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THE F-DEPTH of an IDEAL on a MODULE 1. Introduction Let (R, M) PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 7, Pages 1905{1912 S 0002-9939(01)06269-4 Article electronically published on December 27, 2001 THE F-DEPTH OF AN IDEAL ON A MODULE RENCAI LU¨ AND ZHONGMING TANG (Communicated by Wolmer V. Vasconcelos) Abstract. Let I be an ideal of a Noetherian local ring R and M afinitely generated R-module. The f-depth of I on M is the least integer r such that r the local cohomology module HI (M) is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules. 1. Introduction Let (R; m) be a (commutative) Noetherian local ring, I a proper ideal of R and M a finitely generated R-module. It is well known that the depth, depth(I;M), of I on M, i.e., the length of a maximal M-regular sequence in I, is the least integer r 6 r such that the local cohomology module HI (M) = 0. Faltings [2] proved that the r least integer r such that HI (M) is not finitely generated is minfdepth(Mp) + ht((I + p)=p)jp 6⊇ Ig: r Now we consider the problem of what is the least integer r such that HI (M)is not Artinian. In [4], Melkersson showed that, when Supp(M=IM) 6⊆ fmg,theleast r integer r such that HI (M) is not Artinian is minfdepth(IRp;Mp)jp 2 Supp(M=IM) nfmgg; and called this integer the f-depth (filter depth) of I on M. In this paper, for any proper ideal I, we define the f-depth of I on M as the length of a maximal M-filter regular sequence in I. Then it turns out that our f-depth of I on M is the least r integer r such that HI (M) is not Artinian for any proper ideal I and, in the case Supp(M=IM) 6⊆ fmg, our f-depth coincides with the one of Melkersson. After summarizing some results about filter regular sequences in section 2, we define f-depth in section 3. Then, we characterize f-depth by Ext, Koszul complexes and local cohomology modules. Section 4 contains an equivalent condition using f-depth for an R-module to be an f-module which is similar to a Cohen-Macaulay module. Received by the editors July 26, 2000 and, in revised form, January 16, 2001. 2000 Mathematics Subject Classification. Primary 13C15, 13D45, 14B15. Key words and phrases. f-depth, f-modules, local cohomology modules. This work was supported by the National Natural Science Foundation of China. c 2001 American Mathematical Society 1905 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1906 RENCAI LU¨ AND ZHONGMING TANG 2. Preliminaries on filter regular sequences Throughout the paper, let (R; m) be a (commutative) Noetherian local ring and M a finitely generated R-module. For any submodule N of M,weuseN :M hmi to denote the submodule fm 2 Mjmnm ⊆ N for some n>0g. Definition 2.1. Let x1;::: ;xn 2 m. If, for i =1;::: ;n, (x1;::: ;xi−1)M :M xi ⊆ (x1;::: ;xi−1)M :M hmi; then we say that x1;::: ;xn is an M-filter regular sequence. [ Notice that x 2 m is M-filter regular if and only if x 62 p and p2AssR(M)nfmg x1;x2;::: ;xn is an M-filter regular sequence if and only if x1 is M-filter regular and x2;::: ;xn is an M=x1M-filter regular sequence. For a finitely generated R- module N, the length of N is denoted by `(N). Then `(N) < 1 if and only if dim(N) ≤ 0. Thus x1;::: ;xn is an M-filter regular sequence is equivalent to `((x1;::: ;xi−1)M :M xi=(x1;::: ;xi−1M)) < 1;i=1;::: ;n: On the other hand, we are reminded that y1;::: ;ys 2 R is a poor M-regular sequence if (y1;::: ;yi−1)M :M yi =(y1;::: ;yi−1)M;i =1;::: ;s; and, if furthermore (y1;::: ;ys)M =6 M,wecally1;::: ;ys an M-regular sequence. Then x1;::: ;xn is an M-filter regular sequence if and only if, for any p 2 Supp(M)n fmg;x1=1;::: ;xn=1isapoorMp-regular sequence, and, if x1;::: ;xn is an M- filter regular sequence, then x1=1;::: ;xn=1isanMp-regular sequence for any p 2 Supp(M) nfmg with x1;::: ;xn 2 p. Since, for any poor M-regular sequence y1;::: ;ys