DSpace at VNU: Extension of a Result of Huneke and Miller

/ N U J O U R N A L O F S C IE N C E , M a t h e m a t i c s - Physics T.xx, N - 2004 E X T E N S IO N OF A RESU LT OF H U N E K E A N D M ILLER D a m V an N h i, L u u B a T h a n g Pedagogical University Ha Noij Vietnam A b s t r a c t Let k be the ground field k and X = (x o , , x n ) be indeterminates Let I be a graded ideal of k[x] In [2], [3] there are formulas to determine the Betti numbers and multiplicity of R / I Now we want togive an extension and a new simple proof about a result of Huneke and Miller and we alsoconsider the algebra withminimal multiplicity.1 In tro d u c tio n Let R = k[x , , x n \ be the polynomial ring over the field k Let I be a graded ideal >f R R / I is said to have a pure resolution of type (rfi, , d p) if its minimal resolution lias the form — >0 R ( - d p ) — ■>-> ® ji R ( - d i ) — > R — * R / I — » 0, fil < • • • < dp 3=1 3=1 111 [2] Herzog anti Kill have given a formul to determ ine the B etti num bers of R / I The multiplicity of R / I is given by Huneke and Miller, see [3] This paper presents an extension iind a new simple proof ab o u t a result of Huncke and Miller and we also consider the algebra w ith minimal multiplicity E x t e n s i o n o f a H u n e k e a n d M i l l e r ’s r e s u l t We collect here cl num ber of more or less s tan d a rd definitions, results and notations of graded modules Let R = Q>i>oR, be a graded ring, where Ro is the ground-field k, and M = © tez M t a finitely generated graded i?-m odule of dim ension d For evry i G z , we denote by M( i ) the graded R -m odule w ith coincides w ith M as th e underlying 7?-module and whose grading is given by M(i,)j = M i+ j for all j € z Set i ( Mt ) = dini/fc Mị Let h ( M, t) and ì i m ( z ) denote the Hilbert functions an d th e H ilbert series of M, which are defined to be h { M , t ) = £( Mt ) for all t e z , h M (z) = Y ^ h ( M , t ) z t t ez 1T h e authors are partially su p p o r te d by the N a tio n a l Basic Research Program Typeset by 23 D a m Van N h i , L u u B a Thang 24 It is well known th a t h ( M , t ) = The multiplicity of M is defined as follows e (M ) = j J) w ith ej € eo if d > , £( M) if rf = Suppose th a t — > ® j L i R ( - d p j ) — > z and Vi > > — » ® j l i R ( ~ d 0j) — * M — minimal graded free resolution of M Since hji(z) = {t+n~[i ) zt = (i-~ )n and h M (z) = £ ( - )* £ > « ( - * , ) ( * ) = x > i r ( i—0 j —1 i —0 (1 - z ) n ^ '1 2— v i=00 jj==l l (1 - ) v d with g(z) G /c[z] ' there is (1 - z ) n~dg(z) = XZi=o(—1 )i (53Ít=i *dij) := S m ( z ), see [1] T h e o r e m [2, Corollary 4.1.14] I f M is finitely generated graded R -m o d u le o f dimen­ sion d , then ( _ \ n — d Q ( n ~ d + j ) ( -i \ / ' _ \™ — d o ( n ~ d ) / « \ = ( ir _ J ) and e(M ) = ( (n — d + j)! j M (n — d)! ( ■ Let M be a finitely generated graded jR-module M is said to have a pure resolution of type ( c/oj ^ I j • • • >^p) if its minimal resolution has the form - — > R ( - d p) — > > © t\ ? (-d i) — > © 7= j=l Íq R ( - d o ) — > M — > 0, < ■• ■< dp j=l The following theorem shows th a t the B etti numbers and multiplicity of th e