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


Authors: Shiro Goto, Futoshi Hayasaka and Shin-ichiro Iai
Journal: Proc. Amer. Math. Soc. 131 (2003), 87-94
MSC (2000): Primary 13H05; Secondary 13H10
DOI: https://doi.org/10.1090/S0002-9939-02-06635-2
Published electronically: May 22, 2002
MathSciNet review: 1929027
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Abstract: Let $A$ be a regular local ring and let $\mathcal{F} = \{F_{n}\}_{n \in \mathbb{Z} }$ be a filtration of ideals in $A$ 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 $a$-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.


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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki, 214-8571 Japan
Email: goto@math.meiji.ac.jp

Futoshi Hayasaka
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki, 214-8571 Japan
Email: ee68048@math.meiji.ac.jp

Shin-ichiro Iai
Affiliation: Department of Mathematics, Hokkaido University of Education, Sapporo, 002-8502 Japan
Email: iai@sap.hokkyodai.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-02-06635-2
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
Received by editor(s): August 25, 2001
Published electronically: May 22, 2002
Additional Notes: The first author was supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No. 11640049).
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society

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