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Locally Unmixed Modules and Ideal Topologies
Journal of Algebra 236, 768᎐777 Ž2001. doi:10.1006rjabr.2000.8518, available online at http:rrwww.idealibrary.com on
Locally Unmixed Modules and Ideal Topologies R. Naghipour Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran; and Department of Mathematics, Uni¨ ersity of Tabriz, Tabriz, Iran E-mail: [email protected] Communicated by Susan Montgomery Received March 3, 2000
Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. In this paper, the main result asserts that N is locally unmixed if and only if, for any N-proper ideal ᑾ of R generated by ht N ᑾ elements, the topology defined by Ž ᑾ N .Ž n., n G 0, is equivalent to the ᑾ-adic topology. 䊚 2001 Academic Press
1. INTRODUCTION Throughout this paper, all rings considered will be commutative and Noetherian and will have non-zero identity elements. Such a ring will be denoted by R and a typical ideal of R will be denoted by ᑾ. Let N be a non-zero finitely generated module over R. We denote by R the Rees ring Rw u, ᑾ t x [ [ng ⺪ ᑾ n t n of R w.r.t. ᑾ, where t is an indeterminate and u s ty1 . Also, the graded Rees module N w u, ᑾ t x [ [ng ⺪ ᑾ n N over R is denoted by N , which is a finitely generated R-module. If Ž R, ᒊ . is local, then RU Žresp. N U . denotes the completion of R Žresp. N . w.r.t. the ᒊ-adic topology. In particular, for any ᒍ g Spec R, we denote RUᒍ and NᒍU the ᒍ R ᒍ-adic completion of R ᒍ and Nᒍ , respectively. For any multiplicatively closed subset S of R, the nth Ž S .-symbolic power of ᑾ w.r.t. N, denoted by SŽ ᑾ n N ., is defined to be the union of ᑾ n N : N s where s varies in S. The ᑾ-adic filtration ᑾ n N 4nG 0 and the Ž S .-symbolic filtration SŽ ᑾ n N .4nG 0 induce topologies on N which are called the ᑾ-adic topology and the Ž S .-symbolic topology, respectively. These two topologies are said to be equivalent if, for every integer m G 0, there is an integer n G 0 such 768 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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that SŽ ᑾ n N . : ᑾ m N. In particular, if S s R_ j ᒍ g m Ass R Nrᑾ N 4 , where m Ass R Nrᑾ N denotes the set of minimal prime ideals of Ass R Nrᑾ N, the nth Ž S .-symbolic power of ᑾ w.r.t. N is denoted by Ž ᑾ N .Ž n., and the topology defined by the filtration Ž ᑾ N .Ž n.4 is called the symbolic topology. The purpose of the present paper is to show that N is locally unmixed if and only if, for each N-proper ideal ᑾ that is generated by ht N ᑾ elements, the ᑾ-adic and the symbolic topologies are equivalent. Schenzel has characterized unmixed local rings w8, Theorem 7x in terms of comparison of the topologies defined by certain filtrations. Also, Katz w4, Theorem 3.5x and Verma w11, Theorem 5.2x have proved a characterization of locally unmixed rings in terms of s-ideals. Equivalence of ᑾ-adic topology and Ž S .-symbolic topology has been studied, in the case N s R, in w4, 6᎐10x, and has led to some interesting results. Let ᒍ g SuppŽ N .. Then the N-height of ᒍ, denoted by ht N ᒍ, is defined to be the supremum of lengths of chains of prime ideals of SuppŽ N . terminating with ᒍ. We have ht N ᒍ s dim R ᒍ Nᒍ . We shall say that an ideal ᑾ of R is N-proper if Nrᑾ N / 0, and, when this is the case, we define the N-height of ᑾ Žwritten ht N ᑾ . to be inf ht N ᒍ : ᒍ g SuppŽ N . l V Ž ᑾ . 4
Ž s inf ht N ᒍ : ᒍ g Ass R Ž Nrᑾ N . 4 . . For any N-proper ideal ᑾ of R, denote by gradeŽ ᑾ, N . the maximum length of all N-sequences contained in ᑾ. Suppose for the moment that Ž R, ᒊ . is local. It follows from Nakayama’s lemma that every proper ideal of R is N-proper. N is said to be a Cohen᎐Macaulay module Žabbreviated as CM module. if and only if gradeŽ ᒊ, N . s ht N ᒊ. More generally, if R is not necessarily local and N is non-zero and finitely generated, N is said to be CM if and only if Nᒊ is a CM R ᒊ -module in the above sense for each maximal ideal ᒊ g SuppŽ N .. In the following we refer to w2, 3, 5x for the basic results about CM modules. For any ideal ᑾ of R, the radical of ᑾ, denoted by RadŽ ᑾ ., is defined to be the set x g R : x n g ᑾ for some n g ⺞4 . For any R-module L, we denote by m Ass R L the set of minimal prime ideals of Ass R L. In the second section, we characterize the CM property of a non-zero finitely generated R-module N in terms of the equalities ᑾ n N and Ž ᑾ N .Ž n. for certain N-proper ideals ᑾ of R. More precisely we prove the following result: THEOREM 1.1. The following conditions are equi¨ alent: Ži. N is CM. Žii. For any N-proper ideal ᑾ of R that is generated by ht N ᑾ elements, ᑾ n N s Ž ᑾ N .Ž n. for all n g ⺞.
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The result of 1.1 is proved in 2.3. Let Ž R, ᒊ . be local and let N be a non-zero finitely generated R-module. N is said to be an unmixed module if for any ᒍ g Ass RU N U , dim RUrᒍ s dim N. More generally, if R is not necessarily local and N is non-zero finitely generated, N is a locally unmixed module if for any ᒍ g SuppŽ N ., Nᒍ is an unmixed R ᒍ-module. As the main result of Section 3 we characterize the locally unmixed property of a non-zero finitely generated R-module N in terms of the equivalence of the topologies defined by ᑾ n N 4nG 0 and Ž ᑾ N .Ž n.4nG 0 , for certain N-proper ideals ᑾ of R. More precisely we shall show that: THEOREM 1.2. The following conditions are equi¨ alent: Ži. N is locally unmixed. Žii. For each N-proper ideal ᑾ of R that is generated by ht N ᑾ elements, the topology gi¨ en by Ž ᑾ N .Ž n.4nG 0 is equi¨ alent to the ᑾ-adic topology on N. The proof of Theorem 1.2 is given in 3.12. 2. COHEN᎐MACAULAY MODULES AND SYMBOLIC POWERS Let N be a non-zero and finitely generated R-module and let ᑾ be an N-proper ideal of R. The following theorem is well known when N s R. The proof in w2, 5x can be easily carried over to a module, so we omit the proof. THEOREM 2.1.
Let N and ᑾ be as abo¨ e. Then the following hold:
Ži. ht N ᑾ s htŽ ᑾ q Ann R NrAnn R N .. Žii. Ž Krull’s principle ideal theorem. If x s x 1 , . . . , x n is a sequence of elements of R, then ht N ᒍ F n for all ᒍ g m Ass R NrxN. Furthermore, if x is an N-sequence, then any minimal element of Ass R NrxN has N-height n. In particular, ht N x s n. Žiii. If ht N ᑾ s n, then there exist x 1 , . . . , x n in ᑾ such that ht N Ž x 1 , . . . , x i . s i for i s 1, 2, . . . , n. Živ. N is CM if and only if ht N ᑿ s gradeŽ ᑿ, N . for any N-proper ideal ᑿ of R. PROPOSITION 2.2. Let N be a non-zero finitely generated R-module. Then the following conditions are equi¨ alent: Ži. N is CM. Žii. For e¨ ery N-proper ideal ᑾ of R generated by ht N ᑾ elements, m Ass R Ž Nrᑾ N . s Ass R Ž Nrᑾ N ..
