Necessary and Sufficient Conditions for the Cohen Macaulayness of Form Rings

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1 Joural of Algebra 212, Artcle ID jabr , avalable ole at o Necessary ad Suffcet Codtos for the CoheMacaulayess of Form Rgs Eero Hyry Natoal Defese College, Satahama, FIN-00860, Helsk, Flad E-mal: eero.hyry@helsk.f Commucated by Crag Hueke Receved December 18, 1997 Let A be a CoheMacaulay local rg wth a fte resdue feld ad let I A be a deal of heght h 0. Supposg that depth AI s hgh eough for certa values of, we gve ths paper ecessary ad suffcet codtos for the form rg gr Ž I. A to be CoheMacaulay wth a-varat h or h Academc Press INTRODUCTION Let A be a CoheMacaulay local rg of dmeso d wth a fte resdue feld ad let I A be a deal of postve heght h ad aalytc 1 spread l. Let G deote the form rg gra I I I.Itsa mportat questo commutatve algebra to fd out whe G s CoheMacaulay. I recet years may authors have studed ths questo the case I s geercally a complete tersecto or has geercally reducto umber oe. Recall here that a deal J I s sad to be a reducto of I f I 1 JI for 0 ad that the reducto umber r Ž I. J s the smallest teger wth ths property. Startg from the assumpto that depth AI s hgh eough for certa values of, they have foud suffcet codto for the CoheMacaulayess of G terms of local reducto umbers of I. For the latest developmet of ths research, tated by Huckaba ad Hueke 10, 11, see, e.g., 12, 1, 2, 6. The purpose of ths paper s to provde ecessary ad suffcet codtos for the CoheMacaulayess of G the above settg. At the same tme we wat to pot out how the depth of G depeds o vashg of homogeeous compoets of the graded local cohomology wth respect to $30.00 Copyrght 1999 by Academc Press All rghts of reproducto ay form reserved.

2 18 EERO HYRY the rrelevat deal. Whe the deal I s equmultple,.e., l h, ecessary ad suffcet codtos for the CoheMacaulayess of G has bee gve by Valabrega ad Valla 18. Ths well-kow result says that f Ž a,...,a. 1 l s a mmal reducto of I, the G s CoheMacaulay f ad 2 oly f a 1,...,al s a regular sequece ad a 1,...,al I Ž a. 1,...,al I 1 for all 0. We ow descrbe the results of ths paper more precsely. By the theorem of Serre we kow that the local cohomology wth respect to the rrelevat deal vashes all hgh degrees. Supposg that ths vashg happes degrees where 0, we show our Theorem 2.3 that f depth AI d q for some q h 1 ad all 1,...,d q 1, the depth G mž d q, d.. The proof of ths result reles o Lemma 2.1, whch deals wth vashg propertes of the sheaf cohomology wth supports. By utlzg the theory of flter-regular sequeces, 4 we ca the show Corollary 2.5 that f q h 1, h ad depth AI d q for 1,...,l q, the G s CoheMacaulay wth a-varat h or h 1 f ad oly f some Ž ad the also every. mmal reducto J I satsfes the codtos: Ž. 1 There exsts a geeratg set a,...,a 4 1 l of J such that Ž q. q a a,...,a I : a I Ž a,...,a. I q1 Ž ,...,l. Ž b. Ž a,...,a. I q I q2 Ž a,...,a. I q1 Ž 1 1 q,..., l 1;. Ž. 2 r Ž I. J l q. Fally, we expla Remark 2.8 why the suffcet codtos gve 12, 1, 2, 6 mply the codtos of Corollary PRELIMINARIES I the followg we assume that all rgs are Noethera. Let S be a stadard graded rg defed over a local rg Ž A,.. By the word stadard we mea here that S S s ftely geerated over S0 A by elemets of S 1. The rrelevat deal of S s deoted by S as usual. Let be the homogeeous maxmal deal of S. Let X Proj S ad E X A. Let M be a graded S-module ad let M A deote the assocated sheaf of OX-modules. We eed to compare the graded local cohomology wth the sheaf cohomology wth supports E. Recall from 15 that ths ca be doe by meas of the Sacho de Salas sequece H Ž M. H Ž M. H X, M. E

3 COHENMACAULAYNESS OF FORM RINGS 19 Note that f M 0 for some, the H Ž M. H ŽX, M. for all 0. Let S be a homogeeous deal. Let 0. The th a-arat of S wth respect to s 1 E ½ 5 a, S sup H S 0. Of course we thk here maly the case S or. I the latter case we call a Ž, S. the a-arat of S ad deote t by as dm S as usual. I comparg the a-varats wth respect to S ad we eed the followg lemma, whch s closely related to 15, Theorem LEMMA. Let S be a stadard graded rg defed oer a local rg Ž A,.. Let be the homogeeous maxmal deal of S. Let X Proj S ad E X A A. Let M be a graded S-module. Let. The the followg codtos are equalet: Ž. 1 H Ž M. 0 for all 0; Ž. 2 The caocal homomorphsm H Ž M. H ŽX, M. E s a somorphsm for all 0; Ž. 3 The caocal homomorphsm M ŽX, M. s a somor- phsm ad H ŽX, M. 0 for 0; Ž. 4 H Ž M. 0 for all 0. Proof. sequece S The equvalece of Ž. 1 ad Ž. 2 follows from the Sacho de Salas H Ž M. H Ž M. H X, M E whereas the equvalece of Ž. 3 ad Ž. 4 s mmedate from the exact sequece H M M X, M H S S M 0 1 Ž. S ad the somorphsms H M H X, M 1. The proof ca ow be completed by usg the followg lemma whch s a modfcato of 15, Lemma 4.2 ad ca be proved the same way LEMMA. Wth the otato of Lemma 1.1 the followg codtos are equalet for all. Ž. 1 The caocal morphsm R H Ž M. R ŽX, M. s a somorphsm; Ž. 2 The caocal morphsm M RŽX, M. s a somorphsm. 0 E

4 20 EERO HYRY! Moreoer, f A has a ormalzed dualzg complex R, ad R f R X where f: X Spec A s the caocal projecto s the duced dualzg complex o X, the both codtos are equalet to Ž. 3 The caocal morphsm s a somorphsm. Ž. R Hom MŽ., R R Hom R X, MŽ., R X X A R Hom A Ž M, R. It s coveet to troduce the followg varat Žcf. 13. : 1.3. DEFINITION. Let S be a stadard graded rg defed over a local rg Ž A,.. Set 4 S max a S, S 0,...,l S. Here ls dm SS deotes the aalytc spread of S. Note that we always have H Ž S. 0 for ls Žuse, for example, S 7, Corollare Ž PROPOSITION. Let S be a stadard graded rg defed oer a local rg Ž A,.. The až S. Ž S.. If S s CoheMacaulay, the až S. Ž S.. Proof. The equalty as Ž S. mmedately follows from the spectral sequece p, q p q pq E H H 2 S S H S, where s the homogeeous maxmal deal of S. IfS s CoheMacaulay, the H Ž S. 0 for all 0 ad as. Accordg to Lemma 1.1 ths meas that also H Ž S. 0 for all 0 ad as. S Ths mples that a ŽS, S. as for all 0 as eeded Remark. Let S be a stadard graded rg defed over a local rg A,.IfS s CoheMacaulay, the as ht S. Whe I A s a deal of postve heght such that gr Ž I. A s CoheMacaulay, ths gves partcular that agr Ž Ž I.. A ht I. To see ths, ote frst that by movg to the fathfully flat exteso At Atwhere t s a varable, we ca assume that the resdue feld of A s fte. Set h ht S ad choose a regular sequece Ž s,...,s. 1 h S1 Žsee 4, Proposto Set T SŽ s,...,s..as at as 1 h h Žsee 19, Remark , t s eough to show that at 0. Cosder the exact sequece 0 K T A 0.

