃舳川・んα・θB＊＝λ＊． 62ソバ・∫α・左・ヅ1α吻・1…d伽B，1・θ．，ψ6’∈λω肋6，6’∈B一（0）フ海η. 6∈λαη〃∈λ．比ηC吋Bづ8αひFDフ抗㎝80乞5λ． 5.
(8) 6. CHAPTER2．MODσLE DER∫帆nONS 63ソ山・αlg伽α1・α吻・1・・θ♂伽B．∬θη・θぴ引・伽舌θgm吻・1…∂，流㎝・・ 乞5λ．. ω耽伽伽舳ηδ・娩ηd”物的1・α伽伽α右1・ηδQ（写）㎝Q（B）㎝∂ ωeんωθQ（λ）＝KerδQ（B）・. C5フ4δ（b）∈6Bω肋6∈B，伽η6∈A −Prooヅ For（1）一（5），see respective1y Princip1e1；（b）of［91，Princip1e1，（a）of. ［9］，Lemma1．3of［4］，Princip1e g of［9］，and−Goro11ary1．20of［8］．. 口. A derivationδ∈LND（B）de丘nes anλ一a1gebra homomorphism仰from B to B［亡］and一λ一a1gebra au七〇morphismsψαof B as fo11ows．. lLemma2．1．3．〃Bわe㎝αlg伽αoηerα伽”ん。プ。んαη㏄亡θヅ前。鮒。， δ∈LND（B）フαηdλ二Kerδ．工θ亡仰：B→B［t］うεdψηεd∼. oo 洲一Σ去1w， 4＝0. 舳舳帥11・αρ・伽・mlα1・伽g…ザB』んθ岬士1・㎝んα1g伽αん・m・m・・一 ψ5m．ハ0rθαCんα∈んフ1θ｝α：B→B6ε加伽εd切 oo ρ一（1）一Σ去1w・ づ＝0. 肋θη％づ・㎝んαlg伽ααm亡・肌・ψ1・m・α挑伽ηgψ0＝id．㎝∂％十β＝ ψαOψβ・. ProoヅSeePropositions2．アand3，2of［4］．口 Next we de丘ne a s1ice and−summarize the property of a s1ice．. DEFINITI0N2．1．4．Let B be a ring and一δ∈LND（B）．An e1ement u of B is ・・11・d・・1i・・一. Efδifδ（・）：1．. lLemma2．1．5．〃3ろθαヅ初g c㎝6α伽伽g Q，δ∈LND（3）α〃λ＝Kerδ． 舳〃05θ伽右δんα8α51乞Ceu∈B．珊㎝ωθんαのθj （1）B＝小1αηれ1・かαη・・㎝δ㎝右α1・り・・B．. （2）D・伽・α“一αlg伽α㎝d・m・・吻・mダ。：B→B的 oo ψ一一（・）一Σ去州（一・）1・.
(9) 2．ユ．BAS∫σPROPERT工ES OF LOCALLYNILPOTENT DER∫WしnONS7 珊θ一＝ψ一、、（B）．∫ηα棚㎝1α・，ゲBl・αん一α1g伽αg㎝θm舌θd物ろ。，．．．，6、∈. B，1．ε．，B＝ん［6。，＿，6，1フω加ヅθハ・αμ∂・プ・んαη㏄工θ舳牝鮒・，伽η λ：んレ＿u（61），、．．，ψ＿仙（わ、）1．. Pザ。oヅSee Theorem2．8of［4］for（1）and see Coro11ary1．3．23of［7］for（2）． □ Next we consider a ring of fractions of B．. Lemma2．1．6．〃δ∈LND（B）㎝d lθωうθαmu物1乞。α左伽吻。105ed舳う8e亡 。∫Kerδ．工θ左8山1δ：8−1B→3−1Bろe♂φ鵬dろμ ・一・／. i1ド）. τ加ηωθんα〃θ3−1δ∈LND（8−1B）α棚Ker（3■1δ）＝8−1（Kerδ）．. ProoヅSeePrincip1egof［91．口 We de丘ne aδ一idea1and summarize the properties ofaδ一idea1．. DEFINITI0N2．1．7．Letδ∈Der（B）．An idea1∫of B is ca11ed aδ一idea1if δ（∫）⊂∫．Then we can d−e丘neδ∈LND（B／∫）in a natura1way，. Miyanishi［ユ51proves the fo11owing assertions．. I−emma2．1．8．〃Bろθα肌。e伽れ㎝domα伽。㎝6α伽伽g Q，δ∈LND（B）， αηd∫αδ＿づdθαz． 71んθηtoeんα①θj. ω肋㈹帥mθ舳・・ヅ・∫ハ・αδ一助α1． C2ソ別・ψ・・1α左θd帥mαザμ・岬㎝θη6・μ1・αδ一渡α1．. 例耽舳・α1市1・αδ一棚・α1． The above1emma is proved−by making use of the fo11owing．. lLemma2．1．9．〃B6εαηαlgかαo附α加〃ん。∫cんαmc右erづ5枕鮒。， δ∈LND（B），αηdλ＝Kerδ．W挑ん挽θ肌。亡α切。ηoゾエemmα2．ノ．3，αη〃θαZ∫. ・∫引・αδ一助αけ㎝d㎝佃れα（∫）⊂∫∫・・㎝Vα∈ん． Next we introduce the notion of丘xed−point freeness．. DEFINITI0N2．1．1O．Letδ∈LND（B）．We say thatδis丘xed−point free if the idea1of B generated byδ（B）is a unit ideaI．.
(10) CHAPTER2．M−ODσLE DER工帆nONS The丘xed−point freeness is characterized as foユ1ows．. Lemma2．1．11．〃B6θαザ切。㎝亡α伽伽g Q．肋㎝δ∈LND（B）づ8伽e∂一 ρ0加古伽θぴ㎝d㎝1いア海re4”0mα切mα1乞伽Z0ゾBω肋Cん必αδ一助α1．. Pザ。oψSeePrincip1e4of［9］．口 丁ρend this section，we consid−er some specia1ized cases、. ：Lemma2．1．12．〃B＝ψ］6eαρo伽。m4α1ヅ伽g o〃εヅαμハ。μんαr㏄一 加れ5切。 zeroαηd Zθ亡δ∈LND（B）．r加ηδ＝α岳∫or80meα∈ん． Pザ。oヅSee Princip1e8of［8］、. □. lLemma2．1．13．〃B6eα刀。伽。mづαけ伽g加地。りαグ台α6Ze50のeグα〃” o！cんαmc施r乞5切。 zemαηd Zεオδ∈LND（B）、Tんeη流6ナe e切5左”，μ∈Bαη∂. ∫∈ψ1舳舳α㍑一仏μ1α舳一∫品・ The above1emma is due to Rentsch1er（see［18］）．We can easi1y deduce the fo11owing coro11arγ Coro11ary2．1．14．工θ亡B，δ6θα5加工θmmα2．ノ．ノ3．舳〃08θ仇α￠δづ8ηoηzθヅ。. ㎝d伽e抑0舳伽θ．珊㎝δんα5α51乞Cθ． The above coro11ary was genera1ized by Ka1iman151in dimgnsion3．His reSu1t StateS：. lLemma2．1．15．〃B6εαρo伽。mづα用ηg伽伽eεりαヅ肋Zε50りεrα伽” ○ゾ。んαmc加桐曲。 zθroα〃ZθCδ∈LND（B）。8伽〃03e挽α之δ乞3ηo肌θηoαη∂ 伽e∂一ρ0伽亡伽ε。τ加ηδんα8α311Cθ。. 2．2 δ一mod．u1es Let B be a ring，δ∈LND（B）andλ＝Kerδ．Genera1izingδ一idea1s and−the residue rings of B byδ一idea1s，we sha11introduce the notion ofδ一modu1es．. DEFINITI0N2．2．1．Let M「be a B−mod−u1e with anλ一mod−u1e endomorphism δM・：M→M．Apair（M一，δM）isca11eda〃e一（B，δ）一mo仇Zθ（abbreviated−as aρヅe一δ一mo∂uZθ）if for any6∈一B andητ∈M，、 δM（6m）＝・δ（b）m＋6δM（m）．.
