Teorema Krulya pro golovnij ideal vazhlive tverdzhennya u komutativnij algebri yake razom zi svoyimi naslidkami ye osnovoyu dlya oznachennya rozmirnosti v algebri i algebrichnij geometriyi Teorema nazvana na chest avstrijskogo matematika Volfganga Krulya Zmist 1 Tverdzhennya teoremi 2 Dovedennya 2 1 Teorema Krulya pro golovnij ideal 2 2 Teorema Krulya pro visotu 3 Div takozh 4 LiteraturaTverdzhennya teoremi red Nehaj A kilce Neter a A displaystyle a in A nbsp element kilcya sho ne ye oborotnim chi dilnikom nulya i p displaystyle mathfrak p nbsp minimalnij prostij ideal kilcya nad golovnim idealom aA Todi visota idealu dorivnyuye 1 Naslidkom teoremi ye tak zvana teorema Krulya pro visotu yaksho minimalna kilkist elementiv sho porodzhuyut deyakij ideal neterovogo kilcya rivna m to visota cogo idealu ne bilsha nizh m Dovedennya red Teorema Krulya pro golovnij ideal red Oskilki nas cikavlyat lishe prosti ideali q p displaystyle mathfrak q subseteq mathfrak p nbsp mozhna zaminiti A na jogo lokalizaciyu A p displaystyle A mathfrak p nbsp Dijsno vsi prosti ideali kilcya A p displaystyle A mathfrak p nbsp mayut viglyad q A p displaystyle mathfrak q A mathfrak p nbsp de q p displaystyle mathfrak q subseteq mathfrak p nbsp prostij ideal kilcya A Otzhe nadali pripustimo sho kilce A ye lokalnim z yedinim maksimalnim idealom p displaystyle mathfrak p nbsp i a q displaystyle a not in mathfrak q nbsp dlya kozhnogo prostogo idealu q p displaystyle mathfrak q neq mathfrak p nbsp Zaminyuyuchi A displaystyle A nbsp na A 0 displaystyle A sqrt 0 nbsp mozhna takozh pripustiti sho A redukovane ne mistit nilpotentiv abo sho te same 0 displaystyle 0 nbsp radikalnij ideal Rozglyanemo jogo prostij rozklad tobto minimalni prosti ideali peretin yakih rivnij nulovomu idealu dlya neterovih kilec ci ideali utvoryuyut skinchennu mnozhinu 0 i 1 s p i displaystyle 0 bigcap i 1 s mathfrak p i nbsp Oskilki dobutok idealiv kilcya ye pidmnozhinoyu peretinu cih idealiv to takozh i 1 s p i 0 displaystyle prod i 1 s mathfrak p i 0 nbsp otzhe p displaystyle mathfrak p nbsp mistit dovilnij ideal p i displaystyle mathfrak p i nbsp ale a p i displaystyle a not in mathfrak p i nbsp oskilki vsi elementi z p i displaystyle mathfrak p i nbsp dilniki nulya Tomu ht p gt 0 displaystyle operatorname ht mathfrak p gt 0 nbsp Pripustimo sho p q displaystyle mathfrak p supset mathfrak q nbsp de q displaystyle mathfrak q nbsp prostij ideal Rozglyanemo faktor kilce A a A displaystyle A aA nbsp Vono maye yedinij prostij ideal p a A displaystyle mathfrak p aA nbsp otzhe ye artinovim Ce oznachaye sho bud yakij spadnij lancyuzhok idealiv A yaki mistyat a stabilizuyetsya Zokrema ce virno dlya lancyuzhka sho skladayetsya z idealiv a A q k displaystyle aA mathfrak q k nbsp de q k displaystyle mathfrak q k nbsp poznachaye simvolichnij stepin ideala Otzhe isnuye cile k take sho a A q k a A q k 1 displaystyle aA mathfrak q k aA mathfrak q k 1 