CohenMacaulay module M are completely determined by the twists di and Betti num ber ỂQ T h e o r e m 1.2 Let M be a finitely generated graded R -m o d u le o f dimension d I f M is a Cohen-Macaulay m odule and has a pure resolution o f type (do, d ị , j dp), then 0— —(( 1)1\iT-TPnj=i(4> ~dj) ' ^ • • • >p> e(M (Kf)\ — _ (—1)P^0rr,, / , _ \ ’?' “ liiz = o i^ iv a i aj) V- JN dj) , =1 Proof Since M is a Cohen-Macaulay module, there is p = n —d and a/(z) = (1—z)p ' E L o V i d ị = 0, (A)< 1 ■■ ■ I p J I E L o y i d pi = g { i ) - Consider the following equation g[l) 5(1) ( i - do){x - di ) ■• • (x - dp) _ _ Xị Xọ X - X - x2 + dỵ X - d2 Then (D) : g( 1) = xq( x — d\) • • • (x — dp) + x \ ( x — do)(x d0){x - d p - i ) For X = d , x = ( * ! , • • • , x = dp, (Ỉ2 ) • •' i t can b e dp) + • • • + xp( x v e r i f i e d th a t ỡ(i) x - £1 = (C) X'? = « _ (d o -d i)- (d o -d p )’ gO) ( d i- d o ) - ( d i- d p ) ’ _ gill _ ( d - d o ) ( d - d i ) - ( d2 - d p ) ’ _ gili _ p [ d p —do ){dp d \ ) •••[dp —d p —I ) The ith power sum and the ith elementary symmetric polynomial of d o , ,dp will be denot'd by Mt = ^0 + ■■■~t~ identity Et — fỉ'' ;r/J • • • , X, f r o m the above a n d Et — Eị ( d o , ,dp) Set E t 1J — (Et)di=0 - Use the iri8.tlioniiitiCiil induction to CcLCul&tG the cocfficicnts of e q u a t i o n ( £?) i t f o l l o w s t h a t f E f = o ^ = ( ^ ) < j = , • ■• ( x q ( ỉ ,q + X i d i H 4- X p d P = ( ) From (A ),(C ) and (Ơ) it follows th a t 2/j = Xj,i = , ,p Since ( ) = x o Y l j =1(do - dj), there are n ;=0 -4) Consider x(x - 1) • • • (x - p + 1) _ Zo Zi x ( x — ) • • • (x - p + ) X — d, ( x — d o ) ■■• ( x - d p ) X - X - d\ zo(x - di) • • ■(x - dp) + Zi (x - do)(x - d2) ■■■{x - dp) + ■■■+ zp (x - do) ■■• (x d p - ) There is z0 + - h zp = and for X = d0, ■■■ , X = dp we obtain c io (d o- l) - - (do- dl)di(di- l)- - I ( d o—p-f-1) ■■(do-dp ) ’ ■( d i - p + ) ( i l l —do)- - ( d i - d p ) d „ ( d v - D - ( d p -p + ) (dp-do)" ( d p —dp —1 ) ’ D a m V a n N h i , L u u B a Thang 26 Since e ( M ) = — y y( 1) , therefore C[M) = ( - i K S f f ( i ) = ( - K £ i - y e M d - ) = 1— P! fV ' ảí p + 1) - ) ■■• ( P - d j) 1, , p - Tl + d wt; tcM t c i i i i iin iie e aall i l Cị when w xien £ t oq, , ,, C p - n + d aare re given Ui1 ud ceterm £p_n+(i Let I be a homogeneous ideal of R R / I has a pure resolution of type ( d i , , dp) with its m inim al resolution of th e form ii R ( - d p ) — > • • • — > ( Ị ỳ R ( - d i ) — > R — * R / — > 0, d\ < • • • < dp j=1 j= i T he following result was proved by Huneke and Miller using residues theory of complex function, see [3] As an im m ediate consequence of Theorem 1.2 we now want to give an an oth er simple proof E xten sion o f a result o f H u n eke and M ille r 27 C o r o l l a r y 1.4 [3, T heorem 1.2] L et I be a homogeneous ideal o f R If R / I is CohenMacaulay and has a p u re resolution o f typ e ( d i , , dp), then i#j Proof ủ" J j= Here Ĩ.Q £o = = 1l,d ,^ o = 0u By b y TIhheorem e o re n 1.2, we have g ( l ) = d \ d p Hence t j = ( - I >i+ r u i r f h ; ) and Note t hhat a t T hheorem an extension of Corollary 1.4 eo rem 1.2 is as considered consider R e m a r k 1.5 I f we present (x - l ) ( x - 2) ■■■( x - p + I) = x p - Sị X?