Finite homological dimension and primes associated to integrally closed ideals

Authors:
Shiro Goto and Futoshi Hayasaka

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3159-3164

MSC (2000):
Primary 13H05; Secondary 13H10

DOI:
https://doi.org/10.1090/S0002-9939-02-06436-5

Published electronically:
March 14, 2002

MathSciNet review:
1912992

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an integrally closed ideal in a commutative Noetherian ring . Then the local ring is regular (resp. Gorenstein) for every if the projective dimension of is finite (resp. the Gorenstein dimension of is finite and satisfies Serre's condition (S)).

**[A]**I. M. Aberbach,*Tight closure in F-rational rings*, Nagoya Math. J.**135**(1994), 43-54. MR**95g:13020****[Au]**M. Auslander,*Anneaux de Gorenstein et torsion en algèbre commutative*, Séminaire d'algèbre commutative dirigé par Pierre Samuel 1966/67, École Normale Supérieure de Jeunes Fillies, 1967. MR**37:1435****[B]**L. Burch,*On ideals of finite homological dimension in local rings*, Proc. Camb. Phil. Soc.**64**(1968), 941-948. MR**37:5208****[CHV]**A. Corso, C. Huneke, and W. V. Vasconcelos,*On the integral closure of ideals*, Manuscripta Math.**95**(1998), 331-347. MR**99b:13010****[G1]**S. Goto,*Vanishing of*, J. Math. Kyoto Univ.**22**(1982), 481-484. MR**84c:13019****[G2]**S. Goto,*Integral closedness of complete-intersection ideals*, J. Alg.**108**(1987), 151-160. MR**88d:13015****[GH1]**S. Goto and F. Hayasaka,*Finite homological dimension and primes associated to integrally closed ideals II*, Preprint 2001.**[GH2]**S. Goto and F. Hayasaka,*Gorenstein integrally closed**-primary ideals*, in preparation.**[GHI]**S. Goto, F. Hayasaka, and S.-I. Iai,*The**-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings*, Proc. Amer. Math. Soc. (to appear).**[V]**W. Vasconcelos,*Ideals generated by**-sequences*, J. Alg.**6**(1967), 309-316. MR**35:4209****[W1]**J. Watanabe,*-full ideals*, Nagoya Math. J.**106**(1987), 101-111. MR**88g:13003****[W2]**J. Watanabe,*The syzygies of**-full ideals*, Math. Proc. Cambridge Phil. Soc.**109**(1991), 7-13. MR**92b:13019****[YW]**K. Yoshida and K. Watanabe,*Hilbert-Kunz multiplicity, McKay correspondence, and good ideals in two-dimensional rational singularities*, Manuscripta Math.**104**(2001), 275-294. CMP**2001:11**

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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

**Futoshi Hayasaka**

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

Email:
ee68048@math.meiji.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-02-06436-5

Keywords:
Projective dimension,
Gorenstein dimension,
integrally closed ideal,
$\mathfrak{m}$-full ideal,
regular local ring,
Gorenstein local ring

Received by editor(s):
January 1, 2001

Received by editor(s) in revised form:
June 8, 2001

Published electronically:
March 14, 2002

Additional Notes:
The first author was supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No. 13640044).

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2002
American Mathematical Society