On the almost Gorenstein property in Rees algebras of contracted ideals

TL;DR

This paper investigates when Rees algebras of contracted ideals in two-dimensional regular local rings are almost Gorenstein, revealing that the almost Gorenstein property depends on the order of the ideal.

Contribution

It characterizes the almost Gorenstein property of Rees algebras for contracted ideals based on their order, providing both positive and negative cases.

Findings

Rees algebra of contracted ideals with order ≤ 2 is almost Gorenstein.

Rees algebra of contracted ideals with order ≥ 3 may not be almost Gorenstein.

The property depends on the order of the ideal, not just on being contracted or stable.

Abstract

The question of when the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ 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.