### 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 pd_{R}I of I is finite. Let q = (a_{1}, a_{2},..., a_{t})R be an ideal in a Noetherian local ring R generated by a maximal R-regular sequence a_{1}, a_{2},..., a_{t} 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 I^{2} = qI, unless A is a regular local ring. Consequences are discussed.

Original language | English |
---|---|

Pages (from-to) | 631-639 |

Number of pages | 9 |

Journal | Kyoto Journal of Mathematics |

Volume | 42 |

Issue number | 4 |

Publication status | Published - 2003 |

Externally published | Yes |

### Fingerprint

### 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

*Kyoto Journal of Mathematics*,

*42*(4), 631-639.

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

Research output: Contribution to journal › Article

*Kyoto Journal of Mathematics*, vol. 42, no. 4, pp. 631-639.

}

TY - JOUR

T1 - Finite homological dimension and primes associated to integrally closed ideals, II

AU - Goto, Shiro

AU - Hayasaka, Futoshi

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - Associated graded ring

KW - Cohen-Macaulay local ring

KW - Gorenstein local ring

KW - Integrally closed ideal

KW - m-full ideal

KW - Projective dimension

KW - Rees algebra

KW - Regular local ring

UR - http://www.scopus.com/inward/record.url?scp=0038171257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038171257&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0038171257

VL - 42

SP - 631

EP - 639

JO - Journal of Mathematics of Kyoto University

JF - Journal of Mathematics of Kyoto University

SN - 0023-608X

IS - 4

ER -