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Proof. Ži. « Žii. Let ht N ᑾ [ n and let ᑾ s Ž x 1 , . . . , x n .. Suppose that ᒍ, ᒎ g Ass R Nrᑾ N, and let ᒊ be a maximal ideal of R such that ᒎ : ᒍ : ᒊ. It is enough to show that ᒎ s ᒍ. To this end, we have ᒍ R m , ᒎ R m g Ass R ᒊ Ž Nᒊ rᑾ Nᒊ . . ht N ᑾ s n and Nᒊ is CM w2, Theorem 2.1.2x imply that x 1 , . . . , x n is an Nᒊ -sequence. Therefore by w2, Theorem 2.1.3x Nᒊ rŽ x 1 , . . . , x n . Nᒊ is a CM R ᒊ -module. When ᒍ R ᒊ s ᒎ R ᒊ and so ᒍ s ᒎ, as required. In order to prove the implication Žii. « Ži. assume that ᑿ is an arbitrary N-proper ideal of R. In view of Theorem 2.1Živ., it is enough to show that ht N ᑿ s gradeŽ ᑿ, N .. To achieve this, suppose x 1 , . . . , x n in ᑿ are such that ht N Ž x 1 , . . . , x i . s i for all 1 F i F n, where n s ht N ᑿ. Thus we have x i f D ᒍ g m Ass R Nr Ž x 1 , . . . , x iy1 . N 4 ,
for all 1 F i F n.
So the condition Žii. implies that x i f D ᒍ g Ass R NrŽ x i , . . . , x iy1 . N 4 for all 1 F i F n. That is, x 1 , . . . , x n in ᑿ is an N-sequence. Now the assertion follows. We are now ready to state and prove the main theorem of this section. THEOREM 2.3. Let N be a non-zero finitely generated R-module. Then the following conditions are equi¨ alent: Ži. N is CM. Žii. For any N-proper ideal ᑾ of R generated by ht N ᑾ elements, Ž ᑾ N .Ž n. s ᑾ n N for all n g ⺞. Proof. Ži. « Žii. Suppose N is CM and let ᑾ be an arbitrary N-proper ideal of R generated by ht N ᑾ elements. Then by Proposition 2.2, we have Ass R Nrᑾ N s m Ass R Nrᑾ N. Thus, by assumption Ži. and w3, Theorem 125 and Exercise 13, p. 103x, it follows that Ass R Nrᑾ n N s m Ass R Nrᑾ n N for any n g ⺞. ŽNote that m Ass R Nrᑾ n N s m Ass R Nrᑾ N.. Now, by considering a normal primary decomposition for ᑾ n N and using the definition of Ž ᑾ N .Ž n., it is straightforward to check that Ž ᑾ N .Ž n. s ᑾ n N for all n g ⺞, as required. Žii. « Ži. Let ᑾ be an ideal of R generated by ht N ᑾ elements. In view of Proposition 2.2, it is enough to show that the associated prime ideals of Nrᑾ N are minimal. Suppose this is not the case and let ᒎ be an element of Ass R Nrᑾ N which does not belong to m Ass R Nrᑾ N. Then ᒎ s ᑾ N : R x for some x g N _ ᑾ N and q D ᒍ g m Ass R Nrᑾ N 4 . Let s g ᒎ be such that s f D ᒍ g m Ass R Nrᑾ N 4 . Then sx g ᑾ N and, so x g ᑾ N : N s. Now, the condition Žii. provides a contradiction.
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3. LOCALLY UNMIXED MODULES AND COMPARISON OF TOPOLOGIES The purpose of this section is to prove that a non-zero finitely generated module over a Noetherian ring R is locally unmixed if and only if, for any N-proper ideal ᑾ of R that can be generated by ht N ᑾ elements, the topologies ᑾ-adic and symbolic, on N, are equivalent. We begin with DEFINITION 3.1. Let N be a non-zero finitely generated R-module and let ᑾ be an ideal of R. A prime ideal ᒍ of R is called a quintessential prime ideal of ᑾ w.r.t. N precisely when there exists ᒎ g Ass RUᒍ NᒍU such that RadŽ ᑾ RUᒍ q q . s ᒍ RUᒍ . The set of quintessential primes of ᑾ w.r.t. N is denoted by Q Ž ᑾ, N .. It is easy to see that Q Ž ᑾ, N . s Q Ž ᑾ q Ann R N, N .. LEMMA 3.2. Let N be a non-zero finitely generated R-module and let ᑾ be an ideal of R. Suppose ᒍ g SuppŽ Nrᑾ N ._ m Ass R Ž Nrᑾ N . is such that the topology defined by Ž ᑾ N .Ž n., n G 1, is finer than the topology defined by Ž ᒍ N .Ž n., n G 1. Then ᒍ f Q Ž ᑾ, N .. Proof. Suppose the contrary; i.e., ᒍ g Q Ž ᑾ, N .. Let k G 1 be as w1, Proposition 3.12x. Then, for such k, there exists an integer n G 1 such that Ž ᑾ N .Ž n. : Ž ᒍ N .Ž k .. Again, from w1, Proposition 3.12x, it follows that ᒍ g Ass R Ž NrŽ ᑾ N .Ž n. .. Since Ass R NrŽ ᑾ N .Ž n. s q g Ass R Nrᑾ n N : ᒎ l S s 4 , where S s R_ j ᒎ g m Ass R Nrᑾ N 4 , it yields ᒍ : ᒎ for some ᒎ g m Ass R Nrᑾ N. Consequently, ᒍ s ᒎ, and so ᒍ g m Ass R Nrᑾ N, which is a contradiction. PROPOSITION 3.3. tions are equi¨ alent: Ži. Žii.
Let N and ᑾ be as abo¨ e. Then the following condi-
Q Ž ᑾ, N . s m Ass R Nrᑾ N. The symbolic topology is equi¨ alent to the ᑾ-adic topology.
Proof. Ži. « Žii. Let n G 1, and let ᑾ nN s Ž ᑾ N .
Ž n.
l Q 2 l ⭈⭈⭈ l Q r
where Q i is ᒍ i-primary in N
Ž2 F i F r . be a normal primary decomposition of ᑾ n N. Then, ᒍ i g SuppŽ Nrᑾ N ._ m Ass R Nrᑾ N for all 2 F i F r. It is easy to see that Ž ᒍ i N .Ž k i . : Q i for sufficiently large k i Ž2 F i F r .. Furthermore, by assumption Ži. and Lemma 3.2, it follows that Ž ᑾ N .Ž m i . : Ž ᒍ i N .Ž k i . for sufficiently large m i with 2 F i F n. Letting m be the maximum of m 2 , . . . , m r , we see that Ž ᑾ N .Ž m. : Ž ᒍ i N .Ž k i . Ž2 F i F r .. Thus Ž ᑾ N .Ž m n. : ᑾ n N, as required.
