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arXiv 1604.04747math.AC
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关于收缩理想Rees代数的几乎Gorenstein性质

On the almost Gorenstein property in Rees algebras of contracted ideals

Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, Ken-ichi Yoshida

备注：
15 pages
发表信息：
Kyoto J. Math. 59, no. 4 (2019), 769-785
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摘要：

研究了$I$的Rees代数${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$何时是几乎Gorenstein分次环的问题，其中$R$是二维正则局部环，$I$是$R$的收缩理想.对于$R$的每个整闭理想$I$，${\mathcal R} (I)$是几乎Gorenstein分次环.本文的主要结果表明：若$I$是具有$\mathrm{o}(I) \le 2$的收缩理想，则${\mathcal R} (I)$是几乎Gorenstein分次环;若$\mathrm{o}(I) \ge 3$是几乎Gorenstein分次环，则即使$I$是收缩稳定理想，${\mathcal R} (I)$也不一定是几乎Gorenstein分次环.因此，既有肯定的答案，也有否定的答案。

英文摘要：

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.

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