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 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 |
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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 journal › Article
}
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 -