Lifting Ore Sets of Noetherian Filtered Rings and Applications

JOURNAL OF ALGEBRA 179, 686᎐703Ž. 1996 ARTICLE NO. 0031

Li Huishi*

Department of Mathematics, Shaanxi Normal Uni¨ersity, 710062 Xian, People’s Republic of China

CORE Communicated by Kent R. Fuller Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Received November 30, 1993

Let R be a filtered ring such that the Rees ring R˜ of R is left Noetherian. It is observed that if T is a left Ore set of the associated graded ring GRŽ.consisting of homogeneous elements then the saturated set S s ÄŽ.4s g R, ␴ s g T is a left Ore set of R. This lifting property is used to obtain local-global results on R which simplify some microlocal-global results in the literature. If GRŽ.is positively graded and commutative then certain structure sheaves on R are constructed. ᮊ 1996 Academic Press, Inc.

INTRODUCTION

The theory of algebraic microlocalization, which has its roots in the algebraic study of systems of linear differential equations with holomor- phic coefficientswx KKS, GQS , has been developed by several authors in recent years, e.g.,wx Spr, Gin, VE1, WK, AVVO . Roughly speaking, if R is a filtered ring and ⌺ is a multiplicatively closed subset of R such that the corresponding subset ␴ Ž.⌺ of homogeneous elements in the associated graded ring GRŽ.is a left Ore set, then ⌺ is generally not a left Ore set of R and hence one cannot localize R at ⌺. Nevertheless, one may construct ␮ a complete filtered ring QR⌺ Ž.which is called the microlocalization of R at ⌺ such that there exists aŽ. canonical filtered ring homomorphism ␮Ž. Ž.Ž. ␮Ž. ␸:RªQR⌺ with the properties: 1 ␸ s is a unit of QR⌺ for every sg⌺;2ifŽ. f:RªAis a filtered ring homomorphism, where A is complete and fsŽ.is a unit of A for each s g ⌺, then there exists a ␮Ž. unique filtered ring homomorphism g: QR⌺ ªAsuch that g (␸ s f. Ä Ž. Ž .4ŽŽ. Moreover, if ⌺sat s r g R, ␴ r g ␴ ⌺ see the definition of ␴ r in Section 2. then QR␮ Ž. QR␮Ž.Žcf. Sao. , and if ⌺ is a left Ore set of ⌺⌺sat s wx

*Supported by a grant from the Education Committee of P.R. China.

686

0021-8693r96 $12.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. LIFTING ORE SETS 687

$ ␮Ž. y1 y1 Rthen QR⌺ s⌺Rwhere the latter is the completion of ⌺ R with respect to the localized filtration on ⌺y1 R Žcf.wx AVVO. . However, in order to obtain some localization-like properties of microlo- ␮Ž. calizations, such as the important exactness of the functor QR⌺ myR in wxAVVO , the microlocal description of modules with regular singularities inwx VE2, LVO1 and the microlocal description of dimension theory and the Auslander regularity of filtered rings inwx Sao , the Noetherian condi- tionŽ even the Zariskian condition in the sense ofwx LVO2. has been assumed on the Rees ring R˜ of R Žalthough the ⌺-Noetherian condition on a filtered ring was used inwx VE2 , it was proved that this condition is equivalent to the Zariskian condition; seewx LVO2. . Under the assumption that R is left Zariskian the microlocalization theory has also been used in the construction of sheaves of quantum sections which has been connected to the study of noncommutative algebraic geometry inwxw SVO and LBVO x . In the present paper, after giving some necessary preliminaries we first observe the fact that if the Rees ring R˜ of a filtered ring R is left Noetherian, then any left Ore set T of GRŽ.consisting of nonzero homogeneous elements may be lifted to aŽ. saturated left Ore set in R˜, resp. in R. ␮ Consequently the microlocalization QR⌺ Ž.of R at a multiplicatively closed subset ⌺ is nothing but the completion of a localization of R at an Ore setŽ. Section 2 . This fact motivates us to revisit the microlocal-global results ofwx VE2, LVO1, Sao and to obtain a local-global analogy by using localizations at Ore setsŽ. Sections 3, 4 . Moreover, if GR Ž.is positively graded and commutative we lift the classicalŽ. graded structure sheaves O g & Y ŽŽ.. &gFŽ.Ž. G F and OY over Y s Proj GR to sheaves OY, OY , OY 0 , OY 0 on R˜and Rrespectively in a natural wayŽ. Section 5 .

1. PRELIMINARIES

All rings considered in this paper are associative rings with identity 1. Unless otherwise stated, module means left unitary module. For the readers convenience we recall some basic notions on filtered rings and filtered modules. By a filtered ring we mean a ring R with a filtration Ä4 FR s FRn ,ngޚ consisting of additive subgroups of R such that R s

D ng ޚ FRn ,1gFR0 ,FRnn:FRq1and FRFRnm:FR nqmfor all n, m g ޚ. By a filtered R-module we mean an R-module with a filtration FM s Ä4 FMn , ngޚ consisting of additive subgroups of M such that M s ޚ D ng ޚ FMnn, FM:FM nq1, and FRFMnm:FM nqm for all n, m g . The category of filtered left R-modules and filtered R-morphisms of degree zero is denoted by R-filt. If R is a filtered ring with filtration FR, 688 LI HUISHI then there are two associated graded rings with respect to FR, namely, the ˜ Ž. Rees ring R s[ngޚFRn and the associated graded ring GRs [n ޚ FRnnrFR1. For any M g R-filt with filtration FM the Rees mod- g ˜˜y ule M s[ng ޚ FMn , which is a graded R-module, and the associated Ž. Ž. graded module GM s[ng ޚ FMnnrFMy1, which is a graded GR-module, are also defined in a natural way. If X denotes the central regular element of degree 1 which is represented by the identity 1 of R in R˜1, then we have R ( R˜˜rŽ.1 y XRas filtered rings where the latter has the dehomogenized filtration, GRŽ.(R˜˜rXR as graded rings where the latter 1 1 has the quotient gradation. Moreover, RX˜wxwy (Rt,ty xas graded rings where the latter has the natural gradation and RX˜wxy1 is the localization of R˜ atÄ4 1, X, X 2, . . . . For any filtered R-module M with filtration FM we also have M ( M˜˜rŽ.1 y XMas filtered modules where the latter has the dehomogenized filtration GMŽ.(M˜˜rXM as graded modules where 1 1 the latter has the quotient gradation. Moreover, MX˜wxy (Mt w,ty xas graded modules where the latter has the natural gradation and MX˜wxy1 is the localization of M˜ atÄ4 1, X, X 2, . . . . A filtration FM on an R-module

M is said to be good if there are k1,...,ks gޚ and ␰ 1,...,␰s gM such that FM Ýs FR␰for all n ޚ. FM is good if and only if M˜ is a nis s1nykii g finitely generated R˜-module. Finally we recall that a filtered ring R with filtration FR is called a left ˜ Zariskian ring in the sense ofwx LVO2 if R is left Noetherian and FRy1 is contained in the Jacobson radical JFRŽ.00of FR Žor equivalently, X is contained in the graded Jacobson radical JRgŽ.˜˜of R which is the largest proper graded ideal of R˜˜such that its intersection with R0 is contained in the Jacobson radical in R˜0 .. For some basic properties of a Zariskian filtered ring we refer the reader towxwx LVO2 and LVOW . We also refer the reader towx LVO2 for some typical examples of Zariskian filtered rings. In particular, if a filtered ring R has discrete or positi¨e filtration such that GRŽ.is Noetherian then R is a Zariskian ring, hence R˜ is Noetherian.