(11) 2．2．δ一M．ODσLE8. 9. Then we say thatδM is a modu1e derivation．A pre一δ一modu1e（M「，δM）is ca11ed a（B，δ）一modωθ（abbreviated as aδ一η70dωe）ifδM is1oca11y ni1potent， i．e．，．for eachη7∈ハ∫，there exists an integerηsuch thatδvη（η7）＝0．If there. is no fear of confusion，we denoteδM simp1y byδ．Whenever we consider a pre一δ一modu1e structure on a B−modu1e，the derivationδon B is the same and丘xed once for a11，. Let（M’，δM）be aδ一modu1e．A B−submodu1e W of M is ca11ed−aδ一. submodu1e of（M，δM）ifδM（W）⊂W．Then（N，δN）is aδ一modu1e，whereδN is the restriction ofδM．The quotient modu1e〃／N is a1so aδ一modu1e with δM／Nd・丘n・din・n・tu・・1f・・hi…. If（M，δM）is aδ一modu1e，then Mo：＝KerδM＝｛m∈M lδ〃（m）＝0｝ is anλ一modu1e．We retain be1ow the notationsλ，Mo，etc．㎜1ess otherwise speciied． We can regard modu1e derivations on modu1es as homogeneous1oca11y. ni1potent derivations on grade（l rings as fo11ows． Letδ∈LND（B）and. 1et M be aδ一modu1e，We de丘ne B−modu1es刀（M）（乞∈N）inductive1y. by乃（M）＝B andη（M）＝ルτ⑳Bη＿1（M）． Then the刀（M）areδ一 modu1es in a natura1fashion by（1）of Lemma2．4．3be1ow．Hence T（M一）：： ㊥二〇刀（M’）is aδ一modu1e．Let∫be the two−sidedid−ea1ofT（M「）generated by ｛m⑳几＿η⑧m I m，η∈M’｝、Since∫is aδ一submoduIe ofT（M），it fo11ows that 3（M「）：：T（M「）／∫is aδ一modu1e．Thenδ5（M）is a homogeneous1oca11y ni1po− tent derivation on a graded ring3（M）＝㊥二〇η（M一）／（∫∩η（M’））．More−. over，we have T1（M）／（∫∩T1（M））：T1（M）＝M andδ8（〃）IM：M→M. isequa1toδM・Converse1y，1etR二㊥二〇兄beagradedringsuchthat Ro＝B and−1etδbe a homogeneous1ocaIIy ni1potent deriva七ion on R．Then δ：＝δIR。∈LND（B）．lMoreover，兄is a（B，δ）一modu1e for each乞．Hence modu1e derivations on modu1es correspond−equiva1ent1y to homogeneous1o− ca11y ni1potent derivations on graded−rings R＝㊥こ。兄whenβ1generates. ROVerRO． From this point of view，we can obtain the assertions concemingδ一 modu1es in this section from those for homogeneous1oca11y ni1potent deriva− tions on graded rings． Since we are interested−in the modu1e derivations be1ow，we rewrite the known properties in terms of modu1e derivations．. Lemma2二2．2．ハ。ヅδ∈LND（B），ωθんαりθ伽∫o〃。ω伽gα58θ棚㎝5．. ω〃δMろθαm・〃θ伽1州㎝㎝αB−m・〃θM．Tん・η加工θ伽1・m1・ ん01d8．Wαm吻，∫・ザαημ∈B，m∈M，㎝“・ヅ㎝〃05舳θ加物e用，.
(12) 10. CHAPTER2．M’ODσLE DER∫帆丁工ONS ωeんαりθ. ／小）一. g（1ジ（・）1・壬（・）・. C2μ6川一うθα岬一δ一m・d仙1θ．8W・・θ舳α舳う・θ亡丁・〃g・η・m之θ・Mα・. α3−m0〃θα〃舳〃03e流α6，伽伽んm∈T，伽陀θ地63αη加物θグ 几舳・ん伽亡δMη（m）：0．珊㎝M乞・αδ一m・〃θフ1。θ．，δM1・1・・α吻 η物0右θη古．. 63ソ物θηαηλ一m・〃θM。，1θ6M＝M。軌Bαη〃物θδパパM→M物 δM（m舳）＝m⑭δ（6）』ん㎝（M，δM）1・αδ一m・〃・． ω〃Mう・αρグθ一δ一m・地1θ㎝〃右3ろθαm刎物1乞・α批り吻・1・・θd舳ろ・θ右・∫. B．肋θm0〃θ伽伽α切0ηδMε娩ηゐ舳ψ物60αm0〃θ扮伽肋0η 3’1δM・㎝3■1M6μ. ・一・1・（子）一5δ”（宇（8）m. ∬θη・θ3−1Mづ・α岬一δ一m・d此．伽ρα棚㎝1α・フぴ8⊂λ㎝d〃乞・αδ一 mo伽Zε，挽eη3’1M．乞5α（8−1B，3−1δ）一mo伽Zeα〃砒んα〃θKer（8’1δM） ＝3■1（KerδM）．. Wehavetheresu1tssimi1artoLemma2．1，3andLemma2．1．5foramodu1e deriva．tion．. lLemma2．2．3．舳〃08θ伽τB⊃Q㎝”θ伽岬士：B→靴1α5伽刀emmα 2．ブ．3一〃M6・αδ一m地1θ．λ・舳mθ伽伽・伽伽α…舳η・62ソ㎝“3ソ 流α老δんα8α51乞C舳∈B．τんeη胱んαりe伽∫0110ωηgα55e榊0冊 μノー0e派ηe帖，M・：ハ∫→ル7［左］物 oo. 舳（・）一Σ去1・wl ｛＝0. 肋㈹M［tl＝M晩帥1．珊㎝仰，Ml・㎝λ一m・ω・ん・m・m・ψ1・m ・α亡舳伽岬亡，M（6m）＝ψ士（6）物，M（m）∫・・㎝μろ∈B，m∈M．. 62ソ肌んα・・M＝M。［u1＝M。軌B。∬㎝・・Ml・α伽的ク㎝・m看・dB− m0〃θぴ㎝d㎝いゾM0づ8α伽伽佃9㎝εmτedλ一m0伽e．MOreOUeヅフ M乞5α伽eB−m0伽e乞∫αηd㎝いプM0乞8α伽eんm0〃θ．.
(13) 11. δ一MIODσLE8. 2．2．. C3ソ. 冊∂・伽用λ一m・舳θ〃・m・ψ・W一、，M：M→M’6μ oo. ψ一一、・（・）一Σ去1・｛（・）（一・）1・. 4＝0. 丁加ηωθんα〃θM；o＝KerδM二ψ＿伽，M（M’）．肋ρα彬。刎Zαザ，ゲM台3. g舳m亡θれひm。ゾ．．，m、α・αB−m・〃θ，伽ηM。＝ルー。，M・（m・）十 十ルー、，M・（m、）．. Wesha11app1yLemma2．2．3，（2）tothespecia1cases．FromLemma2．1．12， Coro11ary2．1．14，and Lemma2，1，15，we can easi1y d−ed−uce the fo11owing coro1− 1ary．. Coro11ary2．2．4．工θ左δ∈LND（3），α棚Mαδ一moぬZθ．舳〃03θ流α左δ45. η㎝湖0αηd伽ed一ρ0舳伽θ。〃B乞8αρ0切0m乞α用ηg佃0ηθフω00r伽θe ・αグ州ε・・ωεザαμd・∫・んα・㏄1・榊1・鮒・，挽㎝胱んαωθM＝M。［u1＝. MO軌B∫0r80m舳∈B． Wecana1soapp1yLemma2．2．3，（2）tothecasewhereBisaPID（Coro1− 1ary2．2，6）．In order to show it，we need the fo11owing1emma．. lLemma2．2．5．〃BろθαP∫Dα〃1θ亡δ∈LND（B）．舳〃05θ舳右δ乞8 ηoηzぴ。α〃ガ”ed一ρo伽之介ee。珊eηλ二Kerδづ5α伽〃α〃B＝λ［”1φ5α ρ0切0mづαlr伽g伽㎝“αグ肋1θ． Prooゾ．Since B is a PID，λis a UFD．We sha11丘rst show thatλis a丘e1d，. Assumethe contrary，and1et∫be anonzero non−unit e1ement ofλ．We can take∫to be irreducib1e inλ．Then∫is irreducib1e in B as we11．Hence∫B is. a prime idea1．Since B is a PID，∫B must be a maxima1idea1．Meanwhi1e∫B is aδ一idea1，which contradicts the丘xed−point freeness ofδ（Lemma2ユ．11）． Now it is c1ear that B＝刈”l withδ（”）＝1．. □. Coro11ary2．2．6．工e亡B，δ6θα5加工emmα男．2−5．工e右Mうθαδ一mo仇Ze． 肋εηM’乞8α伽θB−mo〃θω肋αうα5づ3c㎝オα伽θd伽Mo Pヅ。oゾ、Sinceλis a丘e1d，Mo is a freeλ一modu1e．Sinceδhas a s1ice，it fo11ows. from Lemma2．2．3，（2）that〃：Mo⑭λB，which conc1ud−es the proof． □ We can a1so app1y Lemma2．2．3，（2）to the case where B is an a伍ne a1gebra of d−imension one over a丘e1d of characteristic zero（Coro11ary2．2．9）．. We need七he fo11owing1emma（see Miyanishi［16，Theorem2．11）．.
