On the almost Gorenstein property in Rees algebras of contracted ideals

Abstract

The question of when the Rees algebra $\mathcal{R}(I)={⨁}_{n\ge 0}{I}^n$ of $I$ is an almost Gorenstein graded ring is explored, where $R$ is a two-dimensional regular local ring and $I$ a contracted ideal of $R$. It is known that $\mathcal{R}(I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present paper 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 answers and negative answers are given.

15 pages

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