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Signature invariants of monomial ideals

Jovanny Ibarguen, Carlos E. Valencia, Rafael H. Villarreal

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.

Signature invariants of monomial ideals

Abstract

Let be a monomial ideal of a polynomial ring over a field and let be its signature ideal. If is not a principal ideal, we show that the depth of is the depth of , and the regularity of is at most the regularity of . For ideals of height at least , we show that the height and the associated primes of and its signature are the same, and we show that is Cohen--Macaulay (resp. Gorenstein) if and only if is Cohen--Macaulay (resp. Gorenstein), and furthermore we show that the v-number of is at most the v-number of . We give an algorithm to compute the signature of a monomial ideal using \textit{Macaulay}, 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.
Paper Structure (9 sections, 22 theorems, 90 equations)

This paper contains 9 sections, 22 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. The depth and regularity of a monomial ideal and its signature
  3. Procedure to transform $A$ into a tight matrix
  4. The signature invariants
  5. Comparing the free resolutions of $I$ and ${\rm sgn}(I)$
  6. Associated primes and v-number of a monomial ideal and its signature
  7. Examples: Illustrating the shift and polarization operations
  8. Procedures for Macaulay$2$
  9. A List of signature Cohen--Macaulay monomial ideals

Key Result

Lemma 2.1

The minimal set of generators of $I_{\rm sft}$ is where $p+3\leq q_1'\leq\cdots\leq q_{s'}'$ if $I_{\rm sft}$ has a gap at "$x_1^{p+1}x^{\gamma_{k_{p+1},p+1}},x_1^{q_1'}x^{\epsilon_1'}$" and $\{x_1^{q_i'}x^{\epsilon_i'}\}_{i=1}^{s'}$ is empty if $I_{\rm sft}$ has no gaps.

Theorems & Definitions (54)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • ...and 44 more