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Ideals with sliding depth. (English) Zbl 0561.13014
This 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.

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.)
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