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In order to prove that Žii. « Ži., suppose that for any k G 1 there is an integer m G 1 such that Ž ᑾ N .Ž m. : ᑾ k N. Let ᒍ g Q Ž ᑾ, N .. Then Ž ᑾ N .Ž m. : Ž ᒍ N .Ž k .. By virtue of Lemma 3.2, ᒍ g m Ass R Nrᑾ N. Hence, w1, Lemma 3.5x yields the claim. LEMMA 3.4. Let N be a non-zero finitely generated R-module. Suppose that ᒍ g SuppŽ N . and let ᑿ be an Nᒍ-proper ideal of R ᒍ generated by ht N ᒍ ᑿ elements. Then there is an N-proper ideal ᒀ of R generated by ht N ᒀ elements such that ᑿ q Ann R ᒍ Nᒍ s ᒀ R ᒍ q Ann R ᒍ Nᒍ . Proof. Since ht N ᒍ ᑿ s htŽŽ ᑿ q Ann R ᒍ Nᒍ .rAnn R ᒍ Nᒍ ., by Theorem 2.1 Ži., it follows that the ideal Ž ᑿ q Ann R Nᒍ .rAnn R Nᒍ of R ᒍrAnn R Nᒍ ᒍ ᒍ ᒍ is generated by htŽŽ ᑿ q Ann R ᒍ Nᒍ .rAnn R ᒍ Nᒍ . elements. According to Verma w11, Lemma 5.1x, there is an ideal ᑾrAnn R N of RrAnn R N that is a generated by htŽ ᑾrAnn R N . s ht N ᑾ elements, and
Ž ᑿ q Ann R
Nᒍ . rAnn R ᒍ Nᒍ s Ž ᑾrAnn R N . R ᒍrAnn R ᒍ Nᒍ .
ᒍ
The result now follows. PROPOSITION 3.5. Let Ž R, ᒊ . be local and let N be a non-zero finitely generated R-module. Suppose that ᒎ g Ass RU N U with dim RUrᒎ ) 0. Then there exists an N-proper ideal ᑾ of R generated by ht N ᑾ elements, such that Rad Ž ᑾ RU q ᒎ . s ᒊ RU
ht N ᑾ s dim RUrᒎ .
and
Proof. Let ᒎ g Ass RU N U with dim RUrᒎ [ n ) 0. Using Krull’s principal ideal theorem and prime avoidance arguments one constructs elements x 1 , . . . , x n g ᒊ such that xi f
ž
D
ᒍ
ᒍgm Ass R Nr Ž x 1 , . . . , x iy1 . N
j
ž
U
U
D
/ U
ᒍ gm Ass RU N r Ž ᒎq Ž x 1 , . . . , x iy1 . R . N
U
ᒍU l R ,
/
for all 1 F i F n. Select ᑾ s Ž x 1 , . . . , x n .. We need to show that ht N ᑾ s n and ᑾ RU q ᒎ is ᒊ RU-primary. First we show ht N ᑾ s n. We prove this by induction on n. The case n s 1 follows from the principal ideal theorem, together with x 1 f Dᒍ g m Ass R N ᒍ. So let n ) 1 and suppose that the result is true for n y 1. Let ᒍ g m Ass R Nrᑾ N. Because of x n g ᒍ, we have ᒍ f m Ass R NrŽ x 1 , . . . , x ny1 . N. The result now follows from Krull’s
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principal ideal theorem and inductive hypothesis. Now we prove RadŽ ᑾ RU q ᒎ . s ᒊ RU . To this end let ᒎUn be a minimal prime over ᑾ RU q ᒎ. Then from x n g ᒎUn l R, follows that ᒎUn f m Ass RU N U rŽ ᒎ q Ž x 1 , . . . , x ny 1 . RU . N U . Therefore there is a ᒎUny1 g m Ass RU N U rŽ ᒎ q Ž x 1 , . . . , x ny 1 . RU . N U such that ᒎUny1 m ᒎUn , and so on. Thus we have a saturated chain of primes from ᒎ to ᒎUn , as ᒎ s ᒎU0 m ⭈⭈⭈ m ᒎUny1 m ᒎUn , with 1 F i F n, ᒎUi g m Ass RU N U rŽ ᒎ q Ž x 1 , . . . , x i . RU . N U . Because dim RU rᒎ s n, it follows that ᒎUn s ᒊ RU , so that RadŽ ᑾ RU q ᒎ . s ᒊ RU , as desired. DEFINITION 3.6. Let N be a non-zero finitely generated R-module. A prime ideal ᒍ of R is an essential prime of ᑾ w.r.t. N, if ᒍ s ᒎ l R for some ᒎ g Q Ž u R, N .. The set of essential primes of ᑾ w.r.t. N will be denoted by E Ž ᑾ, N .. DEFINITION 3.7. A sequence x s x 1 , . . . , x n of elements of R is called an essential sequence on N if the following conditions are satisfied: Ži. For all 1 F i F n, x i f D ᒍ g E ŽŽ x 1 , . . . , x iy1 ., N .4 . Žii. NrxN / 0. An essential sequence x s x 1 , . . . , x n of elements of R Žcontained in an ideal ᑾ . on N is maximal Žin ᑾ ., if x 1 , . . . , x n , x nq1 is not an essential sequence on N for any x nq 1 g RŽ x nq1 g ᑾ .. It is shown that Žsee w1, Theorem 4.16x. all maximal essential sequences on N in an ideal ᑾ have the same length. This allows us to introduce the fundamental notion of essential grade Žsee w1, Definition 4.17x.. LEMMA 3.8.
Let N be a non-zero finitely generated R-module and let
x s x1, . . . , x n be elements of R which form an essential sequence on N. Then the following hold: Ži. ht N Ž x 1 , . . . , x i . s i for all 1 F i F n. Žii. If N is locally unmixed, then E Ž x, N . s m Ass R NrxN. Proof. In order to prove Ži., it is sufficient to show that if ᒍ g m Ass R Ž Nr Ž x 1 , . . . , x i . N ., then ht N ᒍ s i. To this end recall that m Ass R Nrᑾ N : E Ž ᑾ, N ., for any ideal ᑾ of R, and x 1 , . . . , x i is an essential sequence on N. Putting this together the proof of Ži. follows by induction. For the proof of Žii., it is clearly sufficient to prove that E Ž x, N . : m Ass R NrxN. Let ᒍ g E Ž x, N .. Then by virtue of Ži., ht N ᒍ G n. Thus we need only to show that ht N ᒍ F n, which implies that ᒍ g m Ass R NrxN. To do this, let ht N ᒍ s k and let ᒍ 0 m ⭈⭈⭈ m ᒍ k s ᒍ be a saturated chain of primes of SuppŽ N . with ᒍ 0 g m Ass R N. Then, by
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w1, Lemmas 4.9, 3.2, 4.11x, htŽ ᒍ R ᒍrᒍ 0 R ᒍ . s n and so n G k. This completes the assertion. PROPOSITION 3.9. Let Ž R, ᒊ . be local and let N be a non-zero finitely generated R-module. Then e gradeŽ ᑾ, N . s Min htŽ ᑾ RU q ᒎrᒎ . : ᒎ g AssRU N U 4 . Proof. Let e gradeŽ ᑾ, N . s n, and let x s x 1 , . . . , x n be a maximal essential sequence on N in ᑾ. Then ᑾ : ᒍ for some ᒍ g E Ž x, N ., by w1, Theorem 3.17x. By virtue of w1, Proposition 3.8x, there exists a prime ᒎU in RU such that ᒎU l R s ᒍ and ᒎU g E Ž xRU , N U .. Furthermore, by w1, Proposition 3.6x, there is a ᒎU0 g Ass RU N U such that ᒎU0 : ᒎU and ᒎU rᒎU0 g E Ž xRU q ᒎU0 rᒎU0 , RU rᒎU0 .. Now, by w1, Lemma 4.9x, x 1 q ᒎU0 , . . . , x n q ᒎU0 is an essential sequence in the complete domain RUrᒎ 0 . Therefore, Lemma 3.8 shows that htŽ ᒎU rᒎU0 . s n. As ᑾ RU q ᒎU0 : ᒎU , we have htŽ ᑾ RU q ᒎU0 rᒎU0 . F n. Now it is easy to see that the assertion follows from w1, Theorem 4.16x. THEOREM 3.10. Let N be a non-zero finitely generated R-module and let be an essential sequence on N. Then E Ž x, N . s Q Ž x, N ..
