Matching powers of monomial ideals and edge ideals of weighted oriented graphs

TL;DR

This paper introduces matching powers of monomial ideals, generalizing squarefree powers, and investigates their algebraic properties, especially for edge ideals of weighted oriented graphs, revealing conditions for linear resolutions.

Contribution

It defines matching powers of monomial ideals and analyzes their depth and regularity, providing new insights into their structure and resolutions, especially for weighted oriented graph edge ideals.

Findings

Last nonvanishing quadratic power is polymatroidal with linear resolution

Characterization of linear resolution conditions for matching powers of non-quadratic edge ideals

Matching powers generalize squarefree powers and extend understanding of monomial ideal properties

Abstract

We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$ where $u_1,\dots,u_k$ is a monomial regular sequence contained in $I$. This concept naturally generalizes that of squarefree powers of squarefree monomial ideals. We study depth and regularity functions of matching powers of monomial ideals and edge ideals of weighted oriented graphs. We show that the last nonvanishing power of a quadratic monomial ideal is always polymatroidal and thus has a linear resolution. When $I$ is a non-quadratic edge ideal of a weighted oriented forest, we characterize when $I^{[k]}$ has a linear resolution.