On a-invariant Formulas

Transcription

1 Joural of Algebra 227, do: jabr , avalable ole at o O a-ivarat Formulas Mafred Herrma 1 Mathematsches Isttut der Uerstat zu Kol, Weyertal 8690, Cologe, Germay Eero Hyry Natoal Defece College, Satahama, Helsk, Flad E-mal: eero.hyry@helsk.f ad Thomas Korb Mathematsches Isttut der Uerstat zu Kol, Weyertal 8690, Cologe, Germay E-mal: korb@etcologe.de Commucated by Crag Hueke Receved February 2, 1999 Let GI be the form rg of a deal I of postve heght a local rg A. I ths work we wll provde formulas for the a-varat of GI. Our ma result wll oly eed the assumpto that A s CoheMacaulay ad that GI fulflls Serre s codto Ž S. l where l s the aalytc spread of I. As a cosequece of our formula we wll prove upper bouds for the reducto expoet of I the case that A s a regular local rg. If GI fulflls Serre s codto Ž S. l, the ths upper boud s l 1. Ad the case that GI s eve Goreste, t s l Academc Press 1. INTRODUCTION Let A, be a local rg of dmeso d wth A ad let I A be a deal of postve heght. I the past few years, may papers have bee cocered wth the questo how the CoheMacaulay ad 1 Shortly after the fal verso of ths paper had bee fshed, Professor Mafred Herrma passed away o November 15, $35.00 Copyrght 2000 by Academc Press All rghts of reproducto ay form reserved. 25

2 ON a-invariant FORMULAS 255 Goreste property of the blow-up algebras RA I 0 I ad 1 GA I 0 I I the Rees algebra ad the form rg of A wth respect to Iare related? It fally tured out that the so-called a-varat of the form rg as troduced by Goto ad Wataabe GW s the key to the soluto of ths problem Žsee TI ad Ik.. Ths varat ca be defed as the hghest o-vashg homogeeous degree of the dth local cohomology module of the form rg wth respect to ts maxmal homogeeous deal. So t was atural to look for formulas descrbg the a-varat of the form rg terms of other varats of I A. Goto ad Shmoda observed GS that ths a-varat s related to the reducto expoet rž I. of I. Recall from NR that a deal J I s called a mmal reducto of I f JI I 1 for some teger ad J s mmal w.r.t. cluso amog all deals havg ths property. The reducto expoet of I w.r.t. J, whch s deoted by r Ž I. J, s defed to be the mmal teger such that JI I 1 holds. Fally, the reducto expoet of I s the umber rž I. mr Ž I. J J s a mmal reducto of I. For -prmary deals I, Trug showed Tr1 that a rž I. d f G Ž I. A s CoheMacaulay. A terestg approach to fd a geeral a-varat formula stems from Goto ad Huckaba. I GH they gve a a-varat formula for form rgs of deals of aalytc devato oe the case that G Ž I. A s CoheMacaulay ad I s geercally a complete tersecto. Deotg the aalytc spread of I by lž I. dmžg Ž I. G Ž I.., ther formula s a rž I. lž I. f rž I. A A 0 ad a htž I. f rž I. 0. Goto ad Nakamura the showed a smlar formula for deals of aalytc devato two GNa. But attempts to geeralze these results step by step to deals of hgher aalytc devato dd ot work out. I the last years, may a-varat formulas were prove whch hold specal stuatos Žsee e.g. Tr2, Ta, AHT, SUV, GNN.. I the preset work we wll frst prove a geeral a-varat formula for stadard graded algebras. I the case of the form rg t states Žsee Theorem 3.6. a max rž I. lž I., G Ž I. uder the assumpto that A s CoheMacaulay ad that G Ž I. A fulflls Serre s codto Ž S. oly. Here, ŽG Ž I.. Ž lž I. A see Defto 3.3 ad Remark 3.. s a varat whch was troduced the thess of the thrd author Žsee Ko. where the above formula was observed uder the assumpto that G Ž I. A s CoheMacaulay. Combg our formula wth the result of Lpma L wll show that the reducto expoet of a deal I a regular local rg A whose form rg G Ž I. A fulflls Serre s codto Ž S. s bouded above by lž I. 1 Ž see Theorem 5.1. lž I.. Ths exteds results from AHT where deals of aalytc devato oe ad two A

