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

Shiro Goto; Futoshi Hayasaka; Shin Ichiro Iai

doi:10.1090/S0002-9939-02-06635-2

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

Shiro Goto, Futoshi Hayasaka, Shin Ichiro Iai

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

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 F₁ is a principal ideal and F_n = 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.

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	https://doi.org/10.1090/S0002-9939-02-06635-2
Publication status	Published - Jan 2003
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

Access to Document

10.1090/S0002-9939-02-06635-2

Cite this

@article{80d0e7285df4443c98457b6a9ba16370,

title = "The a-invariant and gorensteinness of graded rings associated to filtrations of ideals in regular local rings",

abstract = "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.",

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

author = "Shiro Goto and Futoshi Hayasaka and Iai, {Shin Ichiro}",

year = "2003",

month = jan,

doi = "10.1090/S0002-9939-02-06635-2",

language = "English",

volume = "131",

pages = "87--94",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "1",

}

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 - Associated graded ring

KW - Filtration of ideals

KW - Gorenstein local ring

KW - Injective dimension

KW - Integrally closed ideal

KW - Rees algebra

KW - Regular local ring

KW - a-invariant

KW - m-full ideal

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

SN - 0002-9939

VL - 131

SP - 87

EP - 94

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this