Spektr kilcya mnozhina prostih vlasnih idealiv p displaystyle mathfrak p kilcya R Zazvichaj na spektri zadayetsya topologiya Zariskogo Inodi rozglyadayut maksimalnij spektr S p e c m R displaystyle mathrm Specm R pidprostir prostoru S p e c R displaystyle mathrm Spec R sho skladayetsya iz zamknutih tochok Vlastivosti RedaguvatiProstir S p e c R displaystyle mathrm Spec R nbsp nese puchok lokalnih kilec O S p e c R displaystyle mathcal O mathrm Spec R nbsp zvanij strukturnim puchkom Dlya tochki p S p e c R displaystyle mathfrak p in mathrm Spec R nbsp shar puchka O S p e c R displaystyle mathcal O mathrm Spec R nbsp nad p displaystyle mathfrak p nbsp ce lokalizaciya R p displaystyle R mathfrak p nbsp kilcya R shodo p displaystyle mathfrak p nbsp Bud yakomu gomomorfizmu kilec f A B displaystyle varphi A rightarrow B nbsp sho perevodit odinicyu v odinicyu vidpovidaye neperervne vidobrazhennya f p f 1 p displaystyle varphi mathfrak p mapsto varphi 1 mathfrak p nbsp Yaksho N nilradikal kilcya A to prirodne vidobrazhennya S p e c R N S p e c R displaystyle mathrm Spec R N mapsto mathrm Spec R nbsp ye gomeomorfizmom topologichnih prostoriv Dlya nenilpotentnogo elementu f R displaystyle f in R nbsp nehaj D f S p e c R V f displaystyle D f mathrm Spec R setminus V f nbsp de V f p S p e c R f p displaystyle V f mathfrak p in mathrm Spec R f in mathfrak p nbsp Todi prostori D f i S p e c R f displaystyle mathrm Spec R f nbsp de R f displaystyle R f nbsp lokalizaciya R vidnosno f ye izomorfnimi Mnozhini D f nazivayutsya golovnimi vidkritimi mnozhinami Voni utvoryuyut bazis topologichnogo prostoru S p e c R displaystyle mathrm Spec R nbsp Tochka p S p e c R displaystyle mathfrak p in mathrm Spec R nbsp zamknuta todi i tilki todi koli p displaystyle mathfrak p nbsp maksimalnij ideal kilcya R Zistavlyayuchi tochci p displaystyle mathfrak p nbsp yiyi zamikannya p displaystyle overline mathfrak p nbsp v S p e c R displaystyle mathrm Spec R nbsp oderzhuyetsya vzayemno odnoznachna vidpovidnist mizh tochkami prostoru S p e c R displaystyle mathrm Spec R nbsp i mnozhinoyu zamknutih nezvidnih pidmnozhin v S p e c R displaystyle mathrm Spec R nbsp Prostir S p e c R displaystyle mathrm Spec R nbsp ye kvazikompaktnim ale yak pravilo ne ye gausdorfovim Rozmirnistyu prostoru S p e c R displaystyle mathrm Spec R nbsp nazivayetsya najbilshe n dlya yakogo isnuye poslidovnist vidminnih zamknutih nezvidnih mnozhin Z 0 Z n S p e c R displaystyle Z 0 subset ldots subset Z n subseteq mathrm Spec R nbsp Bagato vlastivostej kilcya R mozhna oharakterizuvati v terminah topologichnogo prostoru S p e c R displaystyle mathrm Spec R nbsp Napriklad kilce R neterove todi i tilki todi koli S p e c R displaystyle mathrm Spec R nbsp neterovij prostir prostir S p e c R displaystyle mathrm Spec R nbsp ye nezvidnim todi i tilki todi koli kilce R N ye oblastyu cilisnosti rozmirnist S p e c R displaystyle mathrm Spec R nbsp zbigayetsya z rozmirnistyu Krulya kilcya R i t d Dlya kozhnoyi pidmnozhini U S p e c R displaystyle U subset mathop mathrm Spec R nbsp yaka ye odnochasno vidkritoyu i zamknutoyu u topologiyi Zariskogo isnuye yedinij idempotent e R displaystyle e in R nbsp dlya yakogo U D e displaystyle U D e nbsp Takim chinom oderzhuyetsya biyekciya mizh pidmnozhinami U S p e c R displaystyle U subset mathop mathrm Spec R nbsp sho ye odnochasno vidkritimi i zamknutimi i idempotentami e R displaystyle e in R nbsp Nehaj S p e c R U V displaystyle mathop mathrm Spec R U amalg V nbsp de U displaystyle U nbsp i V displaystyle V nbsp ye vidkritimi i vidpovidno takozh zamknutimi pidmnozhinami Todi R R e R 1 e displaystyle R cong R e times R 1 e nbsp e R displaystyle e in R nbsp dlya yakogo U S p e c R e displaystyle U cong mathop mathrm Spec R e nbsp i V S p e c R 1 e displaystyle V cong mathop mathrm Spec R 1 e nbsp Div takozh RedaguvatiProstij ideal Shema matematika Topologiya ZariskogoLiteratura RedaguvatiAtya M Makdonald I Vvedenie v kommutativnuyu algebru Moskva Mir 1972 160 s ros Hartshorn R Algebraicheskaya geometriya M Mir 1981 Shafarevich I R Osnovy algebraicheskoj geometrii M Nauka 1972 Otrimano z https uk wikipedia org w index php title Spektr kilcya amp oldid 37160390
