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Extremal Betti numbers of certain two-dimensional monomial ideals

Nguyen Quang Loc, Nguyen Cong Minh, Phan Thi Thuy

Abstract

In this paper, we shall provide explicit formulas for the extremal Betti numbers of $R/I$, where $I$ is the defining ideal of certain weighted hyperplanes in $\Bbb{P}^{n-1}$ and $R$ is the polynomial ring in $n$ indeterminates over a field. As a consequence, we completely classify such ideals which are pseudo-Gorenstein as in sense of V. Ene, J. Herzog, T. Hibi and S. S. Madani.

Extremal Betti numbers of certain two-dimensional monomial ideals

Abstract

In this paper, we shall provide explicit formulas for the extremal Betti numbers of , where is the defining ideal of certain weighted hyperplanes in and is the polynomial ring in indeterminates over a field. As a consequence, we completely classify such ideals which are pseudo-Gorenstein as in sense of V. Ene, J. Herzog, T. Hibi and S. S. Madani.
Paper Structure (10 sections, 15 theorems, 117 equations)

This paper contains 10 sections, 15 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes and Stanley-Reisner correspondence
  4. Simplicial homology
  5. Degree complexes
  6. Graphs
  7. The main results
  8. The values of $\beta_{n-2,n+a_2}(R/I)$ when it is extremal
  9. The values of $\beta_{n-1, n+a_1}(R/I)$ when it is extremal and $\alpha\geq \beta +3$
  10. Applications

Key Result

Theorem 1.1

A pair $(i, j)$ is a corner of the Betti diagram if and only if $(n-i, j)$ is a corner of the local cohomology diagram of $M$. In this case, we have

Theorems & Definitions (24)

  • Theorem 1.1: Sch
  • Definition 2.1
  • Lemma 2.2: Takayama's formula
  • Lemma 2.3: MN, Lemmas 2.2, 2.3
  • Lemma 2.4: MN, Proposition 2.4
  • Lemma 2.5
  • Theorem 3.1
  • Remark
  • Theorem 3.2
  • Theorem 3.3
  • ...and 14 more