# Castelnuovo-Mumford regularity of Koszul cycles and Koszul homologies

## TL;DR

This paper extends known results on the convexity of Koszul cycle regularity to one-dimensional quotients and establishes a sharp bound for Koszul homologies, linking their regularities.

## Contribution

It introduces a relation between Koszul cycles and homologies and provides a new sharp regularity bound for Koszul homologies in polynomial rings.

## Key findings

- Extended convexity results to one-dimensional quotients.
- Established a sharp regularity bound for Koszul homologies.
- Linked regularities of Koszul cycles and homologies.

## Abstract

We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. By providing a relation between the regularity of Koszul cycles and Koszul homologies we prove a sharp regularity bound for the Koszul homologies of a homogeneous ideal in a polynomial ring under the same conditions.