(14) 12. CHAPTER2．M’ODσLE DER∫V兇nONS. Lemma2．2．7．〃B6e㎜蜥胱αlg伽αo∫肋m㎝3乞。η㎝εoりθrα伽〃。∫ cんαmc加れ3地zθro，δ∈LND（B），α〃λ：Kerδ一3刎〃03e挽α左δ乞5ηoηzθヅ。. ㎝d伽d一ρ・舳伽・。Tん・η≠ん…θ洲川∈B舳・ん伽右B二刈u1フい・ 亡伽・・θη∂・物1・・θ一αηd1一δ（u）乞舳伽亡㎝老．孔ザ伽・m…，川・㎝ル右伽 r伽g．. In fact，we have the fo11owing resu1t．. PI．oposition2．2．8．工e亡B，δうeα5加工θmmα2．2．γ丁加ηδんα5α8Z乞。e．. ProoヅSince q＝1＿δ（u）is ni1potent，we haveゴ＝0for some integgr. r．Let∫∈B and write∫＝αo＋α1u＋＿十α肌”withα乞’∈λ．Then δ（∫）＝（α1＋2α2u＋…十肌α肌u卜1）（1−9）．If we can take∫so thatα1＋. 2α2u＋一十ηαηが一1＝1＋g＋…十g「．1，then we haveδ（∫）＝1，The. existenceoftheαづsatisfyingthisequa1ityisobvious．．口 App1ying Lemma2，2．3，（2），we ded−uce：. Coro11ary2．2．9．工θ舌B，δ6θα8加工εmmα2．2．γ 工θ舌M6eαδ一mo仇Ze．. 珊θηωθんαηθM二M。［u1＝M。⑳λB∫・…m舳∈B． We a1so obtainthe fo11owingproperties ofδ一modu1es．. remma2．2．10．Zθ舌Mうθαδ珊。〃θ．τん㎝ωθんαりθ伽∫o〃。ω伽gα58θr一 向0η5j. μノAnn（M）づ5αδ一〃eαZ oゾB．. 62ソ∬引・αη・伽θ∂ヅ切g，伽η伽6・グ・4㎝ρα吋払。、・ゾM4・αδ一・ωm・〃θ 0∫M．. Pヅ。oヅ（1） Take any6∈Ann（〃）．Then6m＝0for any m∈M．Thus we have0＝δM（6m）：δ（う）m＋ろδM（m）＝δ（6）m for any m∈M，This imp1ies thatδ（6）∈Ann（M）． （2） Let m∈M七。、．Then there exist a nonzero e1ement6∈B such that. 6m＝O，We have0＝δM（らm）：δ（わ）m＋6δM（m）。Mu1tip1yingろ，we have. 62δM・（m）：0．Si…Bi…d…d，62≠0・・dh・…δM（m）∈仏。、． □.
(15) 2．2．. 13. δ一MODσLES. DEFINITI0N2．2．11．Forδ∈LND（B），the degree of仰（6）∈B［亡1，which we denote byレ（6），is ca11ed−theδ一degree of6∈B，Simi1ar1y for aδ一modu1e M，the degree of仰，〃（m）∈M［亡］，which is equa1to the integer r such that. δ「（m）≠0andδ「十1（m）＝0ifm≠0，is ca11ed theδ一物脇。fm∈M，We d−e丘ne the degree of七he zero e1ement of〃［τ］to be＿oo．For m∈〃，we denote theδ一degree ofη7∈M byμM（η7）． We have the fo11owing properties ofδ一degrees．. 1Lemma2．2．12．〃B6θ“伽g c㎝舌α伽伽g Q，δ∈LND（B）㎝∂Mαδ一 m0〃θ．τん㎝ωθんαωθ伽∫0110ω伽gα85θr肋㎝5j. （1）伽岬ら∈Bα棚m∈M，ωθんαりθリM・（うm）≦ひ（6）十レM（m）．耽 榊α物ん0脇乞∫Mづ8B一亡0ヅ8乞㎝一価θ，mOre岬Cづ5吻，ゲM0づ5ん右0グ8乞㎝一価θ 6・・θ工・mmα2．2．ノη。. （2） レM（m＋m’）≦max（レM（m），レM（m！））∫oヅαημm，m’∈M． Pザ。oア（1）is easi1y shown by the Leibniz ru1e（Lemma2．3，12）．（2）is obvious． □. The fo11owing resu1t is an ana1ogue of Lemma2．1．2，（5）、. Lemma2．2．13．〃Bうθαr切。㎝左α伽伽g Qフδ∈LND（B）㎝d Mαδ一 m・〃θ。8W…挽α6〃1・B一乏・ザ・づ㎝一価θ㎝d物6m∈M・α切・伽δM・（m）∈ Bm．τ加ηδM（m）・・0．. Pグ。oヅSupPose that m≠0andδM（m）≠0．ThenδM（m）＝6m for some nonzero e1ement6∈B．We have レ（6）十μ〃（m）＝μ〃（6m）＝レM・（δ〃（m））：ひ〃（m）一1．. Henceμ（わ）：一1，which is absurd．. □. Lemma2．2．13imp1ies the fo11owing resu1t．. Proposition2．2．14．工e亡Bうeαr伽g co肋α伽伽g Q，δ∈LND（B），α棚〃＝. 8・α伽θ3−mo〃θoゾm舳。ηθ．舳〃03e物6M48αδ一mo〃θ．肋θη δM（・）＝0．批η・θM・＝λ・㎝dM＝B⑳λM・． Proposition2．2．14imp1ies that M is id−enti丘ed．with3asδ一modu1es．The equa1ity M＝BMo d−oes not ho1d−for a projective（B，δ）一mod−u1e of rank one. （see Examp1e2．7．1）．We can genera1ize Proposition2．2，14as fo11ows．.
(16) 14. CHAPTER2．M’ODσLE DER工帆丁工ONS. Proposition2．2．15．工θ右B6θαグ初g co枇α伽伽g Q，δ∈LND（B），MαB−. m0〃θ，α〃M’＝M㊥Bθ伽伽θ0右5um0！M㎝れ伽θB−m0〃θ0ゾm舳 ㎝θ．舳〃05θ右んα左M’1・αδ一m・♂uleα棚M乞・αδ一舳うm・〃e・ゾM’．rんθη δMl（θ）∈M．. ProoヅSince M is aδ一submodu1e of M’，it fo11ows that M’／M＝Be is a. δ一modu1e．By Proposition2．2．14，we haveδ〃（e）∈M．. 口. The fo11owing resu1t is an am1ogue of Lemma2．1．2，（2）．. lLemma2．2．16．〃Bろθαグ初g c㎝亡α伽伽g Qノδ∈LND（B）㎝d Mα δ一m・〃・．8W…舳舌Mづ・B一右…1㎝一価θ．τん・いアδM（6m）＝0ω肋 わ∈B一（0）α棚m∈M一（0），枕㎝δ（6）＝0㎝∂δM（m）＝0． Pヅ。oヅ IfδM（6η7）＝0，thenひM（6ηz）＝リ（6）十μM（η7）＝0and henceリ（う）＝0. and一ひM・（m）＝0，which imp1ies七hatδ（b）＝0andδM（m）＝O．. □. In proving the above resu1ts by using一δ一degrees，we have to assume that. M is B−torsion−free．We sha11show that this assumption is equiva1ent to assuming thatルτo isノ」torsion−free．. Lemma2．2．17．〃Bうeαグ伽g c㎝老α伽伽g Qフδ∈LND（B）フ㎝d Mα δ一mo〃θ．肺㎝〃乞8B一右。ヅ5乞㎝一価eぴα〃㎝1リゲM；o乞8ん左。ヅ84㎝一価e． 〃。oポ‘OnIy ifフ’part is c1ear．We prove“if”part．Suppose that Mo isん. torsion−free．For6∈B and m∈M，sippose that6m＝0，6≠0and m。≠0． Let r±リ（6），5＝リM（m）．Then we have. ・一岬・）一Σ（∵）／｛（・）ll（・）一（∵）W（・） ｛十ゴ＝r＋8. Sinceδ「（6）∈λ，δ5（m）∈Mo and一八4o is／1＿torsion＿free，eitherδ「（6）orδ8（η7）. must be zero which is absurd．. 口. Ifδhas a s1ice or if M is a free B−modu1e of rank one，then it ho1ds that. 〃＝Mo⑳λB．This is a1so the case with more geometric conditions． Proposition2．2．18．工θ亡Bうεαηαが胱ηormαZ r伽g oりerαηαZgθ伽α乞。α吻 。Z08θdガe”ん0ゾ。んαmc老θれ8れ。 zeroフδ∈LND（B），αηdα＝δ（B）∩λ，ω肋。ん. 乞5αηづdeαZ oゾλ．工e亡X＝Spec B，γ：Specλフαη∂7r：X一÷γ右んθηα舌urαZ. 仰切㎝右m・・p肋・m．λ舳mθ舳・・dimx（π一1（γ（α））≧2．工θ〃6・α伽的 g㎝θm亡θd（B，δ）一m・〃θ．λ・舳m・伽τdepth（M）≧2㎝“・pth（M。⑳λB）≧ 2．肋θηM一＝・Mo⑳λB．.