3 256 HERRMANN, HYRY, AND KORB regular local rgs cotag a feld were cosdered. I the case that G Ž I. s Goreste ad htž I. A 1 we ca eve prove that ths upper boud ca be decreased by oe Ž see Theorem Fally, we make use of the above formula to descrbe the a-varat of the form rg terms of reducto expoets ad aalytc spreads of localzatos of I at ftely may prme deals oly Ž see Theorem.5.. If A s quas-umxed ad G Ž I. A s CoheMacaulay for all P P A P AssŽG Ž I.., the the statemet s as follows: A a max r I l I P. I the case that G Ž I. A s umxed t turs out that P s exactly the set of asymptotc prmes of I Ž see Proposto.6.. Ths wll also show that our local a-varat formula exteds a a-varat formula of Ulrch Žsee U1, Thm. 1.Ž.. a ad that the so-called geeralzed GotoShmoda theorem Žsee GS, AHT, Thm. 5.1, JK, Thm follows from t. 2. PRELIMINARIES All rgs ths work wll be Noethera ad all local rgs are assumed to have a fte resdue feld. By a stadard graded algebra S defed over a local rg Ž S,. 0 we wll always mea a postvely graded algebra S S such that S S S We deote the rreleat deal ad the maxmal homogeeous deal of S by S ad, respectvely,.e., S 0 S ad S. Let S be a homogeeous deal of S ad let M be a ftely geerated graded S-module. Deotg the graded local cohomology mod ules of M w.r.t. by H Ž M., the so-called a-arat of M wth respect to s defed as ½ 5 a M sup H M 0. I the case that S t s well kow that a Ž M.. Ad f d deotes the dmeso of S, the as a d Ž S. s the classcal a-varat of S as troduced by Goto ad Wataabe GW. We wll wrte S for the caocal module of S. If t exsts, the t s clear that the frst o-vashg homogeeous compoet of lves degree až S.. S The followg lemma s well kow Žsee e.g. GN, 2.10, p. 85. ad wll be used several tmes ths work.

4 ON a-invariant FORMULAS 257 LEMMA 2.1. Suppose that S has a caocal module. The S Ž. for all Supp Ž.. S S S0 S Remark 2.2. If S s quas-umxed, the the statemet of the above lemma holds for all SpecŽ S.. To see ths, let SpecŽ S. 0 0 ad put P S. Sce ŽŽ.. Ž., we see that Supp Ž. S PS S P S S f ad 0 oly f P Supp Ž.. Now recall from Ao, 1.7 that SuppŽ. SpecŽ S. S S S f S s quas-umxed. LEMMA 2.3. Suppose that S s quas-umxed. The a S až S. for all SpecŽ S.. 0 Proof. Observe frst that we may assume S to be complete. Thus we 0 see from Lemma 2.1 together wth Remark 2.2 that we have for all Ž for all SpecŽ S.. S. S 0 Sce the egatve value of the a-varat descrbes the frst o-vashg homogeeous degree of the caocal module, ths fshes the proof. We wll eed the followg result whch was obtaed by Johsto ad Katz Žsee JK, Prop Let SpecŽ S. 0. We wll use the otato M for the maxmal homogeeous deal of S. PROPOSITION 2.. Let be a teger. Suppose that a Ž S. MŽ. for all Spec S ad all. The a Ž S. 0 for all ad ay homogeeous deal S cotag S. The ext lemma s a combato of Ko, Lem. 5. ad KN, Lem. 3.1 ad geeralzes a result of Marley Žsee Ma ad KN for the proof.. It shows a specal property of the a-varats of the form rg G G Ž I. A of a deal I a local rg A of dmeso d. The lemma wll be used the proof of our ma theorem 3.6 ad shows the dfferece betwee arbtrary stadard graded algebras S ad the form rg G. We deote the maxmal homogeeous deal of G by. LEMMA 2.5. Suppose that oe of the followg two codtos holds: Ž. a htž I. 0 ad depthž A. depthž G.. Ž b. lž I. 0 ad A s CoheMacaulay. g g1 The, f g depth G d, we hae a G a Ž G..

