Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity (Q2844843)

Residual intersection generalizes the notion of linkage. Let \(R\) be a Noetherian ring and \(s\) be an integer. Two ideals \(I\) and \(J\) of \(R\) are \textit{linked} if \(J = \mathfrak a : I\) and \(I = \mathfrak a : J\) for some ideal \(\mathfrak a\) of \(R\) which is generated by a regular sequence contained in \(I \cap J\). The ideal \(J\) is an \(s\)-\textit{residual intersection} of I if there exists an \(s\)-generated ideal \(\mathfrak a \subset I\) with \(J = \mathfrak a : I\) and \(\text{ht} J \geq s\). Residual intersections were introduced by \textit{M. Artin} and \textit{M. Nagata} [J. Math. Kyoto Univ. 12, 307--323 (1972; Zbl 0263.14019)], improved by \textit{C. Huneke} (Trans. Am. Math. Soc. 277, 739--763 (1983; Zbl 0514.13011)], and studied by \textit{C. Huneke} and \textit{B. Ulrich} [J. Reine Angew. Math. 390, 1--20 (1988; Zbl 0732.13004)]. Two central questions in residual intersection theory remain open: for a residual intersection \(J\) of \(I\), when is \(R/J\) Cohen-Macaulay, and what is the canonical module for \(R/J\) .NEWLINENEWLINEIt is shown in the present paper that in a Cohen-Macaulay local ring any geometric residual intersection of an ideal which satisfies the sliding depth condition is Cohen-Macaulay. The proof makes use of a finite complex of not necessarily free modules. This approximation complex is also used to establish a bound for the Castelnuovo-Mumford regularity of a residual intersection in terms of the degrees of the generators of the defining ideals. Also, the precise formula is given for the regularity of the residual intersections of a perfect ideal of height 2.