JOURNL OF LGEBR 189, 390405 1997 RTICLE NO. J966885 Connected Graded Gorensten lgebras wth Enough Normal Elements James J. Zhang* Department of Mathematcs, Unersty of Washngton, Seattle, Washngton 98195 Communcated by J. T. Stafford Receved November 1, 1996 We generalze 12, 1.1 and 1.2 to the followng stuaton. Theorem 1. Let be a connected graded noetheran algebra of njecte dmenson d such that eery nonsmple graded prme factor rng of contans a homogeneous normal element of poste degree. Then: Ž 1. s uslandergorensten and CohenMacaulay. Ž 2. has a quas-frobenus quotent rng. Ž 3. Eery mnmal prme deal P s graded and GKdm P d. Ž 4. If, moreoer, has fnte global dmenson, then s a doman and a maxmal order n ts quotent dson rng. To prove the above we need the followng result, whch s a generalzaton of 3, 2.46Ž.. Theorem 2. Let be a connected graded noetheran S-Gorensten algebra of njecte dmenson d. Then: Ž 1. The last term of the mnmal njecte resoluton of s somorphc to a shft of *. Ž 2. For eery noetheran graded -module M, Ext d Ž M,. s fnte dmensonal oer k. 1997 cademc Press 0. INTRODUCTION Let k be a feld. k-algebra s called connected f 0, j j, and 0 k. In ths paper we wll only consder connected left and rght noetheran algebras and graded modules except for the proof * Research supported by an NSF Postdoctoral Fellowshp. E-mal: zhang@math.washngton.edu. 0021-869397 $25.00 Copyrght 1997 by cademc Press ll rghts of reproducton n any form reserved. 390
GRDED GORENSTEIN RINGS 391 of Theorem 3.2. If M M s a noetheran Ž left andor rght. -module, we smply say M s fnte. For every nteger n, M n denotes the submodule n M. Let be the unque maxmal graded deal 1 of. The tral -bmodule s denoted by k because t s somorphc to k as a vector space. Gven a graded module M, the degree shft sm s defned by sm M and s l Ž M. s denoted by Ml n n1 for all l. graded module M M s called left bounded Ž respectvely rght bounded. f M 0 for all 0 Ž respectvely 0.. We say M s locally fnte f dm M for all, where dm s the dmenson of a vector space. Every fnte graded -module s left bounded and locally fnte. If M and N are two left Ž or rght. -modules, we use Hom d Ž M, N. to denote the set of all -module homomorphsms h: M N such that hm N. We set HomŽ M, N. Hom d Ž M, N. d d and denote the correspondng derved functors by Ext Ž M, N.. Gven any -module M, the j-number of M s defned by 4 j M mn Ext M, 0 4. In partcular, f M 0, then jm.if s a noetheran rng wth fnte left and rght njectve dmenson and M s a nonzero left or rght -module, then jm. By 16, Lemma, f has fnte left and rght njectve dmenson, then the left njectve dmenson s equal to the rght njectve dmenson. We wll wrte ths common nteger as njdm. Note that f M s a fnte graded rght Ž respectvely left. -module, then Ext Ž M,. s a fnte graded left Ž respectvely rght. -module for each. n algebra s called uslandergorensten f has fnte left and rght njectve dmenson and, for every fnte graded -module M and for every graded -submodule N Ext Ž M,., jn ;s called CohenMacaulay f, for every fnte graded -module M, jm GKdm M GKdm, where GKdm s the GelfandKrllov dmenson. connected algebra s called S-Gorensten ŽrtnSchelter Gorensten. f has fnte njectve dmenson d and d Ext k, 0 for d and Ext k, k e for some e, Ž E1. where k s vewed as a ether left or rght -module. If s uslandergorensten, then s S-Gorensten 7, 6.3. By nducton and Ž E1., f F s a fnte dmensonal left or rght -module, then Ext d Ž F,. F* Ž e. as graded k-vector spaces, where F* s the graded k-lnear dual Hom Ž F, k. n k n. For more nformaton about the above defntons and related results see 3, 7, and 12.
