On vanishing of certain Ext modules

Abstract

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 ${\text{Ext}}_R^1(R/Q,R)$ does not vanish for every parameter ideal $Q$ in $R$ , if the embedding dimension $\text{v}(R)$ of $R$ is at most $4$ and the ideal ${\text{m}}^2$ kills the $0^{\underset{¯}{th}}$ local cohomology module $H_{\mathfrak{m}}^0(R)$ . The assertion is no longer true unless $\text{v}(R)\le 4$ . Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.