Cohen-Macaulay residual intersections and their Castelnuovo-Mumford regularity

Abstract:

In this article we study the structure of residual intersections via constructing a finite complex of not necessarily free modules. The complex provides information about an ideal which coincides with the residual intersection in the geometric case; and is closely related to it in general. A new success obtained through studying such a complex is to prove the Cohen-Macaulayness of residual intersections of a wide class of ideals. In particular, it is shown that in a Cohen-Macaulay local ring any geometric residual intersection of an ideal which satisfies the sliding depth condition is Cohen-Macaulay. This is an affirmative answer for one of the main open questions in the theory of residual intersections (Huneke and Ulrich, 1988, Question 5.7).

The complex that we come up with in this article suffices to obtain a bound for the Castelnuovo-Mumford regularity of a residual intersection in terms of the degrees of minimal generators. More precisely, in a positively graded Cohen-Macaulay *local ring $R=\bigoplus _{n \geq 0} R_{n}$, if $J=\mathfrak {a} :I$ is a “geometric” $s$-residual intersection such that $\operatorname {Ht} (I)=g>0$ and $I$ satisfies a sliding depth condition, then $\operatorname {reg}(R/J) \leq \operatorname {reg}( R) + \dim (R_0)+ \sigma ( \mathfrak {a}) -(s-g+1)\operatorname {indeg} (I/\mathfrak {a})-s$, where $\sigma ( \mathfrak {a})$ is the sum of the degrees of elements of a minimal generating set of $\mathfrak {a}$. It is also shown that the equality holds whenever $I$ is a perfect ideal of height 2 and $R_0$ is a field.