392 JMES J. ZHNG Stafford and the author proved the followng result for PI Žpolynomal dentty. rngs 12, 1.1 and 1.2. s usual clkdm denotes the classcal Krull dmenson and Kdm denotes the Krull Ž RentschlerGabrel. dmenson. THEOREM 0.1 12. Let be a connected noetheran PI algebra of njecte dmenson d. Then: Ž. 1 s uslandergorensten and CohenMacaulay. Ž. 2 GKdm Kdm clkdm njdm. Ž. 3 If, moreoer, has fnte global dmenson, then s a doman and a maxmal order n ts quotent dson rng. Recent studes on quantum groups and deformatons of commutatve algebras suggest that we should generalze the above result to non-pi quantzed algebras. s we saw from 5 and other papers on quantum groups, many quantzed algebras are not PI, but satsfy the property defned next. If, for every nonsmple graded prme factor rng P, there s a nonzero homogeneous normal element n Ž P. 1, then we say has enough normal elements. If has a sequence of normal elements x,..., x 4 namely, the mage of x n Ž x,..., x. 1 n 1 1 1 s normal for all such that Ž x,..., x. 1 n s fnte dmensonal, then has enough normal elements. The prme spectrum Spec s called normally separated f, for any par of prmes P Q, QP contans a nonzero normal element of P 5, 1.5. The prme spectrum of a PI rng s normally separated 8, 13.6.4. Many quantzed algebras are non-pi, but ther spectra are normally separated 5. By defnton, f Spec s normally separated, then has enough normal elements. If Ext Ž k, M. s fnte dmensonal over k for all 0 and for all fnte graded rght -modules M, we say satsfes 4, Defnton 3.7. The condton s equvalent to the S-Gorensten condton when has fnte njectve dmenson Žsee 15, 4.3 and Theorem 0.3.. By 4, 8.12Ž. 2, connected algebras wth enough normal elements satsfy the condton. If, moreover, has fnte global dmenson, then s a doman 12, p. 1024. The man result of ths paper s the followng. THEOREM 0.2. Let be a connected noetheran algebra of njecte dmenson d. Suppose that has enough normal elements. Then: Ž. 1 s uslandergorensten and CohenMacaulay. Ž. 2 has a quas-frobenus ungraded quotent rng. Ž. 3 For eery fnte graded left or rght -module M, GKdm M Kdm M ; for eery two-sded graded deal I, GKdm I clkdm I.
GRDED GORENSTEIN RINGS 393 Ž. 4 Eery mnmal prme deal P s graded and GKdm P d. Ž. 5 If, moreoer, has fnte global dmenson, then s a doman and a maxmal order n ts quotent dson rng. The key step s to prove Theorem 0.2Ž. 1 and our basc dea s to modfy the proof of 12, 3.10. The dffculty here s to show that f s an S-Gorensten algebra, then Ext Ž M,. Ext Ž M,. Ž E2. as graded k-vector spaces for all fnte P-modules M and for graded algebra automorphsms utž P., where P s an deal of. Note that f s a graded algebra automorphsm of, then Ž E2. holds for any graded algebra Lemma 2.1Ž. 1. Gven a rght -module M and x M, x s called -torson f x n 0 for some n. The set of torson elements of M forms a submodule, whch s denoted by M. graded module M s called -torson Ž respectvely -torson-free. f M M Žrespectvely M 0.If. M s fnte, then M s the largest fnte dmensonal submodule of M. By usng the recent results n 4, 14, and 15 we can show the followng. THEOREM 0.3. Let be a connected noetheran algebra of njecte dmenson d. Suppose that Ext Ž k,. and Ext Ž k,. are fnte dmensonal for all. Then Ž. 1 ss-gorensten,.e., Ext Ž k,. Ext Ž k,. 0 for d and Ext d Ž k,. Ext d Ž k,. kž e. for some nteger e. Ž. 2 satsfes. Ž. 3 The last term of the mnmal njecte resoluton of Ž or. s * Ž e.. Ž. 4 Ext d Ž M,. s fnte dmensonal for all fnte graded left and rght -modules M and Ext d Ž M,. Ext d Ž M,. Ž M.* Ž e. as graded k-ector spaces. Theorem 0.3Ž. 1 was also proved n 12, 3.8 and 6, 3.5. By usng Theorem 0.3, the local cohomology ntroduced n 14 and 4, Sect. 7, and the Serre dualty 15, we can prove Ž E2.. 1. PROOF OF THEOREM 0.3 LEMM 1.1. Let be a connected algebra wth fnte njecte dmenson. Suppose that F are nonzero fnte dmensonal graded rght -modules. Then Ext Ž F,. 0 for all p Ž respectely for all p. f and only f Ext Ž k,. 0 for all p Ž respectely for all p..
