Bounds on Cohomology and Castelnuovo Mumford Regularity

Transcription

1 JOURAL OF ALGEBRA 185, ARTICLE O. 03 Bouds o Cohomology CasteluovoMumfod Regulaty Chkash Myazak agao atoal College of Techology, 716 Tokuma, agao 381, Japa Wolfgag Vogel Depatmet of Mathematcs, Massey Uesty, Palmesto oth, ew Zeal Commucated by D. A. Buchsbaum Receved Jauay 25, ITRODUCTIO Let X PK be a pojectve scheme ove a algebacally closed feld K. We deote by IX the deal sheaf of X. The X s sad to be m-egula f Ž H P,I Ž m.. 0 fo all 1 Žcf. 17. K X. The CasteluovoMumfod egulaty eg X of X P K, fst toduced by Mumfod by geealzg deas of Casteluovo, s the least such tege m. The teest ths cocept stems patly fom the well-kow fact that X s m-egula f oly f fo evey p 0 the mmal geeatos of the pth syzygy module of the defg deal I of X P occu degee m p Žsee, e.g.,. K 2. Thee ae good bouds some cases f X s assumed to be smooth Žsee 1 the efeeces thee.. Ou teest s to cosde the case whee X s locally CoheMacaulay equdmesoal. Ude ths assumpto thee s a o-egatve Ths autho would lke to thak Massey Uvesty fo facal suppot the Depatmet of Mathematcs fo ts fedly atmosphee whle wtg ths pape. E-mal: myazak@e.agao-ct.ac.jp. E-mal: W.Vogel@massey.ac.z $18.00 Copyght 1996 by Academc Pess, Ic. All ghts of epoducto ay fom eseved. 626

2 BOUDS O COHOMOLOGY 627 tege k such that k Ž 0. K X lz X,..., X H P, I l 0 fo 1 dm X, whee PK Poj K X 0,..., X. I ths case X PK s called a k-buchsbaum scheme. A efed veso s a Ž k,. -Buchsbaum scheme toduced 3, 10, 12 : Let k be teges wth k 0 1 dm X. The we call XP a Ž k,. K -Buchsbaum scheme f, fo all j 0,...,1, X V s a k-buchsbaum scheme fo evey Ž j. -dmesoal complete tesecto V P wth dm Ž X V. dmž X. j Žsee Ž 2.3. see also Ž 2.. K fo a equvalet defto case k 1.. I ecet yeas uppe bouds o the CasteluovoMumfod egulaty of such a vaety X PK have bee gve by seveal authos tems of dmž X., degž X., k,, Žsee, e.g., 1012, 19, 20.. These bouds ae stated as follows; degž X. 1 egž X. CŽ k,, d., codm Ž X. whee d dm X, Ck,,d s a costat depedg o k,, d, s the smallest tege l fo a atoal umbe. I case X s athmetcally CoheMacaulay, that s, k 0, t s well kow that Ck,,d1 Žsee, e.g., 22, 23.. We assume k 1. I case 1 t was Ž d 1 show that Ck,1,d. kd1 12. I 11 Ž 2 a slghtly weake fom. 19 t was mpoved to Ck,1,d Ž 2d1. kd1. Futhe, a bette boud Ck,1,d dk was obtaed 20. The geeal case 1 was fst studed 10, whch was mpoved 12 by showg Ž d 2 that Ck,,d 1k. 2 kd1. The pupose of ths pape s to gve bouds o egž X. tems of dmž X., degž X., k,, whch mpove some of the pevous esults. I the geeal case we show that CŽ k,, d. dk Moeove, case k 1, we show that CŽ 1,, d. Ž Ou method hee s to use a spectal sequece theoy fo gaded modules developed 116 ode to get bouds o the local cohomology the CasteluovoMumfod egulaty. d

3 628 MIYAZAKI AD VOGEL Let R be the coodate g of X PK wth the homogeeous maxmal deal. The we defe a Ž R. sup; H Ž R. 0 fo 0,...,d1, egž R. maxa Ž R. ; 0,...,d1. It s easy to see that egž X. egž R. 1. Amog a Ž R. s, a Ž R. d1, whch s called a a-vaat of R Žcf.. 7, plays a mpotat ole to chaacteze the gaded g R. The esults Ž 2.7. Ž 2.9. o bouds o the cohomology tems of a-vaats ae appled to Ž 3.2. Ž I Secto 2, we cosde bouds o the cohomology fo gaded modules. Fst we expla a spectal sequece theoy ode to get cohomologcal bouds. Theoem Ž 2.5. s the key techcal esult of ths pape. Ths theoem gves seveal mpotat esults stated Ž 2.7. Ž I Secto 3, we descbe bouds o the CasteluovoMumfod egula- ty fo locally CoheMacaulay vaetes X P. Theoem Ž 3.2. K Theoem Ž 3.3. ae ma esults of ths secto. I Secto, we costuct shap examples of Ž 2.8. Ž 2.11., coclude by descbg some ope poblems. 2. BOUDS O LOCAL COHOMOLOGY Thoughout ths pape let R KR 1 be a oethea gaded K-algeba, whee K s a fte feld. We deote by the maxmal homogeeous deal of R. Let M be a gaded R-module. We wte M fo the th gaded pece of M MŽ p. fo the gaded module wth MŽ p. M. We set: p supž M. sup; M 0. If M 0, we set sup M. ow we assume that M s a ftely geeated gaded R-module wth dm M d 0. The we set Ž. a Ž M. sup H Ž M., 0,...,d. also a Ž M. s called the a-vaat of M s wtte as am. d The CasteluovoMumfod egulaty of M s defed as follows: eg Ž M. max a Ž M. ; 0,...,d. I ode to state ou esults we use the otato eg Ž M. max a Ž M. ;,...,d, whee 0 d. The we see egž M. eg Ž M. eg Ž M. 0 depthžm.. Let X be a closed subscheme of PK. Let R be the coodate g of X. The the CasteluovoMumfod egulaty of X s defed as follows. eg Ž X. eg Ž R. 1.

4 BOUDS O COHOMOLOGY 629 Fom ow o we assume that the gaded module M s a geealzed CoheMacaulay module. I othe wods M s locally CoheMacaulay; that s, M s CoheMacaulay fo all o-maxmal homogeeous pme deals of R, M s equ-dmesoal, that s, dm Ž R. d fo all mmal assocated pme deals of M. DEFIITIO 2.1. Let k be a o-egatve tege. The gaded R-modk ule M s called a k-buchsbaum module f H Ž M. 0 fo all d. DEFIITIO , 10, 12. Let k be teges wth k 0 1d. M s called a Ž k,. -Buchsbaum module f fo evey s.o.p. x,..., x of M we have 1 d k H MŽ x,..., x. M 0, 1 j fo all o-egatve teges, j wth j 1 j d. Remak. Ž 0, d. -, Ž 1,d. -, Ž k, 1. -Buchsbaum modules ae the CoheMacaulay, Buchsbaum, k-buchsbuam modules, espectvely. The Ž 1,. -Buchsbaum module, toduced as -Buchsbaum module 1, was studed also 15, 16. Remak. Fo a fxed tege, 1d, evey geealzed Cohe Macaulay gaded module M has the Ž k,. -Buchsbaum popety fo k lage eough; see 12, Ž DEFIITIO Let be teges wth 1 d. Let x 1,..., x be a pat of s.o.p. fo M. We say x 1,..., x s -stad, f fo ay choce x,..., x Ž l1. fo j l d. 1 l Ž x,..., x. H j MŽ x,..., x. M l Remak. Stad s.o.p. was toduced seveal papes; see, e.g., 2. The -stadess s ts geealzato. A smla geealzato was toduced 19, Ž.2.. Fst we ote a chaactezato of Ž 1,. -Buchsbaum modules tems of s.o.p. of degee oe whch mpoves, e.g., 1; Ž 2.5. of. PROPOSITIO 2.. Let R KR 1 be a gaded g oe a fte feld K. Let be the homogeeous maxmal deal of R geeated by X 0,..., X. Let M be a geealzed CoheMacaulay gaded R-module wth dmž M. d. We assume that fo all teges 0, X,..., X s a 1 d 1 d

