Residual Intersections and Linear Powers

TL;DR

This paper explores new Cohen-Macaulay properties of residual intersections in Gorenstein rings, focusing on cases with high powers of ideals that are linearly presented, revealing structural and homological insights.

Contribution

It introduces a novel Cohen-Macaulay property for certain residual intersections of maximal codimension and analyzes their algebraic and homological characteristics.

Findings

Residual intersections of specific ideals are Cohen-Macaulay under new conditions.

High powers of certain ideals have linear free resolutions.

Examples include residual intersections of minors of generic matrices.

Abstract

If I is an ideal in a Gorenstein ring S and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'. However, such statements hold for residual intersections of higher codimension only under very restrictive hypotheses, not satisfied even by ideals as simple as the ideal L_n of minors of a generic 2 x n matrix when n>3. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I. For example, we prove that if K is the residual intersection of L_n by 2n-3 general quadratic forms in L_n, then S/K is integrally closed with isolated singularity and I^{n-3} S/K is a self-dual Maximal Cohen-Macaulay module over S/K with linear free resolution over S. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.