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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].

##### 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.)