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Cohen-Macaulayness of vertex splittable monomial ideals

Marilena Crupi, Antonino Ficarra

Abstract

In this paper, we give a new criterion for the Cohen-Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen-Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen-Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen-Macaulay graphs by the novel criterion for the Cohen-Macaulayness of vertex splittable ideals.

Cohen-Macaulayness of vertex splittable monomial ideals

Abstract

In this paper, we give a new criterion for the Cohen-Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen-Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen-Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen-Macaulay graphs by the novel criterion for the Cohen-Macaulayness of vertex splittable ideals.
Paper Structure (10 sections, 14 theorems, 32 equations, 2 figures)

This paper contains 10 sections, 14 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cohen-Macaulay Property
  4. Vertex Splittable Monomial Ideals
  5. A Cohen--Macaulay Criterion
  6. Families of Cohen--Macaulay Vertex Splittable Ideals
  7. (Vector-Spread) Strongly Stable Ideals
  8. Componentwise Polymatroidal Ideals
  9. Bi-Cohen-Macaulay Graphs
  10. Conclusions and Perspectives

Key Result

Theorem 1

MKA16 Let $I = xI_1+ I_2$ be a vertex splitting for the monomial ideal $I$ of $S$. Then $I = xI_1+ I_2$ is a Betti splitting.

Figures (2)

  • Figure 1: A bi-CM graph.
  • Figure 2: A not bi-CM graph.

Theorems & Definitions (27)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • Corollary 1
  • proof
  • ...and 17 more