5 630 MIYAZAKI AD VOGEL s.o. p. fo M. Let be a tege wth 1 d. The followg ae equalet: Ž. 1 MsaŽ 1,. -Buchsbuam module. Ž. 2 Fo all teges 0 1 d, X,..., X s a -sta- 1 d dad s.o. p. fo M. Poof. It s clea that Ž. 1 mples Ž. 2. ow we wll show that Ž. 2 mples Ž. 1 by ducto o. It s clea case 1. We assume that 1. To pove that M s a Ž 1,. -Buchsbaum module we have oly to show that, fo ay pat of a s.o.p. y 1,..., y fo M, y 1,..., y s -stad, whch s q equvalet to sayg that Ž M. y y s a zeo map fo all 1 q 1 Ž. d1 by 16, 2.3. Also see the otato 16. We set yj Ý0 aj X fo some aj R. Sce X X s -stad fo all q, we see Ž M. s a zeo map by 16, Ž 2.3. X X. By vtue of 1 q 16, 3.. We have that Ž M. y y s a zeo map. Thus the asseto s 1 poved. The followg theoem s the key techcal esult of ths pape. The spectal sequece theoy as developed 116 plays a mpotat ole to pove t. THEOREM 2.5. Let M be a geealzed CoheMacaulay gaded R-module wth dmž M. d 0. Let be teges wth 0 d 1 1d. Let x,..., x be a pat of s.o. p. fo M wth degž x. 1 j ej 1, j 1,...,. We set c maxe e ;1 j 1 j fo j 1 j 1,...,. The we hae: Ž. 1 If d, the sup H Ž M. Ž x 1,..., x. H Ž M. max a Ž M. c, a MŽ x,..., x. M ; j1,...,1. j j1 1 Ž. 2 If d, the sup H Ž M. Ž x 1,..., x. H Ž M. max a Ž M. c ; j1,...,d. j Futhemoe let be a tege wth 1. Assume that the sequece x 1,..., x s -stad. The we hae: Ž. 3 If d, the a Ž M. max a Ž M. c, a MŽ x,..., x. M ; j,...,1. j1 j j1 1

6 BOUDS O COHOMOLOGY 631 Ž. If d, the a Ž M. max a Ž M. c, až M. c ; j,...,d1. j j1 d1 I ode to pove Ž 2.5. we eed to apply the followg spectal sequece coespodg to the gaded R-module M: Let I be the mmal jectve esoluto of M the categoy of the gaded R-modules. Let K be the Koszul complex KŽŽ x,..., x.;r. 1 fo a pat of a s.o.p. x 1,..., x fo M. The we cosde the double complex Ž. Ž. p, q B Hom R K, I. The fltato Ft B Ýpt B gves a spectal p,q sequece E Žsee, e.g.,. 5. The we have the somophsms u E p, q K p Ž x,..., x.; H q Ž M. 1 1 ½ H pq Ž Ž x 1,..., x. ;M., pq pq H Ž B. H pq Ž MŽ x,..., x. M. Ž e e., p q 1 1 p. fo all p, q; see, e.g., 16. ote that the spectal sequece E q u co- pq veges to H Ž B.. The we have the followg lemma by usg the above otato. LEMMA 2.6. Let be a tege wth 1. The the followg codtos ae equalet: Ž. 1 x,..., x s -stad. 1 Ž. p,q s p, q s ps, qs1 s 2 d :E E s a zeo map fo all p, q, s wth q d, 1s. Poof. The lemma follows by ducto o usg 3.3 of 16. ow let us beg to pove 2.5. p, Poof of 2.5. Let us cosde the spectal sequece E q coe- spodg to the gaded R-module M dscussed above fo a pat of s.o.p. x,..., x fo M. We see that 1 E2, H Ž M. Ž x 1,..., x. H Ž M.Ž c. H MŽ x 1,..., x. M Ž c., d H ½ 0, d.

7 632 MIYAZAKI AD VOGEL Sce E p, q 0 fo p, we see E p, q 0 fo p. Thus we see that 1 the gaded R-homomophsm E, H s jectve that E, s a quotet of the gaded R-module E2,. Fst let us pove Ž. 1. Take a tege l satsfyg l a Ž M. c fo all j 1,...,1 l a ŽMŽ x,..., x. M.. We wat to show 1 H M x 1,..., x H M 0. l Ž., Sce H H M x,..., x M 0, we have E lc 1 l lc 0. O the othe h we have somophsms Ž. j1, j j1 j E1 lc K x 1,..., x ; H M lc j H Ž M.Ž e e. 1 j1 11j1 lc j H Ž M. lž e e.. k1 kj1 1k1kj1 j1, j Thus we have E 1 lc 0 fo all j 1,...,1, because lž e e. l c a Ž M. k k j1 j. Ths leads to the equalty 1 j1,,, E E. Hece we have E 2 lc lc 2 lc 0. Thus the asseto s poved. ext let us pove Ž. 2. Take a tege l satsfyg l a Ž M. j cj1 fo all j 1,...,d. Sce we have j1, j j E1 lc H M l ek ek 1k1kj1 j 1 j1 Ž. j1, j as show the poof of 1, we get E 1 lc 0 fo all j 1,...,,, d. Thus we have E E 2 lc lc 0. Hece the asseto s poved. To pove Ž. 3 we take a tege l satsfyg l a Ž M. j cj1 fo all Ž. j,...,1 l a M x,..., x M. Sce H 1 lc Ž.,, H M x 1,..., x M l0, we have E lc 0. Also we see E2 H Ž M.Ž c., because the sequece x,..., x s 1-stad, Sce 1 j1, j j E1 lc H M l ek ek 1k1kj1 1 j1 Ž. j1, j as show the poof of 1, we have E 1 lc 0 fo all j,...,,, 1. By usg Lemma 2.6, we theefoe have E E 2 lc lc. Hece we have H Ž M. l 0. Thus the asseto s poved. Fally let us pove Ž.. Take a tege l satsfyg l a Ž M. j cj1 fo all j,...,d1 l am c. Smlaly we have d1 j1

8 BOUDS O COHOMOLOGY 633,, d1, E 0, E H M c, E d lc 2 1 lc 0 j1, j E 0 fo all j,...,d. By usg Ž lc we theefoe,, have E E. Hece we have H Ž M. 2 lc lc l 0. Thus the as seto s poved. Ths completes the poof of Ž The fst applcato of 2.5 s the followg theoem whch gves bouds o local cohomology. THEOREM 2.7. Let M be a geealzed CoheMacaulay gaded R-module wth dmž M. d 0. Let k 1,...,kd1 be o-egate teges satsfyg k that j H j Ž M. 0 fo j 1,...,d1. Let be a tege wth 1 d. Let x,..., x be a s.o.p. fo M wth degž x. 1 d j 1 fo j 1,...,d. Assume that the sequece x 1,..., x 1 s -stad fo some poste teges 1,...,. The we hae Ž. Ž. d1 1 a M ad M x 1,..., x M d Ý k Ý 1 fo 1,...,d1. Ž. Ž. d 2 a M a M x,..., x M Ý Ž d. 1 d 1 fo d,...,d1. Ž. Ž. d1 3 eg M ad M x 1,..., x M d Ý k Ý fo 1,...,d1. 1 Poof. Fst we wll pove Ž. 2. By vtue of Ž we have Ž. a M a M x 1,..., x d M 1 d 1 fo d,...,d1, because x,..., x d s Ž d. 1 d -stad. Hece we have poved Ž. 1. ote that a Ž M. sup H j Ž M. H j Ž M. k 1 j j Ž. fo j 1,...,d1. see, e.g., 20, 3.3. By vtue of we see a Ž M. sup H Ž M. Ž x,..., x. H Ž M. k 1 1 d max a M jk ; j1,...,d j fo 1,...,d1. ow we pove the followg clam. Let be a tege wth 1 d 1. The we hae CLAIM. t1 až M. max½ajž M. Ž j. Ý k ; jt,...,d5 fo t 1,...,d.

9 63 MIYAZAKI AD VOGEL We wll show ou clam by ducto o t. The case t 1 has aleady bee show. I case a Ž M. t, othe wods, kt 0. the clam follows mmedately fom the hypothess of ducto. I case a Ž M. t, othe wods, kt 0, we have by the hypothess of duc- to t1 ½ j Ý 5 a M max a M j k ; jt,...,d Thus the clam s poved. I patcula, we have ½ t1 max atsž M. sks Ž t. Ý k, t1 ajž M. Ž j. Ý k ; s1,...,dt j t 1,..., d5 t ½ j Ý 5 max a M j k ; jt1,...,d. d1 ½ j Ý 5 a M max a M j k ; jd,...,d Ž. fo 1,...,d1. By usg 2, we have ½ d1 Ý dj Ý Ž dj.; jd,...,d5 a Ž M. max a MŽ x,..., x. M Ž j. k j 1 dj 1 a MŽ x,..., x. M Ž d. d 1 d1 Ý k d Ý 1 fo 1,...,d1. Hece asseto Ž. 1 s poved. Asseto Ž. 3 follows mmedately fom Ž. 1 Ž. 2. Ths completes the poof of Ž 2.7..

