U abstraktnij algebri rozmirnist Krulya kilcya R chislo strogih vklyuchen v maksimalnomu lancyuzi prostih idealiv Rozmirnist Krulya ne obov yazkovo ye obmezhenoyu navit dlya neterovih kilec Zmist 1 Oznachennya 2 Prikladi 3 Vlastivosti 4 Rozmirnist modulya 5 Primitki 6 Div takozh 7 DzherelaOznachennya red Yaksho P0 P1 Pn prosti ideali kilcya taki sho P 0 P 1 P n displaystyle P 0 subsetneq P 1 subsetneq ldots subsetneq P n nbsp to kazhut sho ci ideali utvoryuyut lancyug dovzhini n Rozmirnist Krulya supremum dovzhin lancyugiv golovnih idealiv Prikladi red U kilci Z 8Z x y z mi mozhemo rozglyadati lancyug 2 2 x 2 x y 2 x y z displaystyle 2 subsetneq 2 x subsetneq 2 x y subsetneq 2 x y z nbsp Kozhen z cih idealiv golovnij tak sho rozmirnist Krulya Z 8Z x y z ye yak minimum 3 Faktichno rozmirnist cogo kilcya rivna tochno 3 Dovilne pole k maye rozmirnist Krulya 0 Kilce mnogochleniv k x 1 x n displaystyle k x 1 x n nbsp i kilce formalnih stepenevih ryadiv k x 1 x n displaystyle k x 1 x n nbsp nad deyakim polem k mayut rozmirnist Krulya n Bilsh zagalno dlya dovilnogo neterovogo komutativnogo kilcya R dlya rozmirnosti Krulya vikonuyetsya rivnist dim R x 1 x n dim R x 1 x n dim R n displaystyle dim R x 1 ldots x n dim R x 1 ldots x n dim R n nbsp Dlya dovilnogo komutativnogo kilcya R rozmirnist Krulya kilcya mnogochleniv zadovolnyaye nerivnist dim R 1 dim R x 1 x n 2 dim R 1 displaystyle dim R 1 leqslant dim R x 1 ldots x n leqslant 2 dim R 1 nbsp Dlya kilcya formalnih stepenevih ryadiv u comu vipadku vikonuyetsya lishe nerivnist dim R 1 dim R x 1 x n displaystyle dim R 1 leqslant dim R x 1 ldots x n nbsp Natomist isnuyut kilcya skinchennoyi rozmirnosti Krulya nad yakimi kilce formalnih stepenevih ryadiv maye neskinchennu rozmirnist Zokrema kilce R x displaystyle R x nbsp maye neskinchennu rozmirnist todi i tilki todi koli isnuye prostij ideal p displaystyle mathfrak p nbsp dlya yakogo p x p R x displaystyle mathfrak p x neq sqrt mathfrak p R x nbsp Tut p x displaystyle mathfrak p x nbsp poznachaye formalni stepenevi ryadi iz koeficiyentami iz p displaystyle mathfrak p nbsp a p R x displaystyle sqrt mathfrak p R x nbsp radikal idealu u R x displaystyle R x nbsp porodzhenogo p displaystyle mathfrak p nbsp Zokrema yaksho R kilce rozmirnosti 0 to rozmirnist R x displaystyle R x nbsp rivna abo 1 abo neskinchennosti Prikladom skinchennovimirnih komutativnih kilec dlya yakogo R x displaystyle R x nbsp maye neskinchennu rozmirnist ye kilcya nediskretnogo normuvannya rozmirnosti 1 Inshim prikladom ye R Q x 1 x 2 x 1 n x 2 n n 2 displaystyle R mathbb Q x 1 x 2 ldots x 1 n x 2 n ldots n geqslant 2 nbsp yake ye kilcem rozmirnosti 0 a takozh vsi skinchennovimirni kilcya spektr yakih ne ye neterovim topologichnim prostorom 1 Kilce golovnih idealiv sho ne ye polem maye rozmirnist Krulya 1 Rozmirnist dovilnogo kilcya Artina ye rivnoyu 0 Rozmirnist dovilnogo kilcya Dedekinda ye rivnoyu 1 Lokalne kilce maye nulovu rozmirnist todi i tilki todi koli vsi elementi jogo maksimalnogo idealu ye nilpotentnimi Priklad