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Primary Decomposition: Compatibility, Independence and Linear Growth PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 6, Pages 1629{1637 S 0002-9939(01)06284-0 Article electronically published on November 15, 2001 PRIMARY DECOMPOSITION: COMPATIBILITY, INDEPENDENCE AND LINEAR GROWTH YONGWEI YAO (Communicated by Wolmer V. Vasconcelos) Abstract. For finitely generated modules N ( M over a Noetherian ring R, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if Ass(M=N)=fP1;P2;:::;Psg and Qi is a Pi-primary component of N ( M for each i =1; 2;:::;s,thenN = Q1 \ Q2 \···\Qs; (2) For a given subset X = fP1;P2;:::;Pr}⊆Ass(M=N), X is an open subset of Ass(M=N) if and only if the intersections Q1 \ Q2 \ ···\ 0 \ 0 ∩···\ 0 0 Qr = Q1 Q2 Qr for all possible Pi-primary components Qi and Qi of N ( M; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals I1;I2;:::;It of R there exists a k 2 N such that for any 2 N n1 n2 ··· nt ⊂ n1;n2;:::;nt there exists a primary decomposition of I1 I2 It M M such that every P -primary component Q of that primary decomposition ··· contains P k(n1+n2+ +nt)M. 0. Introduction Throughout this paper R is a Noetherian ring and M =6 0 is a finitely generated R-module unless explicitly stated otherwise. Let N ( M be a proper R-submodule of M. By primary decomposition N = Q1 \ Q2 \···\Qs of N in M,wealways mean an irredundant and minimal primary decomposition, where Qi is a Pi-primary submodule of M, i.e. Ass(M=Qi)=fPig,foreachi =1; 2;:::;s, unless explicitly mentioned otherwise. Then Ass(M=N)=fP1;P2;:::;Psg and we say that Qi is a Pi-primary component of N in M. As a subset of Spec(R)withtheZariski topology, Ass(M=N)S inherits a topology structure. For an ideal I in R,weuse 1 i (N :M I )todenote i(N :M I ). Notation 0.1. Let N ( M be finitely generated R-modules. We use ΛP (N ( M), or ΛP if the R-modules N ( M are clear from the context, to denote the set of all possible P -primary components of N in M for every P 2 Ass(M=N). We know that if P 2 Ass(M=N) is an embedded prime ideal, then ΛP (N ( M) contains more than one element. (Also see the passage following Theorem 2.2 and the reference to [HRS].) Suppose that N = Q1 \ Q2 \···\Qs is a primary ( 2 decomposition of N M such that Qi ΛPi for i =1; 2;:::;s. Then if we choose 0 ⊆ 0 ⊆ a Pi-primary submodule Qi of M such that N Qi Qi for each i =1; 2;:::;s, 0 \ 0 \···\ 0 ( we get a primary decomposition N = Q1 Q2 Qs of N M. For example Received by the editors October 5, 2000 and, in revised form, January 12, 2001. 2000 Mathematics Subject Classification. Primary 13E05; Secondary 13C99, 13H99. Key words and phrases. Primary decomposition, Linear Growth, Artin-Rees number. c 2001 American Mathematical Society 1629 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1630 YONGWEI YAO 0 ! ni we may choose Qi =ker(T M (M=(Pi M + N))Pi ) for all ni 0togetprimary ! ni decompositions N = 1≤i≤s ker(M (M=(Pi M + N))Pi ) for all ni 0. But 00 2 given an arbitrary Qi ΛPi for each i =1; 2;:::;s,wedonotknowaprioriif 00 \ 00 \···\ 00 N = Q1 Q2 Qs . This compatibility question is answered positively in Theorem 1.1: Theorem 1.1 (Compatibility). Let N ( M be finitely generated R-modules and Ass(M=N)=fP1;P2;:::;Psg. Suppose that for each i =1; 2;:::;s, Qi is a 2 \ \···\ Pi-primary component of N in M, i.e. Qi ΛPi .ThenN = Q1 Q2 Qs, which is necessarily an irredundant and minimal primary decomposition. Definition 0.2. Let N ( M be finitely generated R-modules and X a subset of Ass(M=N), say X =fP1;P2;:::;Pr}⊆Ass(M=N)=fP1;P2;:::;Pr;Pr+1;:::;Psg. We say that the primary decompositions of N in M are independent over X,orX- independent, if for any two primary decompositions, say, N = Q1 \ Q2 \···\Qs = 0 \ 0 \···\ 0 ⊂ f 0 }⊆ ⊂ Q1 Q2 Qs,ofN M such that Qi;Qi ΛPi (N M)fori =1; 2;:::;s, \ \···\ 0 \ 0 \···\ 0 we have Q1 Q2 Qr = Q1 Q2 Qr. In this case, we denote the invariant intersection by QX (N ⊂ M), or QX if N ⊂ M is clear from the