VNU JOURNAL OF SCIENCE Nat Sci t XV, n ° l - 1999 ON LENGTH SY STEM F U N C T IO N S D E F IN E D OF P A R A M E T E R S IN BY A LO CAL R IN G S N guyen Thai H oa Faculty of Mathe matics Pedagogical Ins titu te o f Q u y Nhon I, IN T R O D U C T IO N Let (.4,m) be a com m utative Noetherian local ring and A/ be a finitely geiK'iated A- modulo with dim A/ = il We dpnote Q m U ) the subm odule of A/ (kfined by n>0 w h ere X = ( x i , Td) is a s y s t e m o f p a r a m et e r s o f M Note th a t the subm odule Q m { x ) is used for studying the monomial couj('ctui(‘ with respect to the system of param eters X (see [7, 8]) Recall th a t the monomial conjectuiP h o l d s t n i e for t h e s y s t e m o f p a r a m e t e r s X if ị fo' Therefore, the monomial corijectuiP holds true for T if and only if Qhìiỉl) Ỷ " > t) ■ On the other hand, it was shown in [4] th a t Q m { t ) = { x \ , X d ) M provided M is Cohen Macavilay niodtile Conversely, if there is a system of param eters X such that Q,\t{r) = x M then M is Cohen - M acaulay module This fact suggest us to study the length I.a {M /Q m {x) ) T h e purpose of this note is to stu d y the following function of n ( ] M, r{ĩ í ) - IQ m {'L^U-ÌÌ^ where n = {Uị, ,ĩid) is a (1-tuple of positive integers and T{n) = (.r"‘ x y ) Th(>n, a nat ura l q u e s t io n is w h e t h e r qM, A v ) is a p o l y n o m i a l o f ni, ,7i.d for l i sufficiently laip,r (n » ) ? or it is oquivalent to ask w hether the function = ” - ” de(x, A/) - (7a/,£(zi) is a polynomial for n » ? We will give in this note some basic properties of the function Qm ặ r ) in Section and some properties of the function J m ,t Ìĩ1.) in Section II BASIC P R O P E R T IE S O F with thp m axim al ideal m and by A/ a finitely generated A-module with dim A/ = d L('t 22 O n L en g th P u n c t io n s D efin ed by a S y s t e m z = ( ^’i , i’fi) b\- 23 a system of pararnotors of A/ Then the submodule Q m Ụl ) of M is Q m [- D= i J ( ( r y + ', , r " + ' ) A / : , r ’; r " ) , n >0 we put and for 1 = ( » 1, Q m {ĩ1,R) = Q m [ĩ {ĩ i ))T h e functions k, hence it is also injective Therefore (7a/ r(zi) = ii) C an be proved similarly as (i) □ N g u y e n Thai Hoa 24 L e m m a 2.2 Sĩippose tlìHt A is the m-adic co/ijpietioji o f A aijci M is the m- adic completioii o f M Then Qm , A r ) = 'i A/,£ f, for aii n == (7?1, Ufi) Proof Shicp the n a tu l hom om orphism A A is absolutly fiat, then T is a system of param eters of M and Q m ÌĨL.ĨI) = Q A ( T , r i ) A/ Therefore we liavo (ÌM r i ĩ l ) = ỉa { Ỉ ^ Í / Q m {Z^ĨÌ)) = I a { M / Q m {3L^E)) - L e m m a 2.3 I f n > m {i.e n, > mr, i = 1, ,d) then Q a / ( t , n) c Q.\i{£,m )- Proof Let a bo a positive integer We put Q m {o ) = : rr.r.^ r2) ri>0 Si nce Qm( i ) is indepp iid en t o f t h e Older o f t h e spqupru’p X have- o n l y t o s h o w WP th at Ọ A / ( a ) c Ọ A / ( a - l ) C C Q a/(1), w i t h a > In fact, M is Nof'therian t h e n there e x ist 7/0 » ỵ~\ Q m / ^ (^^) - I Or(nn■^^) (-^ TÍO+1 nr,4-l\jir .