# Koszul cycles and Golod rings

## TL;DR

This paper provides an explicit description of Koszul cycles for quotients of polynomial or power series rings, enabling the identification of Golod ideals and applications to various classes of rings.

## Contribution

It introduces a new explicit description of Koszul cycles in terms of free resolutions, advancing the understanding of Golod rings and their ideals.

## Key findings

- Explicit description of Koszul cycles in terms of free resolutions
- Identification of classes of Golod ideals including powers of monomial ideals
- Application to stretched local rings

## Abstract

Let $S$ be the power series ring or the polynomial ring over a field $K$ in the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal which we assume to be graded if $S$ is the polynomial ring. We give an explicit description of the cycles of the Koszul complex whose homology classes generate the Koszul homology of $R=S/I$ with respect to $x_1,\ldots,x_n$. The description is given in terms of the data of the free $S$-resolution of $R$. The result is used to determine classes of Golod ideals, among them proper ordinary powers and proper symbolic powers of monomial ideals. Our theory is also applied to stretched local rings.