The -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 be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal 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