Cohen-macaulay Residual Intersections and Their Castelnuovo-mumford Regularity

In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; 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. For example we show that, in a Cohen-Macaulay local ring, any geometric residual intersection of an ideal that satisfies the sliding depth condition is Cohen-Macaulay; this is an affirmative answer to one of the main open questions in the theory of residual intersection, [20, Question 5.7]. The complex we construct also provides a bound for the Castelnuovo-Mumford regularity of a residual intersection in term of the degrees of the minimal generators. More precisely, in a positively graded Cohen-Macaulay *local ring R = ⊕ n≥0 Rn, if J = a : I is a (geometric) s-residual intersection of the ideal I such that ht(I) = g > 0 and satisfies a sliding depth condition, then reg(R/J) ≤ reg(R)+dim(R0)+σ(a)−(s−g+1) indeg(I/a)− s, where σ(a) is the sum of the degrees of elements of a minimal generating set of a. It is also shown that the equality holds whenever I is a perfect ideal of height 2, and the base ring R0 is a field.