5 COHENMACAULAYNESS OF FORM RINGS 21 Sce dm T dm A, the correspodg log exact sequece of cohomoldm T dm ogy gves us a epmorphsm H T H A Ž A. where s the dm homogeeous maxmal deal of S. As H A Ž A. 0, we see that also dm H T Ž T. 0, whch mples that at 0 as wated. 0 We ext recall the oto of flter-regularty. Let z 1,..., zr S be homogeeous elemets. Let t,...,t, 4. Followg 1 r 3 we say that the sequece Ž z,..., z. s t,...,t -regular f 1 r 1 r Ž z,..., z.: z Ž z,..., z for t Ž 1,...,r.. The sequece Ž z,..., z. 1 r s called flter-regular f t s t,...,t -regular for some t,...,t. We deote 1 r 1 r Ž z,..., z. f m Ž z,..., z. : z 1 r 1 1 Ž z 1,..., z1. for m 4 for 1,...,r ad call,..., the flter-regularty of Ž z,..., z. 1 r 1 r. Flter-regularty s closely coected to the a-varats a ŽS, S LEMMA. Let S be a stadard graded rg defed oer a local rg A, whch has a fte resdue feld. Set l ls. Let Z S be a mmal reducto geerated by elemets of degree oe. Choose a flter-regular sequece Ž z,..., z. 1 l cosstg of degree oe elemets whch geerate Z Žsee, e. g., 17, Lemma The Ž S. t f ad oly f Ž z,..., z. 1 l s t, t 1,...,tl 1-regular ad r ŽS. t l. Proof. We kow by 17, Proposto 2.2 that max Ž z,..., z. j 1,..., Z ½ j 1 l 5 max½ajž S, S. j 1 j 0,...,15 for all 1,...,l. It follows by ducto o that the codtos a ŽS, S. t Ž j 0,...,1. ad Ž z,..., z. t j 1 Ž j 1,...,. j j 1 l are equvalet for all 1,...,l. So ½ 5 max aj S, S j 0,...,l 1 t f ad oly f Ž z,..., z. s t, t 1,...,tl 1 1 l -regular. If ths s the case, the by 17, Proposto 3.2 ½ 5 al S, S l rz S max aj S, S j j 0,...,l 4 max t l 1, a Ž S, S. l l

6 22 EERO HYRY from whch we see that a ŽS, S. t f ad oly f r ŽS. l Z t l. The lemma has therefore bee prove. The followg lemma ca be proved the same way as 9, Lemma LEMMA. Let Ž A, m. be a local rg ad let I A be a deal. Let 2 a,...,a be a mmal reducto of I ad set a a I gr Ž I. 1 l A 1 Ž 1,...,l.. Let t t. The Ža,...,a. s t,...,t 1 l 1 l 1 l -regular f ad oly f the followg codtos are satsfed for 1,..., l ad t : Ž. 1 a 1,...,a1 I : a I a 1,...,a1 I 1 Ž. 2 Ž a,...,a. I I 2 Ž a,...,a. I THE MAIN RESULT We beg wth two lemmas LEMMA. Let S be a stadard graded rg defed oer a local rg Ž A,. whch has a fte resdue feld. Set X Proj S ad E X A A. Let F be a coheret OX-module. Suppose that there exsts m ad p such that H ŽX, F Ž m.. 0 for all p. The H ŽX, F Ž E E.. 0 for all p ad m. Proof. We use ducto o dm Supp F. I the case dm Supp F 1 e., Supp F. the clam s trval. We ca thus assume that F 0. Wrte F M where M s a ftely geerated graded S-module. By prme avodace we ca fd a elemet s S1 such that s P whe P Ass M ad S P. Set Y V Ž s. ad G j*f where j: Y X s the cluso. The dm Supp G dm Supp F 1 ad there s for all a exact sequece 0 FŽ 1. F j G 0. From the correspodg cohomology sequece we obta the exact sequece HE Ž X, FŽ 1.. HE Ž X, F. HE Ž Y, G. H 1 Ž X, FŽ 1... H 1 Y, G E E By takg m, ths sequece frst mples that H ŽY, GŽ m.. E 0 for all p 1. By the ducto hypothess we the have H Ž E Y, GŽ.. 0 for all p 1 ad m. Fally, usg ducto o ad lookg the sequece aga we see that H ŽX, FŽ.. E 0 for all p ad m.

