On vanishing of certain Ext modules

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  • Goto Shiro
    Department of Mathematics, School of Science and Technology, Meiji University
  • Hayasaka Futoshi
    Department of Mathematics, School of Science and Technology, Meiji University
  • Takahashi Ryo
    Department of Mathematical Sciences, Faculty of Science, Shinshu University

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Let R be a Noetherian local ring with the maximal ideal \mathfrak{m} and dim R = 1. In this paper, we shall prove that the module Ext1R (R/Q, R) does not vanish for every parameter ideal Q in R, if the embedding dimension \mathrm{v}(R) of R is at most 4 and the ideal \mathfrak{m}2 kills the 0^{\underline{th}} local cohomology module H\mathfrak{m}0(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.

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