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Refiltering for Some Noetherian Rings Refiltering for some noetherian rings Noncommutative Geometry and Rings Almer´ıa, September 3, 2002 J. Gomez-T´ orrecillas Universidad de Granada Refiltering for some noetherian rings – p.1/12 Use filtrations indexed by more general (more flexible) monoids than Filtration Associated Graded Ring Re-filtering: outline Generators and Relations Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Associated Graded Ring Re-filtering: outline Generators Filtration and Relations Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline Generators Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Re-filtering: outline Use filtrations indexed by more general (more flexible) monoids than New Generators Re- Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring Keep (or even improve) properties of Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring We will show how these ideas can be used to obtain the Cohen-Macaulay property w.r.t. the Gelfand-Kirillov dimension for the quantized enveloping algebras. The results will be published in a joint paper with F. J. Lobillo Refiltering for some noetherian rings – p.2/12 is a free abelian monoid with generators a total monoid ordering on with as minimal element Assume is –filtered by The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢¤£¦¥ ¢ § § § ¡ Let denote a free abelian monoid with generators ¥ , whose ¨ ¨ ¨ © § § § £ ¥ ¡ elements will be considered as vectors ¥ with non-negative integer components. Refiltering for some noetherian rings – p.3/12 a total monoid ordering on with as minimal element Assume is –filtered by The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢ ¢ £ ¡ is a free abelian monoid with generators ¥ ¥ ¡ Let be a monoid total ordering on such that ¨ for every ¡ ¨ . Basic examples are the lexicographical orderings Refiltering for some noetherian rings – p.3/12 Assume is –filtered by The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢ ¢ £ ¡ is a free abelian monoid with generators ¥ ¥ ¡ a total monoid ordering on with as minimal element ¡ ¥ becomes then a well-ordered monoid Refiltering for some noetherian rings – p.3/12 p.3/12 – rings –filtration noetherian some ¥ for ¡ of ¡ ¢ element ¡ Refiltering ¥ An . ¡ ¨ ¢ ¥ £ minimal . ring ¡ as ¨ e ¥ v all rings © with generators ¡ for ¨ with commutati on that all a . amily f er for © v such a o graded monoid is ordering of . algebra and abelian #" an monoid free –algebra ed be ! a total is a a –submodules 3. 2. 4. 1. ¡ on Let Filter . by en v gi is and by algebra –filtered –graded is associated The Assume where The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢ ¢ £ ¡ is a free abelian monoid with generators ¥ ¥ ¡ a total monoid ordering on with as minimal element ¡ ¡ ¨ © Assume is ¥ –filtered by Refiltering for some noetherian rings – p.3/12 p.3/12 – rings . noetherian some for ¡ ¡ ¢ element Refiltering ¨ ( ¥ © ¥ ¢ ¥ £ by minimal % $ en as v ¤' © gi is rings © with generators and ¡ by with % algebra ¤' on $ & –filtered graded monoid –graded ordering ¡ © ¥ ¡ and abelian is % monoid $ free ed a associated total is a ¡ Assume The where Filter n it- are and p.4/12 such – s ’ ratio rings -.2 ¥ –filtration –f.g., –filt + –graded noetherian ¡ ¥ £ some ¥ ¡ an - for are the an s is ’ ¡ is Refiltering an 1 ring, 2 there be /0 elements the ,.-.2 , + ¡ Then © 1 . + noetherian £ ¨ " /0 £ left ,.- em + *) a homogeneous 3 6 4 that © is + some © such © % where for Theor $ , -54 " on 3 6 . 4 and © Let , + ; xtension that e $ 1. 