Deformation of Residual Intersections

TL;DR

The paper proves that in Cohen-Macaulay local rings, generic linkages of an ideal are deformations of arbitrary linkages, extending to residual intersections under certain conditions.

Contribution

It establishes a general deformation relationship between generic and arbitrary linkages and residual intersections without requiring the ideal to be Cohen-Macaulay.

Findings

Generic linkage is a deformation of arbitrary linkage in Cohen-Macaulay rings.

The result extends to s-residual intersections when s is close to the height of I.

Under certain conditions, this principle applies to all s-residual intersections.

Abstract

It is shown that in a Cohen-Macaulay local ring, the generic linkage of an ideal $I$ is a deformation of the arbitrary linkage of $I$. This fact does not need $I$ to be a Cohen-Macaulay ideal. The same holds for $s$-residual intersections of $I$ when $s$ does not exceed the height of $I$ by one. Under some slight conditions on $I$, one further generalizes this principle to encompass any $s$-residual intersection.