The <IMG >-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Authors:	Shiro Goto Futoshi Hayasaka Shin-ichiro Iai

Institution:	Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki, 214-8571 Japan ; Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki, 214-8571 Japan ; Department of Mathematics, Hokkaido University of Education, Sapporo, 002-8502 Japan

Abstract:	Let be a regular local ring and let $\mathcal{F} = \{F_{n}\}_{n \in \mathbb{Z} }$ be a filtration of ideals in such that $\mathcal{R}(\mathcal{F}) = \bigoplus _{n \geq 0}F_{n}$ is a Noetherian ring with $\mathrm{dim} \mathcal{R}(\mathcal{F}) = \mathrm{dim} A + 1$ . Let $\mathcal{G}(\mathcal{F}) = \bigoplus _{n \geq 0}F_{n}/F_{n+1}$ and let $\mathrm{a}(\mathcal{G}(\mathcal{F}))$ be the -invariant of $\mathcal{G}(\mathcal{F})$ . Then the theorem says that $F_{1}$ is a principal ideal and $F_{n} = F_{1}^{n}$ for all $n \in \mathbb{Z}$ if and only if $\mathcal{G}(\mathcal{F})$ is a Gorenstein ring and $\mathrm{a}(\mathcal{G}(\mathcal{F})) = -1$ . Hence $\mathrm{a}(\mathcal{G}(\mathcal{F})) \leq -2$ , if $\mathcal{G}(\mathcal{F})$ is a Gorenstein ring, but the ideal $F_{1}$ is not principal.

Keywords:	Injective dimension integrally closed ideal $\mathfrak{m}$-full ideal regular local ring Gorenstein local ring $a$-invariant Rees algebra associated graded ring filtration of ideals

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