nbsp Beruchi dovilnij b q k displaystyle b in mathfrak q k nbsp oderzhimo sho b a c d displaystyle b ac d nbsp dlya deyakih c A d q k 1 displaystyle c in A d in mathfrak q k 1 nbsp zvidki a c q k displaystyle ac in mathfrak q k nbsp i s a c q k displaystyle sac in mathfrak q k nbsp dlya deyakogo s q displaystyle s not in mathfrak q nbsp vidpovidno do oznachennya simvolichnogo stepenya Ale s a q displaystyle sa not in mathfrak q nbsp otzhe takozh c q k displaystyle c in mathfrak q k nbsp i q k a q k q k 1 displaystyle mathfrak q k a mathfrak q k mathfrak q k 1 nbsp Z lemi Nakayami oderzhuyetsya rivnist q k q k 1 displaystyle mathfrak q k mathfrak q k 1 nbsp Spravdi mayemo a p displaystyle a in mathfrak p nbsp i ideal p displaystyle mathfrak p nbsp ye maksimalnim tozh z lemi Nakayami dlya bud yakogo skinchennoporodzhenogo modulya M z rivnosti p M M displaystyle mathfrak p M M nbsp viplivaye sho M 0 displaystyle M 0 nbsp Yak naslidok dlya skinchennoporodzhenogo modulya N sho ye pidmodulem M z rivnosti p M N M displaystyle mathfrak p M N M nbsp viplivaye sho N M displaystyle N M nbsp Vzyavshi q k q k 1 displaystyle mathfrak q k mathfrak q k 1 nbsp yak M N displaystyle M N nbsp otrimuyemo neobhidnu rivnist Otzhe q k q k 1 displaystyle mathfrak q k mathfrak q k 1 nbsp i q displaystyle mathfrak q nbsp ye minimalnim prostim idealom vidpovidno do vlastivostej simvolichnih stepeniv i ht p 1 displaystyle operatorname ht mathfrak p 1 nbsp Teorema Krulya pro visotu red Spershu dovedemo take tverdzhennya Nehaj q 1 q 2 q m displaystyle mathfrak q 1 mathfrak q 2 mathfrak q m nbsp prosti ideali neterovogo kilcya A i p 0 p 1 p l displaystyle mathfrak p 0 supsetneq mathfrak p 1 supsetneq supsetneq mathfrak p l nbsp lancyuzhok prostih idealiv A takij sho p 0 q i displaystyle mathfrak p 0 not subseteq mathfrak q i nbsp dlya vsih i Todi isnuye lancyuzhok prostih idealiv p 0 p 1 p l 1 p l displaystyle mathfrak p 0 supsetneq mathfrak p 1 supsetneq supsetneq mathfrak p l 1 supsetneq mathfrak p l nbsp takij sho p i q j displaystyle mathfrak p i not subseteq mathfrak q j nbsp dlya vsih i j Mozhna pripustiti sho p l q i displaystyle mathfrak p l subseteq mathfrak q i nbsp dlya vsih i Zaminivshi A na A p l displaystyle A mathfrak p l nbsp vvazhatimemo sho p l 0 displaystyle mathfrak p l 0 nbsp Vikoristovuyuchi indukciyu shodo dovzhini l mozhna takozh pripustiti sho p l 2 q i displaystyle mathfrak p l 2 not subseteq mathfrak q i nbsp dlya vsih i Zgidno lemi pro uniknennya prostih idealiv isnuye a p l 2 displaystyle a in mathfrak p l 2 nbsp takij sho a i 1 m q i displaystyle a not in bigcap i 1 m mathfrak q i nbsp Element a ne ye oborotnim i ne ye dilnikom nulya oskilki za pripushennyam nulovij ideal ye prostim Tomu yaksho p l 1 displaystyle mathfrak p l 1 nbsp minimalnij prostij ideal yakij mistitsya v p l 2 displaystyle mathfrak p l 2 nbsp i mistit a to za teoremoyu Krulya pro golovnij ideal ht p l 1 1 