7 COHENMACAULAYNESS OF FORM RINGS 23 The ext lemma s 14, Lemma LEMMA. Let A be a local rg ad let I A be a deal of poste heght. Let be the homogeeous maxmal deal of gr Ž I. A. Set g depth gr Ž I.. If g depth A, the a Ž, gr Ž I.. a Ž, gr Ž I.. A g A g1 A. We ca ow prove the followg theorem whch relates the varat Žgr Ž I.. of Defto 1.3 to the depth of gr Ž I. A A THEOREM. Let Ž A,. be a CoheMacaulay local rg of dmeso d ad let I A be a deal of poste heght. Set h ht I, l lž I., ad Ž. gra I. Let q h 1. Suppose that depth AI d q for 1,...,dq1. If 0, the depth gr Ž I. mž d q, d. A. Proof. By movg to the fathfully flat exteso A t At where t s a varable, t s possble to assume that the resdue feld of A s fte. Set G gr Ž I. A ad let be the homogeeous maxmal deal of G. Also set X Proj G ad E X A A. Let us frst show that depth G d q 1. As a ŽG, G. for all 0, we kow that H Ž G. G 0 for all 0 ad 1. Lemma 1.1 the mples that also H Ž G. 0 for all 0 ad 1. We therefore eed to show that H Ž G. 0 for 0 d q 1 ad. As 0, ths s the same as H ŽX, O. E X 0 for 0 d q 2 ad. By Lemma 2.1 t s eough to prove that H ŽX, O Ž d q 2.. E X 0 for all d q 2. By Lemma 1.1 we also have H ŽX, O. H Ž G. E X for all 0 ad 1. Ths mmedately mples that H ŽX, O Ž d q 2.. E X 0 for d q 2 d q 2. Suppose d q 2. By lookg at the log exact sequece of cohomology correspodg to the exact sequece 0 I I 1 AI 1 AI 0, we see that the assumpto mples that depth I I 1 d q 1 for 0,...,dq2. Hece depth Gdq2 1, ad we obta H Ž G. dq2 0 as eeded. Suppose the that we would have depth G d q 1 d. Accordg to Lemma 2.2 ths mples that a Ž, G. a Ž, G. dq1 dq so that Ž. dq2 dq1 HE X, OX H G 0. Hece H ŽX, O Ž d q 2.. E X 0 for all d q 2. But t the follows from Lemma 2.1 that Ž. dq1 dq2 H G HE X, OX 0 for all, whch s a cotradcto.

8 24 EERO HYRY We are ow gog to gve ecessary ad suffcet codtos for the CoheMacaulayess of gr Ž I. A. To ths purpose we eed the followg techcal lemma LEMMA. Let A be a CoheMacaulay local rg of dmeso d ad let I A be a deal. Let J Ž a,...,a. 1 l be a mmal reducto of I wth reducto umber r l. Set J Ž a,...,a. 1 for 0,...,l. Let q be a Ž. 1 teger wth 0 q l r. Suppose that J1 I : a I J1 I for 1,...,l ad q. If depth AI d q for 1,...,l q, the depth AI d q 1 for all l q 1,...,dq. Proof. As I JI l 1 for l q 1, we ca prove the clam by showg that depth AJ I d q for all 1,...,l ad q. Assume the cotrary. Take the smallest gvg a couterexample. For ths, choose to be mmal. Cosder the exact sequeces 0 I J1 I 1 AJ1 I 1 AI 0 ad 0 JI J1 I AJ1 I AJ I 0. Ž. 1 The assumpto J I : a I J I for q mples that 1 1 a the epmorphsm I JI J I duces a somorphsm I J I JI J1 I. Whe 1, the above sequeces mply that depth AJ1 I depth AI. But for l q, we have depth AI d q by assumpto. Moreover, the case l q 1 our choce of would gve depth AI depth AJl I 1 d q 1. Ths meas that ecessarly 1. By the choce of ad we ow have depth AJ I 1 d q 1 ad depth AJ1 I d q. As also depth AI depth AJ1 I d q, t thus follows from the above exact se- queces that depth AJ I d q, whch s a cotradcto. The clam has thus bee prove COROLLARY. Let Ž A,. be a CoheMacaulay local rg of dmeso d wth a fte resdue feld ad let I A be a deal of poste heght. 4 Set h ht I ad l l I. Let q h 1, h. Suppose that depth AI d q for 1,...,l q. The gr Ž I. A s CoheMacaulay wth agr Ž Ž I.. q f ad oly f some Ž ad the also eery. A mmal reducto J I satsfes the codtos Ž. 1 There exsts a geeratg set a,...,a 4 1 l of J such that Ž q. q a a,...,a I : a I Ž a,...,a. I q1 Ž ,...,l. Ž b. Ž a,...,a. I q I q2 Ž a,...,a. I q1 Ž 1 1 q,..., l 1;. Ž. 2 r Ž I. J l q. 1