x s x1, . . . , x n
Proof. In view of w1, Theorem 3.17x, it is sufficient to show that E Ž x, N . : Q Ž x, N .. Let ᒍ g E Ž x, N .. By w1, Lemma 3.2x, ᒍ R ᒍ g E Ž xR ᒍ , Nᒍ ., and so e gradeŽ ᒍ R ᒍ , Nᒍ . s n. Thus, by virtue of Proposition 3.9, there is a ᒎ g Ass RUᒍ NᒍU such that dim RUᒍ rᒎ s n. Furthermore, by w1, Lemma 4.9x, x 1 q ᒎ, . . . , x n q ᒎ is an essential sequence on the complete local domain RUᒍ rᒎ. So, Lemma 3.8 implies that htŽ x 1 q ᒎ, . . . , x n q ᒎ . s n. That is, htŽ xRUᒍ q ᒎrᒎ . s n. Hence ᒍ RUᒍ rᒎ is minimal over xRUᒍ q ᒎrᒎ. Consequently, RadŽ xRUᒍ q ᒎ . s ᒍ RUᒍ , as required. COROLLARY 3.11. Let N be a non-zero finitely generated R-module and let x 1 , . . . , x n in R be such that Ž x 1 , . . . , x n . N / N. Suppose N is locally unmixed and ht N Ž x 1 , . . . , x i . s i for 1 F i F n. Then x 1 , . . . , x n is an essential sequence on N. Proof. First assume that n s 1. Since E Ž0 R , N . s Ass R N, we need to show that x 1 f Dᒍ g Ass R N ᒍ. Since N is locally unmixed, Ass R N s ᒍ g SuppŽ N . : ht N ᒍ s 04 . Thus, since ht N Ž x 1 . s 1, x 1 is not a zero-divisor on N. Let n ) 1, and suppose that the result is true for n y 1. Let ᒍ g E ŽŽ x 1 , . . . , x ny1 ., N .. Then, by the inductive hypothesis and Lemma 3.8, we have x n f ᒍ. Therefore, x 1 , . . . , x n is an essential sequence on N, as desired.
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We are now ready to state and prove the main theorem of this section, which is a characterization of locally unmixed modules in terms of comparison of the topologies defined by certain decreasing families of submodules of a finitely generated module over a commutative Noetherian ring. THEOREM 3.12. Let N be a non-zero finitely generated R-module. Then the following conditions are equi¨ alent: Ži. N is locally unmixed. Žii. For e¨ ery N-proper ideal ᑾ of R generated by ht N ᑾ elements, the ᑾ-adic topology is equi¨ alent to the symbolic topology. Proof. First we show Ži. « Žii.. Let ᑾ be an N-proper ideal of R which is generated by ht N ᑾ elements. Suppose that ᒍ g Q Ž ᑾ, N .. We show that ᒍ g m Ass R Nrᑾ N. To do this, it is straightforward to check that the ideal ᑾ R ᒍ of R ᒍ can be generated by ht N ᒍ ᑾ R ᒍ elements, by Theorem 2.1Žii.. Therefore, by w1, Lemma 3.2x, we may assume that Ž R, ᒍ . is local. Let ht N ᑾ s n. By Theorem 2.1, there exist x 1 , . . . , x n in ᑾ with ht N Ž x 1 , . . . , x i . s i for all 1 F i F n. As shown in Corollary 3.11, x 1 , . . . , x n is an essential sequence on N, so e gradeŽ ᑾ, N . s n by Proposition 3.9. Now, analogous to the proof of w3, Theorem 125x, it is easy to see that ᑾ can be generated by an essential sequence of length n. Therefore by Theorem 3.10 and Lemma 3.8Žii., we have ᒍ g m Ass R Nrᑾ N. We can now use Proposition 3.3 and the fact that m Ass R Nrᑾ N : Q Ž ᑾ, N . to complete the proof of Žii.. In order to prove Žii. « Ži., suppose that ᒍ g SuppŽ N .. By Lemma 3.4 and the fact that Q Ž ᑿ, L. s Q Ž ᑿ q Ann R L, L. for any ideal ᑿ of R and any R-module L, and w1, Lemma 3.2x, we may assume without loss of generality that Ž R, ᒍ . is local. If gradeŽ ᒍ, N . s 0, then ᒍ g Ass R N. Thus ᒍ g Q Ž0 R , N .. Therefore by condition Žii. and Proposition 3.3, ᒍ g m Ass R N. So ht N ᒍ s 0; that is, dim N s 0. Whence N is a CM module, and so N is unmixed. We may assume that gradeŽ ᒍ, N . ) 0. Let ᒎ g Ass RU N U be such that dim RUrᒎ s n. We need to show that dim N s n. Because gradeŽ ᒍ, N . ) 0 we have n ) 0. Then, by virtue of Proposition 3.5 there exits an N-proper ideal ᑾ of R generated by ht N ᑾ s n elements such that RadŽ ᑾ RU q ᒎ . s ᒍ RU . Consequently, by Proposition 3.3, ᒍ g m Ass R Nrᑾ N. Hence ht N ᒍ s n and the claim is true.
ACKNOWLEDGMENTS The author is deeply grateful to the referee for his very careful reading of the original manuscript and to Professor H. Zakeri for the many helpful discussions during the preparation of this paper. The author also gratefully acknowledges the helpful leads of Professor L. J.
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Ratliff, Jr. This research is supported by Tabriz University and the Institute of Studies in Theoretical Physics and Mathematics.