5 258 HERRMANN, HYRY, AND KORB We wll ow recall how the reducto expoet of the rrelevat deal S of a stadard graded algebra S s defed. As the local case, let ls dmž S S. 0 be the aalytc spread of S. A deal Z S geerated by 1-forms such that Z S holds for some postve teger ad whch s mmal w.r.t. cluso amog all such deals havg ths property s called a mmal reducto of S. As the local case oe ca show that mmal reductos always exst ad that they are mmally geerated by ls elemets f S0 has a fte resdue feld. If Z S s a mmal reducto of S, the we call the mmal umber for whch Z S the reducto expoet r Ž S. 1 1 Z of S w.r.t. Z. Let A be a local rg ad let J I A be deals. We wll ow cosder the above otos the mportat case that S s the form rg G Ž I. A of A wth respect to I. Suppose that J s geerated by elemets x 1,..., x A ad put Z Ž z,..., z. 1 where z x deotes the tal form of x G G Ž I. A. The J s a mmal reducto of I f ad oly f Z s a mmal reducto of G. Ad we have r Ž I. r Ž G. ad lž I. lg J Z. Recall that for a ftely geerated graded S-module M the umber reg M max a Ž M. S s called the CasteluooMumford regu- larty of M Ž or sometmes: Casteluoo s dex of regularty of M.. The followg relato betwee the reducto expoet of the rrelevat deal of S ad ts CasteluovoMumford regularty s due to Trug Žsee Tr1, Prop ad was also observed by Schezel Žsee Sch, Prop PROPOSITION 2.6. Let Z be a mmal reducto of S. We put l ls. The the followg relato holds: a l Ž S. l r Ž S. regž S.. S Z 3. AN a-invariant FORMULA FOR STANDARD GRADED ALGEBRAS Throughout ths secto let S be a stadard graded algebra of dmeso d defed over a local rg Ž S,. 0. We deote the maxmal homogeeous deal of S by. LEMMA 3.1. We always hae až S. max a Ž S.. S 0,...,lŽ S. Ad f S s CoheMacaulay, the equalty holds. Remark 3.2. It s easy to see that ls max H Ž S. 0 S Ž. see, e.g. Ko, Rem..27. Therefore all relevat as -varats of S appear the above lemma.

6 ON a-invariant FORMULAS 259 Equalty holds the above lemma uder eve weaker assumptos as we wll see the proof of Theorem 3.6. Proof of Lemma 3.1. From the spectral sequeces ž / pq p q pq E2 H HS S H S p we deduce mmedately that as max a Ž S. S 0,...,lŽS.. To see the other equalty, observe frst that the a-varat does ot crease upo localzato at prmes SpecŽ S. 0 by Lemma 2.3. Now the remag equalty easly follows from Proposto 2. sce S s CoheMacaulay. The ext lemma s Ko, Lem..39. We eed the followg defto for ts formulato ad our a-varat formula. DEFINITION 3.3. We put Ž S. max a Ž S.. S 0,...,lŽ S. 1 Remark 3.. Let Z Ž z,..., z. 1 l be a mmal reducto of S, where l ls. It s well kow that the z ca be chose to form a flter-regular sequece. I Ko, Ž S. s troduced as the mmal teger such that Ž z,..., z.: z Ž z,..., z. 1 1 S 1 1 holds for 1,...,l ad Ž S.. For ths reaso, Ž S. s called the Ž mmal. dex-regularty Ko. It cocdes wth the so-called sldg regularty of Aberbach, Hueke ad Trug Žsee AHT, Def. 2.5 ad Tr2.. That our above defto of Ž S. s equvalet to the oe gve Defto 3.3 s show Ko, Lem..33. Suppose ow that S G Ž I. A where A s a local rg ad I A s a deal of postve heght. We kow already that we ca fd elemets x A such that Ž x,..., x. 1 l s a mmal reducto of I ad that the z are the tal forms of the x G Ž I.. As the proof of HHK, Lem A oe ca see that ŽG Ž I.. A ca be characterzed as the mmal teger satsfyg the codtos Ž. a Ž x. 2 1,..., x I I x 1,..., x I 1 Ž. b x,..., x I : x I Ž x,..., x. I Ž. for 1,...,l ad G I. A

