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σHAPTER4．工NF工NITELy MANYI GENERATORS
（・）・ろ十〆＜亡十1・ndρ ＞0
（d） rら十〆＝百十1（rら≧1andρ ≧1in this case）
（e） 杉十〆＞亡十1（rら≧2andρ ≧2in this case）
We note that4＿ω…r≦十〆（mod亡十1）andthe exponents of 2ヅ、．，叫in 岬are a11congruent to左（グら十〆）modu1o f＋1，We on1y consider the case
（e）．The remaining cases can be treated−in a simiIar fashion．In the case（e），
we have4一ω≡rら十ρ 一亡一1with0＜rら十〆一亡一1く亡十1and
（亡十1）τ（岬）＝ψr・1）（η一1）十（1＋1）巾。d・千1）・・…・、dr（t＋1）・一）
・（一1）（1・・）［之（午）1一（t・1）・・
一（亡十1）（β2＋… 十β肌）
＝左（グ。一・ら）（η一1）十（1＋1）τ（μ。q・，・・。∂2・…肌dη）
・（上1）（t・・）ト・パ岩41
一（亡十1）・。十ρ一ω一・。
≧2ψ。一・ら）十ト肘2ρL6＋2（乏十1）（・ら十ρL2）
一（亡十1）・。十P一ω一・。
＝4一ω十2・ら十2ρL6＋2（士十1）（〆一2）十（オー2）γ。
where we use the conditionη1≧4and之≧2to show the inequa1ity．Thus the condition（ii）of（3）ho1ds for mF．This ind−uction comp1etes a proof．□
Chapter5
Further prob1ems and
COmmentS
5．1 Tensor products
Let右1，B2beん一a1gebras andδ1，δ2be1oca11y ni1potent d−erivations on−B1，一B2 respective1γLet B＝。81⑭んB2and deineδ：。B＿＞一B by
δ（6。⑳ろ。）＝δ。（6。）舳。十ら。⑱δ（6。）
for61∈B1andう2∈B2．Thenδis a we11−de丘ned−1oca11y ni1potent deriva．tion on B（cf．Lemma2．4．3）．Letλ1＝Kerδ1，λ2：Kerδ2andλ＝Kerδ．We consider the丘nite genera．tion ofλ1，λ2andλasん一a1gebras and propose the fo11owing Prob1ems．
PR0BLEM5．1．1．46o挽λ1α棚λ2αヅθガη批θ佃9e胱m加d o〃erん，挽eη48λ 伽棚g9㎝em亡εd0りeヅん9
PR0BLEM5．1．2．4λ乞8束η批ε佃9θηem加d o〃εrんフ流eηαreう。流λ1αη∂■42 伽伽ゆ9㎝em亡θd・・θ・ん9
We conjecture that Prob1em5．L2is a伍rmative，i．e、，eitherλ10rλ2is in丘nite1y generated−overんthen so is A The fo11owing is an examp1e such thatλ1is ininite1y generated overんandλis in丘nite1y generated overん．
ExAMPLE5．1．3・Let B1＝ん［ 1，＿， れ，μ1，＿，％斗11．and d−eineδ1byδ1（叫）＝
0，δ。（挑）＝ づ士十1f・・1≦づ≦Hndδ（％。。）二（ 。… η）士．SupP・…ith・・
69
70
CHAPTER5．FσRTHER PROB工EMS ANDσOMlMENTS
几≧3and亡≧2or几≧41and亡≧1．Then by Theorems4．1．3and4．2．1，λ1 contains e1ements of the form
1μ九十14＋（terms of1ower degree inμ肌十1）
for each4≧1．Henceλ1is in丘nite工y generated over々．Let B2is apo1ynomia1 ring overたwith arbitrary1oca11y ni1potent derivationδ2．Thenλis in丘ni七e1y generated overんin a simi1ar fashion．
We conjecture Problem5．1．1is a趾mative．The fo11owing is an examp1e such that a11ofλ1，λ2andλare丘nite1y genera七ed overんbut a strange phenomenon o㏄urs．