394 JMES J. ZHNG Proof. By usng a long exact sequence we see that f Ext Ž k,. 0, then Ext Ž F,. 0 for each. Conversely, we suppose Ext Ž F,. 0 for all p. If Ext Ž k,. 0 for some p, we may assume s mnmal amongst such values. Snce F s fnte dmensonal, we have a short exact sequence 0 K F kž l. 0 for some l and some submodule K F. The short exact sequence above yelds an exact sequence Ext Ž K,. Ext k Ž l., Ext Ž F.. 1 The left term s zero because Ext 1 Ž k,. 0 and the rght term s zero by the hypothess. Hence the mddle term s zero, a contradcton. There Ž k,. 0 for all p. The proof of the other case s fore Ext smlar. In the proof of Theorem 0.3 below and n the next secton we wll use the noton of noncommutatve projectve scheme ntroduced n 4. Let Gr be the category of graded rght -modules. Let Tor be the subcategory of Gr consstng of -torson rght -modules and let QGr denote the quotent category Gr Tor. The canoncal functor from Gr to QGr s denoted by. The functor has a rght adjont functor : QGr Gr Žn 4, 2.2.2. the rght adjont functor of was and then n 4, Sect. 4 t was proved that.. We use scrpt letter M for the object Ž M.. The trple Ž QGr,, s. s called the projecte scheme of and s denoted by Proj, where Ž. and s s the automorphsm of QGr defned by the degree shft. For more detals about Proj, see 4, and for basc propertes about quotent category, see 9. Proof of Theorem 0.3. Ž. 1 In ths proof E Ž M. denotes Ext Ž M,. for any left or rght -module M. Snce both jk and j are fnte, there are and j Ž maybe the same. such that E Ž k. 0 and E j 0. We clam that E Ž k. E 0 except for one. If not, there are two dstnct ntegers l and r such that E l 0 and E r Ž k. 0. Wthout loss of generalty, we may assume that l r and that l mn E 04 and r max E Ž k. 04. By Lemma 1.1, we have E l ŽE r Ž k.. 0. By 12, Ž 3.8.1.there s a convergent spectral sequence p, q p q pq 2 E Ext Ext Ž M,., Ž M., Ž E3. pq where Ž M. 0f pqand 0 Ž M. M. The bdegree of the rth dfferental of Ž E3. s Ž r, r 1.. Let M k n Ž E3.. We have a table of
GRDED GORENSTEIN RINGS 395 E p,q terms, 2 0 0 0 0...... 0 0 0 0 0 0 E E Ž k.,...., j E Ž k.. 0 0 l,0 E d,0 E l, r d, r E4 p, where E q denotes Ext p ŽExt q Ž k,.,.. From Ž E4. we see that every l, r l, boundary map passng through E k s zero, whence E r s the term E l, r. Snce l r, ths s zero, a contradcton. Therefore we have proved our clam that there s an nteger p d such that E p Ž k. 0, E p 0, and E Ž k. E 0 for all p. Let F be the fnte dmensonal left -module E p Ž k..byž E3,E. p Ž F. k. For every fnte dmensonal left -module F, we can prove by nducton that dm E p Ž F. dm E p dm F because E 0 for all p. Hence dm E p Ž k dm F dm E p. Ž F. dm k 1. Therefore dm F 1 and F ke,.e., E p Ž k. ke for some e. So we have E p k E p kž e. Ž e. E p Ž F.Ž e. k Ž e.. Next we wll prove that p d. Snce s a noetheran rng wth fnte njectve dmenson, the complex satsfes the condtons Ž. and Ž. n 14, 3.3. Snce s projectve as ether left or rght -module, the complex satsfes the condton 14, 3.3Ž. and hence s a dualzng complex. We have proved above that E 0 for all p and E p ke, whence the complex Ž e. p satsfes the condton 14, 4.4Ž.. Here n general M n denotes the shft of a complex M by n. By 14, 4.5 and p. 61, Ž e. p s a prebalanced dualzng complex. By 14, 4.10 and 4.13, has a balanced dualzng complex Ž, e. p for some graded algebra automorphsm Žsee 14, p. 48 for the defnton of Ž, e. p.. By 14, 4.18, the local dualty theorem holds. By 14, Ž 4.17. or equvalently 15, 4.2.1, we have graded k-vector space somorphsms pq q q Ext Ž M,.* Ž e. Ext Ž M, Ž e. p. *HŽ M., Ž E5. q q Ž n where H M lm Ext, M.. Lettng q 0 n Ž E5. n, we obtan Ext pq Ž M,. 0. Hence the njectve dmenson of s at most p and thus p d.
