Regularity and Free Resolution of Ideals which are Minimal to $d$-linearity

TL;DR

This paper investigates the properties of certain monomial ideals related to $d$-uniform clutters, focusing on their regularity, free resolutions, and conditions for linearity, with applications to graph theory and combinatorial topology.

Contribution

It introduces classes of $d$-uniform clutters with specific resolution properties and provides explicit minimal free resolutions for circuit ideals of complex topological structures.

Findings

Regularity of union of clutters equals maximum of individual regularities under certain conditions.

Explicit minimal free resolutions for circuit ideals of triangulations of pseudo-manifolds.

Alternative proofs for Fr"oberg's Theorem and properties of generalized chordal hypergraphs.

Abstract

Toward a partial classification of monomial ideals with $d$-linear resolution, in this paper, some classes of $d$-uniform clutters which do not have linear resolution, but every proper subclutter of them has a $d$-linear resolution, are introduced and the regularity and Betti numbers of circuit ideals of such clutters are computed. Also, it is proved that for given two $d$-uniform clutters $\mathcal{C}_1, \mathcal{C}_2$, the Castelnuovo-Mumford regularity of the ideal $I(\bar{\mathcal{C}_1 \cup \mathcal{C}_2})$ is equal to the maximum of regularities of $I(\bar{\C}_1)$ and $I(\bar{\C}_2)$, whenever $V(\mathcal{C}_1) \cap V(\mathcal{C}_2)$ is a clique or ${\rm SC}(\mathcal{C}_1) \cap {\rm SC}(\mathcal{C}_2)=\emptyset$. As applications, alternative proofs are given for Fr\"oberg's Theorem on linearity of edge ideal of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the circuit ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.