displaystyle operatorname ht mathfrak p l 1 1 nbsp Oskilki ht p l 2 2 displaystyle operatorname ht mathfrak p l 2 geqslant 2 nbsp to p l 1 p l 2 displaystyle mathfrak p l 1 neq mathfrak p l 2 nbsp i mi oderzhuyemo neobhidnij lancyuzhok Dovedennya teoremi pro visotu zdijsnyuyetsya indukciyeyu po kilkosti porodzhuyuchih elementiv m Vipadok m 1 viplivaye z teoremi Krulya pro golovnij ideal Rozglyanemo ideal a 1 a 2 a m displaystyle a 1 a 2 a m nbsp de porodzhuyucha mnozhina mistit najmenshu mozhlivi kilkist elementiv i nehaj p displaystyle mathfrak p nbsp vidpovidnij minimalnij prostij ideal Nehaj q 1 q 2 q m displaystyle mathfrak q 1 mathfrak q 2 mathfrak q m nbsp minimalni prosti ideali yaki mistyat ideal I a 1 a 2 a m 1 displaystyle I a 1 a 2 a m 1 nbsp yih kilkist zavzhdi ye skinchennoyu Yaksho p q i displaystyle mathfrak p mathfrak q i nbsp dlya deyakogo i to ht p m 1 displaystyle operatorname ht mathfrak p leqslant m 1 nbsp Pripustimo sho p q i displaystyle mathfrak p neq mathfrak q i nbsp Rozglyanemo bud yakij lancyuzhok prostih idealiv p p 0 p 1 p l displaystyle mathfrak p mathfrak p 0 supsetneq mathfrak p 1 supsetneq supsetneq mathfrak p l nbsp Iz poperednogo mozhna pripustiti sho p l 1 q i displaystyle mathfrak p l 1 not subseteq mathfrak q i nbsp dlya vsih i Poznachimo A A I a a I A q i q i I p i p i I displaystyle bar A A I bar a a I in bar A bar mathfrak q i mathfrak q i I bar mathfrak p i mathfrak p i I nbsp Todi p p 0 displaystyle bar mathfrak p bar mathfrak p 0 nbsp ye minimalnim sered prostih idealiv A displaystyle bar A nbsp sho mistyat a m displaystyle a m nbsp otzhe ht p 1 displaystyle operatorname ht bar mathfrak p leqslant 1 nbsp Oskilki q i displaystyle mathfrak q i nbsp ye vsima minimalnimi prostimi idealami A displaystyle bar A nbsp i p l 1 q i displaystyle bar mathfrak p l 1 not subseteq mathfrak q i nbsp to p displaystyle bar mathfrak p nbsp ye minimalnim sered prostih idealiv A displaystyle bar A nbsp sho mistyat p l 1 displaystyle bar mathfrak p l 1 nbsp Tomu p p l 1 displaystyle mathfrak p mathfrak p l 1 nbsp ye minimalnim sered prostih idealiv u A p l 1 displaystyle A mathfrak p l 1 nbsp yaki mistyat vsi klasi a i p l 1 i 1 m 1 displaystyle a i mathfrak p l 1 i 1 m 1 nbsp Za induktivnim pripushennyam ht p p l 1 m 1 displaystyle operatorname ht mathfrak p mathfrak p l 1 leqslant m 1 nbsp tobto l m displaystyle l leqslant m nbsp Div takozh red Visota teoriya kilec Minimalnij prostij idealLiteratura red Yurij Drozd Vstup do algebrichnoyi geometriyi David Eisenbud Commutative Algebra with a View Toward Algebraic Geometry Springer New York ISBN 0 387 94268 8 10 The Principal Ideal Theorem an Systems of Parameters Michael Francis Atiyah Ian Grant Macdonald Introduction to Commutative Algebra Westview Press New York ISBN 0 201 00361 9 11 Dimension Theory Otrimano z https uk wikipedia org w index php title Teorema Krulya pro golovnij ideal amp oldid 37676249
