A generalization of Wright's inequality

Pub . Ma . UAB Vol . 30 Ng 1 Maiq 1986 A GENERALIZATIONOF WRIGHT'S INEQUALITY J-L . Ga cíaRoig Le R be a commu a i e ingwi h iden i y, E an R-module and x 1 , . . . ,x ER a mul iplici y sys em on E ( see [1], p .295 ) . Then he leng h R(E/(xn1~ . . .,xn )E) is ini e o any posi i e in ege s n 1 , . . .,n , and W igh 'sinequali y ( see [1] p .296 ) says E /(x n  n R(  1, . . .,x )E)  `  n1 . . .n . R( E / (x 1 > . . . . x- )E), o a bi a yn 1 , . . .,n . This inequali y can be w i en as RH 0 K(x 1 1 , . . .,x ¡E) 5 n 1 . . .n " RH 0 K(x 1 , . . .,x JE), whe e  K(x 11 ,. . . . x ¡E) deno es he Koszul . complex de ined by E and he elemen s x11, . . .,x . In his pape we es ablish ha o a Noe he ian modulesimila inequali ies hold o he highe Koszul homo- logy modules, i .e ., o i>,O, H i K(x 11 , . . .,x JE) .~ n1 . . .n * HiK(xl, . . . . x ¡E) . Mo eo e , he same is ue . o he highe Eule -Poinca écha ac e is ics o he Koszulcomplexes K(x 1 1 , . . . . x ¡E),  i .e .,  o 1i0,  we ha e  xi(x11, . . . . x ¡E) n1 . . .n *xi(x1, . . . . x JE), whe e, by de ini ion, xi ( x 19 . . .,x ¡E) = B .(-1) j-1 LHjK(x1, . . .,x ¡E) . j >,i Ac ually he inequali y o xi is an equali y i i=0 (see [1] p .311 ) . We also p o e ha he unc ions  R H i K(x 1 1 , . . .,x JE) and  X i(x11, . . .,x ¡E)  inc ease wi h he exponen s  n l , . . . . n . In wha ollows R deno es a commu a i e ingwi h iden i y, E is a Noe he ian module o e R and x 1 , . . .,x ER is a sys em o múl iplici y on E . This will ensu e ha all he leng hs which appea a e indeed ini e, hough some o he esúl s would also hold wi hou assuming he leng hs o be ini e . We deno e he leng h by R o R, R . Lemma 1 . Le E be an R-module and x 1 , . . . . x ,y be elemen s o R . Then, o any i>,O, p . IV-2  ) ~ RH i K(x 1 , . . .,x JE)  :  R R H i K(x l y,x 2 9 . . . . x JE) . P oo . The inequali y ollows om he exac sequence (c . [2] H i K(x 2 , . . .,x ¡ E) 0-~  Hi K(x 1,x 29 . . .,x ¡E) --~ x 1 H i K(x 2 " . .,x ¡E) -- . (0 :x1)  - H i-l K(x 29 . . .,x ¡E) and he co esponding one o H i K(x 1 y,x 2 , . . .,x ¡E), by obse - ing ha bo h  (O :x )  c (O :x y) 1 Hi-1K(x23 . . .yx ¡E)  1  H i-1 K(x 2 ," ,x JE) and  x 1 H i K(x 21 . . . y x JE) ;?x l yH i K(x 2 , . . . .x 1E) . Bea ing in mind ha he Koszul homology modules do no depend on he o de o he elemen s de ining i , w .e ge he ollowing 92 P oposi ion 2 . Fo any i30, he mapping om QV o U de ined by (ni , . . .,n )~---!LHiK(x11, . . .,x ¡E) is inc easing , i .e ., n 1 I<m l ,.. .,n .<m imply R,H i K(x 11 , . . .,x ~E) : LHiK(x11, . . .,x JE) . Lemma 3 . I aER, hen we ha e : i) a(O :a n ) < n-L(O :a), and E  E ii)  a( E l anE ) 6n .9,(E/aE) . P oo . By induc ionon n . F om he exac sequence n-1 R/aR .a  R /anR - . R / a n-1 R -o 0 i we apply Hom R ( ,E), we ge i), o Hom R ( R/ anR ,E)=0 :a n , E and i we apply .