REFERENCES 1. S. H. Ahn, Asymptotic primes and asymptotic grade on modules, J. Algebra 174 Ž1995., 980᎐998. 2. W. Bruns and J. Herzog, ‘‘Cohen᎐Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, UK, 1998. 3. I. Kaplansky, ‘‘Commutative Rings,’’ Univ. of Chicago Press, Chicago, 1974. 4. D. Katz, Prime divisors, asymptotic R-sequences and unmixed local rings, J. Algebra 95 Ž1985., 59᎐71. 5. H. Matsumura, ‘‘Commutative Ring Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 6. L. J. Ratliff, Jr., The topology determined by the symbolic powers of primary ideals, Comm. Algebra 13 Ž1985., 2073᎐2104. 7. P. Schenzel, Symbolic powers of prime ideals and their topology, Proc. Amer. Math. Soc. 93 Ž1985., 15᎐20. 8. P. Schenzel, Independent elements, unmixedness theorems and asymptotic prime divisors, J. Algebra 92 Ž1985., 157᎐170. 9. P. Schenzel, Finiteness of relative Rees ring and asymptotic prime divisors, Math. Nachr. 129 Ž1986., 123᎐148. 10. P. Schenzel, Explicit computations around the Lichtenbaum᎐Hartshorne vanishing theorem, Manuscripta Math. 78 Ž1993., 57᎐68. 11. J. K. Verma, On ideals whose adic and symbolic topologies are linearly equivalent, J. Pure Appl. Algebra 47 Ž1987., 205᎐212.
Locally Unmixed Modules and Ideal Topologies R. Naghipour Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran; and Department of Mathematics, Uni¨ ersity of Tabriz, Tabriz, Iran E-mail: [email protected] Communicated by Susan Montgomery Received March 3, 2000
Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. In this paper, the main result asserts that N is locally unmixed if and only if, for any N-proper ideal ᑾ of R generated by ht N ᑾ elements, the topology defined by Ž ᑾ N .Ž n., n G 0, is equivalent to the ᑾ-adic topology. 䊚 2001 Academic Press
1. INTRODUCTION Throughout this paper, all rings considered will be commutative and Noetherian and will have non-zero identity elements. Such a ring will be denoted by R and a typical ideal of R will be denoted by ᑾ. Let N be a non-zero finitely generated module over R. We denote by R the Rees ring Rw u, ᑾ t x [ [ng ⺪ ᑾ n t n of R w.r.t. ᑾ, where t is an indeterminate and u s ty1 . Also, the graded Rees module N w u, ᑾ t x [ [ng ⺪ ᑾ n N over R is denoted by N , which is a finitely generated R-module. If Ž R, ᒊ . is local, then RU Žresp. N U . denotes the completion of R Žresp. N . w.r.t. the ᒊ-adic topology. In particular, for any ᒍ g Spec R, we denote RUᒍ and NᒍU the ᒍ R ᒍ-adic completion of R ᒍ and Nᒍ , respectively. For any multiplicatively closed subset S of R, the nth Ž S .-symbolic power of ᑾ w.r.t. N, denoted by SŽ ᑾ n N ., is defined to be the union of ᑾ n N : N s where s varies in S. The ᑾ-adic filtration ᑾ n N 4nG 0 and the Ž S .-symbolic filtration SŽ ᑾ n N .4nG 0 induce topologies on N which are called the ᑾ-adic topology and the Ž S .-symbolic topology, respectively. These two topologies are said to be equivalent if, for every integer m G 0, there is an integer n G 0 such 768 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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that SŽ ᑾ n N . : ᑾ m N. In particular, if S s R_ j ᒍ g m Ass R Nrᑾ N 4 , where m Ass R Nrᑾ N denotes the set of minimal prime ideals of Ass R Nrᑾ N, the nth Ž S .-symbolic power of ᑾ w.r.t. N is denoted by Ž ᑾ N .Ž n., and the topology defined by the filtration Ž ᑾ N .Ž n.4 is called the symbolic topology. The purpose of the present paper is to show that N is locally unmixed if and only if, for each N-proper ideal ᑾ that is generated by ht N ᑾ elements, the ᑾ-adic and the symbolic topologies are equivalent. Schenzel has characterized unmixed local rings w8, Theorem 7x in terms of comparison of the topologies defined by certain filtrations. Also, Katz w4, Theorem 3.5x and Verma w11, Theorem 5.2x have proved a characterization of locally unmixed rings in terms of s-ideals. Equivalence of ᑾ-adic topology and Ž S .-symbolic topology has been studied, in the case N s R, in w4, 6᎐10x, and has led to some interesting results. Let ᒍ g SuppŽ N .. Then the N-height of ᒍ, denoted by ht N ᒍ, is defined to be the supremum of lengths of chains of prime ideals of SuppŽ N . terminating with ᒍ. We have ht N ᒍ s dim R ᒍ Nᒍ . We shall say that an ideal ᑾ of R is N-proper if Nrᑾ N / 0, and, when this is the case, we define the N-height of ᑾ Žwritten ht N ᑾ . to be inf ht N ᒍ : ᒍ g SuppŽ N . l V Ž ᑾ . 4
Ž s inf ht N ᒍ : ᒍ g Ass R Ž Nrᑾ N . 4 . . For any N-proper ideal ᑾ of R, denote by gradeŽ ᑾ, N . the maximum length of all N-sequences contained in ᑾ. Suppose for the moment that Ž R, ᒊ . is local. It follows from Nakayama’s lemma that every proper ideal of R is N-proper. N is said to be a Cohen᎐Macaulay module Žabbreviated as CM module. if and only if gradeŽ ᒊ, N . s ht N ᒊ. More generally, if R is not necessarily local and N is non-zero and finitely generated, N is said to be CM if and only if Nᒊ is a CM R ᒊ -module in the above sense for each maximal ideal ᒊ g SuppŽ N .. In the following we refer to w2, 3, 5x for the basic results about CM modules. For any ideal ᑾ of R, the radical of ᑾ, denoted by RadŽ ᑾ ., is defined to be the set x g R : x n g ᑾ for some n g ⺞4 . For any R-module L, we denote by m Ass R L the set of minimal prime ideals of Ass R L. In the second section, we characterize the CM property of a non-zero finitely generated R-module N in terms of the equalities ᑾ n N and Ž ᑾ N .Ž n. for certain N-proper ideals ᑾ of R. More precisely we prove the following result: THEOREM 1.1. The following conditions are equi¨ alent: Ži. N is CM. Žii. For any N-proper ideal ᑾ of R that is generated by ht N ᑾ elements, ᑾ n N s Ž ᑾ N .Ž n. for all n g ⺞.
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The result of 1.1 is proved in 2.3. Let Ž R, ᒊ . be local and let N be a non-zero finitely generated R-module. N is said to be an unmixed module if for any ᒍ g Ass RU N U , dim RUrᒍ s dim N. More generally, if R is not necessarily local and N is non-zero finitely generated, N is a locally unmixed module if for any ᒍ g SuppŽ N ., Nᒍ is an unmixed R ᒍ-module. As the main result of Section 3 we characterize the locally unmixed property of a non-zero finitely generated R-module N in terms of the equivalence of the topologies defined by ᑾ n N 4nG 0 and Ž ᑾ N .Ž n.4nG 0 , for certain N-proper ideals ᑾ of R. More precisely we shall show that: THEOREM 1.2. The following conditions are equi¨ alent: Ži. N is locally unmixed. Žii. For each N-proper ideal ᑾ of R that is generated by ht N ᑾ elements, the topology gi¨ en by Ž ᑾ N .Ž n.4nG 0 is equi¨ alent to the ᑾ-adic topology on N. The proof of Theorem 1.2 is given in 3.12. 2. COHEN᎐MACAULAY MODULES AND SYMBOLIC POWERS Let N be a non-zero and finitely generated R-module and let ᑾ be an N-proper ideal of R. The following theorem is well known when N s R. The proof in w2, 5x can be easily carried over to a module, so we omit the proof. THEOREM 2.1.