•o 7ỈO ■■■ ' d ' and V A /(i> - 1) = ^-^2 -Ì2 -^d For any rloment a E Qhiick) ^ ^ (a -i)2nov^,2no ^ r;;“ )(.r"''“ 7 ) = for s o m e y i , y d + + r"'> + 'y + + € A/ It follows th at + for soniP Z\ , Zd € A / There fore , a e Q m Ì ũ - 1) + ; r f “ + ‘ 22 + □ + ■ O n Len gth F u n c tio n s D efin ed by a S y s t e m 25 C o r o l l a r y 2.4 T h e fiinctiuii (]M^r{ii) is Hsccndiiig i.e., m , {n, > ÌÌ, fui all i = \ , (I) > ({M r { m) ỈUI II > Proof For n > rji, WP c on s id er the m a p V? : M / Q m (x h ) ỉ^ỉ / Q m (z ^U1), (IcfilK'cl t)V + Q m (L' »)) = (I + m) , for aiiv n e M By Leninia 2.3 the m ap ự) is well (lefiiipd and it is surjpctivo Hence Ia Ì M/ Q m ÌL^Ỉr )) < Ia { M/ Q m {-t ,EÌ) □ T h e o r e m 2.5 (i m A il) < »d c { r M ) Proof We onlv nef'cl to show th at (Im J I ) < c ị x M ) \vv provo this inoquality bv induetion on (I If (Ỉ = bv Loinnia 2.1 (i) vve may assume th a t depth M > Since d ep th M (ỉiin*1/ then M is an A-nioduk' Cohen-Macaulay Hf'nce wo get I ^ ( M / r ị M ) = e{ r \ , M) and QmÌ.ì'1 1) = v i M So wo have done for the case r/ = For (I > and the a s s n tio n is tnio for all A-nioỊ)lvìn^ flu' i n d u c t i o n hvplì(\sis \vo not ỈA{JỈ/Qjỵ{r\ì)) < (i.r\Jĩ) S iiu r /-i is a non-z('ro divisor of M íh rn c{.r',JĨ) = c(r M) Thoiofon*, Ia{M/Qm(,l^, 1) < e { r M ) aiỉíl t h í ‘ t h o o r o n i is provf'd □ III TH E FU N X TIO N ìxỊ^l) Rorall that tho f u n c ti o n is a p o l y n o m i a l wlien II is large e n o u g h (// and only if JmJ is a polynomial for ĨÌ h) - M)-lAÌM/QMÌ.r.ii)) 0) if N g u y e n Thai Hoa P r o p o s i t i o n Suppose that X = ( r i .t'd) IS H s ys tem o f p n m m c t c i s of M and n= ( n i , lid)- Then J m ẶU.) < »1 Proof Let a bo a p o s i t i v e intrger and r ( a ) — Í-2 , '■,/)■ B y Lomrna 2.3, wo o h t a in Q m ( « ) C Q a / ( « - )C c g , „ ( l ) (1 ) for a > Consider the m ap V? : M/ Q MÌ a ) M / Q M Ì a - 1), defined by + Q a /( 0;)) = ^ -h Q m {cí ” 1)' for any e le m e n t o e M B y (1), it is e a s y t o s h o w t h a t t h e m a p ^ is well defined and it is an epim orphism and Ker{ip) = Q m { oc - \ ) / Q m { oc) Consider the map : M / Q m {1) Ker{ip) defined by + Q a / ( )) = + QMÌa), for any element a e M Since T p ^ Q A /( a ) c Q a/ ( ) , we can verify th a t the m ap 'I' is \v-ell defined an d it is a m o n o n i o r p h i s m S i nc e ự> is s urj ec tiv e a n d ^ is injective WP o b t a i n Ia { M / Q m {Oí)) - - ^)) ■f' Í.4 (A'c< {^) > U { M / Q M { a - 1)) + /.4(A//Qa/(1))Applying the induction hypothesis, we get U iM /Q M Ìa - 1)) > ( « - ) ) / ^ ( M / Q a / ( ) ) - Hence /,4(M /Q A /(a)) > Q I.a { M / Q m {1)) Because the proof is independentẬthe order of the sequence X, finally, we have Hence JM.xin) = n i n d e(x, M ) - Ia { M I Q m {x , ti) ) < ni rzrf T he proposition is proved □ O n L e n g t h P u n c t i o n s D e f i n e d by a S y s t e m , 27 T h e o r e m 3.2 Tỉie fiinctioii J m ,r{n) is Hscciidiiig, i.e, JmA ui ) < J m , A r )^ wlien Hi < n Proof: For every Ơ € 5f/ wo have Q m { ji ^r ) = Q m Ì ĩ I ^ ĩ i ì when' ]f_ = {-Tail), ■■■, -I'aid))- Hence, Wf' only need to prove the theorem in the case = >h — ĩhi-i and nifi < ĩiịị \\v* it by induction on d In the case d = 1, wo get J M r i m ) = J m A r ) = For í/ > 1, by Lemma 2.1, (ii), we can assume th a t depth M > and Ti is a nonzerodivisor Lot M = A //:r” *A/ Consider the map • ^"^ỉ / Q ã ĩ Ìl m ) m /Q a í Ìĩ i u i )^ dehned by + Q j ĩ (i \ ĩ r )) for any ('leiiifnit Siiicp M = dim M then M is an A-module ColiPii-Macaulay Hence, QA/(.ri, » i) = l.AÌM /QM Ì-Turn)) = d e p th M and = e (, r ” ‘ , A ; ) = Theif-fore JA /,.n ("i) = In the case d = 2, by Lem m a 2.1 and Leiunia 2.2, without any loss of the gonorality Wf (.all ci.'