(17) 15. 2．2．δ一MODσLES PηooヅLetア，9be the Ox−Modu1e associated to M，M；o⑳λ8respective1〆 Letひ＝7r−1（γ＿γ（α））and1etゴ：σ→X be the natura1open immersion． By the next1emma，we haveゴ、（プ＊ア）皇アand一ゴ、（ゴ＊9）隻9，Note that ゴ＊ア＝ア■σandゴ＊9＝9■σ．There is anatura1morphism∫：9→ア．In ord−er to show that∫is an isomorphism，i七su伍。es七〇show that∫1ひ：91σ→ア1σ is an isomorphism，i．e．，∫π：9π→みis an isomorphism for every”∈σ．If we show that M拒皇（Mo⑳λB）沖for a．ny p∈γ＿γ（α），then for any準∈σ with p＝7r（準），we have. M奉＝（B一羽）一1M＝（B一準）■1（λ一P）一1M ＝（B一郡）一1（ガP）■1（M。⑳。B）＝（M。軌B）邪． Hence it is enough to prove that M拒呈（Mo⑱A B）事for any p∈γ＿γ（α）一If. 仲≠α，then there existsα∈α＿炉．Sinceα∈α，we haveδ（u）＝αfor some u∈B．Then u／αis a s1ice of the derivation on到α■1］ind−uced byδ．Hence. We haVe M［α一11＝（M［α一11）。［ψ1＝M。［α一11［・／α1. ＝M・⑳川ガ11⑳刈、一・］λ［α一111ψ1. ＝M。軌帥λ刈α’11＝（M。軌B）［α一11．. Si…α≠≠，w・h…Mr（M商B）早．. □. In the proof of Proposition2．2．18，we use the fo11owing1emma，which is Coro11ary5．10．6of Chapter IV of EGA［6］．. ＝Lemma2．2．19．〃XうθαηoヅmαZ5cんθme，ひ㎝oφ㎝舳う8θ老。ゾXフ㎝d ゴ：σ→X伽ηα施ml・岬づmm・ザ・1㎝．λ舳mθ流ατ・・dimx（X一σ）≧2． Ze右アうθαco加肥枇0x−mo仇Zθω肋depthx＿σア≧2．珊θηゴ、（ゴ＊（ア））皇ア．. Wegive anexamp1eofaδ一modu1e Mofdepthonewhichdoes not satisfy the equa1ity BルZo＝ルグ（cf．Bonnet［1］）．. ExAMPLE2．2．20．Let B＝Ck，”’，μ，μ’l be a po1ynomia1ring and−1etδ∈ LND（B）be de丘ned byδ（”）＝δ（〆）＝0，δ（g）＝”andδ（μ’）＝”’．Then we can easi1y show thatλ：C［”，”’，”リ’＿”’μ］as fo11ows，Indeed−1et3；. ｛1，”，”2，．．．｝beamu1tip1icative1yc1osedsubsetofB．Since3■1δ（ツ／”）二1，. we have Ker3−1δ：C［”，1μ，〃＿”’g，刈by using Lemma2．1．5，（2）．Since λ。：Ker3−1δ∩B，we haveλ＝C［”，”’，”ノ＿〆μ］．Let M be the idea1of B.
(18) 16. CHAPTER2．M’ODσLE DER∫帆丁∫ONS. generatedby”，ひ，”’，ぴ．ThenMisaδ一idea1andhenceaδ一moduユe．Wec1aim that depth M＝ユ。Ind−eed”is not a zeエ。 divisor on M．For any∫∈B，we. have∫ア＝0andπ≠0in Mμ〃，whereτis the resid−ue c1ass of”∈M−in. M／”M．HencedepthM＝1．WehaveMo＝λ∩M；拙十州十（〃一ψ）λ and−therefore BMo＝”B＋”’B＋（πび＿”’リ）B＝”B＋〆B，which is not equa1to M’．Note thatα：＝δ（B）∩λ＝πλ十〆λa．ndγ（α）contracts to a. point（0，0，0）by the quotient morphismπ：Spec B→Spec A Let M be a B−modu1e generated by m1，．、．，m、∈M一．Suppose that M「is a pre一δ一modu1e．Then the pre一δ一modu1e structure is d−etermined byδM（m｛），. 1≦乞≦r．Further，ifδM・w（m｛）＝0for some N〉0（1≦乞≦r），then M’is a δ一modu1e（see Lemma2．3．12，（2））。With this remark in mind，we deine the triangu1abi1ity ofδM for a free（B，δ）一modu1e M’of丘nite rank．. DEFINITI0N2．2．21．Let M’be a（B，δ）一mod−u1e〃which is a free B−modu1e of丘nite rank，A modu1e d−erivationδM is detemined by theδM（e｛）for a free. Ibasis｛e1，．、．，e、｝of〃．Wesaythat amodu1ederivationδ〃isか乞㎝g砒励1eif there exists a free basis｛θ1，．．。，e、｝such thatδM（e4）∈Bθ1θ．．．㊥Be乞＿1（2≦. 乞≦r）andδ〃（e1）一＝0、’C1ear1y，a triangu1ab1e mod−u1e derivation is1oca11y ni1potent． To end this section，we raise the fo11owing prob1em．. PR0BLEM2．2．22．∫8αmo仇Zε加ヅ伽αれ。ηδM0ηα介θe B−mo此ZθM納αη一 gulαう1θ9. In fact，it has a counterexamp1e（Examp1e2．3．10）．. 2．3 The case where B＝刈μ］ In this section，we consider the case where B：λ［μ］is a po1ynomia1ring in. one variab旦e over a noetherian ri早gλ⊃Q．Suppose that．δ∈LND（B）is nonzero．We assume thatδ∈LNDλ（B）is de丘ned byδ（μ）＝α∈λ＿（O） and Kerδ＝。A This condition is automaticauy satis丘ed ifλis an integra1 d−omain．The case where B is a po1ynomia1ring in七wo variab1es over a丘e1d． ん。f characteristic zero is a specia1case，Indeed，if B三s a po1ynomia1ring in. two variab1es over a丘e1d一ん。f characteristic zero，then there exist”，μ∈B such that B：んk，μ1，δ（”）＝0andδ（μ）・：α∈ん［”1一（0）by Lemma2．1．13．. Thenλ＝Kerδ：ψ］is a PID and B：A［μ］．We sha11investigate the structure of（B，δ）一mod−u1es．.
(19) 17. 2．3．THE CA8E WHERE B：λ［γ］. 1Lemma2．3．1．〃B＝刈V］ろeαρo伽。m乞α用ηg o〃θrαηoθ伽ヅ乞α“omα伽 λαηd dφ胱δ∈LNDλ（B）うツδ（μ）＝αフω加肥α45αηo伽eroθZθme耐。ゾλ．. 〃Mうεα伽的9㎝・耐ε∂（3，δ）一m・〃・舳・ん舳右伽・1θm㎝広αんα舳・. 亡。r5づ㎝伽M．〃M1＝BMo6θ伽B−8ωmo〃θoゾMgeηem右θれVMo． 肋㎝M。づ・α伽・1舳m㊥二。9｛M・㎝∂M・＝BM・ルBM・・∬㎝・θM・1・α 伽的9・η・・αオθ“一m・d仙1・．伽伽㎜・ヅ・，ω＾舳811〃＝8■1M。フωんθηθ 3＝｛1，α，α2，＿｝．. ProoヅWe note thatλ：Kerδ．Suppose m：mo＋岬1＋．．．’十ガm、＝0 with m乞∈M；o．Thenδ「（m）二r！α「m、＝0and hence m、：0by the assumption．By repeating the same argument，we obtain肌＝0for a11乞． This imp1ies that BMo＝㊥二〇ノMo．Since B is a noetherian ring，BMo is. a inite1y generated B−modu1e．Suppose that肌1，＿，η、∈M generate BMo as a B−modu1e．We may assume that eachη4be1ongs to Mo．Then we have. M。＝BM。／ψM。＝ル・十…十ル、． Fina11y we show that3−1M＝8■1M1，Since the derivation3■！δon 3−1B has a s1iceμ／α，we have3−1M＝（3■1M）o［μ／α］＝3−1MoエV／α］＝. 3−1（州μ1）：3−1M。．. □. The fo11owing is an immed−iate consequence of the above proposition．. Coro11ary2．3．2．工θ亡B，λフδ，M，Mo，αηd〃1うθα8加工θmmα2，3．ノ．. 舳〃05e伽亡M0乞8α加eλ一m0〃e。肺5C0η∂肋㎝乞8α伽m舳Cα物5α的βθ∂ ぴ山・α〃D．4｛∫。，．．．，∫η｝1・α伽・λ一6α・1・・∫M・フ流㎝舳・α1・・α伽θ 8一ろα・1・・ゾM。αη＾θη・θα！冊8−18一ろα・1・・ゾ8■1M＝3−1〃。．. In Lemma2．3，1，if（0M・：α）：＝｛m∈M−1αm：0｝≠0，then Mo is not necessari1y a丘nite1y generated。λ一modu1e．This is shown in the fo11owing. 1emma． lLemma2．3．3．〃B二ん［”，V］わθαρo伽。m乞α1ザ初g㎝〃ε伽eδ∈LNDん（B）. 物δ（π）＝0㎝dδ（μ）斗．〃M＝B／・2B。肺η〃1・α（B，δ）一m・〃θ伽 αη仇mけα5肋㎝α〃M04”0≠α伽伽ゆ9εηεm右edλ一m0〃e． ProoナWe can prove Mo＝（ん十畑）／”2B as fo11ows．SupPoseδ（∫）＝功∈ ”2B with∫∈B，where∫リdenotes the partia．1d−erivative of∫with respect to. μ．Then we haveん∈”8，i．e．，∫＝”g＋αfor some g∈B andα∈ん．Hence we have Mo＝（た十”B）／π2B which contains”ぴfor a11乞．This imp1ies that Mo is not a丘nitely genera七edλ一modu1e sinceλ＝ん［”1一. □.
(20) 18. CHAPTER2．MODσLE DER工VAnONS However，there exists a丘ni七e1y generated torsion（3，δ）一modu1e M such. that Mo is a丘nite1y generated一λ一mod−u1e，as s与。wn in the fo11owing1emma．. I．emma2．3．4．〃B：ψ，V］ろθαρo伽。m乞α1ヅ初gαη〃φηeδ∈LNDん（B） 物δ（・）＝0㎝州μ）＝ル）∈ψ1＝A工・亡M＝（B・。㊥B・。）／（胸，μ・。十 ル）・・，∫（・）・ユ）6・α（B，δ）一m・伽・d・伽洲gδ（・。）＝0㎝dδ（・。）＝・。フ舳…. B・。㊥B・。1・α伽・（B，δ）一m・〃・・μα舳地・．4ω・ω伽M＝Bε。十旋。，. 抗㎝ωθんαのθM＝佃十地α〃M。＝佃十λル）百。． Pザ。oヅSinceμ百1＝一∫（”）百2，we have B百1⊂λ百1＋Bε2． Sinceμ百2＝0，. we have B百2⊂ λ百2， Hence we have M ＝ λ百1＋λ百2． Suppose that. m＝。α1官1＋α2言2∈Mo withαづ∈λ．Thenδ（m）is represented byα2e1∈ （μe2，脈1＋∫（”）e2，∫（”）e1）．This imp1ies thatα2∈∫（”）λand−hence Mo：. λ百十・4∫（”）奄．口 Next，．we consider the case where M「is a free B−modu1e．For the sake of. simp1icity，we denote a11ofδ，δM，3−1δ，and3■1δM by the same symbo1δ ifthere is no fear ofconfusion．We extendδto a derivation oh the matrix rings M（η，B）and M（几，8■1B）in a natura1fashion，which is a1so denoted by. δ．We have the fo11owingassertion．. ：Lemma2．3．5．〃B＝刈切うθαρo伽。m乞αけ伽g伽㎝“αr肋Zθ0りer αP∫Dλαηd Zetδ∈LNDλ（3）6e dφηeゴ6μδ（μ）＝α∈λ＿（0）ω乞肋 Kerδ＝λ．工θ右3＝｛1，α，α2ゾ．．｝．工et Mうeα介eθ（B，δ）一mo伽Zeω肋α介eθ うα・1・｛・・，…，・几｝・肋ηδパパ・岬η・州. （1：ll／一δ（パ）・（1），. 肋肌D』α岬θ・（1）D∈Ml（η，3），（2）D−1∈M（η，3一・B）α州3）δ（D一・）D∈. Ml（η，B）．伽αd肋㎝，伽・・岬㎝θ舳・ゾ㎝ザ・ωηθ・舌・ヅ・ゾDんα・川…m−. m㎝伽右0ザ5切B． ProoヅLetルτ1＝BルZo，By Coro11ary2．3．2，there exists a free basis｛∫ユ，．。．，ん｝. of3−1M1＝8−1M such that∫乞∈Mo．Since∫｛∈M’，we have. （二）一・（二）.
(21) 2．3．THE CA肥WHE肥B。。λ［γ］. 19. with0∈M（几，B）．Since both｛θ1ゾ、、，eη｝and｛∫1，＿，ん｝are free bases of. the free8−1B−modu1e3■1M，we have. （二）一パ（二）. and D−1∈M（η，3−1B）．lMoreover，since theヵare in Mo，we have. （ll：ll）一イ）一ψ（1）・. Henceδ（D−1）D∈1M（η，B）．Fina11ywe cIaim that the components ofany row vector of D have n−o common factors in B，i．e．，we have gcd一（dづ1ゾ．、，d伽）≡1. for a11づ，where we denote D＝（d乞ゴ）．Ind−eed，suppose that d｛1＝〃’｛1，＿，d乞η＝. 枷’4η．Then∫4＝dハfor some∫’｛∈M．Since九∈Mo，we have d∈λ＝Kerδ and∫ノ｛∈M；o by Lemma2．2．16．Since｛∫！ゾ、、，ん｝is a free basis of the人 mod−u1e Mo，there existα！，＿，α、、∈λsuch that∫’｛＝α1∫1＋＿十αηん．Then. ∫｛＝∂α1∫1＋．、．十dα｛∫｛十＿十dαη∫ηand hence1＝dα4．Thus d∈λ＊． □. We have the converse o年Lemma2．3．5．. lLemma2．3．6．〃Bフλ，δ㎝d舳θα5加工emmα2．3．5．〃M一＝Be1㊥…θ Beη6eα伽θB−mo〃e．4㎝ηxηmαか虹D8α桃伽8（1）D∈M（几，B），（2） D’1∈M（η，3−1B）α〃（3）δ（D’1）D∈M（η，B）フ伽ηαm・〃・伽1・α批㎝. ㎝M一州伽e〃ひ. （llll）一沖（1）. 乞510Cα吻η物0亡θη亡．∫肌αd肋0η，砒C㎝舳eD50伽右伽C0町㎝㎝右50∫ ㎝ザ。ω〃θc右。ヅ。ゾDんα〃舳。comm㎝∫㏄古。グ5． Pグ。oヅ Let. （二）一・（二）・.
(22) 20. CHAPTER2． M．ODσLE DER工帆nONS. Since0＝δ（万）＝δ（DD■1）＝δ（D）D−1＋Dδ（D’1），it fo11ows thatδ（D）十. Dδ（D’1）D＝0．Hence we have. （llll）一刈・・（llll） 一岬）・咋）・）（二）一（：）・. Thus we have. （ll；lll）一咋）（1）一（：）. for a suf巳。ient1y Iarge in七eger／V since the deriva．tion3’1δon3−1／3「is1oca11y. ni1potent．Next，we c1aim that we can take D：（ん）so that∂｛1ゾ．．，φ肌 have no common factors for each乞、Suppose tha“11＝〃11，．．．，d1几＝〃1几. with d，♂1ゴ∈B．Then∫1：：d∫’1∈KerδM＝Mo with∫’1∈M’．Hence ∂∈Kerδ＝λby Lemma2．2．16．Let ♂11♂12．．．♂1η d21d22．、．d2、、 D’＝. dη1dπ2．．．d㎜. Then D＝五dD’with d. 1 0. 助：＝. 0 ’’。 Since d∈λ，we have δ（D−1）D一δ（D’一1亙。μ）助D’一δ（D’一1）E。／。叩’一δ（Dト1）D’. and Dパ1＝D−1坊∈M（η，3−1B）．Hence D’and−D de丘nethesamemodu1e d−eriva．tion on M，which comp1etes the proof．. 口.
(23) 21. 2．3．THECASEW肥REB＝刈γ1 We a1so have thefo11owi㎎．. I−emma2．3．7．〃伽ヵ∈Moうeα5伽伽〃。oゾ。μemmα2−3．6．肋㎝ 伽＾舳m加8．1M。α・αη8−1λ一m・〃・。 ProoヅIt su伍。es to show that Mo⊂8−1λ∫1＋．．．十3−1λ∫、．Let m＝ わ1e1＋．．．十6几eη∈Mo withろ｛∈B．Then if we write. i二），. 叫・・川 wehave i1）・. ・一州一ル…，州…ψ） Henceδ（61ゾ．．，6、、）十（う1，．．，6、、）δ（D−1）D＝（0，＿，0）．OI1the other hand，. wehave. i二）一町…炉（二） ・一仏…川. and一 δ（（6。，．．．，6几）D’1）＝（δ（61ゾ．．，6几）十（6。，．．．，6几）δ（D−1）D）D−1. ＝（0ゾ、．，0）， whi・himp1i・・th・t・・y・・mp・…t・fth…w・・6t・・（6。，．．．，6几）D−1i・i・. 3−1λ．Thusm∈3■1λ∫1＋＿十3−1λん．I口 We redeine the triangu1abi1ity of a modu1e derivation on a free（B，δ）一 modu1e in a matrix form．. DEFINITI0N2．3．8．Let M’＝Bθ1㊥… ①Be肌be a free（B，δ）一mod−u1e．Sup− pose thatδM is de丘ne（1by. （1：：ll）一・（1）.
(24) 22. CHAPTER2．M’ODσLE DER工帆nONS. withσ∈M（η，B）．WesaythatδM istriangu1ab1eifthere exist aninvertib1e matrix P in lM（η。，B）andら4ゴ∈B such that. 0. 0. 0. 621 0. 0. ろ几1ろ。2. 6れト10. （δ（P）十P0）P−1：. In fact set. （lll）一・（1）・. Then｛e∫1，、、．，e㌦｝is a free basis of〃，and we have. （ll：lll）一δぽ）（1）・・（1：：ll）. 一と岬・町・（lll）・. HenceδM satis丘esδM（e’1）＝0，δM（e’2）∈Bθ’1，＿，δM（e㌦）∈Be’1㊥・・（王） Be㌦一1．. Wegive anexamp1eofatriangu1ab1emodu1ederivationinthecasewhere B＝ん［”，リ1．. ExAMPLE2．3．9．Let B＝ゐ［”，リ1be a po1ynomia1ring in two variab1es over a』丘e1d一ん。f characteristic zero．Letδ∈LND（B）be d−e丘ned−byδ（”）二0and δ（μ）二”．Let3＝｛1，”，”2，．。、｝be a mu1tip1icative1y c1osed−subset of B．. Let M：Be1①Be2be a free3−modu1e and−Iet ・一. i、・｝”、4”）1・（・，・）・.
(25) 2．3．. 23. THE C珊E WHEREB＝λ［γ1. Then we can de丘ne aエ。caエ1y ni1potent mod−u1e deriva七ionδM0n M’by. （沈；）一1（・一1）・（ll） 一（．、々二二．、. μ. ∵”）（ll）・. 2 一μ. Let ・一. iT1一μ）. and−1et （：ll）一・（ll）・. Then it is easy to see that｛θ！1，e！2｝is a free basis of the B一皿。du工e〃such. thatδM（e’1）＝0andδM（e’2）ごe’1． Next we give an examp1e of a non−triangu1ab1e modu1e derivation．. ExAMPLE2．3．10．Let B，δbe the same as in Examp1e2．3．9．Let M「＝・ Be1θBe2be a free B−modu1e and1et. ・一. i㌃1㌘）・. De丘ne a mod−u1e derivationδM by. （1州一／（・一1）・（：1）一（ブ1㌃2）（：1）・ Then（M「，δM）is aδ一modu1e． First we c1aim that Mo is a freeλ一modu1e with a basis｛∫1，∫2｝，where ∫1，∫2a．re de丘ned by. （丑）一・（：1）・. We can show as before that∫！，∫2∈Mo and hence〃1＋〃；⊂Mo．We prove the other inc1usion．Take any e1ement m∈Mo．Then m∈8−1M；o＝ 8−1λ∫1＋8−1λ∫2by Lemma2．3．7．Hence／m＝g1∫1＋g2∫2with Z∈NU｛0｝ and g1，g2∈λ＝ん同．We take l to be the sma11est integer．Then we have ・∼m＝9。∫。十9。∫。：9ユ（（ツ2＋1）・。十桝。）斗9。（胸十∬・。） ＝（9。μ2＋卿十9。）・。十（9。・叶9。・）・。．.
(26) 24. CHAPTER2．M’ODσLE DER工WしnON8. If Z＞0，then”divides g1μ2＋g2μ十g1，i．e．，g1（0）μ2＋g2（0）μ十g1（0）＝O and. therefore g1（O）＝g2（0）：0，which contrad−icts the choice of Z．Thus Z＝O. and henceη7∈λ∫1＋λ∫2as desired，. Next we c1aim thatδM is not triangu1ab1e．For this purpose，we may assume thatんis a1gebraica11y c1osed一．Suppose that there exists a B−basis ｛e’1，θ’2｝ofル7such thatδM（θ’1）＝0andδM（e’2）∈Be’1．Then there exist 91，92∈λ＝ん［”］such that ・’。＝9。∫。十9。∫。＝9。（（μ2＋1）・。十仰。）十9。（μ・。十・θ・） ＝（9。μ2＋9。叶9。）・。十（9。・叶9。・）・。一. Since both｛e1，e2｝a．nd｛e’1，e’2｝are B−bases ofルτ，the B−idea1generated by. 9．V2＋9。叶9。・・dg。榊十9。・i・…itid・・1，i．・．，th・・1…d・・tγ（卿2＋ g2V＋g1，g1”μ十g2”）of SpecB is the empty set．Name1y，there is no so1ution of the system of equations ｛. 9。μ2＋卿十9。：0 91”9＋92”＝0. Here”二0is a」so1ution of the second equa．tion．Then it is easy to see tha．t the equation g1（0）μ2＋g2（0）g＋g1（0）＝O ha．s at1east one so1uti6n ofμ．This iS a COntradiCtiOn．. エn the丘na1ofthis sec七ion，we1ook into the structure of aδ一mod−u1e in the. case where B is a po1ynomia1ringんレ，g］．Given anyδ∈LNDた（B），after a change of coordinates，we may assume thatδ（”）＝0andδ（μ）∈ん同by the theorem of Rentsch1er［18］．. Lemma2．3．11．〃B＝ψ，ひ16θαρo伽。m乞α1ヅ初g，δ∈LNDん（B）フMα 伽的9㎝θm左・d（B，δ）一m・〃θαηdM。。、伽亡…1㎝ραグ吋M・8W・・θ伽左 δ1”㎝鮒・．肺η伽・θθ洲・㎝θ1θmθ伽∈λ舳ん伽亡（M／M。。、）［α一1ト ㊥』B［α一11再ω肋α伽θうα・1・｛百ゾ．．，再｝・3岬・・θ舳・1∈M一ヅ・ρ…㎝お. 百∫・・θ㏄ん乞．〃M’＝Bε。十…十B・η．珊㎝ωθんα・θM’∩M七。、＝0㎝d んθη・・〃㊥M。。、⊂M．〃伽・m・ザθ，M／（〃㊦〃。。、）1・㎝舳伽θれμ伽 ρ0脱ヅ0∫α．. 〃。oヅLet u’be an e1ement of B such thatα：＝δ（u’）is a nonzero e1ement ofλ．Then u＝び／αis a s1ice for the extension ofδon B［α一1］．Furthermore，. by Lemma2．2．3，（1），we have B［α一11＝刈α一11［・1，ハ何。B［ガ11＝M。軌B［α一11，・・d （M／M1七。、）［α■11：（M／M・。。）・［α一11⑳刈、一・］刈α■11［・1.
(27) 2．3．THECA肥W肥肥B＝λ［γ1. 25. Since（M／M一。、）o［α■11is a丘niteユy generate（1，torsion−freeλ［α一1］一modu1e and λ［α■1］is a PID，it is a freeλ［α■1］一modu1e，whence（M／Mt。、）o［ガ1］is a free. 刈α一1］一modu1e with a free basis｛ε1，＿，百九｝and（M／Mt。、）［α■1］is a free B［α■11一皿。du1e with a free basis｛百1ゾ．．，島｝、. Suppose that61e1＋＿十ろηeη∈Mt。、withわ｛∈B．Then there exists a. nonzero e1ement6∈B such that肋1e！＋＿十肋几e。＝0，Weヰaveるる1百十 十肋η可＝0．Since｛百ゾ．、，再｝is a free basis，we have軌：0and hence わ｛＝0for a11乞．Hence M．’∩M1七〇、：0．The rest of the assertion is c1ear． □. We1ook at the structure ofprimeδ一idea1s and primaryδ一idea1s．. lLemma2．3．12．De伽eδ∈LNDん（B）6Vδ（”）＝0㎝dδ（μ）＝∫＝ρ1「1…パ几，. ωん肌θ㏄んパ・α帥m・θ1・m㎝6切ψ1．rん・Hη〃㎝蜥・帥mθδ一助α∼ 5α桃伽50ηθ0μんθ！0〃0ω伽g・. ωP：（ρ）∫・・α帥mθθ1θm・肋ρ∈ψ11 62ハ＝（仰，5）∫・…mθ1，ωん㈹gl・か棚・・伽・1η（ψ1／（ρ｛））［μ1一. ∫ηαグ切㎝1αグ，榊（μ）1・㈹肋，栃㎝αη〃㎝獅・帥mθδ一助α1乞・gθη舳右θ♂. ∼αρ伽θθ1・mθ肌古伽ゆ1．ル〃㎝鮒・帥mα・Vδ一価αlq・α挑伽・㎝・・プ 海！0〃0ω伽g・. 例q＝（〆）伽α師mθε1・m・械ρ∈ψ1㎝dαρ・・舳・加物・川 ωq：（〆，98）∫・…mθづ㎝∼・・舳θ棚・gθW，・，舳肌g乞・伽・幽・肋 伽（ψ1／（ρ4））［μ1．. ∫ηα舳ulα・，1アδ（μ）づ・“肋，伽η㎝〃㎝獅・帥mαザVδ一価α1乞・gθη一 肌6θれμ伽ρ・ωθグ・∫αρザlm・θ1舳・η乏伽ψ1．. Pヅ。o∫．We prove on1y the assertion conceming a primeδ一idea1．First we. consider the case where htρ＝1．Then p＝（ρ）for someρ∈B and hence δ（ρ）∈（ρ）．This imp1ies thatρ∈A. Second−we consider the case where ht p＝2，Then仲。：＝p∩珂”］is a nonzero prime idea1ofんk］一 Indeed，sinceρ ≠ 0andδ（P） ⊂ ρ，there exists a nonzero e1ement inρ∩ん［”1．Hence po：（P）for some prime e1ement. p∈ん［”］．The derivationδon3induces a1oca11y ni1potent derivationδon んk，ツ1／（ρ）＝（ん［π1／（P））［μ1and一百：＝P／（P）is a nonzero primeδ一id−ea1．Since. （ん［”］／（ρ））［μ］is a PID，we have百：＝（す）for some irred−ucib1e e1ementずin.
(28) 26. CHAPTER2．MODσLE DER工帆丁∫ONS. （ψ］／（ρ））［μ］．Weshowthat∫≠（ρ）1eadsto ac㎝tradiction．If∫≠（ρ），then Kerδ＝ん同／（ρ）．Sinceδ（ず）∈（す），we haveす∈ん［”］／（ρ）．Hence p＝（ρ，g）. for some g∈ん［”］，but sinceρand−g are mutua11y prime inん［エ］，we have. （ρ，g）＝B，which is a contradiction．We can prove the assertion concerning a primaryδ一idea1in a simi1ar fashion noting that any primary B−idea1q of height l is of the form（q「）for some prime e1ement g∈3．Indeed，ψ二（g）. for some prime e1ement g∈B．Since R市is a DVR，we haveψψ＝g叩ψ. forsomeグ．Henceq＝qRψ∩R＝g「Rψ∩R＝グR口 Since every prime divisor ofaδ一idea1is aδ一idea1，the above1emma imp1ies that any radica1δ一idea1∫of B is of the form ∫＝（α）∩（〃1，9。）∩…∩（ρ｛亡，9。），. whereα∈ん［”］is not divisib1e by anyρ｛ゴand each gゴis irred−ucib1e in （ん同／（ρ4ゴ））［リ］andif乞τ＝乞。，theng．and−g．are mutua11yprime in（ん［”1／（ハ、））［V1．. Then we have B／∫隻B／（α）・B／（P｛、，9。）・…・B／（P亡，9土）。. Note that an embed−d−ed−primary component of aδ一idea1is not necessari1y a δ一id−eaI．This is shown in the fonowing examp1e．. ExAMPLE2．3．13．De丘neδ∈LNDん（B）byδ（”）＝0and一δ（9）＝”．Then ∫：＝（”2，ψis aδ一idea1．We have a minima1primary decomposition∫＝ （”）∩（エ2，ひ），where（”）is an iso1ated component and一（”2，μ）is an embedded component．Then（”）is aδ一idea1but（”2，μ）is not aδ一idea1．Ind−eed一，δ（μ）。。。. π≠（π2，ひ）．Note that〉てη＝（”，μ）is aδ一idea1．. 2．4 The homo1ogica1property ofδ一modu1es In this section，we describe the homo1ogica1property ofδ一modu1es．. DEFINITI0N2．4．1．Letδ∈LND（B）．Let M，N be a pre一（B，δ）一modu1es. and1et∫：M→W be a B−modu1e homomorphism．We say that∫is a （3，δ）一homomorphism（abbreviated as aδ一homomorphism）if∫（δM一（m））＝ δN（∫（m））for any m∈〃．We denote the set of a11δ一homomorphisms from M’to N by Homδ（M，W）．.
(29) 2．4．. THE HOM−OLOG∫CAL PROPERTγOFδ一MIODσLES. 27. remma2．4．2．〃M，W6e〃一δ一mo〃θ3作3ρ．δ一mo〃θ3ノ，ψ灯：M→ Wαδ一ん。momoη肋8ml T加肌Kerプ，Im∫αηd Coker∫αηθαZ80〃θ一δ一mo仇Zθ8 伽・ρ．δ一m・〃θ・ノ。. Pr00ヅ Straightforward．. Lemma2．4．3． 肌わ舳伽∫0〃0ω伽gα55θ棚㎝5j ω〃M，Wろθδ一m・〃ε・。 Dθ伽θαm0〃e伽4〃αれ・ηδ〃洲0η伽6εη30ヅ. 〃0伽C亡M晩N的 δM洲（m帥）＝δM（肌）帥十m⑱δN（几）． 珊θη（M晩W，δM酬）1・αδ一m・〃θ．. C2ソ〃M，W6θδ一m・〃・・．D・伽εαm・伽θ伽伽α批㎝I㎝伽B−m・〃θ HomB（M，W）o！3−mo伽3θん。momoΨ肋3m3作。m M舌。 N物 δ（∫）（m）＝δN（∫（m））一プ（δM（m））. ∫oグ∫∈HomB（M，N），m∈M「．丁加ηHomB（M，N）乞5α〃θ一δ一mo仇Zθ． 丸竹加ヅmoヅθフケM「加α伽伽伽9θmヅα士eゴB−mo伽Zθフ挽θηHom3（M，M） 乞5αδ一mOdωe。∬θグe挽e加rηθZ0ゾδHomB（M，N）C0η5乞8右50∫（B，δ）一ん0m0−. mOr〃5m3伽mM左0W． 63ソ〃L，M，N6θδ一m・〃・・．3W・・θ右んαπ㎝dMαヅ・伽吻9ε几・mむθd. B−m0〃ε8，Dε伽eαB−m0〃e乞30mOr〃5m σ：HomB（工⑳月M’，N）→Hom．B（ム，HomB（M，W）） うgσ（∫）（Z）（η7）＝∫（Z⑭ηZ）∫0rJ∈ムαηdη7∈ハ∫．τんeησづ8α（B，δ）一. 乞80m0ヅ〃8m． μノ8他ρρ08θ杭α亡（Mλ，δλ）45αδ一modωe∫orθαcん入∈A・丁加η（Dλ∈Aルヘ乞8. αδ一m・〃θ㎝州λ∈。仏1・α〃一δ一m・仇1・伽川商mけα・肋・η・ C5μθ亡｛仏；∫μ入｝6θα伽・・∼・左・m・∫B−m・〃θ・フωんθグθ∫μλ1・αん・m・一. m・グ的・m伽m仏君・Mμ伽λ≦μ・8α〃・・θ挽α右θα・ん（Mλ，δλ）1・α δ一m0地Zeα〃δμ∫μλ＝∫μλδλ∫0rα〃λ≦μ．肋θη抗θ伽θC〃棚亡1imMλ →. 乞8αδ一η70duZθω批んαηα亡刎rαZη70duZθ∂θヅ乞ωα切0η．.
