Artin-Nagata properties and reductions of ideals.

*(English)*Zbl 0821.13008
Heinzer, William J. (ed.) et al., Commutative algebra: syzygies, multiplicities, and birational algebra: AMS-IMS-SIAM summer research conference on commutative algebra, held July 4-10, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 159, 373-400 (1994).

Let \(R\) be a Cohen-Macaulay ring and \(I\) an ideal of \(R\) of height \(g\). Let \(s\) be an integer \(\geq g\). We say that an ideal \(J (\neq R)\) is an \(s\)-residual intersection of \(I\) if \(J = {\mathfrak a} : I\) for some ideal \({\mathfrak a} \subset I\) with ht\( J \geq s \geq \mu ({\mathfrak a}) \), where \(\mu (\;)\) denotes the minimal number of generators. If, moreover, ht\( (I + J) \geq s\), we say that \(J\) is a geometric \(s\)-residual intersection of \(I\).

We say that \(I\) satisfies \(AN_ s\) (resp. \(AN^ -_ s)\) if for every \(g \leq i \leq s\) and every \(s\)-residual intersection (resp. geometric \(\dots\)) \(J\) of \(I\), the ring \(R/J\) is Cohen-Macaulay. The notion of \(s\)- residual intersection was essentially introduced by M. Artin and M. Nagata [J. Math. Kyoto Univ. 12, 307-323 (1972; Zbl 0263.14019)], and generalizes the concept of linkage. In this work the author describes and characterizes \(AN_ s\) and \(AN^ -_ s\) conditions in terms of the depths of \(R/I^ j\) and \(R/I^{(j)}\) for some values of \(j\). – As an application, he extends results by Huckaba, Huneke and Vasconcelos concerning Rees algebras of certain ideals and their reductions.

For the entire collection see [Zbl 0790.00007].

We say that \(I\) satisfies \(AN_ s\) (resp. \(AN^ -_ s)\) if for every \(g \leq i \leq s\) and every \(s\)-residual intersection (resp. geometric \(\dots\)) \(J\) of \(I\), the ring \(R/J\) is Cohen-Macaulay. The notion of \(s\)- residual intersection was essentially introduced by M. Artin and M. Nagata [J. Math. Kyoto Univ. 12, 307-323 (1972; Zbl 0263.14019)], and generalizes the concept of linkage. In this work the author describes and characterizes \(AN_ s\) and \(AN^ -_ s\) conditions in terms of the depths of \(R/I^ j\) and \(R/I^{(j)}\) for some values of \(j\). – As an application, he extends results by Huckaba, Huneke and Vasconcelos concerning Rees algebras of certain ideals and their reductions.

For the entire collection see [Zbl 0790.00007].

Reviewer: H.Matsumura (Fukuoka)

##### MSC:

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

13C40 | Linkage, complete intersections and determinantal ideals |

13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |

13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |