Linear quotients, linear resolutions and the lcm-lattice
Roni Varshavsky
TL;DR
The paper characterizes when monomial ideals have linear quotients and linear resolutions via their lcm-lattices. It proves that an ideal $I$ generated in degree $d$ has linear quotients if and only if its lcm-lattice $L(I)$ is $d$-degree graded and CL-shellable, and that $I$ has a $d$-linear resolution if and only if $L(I)$ is $d$-degree graded and Cohen–Macaulay. It further connects these lattice conditions with the Alexander dual of the associated simplicial complex, detailing correspondences between CL-shellability and shellability in the dual, and provides a lattice-theoretic perspective on edge ideals, including cochordality of graphs. The results unify homological invariants of monomial ideals with combinatorial/topological structures of lcm-lattices and their dual complexes, offering a framework to transfer problems across commutative algebra, combinatorics, and topology. Practical implications include new criteria for detecting linear quotients and linear resolutions from lattice properties and implications for edge ideals and chordality.
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.