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
DOIs
Publication statusPublished - Feb 2003
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Finite homological dimension and primes associated to integrally closed ideals, II'. Together they form a unique fingerprint.

  • Cite this