396 JMES J. ZHNG Ž. 2 Follows from Ž. 1 and 15, 4.3. Ž. 3 By 15, 4.2 and 4.3, Ž e.d1 s a dualzng complex for XProj and X s classcal CohenMacaulay 15, Defnton 2.4. In partcular, Ext M, Ž e. Ext M, Ž e. d1 Ž d1. H Žd1. Ž X, M.*0 for all d 1. Hence the njectve dmenson of Ž e. Ž and of. s at most d 1. Suppose the mnmal njectve resoluton of s 0 I 0 I d 0. By 4, p. 234 the functor :GrQGr s exact and the rght adjont functor s left exact. Hence the mnmal njectve resoluton of s 0 I 0 I d 0, where I ŽI.. We refer to 4, Sect. 4.5 for some basc facts about quotent category and njectve object. Snce the njectve dmenson of s at most d 1, I d 0. Equvalently, I d s an -torson njectve and d consequently I s a drect sum of shfts of *. By 4, 7.7Ž. 1, Ext d Ž k,. d ke mples I * Ž e. and we have proved Ž 3.. Further, by 4, 7.7Ž. 1, Ext k, 0 mples that I s -torson-free for all d and hence has njectve dmenson d 1. Ž. 4 By the proof of Ž. 1, Ž, e.d s a balanced dualzng complex over for some graded algebra automorphsm. Lettng q 0nŽ E5. and takng k-lnear dual of Ž E5., we have Ext d n Ž M,. lm HomŽ, M.* Ž e. Ž M.* Ž e. n as graded k-vector spaces. Hence Ext d Ž M,. s fnte dmensonal for all fnte graded modules M. Smlarly, Ext d Ž M,. Ž M.* Ž e.. Therefore d d Ext Ž M,. Ext Ž M,.. COROLLRY 1.2. Let be a connected noetheran algebra wth fnte njecte dmenson. If Ext Ž k,. 0 for all l and Ext l Ž k,. ke, then s S-Gorensten and l s the njecte dmenson of. Proof. By the hypotheses, we have Ext p ŽExt q Ž k,.,. 0 for all ql. By Ž E3., we have Ext p ŽExt l Ž k,.,. 0 for all p l and Ext l ŽExt l Ž k,.,. k. Snce Ext l Ž k,. ke, Ext Ž k,. 0 for all l and Ext l Ž k,. k Ž e.. Therefore the hypotheses of Theorem 0.3 hold and the statement follows from Theorem 0.3Ž. 1.
GRDED GORENSTEIN RINGS 397 COROLLRY 1.3. Let be a connected noetheran S-Gorensten algebra of njecte dmenson d and e as n Ž E1.. Let M be a fnte graded rght -module and M Ž M.. Then: Ž. 1 Ext Ž M,. Ext Ž M,. for all d 2. Ž. 2 There s an exact sequence of graded k-ector spaces d1 d1 0 Ext M, Ext M, M* e M * e 0. Proof. For the proof of Ž. 1, see 4, 8.1Ž. 5. Ž. 2 pplyng Ext Ž,. to the short exact sequence 0 M n M MM n 0 d1 and usng the fact Ext MM, 0, we obtan an exact sequence n d1 d1 Ž n. 0 Ext M, Ext M, Ext Ž MM,. Ext Ž M,. Ext Ž M,. 0. Ž E6. d d d n By Theorem 0.3Ž. 4, Ext d Ž M,. Ž M.* Ž e. n n 0 for all n 0, Ext d Ž M,. Ž M.* Ž e., and Ext d Ž MM,. Ž MM.* Ž e. n n. Let n go to the nfnty. Then the second term n Ž E6. becomes Ext d1 Ž M,. and the thrd term becomes M* Ž e.. Hence Ž 2. follows. n 2. SIMILR MODULES Let M and N be two graded left or rght -modules. We say M s smlar to N and wrte M N f Ž S1. M N as graded k-vector spaces and Ž S2. Ext Ž M,. Ext Ž N,. as graded k-vector spaces for all. Recall that the Hlbert seres of a graded, left bounded, locally fnte module M M s defned to be the formal power seres Ý H Ž t. dm M t. M The condton Ž S1. s equvalent to H Ž. t H Ž. t and Ž S2. M N s equvalent to H Ž. t H Ž. Ext Ž M,. Ext ŽN,. t. If s a connected noetheran algebra wth fnte global dmenson, then H Ž. M t s determned by Ý Ž 1. H Ž. t 13, 2.3. Hence n ths case Ž S2. mples Ž S1. Ext ŽM,.. It s unclear f Ž S2. mples Ž S1. n general. It s easy to construct two fnte -modules M and N such that Ž S1. holds and Ž S2. fals. If M has a proper
398 JMES J. ZHNG graded submodule M M such that M s smlar to MŽ l. for some l 0, then we say M has a proper smlar submodule. If for every -torson-free module, M, there s a submodule N M such that N has a proper smlar submodule, then we say satsfes the smlar submodule condton Ž or SSC for short.. Graded PI algebras satsfy SSC as we now prove. Let be a connected noetheran PI algebra and M be a fnte graded rght -module. Then there s a nonzero submodule N M such that N s somorphc to a unform rght deal of P for some graded prme deal P 12, 2.1. Thus there exsts a proper submodule of N somorphc Ž and hence smlar. to a shft of N. Therefore satsfes SSC. In the same way we can prove that graded FBN rngs satsfy SSC. Let be a graded algebra automorphsm of. For every graded rght -module M, we defne an -module structure on the twsted module M by m a m a. Then M M defnes an nvertble functor from Gr to tself. Snce every graded projectve module s free, s free and hence as graded rght -modules. LEMM 2.1. Ž. 1 Let be a graded noetheran algebra and a graded algebra automorphsm of. Then M M. Ž. 2 Let be an S-Gorensten noetheran algebra wth fnte njecte dmenson and let M and N be fnte dmensonal graded rght -modules. Then M N f and only f M N as graded k-ector spaces. Ž. Proof. 1 By defnton, M M as graded k-vector spaces. For every, 1 Ext Ž M,. Ext M, Ext Ž M,.. Hence M M. Ž. 2 If MN as graded k-vector spaces, M* N* as graded k-vector spaces. Let d be the njectve dmenson of. By Theorem 0.3Ž. 4, Ext d Ž M,. Ext d Ž N,.. For every d, Ext Ž M,. Ext Ž N,. 0. Hence M N. If satsfes SSC, then we can use nducton on modules effectvely. Frst we prove some good propertes of GK-dmenson. Let be a connected noetheran algebra and M a fnte graded -module. Let f Ž n. dm M for all n. The GK-dmenson of M s equal to M n ž Ý M / GKdm M lm log f Ž.. Ž E7. n Snce dm s addtve, E7 mples that GKdm s exact,.e., n n GKdm M maxgkdm N, GKdm NN 4
GRDED GORENSTEIN RINGS 399 for all N M. Let fž n. be a functon from to. If there exst an nteger t and polynomal functons p Ž n.,..., pž n. n such that fž n. p Ž n. 1 t s for all n s Ž mod t., then fž n. s called a mult-polynomal functon. Defne deg fž n. max deg p Ž n. s1,...,t. 4 LEMM 2.2. Let be a connected noetheran algebra satsfyng SSC and M be a fnte graded -module. Then: Ž. 1 f Ž n. M s a mult-polynomal of n for n 0 and GKdm M deg f Ž n. M 1. Ž. 2 Kdm M GKdm M. Suppose N s somorphc to M as graded k-ector spaces and the sequence 0 K1 NŽ l. M L K2 0 s exact, for some l 0. Then Ž. 3 GKdm M GKdm L 1. If moreoer K1 K2 0, then GKdm M GKdm L 1. Remark. Ths lemma s smlar to 12, 6.1. Note that there s a gap n 12, p. 1022, 1.16 for the nequalty Kdm M GKdm M. However, that does not affect 12, 6.2 Žand hence other theorems n 12. because Kdm M GKdm M holds for uslandergorensten and Cohen Macaulay rngs see the proof of Theorem 3.1Ž. 2. Proof. Ž. 1 and Ž. 2 : We modfy the proof of 12, 6.1. If Kdm M 0, then M s fnte dmensonal and the statements are obvous. Now suppose the statements hold for all modules of Kdm for some 0. Let Kdm M for a fnte graded -module M and we wll prove the statements for M. By the noetheran property and the exactness of GKdm and Kdm, t suffces to show Ž. 1 and Ž. 2 for a nonzero submodule of M. Snce satsfes SSC, we may assume that M