Ideals with sliding depth.

*(English)*Zbl 0561.13014This paper studies a class of ideals of a Cohen-Macaulay local ring R, somewhat intermediate between complete intersections and general Cohen- Macaulay ideals. Let \(I=(x_ 1,...,x_ n)=({\mathfrak x})\) be an ideal of R and denote by \(H_*({\mathfrak x})\) the homology of the ordinary Koszul complex \(K_*({\mathfrak x})\) built on the sequence \({\mathfrak x}\). I is said to satisfy sliding depth if \(depth H_ i({\mathfrak x})\geq \dim (R)-n+i,i\geq 0.\) This definition depends solely on the number of elements in the sequence \({\mathfrak x}\). This property localizes and is an invariant of the even linkage class of I. - These ideals have appeared earlier in two settings: (i) The analysis of the arithmetical properties of the Rees algebra of I and its associated graded ring; (ii) A generalization (with corrections) by C. Huneke [Trans. Am. Math. Soc. 277, 739-763 (1983; Zbl 0514.13011)] of a result of M. Artin and M. Nagata on residual Cohen-Macaulayness [J. Math. Kyoto Univ. 12, 307-323 (1972; Zbl 0263.14019)].

One of the aims is to define the distinction between this property and the stronger condition that the Koszul homology be Cohen-Macaulay. This can be reasonably well analyzed for ideals that differ a bit from being a complete intersection. Another one is to find classes of such ideals in terms of other invariants of the ideals - e.g. the dual of the canonical module for ideals of codimension three. (More recently we have used a computer to examine these properties.) Finally, we prove the naturality of sliding depth in the proof in Huneke’s paper.

One of the aims is to define the distinction between this property and the stronger condition that the Koszul homology be Cohen-Macaulay. This can be reasonably well analyzed for ideals that differ a bit from being a complete intersection. Another one is to find classes of such ideals in terms of other invariants of the ideals - e.g. the dual of the canonical module for ideals of codimension three. (More recently we have used a computer to examine these properties.) Finally, we prove the naturality of sliding depth in the proof in Huneke’s paper.

##### MSC:

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

13A15 | Ideals and multiplicative ideal theory in commutative rings |

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |

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