Betti numbers and almost complete intersection monomial ideals

TL;DR

The paper addresses the problem of determining the Betti numbers and minimal free resolutions of almost complete intersection monomial ideals, along with a Cohen–Macaulayness criterion and extensions to dominant monomial ideals. It employs combinatorial and homological techniques, including the Scarf complex and polarization, to obtain explicit formulas for Betti numbers and to relate these algebraic properties to combinatorial data such as lcms of generators. The key contributions include exact expressions for $\beta_i(R/I)$ in the standard form $I=(u_1,\dots,u_q,v)$ with $v \mid \operatorname{lcm}(u_1,u_2)$, a generalization based on the smallest $s$ with $v$ dividing an $s$-way lcm, and a CM characterization linking unmixedness and the dominant/semi-dominant structure. These results enable rapid construction of minimal free resolutions and provide structural insight into how binomial coefficients governing Betti numbers arise from generator interactions, with implications for related dominant and semidominant ideals.

Abstract

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $I$ be a monomial ideal of $R$. In this paper, we present an explicit formula for the Betti numbers of almost complete intersection monomial ideals, which enables a rapid construction of their minimal free resolutions. In addition, we characterize the Cohen-Macaulayness of these ideals and also we show the same result for dominant monomial ideals.

Theorems & Definitions (27)

• Theorem 1.1
• Definition 1.2
• Example 1.3
• Lemma 1.4
• proof
• Definition 1.5
• Proposition 1.6
• proof
• Proposition 1.7
• Example 1.8