The $a$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings
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- by Shiro Goto, Futoshi Hayasaka and Shin-ichiro Iai PDF
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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.References
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Additional Information
- Shiro Goto
- Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki, 214-8571 Japan
- MR Author ID: 192104
- 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
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1929027