(30) 28. CHAPTER2．M’ODσLE DER工VAT工ONS. C6με亡｛Mλ；ムμ｝6・㎜伽附・θ・蚊θm・μ一m・d仙1・・，ωんθ・θムい・αん・一. m・m・ψづ舳伽m仏左・仏∫・ヅλ≦μ・畑ρ…抗αtθ㏄ん（払，δλ） 1・αδ一m・〃θαηdδλ∫入μ＝∫入μδμ∫・ザα〃λ≦μ・砒η伽伽・θ・・θllm批. 1imM入乞3α〃θ一δ一mo仇Zθω肋αηα施mZ mo伽Zθdeザ伽肋。η． ←. P…ヅ（1）. It is easy to show七hatδM酬is a we1工一de丘nedルmodu1e endo−. morphism． For any m⑳η∈〃⑭B W and6∈B，we have δ（6（m帥））＝δ（（ろm）帥）＝δM（6m）帥十6m⑳δN（η） 二（δ（6）m＋うδM・（m））帥十6m⑧δw（η） ：δ（う）（m帥）十6δ（m帥）．. For each positive integer2，we lhave an equa1ity. 1エ（舳）. p）1（・）llゴ（一）・. HenceδM⑱N is1oca11y ni1potent． （2） First we show thatδ（∫）：〃→N is a B−modu1e homomgrphism．. For any∫∈HomB（M’，W）and b∈B and−m∈M，we have δ（∫）（6m）＝δN（∫（6m））一∫（δM（6m）） ＝δN（6∫（m））一∫（δ（6）m＋ろδM（m）） ＝δ（わ）∫（m）十ろδM（∫（㎡））一δ（わ）∫（m）一ら∫（δM（m））. ＝6δN（∫（m））一ろ∫（δM（m）） ＝6δ（∫）（m）．. I七is easi1y veri丘ed七hatδ（∫）（γπ十η）＝δ（∫）（η7）十δ（∫）（η）．. Next we show that HomB（M，N）is a pre一δ一mod−u1e．For any∫，g∈ HomB（M，W），it is c1earthatδ（∫十g）＝δ（∫）十δ（g）．For any∫∈HomB（M，W）. and6∈3，wehave δ（6∫）（m）＝δW（6∫（m））一6∫（δM・（m））. ＝δ（6）∫（m）十6δW（∫（m））一6∫（δM（m）） ＝（δ（6）∫十ろδ（∫））（m），. and henceδ（6∫）＝δ（6）∫十6δ（プ）．.
(31) 2．4．. 29. THE HOMOLOG工CAL PROPERTY OFδ一M’ODσLES. Fina11y we c1aim thatδH．mB（M・，N）is1ocauy ni1potent provided M is a 丘nite1y generated−B−modu1e．First，we can prove by induction on Z that for any positive integer Z，. 1！（1）（・）一. i吉（一・小・小）・. SupPosethatル7＝Bm1＋… 十3m、．Let∫∈HomB（ルグ、ノV）．Inorderto show thatδ∼（∫）＝0for some integer Z，it is enough to show that there exists an integer Z such thatδj（∫）（mゴ）＝0for a11プ。For each mプ，there exists an. integerらsuch thatδ〃エゴ（mゴ）＝0．Thenδペゴー4∫δM・4（mゴ）＝0for any乞≧4ク． Si…∫（mゴ），∫δM（mゴ）ゾ、．，∫δM互1−1（mゴ）…丘・it・lym・・y・1・m・・t・・fN，it fo11ows that there exists an integer Z’ゴ＞らsuch thatδNJ’グづプδM｛（mゴ）＝O for. 0≦乞≦ら＿1．Hence we have. め（1）（・ゴ）一. i‡（一州δ｛小）…. Let Z∫：max（Z’1ゾ．．，Z’、）．Thenδユ’（∫）（mゴ）：0for a11ゴ．. （3） It is we11known thatσis an isomorphism ofB−modu1es and it is easy. to show thatδσ（∫）二σδ（∫）for any B−mod−u1e homomorphism∫一：ム⑳BM「→ w．. （4） Obvious．. （5） For the disjoint union uλMλ，we d−e丘ne an equiva1ence reIa七ion≡as. fo11ows．For”∈MI ﾉandμ∈〃μ，we say that”≡μif there existsひsuch that λ≦μ，μ≦μand∫。ル）＝んμ（μ）．Then1im仏＝（IJλMλ）／≡．we de丘ne →. a modu1e d−erivationδiim→on1imM入byδIim→（”）＝δλ（”）for”∈M’入、Then it →. is immediate that七his is a we11−de丘ned−1oca11y ni1potent modu1e d−erivation．. （6）We can regard1imM入as the set ← ｛. ^・ （”λ）λ∈■M一λ. ／・！一町（伽）. λ. We de丘ne a modu1e derivationδ1im←on1im仏byδ1im←（（”λ）λ）：（δルλ））λ1 ←. Then it is easy to see thatδ1im←is a we11−de丘ned−mo（iu1e d−erivation．. □.
(32) 30. CHAPTER2． MODσLE DER∫V兇丁工ONS. We sha11de丘ne modu1e deriva七ions on a free reso1ution of aδ一moduIe （Lemmas2−4−5and2．4．6）．We deine an exact sequence ofδ一modu1es and a free reso1ution of aδ一modu1e as fo11ows．. DEFINITI0N2．4．4．Forδ一modu1esム，M and W，an exact sequence of B−. modu1es 0＿→ム」㌧M」㌧W＿→0 is ca11ed anexact sequence ofδ一mod−u1es ifbothψandψareδ一homomorphismsl For aδ一modu1e M，a free reso1ution of the B−modu1e M’ ∫、：… 一一一今片一一一＞p肌＿1一一一＞… 一一一＞P1一一一＞Po一一一＞0. is ca11ed a free reso1ution of中eδ一modu1e M if each片→片＿1and片→M. areδ一homomorphisms． L−emma2．4．5．ハ。rδ一mo〃θ8乙フ”α”N，5ψρ05e肋ατ. 0＿→工」㌧M」㌧W＿→0 乞8㎝舳C左5θμηCθ0プδ一mOd仙1θ8．舳〃05e舳右. Q。：…一合軌一→Q卜r→…一合Q1一＞Qo一＞0， R・：… 一一≒》R肌一一一》R肌＿1一一一》… 一一一＞R1一→∫…＝o一一二》0. づ8伽“e801u右乞0η80∫挽eδ一m0ωθ5工，W欄ρθC伽的．珊㎝伽reθ挑老5α伽θ. 欄01〃㎝0∫伽δ一m0〃eM ∫：＝．：… 一一→片一一→凡＿1一一」》… 一一→P1一一÷》∫＝…〕一一一》0. 8㏄ん挽α左伽∫0〃0ω伽g仇gmm18COmm刎舌α切θ． 0. 0. 0. T 0一一→ Z ・・. P. T. T. 」→〃」㌧ W一一→O 允丁 1・. ｪ. 0一→Q。』→片 」→R。＿→0 ・・. P. 1・T 1・. s. 0一一→Q1 』→P。 」→R。 ・・. s. わT. 1・. s. 一一一. tO.
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