2. LIFTING ORE SETS AND BASIC PROPERTIES

In this section we first discuss the lifting property of a homogeneous Ore set and then, by using the dehomogenization method given inwx LVO3 we derive some basic properties of lifting Ore sets. ˜˜Ž Let R be a filtered ring with filtration FR. We write ␲ 1: R ª Rr 1 y .Ž˜˜˜. XRsR,␲2 :RªRrXR s GR for the natural homomorphisms, and we define the ␴-function on a filtered R-module M as usual: ␴ Ž .␴ Ž. ␴ Ž. :MªGM, msmqFMny1 if m g FMnnyFMy1and m s 0if mgFngޚFMn . LIFTING ORE SETS 689

Suppose that T is a multiplicatively closed subset of GRŽ.consisting of Ž nonzero homogeneous elements i.e., 1 g T and t12, t g T implies tt12g T.. Then the saturated subset

˜˜ Ss½5˜˜rgR,rhomogeneous, ␲ 2Ž.˜r g T is a multiplicatively closed subset of R˜consisting of nonzero homogeneous Ž. elements and ␲ 2 S˜ s T. Moreover, one also easily sees that the saturated subset

S s Ä4r g R, ␴ Ž.r g T is a multiplicatively closed subset of R consisting of nonzero elements and Ž. ␲1S˜ sS. 2.1. LEMMA. Suppose that the ideal XR˜ has the Artin᎐Rees property for graded left ideals and R˜ satisfies the ACC on left annihilators. If T is a left Ore set in GŽ. R consisting of homogeneous elements then S˜ is a left Ore set in R˜ and S is a left Ore set in R. Proof. It is sufficient to prove that S˜˜is a left Ore set of R. Bywx NVO we only have to check the Ore conditions for homogeneous elements. Using the commutative diagram

6 n

RR˜˜˜rXR

6 6

R˜˜rXR, a similar argument to that in the proof ofwx AVVO Lemma 3.2 shows that n the image of S˜˜in RrXR˜˜, where n G 1, is a left Ore set. Let ˜r g R be a homogeneous element and ˜˜s g S˜˜˜. Then there exist rЈ g R, ˜sЈ g S such n that ˜˜˜˜˜˜rЈs y sЈr g Ž.Rs˜˜q Rr l XR˜˜˜:XRs Ž. ˜˜qRr for some n G 1 because XR˜has the Artin᎐Rees property for graded left ideals. Hence ˜˜rЈs y ˜˜sЈr s Ž.Ž.Ž. Xrs˜˜12q ˜˜rr and ˜rЈ y Xr ˜1 ˜s s ˜sЈ q Xr ˜2 ˜r. Note that ˜sЈ q Xr ˜2g S˜,it follows that the first Ore condition holds. Since R˜satisfies the ACC on left annihilators it is well known that the second Ore condition follows from the first Ore conditionŽ e.g.,wx GW, Proposition 9.9. .

2.2. COROLLARY. Let R˜ be left Noetherian. If T is a left Ore set in GŽ. R consisting of homogeneous elements, then S˜˜ is a left Ore set in R and S is a left Ore set in R. From now on we assume that R˜ is left Noetherian. 690 LI HUISHI

Let T be a left Ore set in GRŽ.consisting of homogeneous elements, and S˜, S be the saturated Ore sets obtained from T. Using the notations as before and considering the corresponding localizations we obtain

y1 y1 y1 y1 SR˜˜˜rSŽ.1yXR ˜(␲1Ž.SR ˜Ž. ˜rŽ.1yXR ˜sSR

y1 y1y1 y1 SR˜˜˜rSXRŽ. ˜(␲2 Ž.SR ˜Ž. ˜rXR ˜s TGRŽ.. Note that SR˜y1˜is a ޚ-graded ring and X is again a central regular element of degree 1 in SR˜y1 ˜; it follows fromwx LVO3 that the following proposition holds.

2.3. PROPOSITION. Let the notations be as abo¨e. Ž.1 Sy1 R is the dehomogenization of S˜y1 R˜ with respect to X in the sense of wxLVO3 , hence Sy1 R is a filtered ring with filtration

y1 Ž.SR˜ ˜ nŽ.1 X y1 q y FSnŽ. Rs , ngޚ. Ž.1yX & 1 1 1 1 Ž.2 GS Žy R .(TGRy Ž .;SRy(SR˜˜y. 2.4. COROLLARY. Let ⌺ be a multiplicati¨ely closed subset of R such that Ts␴Ž.⌺is a left Ore set of GŽ. R . Let S˜ and S be the saturated Ore sets obtained from T as before. Consider the microlocalizations at S˜, resp. at ⌺, as ␮ constructed in wxAVVO and adopt the notations in loc. cit. We ha¨eQ⌺Ž. R $ 1 1 sSy R where the latter is the completion of Sy R with respect to its filtration $ & ŽŽ..Ž␮ y1 . Žy1 .y1 Ž. ␮ topology; GQ⌺ R sGS R sGS R (TGR;QR⌺Ž.s & ␮Ž. Žy1 .nyXX Ž1 .n QRSX˜ ˜˜sSR sS R˜ where n denotes the graded completion with respect to the XR˜˜-adic filtration on R. Let M be a filtered R-module with filtration FM Žnote that any R-module may be filtered with respect to FR.. If we consider the Rees module M˜ of M then we also have:

y1 y1 y1 y1 SM˜˜˜rSŽ.1yXM ˜(␲1Ž.SM ˜Ž. ˜rŽ.1yXM ˜sSM

y1 y1y1 y1 SM˜˜˜rSXMŽ. ˜(␲2 Ž.SM ˜Ž. ˜rXM ˜s TGMŽ..

