Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Published electronically: March 14, 2002
MathSciNet review: 1912992
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $I$ be an integrally closed ideal in a commutative Noetherian ring $A$. Then the local ring $A_{\mathfrak{p}}$ is regular (resp. Gorenstein) for every $\mathfrak{p} \in \mathrm{Ass}_{A} A/I$ if the projective dimension of $I$ is finite (resp. the Gorenstein dimension of $I$ is finite and $A$ satisfies Serre's condition (S$_{1}$)).


References [Enhancements On Off] (What's this?)

  • [A] Ian M. Aberbach, Tight closure in 𝐹-rational rings, Nagoya Math. J. 135 (1994), 43–54. MR 1295816
  • [Au] Anneaux de Gorenstein, et torsion en algèbre commutative, Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, 1966/67. Texte rédigé, d’après des exposés de Maurice Auslander, Marguerite Mangeney, Christian Peskine et Lucien Szpiro. École Normale Supérieure de Jeunes Filles, Secrétariat mathématique, Paris, 1967 (French). MR 0225844
  • [B] Lindsay Burch, On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 941–948. MR 0229634
  • [CHV] Alberto Corso, Craig Huneke, and Wolmer V. Vasconcelos, On the integral closure of ideals, Manuscripta Math. 95 (1998), no. 3, 331–347. MR 1612078, 10.1007/s002290050033
  • [G1] Shiro Goto, Vanishing of 𝐸𝑥𝑡ⁱ_{𝐴}(𝑀,𝐴), J. Math. Kyoto Univ. 22 (1982/83), no. 3, 481–484. MR 674605
  • [G2] Shiro Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), no. 1, 151–160. MR 887198, 10.1016/0021-8693(87)90128-1
  • [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 $\mathfrak {m} $-primary ideals, in preparation.
  • [GHI] S. Goto, F. Hayasaka, and S.-I. Iai, The $\mathrm{a}$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings, Proc. Amer. Math. Soc. (to appear).
  • [V] Wolmer V. Vasconcelos, Ideals generated by 𝑅-sequences, J. Algebra 6 (1967), 309–316. MR 0213345
  • [W1] Junzo Watanabe, 𝔪-full ideals, Nagoya Math. J. 106 (1987), 101–111. MR 894414
  • [W2] Junzo Watanabe, The syzygies of 𝔪-full ideals, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 1, 7–13. MR 1075117, 10.1017/S0305004100069528
  • [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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13H05, 13H10

Retrieve articles in all journals with MSC (2000): 13H05, 13H10


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