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The stable set of associated primes of a complementary edge ideal

Antonino Ficarra

Abstract

We explicitly determine the associated primes of every power of a complementary edge ideal, prove that they satisfy the persistence property, and compute the $\text{v}$-function. In the course of the proofs, we completely describe the homological properties of all powers of squarefree monomial ideals generated in degrees large relative to the number of variables defining them.

The stable set of associated primes of a complementary edge ideal

Abstract

We explicitly determine the associated primes of every power of a complementary edge ideal, prove that they satisfy the persistence property, and compute the -function. In the course of the proofs, we completely describe the homological properties of all powers of squarefree monomial ideals generated in degrees large relative to the number of variables defining them.
Paper Structure (4 sections, 12 theorems, 51 equations)

This paper contains 4 sections, 12 theorems, 51 equations.

Table of Contents

  1. Criteria for the persistence property
  2. Squarefree monomial ideals generated in big degrees
  3. The stable set of associated primes of $I_c(G)$
  4. Consequences

Key Result

Theorem A

Let $G$ be a finite simple graph on the vertex set $[n]$ with $n\ge3$. Then $I_c(G)$ satisfies the persistence property: Moreover, and $\operatorname{astab}(I_c(G))\le n-2$.

Theorems & Definitions (21)

  • Theorem A
  • Theorem B
  • Theorem C
  • proof : Proof of Theorem \ref{['ThmB']}
  • Theorem 1.1
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • ...and 11 more