Almost Gorenstein rings – towards a theory of higher dimension

Gotō, Shirō; Takahashi, Ryo; Taniguchi, Naoki

doi:10.1016/j.jpaa.2014.09.022

Cited by 109 publications

(158 citation statements)

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“…The reader may consult [4] for examples of analytically ramified almost Gorenstein local rings. In 2015 S. Goto, R. Takahashi, and N. Taniguchi [6] finally gave the definition of almost Gorenstein graded/local rings of higher dimension. In [6] the authors developed a very nice theory, exploring many concrete examples.…”

Section: Introductionmentioning

confidence: 99%

“…In 2015 S. Goto, R. Takahashi, and N. Taniguchi [6] finally gave the definition of almost Gorenstein graded/local rings of higher dimension. In [6] the authors developed a very nice theory, exploring many concrete examples. We recall here the definition of almost Gorenstein graded rings given in [6].…”

Section: Introductionmentioning

confidence: 99%

“…In [6] the authors developed a very nice theory, exploring many concrete examples. We recall here the definition of almost Gorenstein graded rings given in [6]. We refer to [6,Definition 1.1] for the definition of almost Gorenstein local rings.…”

Section: Introductionmentioning

confidence: 99%

“…We recall here the definition of almost Gorenstein graded rings given in [6]. We refer to [6,Definition 1.1] for the definition of almost Gorenstein local rings. Definition 1.3 ([6, Definition 1.5]).…”

Section: Introductionmentioning

confidence: 99%

“…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. …”

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confidence: 99%

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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: 99%