Nagati Nehaj R k x 1 x n displaystyle R k x 1 x n nbsp kilce mnogochleniv zi zlichennoyu kilkistyu zminnih Rozglyanemo poslidovnist prostih idealiv p 1 x 1 p 2 x 2 x 3 p 3 x 4 x 5 x 6 displaystyle mathfrak p 1 x 1 mathfrak p 2 x 2 x 3 mathfrak p 3 x 4 x 5 x 6 ldots nbsp Todi S R p i displaystyle S R setminus cup mathfrak p i nbsp ye multiplikativnoyu mnozhinoyu i mozhna rozglyanuti lokalizaciyu A S 1 R displaystyle A S 1 R nbsp Nehaj takozh m i S 1 p i displaystyle mathfrak m i S 1 mathfrak p i nbsp Mnozhina m i displaystyle mathfrak m i nbsp ye mnozhinoyu maksimalnih idealiv kilcya A Spravdi ideali kilcya A ye u biyektivnij vidpovidnosti iz idealami kilcya R sho mistyatsya u p i displaystyle cup mathfrak p i nbsp Yaksho a displaystyle mathfrak a nbsp ye takim nenulovim idealom to a p i displaystyle mathfrak a subset mathfrak p i nbsp dlya deyakogo i Spravdi yaksho ce ne tak to z zapisu p i p 1 p n k gt n p k displaystyle bigcup mathfrak p i mathfrak p 1 cup ldots mathfrak p n cup bigcup k gt n mathfrak p k nbsp i lemi pro uniknennya prostih idealiv viplivaye sho a k gt n p k displaystyle mathfrak a subset bigcup k gt n mathfrak p k nbsp dlya vsih n Ale peretin takih mnozhin ye rivnim nulya sho superechit pripushennyu Bud yakij nenulovij element kilcya A nalezhit lishe skinchennij kilkosti maksimalnih idealiv m i displaystyle mathfrak m i nbsp adzhe bud yakij nenulovij element kilcya R nalezhit lishe skinchennij kilkosti idealiv p i displaystyle mathfrak p i nbsp sho viplivaye z togo sho bud yakij element kilcya R ye elementom deyakogo pidkilcya zi skinchennoyu kilkistyu zminnih i tomu ne mozhe mistiti porodzhuyuchih elementiv dlya vsih p i displaystyle mathfrak p i nbsp Kozhna lokalizaciya A m i R p i displaystyle A mathfrak m i R mathfrak p i nbsp ye neterovim kilcem Dijsno yaksho p i x k x k i 1 displaystyle mathfrak p i x k ldots x k i 1 nbsp to R p i K x k x k i 1 x k x k i 1 displaystyle R mathfrak p i simeq K x k ldots x k i 1 x k ldots x k i 1 nbsp de K pole chastok pidkilcya mnogochleniv u R sho ne mistyat zminnih x k x k i 1 displaystyle x k ldots x k i 1 nbsp Tverdzhennya otrimuyetsya z togo sho kilce mnogochleniv nad polem zi skinchennoyu kilkistyu zminnih i bud yaka jogo lokalizaciya ye neterovimi kilcyami Dlya dovilnogo komutativnogo kilcya R yaksho kozhen jogo nenulovij element mistitsya lishe u skinchennij kilkosti maksimalnih idealiv i lokalizaciya po kozhnomu maksimalnomu kilci ye kilcem Neter to i R kilce Neter Spravdi dlya dovilnoyi zrostayuchoyi poslidovnosti idealiv dovilnij element yakogos iz idealiv nalezhit lishe skinchennij mnozhini maksimalnih idealiv Ale todi i kozhen ideal zrostayuchoyi poslidovnosti ye pidmnozhinoyu ciyeyi skinchennoyi mnozhini maksimalnih idealiv Tomu isnuye deyakij maksimalnij ideal yakomu nalezhit neskinchenna kilkist idealiv poslidovnosti Oskilki pri perehodi do lokalizaciyi po comu maksimalnomu idealu pidposlidovnist stabilizuyetsya to ce zh ye spravedlivim i dlya pochatkovoyi pidposlidovnosti a tomu vsiyeyi poslidovnosti Otzhe