Signature invariants of monomial ideals

TL;DR

This paper investigates the properties of signature ideals of monomial ideals, establishing their relationships with depth, regularity, and Cohen--Macaulayness, and provides algorithms for their computation and analysis.

Contribution

It introduces new theoretical results linking monomial ideals with their signature ideals and offers algorithms for computing and analyzing these signatures using Macaulay2.

Findings

Depth of $R/I$ equals depth of $R/{ m sgn}(I)$ for non-principal ideals

Regularity of $R/{ m sgn}(I)$ is at most that of $R/I$

Algorithms for computing signature ideals and testing Cohen--Macaulay or Gorenstein properties

Abstract

Let $I$ be a monomial ideal of a polynomial ring $R=K[x_1,\ldots,x_n]$ over a field $K$ and let ${\rm sgn}(I)$ be its signature ideal. If $I$ is not a principal ideal, we show that the depth of $R/I$ is the depth of $R/{\rm sgn}(I)$, and the regularity of $R/{\rm sgn}(I)$ is at most the regularity of $R/I$. For ideals of height at least $2$, we show that the height and the associated primes of $I$ and its signature ${\rm sgn}(I)$ are the same, and we show that $I$ is Cohen--Macaulay (resp. Gorenstein) if and only if ${\rm sgn}(I)$ is Cohen--Macaulay (resp. Gorenstein), and furthermore we show that the v-number of ${\rm sgn}(I)$ is at most the v-number of $I$. We give an algorithm to compute the signature of a monomial ideal using \textit{Macaulay}$2$, and an algorithm to examine given families of monomial ideal by computing their signature ideals and determining which of these are either Cohen--Macaulay or Gorenstein.