&E, we ge ii), o ( R / anR )RE'z E /a nE . # P oposi ion 4 . The ollowing inequali y holds o all i10, RH i K(xi,x 2 , . . .,x JE) <n- HiK(x1,x2, . . .,x ¡E) . P oo . F om he exac sequences (see [2j p .IV-2 ) 0 -!H 0 K(ajH i K(x 2 , . . . y x JE))-1-H i K(a,x 2 , . . . y x ¡E) wi h a=x 1 o x1 , we ge - i H 1 K(ajH i-l K(x 2y . . . y x ¡E))-- 0, n  HiK(x2% . . .,x ¡E) g,H i K(x 1 ,x2 , . . .,x JE)=« n )+k(O :xi) x1HiK(x2, . . .,x JE)  H i-1 (x2 ," ,x ¡E) HiK(x2* . . . . x JE) x 1 H i K(x 2 , . . . y x E)  i_l  2  n . H i K(x 1,x 2 , . . . .x /E), he inequali ies being by i ue o lemma 3 . # Again by he independence o he Koszul homology moduleswi h espec o he o de o he elemen s, P oposi ion 4 yields he ollowing heo em which gene alizesW igh 's inequali y . Theo em 5 . Fo any i,>O, andany n1, . . .,n ,O, we ha e 94 2H i K(x 11 , . . .,x ¡E) < n1 . . .n -RHiK(x1, . . .,x ¡E) . # Le >>s conside now he highe Eule -Poinca é cha ac e is ics . Obse e i s ha xo (x 11 , . . .,x ~E) _ n 1 -n -x 0(x 1 , . . .,x IE) and ha x 0 (x1 , . . .,x JE)%0 (c . [1] p .311 ) . Fo he highe cha ac e is ics we ha e P oposi ion 6 . Fo all i,0, he mapping (n1, . . .,n ) I--0 Xi (x11 , . . .,x JE) om IN o IN is inc easing, i .e ., n1<m1, . . .,n <m , imply X i (x11 , .. .,x JE),< Xi(x11, . . .,x ¡E) . P oo . By [2] p .IV-56, we ha e xi(a,x2* . . . . x JE)  =  RH1K(ajHi_lK(x21 . . .,x ¡E))+ + xo(aIxi(x29 . . .,x ¡E)) . Se inga = x 1 o x 1 y and using hemul iplica i i y o£ ( see [1] p .309 Thm .7 ), we deduce xi (x1 ,x 21 . . .,x JE) : xi(xly,x2, . . .9x ¡E) . F om his we ge , o n n, ha xi(x1,x2, . . . . x ¡E)  -< xi(x1,x2, . . .,x ¡E) . P oceeding equally wi h he o he a iables ( x i does no depend on he o de o he elemen s), we ge he esul . # We inish wi h a heo em on highe Eule -Poinca é cha ac e is ics simila o heo em 5 . Theo em 7 . Fo any i30, andany ni, . . .,n > .0, we ha e xi(x 1 1 ,. . .,x ¡E)  :n 1 - n " xi(x1 ... .,x ¡E) . P oo . I is enough o p o e n xi(x1,x2% . . . . x ¡E)  , n " xi(x1,x2, . . . . x ¡E), and his can be done by conside ing he o mulaused in he p oo o he p eceding p oposi ion wi h a=x 1 o x1 . We ge x i (x1 x2, .. .,x /E) _ -_ R(0 :x n )  + x(xn1xi(x2, . . .,x ¡E)) Hi-lK(x2, . . .0x /E) o n R(o :x i)  + n-x(x l lxi(x2 , . . ., x JE)) _ H1-1K(x2, . . .,x ¡E) 0 o = n-xi(x1,x2, . . . . x ¡E), he inequali ybeingjus i iedby lemma 3 . # Re e ences [1] No hco , D .G . Lessons on Rings, Modules and Mul ipli - ci ies , Camb idge U .P .(1968) . [2] Se e, J-P . Algéb e locale : Mul iplici és , Lec u e No es in Ma h . No .11, Sp inge , Be lin(1975) . Rebu el 14 de no embne del 1985 C uzRoja, 26 3s 4a . Hospi ale Ba celona ESPANYA 9 6