R kilce Neter Zokrema i chastkovij vipadok R k x 1 x n displaystyle R k x 1 x n nbsp ye neterovim kilcem oskilki vkazani umovi vikonuyutsya Natomist u m i displaystyle mathfrak m i nbsp isnuye lancyug prostih idealiv dovzhini i displaystyle i nbsp Oskilki i displaystyle i nbsp ye neobmezhenim chislom to R displaystyle R nbsp maye rozmirnist rivnu neskinchennosti i ye prikladom neskinchennovimirnogo neterovogo kilcya Natomist dovilne napivlokalne neterove kilce maye skinchennu rozmirnist Vlastivosti red Rozmirnist Krulya kilcya R rivna supremumu visot vsih prostih idealiv R Zokrema oblast cilisnosti maye rozmirnist Krulya 1 koli kozhen vidminnij vid nulya prostij ideal ye maksimalnim idealom Oblast cilisnosti ye polem yaksho i tilki yaksho jogo rozmirnist Krulya rivna nulyu Rozmirnist Krulya kilcya ye rivnoyu rozmirnosti bud yakogo jogo cilogo rozshirennya Dlya kilcya R i prostogo idealu p R displaystyle mathfrak p in R nbsp vikonuyetsya nerivnist ht p coht p dim R displaystyle operatorname ht mathfrak p operatorname coht mathfrak p leqslant dim R nbsp Nerivnist mozhe buti strogoyu navit dlya neterovih kilec Nehaj napriklad A k X Y Z displaystyle A k X Y Z nbsp kilce formalnih stepenevih ryadiv vid troh zminnih nad polem k I ideal porodzhenij XY i XZ i R A I Todi dim R 2 displaystyle dim R 2 nbsp Yaksho poznachati Y Z displaystyle bar Y bar Z nbsp obrazi Y Z displaystyle Y Z nbsp u R to visota idealu Y Z displaystyle bar Y bar Z nbsp ye rivnoyu 0 a oskilki R Y Z k X displaystyle R bar Y bar Z simeq k X nbsp to coht Y Z dim k X 1 displaystyle operatorname coht bar Y bar Z dim k X 1 nbsp Tomu ht Y Z coht Y Z 1 lt dim R displaystyle operatorname ht bar Y bar Z operatorname coht bar Y bar Z 1 lt dim R nbsp Rozmirnist modulya red Yaksho R komutativne kilce i M R modul rozmirnist Krulya M viznachayetsya yak rozmirnist Krulya faktorkilcya po anulyatoru modulya dim R M dim R Ann R M displaystyle operatorname dim R M operatorname dim R operatorname Ann R M nbsp de AnnR M yadro vidobrazhennya R EndR M sho zistavlyaye elementu kilcya mnozhennya na cej element Takozh mozhna dati oznachennya za dopomogoyu rivnostej dim R M sup p Supp M coht p sup p Ass M coht p displaystyle operatorname dim R M sup mathfrak p in operatorname Supp M operatorname coht mathfrak p sup mathfrak p in operatorname Ass M operatorname coht mathfrak p nbsp de Supp M displaystyle operatorname Supp M nbsp nosij modulya a Ass M displaystyle operatorname Ass M nbsp mnozhina asocijovanih prostih idealiv modulya Primitki red Jimmy T Arnold 1973 Krull dimension in power series rings Transactions of the American Mathematical Society 177 299 304 doi 10 1090 s0002 9947 1973 0316451 8 Div takozh red Visota teoriya kilec Dzherela red R Gordon J Ch Robson Krull dimension American Mathematical Society 1978 ISBN 0 8218 1833 3 J C McConnell J C Robson Lance W Small Noncommutative Noetherian Rings American Mathematical Society 2001 ISBN 0 8218 2169 5 Otrimano z https uk wikipedia org w index php title Rozmirnist Krulya amp oldid 36746717
