Koszul complexes of embedded systems of polynomials and duality

TL;DR

This paper investigates how Koszul complexes and their duals depend on each other when one polynomial system is contained within another, especially in the context of non-homogeneous polynomials and their duality relations.

Contribution

It establishes the dependence relations of Koszul complexes for embedded polynomial systems and applies these results to prove homotopic equivalence in 0-dimensional cases.

Findings

Dependence of Koszul complexes when one system is part of another

Dependence of dual Koszul complexes under linear relations

Homotopic equivalence of Koszul and dual complexes for 0-dimensional ideals

Abstract

The object of the paper is the dependence of Koszul complexes and dependence of dual Koszul complexes of two systems of non-homogeneous polynomials, when one system is a part of other system, in connection with the duality in a Koszul complex established by author earlier. Whence, the dependence of Koszul complexes and dependence of dual Koszul complexes follow when one system is linearly expressed through other system. Obtaned results are used in the proof of homotopic equivalence, formulated earlier by the author, of the Koszul complex and dual Koszul complex of a system of non-homogeneous polynomials, what happens when the ideal of these polynomials is 0-dimensional.