context. It is well known that primary decompositions are independent over open subsets of Ass(M=N). (See Observations 0.3 below.) Actually it turns out that indepen- dence property characterizes open subsets of Ass(M=N): Theorem 2.2. Let N ( M be finitely generated R-modules and X ⊆ Ass(M=N) be a subset of Ass(M=N). Then the primary decompositions of N in M are inde- pendent over X if and only if X is an open subset of Ass(M=N). In Section 3 we use Artin-Rees numbers to prove the following: Theorem 3.3. Let R be a Noetherian ring, M a finitely generated R-module and I1;I2;:::;It ideals of R. Then there exists a k 2 N such that for all n1;n2;:::;nt 2 N and for all ideals J ⊂ R, j j 1 k n n1 n2 ··· nt \ n1 n2 ··· nt n1 n2 ··· nt (J M + I1 I2 It M) (I1 I2 It M :M J )=I1 I2 It M; where jnj := n1 + n2 + ···+ nt. As a corollary of Theorem 3.3, we have a new proof of the `Linear Growth' property, which was first proved by I. Swanson [Sw] and then by R. Y. Sharp using different methods and in a more general situation [Sh2]: Corollary 3.4 (Linear Growth; [Sw] and [Sh2]). Let R be a Noetherian ring, M a finitely generated R-module and I1;I2;:::;It ideals of R. Then there exists a k 2 N such that for any n1;n2;:::;nt 2 N, there exists a primary decomposition of n1 n2 ··· nt ⊆ I1 I2 It M M In1 In2 ···Int M = Q \ Q \···\Q ; 1 2 t n1 n1 nrn where the Qi's are Pi-primary components of the primary decomposition such that kjnj ⊆ ··· j j Pi M Qi for all i =1; 2;:::;nrn ,wheren =(n1;n2; ;nt) and n = n1 + n2 + ···+ nt. Before ending this introductory section, we make the following well-known ob- servations, which is to the effect of saying that primary decompositions are inde- pendent over open subsets. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PRIMARY DECOMPOSITION 1631 Observations on independence 0.3. Suppose N = Q1 \ Q2 \···\Qs is a primary decomposition of N in a finitely generated R-module M such that Qi is Pi-primary for each i =1; 2;:::;s. T (1) For any ideal I ⊆ R, the intersection Q =(N : I1) is independent I6⊆Pi i M of the particular primary decomposition of N in M (cf. D. Eisenbud [Ei], page 101, Proposition 3.13). This means that the primary decompositions of N ( M are independent over X = fP 2 Ass(M=N) j I 6⊆ P g and QX = 1 (N :M I ). (2)T Alternatively, for any multiplicatively closed set W ⊂ R, the intersection Q =ker(M ! (M=N) ) is independent of the particular primary Pi\W =; i W decomposition (cf. D. Eisenbud [Ei], page 113, Exercise 3.12). That is to say that the primary decompositions of N ( M are independent over Y = fP 2 Ass(M=N) j P \ W = ;g and QY =ker(M ! (M=N)W ). 1. Compatibility The main theorem in this section is to show that all the primary components of R-modules N ( M are totally compatible in forming the primary decompositions of N ( M. Theorem 1.1 (Compatibility). Let N ( M be finitely generated R-modules and Ass(M=N)=fP1;P2;:::;Psg. Suppose that for each i =1; 2;:::;s, Qi is a 2 ( \ \ Pi-primary component of N in M, i.e. Qi ΛPi (N M).ThenN = Q1 Q2 ···\Qs, which is necessarily an irredundant and minimal primary decomposition. Proof. We induct on s, the cardinality of Ass(M=N). If s =1,thenN = Q1 and the claim is trivially true. Suppose s ≥ 2. By rearranging the order of P1;P2;:::;Ps, we may assume that 2 Ps is a maximal prime ideal in Ass(M=N). Since Qi ΛPi for i =1; 2;:::;s,we can find s specific primary decompositions N = Q(i;1) \ Q(i;2) \···\Q(i;i) \···\Q(i;s) for i =1; 2;:::;s; S 2 n where Q(i;j) ΛPj and Q(i;i) =Qi for all i; j =1; 2;:::;s.LetW =R 1≤i≤s−1 Pi. By Observation 0.3(2) and our assumption on Ps, we know that the primary de- compositions of N ( M are independent over X = fP 2 Ass(M=N) j P \ W = ;g = fP1;P2;:::;Ps−1g with QX =ker(M ! (M=N)W ). That is to say, QX =ker(M ! (M=N)W )=Q(i;1) \ Q(i;2) \···\Q(i;s−1) for i =1; 2;:::;s; are all primary decompositions of QX ( M and in particular Qi = Q(i;i) 2 ⊂ − ΛPi (QX M)fori =1; 2;:::;s 1.
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