and any integers ij > 0;j = i1 is 1;::: ;s, y1 ;::: ;ys is also a poor M-regular sequence, it follows that if x1;::: ;xn i1 is an M-filter regular sequence, then, for any integers ij > 0;j=1;::: ;n, x1 ;::: ; in xn is also an M-filter regular sequence. However, a permutation of a filter regular sequence is not necessarily filter regular again. For example, let K[x1;x2]bea ⊕ 2 polynomial ring and K a field. Set R = K[x1;x2](x1;x2) and M = R R=(x2). Then, ⊆ 2 as 0 :M x1 =0and0:M=x1M x2 R=(x1;x2) has finite length, it follows that x1;x2 is an M-filter regular sequence. But, as the prime ideal (x2) 2 AssR(M)nf(x1;x2)g and x2 2 (x2), we see that x2 is not M-filter regular, hence, x2;x1 is not an M-filter regular sequence. For any x1;::: ;xn 2 R,letHi(x1;::: ;xn; M)bethei-th homology module of the Koszul complex K:(x1;::: ;xn; M)ofM with respect to x1;::: ;xn. Then, we have the following: Proposition 2.2. If x1;::: ;xn is an M-filter regular sequence, then `(Hi(x1;::: ;xn; M)) < 1; for any i>0: Proof. By induction on n.Ifn =1,thenH1(x1; M)=0:M x1 has finite length by definition. Now, assume that n>1. From the long exact sequence ···!Hi(x1;::: ;xn−1; M) ! Hi(x1;::: ;xn; M) ! Hi−1(x1;::: ;xn−1; M) i−1 (−1) xn ! Hi−1(x1;::: ;xn−1; M) !···!H0(x1;::: ;xn; M) ! 0; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE F-DEPTH OF AN IDEAL ON A MODULE 1907 we see that `(Hi(x1;::: ;xn; M)) < 1 for all i>1 from the induction assumption that `(Hi(x1;::: ;xn−1; M)) < 1 and `(Hi−1(x1;::: ;xn−1; M)) < 1.Forthe case i = 1, we have an exact sequence H1(x1;::: ;xn−1; M) ! H1(x1;::: ;xn; M) ! H0(x1;::: ;xn−1; M) xn = M=(x1;::: ;xn−1)M ! M=(x1;::: ;xn−1)M: As H1(x1;::: ;xn−1; M)and0:M=(x1;::: ;xn−1)M xn have finite length, we see that `(H1(x1;::: ;xn; M)) < 1. Hence, for all i>0, `(Hi(x1;::: ;xn; M)) < 1. Notice that the converse of Proposition 2.2 is not true because the conditions `(Hi(x1;::: ;xn; M)) < 1 for all i>0 do not depend on the order of x1;::: ;xn, but the filter regularity does. 3. Characterizations of f-depth In order to show that f-depth is well-defined, we need the following: Lemma 3.1. Let I ⊆ m be an ideal. If `(HomR(R=I; M)) < 1, then there exists x 2 I which is M-filter regular. [ Proof. Assume the contrary. Then I ⊆ p,sothatI ⊆ p for some p2AssR(M)nfmg p 2 Ass (M) nfmg. Hence p =ann (m)forsomem 2 M,sodim(Rm) > 0. But R ∼ R Im =0,soRm ⊆ 0:M I = HomR(R=I; M). Thus dim(HomR(R=I; M)) > 0, a contradiction. Proposition 3.2. Let I ⊆ m be an ideal and n>0 an integer. Then the following are equivalent: i 1 (1) `(ExtR(R=I; M)) < , for all i<n; (2) I contains an M-filter regular sequence of length n. When x1;::: ;xn 2 I is an M-filter regular sequence, n ∼ 2 nf g ExtR(R=I; M)p = HomR(R=I; M=(x1;::: ;xn)M)p; for any p Spec(R) m : i 1 Proof. Assume that `(ExtR(R=I; M)) < , for all i<n. We use induction on n to show that I contains an M-filter regular sequence of length n.Ifn =1,thenI contains an M-filter regular element by Lemma 3.1. Now assume that n>1and theresultistrueforn − 1. Then, by Lemma 3.1 again, there is x1 2 I which is M-filter regular. From the short exact sequences x1 0 ! 0:M x1 ! M ! x1M ! 0; 0 ! x1M ! M ! M=x1M ! 0; we get the long exact sequences x1 0 ! HomR(R=I; 0:M x1) ! HomR(R=I; M) ! HomR(R=I; x1M) ! 1 !··· ExtR(R=I; 0:M x1) ; 0 ! HomR(R=I; x1M) ! HomR(R=I; M) ! HomR(R=I; M=x1M) ! 1 !··· ExtR(R=I; x1M) : 1 i 1 ≥ As `(0 :M x1) < ,wehavethat`(ExtR(R=I; 0:M x1)) < for any i 0. Then i 1 we see that `(ExtR(R=I; x1M)) < for all i<nfrom the first long exact sequence. i 1 From the second long exact sequence, we get that `(ExtR(R=I; M=x1M)) < for all i<n− 1. Then, by the induction assumption, there exist x2;::: ;xn 2 I which License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1908 RENCAI LU¨ AND ZHONGMING TANG is an M=x1M-filter regular sequence.
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