7 260 HERRMANN, HYRY, AND KORB LEMMA 3.5. Let Z be a mmal reducto of S. The max r Ž S. lž S., Ž S. max a Ž S.. Z S 0,...,lŽ S. Proof. We put r r S, l ls, S, ad a a Ž S. Z S. Suppose frst that a. Ths meas that a maxa l l 0,...,l. Therefore we have to show ths case that r l a l. But ths follows drectly from Proposto 2.6 sce we have al l max a 0,...,l by the above. Suppose ow that al. The t s eough to show that r l. Assume the cotrary,.e., assume that r l. The we get r maxa l maxa. 0,...,l1 0,...,l1 Usg Proposto 2.6 aga, we deduce from ths that r a l. But ths l meas by our assumpto that a, whch s a cotradcto. l We are ow prepared to gve the aouced a-varat formula. THEOREM 3.6. Suppose that S0 s quas-umxed ad that S fulflls Serre s codto Ž S. where lž S. lžs.1 0. The we hae the formula for the a-arat of S, až S. max r Ž S. lž S., Ž S., Z where Z s ay mmal reducto of S. I the case that S G Ž I. A where A s a CoheMacaulay local rg ad I A a deal of poste heght the same formula holds uder the Serre codto Ž S. oly,.e., lž I. agž I. max r Ž I. lž I., G Ž I., A J A where J s ay mmal reducto of I. Proof. Observe frst that f S s CoheMacaulay, we oly have to combe Lemma 3.1 ad Lemma 3.5 to see the asserto of the theorem. By Lemma 3.1 we have as max a Ž S. S 0,...,lŽS.. Recallg Lemma 3.5 aga t s therefore eough to show that also až S. max a Ž S. S 0,...,lŽ S. holds. To see ths we wll prove the followg more geeral clam, a S as for all ad all SpecŽ S., ŽS. 0 by ducto o htž.. Note that lžž S.. ls. If ht 0, the lžž S.. dmž S. ad thus S s CoheMacaulay, because of the Serre

8 ON a-invariant FORMULAS 261 codto whch we assume. Now let ht 0. By ducto we have a S a S for all ad all SpecŽ S.. ŽS. 0 Observe that S satsfes Serre s codto Ž S. Žresp. Ž S. lžž S..1 lžž S.. the case of the form rg.. Chagg the otato we may assume that. The we have a S as for all ad all SpecŽ S.. Ž 1. ŽS. 0 Now choose as. From HIO, Thm , Lem we see that the property of beg quas-umxed passes from S0 to S sce S fulflls Serre s codto Ž S. 2. Therefore we may apply Lemma 2.3 to see that as as for all SpecŽ S.. Usg ths we obta from Ž 1. 0 that Ž H Ž S.. 0 for all ad SpecŽ S. S 0. The S0-module H Ž S. s well kow to be ftely geerated Žsee EGA, Prop S. Thus H Ž S. s of fte legth ad therefore the spectral sequece S degeerates showg that ž / pq p q pq E2 H HS S H S p H S S H S for all Z ad all a S. 2 If S fulflls Serre s codto S, the H Ž S. 0 for ls lžs.1. Because of ths the clam follows from Ž. 2. Cosder ow the stuato where S G G Ž I. A ad G fulflls Serre s codto Ž S. oly, where l lž I. lg l. Deote the maxmal homoge- eous deal of G by. We may assume that depthž G. l ad that l dmž G. Ž otherwse the above argumets work.. From Ž 2. we obta that l1 l1 H G 0 for ag. Ths meas that a Ž G. ag. Sce A s CoheMacaulay, we have l depthž G. depthž A. ad therefore t ow follows from Lemma 2.5 that a l Ž G. a l1 Ž G. ag. Ž. Together wth 2 we get from ths that l l H G G H G 0 for all ag. Ad for l the H Ž G. vash by Ž 2. for ag sce depthž G. G l. Ths fshes the proof.

9 262 HERRMANN, HYRY, AND KORB. A LOCAL a-invariant FORMULA FOR THE FORM RING We wll ow see how the a-varat formula of the prevous secto ca be used to prove a a-varat formula for form rgs of a deal I a local rg A whch oly volves reducto expoets ad aalytc spreads of localzatos of I at ftely may prme deals Ž Theorem.5.. Throughout ths secto let Ž A,. be a local rg ad let I A be a deal of postve heght. We put G G Ž I. A ad deote the maxmal homogeeous deal of G by. PROPOSITION.1. We put ½ ŽG. Ž. 5 Ž G. max a G VŽ I.,, lž I.. If depth G l I, the we hae G G. Proof. We wll frst show that Ž G. Ž G.. To see ths, let Ž G. be a teger ad let be a prme VŽ I.. The we get from the defto of Ž G. that ž / HŽG. G H G G 0 for l I. Ths clearly meas that a Ž G. Ž G. for all lž I. ad thus Ž G. ŽG. Ž G.. It remas to show that also Ž G. Ž G.. For ths, choose ow a teger Ž G. ad cosder aga the spectral sequece ž / pq p q pq E2 H HG G H G. p Now, f l I, we get from the defto of G that ž / 0 HŽG. G H G G, where H Ž G. s a ftely geerated AI-module Žsee G EGA, Prop Sce was chose to be a arbtrary prme VŽ I., we thus see that H Ž G. G s of fte legth as a AI-module. Thus the above spectral sequece degeerates showg that H G G H G for l I. 3 Because of our assumpto depthž G. lž I., the module o the rght Ž 3. vashes. Sce we chose G, ths mples that a Ž G. Ž G. G for all lž I.. But ths meas that Ž G. Ž G. whch was to be show.