7 er : © -54 v Ore 3 em o such ) 0 ¡ g. 89 Refiltering erated Theor f. that on A p.5/12 – rings noetherian some for . Refiltering -F4 ¤' H E I BDC A G 4 © © ¨ 4 4 - that with such ¡ @ polynomials 2 e r H ?=?=?=? ¨ ¨ ¨5> £ G . 2 4 <=<=<=< they’ © choose oof: Set and Pr that from get we , of . of –basis a is –basis a is , en v Gi Since p.5/12 – rings + noetherian some for W T Refiltering ¤' H I ¥ X T G 4 W % © T $ S P Q O -54 LNM in K ¥ U¦V -54 E © T C B A xpression © e polynomials S an ¨ PRQ 4 O e e v LNM J r ¥ ha ' K ¨ 2 I we § ¥ , § 9 § they’ ¨ £ ¥ 9 © en v oof: Gi Pr that from get we , of . of –basis a is –basis a is , en v Gi Since p.5/12 – rings + noetherian some for W T Refiltering ¤' H 2 \ I ¥ ¥ X T G 4 W % © T ¥ £ $ S \ P Q O © -54 LNM in K ¥ .\ U¦V -54 E © T C B A xpression T[Z © e - 2 § § polynomials S § an ¨ PRQ 4 O e e £ v LNM J T Y r - ¥ ha ' K ¨ 2 I we § ¥ , § 9 § they’ ¨ £ ¥ 9 © en v oof: Gi Pr that from get we , of . of –basis a is –basis a is , en v Gi Since p.5/12 – rings + noetherian some S W for T P Q O LNM ¥ X T Refiltering ' K ¤' H W T I S I P Q O ] G LNM T 4 © W K T S U V P Q O -54 T LNM ¥ K © -54 U¦V E C B © T A S P Q © O polynomials LNM ¨ S 4 ' K PRQ e O J r I LNM ¥ ¨ ' K 2 9 I § ¥ , § § to 9 they’ ¨ £ ¥ 9 © leads en v oof: Gi This Pr that from get we , of . of –basis a is –basis a is Since p.5/12 – rings + noetherian some W for T + ¥ X T Refiltering ¤' H W T S I _ P Q T O G LNM 4 © K U V ¥ ] -54 T T _ ¥ T © S -54 that E PRQ O C B ws M L A K S sho P Q U ^ © O polynomials 9 LNM ¨ E © T 4 ' K e BDC J r A I 9 ¥ ¨ 2 9 on § ¥ , § § they’ ¨ £ ¥ 9 © induction en v oof: Gi An Pr that from get we , of . of –basis a is –basis a is Since p.5/12 – rings + that ! noetherian some W ! for T ¥ X from T Refiltering ¤' H W T get S + I ¥ P Q O we G LNM , 4 © _ K T U V % $ -54 T ¥ © of ¥ ] -54 . T E _ T C B S A PRQ of S –basis O + P Q © O LNM polynomials a LNM K ¨ 4 ' K U ^ is e –basis + J r I ¥ T © a 2 ¨ 2 9 is 9 § ¥ , § \ § 2 they’ ) X ¨ £ ¥ T 9 \ © en ) ] v T oof: Gi Since Pr p.6/12 and – + rings b H _ noetherian some for that ' H -54 Refiltering b H _ I © _ G 4 b H 4 - _ _Nc from G 4 get © we -F4 a b H _ elations § ¥ r § § _Nc ! ¥ - 4 © ` , + getting _ en v oof: Gi Pr ery v e for p.6/12 and – + is rings b ' H H _ noetherian some for that that e ' H -54 Refiltering v b H _ I ha © _ we G 4 b H 4 - _ f.g., is _Nc from G 4 get © H we -F4 a b H _ and elations § ¥ r § § _Nc ! ¥ - 4 © noetherian ` , + left getting is _ + en v oof: noetherian. Gi Since Pr ery v e for and p.6/12 and – + d 4 rings b \ H _ noetherian some + for for ¨ 4 that e _ ' H -54 Refiltering b H _ I © ery \ v _ e G 4 b that H 4 - _ for _Nc ] such from T G _ 4 2 T get © f H of d we I 4 T -F4 a b H _ elations G § ¥ 4 r b § H _ § _Nc subset ! © ¥ - 4 © _ G finite ` 4 a , + is getting _ there en v oof: Gi Thus Pr ery v e for , a m p.6/12 – p rings 3 d `on 4 noetherian m + some ! \ for _ for ery 3 Refiltering ¨ v e I ery ¥ ] T v e for ¨ W 3 T 4 - e for f Hq -54 3 6 4 I T ] © T \ 3 _ G T -54 3 G that H i 4 3 - H h j k l 6 g g 4 © I such elations G G 2 4 3 r relation 4 G b H _ of 3 the © d 4 _ G from 4 , getting subsets get finite oof: Analogously for we Pr p.6/12 – rings noetherian or + some for _ t x le m Refiltering ¥ ] T ery W \ v T e r q l for i H H q h and g l k j ] g T I _ by T t t © e 3 G H 2 i H h G j k l g 4 3 g on 6 I \ \ 4 © elations G r 4 G b 4 vw H 3 _ G ordering © _ sut G 4 monoid \ getting the oof: Define Pr p.6/12 – rings noetherian or + some for t _ x le m Refiltering ¥ ] T ery \ W v T e r y q i for H q h and x y H g g k ] I T _ 3 T G t t © e i G xzy H H h 4 3 g g k 6 I 4 © \ \ elations G G 4 r b 4 3 H _ G vw © _ ordering sut G 4 \ getting monoid the oof: for Pr Then is the maximum of w.
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