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On Annihilators and Associated Primes of Local Cohomology Modules M Journal of Pure and Applied Algebra 153 (2000) 197–227 www.elsevier.com/locate/jpaa CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector On annihilators and associated primes of local cohomology modules M. Brodmanna;1;2, Ch. Rotthausb;2;3, R.Y. Sharpc;3;∗ aInstitut fur Mathematik, Universitat Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland bDepartment of Mathematics, Michigan State University, East Lansing, MI 48824, USA cDepartment of Pure Mathematics, University of Sheeld, Hicks Building, Sheeld S3 7RH, UK Received 19 April 1999 Communicated by C.A. Weibel Abstract We establish the Local-global Principle for the annihilation of local cohomology modules over an arbitrary commutative Noetherian ring R at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 4. We explore interrelations between the principle and the Annihilator Theorem for local cohomology, and show that, if R is universally catenary and all formal ÿbres of all localizations of R satisfy Serre’s condition (Sr), then the Annihilator Theorem for local cohomology holds at level r over R if and only if the Local-global Principle for the annihilation of local cohomology modules holds at level r over R. Moreover, we show that certain local cohomology modules have only ÿnitely Smany associated primes. This provides motivation for the study of conditions under which the set Ass(M=(xm;yn)M) (where M is a ÿnitely generated R-module and x; y ∈ R) is ÿnite: an m;n∈N example due to M. Katzman shows that this set is not always ÿnite; we provide some sucient conditions for its ÿniteness. c 2000 Elsevier Science B.V. All rights reserved. MSC: 13D45; 13E05; 14B15; 13C15 ∗ Corresponding author. E-mail addresses: [email protected] (M. Brodmann), [email protected] (Ch. Rotthaus), [email protected] (R.Y. Sharp). 1 The ÿrst author was partially supported by the UK Science and Engineering Research Council (Grant number GR=E=43669). 2 The ÿrst and second authors were partially supported by the US National Science Foundation (Grant number NSF DMS 9502708). 3 The second and third authors were partially supported by the Swiss National Foundation (Project number 2000-042 129.94=1). 0022-4049/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S0022-4049(99)00104-8 198 M. Brodmann et al. / Journal of Pure and Applied Algebra 153 (2000) 197–227 0. Introduction Let R be a commutative Noetherian ring, let a be an ideal of R and let M be a ÿnitely generated R-module. We begin by recalling G. Faltings’ Local-global Principle i for the ÿniteness of local cohomology modules Ha (M)ofM with respect to a. Faltings’ Local-global Principle for the ÿniteness of local cohomology modules 0.1 [7; Satz1]. Let r be a positive integer. Then the R -module H i (M ) is ÿnitely gen- p aRp p i erated for all i ≤ r and for all p ∈ Spec(R) if and only if the R-module Ha (M) is ÿnitely generated for all i ≤ r. By the Flat Base Change Theorem for local cohomology, for each p ∈ Spec(R) and each i ∈ N0 (we use N and N0 to denote the sets of positive and non-negative integers respectively), there is an R -isomorphism H i (M ) ∼= (H i(M)) . Thus 0.1 states that p aRp p a p 0 1 r the modules Ha (M);Ha (M);:::;Ha (M) are all ÿnitely generated if and only if they are all locally ÿnitely generated. Another formulation of Faltings’ Local-global Principle, particularly relevant for this paper, is in terms of the ÿniteness dimension fa (M) of M relative to a, where i fa (M):=inf {i ∈ N: Ha (M) is not ÿnitely generated}; with the usual convention that the inÿmum of the empty set of integers is interpreted as ∞. We can restate 0.1 in the form (M ) ¿r for all p ∈ Spec(R) ⇔ f (M) ¿r: (†) faRp p a It is well known that p i fa (M) = inf {i ∈ N: a * (0: Ha (M))} n i = inf {i ∈ N: a Ha (M) 6= 0 for all n ∈ N} n i = inf {i ∈ N0: a Ha (M) 6= 0 for all n ∈ N}; 0 see [4, 9:1:2] (and note that Ha (M) is ÿnitely generated). Now let b be a second ideal b of R. We deÿne the b-ÿniteness dimension fa (M) of M relative to a by p b i fa (M) = inf {i ∈ N0: b * (0: Ha (M))} n i = inf {i ∈ N0: b Ha (M) 6= 0 for all n ∈ N}: So it is rather natural to ask whether Faltings’ Local-global Principle, as stated in b (†), generalizes in the obvious way to the invariants fa (M). In other words, is the statement fbRp (M ) ¿r for all p ∈ Spec(R) ⇔ fb(M) ¿r (††) aRp p a true for each r ∈ N? We shall say that the Local-global Principle (for the annihilation of local cohomology modules) holds at level r (over the ring R)if(††) is true (for the given r) for every choice of ideals b; a of R and every choice of M. In [14], Raghavan M. Brodmann et al. / Journal of Pure and Applied Algebra 153 (2000) 197–227 199 showed that this Local-global Principle holds at level 1; he also deduced from Faltings’ Annihilator Theorem [6] that if R is a homomorphic image of a regular (commutative Noetherian) ring, then the Local-global Principle holds at all levels r ∈ N. Our main result in Section 2 is that the Local-global Principle holds at level 2, while in Section 3 we show that the Local-global Principle holds at all levels r ∈ N when dim R ≤ 4. b Our work in Section 3 involves use of the b-minimum a-adjusted depth a (M) of M, deÿned by b a (M):=inf {depth Mp +ht(a + p)=p: p ∈ Spec(R)\Var(b)}: b b It is always the case that fa (M) ≤ a (M); Faltings’ Annihilator Theorem [6] states that if R is a homomorphic image of a regular (commutative Noetherian) ring, then b b fa (M)=a (M). We extend the terminology and say that, for r ∈ N, the Annihilator Theorem (for local cohomology modules) holds at level r over R if, for every choice of the ÿnitely generated R-module M and for every choice of the ideals b; a of R,it is the case that b b a (M) ¿r ⇔ fa (M) ¿r: Our main result in Section 4 is that, if R is universally catenary and all the formal ÿbres of all its localizations satisfy Serre’s condition (Sr), then the Annihilator Theorem holds at level r over R if and only if the Local-global Principle holds at level r over R. Section 5 is rather technical. We wish to use (in Section 6) the above-mentioned main result of Section 4 in the special case in which r = 1, and we also wish to know that the property that ‘all formal ÿbres of all localizations of R satisfy (S1)’ is inherited by the polynomial ring R[X ]: Section 5 is concerned with this point. i Our work in Section 2 motivates a search for conditions under which AssR(Ha (M)) is ÿnite. In view of the fact that, if a can be generated by x1;:::;xh, then there is an isomorphism h ∼ n n Ha (M) = lim M=(x1;:::;xh)M → n∈N (see [4, 5:2:9]), we are interested in conditions which ensure that [ n n AssR(M=(x1;:::;xh)M) n∈N is ÿnite. In general, this set need not be ÿnite: in [10], Katzman produced an example of a (commutative Noetherian) algebra R over a ÿeld of characteristic p¿0 and elements S pe pe x; y ∈ R with e∈N AssR(R=(x ;y )) inÿnite. In view of this, we consider that it is of interest to determine conditions on R, elements x; y ∈ R and the ÿnitely generated R-module M which ensure that [ m n AssR(M=(x ;y )M) m;n∈N 200 M. Brodmann et al. / Journal of Pure and Applied Algebra 153 (2000) 197–227 is ÿnite. Our main result of Section 6 is that this set is ÿnite if R is universally catenary, all formal ÿbres of all the localizations of R satisfy (S1), and ht((x; y)+p)=p ¿ 1 for all p ∈ AssR(M)\Var((x; y)). 1. Some preliminaries Notation 1.1. Throughout the paper, R will denote a commutative Noetherian ring (with non-zero identity), and a and b will denote ideals of R. We shall only assume that b ⊆ a when this is explicitly stated. We use Z to denote the set of all integers. The variety Supp(R=b)ofb will be denoted by Var(b). Deÿnition 1.2. Fix r ∈ N. (i) We say that the Local-global Principle (for the annihilation of local cohomology modules) holds at level r for the ideals b; a of R if, for every ÿnitely generated R-module M, it is the case that fbRp (M ) ¿r for all p ∈ Spec(R) ⇔ fb(M) ¿r: (∗) aRp p a (ii) We say that the Local-global Principle (for the annihilation of local cohomology modules) holds at level r (over the ring R) if the principle of (i) holds at level r for every choice of ideals b; a of R.
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