Good ideals in Gorenstein local rings

Authors:
Shiro Goto, Sin-Ichiro Iai and Kei-ichi Watanabe

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2309-2346

MSC (2000):
Primary 13A30; Secondary 13H10

DOI:
https://doi.org/10.1090/S0002-9947-00-02694-5

Published electronically:
November 29, 2000

MathSciNet review:
1814072

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Abstract | References | Similar Articles | Additional Information

Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.

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Additional Information

**Shiro Goto**

Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan

Email:
goto@math.meiji.ac.jp

**Sin-Ichiro Iai**

Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan

Email:
s-iai@math.meiji.ac.jp

**Kei-ichi Watanabe**

Affiliation:
Department of Mathematics, Nihon University, 156-8550 Japan

Email:
watanabe@math.chs.nihon.-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-00-02694-5

Keywords:
Rees algebra,
associated graded ring,
Cohen-Macaulay ring,
Gorenstein ring,
$\mathrm{a}$-invariant

Received by editor(s):
July 25, 1999

Published electronically:
November 29, 2000

Additional Notes:
The first and third authors are supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No. 11640049 and 10640042, respectively)

Article copyright:
© Copyright 2000
American Mathematical Society