ExAMPLE5．1．4．Let B1＝ん［ 1ゾ．．， 41and B2二ん［μ1，μ21．De丘neδ1by δ1（ 1）＝0，δ1（ 2）＝一 12，δ1（ 3）＝吻andδ1（ 4）＝ 3． De丘neδ2by δ2（μ1）＝0andδ2（μ2）＝μ12．Then We can regard B as a po1ynomia1ring ψ1ゾ．、， 4，μ1，吻］containi㎎B1and B2as subrings．We can con丘rm tha七 a11ofλ1，λ2and一λare丘nite1y generated−overんby Essen，s a1gorithm（see
［7，S・・ti・n1．41）．
On the other hand，1et B＝B／（ 1＿ひ1），δ∈LNDん（B）a derivation in−
ducedbyδandλ二Kerδ．WeregardBasapo1ynomia1ringψ1，＿， 4，μ2］．
Then万is de丘ned byδ（ 1）：O，δ（π2）＝ 13，δ（ 3）： 2，δ（ 4）＝ 3，and δ（μ2）＝ 12．The紅a1gebraλcontains e1ements of the form
・。腕4＋（t・㎜・・f1・w・・d・g…i・V。）
for each−e≧1（see［9，Lemma7．5］）．Henceλis in丘niteIy generated overん．
We can consider simi1ar prob1ems forδ一modu1es．Letδ∈LNDん（B），
λ＝Kerδand−M CNδ一modu1es．Thenム：＝M一QBW is a（B，δ）一modu1e with the induced modu1e d−erivation．Let Mo，No and工。 be the kerne1s of the respective modu1e derivations on M，W andム．The fo11owing are unso1ved probIems．
PR0BLEM5．1．5．〃ろ。流M；oα棚ノVoαヅeガη批e砂9eηθヅαted o〃erλ，挽eη乞5Lo 伽伽佃9㎝εm亡・d・ω・ザ〃
PR0BLEM5．1．6．4ム。乞5ガη伽妙9eηem老θd o〃erλ，統θηαヅθろ。抗Moαηd No αre伽伽佃9εηemfεd0りer〃
5．2．DσALM ODσLE8
715．2 Dua1modu1es
Letδ∈LNDん（B），λ：Kerδand M aδ一modu1e．Then the dua1B−mod−u1e W：＝HomB（M ，B）of M is aδ一modu1e with the ind−uced modu1e derivation
（Lemma2．4．3）．Let M；o and−No be the keme1s of the respective mod−u1e derivations on M and W．Wb propose the fouowing prob1ems．
PR0BLEM5．2．1．4Mo乞5ガη伽妙geηθm之ed oのerλ，抗θ肌づ8No力η伽佃geη一
εrα亡θd0りθザノ49
PR0BLEM5．2．2．4No45力η伽ψgeηθmむε♂o〃erλフ6加η乞3Mo力η批εψgε作 erα舌ed0りeザノ19
IfIM is atorsion8−moduIe and B is anintegra1d−omain，thenHom−B（M「，B）
is equa1to0．Indeed，for any∫∈HomB（M ，B）and−m∈M「，there exists nonzeroわ∈B such that6m＝0and henceげ（m）＝∫（ひm）：∫（0）＝0．This imp1ies∫（m）＝0．Hence in this case we can construct a counterexampエe to Prob1em5，2．2easi1y which is shown by the fo11owing．
ExAMPLE5．2．3．Let B＝ん［ ，μ1be a po1ynomia1ring and−d．e丘neδ∈
LNDた（B）byδ（ ）＝0and一δ（V）＝ 、Let M＝B／（ ）and1etδM be a derivation induced byδ．Thenλ：＝Kerδ二ん［ ］and Mio：＝Kerδ〃＝M is in丘nite1y generated−overλ．On the other hand，sinceルτis a torsion B−
modu1e，we have N＝0and thus No＝0is a丘nite1y generatedλ一modu1e．
In the rest of this section we consid−er the case where M is a free B−
modu1e．We prove that the matrix de丘ningδw is（＿1）times the transpose of the matrix de丘ningδM．Name1y，we obtain the fo11owing assertion．
Lemma5．2．4．〃M＝8θ1㊥＿3e几ろθα伽θ（3，δ）一mo〃舳肋α伽e6α一
5乞8｛e1ゾー．，eη｝α棚Ze右｛e1＊，＿，eη＊｝うeα仇αZ介eθうα5乞5N＝HomB（M，B）．