10 ON a-invariant FORMULAS 263 COROLLARY.2. Suppose that A s CoheMacaulay ad that G fulflls Serre s codto Ž S.. The we hae lž I. ag max rž I. lž I., ag VŽ I.,. Proof. As the proof of Theorem 3.6 we see that a Ž G. ag ŽG. for all ad VŽ I.. By Proposto.1 ths meas that Ž G. maxag VŽ I.,. Puttg ths to the formula of Theorem 3.6 yelds ag max rž I. lž I., ag VŽ I.,. To see that the above equalty also holds the other drecto we use Theorem 3.6 aga to see that ag rž I. lž I.. Ad ag ag Ž. follows from Lemma 2.3 Žsce the property of beg quas-umxed passes from A to G by HIO, Cor DEFINITION.3. R Ž I. ad put A We cosder G as a module over the Rees algebra P PŽ G. P A P AssŽ G.. We wll show Proposto.6 how the elemets of P ca be characterzed. PROPOSITION.. Suppose that A s quas-umxed. The we hae ag max ag P. Proof. Choose a P. Aga, the property of beg quas-umxed passes from A to G Žsee HIO, Cor Thus we ca apply Lemma 2.3 to see that ag ag. To see that ag maxag P t remas to prove that ag ag holds, too. Observe that we ca complete order to show ths. Choose a teger such that až G. for all P. The we have G 0 for all P. Ž. We clam that ths already mples that G 0. Ths would mea by our choce of that až G. ag, as wated. So t remas to prove the clam. Suppose the cotrary ad choose 0 x. The a Ž x. G G P P, where P,...,P AssŽ. AssŽ G.. The a Ž x. 1 1 G G s cotaed oe of the P, say P 1. Put 1 P1 A P. We obta that a Ž x. whch mples A I 1 0 x Ž. Ž 5. G. G

11 26 HERRMANN, HYRY, AND KORB Here we used Lemma 2.1 recallg Remark 2.2. But Ž. 5 s a cotradcto to Ž. ad thus fshes the proof. We are ow prepared to prove the local a-varat formula for the form rg. THEOREM.5. Let A be quas-umxed. Suppose that G s Cohe Macaulay for all P. The Proof. ag max r I l I P. Actually, we wll prove the followg more geeral clam: ag max r I l I P such that for all VŽ I.. The theorem follows from ths by takg. We wll prove the clam by ducto o htž I. ad start wth the case that htž I. 0. The I s a A-prmary deal ad G s Cohe Macaulay by our assumptos sce P as we wll see Proposto.6. Thus, the a-varat formula of Trug Ž see Proposto 2.6. tells us that ag rž I. lž I.. Suppose ow that htž I. 0. The maxrž I. lž I., ag P, VŽ I. ag max ag P Ž by Prop... Ž by Cor..2. max r I l I, r I l I P, P max r I l I P. Ths was to be show. Ž by ducto. We wll ow gve a characterzato of the elemets of P whch shows partcular that P s the set of asymptotc prmes of I the case that A s quas-umxed ad G s umxed Žsee Mc, Prop..1.. PROPOSITION.6. Let SpecŽ A.. If lž I. htž., the P. The coerse holds f A s quas-umxed ad G s umxed. Proof. By passg to the rg A we may assume that, the maxmal deal of A. Note that P s equvalet to gradež, G. 0. The frst clam the easly follows from the equalty of Burch Žsee Bu. accordg to whch lž I. dmž A. gradež, G..