.biiiiu- U ia( A - Let M r , Ầ iiitpgor n wo set r{n) = (.r;’, M / i ' l M V.'C \iav (' d i m A /„ - 1- Fov a n y p o r i t i v ( ' and r!_(n) = (.r^) to bo a system of paraiiu'tors of A/„ Thoip is an exact sequpnco of A-niO(lulrs and A-honiomorpliisin - where K c rM - K L/Q m A t I) ^ M IQM {x{n)) - (3) is defined by V?(T7 + Qa/„ (■'?'2 )) “ ^ Q m ÌLÌ^'^))' for an y 77 € Mrr F o l lo w in g [1], we c an c h o o s e Xị so t h a t and the length OÍ H ^ { M ) / X ị H ^ { A Ĩ ) is finite and indppendent of 7/ when V is large (>noui>h By (3), it follows th a t I a Ì M u / Q m A ^ Ĩ ) ) = l■A{^>'er{ip)) + l A { M/ QMÌ z { n ) ) ) We get O n Len gth F u nc tio ns D e fin ed by a S y s t e m 29 ■ h i A n ) = v ^ e { x , M ) - I a { M I Q m {x {ĩ >))) = e(.r ^, A/ „) - / ,(A /„ /(? A, „ ( r") ) + / , ( / ( A / ) / r / / ^ ( M ) ) is a constant for 71 3> Applying Thporom 3.2, the theorem is provod □ REFERENCES 1] X T Cuong and V.T Khoi Moclulos whose local cohomology modules have CohenMacaulay M atlis duals Proc of Hanoi Conference 1995, Springer-Verlag, 223-231 2] \ T Cuong and N.D Minh On the length of Koszul homology and generalized fractions, Math Proc Cambridge Phil Soc 119 (1)(1996), 31- 42 3] N.T Cuoiig and N.D Minh, Length of generalized fractions of rings with polvnoinial type < Vietnam J Math 26 1(1998) 87 - 90 '4] R.Y HaitshoniiP A property of A-sequence Bull Soc Math France, 4(1966), 61-66 5] H M atsurnura Commutative algebra Second edition, London: Beii.janiin 1980, G] N D Miiili O n t h e least clpgTPP o f p o l y n o m i a l s b o u n d i n g a b o v e tlu' differences Ix-twpon m u lt ip lic it ie s and len g th o f gpiioralizpd fractions A c t a Ma t h Vi etnam 20 (1)(1995), 115 - 128 7] R.Y Sharp and H Zakeri Modules of gpiipializod fractions, MathemaUka 29( 1982) 32 - 41 8] R.Y Sharp and H Zakori Lengths of cortain gonoralized fractions, J.Pure AppL Mg 38(1985), 323 - 336, TAP CHI KHOA HOC ĐHQGHN, KHTN, t XV n ° l - 1999 VE XHỬNG HÀM ĐỌ DÀI XÁC ĐỊNH BƠI HẸ TH A M so T R O N G VÀNH ĐỊA PPỈƯƠNG N guyễn KhoH Tốiì Đại ỈIỌC Thái Hòa Sư phạm Qiiv Nhơii Trong hài chúng tòi (lịnh nghĩa liai hàm độ dài qM,r{n) JM,r{ĩl) •‘hf’O biến l i = { V ỵ , lièn kếr với hệ th am số X = ( t ] T r f ) cùa A - niòđun M Một số tính chất cùa hàm nôu ... Cuong and V.T Khoi Moclulos whose local cohomology modules have CohenMacaulay M atlis duals Proc of Hanoi Conference 1995, Springer-Verlag, 223-231 2] T Cuong and N.D Minh On the length of. .. homology and generalized fractions, Math Proc Cambridge Phil Soc 119 (1)(1996), 31- 42 3] N.T Cuoiig and N.D Minh, Length of generalized fractions of rings with polvnoinial type < Vietnam J Math 26... '4] R.Y HaitshoniiP A property of A- sequence Bull Soc Math France, 4(1966), 61-66 5] H M atsurnura Commutative algebra Second edition, London: Beii.janiin 1980, G] N D Miiili O n t h e least clpgTPP