Let N and ᑾ be as abo¨ e. Then the following hold:
Ži. ht N ᑾ s htŽ ᑾ q Ann R NrAnn R N .. Žii. Ž Krull’s principle ideal theorem. If x s x 1 , . . . , x n is a sequence of elements of R, then ht N ᒍ F n for all ᒍ g m Ass R NrxN. Furthermore, if x is an N-sequence, then any minimal element of Ass R NrxN has N-height n. In particular, ht N x s n. Žiii. If ht N ᑾ s n, then there exist x 1 , . . . , x n in ᑾ such that ht N Ž x 1 , . . . , x i . s i for i s 1, 2, . . . , n. Živ. N is CM if and only if ht N ᑿ s gradeŽ ᑿ, N . for any N-proper ideal ᑿ of R. PROPOSITION 2.2. Let N be a non-zero finitely generated R-module. Then the following conditions are equi¨ alent: Ži. N is CM. Žii. For e¨ ery N-proper ideal ᑾ of R generated by ht N ᑾ elements, m Ass R Ž Nrᑾ N . s Ass R Ž Nrᑾ N ..
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Proof. Ži. « Žii. Let ht N ᑾ [ n and let ᑾ s Ž x 1 , . . . , x n .. Suppose that ᒍ, ᒎ g Ass R Nrᑾ N, and let ᒊ be a maximal ideal of R such that ᒎ : ᒍ : ᒊ. It is enough to show that ᒎ s ᒍ. To this end, we have ᒍ R m , ᒎ R m g Ass R ᒊ Ž Nᒊ rᑾ Nᒊ . . ht N ᑾ s n and Nᒊ is CM w2, Theorem 2.1.2x imply that x 1 , . . . , x n is an Nᒊ -sequence. Therefore by w2, Theorem 2.1.3x Nᒊ rŽ x 1 , . . . , x n . Nᒊ is a CM R ᒊ -module. When ᒍ R ᒊ s ᒎ R ᒊ and so ᒍ s ᒎ, as required. In order to prove the implication Žii. « Ži. assume that ᑿ is an arbitrary N-proper ideal of R. In view of Theorem 2.1Živ., it is enough to show that ht N ᑿ s gradeŽ ᑿ, N .. To achieve this, suppose x 1 , . . . , x n in ᑿ are such that ht N Ž x 1 , . . . , x i . s i for all 1 F i F n, where n s ht N ᑿ. Thus we have x i f D ᒍ g m Ass R Nr Ž x 1 , . . . , x iy1 . N 4 ,
for all 1 F i F n.
So the condition Žii. implies that x i f D ᒍ g Ass R NrŽ x i , . . . , x iy1 . N 4 for all 1 F i F n. That is, x 1 , . . . , x n in ᑿ is an N-sequence. Now the assertion follows. We are now ready to state and prove the main theorem of this section. THEOREM 2.3. Let N be a non-zero finitely generated R-module. Then the following conditions are equi¨ alent: Ži. N is CM. Žii. For any N-proper ideal ᑾ of R generated by ht N ᑾ elements, Ž ᑾ N .Ž n. s ᑾ n N for all n g ⺞. Proof. Ži. « Žii. Suppose N is CM and let ᑾ be an arbitrary N-proper ideal of R generated by ht N ᑾ elements. Then by Proposition 2.2, we have Ass R Nrᑾ N s m Ass R Nrᑾ N. Thus, by assumption Ži. and w3, Theorem 125 and Exercise 13, p. 103x, it follows that Ass R Nrᑾ n N s m Ass R Nrᑾ n N for any n g ⺞. ŽNote that m Ass R Nrᑾ n N s m Ass R Nrᑾ N.. Now, by considering a normal primary decomposition for ᑾ n N and using the definition of Ž ᑾ N .Ž n., it is straightforward to check that Ž ᑾ N .Ž n. s ᑾ n N for all n g ⺞, as required. Žii. « Ži. Let ᑾ be an ideal of R generated by ht N ᑾ elements. In view of Proposition 2.2, it is enough to show that the associated prime ideals of Nrᑾ N are minimal. Suppose this is not the case and let ᒎ be an element of Ass R Nrᑾ N which does not belong to m Ass R Nrᑾ N. Then ᒎ s ᑾ N : R x for some x g N _ ᑾ N and q D ᒍ g m Ass R Nrᑾ N 4 . Let s g ᒎ be such that s f D ᒍ g m Ass R Nrᑾ N 4 . Then sx g ᑾ N and, so x g ᑾ N : N s. Now, the condition Žii. provides a contradiction.
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3. LOCALLY UNMIXED MODULES AND COMPARISON OF TOPOLOGIES The purpose of this section is to prove that a non-zero finitely generated module over a Noetherian ring R is locally unmixed if and only if, for any N-proper ideal ᑾ of R that can be generated by ht N ᑾ elements, the topologies ᑾ-adic and symbolic, on N, are equivalent. We begin with DEFINITION 3.1. Let N be a non-zero finitely generated R-module and let ᑾ be an ideal of R. A prime ideal ᒍ of R is called a quintessential prime ideal of ᑾ w.r.t. N precisely when there exists ᒎ g Ass RUᒍ NᒍU such that RadŽ ᑾ RUᒍ q q . s ᒍ RUᒍ . The set of quintessential primes of ᑾ w.r.t. N is denoted by Q Ž ᑾ, N .. It is easy to see that Q Ž ᑾ, N . s Q Ž ᑾ q Ann R N, N .. LEMMA 3.2. Let N be a non-zero finitely generated R-module and let ᑾ be an ideal of R. Suppose ᒍ g SuppŽ Nrᑾ N ._ m Ass R Ž Nrᑾ N . is such that the topology defined by Ž ᑾ N .Ž n., n G 1, is finer than the topology defined by Ž ᒍ N .Ž n., n G 1. Then ᒍ f Q Ž ᑾ, N .. Proof. Suppose the contrary; i.e., ᒍ g Q Ž ᑾ, N .. Let k G 1 be as w1, Proposition 3.12x. Then, for such k, there exists an integer n G 1 such that Ž ᑾ N .Ž n. : Ž ᒍ N .Ž k .. Again, from w1, Proposition 3.12x, it follows that ᒍ g Ass R Ž NrŽ ᑾ N .Ž n. .. Since Ass R NrŽ ᑾ N .Ž n. s q g Ass R Nrᑾ n N : ᒎ l S s 4 , where S s R_ j ᒎ g m Ass R Nrᑾ N 4 , it yields ᒍ : ᒎ for some ᒎ g m Ass R Nrᑾ N. Consequently, ᒍ s ᒎ, and so ᒍ g m Ass R Nrᑾ N, which is a contradiction. PROPOSITION 3.3. tions are equi¨ alent: Ži. Žii.
Let N and ᑾ be as abo¨ e. Then the following condi-
Q Ž ᑾ, N . s m Ass R Nrᑾ N. The symbolic topology is equi¨ alent to the ᑾ-adic topology.
Proof. Ži. « Žii. Let n G 1, and let ᑾ nN s Ž ᑾ N .
