Residual Intersections and the Annihilator of Koszul Homologies

TL;DR

This paper explores the properties of algebraic residual intersections, establishing conjectures and constructing complexes to understand their Cohen-Macaulayness, unmixedness, and canonical modules, revealing a deep connection with Koszul homologies.

Contribution

It introduces a family of approximation complexes for residual intersections and links their properties to the annihilators of Koszul homologies, advancing the theoretical understanding.

Findings

Established conjectures for large classes of residual intersections.

Constructed approximation complexes to analyze properties.

Linked residual intersections to annihilators of Koszul homologies.

Abstract

Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for large classes of residual intersections without restricting local number of generators of the ideals involved. A family of approximation complexes for residual intersections is constructed to determine the above properties. Moreover some general properties of the symmetric powers of quotient ideals are determined which were not known even for special ideals with a small number of generators. Acyclicity of a prime case of these complexes is shown to be equivalent to find a common annihilator for higher Koszul homologies. So that, a tight relation between residual intersections and the uniform annihilator of positive Koszul homologies is unveiled that sheds some light on their structure.