Good ideals in Gorenstein local rings
Authors:
Shiro Goto, SinIchiro Iai and Keiichi Watanabe
Journal:
Trans. Amer. Math. Soc. 353 (2001), 23092346
MSC (2000):
Primary 13A30; Secondary 13H10
Published electronically:
November 29, 2000
MathSciNet review:
1814072
Fulltext PDF Free Access
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Abstract: 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 finitedimensional 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 RiemannRoch theorem, we are able to classify the good ideals in twodimensional 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, 2148571 Japan
Email:
goto@math.meiji.ac.jp
SinIchiro Iai
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, 2148571 Japan
Email:
siai@math.meiji.ac.jp
Keiichi Watanabe
Affiliation:
Department of Mathematics, Nihon University, 1568550 Japan
Email:
watanabe@math.chs.nihon.u.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994700026945
PII:
S 00029947(00)026945
Keywords:
Rees algebra,
associated graded ring,
CohenMacaulay 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 GrantinAid for Scientific Researches in Japan (C(2), No. 11640049 and 10640042, respectively)
Article copyright:
© Copyright 2000
American Mathematical Society
