On vanishing of certain Ext modules

Shiro Goto; Futoshi Hayasaka; Ryo Takahashi

doi:10.2969/jmsj/06041045

On vanishing of certain Ext modules

Shiro Goto, Futoshi Hayasaka, Ryo Takahashi

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

Let R be a Noetherian local ring with the maximal ideal m and dimR = 1. In this paper, we shall prove that the module Ext¹_R does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m² kills the O^th local cohomology module H^o_m(R). The assertion is no longer true unless v(R) ≤ 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

Original language	English
Pages (from-to)	1045-1064
Number of pages	20
Journal	Journal of the Mathematical Society of Japan
Volume	60
Issue number	4
DOIs	https://doi.org/10.2969/jmsj/06041045
Publication status	Published - Oct 2008
Externally published	Yes

Keywords

G-dimension
Parameter ideal
Vanishing of Ext

ASJC Scopus subject areas

Mathematics(all)

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10.2969/jmsj/06041045

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abstract = "Let R be a Noetherian local ring with the maximal ideal m and dimR = 1. In this paper, we shall prove that the module Ext1R does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m2 kills the Oth local cohomology module Hom(R). The assertion is no longer true unless v(R) ≤ 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.",

keywords = "G-dimension, Parameter ideal, Vanishing of Ext",

author = "Shiro Goto and Futoshi Hayasaka and Ryo Takahashi",

year = "2008",

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language = "English",

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T1 - On vanishing of certain Ext modules

AU - Goto, Shiro

AU - Hayasaka, Futoshi

AU - Takahashi, Ryo

PY - 2008/10

Y1 - 2008/10

N2 - Let R be a Noetherian local ring with the maximal ideal m and dimR = 1. In this paper, we shall prove that the module Ext1R does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m2 kills the Oth local cohomology module Hom(R). The assertion is no longer true unless v(R) ≤ 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

AB - Let R be a Noetherian local ring with the maximal ideal m and dimR = 1. In this paper, we shall prove that the module Ext1R does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m2 kills the Oth local cohomology module Hom(R). The assertion is no longer true unless v(R) ≤ 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

KW - G-dimension

KW - Parameter ideal

KW - Vanishing of Ext

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JF - Journal of the Mathematical Society of Japan

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ER -

On vanishing of certain Ext modules

Abstract

Keywords

ASJC Scopus subject areas

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