Linear resolutions of $t$-spread lexsegment ideals via Betti splittings

TL;DR

This paper characterizes all $t$-spread lexsegment ideals with linear resolutions using Betti splittings, providing formulas for their Betti numbers and characterizing those with linear quotients.

Contribution

It introduces a complete characterization of $t$-spread lexsegment ideals with linear resolutions and derives explicit Betti number formulas.

Findings

Identifies all $t$-spread lexsegment ideals with linear resolution.

Provides explicit formulas for Betti numbers of these ideals.

Characterizes $t$-spread lexsegment ideals with linear quotients.

Abstract

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely $t$-spread lexsegment ideals with linear quotients.