s Kdm-crtcal and has a proper smlar submodule,.e., there s a fnte graded -module M such that M M and M Ž l. M for some l 0. Then we have an exact sequence 0 M Ž l. M M 0, where M MM Ž l.. Snce M s Kdm-crtcal, Kdm M Kdm M. By nducton hypothess, f Ž n. M s a mult-polynomal functon for n 0. Snce M M, f Ž n. f Ž n. and hence f Ž n. f Ž n. f Ž nl. M M M M M. Thus f Ž n. M s a mult-polynomal functon for n 0 wth degree equal to deg f Ž n. 1. By Ž E7., GKdmŽ M. deg f Ž n. 1. Snce deg f Ž n. M M M deg f Ž n. M 1, GKdm M GKdm M 1. s a consequence of ths equalty and the nducton hypothess, Kdm M Kdm M 1 GKdm M 1 GKdm M. s
400 JMES J. ZHNG Ž. 3 By the addtvty of vector space dmenson we have f Ž n. f Ž nl. f Ž n. f Ž nl. f Ž n. f Ž n. f Ž n. M M M N L K1 K2 flž n.. Hence deg f Ž n. deg f Ž n. 1 and by Ž 1. M L, GKdm M GKdm L 1. If K K 0, f Ž n. f Ž nl. f Ž n.. Hence deg f Ž n. deg f Ž n. 1 2 M M L M L 1 and by Ž. 1, GKdm M GKdm L 1. Next we wll show that S-Gorensten algebras wth enough normal elements satsfy SSC. Let P be a graded deal of and x be a regular normal element n Ž P. 1. Then x nduces a graded algebra automorphsm by xa Ž ax.. Let M be a graded P-module. The twsted module M s defned by m a m a. Then M M defnes an nvertble functor from Gr P to tself. It nduces an nvertble functor M M from QGr P to QGr P. It s easy to see that Ž P. Ž P. because Ž P. P. PROPOSITION 2.3. Ž. 1 Suppose that s a connected noetheran S- Gorensten algebra of njecte dmenson d. Let P be an deal of, be a graded algebra automorphsm of P, and M be a fnte rght P-module. Then M s smlar to M. Ž. 2 Let be a connected noetheran algebra of njecte dmenson d. Suppose that has enough normal elements. Then s S-Gorensten and satsfes SSC. Proof. Ž. 1 By 15, 4.3, Ž e. s the dualzng sheaf for X Proj and Ž. Ž d1 X s classcal CohenMacaulay,.e., Ext M, e H Ž X, M..* for all and M. Let Y be the projectve scheme Proj P. Snce M s an Ž. d1 d1 P-module, by 4, 8.3 3, H X, M H Ž Y, M.. Hence we have graded k-vector space somorphsms d1 d1 Ext Ž M,. H X, M Ž e. * H Y, M Ž e. *. Ž. d1 Smlarly, Ext M, H ŽY, M Ž e..*. Snce s an automorphsm of P, we have Ž P. 1 Ž P. and hence d1 Ž. d1 H Y, M e H ŽY, M Ž e... By Corollary 1.3Ž 1. and above, we have graded k-vector space somorphsms d1 Ext Ž M,. Ext Ž M,. H Y, M Ž e. * d1 H Y, M Ž e. *Ext Ž M,.
GRDED GORENSTEIN RINGS 401 d1 d1 Ž for d 2. If d 1, we stll have Ext M, Ext M,.. It s easy to see that Ž M.Ž. * e M* Ž. e and Ž M.Ž. * e Ž M.Ž. * e.by Ž. d1 d1 Ž Corollary 1.3 2, we have Ext M, Ext M,.. By Theorem Ž. d d Ž. 0.3 4, Ext M, Ext M,. Therefore M M. Ž. Ž. op 2 By 4, 8.12 2, and satsfy and hence Ext Ž k,. and Ext Ž k,. are fnte dmensonal. By Theorem 0.3Ž 1., s S-Gorensten. Let N be a fnte torson-free graded -module. There s a Kdmcrtcal submodule M N such that Ž. annž M. P s a prme deal of and Ž. M s a fully fathful P-module. By the hypothess, there s a nonzero normal element x Ž P. 1. Hence Mx s a proper submodule of M and Mx M Ž l., where l deg x and xa Ž ax.. By Ž 1., Mx M Ž l. MŽ l. and Ž 2. follows. It s not dffcult to construct a connected noetheran algebra and a fnte graded rght -module M such that Kdm M GKdm M. By Lemma 2.2Ž. 2, such a graded rng does not satsfy SSC. On the other hand, some connected algebras wthout enough normal elements satsfy SSC. The