Quasi-socle ideals in local rings with Gorenstein tangent cones

Gotō, Shirō; Kimura, Satoru; Matsuoka, Naoyuki; Phuong, Tran Thi

doi:10.1216/jca-2009-1-4-603

Cited by 5 publications

(9 citation statements)

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“…We put I = Q : m q and refer to those ideals as quasi-socle ideals in A. In this paper we are interested in the following question about quasi-socle ideals I, which are also the main subject of the researches [1][2][3]. The present research is a continuation of [1][2][3] and aims mainly at the analysis of the case where A is a complete intersection with dim A = 1.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…In this paper we are interested in the following question about quasi-socle ideals I, which are also the main subject of the researches [1][2][3]. The present research is a continuation of [1][2][3] and aims mainly at the analysis of the case where A is a complete intersection with dim A = 1. Following Heinzer and Swanson [4], for each parameter ideal Q in a Noetherian local ring (A, m) we define g(Q ) = max{q ∈ Z | Q : m q ⊆ Q } and call it the Goto number of Q .…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…As is suggested in [1], the one-dimensional case is substantially different from higher-dimensional cases and more complicated to control. This observation has led Goto, Kimura, Matsuoka, and Phuong to the researches [2,3]. In [2] quasi-socle ideals with respect to monomial parameter ideals in Gorenstein numerical semigroup rings over fields are explored.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…In [2] quasi-socle ideals with respect to monomial parameter ideals in Gorenstein numerical semigroup rings over fields are explored. In [3] the authors assumed that G(m) = n≥0 m n /m n+1 is a Gorenstein ring and parameter ideals Q are diagonal, that is, Q is generated by a system of parameters of the form x…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…A which generate a reduction of the maximal ideal m. The present research is a continuation of [1][2][3] and the main purpose is to go beyond the restriction in [3] that the parameter ideals Q = (x a 1 1 , x a 2 2 , . .…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Quasi-socle ideals and Goto numbers of parameters

Gotō

Kimura

Phuong

et al. 2010

Journal of Pure and Applied Algebra

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a b s t r a c tGoto numbers g(Q ) = max{q ∈ Z | Q : m q is integral over Q } for certain parameter ideals Q in a Noetherian local ring (A, m) with Gorenstein associated graded ring G(m) = n≥0 m n /m n+1 are explored. As an application, the structure of quasi-socle ideals I = Q : m q (q ≥ 1) in a one-dimensional local complete intersection and the question of when the graded rings G(I) = n≥0 I n /I n+1 are Cohen-Macaulay are studied in the case where the ideals I are integral over Q .

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Quasi-socle ideals and Goto numbers of parameters

Gotō

Kimura

Phuong

et al. 2010

Journal of Pure and Applied Algebra

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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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Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

Bryant

2010

Communications in Algebra

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The Goto number of a parameter ideal Q in a Noetherian local ring (R, m) is the largest integer q such that Q : m q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x a1 , x a2 , . . . , x aν ]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x a1 R to be Gorenstein are also given, again using the semigroup of R.Date: October 31, 2018.

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Quasi-socle ideals in local rings with Gorenstein tangent cones

Cited by 5 publications

References 23 publications

Quasi-socle ideals and Goto numbers of parameters

Quasi-socle ideals and Goto numbers of parameters

Quasi-socle ideals in Buchsbaum rings

Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

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