10 BOUDS O COHOMOLOGY 635 Ou Theoem Ž 2.7. has a mpotat coollay whch s used ode to pove Ž COROLLARY 2.8. Let M be a geealzed CoheMacaulay gaded R- module wth dmž M. d 0. Let k be teges wth k 1 1d. Assume that M s a Ž k,. -Buchsbaum module. Fo 1,...,d, eg Ž M. a MŽ x,..., x. M Ž d. Ž d. k d 1 fo ay pat of s.o. p. x,..., x fo M wth deg x 1, j 1,...,. 1 j k k Poof. ote that x,..., x s -stad. By we have 1 d1 Ý Ý eg Ž M. a MŽ x,..., x. M Ž d. k k d 1 1 a MŽ x,..., x. M Ž d. Ž d. k. d 1 Remak. Example Ž.2. shows that the boud stated Ž 2.8. s shap case k 1. Aothe applcato of 2.5 s the followg esult. THEOREM 2.9. Let M be a geealzed CoheMacaulay gaded R-module wth dmž M. d 0. Let be a tege wth 1 d. Let x 1,..., xd be as.o.p. fo M wth degž x. j 1 fo j 1,...,d. Ž. 1 Assume that x,..., x s mž, d. 1 d -stad fo a fxed tege 0,...,d1. The we hae d až M. až MŽ x 1,..., xd. M. 1. Ž. 2 If x,..., x s -stad, the 1 d d eg Ž M. až MŽ x 1,..., xd. M. 1 fo 0,...,d1.

11 636 Poof. By Ž we have MIYAZAKI AD VOGEL a Ž M. max a Ž M. Ž j. 1, a MŽ x,..., x. M ; j 1 d j,...,d1 max a Ž M. Ž j. 2, a MŽ x,..., x. M, j 1 d a Ž MŽ x,..., x. M. 1; j 2,...,d1 1 d max a Ž M. Ž j. 2, a MŽ x,..., x. M 1; j 1 d By epeatg ths step we fally have ½ d až M. max ajž M. Ž j. 1, j2,...,d1. d až MŽ x 1,..., xd. M. 2; d j 1,...,d15 ž / d až MŽ x 1,..., xd. M. 1 fo 0,...,d1. Hece asseto Ž. 1 s poved. Asseto Ž. 2 s a easy cosequece of Ž. 1. COROLLARY Let M be a geealzed CoheMacaulay gaded R- module wth dmž M. d 0. Let be a tege wth 1 d. Assume that M s a Ž 1,. -Buchsbaum module. Fo 0,...,d1, d eg Ž M. adž MŽ x 1,..., x. M. Ž d. 1 fo ay pat of s.o. p. x,..., x wth deg x 1, j 1,...,. 1 j Poof. It follows mmedately fom 2.9. Remak. We have assumed that the degee of a s.o.p. x 1,..., xd s oe Ž 2.7. Ž We took that assumpto fo smplcty, although geealzed esults ae poved smlaly. Futhe, we have two moe esults of Theoem 2.5.

12 BOUDS O COHOMOLOGY 637 COROLLARY Let M be a k-buchsbaum gaded R-module wth dm M d 0. The we hae fo 1,...,d. eg Ž M. až M. dkž d. Poof. It follows mmedately fom the clam the poof of 2.7. Remak. Coollay Ž was fstly obtaed 20, Ž 3... We wll show the pape that the equalty of Ž s shap fo all d k, eve the case M R s a tegal doma; see Ž.1.. PROPOSITIO Let M be a Ž 1,. -Buchsbaum gaded R-module wth dmž M. d 0. The we hae fo 1,...,d. eg Ž M. až M. d d Poof. The poof s smla to that of 2.9 by usg Remak. Let us cosde 2.12 the specal case that M s Buchsbaum, that s, d. The we get fo 1 eg Ž M. až M. d1. Ths esult ca be obtaed also by the stuctue theoem of maxmal Buchsbaum modules. Let us expla ths appoach suggested to us by S. Goto: Fst we take a polyomal g T such that M s a maxmal T-module. By the stuctue theoem 6, we see that M as a gaded T-module s a dect sum of some twstgs of th syzygy modules E.O the othe h we kow that eg Ž E. ae 1 fo 1,..., d1. Hece we have fo all 1. eg Ž M. až M. d1 3. BOUDS O CASTELUOVOMUMFORD REGULARITY Let K be a algebacally closed feld. Let X be a o-degeeate closed subvaety of P wth dmž X. K d. Let R be the coodate g of X wth dmž R. d 1. We assume that X s educble educed, that s R s a tegal doma.

