Homology of the Koszul complex of a system of polynomial equations

TL;DR

This paper constructs an explicit duality morphism for the Koszul complex associated with a system of polynomial equations, establishing a homotopic equivalence in the zero-dimensional ideal case.

Contribution

It introduces an explicit complex morphism linking the dual and original Koszul complexes, proving homotopic equivalence for zero-dimensional polynomial ideals.

Findings

Explicit duality morphism constructed

Homotopic equivalence established for 0-dimensional ideals

Provides a new tool for analyzing polynomial systems

Abstract

For a system of non-homogeneous polynomials it was constructed explicit complex morphism of a dual complex to the Koszul complex into the Koszul complex. If the ideal of these polynomials is 0-dimensional, then this mapping is a homotopic equivalence, thus in this case it was obtained explicit duality of the Koszul complex.