Signature invariants of monomial ideals
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arXiv
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| Format: | Preprint |
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2026
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| author | Ibarguen, Jovanny Valencia, Carlos E. Villarreal, Rafael H. |
| author_facet | Ibarguen, Jovanny Valencia, Carlos E. Villarreal, Rafael H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03208 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Signature invariants of monomial ideals Ibarguen, Jovanny Valencia, Carlos E. Villarreal, Rafael H. Commutative Algebra Combinatorics 13F55, 05C69, 13F20 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. |
| title | Signature invariants of monomial ideals |
| topic | Commutative Algebra Combinatorics 13F55, 05C69, 13F20 |
| url | https://arxiv.org/abs/2601.03208 |