Minimal free resolution of monoimal ideals by iterated mapping cone

TL;DR

This paper investigates minimal free resolutions of certain monomial ideals, providing conditions for minimality, and computes Betti numbers for specific classes like max-path ideals of rooted trees and ideals with squared variables.

Contribution

It introduces a sufficient condition for minimality of resolutions via the mapping cone and applies it to compute Betti numbers for new classes of monomial ideals.

Findings

Derived Betti numbers for max-path ideals of rooted trees

Established minimality criteria for resolutions obtained by the mapping cone

Analyzed resolutions of ideals combining hypergraph edge ideals with squared variables

Abstract

In this paper we study minimal free resolutions of some classes of monomial ideals. we first give a sufficient condition to check the minimality of the resolution obtained by the mapping cone. Using it, we obtain the Betti numbers of max-path ideals of rooted trees and ideals containing powers of variables. In particular, we discuss about resolutions of ideals of the form $J_{\mathcal{H}}+(x_{i_1}^2,\ldots, x_{i_m}^2)$ where $J_{\mathcal{H}}$ is the edge ideal of a hypergraph $\mathcal{H}$.