### Abstract

Let A be a regular local ring and let ℱ = {F_{n}}_{n∈Zdbl;} be a filtration of ideals in A such that ℛ(ℱ) = ⊕_{n≥0} F_{n} is a Noetherian ring with dim ℛ(ℱ) dim A + 1. Let script G sign (ℱ) = ⊕_{n≥0} F_{n}/F_{n+1} and let a(script G sign(ℱ)) be the a-invariant of script G sign(ℱ). Then the theorem says that F_{1} is a principal ideal and F_{n} = F_{1}^{n} for all n ∈ Zdbl; if and only if script G sign (ℱ) is a Gorenstein ring and a script G sign(ℱ)) = -1. Hence a script G sign(ℱ)) ≤ -2, if script G sign (ℱ) is a Gorenstein ring, but the ideal F_{1} is not principal.

Original language | English |
---|---|

Pages (from-to) | 87-94 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 131 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

- a-invariant
- Associated graded ring
- Filtration of ideals
- Gorenstein local ring
- Injective dimension
- Integrally closed ideal
- m-full ideal
- Rees algebra
- Regular local ring

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**The a-invariant and gorensteinness of graded rings associated to filtrations of ideals in regular local rings.** / Goto, Shiro; Hayasaka, Futoshi; Iai, Shin Ichiro.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 131, no. 1, pp. 87-94. https://doi.org/10.1090/S0002-9939-02-06635-2

}

TY - JOUR

T1 - The a-invariant and gorensteinness of graded rings associated to filtrations of ideals in regular local rings

AU - Goto, Shiro

AU - Hayasaka, Futoshi

AU - Iai, Shin Ichiro

PY - 2003/1

Y1 - 2003/1

N2 - Let A be a regular local ring and let ℱ = {Fn}n∈Zdbl; be a filtration of ideals in A such that ℛ(ℱ) = ⊕n≥0 Fn is a Noetherian ring with dim ℛ(ℱ) dim A + 1. Let script G sign (ℱ) = ⊕n≥0 Fn/Fn+1 and let a(script G sign(ℱ)) be the a-invariant of script G sign(ℱ). Then the theorem says that F1 is a principal ideal and Fn = F1n for all n ∈ Zdbl; if and only if script G sign (ℱ) is a Gorenstein ring and a script G sign(ℱ)) = -1. Hence a script G sign(ℱ)) ≤ -2, if script G sign (ℱ) is a Gorenstein ring, but the ideal F1 is not principal.

AB - Let A be a regular local ring and let ℱ = {Fn}n∈Zdbl; be a filtration of ideals in A such that ℛ(ℱ) = ⊕n≥0 Fn is a Noetherian ring with dim ℛ(ℱ) dim A + 1. Let script G sign (ℱ) = ⊕n≥0 Fn/Fn+1 and let a(script G sign(ℱ)) be the a-invariant of script G sign(ℱ). Then the theorem says that F1 is a principal ideal and Fn = F1n for all n ∈ Zdbl; if and only if script G sign (ℱ) is a Gorenstein ring and a script G sign(ℱ)) = -1. Hence a script G sign(ℱ)) ≤ -2, if script G sign (ℱ) is a Gorenstein ring, but the ideal F1 is not principal.

KW - a-invariant

KW - Associated graded ring

KW - Filtration of ideals

KW - Gorenstein local ring

KW - Injective dimension

KW - Integrally closed ideal

KW - m-full ideal

KW - Rees algebra

KW - Regular local ring

UR - http://www.scopus.com/inward/record.url?scp=0037242750&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037242750&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-02-06635-2

DO - 10.1090/S0002-9939-02-06635-2

M3 - Article

AN - SCOPUS:0037242750

VL - 131

SP - 87

EP - 94

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -