# Annihilators of Koszul Homologies and Almost Complete Intersections

## TL;DR

This paper investigates the annihilators of positive Koszul homologies in almost complete intersections, establishing new results that connect to the Monomial Conjecture and provide insights into residual approximation complexes.

## Contribution

It proves the affirmative answer for the first Koszul homology in any almost complete intersection and for certain systems of parameters in rings with small multiplicities, linking to the Monomial Conjecture.

## Key findings

- Affirmative answer for first Koszul homology in all almost complete intersections.
- Positive Koszul homologies vanish for specific systems of parameters in rings with small multiplicities.
- Connection established between the annihilators of Koszul homologies and the Monomial Conjecture.

## Abstract

In this article, we propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection $R$. The question can be stated in terms of the acyclicity of certain (finite) residual approximation complexes whose $0$-th homologies are the residue field of $R$. We show that our question has an affirmative answer for the first Koszul homology of any almost complete intersection, as well as for all positive Koszul homologies of certain system of parameters which exist in some almost complete intersection rings with small multiplicities. The statement about the first Koszul homology is shown to be equivalent to the Monomial Conjecture and thus follows from its validity.