2.5. COROLLARY. Let the notations be as abo¨e. Ž.1 Sy1 M is the dehomogenization of S˜y1 M˜ with respect to X, hence Sy1 M is a filtered Sy1 R-module with filtration

y1 Ž.SM˜ ˜ nŽ.1 X y1 q y FSnŽ. Ms , ngޚ. Ž.1yX LIFTING ORE SETS 691

& 1 1 1 1 Ž.2 GS Žy M .(TGMy Ž .;SMy (SM˜˜y . $ Ž. ␮Ž. y1 3 Let ⌺, S˜, and S be as in Corollary 2.4. Then Q⌺ M s S M where ␮ Q⌺ Ž. M is the microlocalization of M at ⌺ as constructed in wxAVVO and $ Sy1 M is the completion of Sy1 M with respect to its filtration topology, $ & Ž␮ Ž..Žy1 . Žy1 .y1 Ž. ␮␮Ž. GQ⌺ M sGS M sGS M (T G M and Q⌺Ž. M s QMS˜˜ & Ž.y1 nyXX Ž1 .n sSM sS˜˜ M where nX denotes the graded completion with respect to the XR˜˜-adic filtration on M.

2.6. PROPOSITION. Let M be a filtered R-module with filtration FM.

Ž.1 If FM is a good filtration on M then FŽ Sy1 M. is a good filtration on Sy1 M. 1 Ž.2 The canonical map ␸: M ª Sy M is a filtered morphismŽ of degree zero.Ž., and if G M is T-torsionfree then ␸ is a strict filtered map, i.e., Ž y1 .Ž.Ž. FSnn Ml␸Ms␸FM,ngޚ. Proof. Ž.1IfFM is good on M then M˜is a finitely generated & 1 1 R˜˜-module. It follows from Corollary 2.5 that SMy sSMy ˜is a finitely generated SR˜y1 ˜-module and hence FSŽ.y1 M is good. Ž.2 This may be directly checked.

2.7. COROLLARY wxAVVO Corollary 3.20 . Let ⌺ be as in Corollary 2.4. Ž. ␮Ž. 1The functor Q⌺ R mR ᎏpreser¨es strict maps and is exact on R- modules. Ž. ␮Ž. ␮Ž. 2If M g R-filt has a good filtration FM then Q⌺ R mRM ( QM⌺ ␮ ␮ as filtered R-modulesŽ also as filtered Q⌺⌺Ž. R -modules ., and as a QŽ. R - ␮␮ module Q⌺⌺Ž. M has good filtration FŽ QŽ.. M . Proof. Let S˜and S be as in Corollary 2.4. Then by the foregoing $ $ ␮Ž. y1 ␮Ž. y1 results we have QR⌺⌺sSRand QMsSM. $ Ž.1 Since SRy1is a flat SRy1-moduleŽ cf.wx LVO2. and SRy1is a flat $ 1 R-module, it follows that SRy is a flat R-module. If f: M ª N is a strict filtered morphism in R-filt then f˜˜: M ª N ˜has X-torsionfree cokernel. 1 1 1 Consequently, Sf˜y Ž.˜˜:SMy˜˜ªSNy˜has an X-torsionfree cokernel. Therefore Sfy1Ž.:SMy1 SNy1 is strict and hence $ $ $ ª 1 1 1 Sfy Ž.:SMy ªSNy is strict. Ž.2IfFM is good on M then FSŽ y1 M.is good by Proposition 2.6. It $ $ y1 y1 y1 follows fromwx LVO2 that SR y1Ž.SR M SM. Now it is $ mSRm R ( clear that FSŽ.y1 M is good. 692 LI HUISHI

Let us finish this section by giving an interesting property of lifting Ore sets that fails for general Ore sets.

2.8. PROPOSITION. Let T be a left Ore set of GŽ. R consisting of homogeneous elements and S the standard Ore set of R obtained from T as before. Then Sy1 R is a left Zariskian filtered ringŽ. although R is not Zariskian . & 1 1 Proof. Since SR˜˜y sSRy the latter is a left Noetherian ring. It remains to show that X is contained in the graded Jacobson radical JSRg Ž.˜y1 ˜˜. Recall fromwx NVO that the graded Jacobson radical of SRy1˜is the largest proper graded ideal of SR˜y1 ˜ such that its intersection with Ž.y1 Ž.y1y1 SR˜ ˜˜00is contained in the Jacobson radical of SR˜. Now if Xs˜˜ r g y1Ž.y1 y1 Ž.y1 XS˜ R˜˜l SR˜01then we have to prove that for all ˜˜sr1gSR˜˜0the Ž.Ž.y1y1y1 element 1 y Xs˜˜˜˜ r s11 r is invertible in SR˜˜˜.If˜˜˜˜sr2sas 1with a ˜g R Žy1 .Ž.y1 Ž.Ž.Ž.Žy1 y1 and ˜˜s21g S˜, then 1 y Xs˜ r˜ s˜˜ r1s1yXss2˜ ar˜˜1s˜ss2˜˜ ss2˜ . yXar˜˜12. Since S˜˜is saturated ˜ss ˜yXar ˜˜1g S, it follows that 1 y y1y1 XsŽ.Ž.˜˜˜˜ r s11 r is invertible.

3. HOLONOMIC MODULES WITH REGULAR SINGULARITIES REVISITED

In a purely algebraic study of holonomic D-modules with regular singularities, the stalk Ep of analytic micro-local differential operators at a point p has been replaced by an algebraic microlocalization ERpŽ.of a g filtered E-ring R at a prime ideal p g Spec ŽŽ..GR , and a microlocal characterization of holonomic modules with regular singularities has been given first inwx VE2 by using microlocalizations of E-rings, then inw LVO1 x by using microlocalizations of strongly filtered rings. In view of Section 2 the aim of this section is to give a local-global description for holonomic modules with regular singularities by using localizations at lifting Ore sets. Inwx VE2 and w LVO1 x one of the key steps to arrive at the final result is to microlocally study finitely generated FR0 -modules. To this end, an isomorphism ␹ : ␴ ŽSQn .y1 Ž,N .Qn Ž,Np Ž .. has to be employedŽ see the p 0 ª proof ofwxwx VE2, Corollary 4.12 , and the proof of LVO1 Lemma 4.6. . However, the proof of the fact that ␹ is an isomorphism given in both wxVE2 and w LVO1 x is rather complicated because the localized pseudo- normal has been heavily used. In our case the exactness of the localization functor will enable us to obtain a similar isomorphismŽ. Lemma 3.5 and to locally study finitely generated FR0 -modulesŽ. Lemma 3.6᎐Proposition 3.8 quite easily. After Theorem 3.9 we may complete our approach by mimick- ing the argument ofwx LVO1 . To start with, we recall that a bideri¨ation on a commutative ring A is a Ž.Ž. ޚ-bilinear map ␦: A = A ª A satisfying ␦ aa12,b sa 1␦ a 2,b q LIFTING ORE SETS 693

