Linear quotients, linear resolutions and the lcm-lattice

TL;DR

This paper characterizes linear quotients and linear resolutions of monomial ideals using the lcm-lattice, clarifying their relationship and extending existing characterizations, with applications to edge ideals.

Contribution

It provides a complete characterization of linear quotients and linear resolutions via the lcm-lattice, linking these properties explicitly and extending prior duality-based results.

Findings

Characterization of linear quotients in terms of the lcm-lattice

Explicit relationship between linear quotients and linear resolutions

Applications to edge ideals

Abstract

Linear resolutions and the stronger notion of linear quotients are important properties of monomial ideals. In this paper, we fully characterize linear quotients in terms of the lcm-lattice of monomial ideals. We also formulate an analogous characterization for monomial ideals with linear resolutions, making explicit a relationship that is implicit in the existing literature. These results complement characterizations of these two properties in terms of the Alexander dual of the corresponding Stanley-Reisner simplicial complex. In addition, we discuss applications to the case of edge ideals.