Ž n.
l Q 2 l ⭈⭈⭈ l Q r
where Q i is ᒍ i-primary in N
Ž2 F i F r . be a normal primary decomposition of ᑾ n N. Then, ᒍ i g SuppŽ Nrᑾ N ._ m Ass R Nrᑾ N for all 2 F i F r. It is easy to see that Ž ᒍ i N .Ž k i . : Q i for sufficiently large k i Ž2 F i F r .. Furthermore, by assumption Ži. and Lemma 3.2, it follows that Ž ᑾ N .Ž m i . : Ž ᒍ i N .Ž k i . for sufficiently large m i with 2 F i F n. Letting m be the maximum of m 2 , . . . , m r , we see that Ž ᑾ N .Ž m. : Ž ᒍ i N .Ž k i . Ž2 F i F r .. Thus Ž ᑾ N .Ž m n. : ᑾ n N, as required.
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In order to prove that Žii. « Ži., suppose that for any k G 1 there is an integer m G 1 such that Ž ᑾ N .Ž m. : ᑾ k N. Let ᒍ g Q Ž ᑾ, N .. Then Ž ᑾ N .Ž m. : Ž ᒍ N .Ž k .. By virtue of Lemma 3.2, ᒍ g m Ass R Nrᑾ N. Hence, w1, Lemma 3.5x yields the claim. LEMMA 3.4. Let N be a non-zero finitely generated R-module. Suppose that ᒍ g SuppŽ N . and let ᑿ be an Nᒍ-proper ideal of R ᒍ generated by ht N ᒍ ᑿ elements. Then there is an N-proper ideal ᒀ of R generated by ht N ᒀ elements such that ᑿ q Ann R ᒍ Nᒍ s ᒀ R ᒍ q Ann R ᒍ Nᒍ . Proof. Since ht N ᒍ ᑿ s htŽŽ ᑿ q Ann R ᒍ Nᒍ .rAnn R ᒍ Nᒍ ., by Theorem 2.1 Ži., it follows that the ideal Ž ᑿ q Ann R Nᒍ .rAnn R Nᒍ of R ᒍrAnn R Nᒍ ᒍ ᒍ ᒍ is generated by htŽŽ ᑿ q Ann R ᒍ Nᒍ .rAnn R ᒍ Nᒍ . elements. According to Verma w11, Lemma 5.1x, there is an ideal ᑾrAnn R N of RrAnn R N that is a generated by htŽ ᑾrAnn R N . s ht N ᑾ elements, and
Ž ᑿ q Ann R
Nᒍ . rAnn R ᒍ Nᒍ s Ž ᑾrAnn R N . R ᒍrAnn R ᒍ Nᒍ .
ᒍ
The result now follows. PROPOSITION 3.5. Let Ž R, ᒊ . be local and let N be a non-zero finitely generated R-module. Suppose that ᒎ g Ass RU N U with dim RUrᒎ ) 0. Then there exists an N-proper ideal ᑾ of R generated by ht N ᑾ elements, such that Rad Ž ᑾ RU q ᒎ . s ᒊ RU
ht N ᑾ s dim RUrᒎ .
and
Proof. Let ᒎ g Ass RU N U with dim RUrᒎ [ n ) 0. Using Krull’s principal ideal theorem and prime avoidance arguments one constructs elements x 1 , . . . , x n g ᒊ such that xi f
ž
D
ᒍ
ᒍgm Ass R Nr Ž x 1 , . . . , x iy1 . N
j
ž
U
U
D
/ U
ᒍ gm Ass RU N r Ž ᒎq Ž x 1 , . . . , x iy1 . R . N
U
ᒍU l R ,
/
for all 1 F i F n. Select ᑾ s Ž x 1 , . . . , x n .. We need to show that ht N ᑾ s n and ᑾ RU q ᒎ is ᒊ RU-primary. First we show ht N ᑾ s n. We prove this by induction on n. The case n s 1 follows from the principal ideal theorem, together with x 1 f Dᒍ g m Ass R N ᒍ. So let n ) 1 and suppose that the result is true for n y 1. Let ᒍ g m Ass R Nrᑾ N. Because of x n g ᒍ, we have ᒍ f m Ass R NrŽ x 1 , . . . , x ny1 . N. The result now follows from Krull’s
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principal ideal theorem and inductive hypothesis. Now we prove RadŽ ᑾ RU q ᒎ . s ᒊ RU . To this end let ᒎUn be a minimal prime over ᑾ RU q ᒎ. Then from x n g ᒎUn l R, follows that ᒎUn f m Ass RU N U rŽ ᒎ q Ž x 1 , . . . , x ny 1 . RU . N U . Therefore there is a ᒎUny1 g m Ass RU N U rŽ ᒎ q Ž x 1 , . . . , x ny 1 . RU . N U such that ᒎUny1 m ᒎUn , and so on. Thus we have a saturated chain of primes from ᒎ to ᒎUn , as ᒎ s ᒎU0 m ⭈⭈⭈ m ᒎUny1 m ᒎUn , with 1 F i F n, ᒎUi g m Ass RU N U rŽ ᒎ q Ž x 1 , . . . , x i . RU . N U . Because dim RU rᒎ s n, it follows that ᒎUn s ᒊ RU , so that RadŽ ᑾ RU q ᒎ . s ᒊ RU , as desired. DEFINITION 3.6. Let N be a non-zero finitely generated R-module. A prime ideal ᒍ of R is an essential prime of ᑾ w.r.t. N, if ᒍ s ᒎ l R for some ᒎ g Q Ž u R, N .. The set of essential primes of ᑾ w.r.t. N will be denoted by E Ž ᑾ, N .. DEFINITION 3.7. A sequence x s x 1 , . . . , x n of elements of R is called an essential sequence on N if the following conditions are satisfied: Ži. For all 1 F i F n, x i f D ᒍ g E ŽŽ x 1 , . . . , x iy1 ., N .4 . Žii. NrxN / 0. An essential sequence x s x 1 , . . . , x n of elements of R Žcontained in an ideal ᑾ . on N is maximal Žin ᑾ ., if x 1 , . . . , x n , x nq1 is not an essential sequence on N for any x nq 1 g RŽ x nq1 g ᑾ .. It is shown that Žsee w1, Theorem 4.16x. all maximal essential sequences on N in an ideal ᑾ have the same length. This allows us to introduce the fundamental notion of essential grade Žsee w1, Definition 4.17x.. LEMMA 3.8.