Ž. Ž . Ž.Ž. ␦a12,ba,␦a,bb 121sb␦a,b 2q␦a,bb 12, for all a 1212, a , b , b g A. An ideal I in A is ␦-stable if ␦Ž.a, b g I for all a, b g I.If Ris a filtered ring with filtration FR such that GRŽ.is commutative then we can define aޚ-bilinear map on GRŽ.as followsÄ4 , GRŽ.=GR Ž.ªGR Ž.Ž, f,g . Ä4 Ž. Ž. Ä4 ¬f,g, where f s ␴ a , g s ␴ b for some a g FRnm,bgFR, f,g Ž . Ä4 swxa,bqFRanqmy2 wx,bbeing the Lie commutator . Clearly , is a biderivation on GRŽ.and it is called the Poisson product inwx Ga . An ideal Iof GRŽ.is said to be inölutië if I isÄ4 , -stable. 3.1. THEOREM ŽGabber’s Theoremwx Ga. . Let R be a filtered ring with filtration FR such that GŽ. R is a Noetherian commutatië algebra oër ޑ where ޑ is the field of rational numbers. Let M be a finitely generated R-module with a good filtration FM, thenÄ JŽ.Ž. M , JM4:JŽ. M where Ž. JM s'AnnGŽR.GŽ. M is the characteristic ideal of M.

3.2. DEFINITION. Let R be a filtered ring with filtration FR such that GRŽ.is a Noetherian commutative ޑ-algebra. Let I be a graded involutive radical ideal of GRŽ.Ži.e., I s 'K for some graded ideal K; for Ž. . example, we may take I s JM s'AnnGŽR.GMŽ.,MgR-filt. M is said to have regular singularities along IŽ. M has R.S. along I if M Ž. possesses a good filtration FM such that I : AnnGŽ R. GM.Mis said to have regular singularitiesŽ. M has R.S. if M has R.S. along I s JMŽ., i.e., Ž. Ž.Ž Ž. JM sAnnGŽ R. GM note that JM is independent of the choice of good filtrations on M .. Before Theorem 3.10 we always assume that R is a left Zariskian filtered ring with filtration FR satisfying the following two conditions: Ž. 1Ris strongly filtered in the sense ofwx LVO1 , i.e., FRFRnmsFR nqm for all n, m g ޚ, or equivalently, GRŽ.is strongly graded; and Ž.2 GR Ž .is a commutative ޑ-algebra. Ž. Ä4 Let I be as in Definition 3.2. Put G I s ␶ g FR10,␶qFRgI.We first recall fromwx LVO1, Proposition 4.4 the following 3.3. PROPOSITION. Let M g F-filt be finitely generated. The following statements are equi¨alent: Ž.1 M has R.S. along I; Ž.␶ Ž . Ž . Ýϱ ␶i 2 For e¨ery m g M, g G I , the F0 R-module P␶ m s is00FR m is finitely generated. Let R be as above. Let Spec g ŽŽ..GR be the set of all graded prime ideals of GRŽ.and Spec ŽŽ..GR00the set of prime ideals of GR Ž.. For ŽŽ.. Ž. gŽŽ.. each p00g Spec GR we have GRp 0spgSpec GR . Let Tpbe the multiplicatively closed subset in GRŽ.consisting of homogeneous 694 LI HUISHI

Ž. Ž. elements of GRypand Tp s GR00yp. Using the notations as in 0 Ä Section 2 we obtain by Lemma 2.1 the corresponding Ore set Sp s r g R, ␴Ž.r T4Äin R and the Ore set S r FR,␴Ž.r T 4in FR g pp00s g 0g p0 because FRis also a left Zariskian ring. It is easily seen that S S . 0 pp0: Moreover, a slight modification of the proof ofwx LVO1, Proposition 3.5 yields the following Ž. 3.4. PROPOSITION.1For any s g Spp there is an sЈ g S such that sЈsS. gp0 Ž.2 S is also an Ore set in R. p 0 Ž.3 SRy1 SRy1 ,Sy1 R is strongly filtered, and FŽ Sy1 R. SFRy1 , ppps 0 0pps 00 FSŽy1 R.SFRy1,n ޚ. nps p0 n g Ž.4TGRy1 Ž. TGRy1 Ž.,Ty1 GŽ. R is strongly graded andŽ Ty1 GŽ.. R ppps 0 p0 TGRy1Ž., ŽTGRy1 Ž.. TGRy1 Ž.,k ޚ. sp000 pkpks g Ž.5GS Ž y1 R.ŽTGRy1Ž.. TGRy1 Ž.,GS Žy1 R.ŽTGRy1Ž.. p 0s p 0sp0 0 pks p k TGRy1Ž.,k ޚ. spk0 g Let M be an R-module with good filtration FM and N an arbitrary

FR0 -submodule of M. Define

FMn lN QnŽ.,N s , ngޚ. FMny1 lN Ž. Ž. Ž.Ž.Ž. Clearly Qn,N is a GR0-submodule of GM, Qn,N :GMn. y1 Ž. Moreover, if ␸pp: M ª SMis the canonical map then we define Npas y1 y1 the FS0Žpp R.-submodule in SMgenerated by the elements ␸ pŽ.m with mgN, and we put

y1 FSnpŽ.MlNpŽ. QnŽ.,NpŽ. , n ޚ. s y1 g FSMny1Ž.p lNpŽ.

y1 y1 Then QnŽ,Np Ž ..is a GS Ž p R.Ž0-submodule of GSp M.. On the other hand, QnŽ,Np Ž..is a GR Ž.00-module via the canonical morphism GRŽ. TGRy1Ž. ŽGSy1 R.. ªp 0 0 s p 0 3.5. LEMMA.1Ž.Np Ž . SNy1 . s p0 Ž.2There exists an isomorphism of additi e groups ␹ : SQny1 Ž.,N ¨ p0 ª QnŽŽ..,Np. Ž.3TQny1 Ž.,N 0if and only if Sy1 QŽ. n, N 0. pp00s s Proof. Ž.1 By Proposition 3.4 it is easily seen that NpŽ. SNy1 . s p0 Ž.2 For each n g ޚ we have the isomorphism of additive groups ␥:SFMy1 Ž.N SFMy1 SNy1. Hence we obtain the following np00 nlªpnl p0 LIFTING ORE SETS 695 commutative diagram of exact rows and columns:

