Signature invariants of monomial ideals

Let $I$ be a monomial ideal of a polynomial ring $R=K[x_1,\dots ,x_n]$ over a field $K$ and let $\mathrm{s}\mathrm{g}\mathrm{n}(I)$ be its signature ideal. If $I$ is not a principal ideal, we show that the depth of $R\mathrm{/}I$ is the depth of $R\mathrm{/}\mathrm{s}\mathrm{g}\mathrm{n}(I)$, and the regularity of $R\mathrm{/}\mathrm{s}\mathrm{g}\mathrm{n}(I)$ is at most the regularity of $R\mathrm{/}I$. For ideals of height at least $2$, we show that the height and the associated primes of $I$ and its signature $\mathrm{s}\mathrm{g}\mathrm{n}(I)$ are the same, and we show that $I$ is Cohen–Macaulay (resp. Gorenstein) if and only if $\mathrm{s}\mathrm{g}\mathrm{n}(I)$ is Cohen–Macaulay (resp. Gorenstein), and furthermore we show that the v-number of $\mathrm{s}\mathrm{g}\mathrm{n}(I)$ is at most the v-number of $I$. We give an algorithm to compute the signature of a monomial ideal using $\mathit{\text{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.