When does the equality I2 = QI hold true in Buchsbaum rings?

Gotō, Shirō; Sakurai, Hideto

doi:10.1201/9781420028324.ch9

Cited by 15 publications

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“…Theorem 1.2 (see [CHV], [CP1], [CP2], [CPV], [G2] This result has led to two directions of research to better understand the quasi-socle ideals I = Q : m q in arbitrary local rings. One direction is to weaken the assumption on base rings A, which was performed by the first and the third authors (see [GSa1], [GSa2], [GSa3]). They explored the socle ideals I = Q : m inside Buchsbaum local rings A and showed that I 2 = QI and that G(I) is a Buchsbaum ring if e 0 m (A) ≥ 2 and if Q is contained in a sufficiently high power of the maximal ideal m. The other direction was independently performed by Wang [Wan] and also by the first author, Matsuoka, Takahashi, Kimura, Phuong, and Truong (see [GMT], [GKM], [GKMP], [GKPT]).…”

Section: Here M = Mg(i) + G(i) + and [H I M (G(i))] N (I N ∈ Z) Denomentioning

confidence: 99%

Quasi-socle ideals in Buchsbaum rings

Gotō¹,

Horiuchi²,

Sakurai³

2010

Nagoya Math. J.

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Abstract. Quasi-socle ideals, that is, ideals of the form I = Q : m q (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A, m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) = n≥0 I n /I n+1 of I is Buchsbaum for all n 0, where A ( * ) denotes the length of the module. With this notation, the purpose of this article is to prove the following. The associated graded ring G(I) = n≥0 I n /I n+1 of I is a Buchsbaum ring withas A-modules for all i < d anddenotes the homogeneous component with degree n in the ith graded local cohomology module H i M (G(I)) of G(I) with respect to M . Thus, the quasi-socle ideals I = Q : m q behave very well, inside Buchsbaum rings also, under the conditions stated in Theorem 1.1. Notice that, because A is a Buchsbaum ring, the Hilbert coefficients e i Q (A) of the parameter ideal Q are given by the formulaand one has the equalityfor all n ≥ 0 (see [Sch, Korollar 3.2]), so that {e i Q (A)} 1≤i≤d are independent of the choice of Q and are invariants of A. The crucial point in Theorem 1.1 is the equality I 2 = QI; assertions (1), (2), and (3) a d = ab for some a ∈ m q and b ∈ m is rather technical, but at this moment, we do not know whether this additional condition is superfluous.We now briefly explain the background of Theorem 1.1. Our research dates back to works of A. Corso, C. Polini, C. Huneke, W. V. Vasconcelos, and the first author in which the socle ideals Q : m for parameter ideals Q in Cohen-Macaulay rings A were explored with the following result. provided that depth G(m) ≥ 2.Since Buchsbaum rings are very akin to Cohen-Macaulay rings, it seems quite natural to expect that similar results of the Cohen-Macaulay case, such as Theorem 1.3, should be true also in the Buchsbaum case after suitable modifications of the corresponding conditions, which we now report in Theorem 1.1.The proof of Theorem 1.1 is given in Section 2, which we divide into two parts. The first part shows that m q I = m q Q. The second part proves that I 2 = QI. Since A is not necessarily a Cohen-Macaulay ring, the equality I 2 = QI does not readily follow from the fact that m q I = m q Q. We carefully analyze this phenomenon in Section 2. A similar but more restricted result also holds true in the case where G(m) is a Buchsbaum ring with depth G(m) = 1, which we discuss in Section 3. In Section 4, we give exam- Let F denote the set of all the productswhere the family {f i } 1≤i≤q of elements in m \ m 2 is assumed to satisfy the conditions stated above. Let us writeWe put g j = 1≤k≤q,k =j f k for each 1 ≤ j ≤ q, and we choose g ∈ m \ m 2 so that g * , f * j is a regular sequence in G(m) for all 1 ≤ j ≤ q. Letwhence, for all 1 ≤ i ≤ d and 1 ≤ j ≤ q, we get Hence,and soWe may assume that k > 1 and that our assertion holds true for k − 1. Hence,Let y ∈ A, and assume thatThanks to Claim 1, we getWe need the following to show the equality I 2 = QI. Proof. The former assertion is proved exactly in the same way as [GSa3, Lemma 2.3], wher...

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Section: Here M = Mg(i) + G(i) + and [H I M (G(i))] N (I N ∈ Z) Denomentioning

confidence: 99%

Quasi-socle ideals in Buchsbaum rings

Gotō¹,

Horiuchi²,

Sakurai³

2010

Nagoya Math. J.

Self Cite

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“…Let M be a finitely-generated d-dimensional A-module with finite local cohomologies. Then we have the following inequalities: In two recent papers [8,9], Goto with H. Sakurai has returned to the study of the index of reducibility of parameter ideals in order to investigate when the equality I 2 = QI holds for a parameter ideal Q in A, where I = (Q : A m). According to earlier research of A. Corso, C. Huneke, C. Polini, and W. Vasconcelos [2][3][4], this equality holds for all parameter ideals Q in case A is a Cohen-Macaulay ring which is not regular.…”

Section: Introductionmentioning

confidence: 98%

The index of reducibility of parameter ideals in low dimension

Rogers

2004

Journal of Algebra

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In this paper we present results concerning the following question: if M is a finitely-generated module with finite local cohomologies over a Noetherian local ring (A, m), does there exist an integer such that every parameter ideal for M contained in m has the same index of reducibility? We show that the answer is yes if dim M = 1 or if dim M = 2 and depth M > 0. This research is closely related to work of Goto-Suzuki and Goto-Sakurai; Goto-Sakurai have supplied an answer of yes in case M is Buchsbaum.

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“…Set I = (Q : R n). Then according to [GSa,Proposition 2.3], we have that nI = nQ. Since a ∈ I, na ⊆ nI = nQ = nay 1 .…”

Section: Proposition 41 [Eg Theorem A]mentioning

confidence: 99%

“…In 2003, Goto and H. Sakurai showed that if M is Buchsbaum (i.e., m is a standard ideal for M), then there exists a power of m inside which every parameter ideal for M has the same index of reducibility on M [GSa,Corollary 3.13], necessarily equal to the lower bound of the Goto-Suzuki type. We refer to this by saying M has eventual constant index of reducibility of parameter ideals.…”

Section: Introductionmentioning

confidence: 99%

The Index of Reducibility of Parameter Ideals and Mostly Zero Finite Local Cohomologies

Liu

Rogers

2006

Communications in Algebra

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Abstract. In this paper we prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring (A, m), and if H i m (M ) = 0 except possibly for i ∈ {0, r, d} with some 0 ≤ r ≤ d, then there exists an integer ℓ such that every parameter ideal for M contained in m ℓ has the same index of reducibility. This theorem generalizes earlier work of the second author and is closely related to recent work of GotoSuzuki and Goto-Sakurai; Goto-Sakurai have supplied an answer of yes in case M is Buchsbaum.

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When does the equality I2 = QI hold true in Buchsbaum rings?

Cited by 15 publications

References 4 publications

Quasi-socle ideals in Buchsbaum rings

Quasi-socle ideals in Buchsbaum rings

The index of reducibility of parameter ideals in low dimension

The Index of Reducibility of Parameter Ideals and Mostly Zero Finite Local Cohomologies

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