6 6 6 6 6 6 0SFMy1Ž.Ž.Ž.NSFMy1NSQny1,N 0 pn000y1l pnl p

␥␥ 6 6 n1n␹ y 6 6 6 6

0 SFy1 M SNy1SFMy1SNy1ŽŽ..6 pn0000y1lppnpl Qn,Np 0

6 6 00

Consequently ␹ is an isomorphism. Ž. Ž . 3 This follows from the fact that sxqFMny1 lNs0 if and only if ␴ Ž.Žsx FMN. 0 for all s S and x FMN qny1 l s g pn0q y1lg QnŽ.,N. To continue our approach we need a few more notations. g First, put I s Ä p g Spec ŽŽ..GR such that p is involutive in GRŽ.4and ␮ Ä Ž.4 Ž. RpssupgI ht p where ht p denotes the height of p.If Mis an R-module with good filtration FM then we let MinŽŽJM ..denote the set of all minimal prime divisors of JMŽ.. By Gabber’s theorem it is clear that Ž Ž .. Ž . Ž Ž .. Min JM :Iand ht p F ␮R for all p g Min JM . Second, let M have a good filtration FM and let N be an arbitrary

FR0 -submodule of M. Since GRŽ.is strongly graded and commutative if Ž. Ž . Ž. we put InsAnnGŽR.Qn,N and Jns'InŽ.for all n g ޚ, then Ž .Ž.Ž.Ž.Ž.0 as inwx LVO1 for a certain n000we have InsIn and JnsJn for Ž. Ž. Ž. Ž. all n G n00. Hence if we put JN sJn then JN sDngޚJn. Fur- thermore, let MinŽIn Ž .., resp. Min ŽJN Ž .., be the set of all minimal prime divisors of InŽ., resp. JN Ž ., in Spec ŽGR Ž ..0 . 3.6. LEMMA. With notations as abo¨e, if N : M is not finitely generated as an F0 R-module then the following properties hold: Ž.1 Min ŽŽIn ../л,or in other words QŽ. n, N / 0 for all n g ޚ. Ž. ŽŽ .. Ž . 2 Let p00g Min I n then p s G R p satisfies ht p F ␮R. Ž. ŽŽ .. ŽŽ .. Ž. 3 Min JN /л. If p00g Min J N then for p s GRp, the y1 FS0Žp R.Ž.-module N p is not finitely generated. Proof. First of all, note that FR is Zariskian. It follows that every good filtration FM on M is separated. Hence if N / 0 then N FMn for some n g ޚ. The proof ofŽ. 1 , resp. Ž. 2 , is the same as the proof ofw LVO1, Lemma 4.6Ž. 1, 2x . 696 LI HUISHI

Ž. ŽŽ .. ŽŽ .. Ž. 3 Since Min JN sMin Jn00for some n G 0 it follows from 1 Ž Ž .. Ž Ž .. Ž. that Min JN /л. Take p00g Min JN and put p s GRp. Sup- y1 pose that NpŽ.is a finitely generated FS0Žp R.Ž-module, then Qn,NpŽ.. s0 for some nЈ G 0 and all n G nЈ. Therefore, for all n G nЈ, TQny1 Ž.,N 0 by Lemma 3.5. However, JnŽ.Ž.JN p for all n 0. p 0 s : : 0 G Hence InŽ. p and this yields TQny1 Ž.,N/0, a contradiction. There- : 0 p 0 fore NpŽ.is not finitely generated. Now let us turn to holonomic modules. Recall fromwx VE2 that a filtered Ž. R-module M with good filtration FM is said to be holonomic if ht p s ␮R for all p g MinŽŽJM ..or else if M s 0. Clearly, if there is a nonzero holonomic module then ␮R is finite because GRŽ.is Noetherian.

3.7. COROLLARY. Let M be a holonomic R-module and N an F0 R-sub- ŽŽ .. module of M that is not finitely generated. If p0 g Min J N then p s Ž. ŽŽ.. GRp0gMin JM . Proof. By Lemma 3.6Ž. 1 , Qn Ž,Np Ž ../0 for all n G 0, hence y1 Ž. Žy1 .Ž. TGMppsGS M /0 and thus JM :pand pЈ : p for some ŽŽ .. pЈgMin JM . The assumption ht pЈ s ␮RRthen leads to ht p G ␮ and Ž. Ž . it follows from Lemma 3.6 2 that p s pЈ because ␮R is finite! .

3.8. PROPOSITION. Let M be holonomic. The following statements are equi¨alent:

Ž.1 N is a finitely generated F0 R-submodule of M; Ž. Ž . Ž y1 . y1 2 N p is a finitely generated F0 Spp R -submodule of S M for e¨ery pgMinŽŽJM ... Proof. This follows immediately from Lemma 3.5Ž. 1 , Lemma 3.6, and Corollary 3.7.

3.9. THEOREM. Let M be a holonomic R-module with good filtration FM. The following statements are equi¨alent: Ž.1 M has R.S. as an R-module; Ž. y1 y1 ŽŽ .. 2Spp M has R.S. as a S R-module for all p g Min JM ; Ž. y1 y1 gŽŽ.. 3Spp M has R.S. as a S R-module for all p g Spec GR . Proof. It is sufficient to proveŽ. 2 « Ž.1 . By Proposition 3.3. we need to i show that N s Ýi FR00␶mis a finitely generated FR-module for every mgM,␶gGŽŽJM ... But by Proposition 3.8 we only have to show that Ž. Žy1 .ŽŽ.. Np is a finitely generated FS0 p R-module for every p g Min JM . Ž. ŽŽy1 .Ž.Žy1.Ž.i Ž. Since ␸␶ppgGJS M and NpsÝiFS0ppp R␸␶␸mit follows again from Proposition 3.3 that NpŽ.is indeed a finitely generated y1 FS0Ž.p R-module. LIFTING ORE SETS 697

Finally, using the same technique as that inwx LVO1Ž i.e., a dictionary between strongly filtered rings and filtered rings. and replacing microlocalizations QR␮Ž.by localizations SRy1 in VLO1, Sections 5, 6 we can Spp wx announce the main result of this section as follows.

3.10. THEOREM. Let R be a left Zariskian filtering ring with filtration FR Ž. such that G R is a commutatië ޑ-algebra and, moreoër, ␮RRs ␯ , where Ä ŽŽ.. Ž.4 ␯R ssup ht p, p g Spec GR , pisinölutiëinGR.Let M be a holonomic R-module with good filtration FM. Then the following statements are equiälent: Ž.1 M has R.S. as an R-module; Ž. y1 y1 ŽŽ .. 2Spp M has R.S. as a S R-module for all p g Min JM ; Ž. y1 y1 gŽŽ.. 3Spp M has R.S. as a S R-module for all p g Spec GR .