followng can be proved by usng the structure of algebras and the proof s omtted. For detals on S-regular algebras, see 2 and 3, and on the Sklyann algebra, see 10. PROPOSITION 2.4. Connected S-regular algebras of dmenson three and the Sklyann algebra of dmenson four satsfy SSC. 3. PROOF OF THEOREM 0.2 THEOREM 3.1. Let be a connected noetheran S-Gorensten algebra of njecte dmenson d. Suppose that satsfes SSC. Then: Ž. 1 s uslandergorensten and CohenMacaulay and GKdm njdm. Ž. 2 For eery fnte graded -module M, GKdm M Kdm M. Proof. Ž. 1 We wll use the proof of 12, 3.10 wth some modfcatons. Frst we replace Kdm by GKdm and second we use only graded modules as n 12, 6.2. s n the proof of 12, 3.10, t suffces to prove that the followng propertes hold for all fnte graded left and rght -modules M. Ž. a jm GKdm M d; Ž b. GKdm Ext j Ž M,. GKdm M, where j jm; Ž. c For all jm d, GKdm Ext Ž N,. d for all fnte graded modules N. The nequalty n Ž c. s equvalent to GKdm Ext Ž N,. mngkdm N, d 4 because Ext Ž N,. 0 when GKdm N d
402 JMES J. ZHNG see Ž. a. We nduce on GKdm M. If GKdm M 0, then M s fnte dmensonal. Ž. a and Ž. b are obvous and Ž. c holds by Theorem 0.3Ž. 4. Suppose Ž. a, Ž. b, and Ž. c hold for all modules wth GKdm. We now consder a fnte module M wth GKdmŽ M.. By usng the same arguments as n the proof of 12, 3.10 we see that f Ž. a and Ž. b can be proved for the modules M1 and M2 wth GKdm, and 0 M1 MM0 s exact, then Ž. a and Ž. 2 b also hold for M. By the noetheran property and the fact above, to prove that Ž. a and Ž. b hold for a noetheran module M t suffces to show Ž. a and Ž. b hold for some nonzero submodule of M. By the SSC hypothess, we may assume that M has a proper smlar submodule,.e., there s an exact sequence 0 M Ž l. M M 0 Ž E8. for some M smlar to M and for some l 0, where M MM Ž l..by Lemma 2.2Ž. 3, GKdm M GKdm M 1. pplyng Ext Ž,. to Ž E8. we have an exact sequence j j j j1 Ext Ž M,. Ext Ž M,. Ext Ž M Ž l.,. Ext Ž M,.. Ž E9. If j d, by nducton hypothess Ž. a, the left and rght terms of Ž E9. j j are zero. Snce M M, Ext Ž M,. Ext Ž M,. as graded k-vector spaces. Snce Ext j Ž M,. s left bounded, Ž E9. mples that Ext j Ž M,. 0. Thus jm d. If GKdm Ext d Ž M,., then, by nducton hypothess Ž. c, GHdm Ext s Ž M,. for all s. pplyng nducton hypotheses Ž. a and Ž. c to the modules Ext s Ž M,. for all s, we obtan that the GK-dmenson of Ext p ŽExt q Ž M,.,. s less than for all p, q. Then the spectral sequence Ž E3. mples that GKdm M, a contradcton. Hence GKdm Ext d Ž M,. and, consequently, jm d. Thus we have proved Ž. a and Ž. b. It remans to prove Ž. c. By the noetheran property and the long exact sequence on Ext Ž,. and the SSC hypothess, we may assume that N has a proper smlar submodule and that there s a short exact sequence smlar to Ž E8. for N. Lettng j d and M N n Ž E9., we obtan an exact sequence d d d1 Ext Ž N,. Ext Ž N Ž l.,. Ext Ž N,.. Ž E10. d1 By nducton hypothess Ž. c, GKdm Ext Ž N,. 1. Snce N N as chosen, Ext Ž N,. Ext Ž N,. as graded k-vector spaces. pplyng Lemma 2.2Ž. 3 to Ž E10. we have GKdm Ext d Ž N,.. Therefore Ž c. follows and we have fnshed our proof of Ž. 1. Ž 2. By Ž 1., GKdm M d jm M, and by 7, 4.5, GKdm s fntely parttve n the sense of 8, 8.3.17. Hence by 8, 8.3.18, GKdm M Kdm M. Combnng ths nequalty wth Lemma 2.2Ž.Ž. 2, 2 follows.
