Signature invariants of monomial ideals

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 associated primes of $I$ and ${\rm sgn}(I)$ 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 $I$ and compare the irreducible decompositions of $I$ and ${\rm sgn}(I)$. 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 ideals by computing their signature ideals and determining which of these are Cohen--Macaulay or Gorenstein.