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Matchings, Squarefree Powers and Betti Splittings

Marilena Crupi, Antonino Ficarra, Ernesto Lax

Abstract

Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. In this article, we deeply investigate the squarefree powers of $I(G)$ by means of Betti splittings. When $G$ is a forest, it is shown that the normalized depth function of $I(G)$ is non-increasing. Furthermore, we compute explicitly the regularity function of squarefree powers of $I(G)$ with $G$ a forest, confirming a conjecture of Erey and Hibi.

Matchings, Squarefree Powers and Betti Splittings

Abstract

Let be a finite simple graph and let be its edge ideal. In this article, we deeply investigate the squarefree powers of by means of Betti splittings. When is a forest, it is shown that the normalized depth function of is non-increasing. Furthermore, we compute explicitly the regularity function of squarefree powers of with a forest, confirming a conjecture of Erey and Hibi.
Paper Structure (3 sections, 17 theorems, 58 equations, 4 figures)

This paper contains 3 sections, 17 theorems, 58 equations, 4 figures.

Table of Contents

  1. Betti splittings for squarefree powers
  2. Squarefree powers of forests
  3. The Erey--Hibi Conjecture

Key Result

Theorem 1.1

(EK) Let $J,L\subset S$ be non--zero monomial ideals with $J\subset L$. Suppose there exists a map $\varphi:\mathcal{G}(J)\rightarrow \mathcal{G}(L)$ such that for any $\emptyset\ne\Omega\subseteq \mathcal{G}(J)$ we have where $\mathfrak{m}=(x_1,x_2,\dots,x_n)$. Then the inclusion map $J\rightarrow L$ is $\operatorname{Tor}$-vanishing.

Figures (4)

  • Figure 1: A forest $G$ with distant edge $\{w,v\}$
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 1.2
  • Corollary 1.3
  • Lemma 1.4
  • proof
  • Proposition 1.5
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.4
  • ...and 21 more