Monomial Cycles in Koszul Homology

TL;DR

This paper characterizes monomial cycles in Koszul homology over monomial rings, introduces the boundary ideal, and classifies Golod monomial ideals in four variables, with applications to symmetric and shifted ideals.

Contribution

It introduces the boundary ideal to characterize monomial cycles and classifies Golod monomial ideals in four variables, extending to symmetric and shifted ideals.

Findings

Monomial cycles are boundaries iff contained in the boundary ideal.

Classified Golod monomial ideals in four variables.

Identified monomial cycle bases for ideals with linear quotients.

Abstract

In this paper we study monomial cycles in Koszul homology over a monomial ring. The main result is that a monomial cycle is a boundary precisely when the monomial representing that cycle is contained in an ideal we introduce called the boundary ideal. As a consequence, we obtain necessary ideal-theoretic conditions for a monomial ideal to be Golod. We classify Golod monomial ideals in four variables in terms of these conditions. We further apply these conditions to symmetric monomial ideals, allowing us to classify Golod ideals generated by the permutations of one monomial. Lastly, we show that a class of ideals with linear quotients admit a basis for Koszul homology consisting of monomial cycles. This class includes the famous case of stable monomial ideals as well as new cases, such as symmetric shifted ideals.