Extremal Betti Numbers of t-Spread Strongly Stable Ideals

Amata Amata,Crupi Crupi

doi:10.3390/math7080695

Abstract

Let K be a field and let S = K[x1, . . . , xn] be a polynomial ring over K. We analyze the extremal Betti numbers of special squarefree monomial ideals of S known as the t-spread stronglystable ideals, where t is an integer ≥ 1. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the t-spread strongly stable idealswith the maximal number of extremal Betti numbers when t = 2.

Highlights

Let us consider the polynomial ring S = K [ x1, . . . , xn ]
We determine the structure of the t-spread strongly stable ideals with the maximal number of extremal Betti numbers when t = 2
We study the extremal Betti numbers [4] of t-spread strongly stable ideals for t ≥ 0 (Definition 2)

Summary

Introduction

Let us consider the polynomial ring S = K [ x1 , . The class of squarefree monomial ideals is a meaningful object in commutative algebra, due to its strong connections to combinatorics and topology. Many authors have focused their attention toward problems and questions involving such a class of ideals. Ene, Herzog, and Qureshi have introduced the notion of t-spread monomial ideal [1] (see [2,3]), where t is a non–negative integer. If t ≥ 0 is an integer, a monomial xi xi2 · · · xid with 1 ≤ i1 ≤ i2 ≤ · · · ≤ id ≤ n is called t-spread, if i j − i j−1 ≥ t for 2 ≤ j ≤ d

Objectives

Discussion

Conclusion