4δM4・加伽eれひ
（二1111）一（∴，ll）（1），
栃θηδw乞8de力ηεd6v
（lllll）一（二1∴）（lll）
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σHAPTER5．FσRTHER PROBLEMS ANDσOMlMEWTS
Pヅ。oゾ、For any1≦乞≦η，we have
δN（・｛＊）（・｛）＝δ（・｛＊（・壱））一・｛＊（δM（・｛））＝δ（1）一6｛乞・・一6か
For any1≦4≠ゴ≦η，we have
δw（・1＊）（・ゴ）＝δ（θ1＊（・ゴ））一・1＊（δM・（θゴ））＝δ（0）一6ゴ1＝一6ゴ1．
Hence we haveδN（e4＊）：一61｛e1＊一…一6ηづe肌＊． □ The fo11owing is an examp1e such that both M；o and Mo are in丘nite1y generated OVerメし．
Theorem5．2．5．〃B＝ψ1，＿，叫，眈，＿，％十11うθαρo伽。m乞αZれηgI㎝∂
dφηεδ∈LNDん（B）勿δ（軌）＝0，δ（ω＝物2∫oブ1≦づ≦ηα棚δ（％十1）＝
1… η・工θtλ＝Kerδ・工助M＝ΩB／ん6e挽θd蜥θrθη切αZ moduZθα〃dψηe δM伽αηαωmけα5肋。η68eθ§3・3ソ・工θ6N＝HomB（ωB／ん，B）＝Derん（B）αηd 加伽θδWα5αう0〃e．肋θれ0流M0α〃N0αヅθ伽伽伽1ひ9e胱m右ε∂0りθrλ．
ProoヅBy Theorem4，1，5，it su伍。es to show that No is infinite1y generated over／L δM is de丘ned byδ〃（d軌）＝0for1≦乞≦ηand
2 1
δM （吻・） 2 、 0曲・
d吻
δM（吻肌）0 2 、
δM（∂μη十1） d れ π2 叫 1 3 n 1
Thusδw is de丘ned byδw（d挑＊）＝0and
物1＊
雌1：；一2千1．2 、0二11111㍗、伽・
δ。（・ 、・） 0 一・ 、一 1．．． 、．、 ∂μ1＊
払十1＊
For61，c1，cη十1∈B，we have
δ（6。伽＊十・凶。＊十・れ十。dμ肌十。＊）
＝δ（6・）伽＊十（一2・・6・十δ（・・））軌＊十（一∬・…ψ。十δ（・肌十。））払。。＊．
5．2．DσAL−MODσLES
73Let61＝ 1V、、斗1乏十（terms of1ower d−egree in％十1）∈λ：＝Kerδwhich we construct in the proof of Theorem4．1．3．The way of construction imp1ies that each monomia1inわ1is divisib1e by either 10r 2＿ η．Thus any
monomia1in2 1う1and吻… 肌61is divisib1e by eitherηfor some1≦
1≦η…。…叫．Sin・・δ（B）＝・。2B＋…十・几2B＋・。…・、B，th…
existS c1，cη十1∈B such thatδ（c1）＝2 161andδ（c九十1）二 2… 叫61．Then 61伽1＊十。1物1＊十。肌十1吻n＋1＊is be1ong to Mo and it is of the form
（ 1μ、斗1老十（terms of1ower degree車μ、斗1））伽1＊
十（termS nOt COntaining伽1＊）
By the same reasoning as Theorem3，3．4，we conc1ud−e that No is ininite1y
generated overλ． □
Acknow1ed−gement
The author wou1d1ike to expresshis gratitud−e to Professors Noriaki Kawanaka，
Yaichi Shinohara，Masayoshi Miyanishi and Kayo Masuda for examining his doctora1dissertation．
Especia11y，he wou1d−1ike to express his indebtedness to Professor Masayoshi Miyanishi for1eading through1ectures and seminars for six years since he is
undergraduate．Withouthispatient1eading，thisdissertationwou1dnot have
been accomp1ished一．
He a1so thanks to Professor Kayo Masuda for organizing seminars on Osaka Umeda Campus，and appreciate the members of the seminars for d−is−
CuSSiOnS．
74
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75