-1 + S2 XP~ - f ( - l ) p - Sp_ix, then (-1 ỹiidiidi - l ) - - - ( d l - p + l ) = (-1 )^ i[< - S y d pr l + s d p ~ + ( - y - ' s p - i d i } Since £ ? =1 y%4 = 0, j = , , p - 1, and since = 0, we obtain p — -L = - by T h eo rem 1.1 ( n - d + i )\ (p + j ) ! = 7T ~ ~ \T F ( - l )% d i(d i - ) ■• • (di - p - j + ) (p + JV-frt = T i - i y w d i - ) • • • (di - p - j +1) (p + j)! ^ (p + i Y Ế Í III particular, for j = there is e { R / 1) = Ặ E L i ( - )P+i^ d i » (form ula o f Peskine-Szpiro), see [3] Assume th a t / ^ is a homogeneous ideal of i? D enote by v ( R / 1) = h ( R / / ; 1) the embedding dimension of R / I Abhyankax proved th a t if R / I is C ohen-M acaulay then v { R / I ) - dim R / I + □ e { R / I ) Recall th a t a C ohen-M acaulay local ring R / I is called a ring with m in im a l m ultiplicity if v ( R / I ) - dim R / I + = e ( R / I ) We will say th a t R / I has h-linear resolution if R / I has the pure resolution of ty pe (h, h + I , , h + p — ) P r o p o s i t i o n 1.6 A s s u m e th ã t the ring R / I is Cohen-AIãCãuIãy ãiid hãs a h-hneãr res­ olution o f ty p e (h, h + 1, • ■• , h + p - 1) ,p = - dim R / I R / I is the ring with m inim al m ultiplicity i f a n d only i f h = or h = Proof Since R is th e Cohen-M acaulay ring, there is ht(7) = Corollary 2.1.4] Because R / I has a /i-linear resolution of type n - dim R / = p by [1, ( h , h + 1,• • • , h + p - 1), D a m Van N h i , L u u B a T h a n g therefore Ij = for all j < h and dim/c I h = *) If h = 1, then e { R / I ) = = and u( R/ 1) = (n + 1) “ dirrifc/i = n — p Hence / ? / / is the ring with minimal multiplicity, because v ( R / I ) — dim ?// + = (n —p) — (n —p) + = e ( R/ I ) / Theorem 1.2, we have c { m = A h h ± ± L ■: + p - 1> > Ĩ ± ± ± > = p + Ị p! = ^ ^ p! ^ — (n —p) + = v ( R / I ) — dim R / I + ^ so, in the case h > , the Cohen-Macaulay local ring R / I is a ring with minimal multiplicity if and only if h = Hence, the Cohen-Macaulay ring i ? / / , which has a hliiear resolution of type ( h ì h + 1, • • • , h + p — l ) , p = n —d i m R / I , is the ring with minimal nultiplicity if and only if h — or h = references D Einsenbud, s Goto, Linear free resolution and minimal multiplicity, J Algebra 88(1984), 89-133 J Herzog, E Kunz, Der kanonische Module eines Cohen-Macaulay-Rings, Lecture Notes in Math., Vol 238, Springer-Verlag 1971 c Huneke, M Miller, A note on the multiplicity of Cohen-Macaulay algebras with pure resolution, Canad J Math., 37(1985), 1149-1162  ... dimension of R / I Abhyankax proved th a t if R / I is C ohen-M acaulay then v { R / I ) - dim R / I + □ e { R / I ) Recall th a t a C ohen-M acaulay local ring R / I is called a ring with m in im a. .. Cohen-Macaulay-Rings, Lecture Notes in Math., Vol 238, Springer-Verlag 1971 c Huneke, M Miller, A note on the multiplicity of Cohen-Macaulay algebras with pure resolution, Canad J Math., 37(1985),... following result was proved by Huneke and Miller using residues theory of complex function, see [3] As an im m ediate consequence of Theorem 1.2 we now want to give an an oth er simple proof E xten