13 638 MIYAZAKI AD VOGEL Befoe statg ou ma esults of ths secto we eed the followg Ž. lemma whch s well kow see, e.g., 18, Coollay 2; 20,.6 b. LEMMA 3.1. Ude the aboe codto, we hae degž X. 1 až R. d1. codmž X. The followg s ou ma esult of ths secto, whch exteds 20, Ž.8. mpoves 10, Ž , Ž THEOREM 3.2. Let X be a o-degeeate closed subaety of PK wth dmž X. doe a algebacally closed feld K. Let R be the coodate g of X. Assume that X s a Ž k,. -Buchsbaum aety fo some tege k wth k 1 1 d. The we hae deg Ž X. 1 egž X. Ž d1depthž R.. k1. codm Ž X. Poof. It follows mmedately fom Remak. Compag wth the equalty of 12, Ž 3.6., we see that ou esult mpoves the esult all cases. We have oly to check that Fst we ca easly show that ž ž / / d 2 dk 1 k d. 2 d 2 d 1. 2 ž / So we have oly to study the case k 1, whch s left to the eades. The ext theoem geealzes kow esults fo quas-buchsbaum vaetes; see, e.g., 11, 19. THEOREM 3.3. Let X be a o-degeeate closed subaety of PK wth dmž X. doe a algebacally closed feld K. Let R be the coodate g of X. Assume that X s a Ž 1,. -Buchsbaum aety fo some tege wth 1 d. The we hae degž X. 1 d 1 depthž R. egž X.. codmž X. Poof. It follows mmedately fom

14 BOUDS O COHOMOLOGY 639. EXAMPLES AD OPE PROBLEMS The pupose of ths secto s to descbe shap examples of Ž Ž Moeove, we coclude wth some ope poblems. Let X P be a pojectve vaety wth dmž X. K d 1 ove a algebacally closed feld K of chaactestc 0. Let R be the coodate g of X. Assume that X s Ž k,. -Buchsbaum fo some teges k wth k 1 1 d. Let L be a -plae P K wth dmž X L. d. Let R be the coodate g of X L. Ude the above codto, Coollay Ž 2.11., Coollay Ž 2.8., Theoem Ž 3.2. ae stated as follows: ote that: Ž egž X. až R. Ž d1. kd 1, Ž 2.8. egž X. až R. Ž d1. kd 1 degž X. 1 Ž 3.2. egž X. kd 1. codmž X. degž X. 1 až R. Ž d1. až R. Ž d1.. codmž X. Hece Ž 3.2. follows fom Ž Moeove, Ž gves the followg boud case 1: egž X. ar Ž. dkd 1. But ths esult s mpoved Ž ow we wll gve shap examples of Ž Ž 2.8. Examples Ž.1. Ž.2., espectvely. These examples ae based o deas of 1, Ž , Ž The fst example shows that the equaltes of Ž ae shap fo all d k. EXAMPLE.1. Let d be a tege wth d 1. Let Yj be the pojectve le PK 1 ove a algebacally closed feld K fo j 1,...,d1. Let Y be the Sege poduct of Y Ž j1,...,d1. j, that s, Y Y1 Y d1. Let p j:y Yj be the pojecto fo j 1,...,d1. We wte a vet- ble sheaf p O 1 p O 1Ž. o Y as O Ž,...,. 1 P 1 d1 P d1 Y 1 d1 d1 though the somophsms Pc Y Pc Y1 Pc Yd1 Z. O the othe h Y s embedded the pojectve space PK, whee d The, fo ay Z, O Ž. O Ž,...,. P Y Y. Thee exsts K a educble smooth effectve dvso X of Y coespodg to the vetble sheaf O Ž,...,. Y 1 d1 fo all postve teges 1,..., d1. ŽSee, e.g., 9, p Let k be a postve tege. Let us take teges 1Ž k1.ž j1. fo j 1,...,d1. The we ca take a oj