4. LOCAL DESCRIPTION OF DIMENSION THEORY AND AUSLANDER REGULARITYŽ. GORENSTEIN PROPERTY

Inwx Sao some microlocal-global results concerning various dimensions and the Auslander regularityŽ. Gorenstein property of a Zariskian filtered ring R with commutative associated graded ring GRŽ.have been obtained. By using the lifting Ore sets we show in this section that under the same assumptions as those inwx Sao there exist local-global descriptions for various dimensions and the Auslander regularityŽ. Gorenstein property of R. Our approach mainly depends on the faithful flatness of the R-module y1 Ž. [SRp Lemma 4.1 . Throughout this section R is a left and right Zariskian filtered ring with g filtration FR such that GRŽ.is commutati¨e. Let Spec ŽŽ..GR be the set g of all graded prime ideals of GRŽ.. For every p g Spec ŽŽ..GR it follows from Section 2 that we may obtain the saturated Ore set Sp corresponding Ž. to the Ore set Tp consisting of homogeneous elements of GRyp. Clearly, Sp is a left and right Ore set of R because GRŽ.is commutative.

y1 EMMA g 4.1. L . With notations as abo¨e, then [pgSpec ŽGŽ R.. Sp Risa faithfully flat left and right R-module. If we replace Spec g ŽŽ..G R by the set of all maximal graded ideals of GŽ. R then the same result holds.

y1 Proof. Obviously [ppS R is a flat R-module. To prove the faithful flatness let M be any finitely generated R-module. Take a good filtration FM on M, then FM is separated because R is Zariskian. Now suppose Ž y1 . y1 y1 that [ppSRm RMs0. Then SRpm RMsSM p s0 for all p g gŽ Ž .. Ž y1 . y1 Ž. Spec GR . Hence by Corollary 2.5 we have GSpp M sTGMs0. Since GMŽ.is a graded GRŽ.-module and this holds for all p g 698 LI HUISHI

g Spec ŽŽ..GR it follows that GMŽ.s0. Consequently the separability of y1 FM yields that M s 0. This proves the faithful flatness of [ppSR.

Suppose that S is a left and right Ore set of R, then it is well known 1 1 that Sy R s RSy . Since R is Noetherian, if N is a finitely generated n Ž. y1 R-module then it is also well known that Ext RRN, R m SR( nŽy1 y1. Ext SRy1SRmN,SRfor every n G 0.

4.2. THEOREM. Let the notations be as abo¨e.

Ž.1 If inj.dim R denotes the injecti¨e dimension of R then inj.dim R s Ä y1 g ŽŽ..4Ä y1 sup inj.dim SRp ,pgSpec GR ssup inj.dim SRM ,Mmaximal in Spec g ŽŽ..GR 4. Ž.2 If R has finite global dimension gl.dim R then gl.dim R s Ä y1 g ŽŽ..4Ä y1 sup gl.dim SRp ,pgSpec GR ssup gl.dim SRM ,Mmaximal in Spec g ŽŽ..GR 4. Ž.3 If G Ž R . is a gr-semilocal ring, i.e., it has only finite number of Ä y1 maximal graded ideals, then K.dim R s sup K.dim SRM ,Mmaximal in g ŽŽ..4Ä y1 gŽŽ..4 Spec GR ssup K.dim SRp ,pgSpec G R where K.dim Rde- notes the Krull dimension of R in the sense of Gabriel᎐Rentschler.

Proof. Ž.1 We only need to show the inequality inj.dim R F Ä y1 g ŽŽ..4 sup inj.dim SRp ,pgSpec GR . If the latter is infinite then there is nothing to prove. Suppose that the latter is finite, say ␮. Then for any y1 ␮q1Ž. finitely generated R-module M we have SpRExt M, R s ␮q1y1y1 g Ext y1 ŽSM,SR.0 for all p Spec ŽŽ..GR and hence Lemma 4.1 SRp p p s g ␮q1Ž. entails Ext R M, R s 0. This proves that inj.dim R F ␮. Ž.2 Again we only need to show the inequality gl.dim R F Ä y1 g ŽŽ..4 sup gl.dim SRp ,pgSpec GR . Suppose that gl.dim R s ␮ is finite. ␮ Then there is a finitely generated R-module M such that Ext R Ž.M, R / 0. g But then by Lemma 4.1 there must be some p g Spec ŽŽ..GR such that y1 ␮ ␮ y1 y1 SExt Ž.M, R Ext y1ŽSM,SR./0 as desired. pR s SRppp Ž. Ž . y1 3 Since GR is gr-semilocal the direct sum [SRM is a finite sum where M ranges over all maximal graded ideals of GRŽ.. Now the equality Ž y1 . follows from Lemma 4.1 and the well known fact that K.dim [SRM s Ä y1 gŽŽ..4 sup K.dim SRM ,Mmaximal in Spec GR .

Recall that a left and right Noetherian ring R is said to be an Auslander᎐Gorenstein Ž.Auslander regular ring if R has finite injective dimensionŽ. global dimension , say ␮, and satisfies the Auslander condition, i.e., for any finitely generated R-module M, any 0 F k F ␮, and any k Ž. nŽ. submodule N : Ext RRM, R we have Ext N, R s 0 where n - k. LIFTING ORE SETS 699

4.3. THEOREM. Let the notations be as abo¨e.

y1 Ž.1 R is Auslander᎐Gorenstein if and only if Sp R is Auslander᎐ g ŽŽ.. y1 Gorenstein for e¨ery p g Spec GR , if and only SM R is g Auslander᎐Gorenstein for all maximal graded M g Spec ŽŽ..GR . Ž.2 Suppose that R has finite global dimension, then R is Auslander y1 g ŽŽ.. regular if and only if Sp R is Auslander regular for e¨ery p g Spec GR ,if y1 and only if S M R is Auslander regular for all maximal graded M g Spec g ŽŽ..GR .

Proof. By Theorem 4.2 we only need to prove that R satisfies the y1 Auslander condition if and only if Sp R satisfies the Auslander condition g for every p g Spec ŽŽ..GR . y1 Suppose that R satisfies the Auslander condition. Let N be any Sp R- y1 k k y1 y1 submodule of S Ext Ž.M, R Ext y1ŽSM,SR.where M is a pR s SRppP finitely generated R-module, then there is an R-submodule NЈ of k Ž. y1 Ext RpM, R such that N s SNЈ. Hence for n - k we have 0 s y1nn y1 y1 n y1 SExt Ž.NЈ, R Ext y1ŽSNЈ,SR.Ext y1ŽN, SR.. p R s SRpp p p s SR p y1 Conversely, suppose that Sp R satisfies the Auslander condition for g every p g Spec ŽŽ..GR .Let N be any R-submodule k y1 of Ext RŽ.M, R where M is a finitely generated R-module. Since SNp is a y1 y1 k k y1 y1 SR-submodule of S Ext Ž.M, R Ext y1ŽSM,SR., for n - k ppRSs pRpp y1nn y1 y1 we have S Ext Ž.N, R Ext y1ŽSN,SR.0forall p pR s SRppp s g g n Spec ŽŽ..GR . It follows from Lemma 4.1 that Ext Ž.N, R s 0 as wanted.