GRDED GORENSTEIN RINGS 403 Ž. 1 and Ž. 2 of the followng theorem were also proved n 1 under some weaker hypotheses and a part of Ž. 1 was proved n 7, 5.3. Recall that a rng s called quas-frobenus f t s left and rght artnan and selfnjectve. THEOREM 3.2. Let be a connected noetheran, uslandergorensten, and CohenMacaulay algebra of njecte dmenson d. Then: Ž. 1 has a quas-frobenus ungraded quotent rng. Ž. 2 For eery mnmal prme deal P, GKdm P d. Ž. 3 If, moreoer, has fnte global dmenson, then s a doman and a maxmal order n ts quotent dson rng. Proof. Ž. 1 and Ž. 2. By 7, 3.1 and 5.8, s uslandergorensten and CohenMacaulay as an ungraded algebra,.e., the uslandergorensten and the CohenMacaulay condtons hold for fnte ungraded -modules. Let N N be the ntersecton of all prme deals of, whch s called the prme radcal of. By 8, 8.3.14, N s left and rght nvarant wth respect to GKdm n the sense of 8, 6.8.13. In partcular N s left and rght weakly nvarant wth respect to GKdm Žfor defnton, see 8, 6.8.13.. By the CohenMacaulay condton, GKdm M GKdm for all nonzero submodules M,.e., s homogeneous wth respect to GKdm n the sense of 8, 6.8.8. By 8, 6.8.15, has a left and rght artnan quotent rng Q. Let I be an deal of a rng R and let CŽ I. denote the set of elements n R whch are regular n RI. By 8, 4.1.3 and 4.1.4, we obtan CŽ. 0 CŽ N., QN s the prme radcal of Q, and QQN QŽ N., where QŽ N. s the quotent rng of the semprme noetheran rng N. Let P be a mnmal prme deal of. Snce N s nlpotent, PN s a mnmal prme of N. Suppose that GKdmŽ P. d. By 8, 6.8.14Ž. and 6.8.15, there s a regular element c CŽ. 0 CŽ N. such that c P. Thus P QŽ N. 0 and then HomŽ P, N. 0. Ths contradcts the fact that PN s a mnmal prme of N. Therefore GKdmŽ P. d and by the CohenMacaulay property, HomŽ P,. 0. It remans to show that Q s self-njectve. Snce Q s a localzaton of, n njdm Q njdm. ssume on the contrary that n 0. Snce QQN QŽ N., there s a one-to-one correspondence between mnmal prme deals of and prme deals of Q va P PQ. Snce Q s artnan, every smple Q-module M has a fnte drect sum somorphc to QPQ for some mnmal prme deal P. Hence, for some l, HomŽ M l, Q. HomŽ QPQ, Q. HomŽ P Q, Q. 0
404 JMES J. ZHNG because HomŽ P,. 0. Consequently, HomŽ M, Q. 0. Therefore HomŽ L, Q. 0 for every nonzero fnte Q-module L. By the spectral 0, n sequence E3, whch holds for ungraded rngs 7, 2.2, E2 HomŽExt n Ž L, Q., Q. 0. Ths mples that Ext n Ž L, Q. 0 for all fnte modules L. Hence the njectve dmenson of Q s less than n, a contradcton. Therefore n 0 and Q s self-njectve. Ž. 3 Follows from 11, 2.10. Now we are ready to fnsh our proof of Theorem 0.2. Proof of Theorem 0.2. If has enough normal elements, then, by Proposton 2.3, s S-Gorensten and satsfes SSC. Hence most of Theorem 0.2 follows from Theorems 3.1 and 3.2. It remans to show that GKdm I clkdm I for all graded deals I and that every mnmal prme deal of a connected algebra s graded. We prove the second statement frst. It s easy to see that graded prme s prme and that for every nonnlpotent element x, there s a graded prme deal P such that x n P for all n. Hence the ntersecton of all graded mnmal prme deals s the prme radcal N. However, every Ž ungraded. mnmal prme deal must appear n the ntersecton. Therefore every mnmal prme deal s graded. By 8, 6.4.5 and Theorem 3.1Ž 2., clkdm I Kdm I GKdm I. We show next that clkdm I GKdm I for all graded deals I. Snce s noetheran and every mnmal prme deal of I s graded as proved n the last paragraph, we only need to prove the nequalty when I s a graded prme deal. Pck a nonzero normal and regular element x Ž I.. By nducton we have 1 clkdm I clkdm Ž I x. 1 GKdm Ž I x. 1 GKdm I. Therefore clkdm I GKdm I. Several famles of quantum algebras lsted n 5 can be constructed by localzng some specal normal elements n connected noetheran algebras wth enough normal elements. The base connected algebras have also fnte global dmenson Ž or fnte njectve dmenson.. Hence by Theorem 0.2 these algebras are uslandergorensten and CohenMacaulay. By 1, the uslandergorensten and CohenMacaulay propertes are preserved under localzaton. If, n addton, the prme spectra of these algebras are normally separated, then 5, 1.6 mples that these algebras are catenary n the sense that, for any two prme deals P Q of, all saturated chans of prme deals between P and Q has the same length. In partcular the followng s a consequence of 5, 1.6 and Theorem 0.2.
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