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Lifting Ore Sets of Noetherian Filtered Rings and Applications JOURNAL OF ALGEBRA 179, 686]703Ž. 1996 ARTICLE NO. 0031 Lifting Ore Sets of Noetherian Filtered Rings and Applications 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 RÄ 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 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. Q 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 S is a multiplicatively closed subset of R such that the corresponding subset s Ž.S of homogeneous elements in the associated graded ring GRŽ.is a left Ore set, then S is generally not a left Ore set of R and hence one cannot localize R at S. Nevertheless, one may construct m a complete filtered ring QRS Ž.which is called the microlocalization of R at S such that there exists aŽ. canonical filtered ring homomorphism mŽ. Ž.Ž. mŽ. w:RªQRS with the properties: 1 w s is a unit of QRS for every sgS;2ifŽ. f:RªAis a filtered ring homomorphism, where A is complete and fsŽ.is a unit of A for each s g S, then there exists a mŽ. unique filtered ring homomorphism g: QRS ªAsuch that g (w s f. Ä Ž. Ž .4ŽŽ. Moreover, if Ssat s r g R, s r g s S see the definition of s r in Section 2. then QRm Ž. QRmŽ.Žcf. Sao. , and if S is a left Ore set of SSsat s wx *Supported by a grant from the Education Committee of P.R. China. 686 0021-8693r96 $12.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. LIFTING ORE SETS 687 $ mŽ. y1 y1 Rthen QRS sSRwhere the latter is the completion of S R with respect to the localized filtration on Sy1 R Žcf.wx AVVO. However, in order to obtain some localization-like properties of microlo- mŽ. calizations, such as the important exactness of the functor QRS 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 RÄ of R Žalthough the S-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 RÄ 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 RÄ, resp. in R. m Consequently the microlocalization QRS Ž.of R at a multiplicatively closed subset S 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 G .Ž F . and OY over Y s Proj GR to sheaves OY, OY , OY 0 , OY 0 on RÄand 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 ,ngZ consisting of additive subgroups of R such that R s D ng Z FRn ,1gFR0 ,FRnn:FRq1and FRFRnm:FR nqmfor all n, m g Z. By a filtered R-module we mean an R-module with a filtration FM s Ä4 FMn , ngZ consisting of additive subgroups of M such that M s Z D ng Z 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 Ä Ž. Rees ring R s[ngZFRn and the associated graded ring GRs [n Z FRnnrFR1. For any M g R-filt with filtration FM the Rees mod- g ÄÄy ule M s[ng Z FMn , which is a graded R-module, and the associated Ž. Ž. graded module GM s[ng Z FMnnrFMy1, which is a graded GR-mod- ule, 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 RÄ1, then we have R ( RÄÄrŽ.1 y XRas filtered rings where the latter has the dehomogenized filtration, GRŽ.(RÄÄrXR as graded rings where the latter 1 1 has the quotient gradation. Moreover, RXÄwxwy (Rt,ty xas graded rings where the latter has the natural gradation and RXÄwxy1 is the localization of RÄ atÄ4 1, X, X 2, . For any filtered R-module M with filtration FM we also have M ( MÄÄrŽ.1 y XMas filtered modules where the latter has the dehomogenized filtration GMŽ.(MÄÄrXM as graded modules where 1 1 the latter has the quotient gradation. Moreover, MXÄwxy (Mt w,ty xas graded modules where the latter has the natural gradation and MXÄwxy1 is the localization of MÄ atÄ4 1, X, X 2, . A filtration FM on an R-module M is said to be good if there are k1,...,ks gZ and j 1,...,js gM such that FM Ýs FRjfor all n Z. FM is good if and only if MÄ is a nis s1nykii g finitely generated RÄ-module. Finally we recall that a filtered ring R with filtration FR is called a left Ä 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.ÄÄof R which is the largest proper graded ideal of RÄÄsuch that its intersection with R0 is contained in the Jacobson radical in RÄ0 .. 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 RÄ 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. ÄÄŽ Let R be a filtered ring with filtration FR. We write p 1: R ª Rr 1 y .ŽÄÄÄ. XRsR,p2 :RªRrXR s GR for the natural homomorphisms, and we define the s-function on a filtered R-module M as usual: s Ž .s Ž. s Ž. :MªGM, msmqFMny1 if m g FMnnyFMy1and m s 0if mgFngZFMn . 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 ÄÄ Ss½5ÄÄrgR,rhomogeneous, p 2Ž.Är g T is a multiplicatively closed subset of RÄconsisting of nonzero homogeneous Ž. elements and p 2 SÄ s T. Moreover, one also easily sees that the saturated subset S s Ä4r g R, s Ž.r g T is a multiplicatively closed subset of R consisting of nonzero elements and Ž. p1SÄ sS. 2.1. LEMMA. Suppose that the ideal XRÄ has the Artin]Rees property for graded left ideals and RÄ satisfies the ACC on left annihilators. If T is a left Ore set in GŽ. R consisting of homogeneous elements then SÄ is a left Ore set in RÄ and S is a left Ore set in R. Proof. It is sufficient to prove that SÄÄis a left Ore set of R. Bywx NVO we only have to check the Ore conditions for homogeneous elements.
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