5. LIFTING STRUCTURE SHEAVES

Let R denote a filtered ring R with filtration FR such that its Rees ring R˜ is left Noetherian and GRŽ.is commutatië and positiëly graded. Write Ž. Ä Ž. Ž.4 YsProj GRs p, graded prime ideal of GR, prGRq where Ž. Ž. GRqs[n) 0GRn. Our aim in this section is to lift the classical Ž. gg& graded structure sheaves OY and OY over Y to sheaves OY, FgŽ.Ž.& F OY ,OY 0 ,OY 0 on Rãnd R respectively such that the obtained sheaves have nice stalks at p g Y. As in the foregoing sections for each p g Y let Tp be the multiplicatively closed subset of GRŽ.consisting of homogeneous elements of Ž. GRyp. It follows from Section 1 that we have the Ore set S˜˜p in R, y1 resp. the Ore set Sppin R, and the corresponding localization SR˜ ˜, resp. y1 Sp R. Let us first show that the stalks of the sheaves we are going to construct are nice rings. 700 LI HUISHI

5.1. LEMMA. With notation as in Section 1, we ha¨e y1 y1 Ž.1 S˜pp R˜˜ is a graded local ring, henceŽ. S R˜0 is a local ring; y1 y1 Ž.2 F0 Ž Spp R. is a local ring, and S R is a left Zariskian filtered ring. Ž. y1y1 Ž.y1 Proof. 1 From Section 1 we know that SR˜pp˜˜rXS R˜s GS p R s y1 y1 TGRp Ž.. But the latter is a graded local ring, hence SR˜p ˜is a graded y1 local ring and Ž.SR˜p ˜ 0 is a local ring. Ž. Ž y1 .Žy1.Žy1.Žy1Ž.. 2 Since FS0 p RrFSRy1 ppsGS R0s TGRp0it follows y1 y1 that FS0Žpp R.is a local ring. The fact that SRis a left Zariskian filtered ring follows from Lemma 2.8. Now we proceed to construct our sheaves as mentioned in the beginning of this section.

Ž. &g i The construction of OY Ž.&g For every open subset U g Y, put ⌫ U, OY to be the set of all functions ␽ ˜y1 ˜ ␽ Ž. y1 ˜ ␽ from U to @ pgUpSRsuch that p g SRp and is locally a quotient of elements of R˜, i.e., for every p g U there exists an open neighborhood V of p, V : U, and elements a, b g R˜ such that for each Ž. y1 qgV,bgS˜˜qqand ␽ q s arb in SR˜. For open subsets V : U in Y UgŽ.Ž&& g . we define ␳ V : ⌫ U, OY ª ⌫ V, OY to be the natural restriction map. 5.2. PROPOSITION. Let the notations be as aboë. Ž. Ž&g . 1 ⌫U,OY is a ring with identity 1 where the sums and products are the U usual sums and products of functions, ␳ V is a ring homomorphism. Ž.&g Ž . 2 OY is a sheaf of noncommutatië rings. & Ž. g y1 3 For eëry p g Y, the stalk OY, p is isomorphic to S˜˜p R. Proof. Ž.1 Let p g Y and let V and W be two open subsets of Y Ž.&g containing p. Suppose ␽12, ␽ g ⌫ U, OYand ␽12s arf on V and ␽ s brg y1 y1 on W, where arf g SR˜qq˜˜for all q g V and brg g SR˜for all q g W.If arfqbrgssa12qsbruwith u s sf12ssggS˜p, then it is easily seen that s12, s g S˜p. Hence ␽1212q ␽ s saqsbruon the open subset V l W Ž Ž .. Ž Ž .. Ž Ž .. Ž Ž .. l D ␴ f l D ␴ g l D ␴ s12l D ␴ s that is obviously a neighborhood of p, where DŽŽ..␴ f consists of the primes in Y not containing

␴Ž.f. Similarly we may prove that ␽␽12is is locally a quotient of elements of R˜. Ž.2 This may be verified directly. Ž.3 As in the commutative case we construct the isomorphism ␺ from &&& gy1 ggŽ. OY,pto SR˜˜p as follows: for every germ ␽ g OY, p, where ␽ g ⌫ U, OY ,we put ␺␽Ž.s␽ Ž.p. Then by the definition of ␽ it is easily seen that ␺ is LIFTING ORE SETS 701

y1 well defined. Since any homogeneous element of SR˜p ˜ can be represented as a quotient arf where a and f are homogeneous elements of R˜ and ŽŽ.. fgS˜p, the set D ␴ f is an open neighborhood of p, and arf defines a &g ŽŽ.. section of OY over D ␴ f whose value at p is the given element. To Ž.&g prove ␺ is injective, let U be an open neighborhood of p, ␽ , ␩ g ⌫ U, OY having the same value ␽ Ž.p s ␩ Ž.p at p. By shrinking U if necessary, we may assume that ␽ s rrf and ␩ s brg on U, where a, b, f, g are homogeneous elements of R˜˜and f, g g Sp. Since arf and brg have the same y1 image in SR˜p ˜˜, it follows that there are t12, t g Spsuch that ta12stb, y1 tf12stgin R˜˜. Therefore arf s brg in every SRq˜such that f, g, t12, t g Ž Ž .. Ž Ž .. Ž Ž .. S˜q. But the set of such q is the open set D ␴ f l D ␴ g l D ␴ t1 ŽŽ.. lD␴t2 , which contains p. Hence ␽ s ␩ in a whole neighborhood of p, so they have the same stalk at p. This proves that ␺ is an isomorphism.

