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Higher Derivations and Local Cohomology Modules*

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Higher Derivations and Local Cohomology Modules* JOURNAL OF ALGEBRA 201, 363]372Ž. 1998 ARTICLE NO. JA977245 Higher Derivations and Local Cohomology Modules* Caijun Zhou² Department of Mathematics, Shanghai Normal Uni¨ersity, Shanghai, 200234, People's Republic of China Communicated by D. A. Buchsbaum View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received August 2, 1996 provided by Elsevier - Publisher Connector 1. INTRODUCTION Throughout this paper R is a commutative noetherian ring with unit. If i Mis an R-module and I an ideal of R, we denote by HMI Ž.the ith local cohomology module of M with support in VIŽ.sÄPP< >I,PgSpec A4. The local cohomology modules have been widely studied by a number of authors. Yet despite much effort, the structure of these modules is still full of mystery. In recent years, some important properties of the local coho- i mology module HRI Ž.were established when R is a regular ring contain- ing a field. Let us recall some of these achievements in the following. If R is a regular ring containing a field K and I an ideal of R, then the local cohomology modules of R have the following properties: j i Ž.1 HHRmIŽ Ž..is injective, where m is any maximal ideal of R; i i Ž.2idRIHRŽ.Fdim HRIŽ.; i Ž.3 The set of the associated primes of HRI Ž.is finite; i Ž.4 All the Bass numbers of HRI Ž.are finite. i i i Here id RIHRŽ.stands for the injective dimension of HRIŽ., dim HRIŽ. i stands for the dimension of the support of HRI Ž.in Spec R, and the jth i Bass number of HRI Ž.with respect to a prime P of R is defined as j i lengthKRŽ Ext ,pIPŽŽK, HRŽ . ..., where K is the fraction field of RrP wxBa . The above results were established by Huneke and Sharpwx Hu-Sh when Kis of positive characteristic p and by Lyubeznikwx Ly when K is of characteristic zero. The method of Lyubeznik is completely different from * Partially supported by the NSF of China and STF of Shanghai Higher Education. ² E-mail: [email protected]. 363 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. 364 CAIJUN ZHOU that of Huneke and Sharp. While Huneke and Sharp use the Frobenius functor, Lyubeznik uses the theory of D-modules developed inwx Bj . In this paper, we will use higher derivations to study the local cohomol- ogy modules of a unramified regular ring, especially the local cohomology modules of the formal power series ring KXww 12,X,..., Xnxx over either a field or a complete p-ring K. We can give a uniform proof for resultsŽ. 1 andŽ. 2 cited above. The treatment of the paper should be thought as a natural extension of that of Lyubeznikwx Ly . One important thing of our method is that it gives one an insight to the study of the local cohomology modules of unramified regular local rings containing no field. In the paper, we can prove that if R is a unramified regular local ring containing no field and m is its unique maximal ideal, then i j Ž.iidRmHHRŽ IŽ..F1 for all ideal I and for all i, j; i i Ž.ii id RIHRŽ.Fdim HRIŽ.q1 for all ideal I and for all i. 2. HIGHER DERIVATIONS It is well known that, in the context of positive characteristic p, the concept of derivation, thus the D-modules, has less interest, and a more meaningful concept is that of higher derivation. The theory of higher derivation was initiated by Hasse and Schmidtwx Ha-Sc . In this section, we recall a few standard facts about higher derivations and give some special examples needed in the rest of the paper. Let K be a subring of the ring R. The notation DerK Ž.R stands for the set of K-derivations of R. A higher K-derivation from R to R is a sequence D s Ž.D012, D , D ,... of K-linear maps such that D0is the identity map of R and for all n G 1, x, y g R DxyniŽ .sÝDxDy Ž.j Ž.. iqjsn These conditions are equivalent to saying that the map Et: R ª Rtww xx Ž t an indeterminate. defined by ` i ExtiŽ.sÝDxt Ž. is0 is a K-algebra homomorphism with ExtŽ.sxmod Ž.t . We will use HDerK Ž.R to denote the set of all higher K-derivations of R. Remark 2.1.