Cohen-Macaulay residual intersections and their Castelnuovo-Mumford Regularity

TL;DR

This paper introduces a new complex to analyze residual intersections, proving their Cohen-Macaulayness under certain conditions and providing bounds on their Castelnuovo-Mumford regularity.

Contribution

It constructs a finite acyclic complex to study residual intersections, proving Cohen-Macaulayness for a broad class of ideals and establishing regularity bounds.

Findings

Residual intersections are Cohen-Macaulay under sliding depth conditions.

The constructed complex provides bounds on Castelnuovo-Mumford regularity.

Affirmative answer to a key open question in residual intersection theory.

Abstract

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. 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.