Ž. F ii The construction of OY Ž F . For every open subset U g Y, put ⌫ U, OY to be the set of all ␽ y1 ␽ Ž. y1 ␽ functions from U to @ pgUpSRsuch that p g SRp and is locally a quotient of elements of R, i.e., for every p g U there exists an open neighborhood of p, V : U, and elements a, b g R such that for Ž. y1 each q g V, b g Sqqand ␽ q s arb in SR. For open subsets V : U in UŽF.ŽF. Ywe define ␳ V : ⌫ U, OY ª ⌫ V, OY to be the natural restriction map. Using Proposition 1.3, a similar argument as that in the proof of Proposi- tion 4.2 yields the following

5.3. PROPOSITION. Let the notations be as abo¨e. Ž. Ž F. 1 ⌫U,OY is a ring with identity 1 where the sums and products are the U usual sums and products of functions, and ␳ V is a ring homomorphism. Ž. F Ž. 2OY is a sheaf of noncommutati¨e rings. Ž.3 For e ery p Y, the stalk O F is isomorphic to Sy1 R. ¨ g Y, p p

Ž. Ž.&g iii The Construction of OY 0 ŽŽ..&g For every open subset U g Y, put ⌫ U, OY 0 to be the set of all ␽ Ž.˜y1 ˜˜␽ Ž.Ž.y1˜␽ functions from U to @ pgUpSR0such that p g SRp 0and is locally a quotient of elements of R˜, i.e., for every p g U there exists an open neighborhood V of p, V : U, and homogeneous elements a, b with the same degree in R˜˜such that for each q V, b S and ␽ Ž.q a b g g q &s r Ž.˜˜y1 UgŽŽ.. in SRq 0. For open subsets V : U in Y we define ␳ V : ⌫ U, OY 0 ª ŽŽ..&g ⌫V, OY 0 to be the natural restriction map. By an argument similar to that in the proof of Proposition 4.2 we have the following 702 LI HUISHI

5.4. PROPOSITION. Let the notations be as aboë. Ž. Ž Ž&g .. 1 ⌫U,OY 0 is a ring with identity 1 where the sums and products are U the usual sums and products of functions, and ␳ V is a ring homomorphism. Ž.Ž&g . Ž . 2 OY 0 is a sheaf of noncommutatië rings. & Ž. Ž.g Ž.y1 3 For eëry p g Y, the stalk OY 0, ppis isomorphic to S˜˜ R 0.

Ž. Ž F. iv The Construction of OY 0 ŽŽF.. For every open subset U g Y, put ⌫ U, OY 0 to be the set of all ␽ Ž y1 .Ž.Ž␽ y1.␽ functions from U to @ pgU FS0 p Rsuch that p g FS0 p Rand is locally a quotient of elements of R, i.e., for every p g U there exists an open neighborhood V of p, V : U, and elements a, b g R such that for Ž. Žy1 . each q g V, b g Sq and ␽ q s arb in FS0 q R. For open subsets U ŽŽF.. Ž Ž F.. V:Uin Y we define ␳ V : ⌫ U, OY 0 ª ⌫ V, OY 0 to be the natural restriction map. Again, by an argument similar to that before we may obtain the following

5.5. PROPOSITION. Let the notations be as aboë. Ž. Ž Ž F.. 1 ⌫U, OY 0 is a ring with identity 1 where the sums and products are U the usual sums and products of functions, and ␳ V is a ring homomorphism. Ž.Ž F.. Ž . 2 OY 0 is a sheaf of noncommutatië rings. Ž. Ž F.Žy1. 3for eëry p g Y, the stalk OY 0, p is isomorphic to F0 Sp R .

ACKNOWLEDGMENTS

The author thanks the referee for his helpful suggestions and comments.

REFERENCES wxAVVO M. J. Asensio, M. Van den Bergh, and F. Van Oystaeyen, A new algebraic approach to microlocalization of filtered rings, Trans. Amer. Math. Soc. 316, No. 2Ž. 1989 , 537᎐555. wxGa O. Gabber, The integrability of characteristic varieties, Amer. J. Math. 103 Ž.1981 , 445᎐468. wxGin V. Ginsburg, Characteristic varieties and vanishing cycles, In¨ent. Math. 84 Ž.1986 , 327᎐402. wxGQS V. Guillemin, D. Quillen, and S. Sternberg, The integrability of characteristics, Comm. Pure Appl. Math. 23 Ž.1970 , 39᎐77. wxGW K. R. Goodear and R. B. Warfield, ‘‘An Introduction to Noncommutative Noethe- rian Rings,’’ London Math. Soc. Student Texts, Vol. 16, Cambridge Univ. Press, Cambridge, 1989. LIFTING ORE SETS 703 wxKKS M. Kashiwara, T. Kawai, and M. Sato, Microfunctions and pseudo-differential equations, Lecture Notes in Math. 287 Ž.1973 , 264᎐529. wxLBVO L. Le Bruyn and F. Van Oystaeyen, Quantum sections and gauge algebras, Publ. Mat. Uni¨. Autonoma´ de Barcelona 36 Ž.1992 , 693᎐714. wxLVO1 H. Li and F. Van Oystaeyen, Strongly filtered rings, applied to Gabber’s integrability theorem and modules with regular singularities, in ‘‘Proc. Sem. Algebra Dubreil-Malliavin,’’ Lecture Notes in Mathematics, Vol. 1404, pp. 296᎐321, Springer-Verlag, New YorkrBerlin, 1988. wxLVO2 H. Li and F. Van Oystaeyen, Zariskian filtrations, Comm. Algebra 17 Ž.1989 , 2945᎐2970. wxLVO3 H. Li and F. Van Oystaeyen, Dehomogenization of gradings to Zariskian filtrations and applications to invertible ideals, Proc. Amer. Math. Soc. 115, No. 1Ž. 1992 , 1᎐11. wxLVOW H. Li, F. Van Oystaeyen, and E. Wexler-Kreindler, Zariski rings and flatness of completion, J. Algebra 138 Ž.1991 , 327᎐339. wxNVO C. Nastasescuˇˇ and F. Van Oystaeyen, ‘‘Graded Ring Theory,’’ Math. Library, Vol. 28, North-Holland, Amsterdam, 1982. wxRot J. J. Rotman, ‘‘An Introduction to Homological Algebra,’’ Academic Press, New York, 1979. wxSao M. Saorin, Microlocal properties of filtered rings, J. Algebra 140 Ž.1991 , 141᎐159. wxSpr T. Springer, Microlocalization algebrique,´ in ‘‘Proc. Sem. Algebra Dubreil᎐ Malliavin,’’ Lecture Notes in Mathematics, Vol. 1146, pp. 299᎐316, Springer-Verlag, New YorkrBerlin, 1983. wxSVO R. Sallam and F. Van Oystaeyen, A microstructure sheaf and quantum section over a projective scheme, J. Algebra 158 Ž.1993 , 201᎐225. wxVE1 A. Van den Essen, Algebraic microlocalization, Comm. Algebra 14 Ž.1986 , 971᎐1000. wxVE2 A. Van den Essen, Modules with regular singularities over filtered rings, in ‘‘Publications Research Institute for Mathematical Science,’’ Vol. 22, No. 5, pp. 1᎐32, Kyoto Univ., Kyoto, 1986. wxWK E. Wexler-Kreindler, Microlocalization platitude et theorie de torsion, Comm. Algebra 16 Ž.1988 , 1852᎐1913.