The almost Gorenstein Rees algebras of parameters

Gotō, Shirō; Matsuoka, Naoyuki; Taniguchi, Naoki; Yoshida, Ken

doi:10.1016/j.jalgebra.2015.12.022

Cited by 23 publications

(18 citation statements)

References 16 publications

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“…In Section 3 we explore the case where the ideals are linearly presented over power series rings. The result (Theorem 3.1) seems to suggest that almost Gorenstein Rees algebras are still rare, when the dimension of base rings is greater than two, which we shall discuss in the forthcoming paper [8]. In Section 4 we study the Rees algebras of socle ideals Q : m in a two-dimensional regular local ring (R, m) and show that Rees algebras are not necessarily almost Gorenstein graded rings even for these ideals (Corollary 4.5).…”

Section: Introductionmentioning

confidence: 92%

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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Section: Introductionmentioning

confidence: 92%

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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“…A generalized Gorenstein ring [11] is one of the generalization of a Gorenstein ring, defined by a certain embedding of the rings into their canonical modules; see 2.2 for the precise definition. The class of generalized Gorenstein rings is a new class of Cohen-Macaulay rings, which naturally covers the class of Gorenstein rings and fills the gap in-between Cohen-Macaulay and Gorenstein properties; see [6,8,11,12,13,15,16,17,18,21,23,24,26,34]. In fact such rings extend the definition of almost Gorenstein rings which were initially defined by Barucci and Fröberg [4] over one-dimensional analytically unramified local rings, and further developed and defined by Goto, Matsuoka, and Phuong [13] over arbitrary Cohen-Macaulay local rings of dimension one.…”

Section: Introductionmentioning

confidence: 99%

Generalized Gorenstein Arf rings

Celikbas

Gotō

et al. 2019

Arkiv För Matematik

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In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fröberg concerning Arf numerical semigroup rings.2010 Mathematics Subject Classification. 13H10, 13H15, 13A30.

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“…Since b 2 = ac for some c ∈ I, we notice that ν(c) = 10. Let us write c = aρ + bη where ρ, ξ ∈ m. We then have c ∈ t 12 V , which is impossible. Consequently, ν(b) = 10, 11 as claimed.…”

Section: Remember That Extmentioning

confidence: 99%

Ulrich ideals and 2-AGL rings

2020

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The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on 2-AGL rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second one is the study of the question of when certain fiber products, so called amalgamated duplications are 2-AGL rings. We also explore Ulrich ideals in 2-AGL rings, mainly two-generated ones.

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The almost Gorenstein Rees algebras of parameters

Cited by 23 publications

References 16 publications

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

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

Generalized Gorenstein Arf rings

Ulrich ideals and 2-AGL rings

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