One-Dimensional Almost Gorenstein Rings

Barucci, Valentina; Fröberg, Ralf

doi:10.1006/jabr.1996.6837

Cited by 165 publications

(185 citation statements)

References 17 publications

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“…In [8] Barucci and Fröberg describe a notion for a one-dimensional ring to be 'almost' Gorenstein, and give R = k[X, Y, Z]/(XY, XZ, Y Z) as an example of an almost Gorenstein ring in their sense. However, it is not hard to show that curv R ω = curv R k, in other words, g(R) = 1 (R is in fact a Golod ring).…”

Section: Non-extremalitymentioning

confidence: 99%

On the growth of the Betti sequence of the canonical module

Jorgensen

Leuschke

2007

Math. Z.

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Abstract. We study the growth of the Betti sequence of the canonical module of a Cohen-Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen-Macaulay ring possessing a canonical module to be Gorenstein.

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Section: Non-extremalitymentioning

confidence: 99%

On the growth of the Betti sequence of the canonical module

Jorgensen

Leuschke

2007

Math. Z.

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“…The notion of these local rings dates back to the paper [2] of V. Barucci and R. Fröberg in 1997, where they dealt with one-dimensional analytically unramified local rings. Because their notion is not flexible enough to analyze analytically ramified rings, in 2013 S. Goto, N. Matsuoka, and T. T. Phuong [4] extended the notion to arbitrary (but still of dimension one) Cohen-Macaulay local rings.…”

Section: Introductionmentioning

confidence: 99%

The almost Gorenstein Rees algebras over two-dimensional regular local rings

Gotō

Matsuoka

Taniguchi

et al. 2016

Journal of Pure and Applied Algebra

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Abstract. Let (R, m) be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra R(I) = n≥0 I n of I is an almost Gorenstein graded ring in the sense of [6] for every m-primary integrally closed ideal I in R. IntroductionThe main purpose of this paper is to prove the following. Theorem 1.1. Let (R, m) be a two-dimensional regular local ring with infinite residue class field and I an m-primary integrally closed ideal in R. Then the Rees algebra R(I) = n≥0 I n of I is an almost Gorenstein graded ring.As a direct consequence one has the following.Corollary 1.2. Let (R, m) be a two-dimensional regular local ring with infinite residue class field. Then R(m ℓ ) is an almost Gorenstein graded ring for every integer ℓ > 0.The proof of Theorem 1.1 depends on a result of J. Verma [19] which guarantees the existence of joint reductions with joint reduction number zero. Therefore our method of proof works also for two-dimensional rational singularities, which we shall discuss in the forthcoming paper [7]. Here, before entering details, let us recall the notion of almost Gorenstein graded/local ring as well as some historical notes about it.Almost Gorenstein rings are a new class of Cohen-Macaulay rings, which are not necessarily Gorenstein, but still good, possibly next to the Gorenstein rings. The notion of these local rings dates back to the paper

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“…This trend is explained by its extensive use in related areas such as number theory or algebraic geometry, where numerical semigroups appear naturally, as mentioned in the introduction of this paper. This is the case of the valuation ring, K[[S]], of a numerical semigroup S, which is of type Gorenstein or Kunz when S is irreducible (see [4]). Decomposing a numerical semigroup into irreducible becomes particularly useful in the case of valuation rings since Downloaded 02/25/16 to 150.214.182.169.…”

Section: Minimal Decomposition Of S Into M-irreducible Numerical Semimentioning

confidence: 99%

An Application of Integer Programming to the Decomposition of Numerical Semigroups

Blanco¹,

Puerto²

2012

SIAM J. Discrete Math.

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Abstract. This paper addresses the problem of decomposing a numerical semigroup into mirreducible numerical semigroups. The problem originally stated in algebraic terms is translated, introducing the so-called Kunz-coordinates, to resolve a series of several discrete optimization problems. First, we prove that finding a minimal m-irreducible decomposition is equivalent to solve a multiobjective linear integer problem. Then, we restate that problem as the problem of finding all the optimal solutions of a finite number of single objective integer linear problems plus a set covering problem. Finally, we prove that there is a suitable transformation that reduces the original problem to find an optimal solution of a compact integer linear problem. This result ensures a polynomial time algorithm for each given multiplicity m. We have implemented the different algorithms and have performed some computational experiments to show the efficiency of our methodology.

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One-Dimensional Almost Gorenstein Rings

Cited by 165 publications

References 17 publications

On the growth of the Betti sequence of the canonical module

On the growth of the Betti sequence of the canonical module

The almost Gorenstein Rees algebras over two-dimensional regular local rings

An Application of Integer Programming to the Decomposition of Numerical Semigroups

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