Candidates for non-zero Betti numbers of monomial ideals

TL;DR

This paper investigates the structure of Betti numbers of monomial ideals, providing conditions that restrict their non-zero entries and offering new proofs for existing theorems in combinatorial commutative algebra.

Contribution

It introduces a novel degree-shift property for syzygies of monomial ideals and applies it to give alternative proofs of key theorems and to classify potential non-zero Betti numbers.

Findings

Established a degree-shift property for syzygies of monomial ideals.

Provided alternative proofs for Green-Lazarsfeld index and Fröberg's theorem.

Characterized the possible indices for non-zero Betti numbers.

Abstract

Let $I$ be a monomial ideal in the polynomial ring $S$ generated by elements of degree at most $d$. In this paper, it is shown that, if the $i$-th syzygy of $I$ has no element of degrees $j, \ldots, j+(d-1)$ (where $j \geq i+d$), then $(i+1)$-syzygy of $I$ does not have any element of degree $j+d$. Then we give several applications of this result, including an alternative proof for Green-Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fr\"oberg's theorem on classification of square-free monomial ideals generated in degree two with linear resolution. Among all, we describe the possible indices $i, j$ for which $I$ may have non-zero Betti numbers $\beta_{i,j}$.