Let N be a non-zero finitely generated R-module and let
x s x1, . . . , x n be elements of R which form an essential sequence on N. Then the following hold: Ži. ht N Ž x 1 , . . . , x i . s i for all 1 F i F n. Žii. If N is locally unmixed, then E Ž x, N . s m Ass R NrxN. Proof. In order to prove Ži., it is sufficient to show that if ᒍ g m Ass R Ž Nr Ž x 1 , . . . , x i . N ., then ht N ᒍ s i. To this end recall that m Ass R Nrᑾ N : E Ž ᑾ, N ., for any ideal ᑾ of R, and x 1 , . . . , x i is an essential sequence on N. Putting this together the proof of Ži. follows by induction. For the proof of Žii., it is clearly sufficient to prove that E Ž x, N . : m Ass R NrxN. Let ᒍ g E Ž x, N .. Then by virtue of Ži., ht N ᒍ G n. Thus we need only to show that ht N ᒍ F n, which implies that ᒍ g m Ass R NrxN. To do this, let ht N ᒍ s k and let ᒍ 0 m ⭈⭈⭈ m ᒍ k s ᒍ be a saturated chain of primes of SuppŽ N . with ᒍ 0 g m Ass R N. Then, by
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w1, Lemmas 4.9, 3.2, 4.11x, htŽ ᒍ R ᒍrᒍ 0 R ᒍ . s n and so n G k. This completes the assertion. PROPOSITION 3.9. Let Ž R, ᒊ . be local and let N be a non-zero finitely generated R-module. Then e gradeŽ ᑾ, N . s Min htŽ ᑾ RU q ᒎrᒎ . : ᒎ g AssRU N U 4 . Proof. Let e gradeŽ ᑾ, N . s n, and let x s x 1 , . . . , x n be a maximal essential sequence on N in ᑾ. Then ᑾ : ᒍ for some ᒍ g E Ž x, N ., by w1, Theorem 3.17x. By virtue of w1, Proposition 3.8x, there exists a prime ᒎU in RU such that ᒎU l R s ᒍ and ᒎU g E Ž xRU , N U .. Furthermore, by w1, Proposition 3.6x, there is a ᒎU0 g Ass RU N U such that ᒎU0 : ᒎU and ᒎU rᒎU0 g E Ž xRU q ᒎU0 rᒎU0 , RU rᒎU0 .. Now, by w1, Lemma 4.9x, x 1 q ᒎU0 , . . . , x n q ᒎU0 is an essential sequence in the complete domain RUrᒎ 0 . Therefore, Lemma 3.8 shows that htŽ ᒎU rᒎU0 . s n. As ᑾ RU q ᒎU0 : ᒎU , we have htŽ ᑾ RU q ᒎU0 rᒎU0 . F n. Now it is easy to see that the assertion follows from w1, Theorem 4.16x. THEOREM 3.10. Let N be a non-zero finitely generated R-module and let be an essential sequence on N. Then E Ž x, N . s Q Ž x, N ..
x s x1, . . . , x n
Proof. In view of w1, Theorem 3.17x, it is sufficient to show that E Ž x, N . : Q Ž x, N .. Let ᒍ g E Ž x, N .. By w1, Lemma 3.2x, ᒍ R ᒍ g E Ž xR ᒍ , Nᒍ ., and so e gradeŽ ᒍ R ᒍ , Nᒍ . s n. Thus, by virtue of Proposition 3.9, there is a ᒎ g Ass RUᒍ NᒍU such that dim RUᒍ rᒎ s n. Furthermore, by w1, Lemma 4.9x, x 1 q ᒎ, . . . , x n q ᒎ is an essential sequence on the complete local domain RUᒍ rᒎ. So, Lemma 3.8 implies that htŽ x 1 q ᒎ, . . . , x n q ᒎ . s n. That is, htŽ xRUᒍ q ᒎrᒎ . s n. Hence ᒍ RUᒍ rᒎ is minimal over xRUᒍ q ᒎrᒎ. Consequently, RadŽ xRUᒍ q ᒎ . s ᒍ RUᒍ , as required. COROLLARY 3.11. Let N be a non-zero finitely generated R-module and let x 1 , . . . , x n in R be such that Ž x 1 , . . . , x n . N / N. Suppose N is locally unmixed and ht N Ž x 1 , . . . , x i . s i for 1 F i F n. Then x 1 , . . . , x n is an essential sequence on N. Proof. First assume that n s 1. Since E Ž0 R , N . s Ass R N, we need to show that x 1 f Dᒍ g Ass R N ᒍ. Since N is locally unmixed, Ass R N s ᒍ g SuppŽ N . : ht N ᒍ s 04 . Thus, since ht N Ž x 1 . s 1, x 1 is not a zero-divisor on N. Let n ) 1, and suppose that the result is true for n y 1. Let ᒍ g E ŽŽ x 1 , . . . , x ny1 ., N .. Then, by the inductive hypothesis and Lemma 3.8, we have x n f ᒍ. Therefore, x 1 , . . . , x n is an essential sequence on N, as desired.
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We are now ready to state and prove the main theorem of this section, which is a characterization of locally unmixed modules in terms of comparison of the topologies defined by certain decreasing families of submodules of a finitely generated module over a commutative Noetherian ring. THEOREM 3.12. Let N be a non-zero finitely generated R-module. Then the following conditions are equi¨ alent: Ži. N is locally unmixed. Žii. For e¨ ery N-proper ideal ᑾ of R generated by ht N ᑾ elements, the ᑾ-adic topology is equi¨ alent to the symbolic topology. Proof. First we show Ži. « Žii.. Let ᑾ be an N-proper ideal of R which is generated by ht N ᑾ elements. Suppose that ᒍ g Q Ž ᑾ, N .. We show that ᒍ g m Ass R Nrᑾ N. To do this, it is straightforward to check that the ideal ᑾ R ᒍ of R ᒍ can be generated by ht N ᒍ ᑾ R ᒍ elements, by Theorem 2.1Žii.. Therefore, by w1, Lemma 3.2x, we may assume that Ž R, ᒍ . is local. Let ht N ᑾ s n. By Theorem 2.1, there exist x 1 , . . . , x n in ᑾ with ht N Ž x 1 , . . . , x i . s i for all 1 F i F n. As shown in Corollary 3.11, x 1 , . . . , x n is an essential sequence on N, so e gradeŽ ᑾ, N . s n by Proposition 3.9. Now, analogous to the proof of w3, Theorem 125x, it is easy to see that ᑾ can be generated by an essential sequence of length n. Therefore by Theorem 3.10 and Lemma 3.8Žii., we have ᒍ g m Ass R Nrᑾ N. We can now use Proposition 3.3 and the fact that m Ass R Nrᑾ N : Q Ž ᑾ, N . to complete the proof of Žii.. In order to prove Žii. « Ži., suppose that ᒍ g SuppŽ N .. By Lemma 3.4 and the fact that Q Ž ᑿ, L. s Q Ž ᑿ q Ann R L, L. for any ideal ᑿ of R and any R-module L, and w1, Lemma 3.2x, we may assume without loss of generality that Ž R, ᒍ . is local. If gradeŽ ᒍ, N . s 0, then ᒍ g Ass R N. Thus ᒍ g Q Ž0 R , N .. Therefore by condition Žii. and Proposition 3.3, ᒍ g m Ass R N. So ht N ᒍ s 0; that is, dim N s 0. Whence N is a CM module, and so N is unmixed. We may assume that gradeŽ ᒍ, N . ) 0. Let ᒎ g Ass RU N U be such that dim RUrᒎ s n. We need to show that dim N s n. Because gradeŽ ᒍ, N . ) 0 we have n ) 0. Then, by virtue of Proposition 3.5 there exits an N-proper ideal ᑾ of R generated by ht N ᑾ s n elements such that RadŽ ᑾ RU q ᒎ . s ᒍ RU . Consequently, by Proposition 3.3, ᒍ g m Ass R Nrᑾ N. Hence ht N ᒍ s n and the claim is true.
ACKNOWLEDGMENTS The author is deeply grateful to the referee for his very careful reading of the original manuscript and to Professor H. Zakeri for the many helpful discussions during the preparation of this paper. The author also gratefully acknowledges the helpful leads of Professor L. J.
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Ratliff, Jr. This research is supported by Tabriz University and the Institute of Studies in Theoretical Physics and Mathematics.
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