On the almost Gorenstein property in the Rees algebras of contracted ideals

Abstract

We explore the question of when the Rees algebra $\mathcal{R}(I)={\oplus }_{n\ge 0}{I}^n$ of $I$ is an almost Gorenstein graded ring, where $R$ is a 2-dimensional regular local ring and $I$ is a contracted ideal of $R$. Recently, we showed that $\mathcal{R}(I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present article show that if $I$ is a contracted ideal with $\mathrm{o}(I)\le 2$, then $\mathcal{R}(I)$ is an almost Gorenstein graded ring, while if $\mathrm{o}(I)\ge 3$, then $\mathcal{R}(I)$ is not necessarily an almost Gorenstein graded ring, even though $I$ is a contracted stable ideal. Thus both affirmative and negative answers are given.