12 ON a-invariant FORMULAS 265 I order to prove the secod asserto, assume that P,.e., P A for some P AssŽ G.. Suppose that we would have lž I. dmž A.. Sce lž I. dmž GG. ad G s quas-umxed, ths mples htž G. 0. As AssŽ G. MŽ G., t follows that G caot be cotaed a assocated prme of G whch s a cotradcto to P. 5. TWO APPLICATIONS TO REDUCTION EXPONENTS We wll ow show how our a-varat formulas ca be used to prove ew upper bouds for the reducto expoet of a deal. For ths, let A be a local rg ad let I A be a deal of postve heght. We put G G Ž I. A. Our frst result exteds Theorem 8. ad Theorem 8.5 AHT where deals of aalytc devato oe ad two regular local rgs cotag a feld were cosdered. THEOREM 5.1. Suppose that A s CoheMacaulay ad that A s regular for all P. If G fulflls Serre s codto Ž S., the r Ž I. lž I. lž I. J 1 for ay mmal reducto J of I. Proof. Choose a prme P ad wrte P I G. Sce GP Ž G. PG ad PG s the maxmal homogeeous deal of G we frst observe that dmž G. dmž G. so that ht htž P. P. Note also that lž I. lž I.. Sce G fulflls Serre s codto Ž S. we have lž I. ml I,ht P depthž G. P mlž I.,ht ht Ž by Prop..6. htž P. dmž G P.. Thus GP s CoheMacaulay ad so also G. Sce A s a regular local rg, we kow from L, Thm..5 that the Rees algebra R Ž I. A s CoheMacaulay, too. Thus ag 0 by TI, Thm Now we ca apply Proposto. to see that ag 0. Ad Theorem 3.6 the tells us that r Ž I. lž I. ag lž I. 1. J I the case that G s Goreste, we ca say eve more. THEOREM 5.2. Let htž I. 2 ad suppose that A 0 s regular for some prme deal mž AI.. If G s Goreste, the r Ž I. lž I. 0 J 2 for ay mmal reducto J of I.

13 266 HERRMANN, HYRY, AND KORB Proof. Put a ag.asg s Goreste we have Ga. G Hece G Ž a. by Lemma 2.1. But ths meas that a ag G. Sce I s A -prmary we kow from Proposto 2.6 that a rž I. dmž A Therefore we get from HHR, Thm. 2.5 that a 2 sce A 0 s regular. Usg ths the a-varat formula of Theorem 3.6 yelds r Ž I. lž I. J a lž I. 2. Remark 5.3. Let A be a regular local rg ad suppose that htž I. 2. We put R R Ž I. A. The proof of Theorem 5.2 shows that the Goresteess of G mples that ag 2. We therefore obta from HRS, Thm. 2.3 ad HRZ, Cor. 2.7 that f G s CoheMacaulay, the the followg two codtos are equvalet: Ž. a G s Goreste. Ž b. R ŽI. A s Goreste for some teger 0. It also follows from HRS that ag 1. Moreover ag htž I. wheever G s CoheMacaulay Žsee e.g. Ko, Rem Therefore we see that for deals I of htž I. 2 regular local rgs the Goresteess of G s equvalet to the Goresteess of R. REFERENCES AHT I. M. Aberbach, C. Hueke, ad N. V. Trug, Reducto umbers, BraçoSkoda theorems ad the depth of Rees rgs, Comp. Math. 97 Ž 1995., 033. Ao Y. Aoyama, Some basc results o caocal modules, J. Math. Kyoto. U Ž 1983., 859. Bu L. Burch, Codmeso ad aalytc spread, Proc. Cambrdge Phlos. Soc. 72 Ž 1972., EGA A. Grothedeck ad J. Deudoe, Elemets de Geometre Algebrque, III, Publ. Math. IHES 11 Ž GH S. Goto ad S. Huckaba, O graded rgs assocated to aalytc devato oe deals, Amer. J. Math. 116 Ž 199., GNa S. Goto ad Y. Nakamura, CoheMacaulay Rees algebras of deals havg aalytc devato two, Tohoku Math. J. 6 Ž 199., GN S. Goto ad K. Nshda, The CoheMacaulay ad Goreste Rees algebras assocated to fltratos, Mem. Amer. Math. Soc. 526 Ž GNN S. Goto, Y. Nakamura, ad K. Nshda, CoheMacaulay graded rgs assocated to deals. Amer. J. Math. 118 Ž 1996., GS S. Goto ad Y. Shmoda, O the Rees algebras of CoheMacaulay local rgs, LN Pure ad Appled Mathematcs, Vol. 68, pp , Dekker, New York, GW S. Goto ad K. Wataabe, O graded rgs, I, J. Math. Soc. Japa 30 Ž 1978., HHK M. Herrma, E. Hyry, ad T. Korb, Flter-regularty ad CoheMacaulay multgraded Rees algebras, Comm. Algebra 2 Ž 1996.,

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