15 60 MIYAZAKI AD VOGEL sgula subvaety X of Y such that the deal sheaf IX Y s somophc to OY 1,..., d1. Let R be the coodate g of X P K. The R s a tegal doma wth dm R d 1. I ode to get ar a Ž R., 1,...,d, we use the followg exact sequece the somophsms 0 H d1 Ž R. H d1 Y, I Ž l. H d1 Y, O Ž l. 0 XY Y lz lz Ž XY. H Ž R. H Y, I Ž l., 0, d 1, lz because Y s athmetcally CoheMacaulay Žsee, e.g., 1, Ž Sce we have by Kueth s fomula H d1 Y, I Ž l. H 1 O 1Ž l. H 1 O 1Ž l. XY P 1 P d1 d1 1 d1 H Y, O l H O 1 Y P Ž l., d1 we see that H ŽY, I Ž l.. 0f lm Ž 2. X Y 1 j d1 j 1 d1 that H ŽY, O Ž l.. 0 fo l 2. Thus we have ar Y 1 fom the above exact sequece. ext let us take a fxed tege wth 1 d. Smlaly we have by Kueth s fomula H Y, I l H 0 O 1 l H 0 O 1 XY P Ž 1. P Ž l d1. H 1 O 1Ž l. H 1 O 1Ž l.. P d2 P d1 So we see that H ŽY, I Ž l.. X Y 0f d1ld22, that s, Ž k1.ž d1. klž k1.ž d1. 1. Thus we have a Ž R. Ž k1.ž d1. 1 fo 1 d. Also we have eg Ž R. kd Ž 1. d fo 1 d, so egž R. Ž k 1. d. Futhe we ca easly see that R s a k-buchsbaum g. Thus we have a o-sgula pojectve k-buchsbaum vaety X P wth dmž X. d, ar 1, egž X. kd d 1. Hece ths example yelds the equalty stated Ž The secod example shows that the equaltes of Ž 2.8. ae shap case k 1. Also t s possble to gve examples eve case 1 k 1. I fact we have oly to take some hghe dmesoal pojectve 1 space stead of P Ž.2. K. EXAMPLE.2. I Example Ž.1., we take k 1. The the quas-buchsbaum g R gves the equalty of Coollay Ž 2.8. fo all d case

16 BOUDS O COHOMOLOGY 61 k1. I fact, we ca easly see that arhr 1 fo geec elemets h of R. The we see that 1 eg Ž R. 2 d 1 až RhR. d Ž d d 1 fo 1,...,d. Hece ths example yelds the equalty stated 2.8. Fally we coclude by descbg some ope poblems whch ase atually fom ou vestgato. We take the otato assumpto of the begg of ths secto. Poblem 1. Geealze Ž fo Ž k,. -Buchsbuam vaetes fo all 1, o costuct a example of Ž k, d. -Buchsbaum vaetes wth dmž X. d satsfyg the equalty of Ž fo all k d. We ote that the examples Ž.1. ae ot Ž k, d. -Buchsbuam fo d 2 by usg agumets of 1. Poblem 2. Impove the boud stated 2.8 ode to get shap examples fo all k d at least case 1. REFERECES 1. Betam, L. E, R. Lazasfeld, Vashg theoems, a theoem of Seve, the equatos defg pojectve vaetes, J. Ame. Math. Soc. Ž 1991., D. Esebud S. Goto, Lea fee esolutos mmal multplcty. J. Algeba 88, Ž 198., M. Foet W. Vogel, Old ew esults poblems o Buchsbaum modules, I, Sem. Geom., Uv. Stud Bologa , pp. 5361, Bologa, A. Geamta J. Mgloe, A geealzed laso addto, J. Algeba 163 Ž 199., R. Godemet, Topologe algebque et theoe des fasceaux, Hema, Pas, S. Goto, Maxmal Buchsbuam modules ove egula local gs a stuctue theoem fo geealzed CoheMacaulay modules, Advaced Studes Pue Mathematcs o. 11, Commutatve Algeba Combatocs, pp. 396, Kokuyaoth-Holl, Tokyo, S. Goto K.-I. Wataabe, O gaded gs, I, J. Math. Soc. Japa 30 Ž 1978., P. Gffths J. Has, Resdues zeo-cycles o algebac vaetes. A. of Math. Ž Ž 1978., R. Hatshoe, Algebac Geomety, Gaduate Texts Mathematcs, Vol. 52, Spge-Velag, Bel, L. T. Hoa, R. M. Mo-Rog ` W. Vogel, O umecal vaats of locally CoheMacaulay schemes P, Hoshma Math. J. 2 Ž 199., L. T. Hoa C. Myazak, Bouds o CasteluovoMumfod egulaty fo geealzed CoheMacaulay gaded gs, Math. A. 301 Ž 1995.,

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