Finite homological dimension and primes associated to integrally closed ideals, II

Shiro Goto, Futoshi Hayasaka

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Let R be a Noetherian local ring with the maximal ideal m. Assume that R contains ideals I and J satisfying the conditions (1) I ⊆ J, (2) I : m ⊈ J, and (3) J is m-full, that is mJ : x = J for some x ∈ m. Then the theorem says that R is a regular local ring, if the projective dimension pdRI of I is finite. Let q = (a1, a2,..., at)R be an ideal in a Noetherian local ring R generated by a maximal R-regular sequence a1, a2,..., at and let q̄ denote the integral closure of q. Then, thanks to the theorem applied to the ideals I = q : m and J = q̄, it follows that I2 = qI, unless A is a regular local ring. Consequences are discussed.

Original languageEnglish
Pages (from-to)631-639
Number of pages9
JournalKyoto Journal of Mathematics
Volume42
Issue number4
Publication statusPublished - 2003
Externally publishedYes

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Homological Dimension
Associated Primes
Closed Ideals
Regular Local Ring
Noetherian Ring
Local Ring
Regular Sequence
Integral Closure
Projective Dimension
Maximal Ideal
Theorem
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Keywords

  • Associated graded ring
  • Cohen-Macaulay local ring
  • Gorenstein local ring
  • Integrally closed ideal
  • m-full ideal
  • Projective dimension
  • Rees algebra
  • Regular local ring

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Finite homological dimension and primes associated to integrally closed ideals, II. / Goto, Shiro; Hayasaka, Futoshi.

In: Kyoto Journal of Mathematics, Vol. 42, No. 4, 2003, p. 631-639.

Research output: Contribution to journalArticle

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