9 Proof. COHENMACAULAYNESS OF FORM RINGS 25 Let us frst show that f Ž. 2 holds, the Ž. 1 s equvalet to Ž 1. There exsts a geeratg set a,...,a 4 1 l of J such that Ž. a a 1,...,a1 I : a I a 1,...,a1 I 1 Ž b. Ž a. 2 1,...,a I I a 1,...,a I 1 for all 1,...,l ad q. It s eough to prove that Ž. 1 mples Ž 1.. We use ducto o. Whe q, everythg s clear by assumpto. Note that Ž b. holds ths case trvally for l because of Ž. 2. Suppose thus that q. Cosder frst codto Ž a.. The ducto hypothess ow says that ad Therefore Ž a,...,a. I 1 : a I 1 Ž a,...,a. I Ž a 1,...,a1. I 2 I Ž a 1,...,a1. I 1. Ž 1 1. Ž 1 1. Ž a,...,a. I : a I Ž a,...,a. I 1 : a I Ž a 1,...,a1. I 1 mplyg Ž a.. I order to prove codto Ž b. take x Ž a. 1,...,a I I 2.As Ž a 1,...,a. I I 2 Ž a 1,...,a1. I I 2 Ž a 1,...,a1. I 1 by the ducto hypothess, we ca wrte x 1a1 a 1a1 where 1,...,, 1 I 1. But by usg the ducto hypothess aga, we get Ž a,...,a. I : a I 1 Ž a,...,a. I 1 I Ž a 1,...,a. I. Hece x Ž a,...,a. I 1, whch proves Ž b Set G gra I, ad a* a I G1 for a I as usual. It s well kow that f J I s ay mmal reducto, the t s always possble to fd geerators a,...,a J such that the sequece Ža,...,a. 1 l 1 l s flterregular Žsee 17, Lemma By Lemma 1.7 codto Ž 1. meas that Ža,...,a. s q 1,...,q l 1 l -regular. Accordg to Lemma 1.6 codtos Ž. 1 ad Ž. 2 are the equvalet to Ž G. q. I the case G s CoheMacaulay we kow by Proposto 1.4 that ag Ž G.. It therefore remas to show that Ž G. q mples the CoheMacaulayess of G. Note that by Lemma 2.4 we obta depth AI d q for all 1,...,dq1. Thus we may apply Theorem 2.3 to get the clam.

10 26 EERO HYRY 2.6. Remark. Foret has defed 5 the oto of a relatve regular sequece: Let A be a rg, M a A-module, ad N M a submodule. If a,...,a A, Ž a,...,a. 1 r 1 r s sad to be a relatve M-regular sequece wth respect to N f x N ad axž a,...,a. 1 1 N mples x Ž a,...,a. M for 1,...,r. We thus see that codto Ž 1.Ž a the proof of Corollary 2.5 meas that a 1,...,a s a relatve I -regular sequece wth respect to I for 1,...,l ad q Remark. I the stuato of Corollary 2.5 we always have rž I. 1 for all M AI such that ht h. Ideed, by usg the fact that the a-varat does ot crease by localzato, we get agr Ž Ž I.. A h 1 for all M AI. But the by 17, Proposto 3.2, rž I. agr Ž Ž I.. h 1. A 2.8. Remark. Goto et al. have gve 6, Theorem 1.1 suffcet codtos for the CoheMacaulay property of gr Ž I. A. Ths result covers all the earler results metoed the troducto. I ths remark we wat to expla why these codtos mply those of Corollary 2.5. As before, let Ž A,. be a CoheMacaulay local rg of dmeso d wth a fte resdue feld ad let I A be a deal of postve heght. Set h ht I ad l lž I.. Let J I be a mmal reducto. Gve a geeratg set a,...,a 4 of J, set J Ž a,...,a. 1 l 1. It s possble to choose a,...,a such a way that the followg codtos hold 6, Lemma 2.1 : 1 l Set Ž. 1 J s a reducto of I for ay VŽ I. ad ht l; Ž. 2 a f Ass AJ VŽ I. for ay 1 l. 1 ½ 5 r max r I VŽ I.,ht Ž 1,...,l.. J Let 0 r l h 1. The result of Goto et al. ow says that f the codtos Ž. 1 depth AI d l r for 1 r; Ž. 2 r max0, l r4 for all h l; Ž. 3 AJ : I s CoheMacaulay for h l r 1; Ž. 4 r Ž I. r J are satsfed, the gr Ž I. A s CoheMacaulay. It s show 6, Lemma 3.1 ad Corollary 3.3 that codtos Ž. 1 Ž mply J I JI ad J 1 : a I J1 I for 1,...,l ad l r. Therefore also the codtos of Corollary 2.5 hold for q l r.

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