Ž. i Let D be a higher K-derivation of R. Then D1 g DerK Ž.R . LOCAL COHOMOLOGY MODULES 365 Ž.ii Suppose R contains a field of characteristic zero, and d g 2 n DerK Ž.R . Then D s Ž1, d, d r2!,...,d rn!, . .. is a higher K-derivation of R. Ž.iii Suppose R is a ring of prime characteristic p. Let d g DerK Ž.R , DgHDerK Ž.R . Then for x g R and r an positive integer r ppr p dxŽ.s0, Dxpr Ž .sŽ.Dx1 Ž., and pr r DxnŽ.s0 for all 0 - n - p . Ž.iv Let S be a multiplicative subset of R. Then every K-derivation Ž.resp. higher K-derivation of R extends uniquely to the ring RS. EXAMPLES 2.2.Ž. a Let R s KXwx12,X,..., Xnbe the ring of polynomi- als in n variables over a field K of characteristic p ) 0. Byw Mat2, Theorem 30.2 and 30.3x , one can easily see that R is 0-smooth over K. Thus for any d g DerKKŽ.R there exists a D g HDer Ž.R such that D1 sdbywx Mat2, Theorem 27.1 . Ž.b Let R s KXww12,X,..., Xn xx be the ring of formal power series in n variables over a field K. The K-linear derivations form a free R-module on the n generators d12, d ,...,dnj, where d : R ª R is the partial differential with respect to Xjij, ddsdd ji, and dr iyrd is ­ rr­ Xi n for all i, j and all r g R.IfchKs0, thenŽ 1, dii,...,d rn!, . .. is a higher K-derivations on R.IfchKsp)0, it is known that R is not always 0-smooth over K. In fact, Tanimotowx Tan proved that R is 0-smooth over p K only when ch K s p, and w K : K x is finite. Thus we cannot use Theo- rem 27.1 inwx Mat2 to assert that there exists a higher K-derivation Di such that Ž.Di 1 s di for each i Ž.1 F i F n . However, R is the completion of the polynomial ring S s KXwx12,X,..., Xnwith respect to the ideal m s Ž.X12,X,..., Xniof S, and each d is the natural extension of the partial X X differential diiwith respect to X of S. Let D be a higher K-derivation on XX Xs syk Ssuch that D1 s d . It is easy to check that DmkŽ.;m for all s G k. Hence DX can be extended naturally as a higher K-derivation D on R.In particular, we have proved that, for each di Ž.1 F i F n there exists a higher K-derivation DiKg HDer Ž.R such that ŽDi .1s di. The higher derivations given in Examples 2.2Ž. b will play an important role in the rest of the paper. In fact, we need the following generalization of Examples 2.2Ž. b to treat the case of unramified regular local ring containing no field. 366 CAIJUN ZHOU PROPOSITION 2.3. Let R s Kww X xx be the ring of formal power series in one ¨ariables o¨er a ring K; let d : R ª R be the partial differential with respect to X. Then there exists a higher K-deri¨ation D such that D1 s d. Proof. Set D0 to be the identity map on R. For each positive integer n, Ž.Ž.s s syn Ž define Dnn: R ª R as DX snX for all s here, we use the 0 Ž.s . Ž. convention X s 1, and note that n s 0 for s - n , Dan s0 for agK, and Dn is K-linear on R. We assert that D s Ž.D012, D , D ,... isa higher K-derivation on R. In fact, for any two elements x, y g R, we can write n n x s a01q aXq??? qaXnq??? , y s b01q bXq??? qbXnq??? , n where aij, b g K for all i, j.So xy s c01q cXq??? qcXnq??? , where cnisÝqjsnijab. Thus for each positive integer n we have `` DxyŽ.cD Ž Xkk . cXkyn. nksÝÝns kž/n ks0 ks0 On the other hand, observe that `` rryissyj ÝÝÝÝDxDyijŽ. Ž.s aXr bXs ž/i ž/j ijnijnrž/0 ž/s0 qs qss s `` rs rqsyn sÝÝabrs Ý X ž/i ž/j rs0ss0 iqjsn ` cXk kyn sÝkž/n ks0 Ž.Ž.rrs Žqs . because Ýiqjsn inj s . Hence we have proved DxyniŽ .sÝDxDy Ž.j Ž. iqjsn and thus D is the desired higher K-derivation. The following result is an immediate consequence of Proposition 2.3 because of R s KXw12,X,..., Xn xxsKX ww12,X,..., XXny1 xxwwn xx. COROLLARY 2.4. Let R s KXww 12,X,..., Xnxx be a ring of formal power series in n ¨ariables o¨er a ring K. Let di : R ª R be the partial differential with respect to Xi Ž.1 F i F n .
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